Dynamic Response of Aerospace Materials under Extreme Environments

To improve the performance and safety of future aerospace vehicles, worldwide efforts are being directed towards the development of novel aerospace materials which exhibit superior structural and multifunctional capabilities in extreme environments. Fundamental investigation into the thermo-mechanical response and dynamic failure of the materials is paramount before they can be incorporated into the design of future space access vehicles that can operate reliably in combined, extreme environments. For this purpose, a comprehensive study was conducted to evaluate the performance of variety of aerospace materials such as Functionally Graded Materials (FGMs), Ti2AlC, and Hastelloy X under extreme thermo-mechanical loadings. Starting from the first principles, analytical analyses were conducted to develop thermo-mechanical stress fields for mixed mode dynamic curving cracks in functionally graded materials (FGMs) under steady-state and transient loading conditions. Asymptotic analysis was used in conjunction with displacement potentials to develop the stress fields around propagating cracks in FGMs. Asymptotic temperature fields were developed first for the exponential variation of thermal conductivity and later these temperature fields were used to derive thermo-mechanical stress fields for a curving crack in FGMs. Using these thermo-mechanical stress fields, various components of the stresses were developed and the effect of curvature parameters, transient parameters, temperature and gradation on these stresses were discussed. Finally, using the minimum strain energy density criterion, the effect of curvature parameters, transient parameters, crack-tip speeds, non-homogeneity values and temperature gradients on crack growth directions were determined and discussed. Experimental studies were then conducted to evaluate two different materials, namely, Nanolayered Titanium Aluminum Carbide (a MAX phase material) and Hastelloy X under varying rates of loading and at different temperatures. The dynamic behavior of nanolaminated ternary carbide, Ti2AlC, was characterized under dynamic loading using Split Hopkinson Pressure Bar (SHPB) compression apparatus. The dynamic loading experiments were performed in the strain-rate range of 1500-4200s and at temperatures ranging from room temperature (RT) to 1150°C. At room temperature, the failure stress and strain show little dependence on strain rate, whereas the failure stress drops considerably at temperatures above 900°C. At all strain rates and temperatures, Ti2AlC exhibits softening after failure initiation and a more graceful failure due to delamination and kink band (KB) formation. At temperatures higher than 900°C, grain boundary decohesion is suggested to contribute towards the deecrease in the failure stress. The dynamic constitutive behavior of Hastelloy X (AMS 5754) was studied at room and elevated temperatures under varying rates of loading. A split Hopkinson pressure bar (SHPB) apparatus was used in conjunction with an induction coil heating system for applying dynamic loads at elevated temperatures. Experiments were carried out at different temperatures ranging from room temperature (25°C) to 1100°C at an average strain rate of 5000s. Room temperature experiments were carried out at varying strain rates from 1000s to 4000s. The results show that as the strain rate increases from quasi-static to 4000s, the yield strength increases by approximately 50%. Also, under dynamic loading, the yield stress decreases with temperature up to 700°C, after which it (yield strength) shows a peak at 900°C before beginning to decrease again as the temperature is further increased. The Johnson-Cook model was used to predict the dynamic plastic response under varying rates of loading and at different temperatures. A series of experiments were conducted to study the dynamic response of rectangular Hastelloy X plates at room and elevated temperatures when subjected to shock wave loading. The shock tube apparatus was modified and validated for testing Hastelloy X at elevated temperatures. Propane gas was used as the heating source and it was directed onto the sample via four nozzles. A cooling system was also implemented to prevent the shock tube from reaching high temperatures. High-speed photography coupled with the optical technique of Digital Image Correlation (DIC) technique was used to record the real-time deformation of the specimen under shock wave loading. The DIC technique was used in conjunction with band pass optical filters and a high intensity light source to record the full-field deformation images under shock loading at high temperatures up to 900 °C. In addition, a high speed camera was utilized to record the side-view deformation images. The high temperature DIC system has been validated by comparing with the mid-point deflections obtained from the side-view camera. The dynamic response of Hastelloy X was evaluated as a function of temperature under shock wave loading.

Experimental studies were then conducted to evaluate two different materials, namely, Nanolayered Titanium Aluminum Carbide (a MAX phase material) and Hastelloy X under varying rates of loading and at different temperatures. The dynamic behavior of nanolaminated ternary carbide, Ti 2 AlC, was characterized under dynamic loading using Split Hopkinson Pressure Bar (SHPB) compression apparatus. The dynamic loading experiments were performed in the strain-rate range of 1500-4200s -1 and at temperatures ranging from room temperature (RT) to 1150°C. At room temperature, the failure stress and strain show little dependence on strain rate, whereas the failure stress drops considerably at temperatures above 900°C. At all strain rates and temperatures, Ti 2 AlC exhibits softening after failure initiation and a more graceful failure due to delamination and kink band (KB) formation. At temperatures higher than 900°C, grain boundary decohesion is suggested to contribute towards the deecrease in the failure stress.
The dynamic constitutive behavior of Hastelloy X (AMS 5754) was studied at room and elevated temperatures under varying rates of loading. A split Hopkinson pressure bar (SHPB) apparatus was used in conjunction with an induction coil heating system for applying dynamic loads at elevated temperatures. Experiments were carried out at different temperatures ranging from room temperature (25°C) to 1100°C at an average strain rate of 5000s -1 . Room temperature experiments were carried out at varying strain rates from 1000s -1 to 4000s -1 . The results show that as the strain rate increases from quasi-static to 4000s -1 , the yield strength increases by approximately 50%. Also, under dynamic loading, the yield stress decreases with temperature up to 700°C, after which it (yield strength) shows a peak at 900°C before beginning to decrease again as the temperature is further increased. The Johnson-Cook model was used to predict the dynamic plastic response under varying rates of loading and at different temperatures.
A series of experiments were conducted to study the dynamic response of rectangular Hastelloy X plates at room and elevated temperatures when subjected to shock wave loading. The shock tube apparatus was modified and validated for testing Hastelloy X at elevated temperatures. Propane gas was used as the heating source and it was directed onto the sample via four nozzles. A cooling system was also implemented to prevent the shock tube from reaching high temperatures. High-speed photography coupled with the optical technique of Digital Image Correlation (DIC) technique was used to record the real-time deformation of the specimen under shock wave loading. The DIC technique was used in conjunction with band pass optical filters and a high intensity light source to record the full-field deformation images under shock loading at high temperatures up to 900 °C. In addition, a high speed camera was utilized to record the side-view deformation images. The high temperature DIC system has been validated by comparing with the mid-point deflections obtained from the side-view camera. The dynamic response of Hastelloy X was evaluated as a function of temperature under shock wave loading. Chapter 2 focuses on analytical development of thermo-mechanical stress fields for a mixed-mode dynamic crack growth along an arbitrarily smoothly varying path in functionally graded materials (FGMs). The property gradation in FGMs was considered by varying shear-modulus, mass density, thermal conductivity and coefficient of thermal expansion exponentially along the gradation direction.
Asymptotic analysis in conjunction with displacement potentials was used to develop the stress fields around propagating cracks in FGMs. Asymptotic temperature fields viii are developed first for the exponential variation of thermal conductivity and later these temperature fields were used to derive thermo-mechanical stress fields for a curving crack in FGMs. Using these thermo-mechanical stress fields, various components of the stresses were developed and the effect of curvature parameters, temperature and gradation on these stresses were discussed. Finally, using the minimum strain energy density criterion, the effect of curvature parameters, crack-tip speeds, nonhomogeneity values and temperature gradients on crack growth directions were determined and discussed. This chapter follows the formatting guidelines specified by the International Journal of Solids and Structures.
Chapter 3 provides the details of developing transient thermo-mechanical stress fields for a mixed-mode dynamic crack growth behavior along an arbitrarily smoothly varying path in functionally graded materials (FGMs). Similar approach as explained in chapter 2 was used in developing the stress fields. Asymptotic analysis in conjunction with displacement potentials was used to develop transient thermomechanical stress fields around the propagating crack-tip. Asymptotic temperature field equations are derived for exponentially varying thermal properties, and later these equations were used to derive transient thermo-mechanical stress fields for a curving crack in FGMs. The effect of transient parameters (loading rate, crack-tip acceleration, and temperature change) and temperature gradient on the maximum principal stress and circumferential stress associated with the propagating crack-tip was discussed. Finally, using the minimum strain energy density criterion, the effect of temperature gradient, crack-tip speeds and T-stress on crack growth directions were ix determined and discussed. This chapter follows the formatting guidelines specified by Acta Mechanica.
Chalpter 4 details the experimental study conducted on nanolaminated ternary carbide, Ti 2 AlC, was characterized under dynamic loading using Split Hopkinson Pressure Bar (SHPB) compression apparatus. The dynamic loading experiments were performed in the strain-rate range of 1500-4200s -1 and at temperatures ranging from room temperature (RT) to 1150°C. At room temperature, the failure stress and strain show little dependence on strain rate, whereas the failure stress drops considerably at temperatures above 900°C. At all strain rates and temperatures, Ti 2 AlC exhibits softening after failure initiation and a more graceful failure due to delamination and kink band (KB) formation. At temperatures higher than 900 o C, grain boundary decohesion is suggested to contribute towards the decrease in the failure stress. This chapter will follow the formatting guidelines specified by the Acta Materialia.
Chapter 5 details the experimental study conducted on the dynamic constitutive behavior of Hastelloy X (AMS 5754) at room and elevated temperatures under varying rates of loading. A split Hopkinson pressure bar (SHPB) apparatus was used in conjunction with an induction coil heating system for applying dynamic loads at elevated temperatures. Experiments were carried out at different temperatures ranging from room temperature (25°C) to 1100°C at an average strain rate of 5000s -1 . Room temperature experiments were carried out at varying strain rates from 1000s -1 to 4000s -1 . The results show that as the strain rate increases from quasi-static to 4000s -1 , the yield strength increases by approximately 50%. Also, under dynamic loading, the yield stress decreases with temperature up to 700°C, after which it shows a peak at x 900°C before beginning to decrease again as the temperature is further increased. The Johnson-Cook model was used to predict the dynamic plastic response under varying rates of loading and at different temperatures. This chapter follows the formatting guidelines specified by Journal of Materials Science.
Chapter 6 details the design of novel experimental set up for testing materials under shock loading at extreme temperatures. The shock tube apparatus was modified and validated for testing Hastelloy X at elevated temperatures. Propane gas was used as the heating source and it was directed onto the sample via four nozzles. A cooling system was also implemented to prevent the shock tube from reaching high temperatures. A series of experiments were conducted to study the dynamic response of rectangular Hastelloy X plates at room and elevated temperatures when subjected to shock wave loading. High-speed photography coupled with the optical technique of Digital Image Correlation (DIC) technique was used to record the real-time deformation of the specimen under shock wave loading. The DIC technique was used in conjunction with band pass optical filters and a high intensity light source to record the full-field deformation images under shock loading at high temperatures up to 900 xvii LIST OF TABLES CHAPTER 5 Table 1 [3], and Drake et al. [4] reported that the thermal residual stresses are relaxed in metal-ceramic layered materials by inserting a functionally graded interface layer between the metal and ceramic. Kudora et al. [5] and Takahashi et al. [6] reported that FGM coatings suffer significantly less damage than conventional ceramic coatings, under thermal shock condition.

3
Several studies on the quasi-static fracture behavior of FGMs under thermomechanical loading have been reported. Assuming exponential variation of material properties, Jin and Noda [7] investigated the steady thermal stress intensity factors in functionally graded semi-infinite space with an edge crack subjected to a thermal load.
Using both experimental and numerical techniques, Kokini and Choules [8], and Kokini and Case [9] studied surface and interface cracking in FGM coatings subjected to thermal shocks. Jin and Paulino [10]  al. [13] developed the thermo-mechanical stress fields for a mixed mode propagating crack with a constant velocity along the gradation direction. Also, Chalivendra et al. [14] developed the elastic mechanical stress field solutions for an inclined propagating transient crack in FGMs. Even though the above mentioned studies of dynamic thermo-mechanical asymptotic field equations serve the purpose of obtaining fracture parameters from experimental data, these studies have not incorporated the crack path curvature associated with a propagating crack in these materials. It has been reported 4 in the experimental and numerical studies that the propagating cracks follow curved paths under various loading conditions due to the spatial variation of properties in graded materials [15][16][17][18]. In order to extract the fracture parameters accurately from the experimental data associated with curved cracks, it is essential to consider curvature terms in the field equations. Liu and Rosakis [19]  , where M is an early transition metal, A is a group-A element and X is either C or N. [21][22][23]. Titanium aluminum carbide, Ti 2 AlC, one of the well characterized compounds with M 2 AX chemistry, is lightweight (4.11 g/cm 3 ) material with high Young's modulus (278 GPa) [21], excellent electrical and thermal conductivity, high oxidation and corrosion resistance [24]. Like most MAX phases, Ti 2 AlC is relatively soft (Vickers hardness of ~4 GPa) and easily machinable 5 compared to binary transition metal carbides [25][26]. Because of their nanolaminated structure and kinking mechanism, Ti 2 AlC is also exceedingly damage tolerant and thermal shock resistant and most likely creep and fatigue-resistant like some other MAX phases [27][28][29]. One of the interesting aspects of MAX phases in general, and Ti 2 AlC in particular is unusual deformation behavior that can be traced back to their nanolaminated hexagonal crystal structure with high c/a ratio and presence of the relatively large number of movable dislocations. The dislocations, being confined to only parallel basal planes, give rise to extensive kinking and delamination in MAX phases upon deformation [30][31]. At room temperature, Ti 2 AlC, similar to other MAX phases, shows nonlinear stress-strain behavior with fully reversible hysteresis loops and significant amount of energy dissipation under quasi-static cycling loadingunloading [30,[32][33]. Recently, a microscale model, based on nucleation, growth and annihilation of reversible nest of movable dislocations, so called incipient kink bands (IKBs), has been proposed to explain the, stress-strain hysteresis in MAX phases [30,[34][35]. At higher stresses or higher temperature, the IKBs transform into permanent kink bands (KBs). Formation of KB along with delamination has been argued in literature to be the primary reason for excellent damage tolerance of MAX phases [29]. At higher temperatures, i.e. above 1000 o C, Ti 2 AlC, undergoes a brittle-to-plastic (BTP) transition above and strains as large as 12% can be reach before failure during compressive loading [36].
Hastelloy X is a nickel based superalloy that possesses excellent high temperature strength and oxidation resistance. It is widely used in gas turbine operations, petrochemical and in aircraft parts [37]. Several researchers in the past 6 have investigated the behavior of Hastelloy X. Lai [38] investigated changes in the hardness and impact toughness at room temperature of Hastelloy X after aging up to 1000h. The author found that it exhibits age hardening at 540°C and 650°C. The author also observed a slight increase in hardness at 871°C followed by overaging after 4000h. Yasuo Kondo et al. [39] presented experimental results on the changes in the tensile mechanical properties of Hastelloy X after being used in the liner tube of a HENDEL hot gas duct under high temperature helium gas for about 6000h. They observed that the 0.2% proof stress and total elongation were slightly decreased when Hastelloy X was exposed to high temperature helium. Static tensile stress-strain curves of Hastelloy X were reported in Mil Handbook 5h [40]. Swindeman  [48] Nurick GN, Shave GC (1996) The deformation and tearing of thin square plates subjected to impulsive loads -An experimental study. Int J Impact Eng, 18: 99-116.
The research output on the thermo-mechanical response of FGM's is limited. Hasselman and Youngblood (1978) were among the first to study thermal stresses in nonhomogeneous structures associated with thermo-mechanical loading. By introducing a thermal conductivity gradient, they demonstrated significant reductions in the magnitude of the tensile thermal stress in ceramic cylinders. In other studies, thermal residual stresses were relaxed in metal-ceramic layered materials by inserting a functionally graded interface layer between the metal and ceramic (e.g., Kawasaki and Wantanabe, 1987: Giannakopoulos et al. 1995. In their studies, Kuroda et al. (1993) and Takashashi et al. (1993) reported that when subjected to thermal shocks, FGM coatings suffer significantly less damage than conventional ceramic coatings.

16
In continuation of the above studies, several studies on the quasi-static fracture of FGMs under thermo-mechanical loading have been reported. Assuming exponential variation of material properties, Jin and Noda (1994) investigated the steady thermal stress intensity factors in functionally graded semi-infinite space with an edge crack subjected to thermal load. Later,  also determined the steady thermal stress intensity factor of a FGM layer with a surface crack perpendicular to the boundaries. By further assuming the exponential variation of thermal and mechanical properties of the materials, Jin and Batra (1996) investigated the stress intensity relaxation problem at the tip of an edge crack in a FGM subjected to a thermal shock. By employing a finite element method (FEM), Noda (1997) analyzed an edge crack problem in a zirconia/titanium FGM plate subjected to cyclic thermal loads. Jin and Paulino (2001)  loading conditions due to the spatial variation of properties in graded materials (e.g., Lambros et al. 2000;Tohgo et al. 2005;Tilbrook et al. 2005). In order to extract the fracture parameters accurately from the experimental data associated with curved cracks, it is essential to consider curvature terms in the field equations.

Theoretical Formulation
FGMs can be treated as isotropic non-homogeneous solids because, at a continuum level, the properties at any given point in an FGM can be assumed to be the same in all directions. Spatial variation of elastic properties, mass density, thermal properties and inclination of a curved crack with respect to the gradation direction make analytical solutions to the elastic-dynamic equations extremely difficult. Hence, an asymptotic analysis similar to that employed by  is used to expand the stress field around a propagating curving crack under thermo-mechanical loading conditions.
An isotropic linear elastic FGM containing a propagating curving crack with temperature gradient in the x-y plane is shown in Fig. 1. At time t = 0, the crack-tip is at the origin of Cartesian coordinate system x-y. For any time t > 0, the position of the propagating crack-tip is given by (X(t) and Y(t)) as shown in Fig. 1. Shear modulus ( ) µ , Lamé's constant ( ) λ , density ( ) ρ , thermal expansion ( ) α and heat conductivity ( ) k of the FGM are assumed to vary in an exponential manner as given by Eq. (1a) -1(e), whereas, Poisson's ratio ( ) ν is assumed to be a constant. Experimental studies (e.g., Shukla and Jain, 2004;Yao et al. 2007) reported in the literature support the exponential variation of modulus of elasticity and mass density.
The equations of motion for a plane problem are given by Eq. (2) The relationship between stresses and strains for a plane strain thermo-mechanical problem can be written as Substituting for the stresses and density from (3) and (1) respectively into Eq. (2) and after simplification, the equations of motion are written as where c α is the coefficient of thermal expansion in the vicinity of the instantaneous crack-tip and is assumed to be constant.
Equations (5a) and (5b) can be further written as Now, a new moving coordinate system ) , ( 2 1 ξ ξ is introduced, so that the origin of the new system is at the moving crack-tip. The 1 ξ -axis is tangential to the crack trajectory at the crack-tip and coincides with the direction of the crack growth.
The crack is propagating with constant velocity (c) as a function of time along the local 1 ξ -direction. The angle between 1 ξ -axis and the fixed x-axis are denoted by ) (t β , as shown in Fig. 1. Therefore the relationship between the coordinates in these two Cartesian co-ordinate systems is If the length of the trajectory that the crack-tip travels during the time interval, [0, t], is denoted by s(t), then the magnitude of the crack-tip speed c(t) will be s  (t), and the curvature of the crack trajectory at the crack-tip, k(t), is given by Using the transformations and the relations given below in Eqs. (9)- (11), and also neglecting the transient terms associated with acceleration and rate of curvature, the equations of motion (6a) and (6b) are further written in the crack-tip moving coordinate reference ( 1 ξ , 2 ξ ) as given in Eqs. (12a) and (12b).   In the above formulation, we assume that the stress waves generated by the moving crack-tip are not influenced by the gradation. This assumption is acceptable as the property variation around the crack-tip is small in the zone of the interest.

Temperature Fields around the Crack-tip
In this analysis, it is assumed that the temperature field around the crack-tip also changes asymptotically. The transient effects, the thermo-elastic cooling effects, and the coupling effects ( ) are neglected. It has been shown that for most materials under static loading conditions, this coupling term is small and can be neglected (Sadd, 2009). Several researchers studied the effect of coupling under dynamic loading conditions and found that the quantitative effect of coupling is less important for metals (Awrejcewicz and krys'ko, 2003). Few other authors observed that the difference between the couple and uncoupled solutions are about one percent (McQuillen and Brull, 1970;Shiari et al. 2003;Eslami et al. 1994). Brischetto and Carrera (2010) observed that when a mechanical load is applied, the differences between the coupled and uncoupled analysis are minimum, in terms of stresses. Nowinski (1978) in his book states ''Practically speaking, it is generally possible to discount the coupling and to evaluate the temperature and deformation fields, in this order, separately". So, we establish the uncoupled conduction system and the temperature fields can be determined independent of the stress-field calculations. The developed field equations can be used for situations with small temperature gradient 25 thermal loading conditions. The heat conductivity is assumed to vary exponentially as given by Eq. (1e). The steady state heat conduction equation can be written as Substituting the heat conductivity relation given by Eq. (1e) into Eq. (13) results in  At this stage, the asymptotic analysis is performed to solve the above Eq. (15). The advantages of this approach are: (i) it does not consider finite specimen geometry, and (ii) it does not need the information of external loading boundary conditions. It requires only crack-face and crack-line boundary conditions associated with the propagating crack to be satisfied. In this process, first a new set of scaled coordinates is defined as (16) where ε is an arbitrary parameter and is assumed to be In the scaled coordinates, Eq. (15) can be written as For the asymptotic analysis, T is now represented as a power series expansion in ε.
For Eq. (19) to be valid, the partial differential equations corresponding to each power of ε (ε 1/2 ,ε, ε 3/2 , …) should vanish independently. This leads to the following set of partial differential equations.
Equation (20) and the solution is the same as for a homogenous material (e.g, ). An asymmetric temperature field solution is developed to account for real-time thermal loading situations and the heat flux is assumed to be singular at the crack-tip (Jin and Noda, 1994). The singularity of heat For m=1 where ( ) The solution T 2, for Eq. (21) corresponding to higher powers of ε (m=2) has two parts: a homogeneous and a particular solution. The particular solution can be obtained using a recursive approach (e.g., Parmeswaran and Alternate temperature fields can be easily derived for the conditions such as singular temperature at the crack-tip.
At this stage, it is assumed thatΦ, ψ and T can be represented as a power series expansion in ε.
As discussed above, the temperature field is obtained by ensuring that the heat flux or the derivative of the temperature field near the crack-tip is singular. This means that the potentials for temperature are one power less than the displacement potentials. Substituting Eq. (27) into Eqs. (26a) and (26b) gives the following equations.
Eqs. (29a) and (29b) are similar to those for a homogeneous material where the partial differential equations (30a) and (30b), associated with the higher powers of ε are coupled to the differentials of the lower order functions through the nonhomogeneity parameters ζ and temperature term. Eqs. (29a) and (29b) (i.e. for m=0 and m=1) can be easily reduced to Laplace's equation in the respective complex and the solutions are the same as those for homogenous material (eg., Shukla and Jain, 2004), and can be written as   Using the definition of the dynamic stress intensity factor K ID and K IID for the opening mode and shear mode , the relation between A o and K ID and C o and K IID are obtained.   In order to extract both mode-I and mode-II stress intensity factors accurately from the 33 experimental data when the crack-tip stress field is non-homogenous around the cracktip due to imposed thermal fields and material non-homogeneity, higher-order stress-field equations are essential in addition to the leading terms and Tstress (non-singular stress). The solution for the equations (30a) and (30b), corresponding to higher powers of ε (m = 2), consists of two parts: a solution for the homogeneous equation and a particular solution due to non-homogeneity and temperature and these can be obtained recursively (eg., Parmeswaran and . The solutions φ 2 and ψ 2 obtained are given below. The solutions for φ 0 , ψ 0 , φ 1 and ψ 1 automatically satisfy the compatibility equations because these solutions are the same as those for homogeneous materials. Since the non-homogeneous specific parts of φ 2 and ψ 2 are obtained from φ 0 and ψ 0 , they also automatically satisfy the compatibility equations

35
The above definitions of the displacement potentials are now used in Eq. (4) to obtain the displacements fields. These displacement fields are then used to develop strain fields. These strain fields and Eq. (25) are substituted into Eq. (3) to obtain inplane steady state stress fields around the crack-tip as given by Eq. (38). (38) where ij σ are stress components, T represents the change in temperature in the infinite mediumζ , 1 γ and 2 γ are non-homogeneity constants that have the dimension (length) -1 , c is the crack-tip speed, k is the curvature of the crack trajectory at crack-tip, β is the angle between 1 ξ -axis and the fixed x-axis as shown in Fig.1, β  is the angular velocity as referred in Eq. (7).

Results and Discussion
The thermo-mechanical stress fields developed above are used to study the effect of temperature field, curvature of the crack path, crack-tip velocity, and the nonhomogeneity on the variation of the maximum shear stress, the circumferential stress, maximum principal stress and the crack kinking angle at the crack-tip. The coefficient of thermal expansion (α c ) of 2 ZrB (which is of interest in our experimental research), 5.9×10 -6 / o C is used in the analysis. It is assumed that the resulting temperature range generates elastic deformation around the crack-tip. Some of the terms associated with non-homogeneity and the stress field connected with temperature change impose normal and shear stresses on traction free crack faces. The presence of these normal and shear stresses on the crack face violates one of the boundary conditions and these stresses need to be removed. The removal of these stresses from the crack face is accomplished by superimposing an equal and opposite stress field on the crack face. In particular each point on the crack face is subjected to a line load of specific magnitude such that the normal stresses from the crack face are removed. Jian et al. (1991) proposed a solution for a single line load to derive the stress field around the crack-tip and the solution is extended for multiple line loads applied on the crack face line. The revised in-plane stress-fields which have traction-free crack face conditions were then used in developing various results discussed in section 3.1. In the present analysis, since the stress fields are determined closer to the crack tip, the higher order temperature terms are neglected. The higher order temperature terms can be used when determining stress fields far away from the crack-tip in analyzing the experimental data. When the temperature coefficients and the non-homogeneity factors are set to zero, these equations fall down to the equations derived by Liu and Rosakis. The present analysis assumed constant velocity at the crack-tip. Even though the crack-tip acceleration is an important condition to consider, the stress fields may not significantly affected by the acceleration terms as the acceleration terms will only appear in the higher order terms , k = 20 rad/m, β = 20°, 37 q o = 100 o C/ m 1/2 , γ 2 = 1 and a typical value of radius, r = 0.002m. The choice of stress intensity factor is motivated by the fact that ceramics have low fracture toughness and they often form a part of FGM's. At this point, it is also important to mention that the relative dominance of the temperature or mechanical field will depend on the choice of K and q o values.

Effect of Curvature of the Crack path
Three major stress components were considered to show the variation of stress fields under steady state thermo-mechanical loading: maximum shear stress, circumferential tensile stress and maximum principal stress. These stresses are normalized using      Angular variation of the normalized maximum principal stress and circumferential tensile stress as a function of curvature with and without the temperature field is plotted in Fig. 3 and Fig. 4 respectively. The peak value of the principal and circumferential tensile stresses increase significantly when the curvature is introduced for both homogeneous and FGM systems and the increase in peak value of these stresses is approximately 50% higher for the case of a curved crack. Also the angle at which the peak values of these stresses occur changes slightly with the addition of the curvature parameters. As observed in the Fig. 2 and Fig. 3, the nature of variation of these normalized stresses changes significantly with the addition of curvature terms.
The angular variation of all three of the above normalized stress components for a fixed value of the non-homogeneity parameter, , as a function of curvature with temperature field are shown in Fig.5. As the local curvature of the crack path (k) increases, the peak values of these stresses increase while the angle at which the peak value of these stresses occur changes slightly for all the cases. the shear stress increases significantly and the stress contours tilt away from the crack direction. However, if we keep the curvature value constant at k = 20 and change the angle β to 60°, the shear stress contours tilt towards the crack direction, but there is no significant change in the size of shear stress contours. Hence, for cases discussed in the following sections, a typical value of β = 20° is used in further analysis.

Effect of Temperature
Variation of the normalized maximum principal stress, circumferential stress and the maximum shear stress, is plotted around the crack-tip for different temperature ) as shown in Fig. 7. As the temperature gradient increases, the peak value of both the normalized maximum principal stress and circumferential tensile stress decreases and the angles at which the peak value of these stresses occur change slightly for different temperature gradients. In the case of the maximum shear stress, there is no change in the stress values for different temperature gradients. This is due to the fact that the temperature field gets cancelled when yy σ was subtracted from xx σ as shown in below Eq. 39 and there are no temperature terms in xy τ (Refer Eq. 3c).

Effect of Non homogeneity
The effect of property gradation for a curved crack on the contours of maximum shear stress around a propagating crack-tip for k =20 and β = 20° with no temperature field is shown in Fig. 8. Fig. 8(a) shows the case of a homogeneous material, and Fig. 8 the size of the contours increases above the crack-line and decreases below the crackline. Moreover, these contours tilt towards the crack direction which is the increasing property gradation direction as shown in Fig 8(b). However, for 1 − = ζ , the reverse happens with a decreased size above the crack-line and an increased size below the crack-line and the contours tilt away from the crack direction, which is again in the increasing property gradation direction.
A similar plot to that of Fig.8 is made by considering the thermal property gradation on the maximum principal stress with temperature gradient, 100 0 = q , there is no noticeable difference on the contours of maximum principal stress for different non-homogeneity values (γ 2 ). This is due to the dominance of first term ( 0 q ) in the temperature field equation.

Crack Extension Angle
A dynamically moving crack tends to deviate from its path due to crack-tip instability conditions. The crack-tip instability becomes predominant when the cracks tend to propagate in non-homogeneous materials under thermo-mechanical loading. In the present study, using the derived thermo-mechanical stress field equations, the effects of curvature parameters, temperature, crack-tip velocity and material nonhomogeneity on the crack-tip instability are presented. The theoretical prediction of the crack extension angle is investigated by using the well-known fracture criterion: minimum strain energy density (S-criterion). The strain energy density criterion was chosen because it provides a complete description of material damage by including both the distortional and dilatational effects (Gdoutus, 1990). Both distortional and dilatational vary in proportion, depending on the load history and location due to nonuniformity in stress or energy fields.

Minimum Strain-Energy Density (MSED) Criterion
According to this criterion (Sih, 1974), the crack initiates when the strain energy density achieves a critical value and propagates in the direction of the  49 minimum strain-energy density value. The strain energy density dW/dV near the cracktip for an FGM is given as Fracture takes place in the direction of minimum S, and the condition can be obtained by using Eq. (40) where S c is the critical strain energy density.
Variation of strain energy density around the crack-tip for a thermomechanical loading in an FGM for different temperature gradients, curvatures and non-homogeneity is shown in Fig. 9. The angle at which the strain energy density reaches a minimum value changes with temperature, curvature and the nonhomogeneity parameters as shown in Fig. 9.

Effect of Crack-tip Velocity
The crack extension angles as a function of crack-tip velocity for a typical curvature k = 20 and β = 20 O , temperature gradient,

Effect of Non homogeneity
The effect of the non-homogeneity parameter (

Effect of Temperature
The effect of the temperature on the crack extension angle for a mixed-mode thermo-mechanical loading is shown in Fig. 12 for different non-homogeneity values.
Typical values of k = 20, β = 20°, γ 2 = 1 are assumed for constructing Fig Crack extension angle (degrees) No temperature field ζ ζ ζ Normalized crack tip velocity ( ) 13 Crack extension angle as a function of crack-tip velocity for a mixed-mode thermo-mechanical loading at different temperature fields using MSED criterion

54
The effects of temperature on the crack extension angle as a function of cracktip velocity in a homogeneous material, a FGM with 1 = ζ and a FGM with 1 − = ζ are plotted in Fig. 13 using MSED criterion. For all the three cases, the crack 1 − = ζ are plotted in Fig. 13 using MSED criterion. For all the three cases, the crack extension angle increases as the crack-tip velocity increases for various temperature gradients.

Effect of Curvature
The effect of curvature on the crack extension angle for a crack-tip velocity of 0.5c s for a mixed-mode thermo-mechanical loading is shown in Fig. 14  Normalized crack tip velocity ( ) Normalized crack tip velocity ( ) Normalized crack tip velocity ( )

Effect of T-stress on Curvature and Temperature
In the previous studies, it was reported in the literature that T-stress ( ox σ term) plays a vital role in crack kinking or crack branching for a homogeneous material (e.g., : Fang-Li and Roy-Xu, 2007. In this paper, the effect of T-stress is studied in FGMs under thermomechanical loading. In the above Eq. 42, T is the non-singular stress term. The effect of T-stress ( ox σ term) on the crack extension angle for different curvature values and temperature gradients around a propagating crack under mixed-mode loading in an FGM is shown in Fig. 16 for different non homogeneity values using MSED criterion. A Typical value of was used for plotting. Fig. 16(a) shows the effect of Tstress on the crack extension angle as a function of curvature for a fixed temperature ), it is observed that the crack extension angle decreases when the T- It can be concluded from above analysis ( Fig. 2 to Fig. 16

Conclusions
The stress-fields near the crack-tip for steady state mixed-mode thermomechanical loading in a graded material for a smoothly curving crack are developed using displacement potentials in conjunction with an asymptotic approach. First, asymptotic temperature fields near the crack-tip are developed and then by incorporating this developed temperature field into the mechanical field, the thermomechanical stress fields near the crack-tip are generated. Using these developed stress field equations, the effect of the non-homogeneity parameters, curvature, and temperature is studied on various stress components, on the contours of the maximum shear stress and the crack extension angle using minimum strain energy density criterion. The results are compared with the previous findings of static and dynamic 60 crack propagation in homogeneous and FGM systems under thermo-mechanical loadings (Kidane et al. (2010a(Kidane et al. ( ) & (2010b). When curvature terms are set to zero, the solutions match with the previous findings. This analytical study resulted in the following key findings: • The thermo-mechanical stress fields around the crack-tip are significantly affected by the curvature, temperature and non-homogeneity parameters. As the curvature at the crack-tip increases, the peak value of the stress components also increases.
• For a given curvature conditions, as the temperature gradient increases, the peak value of the stress components decreases.
• The mechanical properties gradation affects the contours of the maximum shear stress considerably while the thermal properties gradations show no significant effect on the stresses.
• Crack-tip velocity plays a significant role in the crack extension angle. For a mixed-mode curving crack under thermo-mechanical loading, the crack extension angle increases with the increase in crack-tip velocity. Similar behavior was reported earlier for the case of homogeneous and FGMs without the curvature parameters (Kidane et al., 2010a and2010b).
• Similar to previous findings (Kidane et al., 2010a and2010b), the increase in the temperature gradient decreases the crack extension angle for different nonhomogeneity values.
• The increase in the value of curvature (k) increases the crack extension angle for different non-homogeneity values.
• For a homogeneous material and FGMs with 1 = ζ and 1 − = ζ , the crack extension angle increases as the crack-tip velocity increases for different values of temperature coefficient ( 0 q ) and curvature (k).
• For an applied T-stress, the increase in the value of curvature (k) increases the crack extension angle while the increase in the temperature gradient decreases the crack extension angle for different non-homogeneity values.
• There is a significant effect on the crack extension angle when the T-stress is

Introduction
Functionally Graded Materials (FGMs) are non-homogeneous composites which have spatially varying microstructure and mechanical/thermal properties to meet desired functional performance [1,2]. FGMs used in high temperature applications have metals and ceramics as their constituents and thus have both thermal and mechanical non-homogeneities. These materials can withstand high temperatures effectively due to superior thermal properties of ceramics, and at the same time high strength can be achieved with the presence of metal. Several studies on the superior inservice advantages of FGMs over traditional duplex coatings under high temperature and high thermal cycling conditions have been reported. In their studies, Kawasaki and Wananabe [3], and Drake et al. [4] reported that the thermal residual stresses are relaxed in metal-ceramic layered materials by inserting a functionally graded interface layer between the metal and ceramic. Kudora et al. [5] and Takahashi et al. [6] reported that FGM coatings suffer significantly less damage than conventional ceramic coatings, under thermal shock condition.
Several studies on the quasi-static fracture behavior of FGMs under thermomechanical loading have been reported. Assuming exponential variation of material properties, Jin and Noda [7] investigated the steady thermal stress intensity factors in functionally graded semi-infinite space with an edge crack subjected to a thermal load.
Later, Erdogan and Wu [8] also determined the steady thermal stress intensity factor of a FGM layer with a surface crack perpendicular to the boundaries. By further assuming the exponential variation of thermal and mechanical properties of the materials, Jin and Batra [9] investigated the stress intensity relaxation problem at the tip of an edge crack in a FGM subjected to a thermal shock. Using both experimental and numerical techniques, Kokini and Choules [10], and Kokini and Case [11] studied surface and interface cracking in FGM coatings subjected to thermal shocks. Jin and Paulino [12] studied transient thermal stresses in an FGM with an edge crack and having a constant Young's modulus and Poisson's ratio, but varying thermal properties along the thickness direction. Walters et al. [13] developed general domain 79 integral methods to obtain stress intensity factors for surface cracks in FGMs under mode-I thermo-mechanical loading conditions. Zhang and Kim [14] used a complex variable approach to characterize the higher order terms for stationary crack-tip fields in FGMs. Clements [15] analyzed the antiplane crack problem for inhomogeneous anisotropic elastic FGMs.
The above studies provide closed form solutions for stress intensity factors under thermo-mechanical loading, however, for extracting fracture parameters from experimental studies, asymptotic expansion of thermo-mechanical stress fields around the crack-tip is essential. Jain et al. [16] developed quasi-static stress and displacement fields for a crack in an infinite FGM medium under thermo-mechanical loading. Lee et al. [17] developed analytical expressions for dynamic crack-tip stress and displacement fields under thermo-mechanical loading in FGMs. Recently, Kidane et al. [18] developed the thermo-mechanical stress fields for a mixed mode propagating crack with a constant velocity along the gradation direction. Also, Chalivendra et al. [19] developed the elastic mechanical stress field solutions for an inclined propagating transient crack in FGMs. Even though the above mentioned studies of dynamic thermo-mechanical asymptotic field equations serve the purpose of obtaining fracture parameters from experimental data, these studies have not incorporated the crack path curvature associated with a propagating crack in these materials. It has been reported in the experimental and numerical studies that the propagating cracks follow curved paths under various loading conditions due to the spatial variation of properties in graded materials [20][21][22][23]. In order to extract the fracture parameters accurately from the experimental data associated with curved cracks, it is essential to consider curvature terms in the field equations. Liu and Rosakis [24] developed the higherorder asymptotic expansion of a non-uniformly propagating dynamic crack along an arbitrary curved path for homogeneous materials and they emphasized the importance of the curvature terms when experimental data points are considered away from the singularity zone to extract fracture parameters. Recently, Chalivendra [25]

Theoretical Formulation
FGMs can be treated as isotropic non-homogeneous solids because, at a continuum level, the properties at any given point in an FGM can be assumed to be the same in all directions. Non-homogeneity at the grain-size level is not considered in the formulation of the problem. Spatial variation of elastic properties, mass density, thermal properties, inclination of the curved crack with respect to the gradation direction and transient conditions make analytical solutions to the elasto-dynamic 81 equations extremely difficult. Hence, an asymptotic analysis similar to that employed by Freund [26] is used to expand the stress field around a propagating crack under thermo-mechanical loading conditions. Isotropic linear elastic FGM, containing a propagating curving crack with temperature gradient in the x-y two-dimensional plane is shown in Fig. 1. At time t = 0, the crack-tip happens to be at the origin of Cartesian coordinate system x-y. For any time t > 0, the position of the propagating crack-tip is supposed to be given by (X(t) and Y(t)) as shown in Fig. 1. The crack is propagating with a varying velocity ( c ) as a function of time. It was identified in previous the experimental studies [27][28] that the variation of mechanical properties of laboratory-fabricated FGMs can be fitted by an exponential curve. Thus, the shear modulus (μ), Lamé's constant (λ), and density (ρ), are assumed to vary in an exponential manner as given by Eq. (1a)-(1c). The thermal properties, namely thermal expansion (α), heat conductivity (k) and heat capacity(C) are assumed to vary exponential as given by Eq. (1d)-(1f). Two different nonhomogeneity parameters were used for the thermal properties, where 1 γ was used for thermal expansion, and 2 γ was used for both heat conductivity and heat capacity.

Fig. 1
Propagating curved crack orientation with respect to property gradation direction and temperature gradient.
Each physical variable can have a different non-homogeneity parameter. However for mathematical simplicity only three non-homogeneity parameters, one for mechanical properties and two for thermal properties, are considered. Let u and v, functions of x, y and t, represent the displacement field in x and y directions, respectively, with t representing the time. The Hooke's law for a plane strain thermo-mechanical problem can be written as For a plane strain deformation, the displacements u and v are derived from the dilatational and shear wave potentials φ and ψ. These potentials can be expressed as Substituting the displacement relations into the stress expressions in Eq. (2), and then substituting these stresses into the equations of motion (Eq. 3), after simplification, we get 2 2 0 0 1  Fig. 1. Therefore the relationship between the coordinates in these two Cartesian co-ordinate systems is If the length of the trajectory that the crack-tip travels during the time interval, [0, t], is denoted by s(t), then the magnitude of the crack-tip speed c(t) will be s  (t), and the curvature of the crack trajectory at the crack-tip, k(t), is given by Using the transformations and the relations given below in Eqs. (8) where i = 1, 2 and j = 1, 2 where 85 2 2 2 2 1 1 2 It is assumed that the crack velocity c is a function of time for transient crack growth, and the potentials φ and ψ explicitly depend on the time in the moving coordinate reference.

Temperature Field around the Crack-tip
In this analysis, it is assumed that the temperature field around the crack-tip changes asymptotically. The general heat conduction equation for isotropic materials can be written as Density (ρ), heat conductivity (k), and heat capacity (C) are assumed to vary exponentially as given by Eqs. For the simplification of Eq. (14), the exponential term on the right-hand side is expanded using Taylor series.
In the vicinity of the crack-tip, higher order terms having exponential 2 or more can be neglected, which leads to the equation Transforming the above equation into the moving crack-tip coordinates, it can be written as At this stage, the asymptotic analysis is performed to solve the above Eq. (17). The advantages of this approach are: (i) it does not consider finite specimen geometry, and (ii) it does not need the information of the external loading boundary conditions. It requires only crack-face and crack-line boundary conditions associated with the propagating crack to be satisfied. In this process, first a new set of scaled coordinates is defined as (18) where ε is an arbitrary parameter and is assumed to be In the scaled coordinates, Eq. (17) can be written as For the asymptotic analysis, T is now represented as a power series expansion in ε.
Substituting the above infinite series expansion in Eq. (19) results in For Eq. (21) to be valid, the partial differential equations corresponding to each power of ε (ε 1/2 , ε, ε 3/2 , ...) should vanish independently. This leads to a set of partial differential equations.
The homogeneous solutions for the temperature field can be written in the for Eqs. (23) and (24) corresponding to the higher powers of ε (m ≥2) consist of two parts: the classical solution and the solution due to non-homogeneity, and the latter can be obtained recursively. The details of the procedure are given in Kidane et al. [18]. The temperature field solution (first 3 terms) including the non-homogeneity parameters and the transient term ( ) ( 0 t q  ) can be written as where ( )

Asymptotic Expansion of Crack-tip Stress Fields
A process similar to the asymptotic approach used for deriving the temperature field around the crack-tip is used for deriving solutions of the displacement potentials of the equations of motion. The scaled coordinates discussed in Eq. (18) are now applied to Eqs. (11a) and (11b) as given below At this point, it is assumed thatφ , ψ and T can be represented in the power series of ε.
For Eqs. (28a) and (28b) to be valid, the partial differential equations corresponding to each power of ε (ε 1/2 , ε, ε 3/2 , ...) should vanish independently. This leads to the following set of partial differential equations.  as those for homogenous materials [26,[29][30][31][32][33], and can be written as  Using the definitions of dynamic stress intensity factors K ID and K IID for the opening mode and shear mode [26] as given in Eq. (32), the relation between A 0 (t) and K ID (t), C 0 (t) and K IID (t) are obtained.
, 2 where μ c is the crack-tip shear modulus, K ID (t) and K IID (t) are the mode-I and mode-II dynamic stress intensity factors, respectively.
In order to extract both mode-I and mode-II stress intensity factors accurately from the experimental data when the crack-tip stress field is non-homogenous due to imposed transient thermo-mechanical fields and material non-homogeneity, higherorder stress-field equations are essential in addition to leading terms and thermal stress. The solution for Eq. (30a) and Eq. (30b) corresponding to higher powers of ε (m = 2) consists of two parts: the solution for the homogeneous equation and a particular solution due to non-homogeneity and temperature, and these can be obtained recursively [29][30][31]. The solutions for 0 φ , 0 ψ , 1 φ and 1 ψ automatically satisfy the compatibility equations because these solutions are same as those for homogeneous materials.
To include the effect of the transient temperature term ( ) ( 0 t q  ), the particular solution for 6 φ is derived. The differential equation is as follows The ratio (1/α l 2 ) can be calculated for different crack speeds c/c s from the knowledge of elastic coefficients. The variation is represented by the approximation given by the following equation, The constant ω can be obtained by curve fitting. Using the relation is expanded using Taylor series in terms of l ρ and l θ [34]. Parmeswaran and Shukla [34] provided a remedial solution to solve this type of equations. When they compared their analytical solutions obtained using this approach with the finite element simulations, they observed a maximum error of 5% [34]. Using the same approach, the equation can be written as The solution for 6 φ can be written as The above definitions of the displacement potentials are now used in Eq. (4) to obtain the displacement fields. These displacement fields are then used to develop the strain fields. These strain fields and Eq. (25) are substituted into Eq. (2) to obtain inplane stress fields around the crack-tip as given in Appendix (A2-A4).

Results and Discussion
To study the effects of the transient terms in both mechanical and thermal loading on crack propagation, the contours of the maximum principal stress and circumferential stress are generated for mixed-mode loading conditions. The asymptotic expansion of the crack-tip stress and displacement fields contain the coefficients of A n (t), B n (t), C n (t) and D n (t), which can be related to the fracture parameters such as the dynamic stress intensity factor and the nonsingular stress components around the crack-tip. The constants A 0 (t), B 0 (t), C 0 (t) and D 0 (t), are related to the mode-I and II stress intensity factors [K ID (t) and K IID (t) respectively] as given in Eq. (32) and (33). The higher order coefficients A 1 (t), B 1 (t), C 1 (t) and D 1 (t) are assumed to be zero. The material properties and fracture parameters of ZrB 2 are used in generating the plots [35]. Since the cracks are more likely to propagate from the ceramic side, we used the properties of ceramics and varied the properties exponentially along the gradation direction. The properties are as follows: Poisson's ratio (ν) = 0.16, shear modulus at the crack-tip (µ 0 ) = 210 GPa, density at the crack-tip (ρ 0 ) = 6000 kg/m 3 , coefficient of thermal expansion at crack-tip (α 0 ) = 5.9x10 -6 /°C, thermal diffusivity at crack-tip (α t0 ) = 1x10 -5 m 2 /s, non-homogeneity parameters, It is assumed that the applied thermal field generates elastic deformation around the crack-tip. Some of the terms associated with non-homogeneity and with the thermal stress field impose normal and shear stresses on the traction free crack faces.
The removal of these stresses from the crack face is accomplished by superimposing an equal and opposite stress field on the crack face. In particular, each point on the crack face is subjected to a line load of specific magnitude such that the normal and shear stresses from the crack face are removed. Jiang et al. [36] proposed a solution for a single line load to derive the stress field around the crack-tip and the solution is extended for multiple line loads applied on the crack face line. The revised in-plane stress fields with traction free crack face conditions are used in analyzing the effect of various parameters.

Effect of Transient Terms on the Maximum Principal Stress
The effect of transient terms on the contours of the maximum principal stress around the crack-tip with no temperature field for a mixed mode loading in a FGM is shown in Fig. 2. In Fig. 2(a), by setting the coefficients associated with transient terms to zero, the contours of maximum principal stress with no transient effects are obtained for relative comparison. Fig. 2(b) shows the effect of the crack-tip acceleration on the contours of maximum principal stress. To study the effect of cracktip acceleration, the value of dc/dt was varied over seven orders of magnitude. Dally and Shukla [37] have shown that the rate of change of velocity at crack initiation could be of the order of 10 7 m/s 2 . As the acceleration is increased, the contours ahead of the crack-tip are compressed. Also, the contour size reduces in the direction perpendicular to the crack surface. The effect of loading-rate on the contours of maximum principal stress is demonstrated by Fig. 2 ºC-m -1/2 s -1 was chosen as a reference value. It is clearly visible in Fig. 2(d) that the stress contours are significantly affected as ) ( 0 t q  is introduced. The size of the stress contours decreases significantly away from the crack-tip, due to the higher order dependence of the stress field on ) ( 0 t q  .

Effect of Transient Terms on the Circumferential Tensile Stress
The effect of transient terms on the contours of circumferential stress around the crack-tip with no temperature field for a mixed mode loading in a FGM is discussed in Fig. 3. The contours of the circumferential stress without transient Parameters are shown in Fig. 3(a). The transient effect due to crack-tip acceleration as shown in Fig. 3(b), decreased the size of the contours of circumferential stress ahead of the crack-tip as the acceleration is increased. Also, Fig. 3(c) shows that the effect of dK ID (t)/dt and dK IID (t)/dt on the contour size in front of the crack-tip is negligible.  For 100 ) ( ºC-m -1/2 s -1 , the size of the contours decreases significantly and the shape changes considerably as shown in Fig. 3(d).

Effects of Temperature
Variation of normalized maximum principal stress and circumferential stress, is plotted around the crack-tip for different temperature gradients ( 0 0 = q , 100 0 = q and 200 0 = q ) as shown in Fig. 4. Typical values of dc/dt =10 7 m/s 2 , dK ID (t)/dt = ºC-m -1/2 s -1 , and a typical value for radius, r = 0.002m are assumed for plotting Fig. 4. As the temperature gradient increases, the peak value of both the normalized maximum principal stress and circumferential tensile stress decreases while the angles at which the peak value of these stresses occur change slightly for different temperature gradients.

Effect of Temperature gradient and Crack-tip Velocity on the Crack Extension Angle
The value of the crack extension angle has been calculated using the minimum strain energy density criterion (S-criterion). The strain energy density criterion is used in order to provide a complete description of material damage by including both the distortional and dilatational effects [38]. The strain energy density dW/dV near the crack-tip for an FGM is given as  The effects of temperature on the crack extension angle as a function of cracktip velocity in a FGM with 2 − = ζ is plotted in Fig. 6 using the minimum strain energy density criterion. The crack extension angle increases as the crack-tip velocity increases for various temperature gradients.

Effects of T-stress on the Crack Extension Angle
Previous studies have shown that the T-stress (σ ox term) plays an important role in crack branching and crack kinking for a homogenous material [39,40]. The effect of T-stress on the stress fields in FGMs has been accounted using the following equation: In the above Eq. (43), T(σ ox ) is the non-singular stress term. The effect of Tstress on the crack extension angle for different temperature gradients around a propagating crack under mixed-mode thermo-mechanical loading in an FGM is shown in Fig. 7 using minimum strain energy criterion. Typical values of σ ox / K ID = -(3/4) m - ºC-m -1/2 and 100 ) ( ºC-m -1/2 s -1 are assumed for constructing Fig. 7. The value of the crack extension angle decreases as the temperature gradient is increased as shown in Fig. 7. The decrease in the crack extension angle is more significant for less temperature gradient. The crack extension angle decreases from -44° to -32° as the value of 0 q is increased from 0 to 200°C/m -1/2 . Therefore, it can be concluded that the application of thermal loading decreases the crack extension angle even in the presence of non-singular stresses. Also, for the same temperature field, the value of the crack extension angle is found to be lower under T-stress when compared to that of the zero T-stress case.

Concluding Remarks
Thermo-mechanical stress and displacement fields for a smoothly curving crack in ).

Introduction
Nanolaminated ternary carbides and nitrides, known as MAX phases, have attracted recent attention because of their potential for application in extreme environments. They have a general formula of M n+1 AX n (n = 1, 2, 3), where M is an early transition metal, A is a group-A element and X is either C or N. 1-3 Titanium aluminum carbide, Ti 2 AlC, one of the well characterized compounds with M 2 AX chemistry, is lightweight (4.11 g/cm 3 ) 1 material with high Young's modulus (278 GPa), 1 excellent electrical and thermal conductivity, 4 high oxidation and corrosion resistance. [5][6][7][8][9] Like most MAX phases, Ti 2 AlC is relatively soft (Vickers hardness of ~ 4 GPa) 10 and easily machinable 11 compared to binary transition metal carbides.
One of the interesting aspects of MAX phases in general, and Ti 2 AlC in particular is unusual deformation behavior that can be traced back to their nanolaminated hexagonal crystal structure with high c/a ratio and presence of the relatively large number of movable dislocations. The dislocations, being confined to only parallel basal planes, give rise to extensive kinking and delamination in MAX phases upon deformation. 15,19 At room temperature, Ti 2 AlC, similar to other MAX phases, shows nonlinear stress-strain behavior with fully reversible hysteresis loops and significant amount of energy dissipation under quasi-static cycling loadingunloading. 15,[20][21][22][23] Recently, a microscale model, based on nucleation, growth and annihilation of reversible nest of movable dislocations, so called incipient kink bands (IKBs), has been proposed to explain the stress-strain hysteresis in MAX phases. 15,[24][25] At higher stresses or higher temperature, the IKBs transform into permanent kink bands (KBs). Formation of KB along with delamination has been argued in literature to be the primary reason for excellent damage tolerance of MAX phases. [16][17]19 At higher temperatures, i.e. above 1000 o C, Ti 2 AlC,undergoes a brittle-to-plastic (BTP) transition and strains as large as 12% can be reached before failure during compressive loading. 26

120
In this letter we demonstrate that Ti 2 AlC exhibits strain rate independent deformation behavior below BTP transition temperature up to very high strain rates under dynamic conditions. The maximum stress in the dynamic true stress-strain curves, corresponding to the failure initialization, is comparable with that measured in quasi-static experiments. Above BTP transition temperature, maximum stress measured in dynamic conditions was found to decrease significantly, in the similar way as in quasi-static testing.

Experimental Details
Dense Ti 2 AlC, with a grain size of ~100 µm, was processed by hot pressing and 1220 mm, respectively, was used to study the dynamic behavior of Ti 2 AlC. A clay pulse shaper of thickness of 2 mm was placed at the impact end of the incident bar to improve force equilibrium conditions at the specimen-bar interface. The theoretical and experimental details of SHPB are given elsewhere. [27][28][29] By varying the velocities of the striker bar, room temperature experiments were carried out at strain rates ranging from 1500-4200/s. Using one-dimensional wave theory, the engineering stress and strain in the specimen can be determined from the transmitted and reflected strain pulses respectively as given in Eqs. 1 and 2.
where, ε r and ε t represent reflected and transmitted strains respectively, c b, E b and A b represent wave speed, Young's modulus and cross sectional area of the bar repestively, σ s , ε s , andand A s , , , L s , represent stress, strain and cross sectional area of the specimen respectively, and t represents the time. Equations (1) and (2) were suitably modified to obtain the true stress and true strain in the material.
For experiments at elevated temperatures, the SHPB apparatus in conjunction with the induction coil heating system was utilized. A special fixture was designed and fabricated to load the specimen. The details of our high temperature SHPB setup can be found in (28,29). High temperature experiments were carried out from RT to 1150°C under identical strain rate of about 3500/s.

Results and Discussion
The dynamic true stress-true strain curves for Ti 2 AlC at room temperature for different strain rates ranging from 1500/s to 4200/s are plotted in Fig 1. For all strain rates, the maximum dynamic compressive stress of 480 -600 MPa is attained around 2% true strain, followed by a stress relaxation up to maximum true strain of 9-16%. Unlike most ceramics, and similar to quasi-static experiments, 30 Ti 2 AlC exhibits a more graceful failure rather than brittle failure under dynamic loading (insert in Figure   122 1). More interestingly, the maximum stress during the deformation of Ti 2 AlC does not depend significantly on the strain rate during dynamic experiments. The slight variation observed in the maximum stress may be due to the difference in pre-existing defect concentration in the samples. The adiabatic nature of dynamic tests may also affect dislocation nucleation and propagation, which in turn can result in decrease in maximum stress with increase in strain rates. There is some evidence that the energy absorbed by the material decreases marginally with increasing strain-rates. This is indicated by the decrease in area under stress-strain curve. However, the material still exhibits graceful failure. These comments notwithstanding, more work is needed to and reaches a maximum true strain of 45-50%. Since the diameter of the specimen for high temperature testing is smaller in comparison to the room temperature testing the resulting stress on the sample for a given loading is higher and this results in larger strains 29 . Even at high temperatures, the material exhibits graceful failure (inset in Figure 2a). Here again, SEM micrographs from the fracture surfaces after dynamic deformation at elevated temperatures, as illustrated in Figures 2(b) and 2(c) for samples tested at 500 o C and 1150 o C respectively, reveal large amount of kinking. It is important to note here that it is difficult to infer from the fracture surfaces whether the amount of plastic deformation by kinking depends on the testing temperature,.
However, the drop in maximum stress clearly indicates that the material softens, i.e.
undergoes brittle-to-plastic transition at temperatures above 900 o C even at deformation rates of 3500/s.
The later is even more obvious when the change in maximum stress with temperature for the dynamic experiments is plotted, Figure 3. For comparison, the change in maximum stress from quasi-static tests 10 at 10 -4 /s is also plotted in the same graph. The similar nature of change in maximum stress for both strain rates again confirms our conclusion that deformation in Ti 2 AlC is quite strain-rate insensitive below BTP transition temperature even when strain rates are as high as 3500/s. More importantly, Figure 3 suggests that the material goes through the same BTP transition at strain rates that are orders of magnitude different from each other.
As far as we are aware this is the first report on dynamic stress-strain behavior of Ti 2 AlC at different strain rates and temperatures, and thus more work is needed to fully understand observed trends. However, taking into account structural and mechanical similarities between the Ti 2 AlC and Ti 3 SiC 2 , for which effect of strain rate (of the order of 10 -7 -10 -4 /s) has been extensively studied, 31 it is not unreasonable to assume that both of them would behave in a qualitatively similar way. Below the BTP transition temperature, Ti 3 SiC 2 exhibits brittle deformation with fully reversible hysteresis at low strains, and maximum stresses that are almost independent on the deformation rate. 31 The hysteretic behavior is shown to be due to reversible motion of dislocations in parallel slip planes, and has been discussed in more details elsewhere. 15,20 Above the BTP transition temperature, the deformation in Ti 3 SiC 2 depends on the strain rate. For example, maximum stress in Ti 3 SiC 2 shows strain rate dependency in the 1100 -1200 o C temperature range for strain rates below 10 -6 /s. However, above a strain rate of 10 -5 /s, there is little effect of strain rate on maximum stress, and Ti 3 SiC 2 behaves as a brittle solid. 31 Hence, like in quasi-static experiments, 15 the primary reason for decrease in stress above BTP can be argued to be grain boundary decohesion and/or extensive delamination. Thus, further work on understanding the role of grain boundaries on the dynamic stress-strain behavior of MAX phases in general, and Ti 2 AlC in particular, will shed more light on the nature of maximum stress decrease above BTP transition temperature.

Conclusions
In conclusion, stress-strain analysis along with microscopic investigation of the fracture surfaces in nanolaminated Ti 2 AlC under dynamic loading reveal that, like quasi-static conditions, the underlying failure mechanism is independent of strain rate at temperatures below 900 o C and that it is predominantly controlled by delamination of the nanolaminated structure and KB formation. At temperatures higher than 900 o C, grain boundary decohesion plays an important role in crack propagation, and subsequently in decrease of failure stress.

Introduction
Hastelloy X is a nickel based superalloy that possesses excellent high temperature strength and oxidation resistance. It is widely used in gas turbine operations, petrochemical and in aircraft parts. Due to its high oxidation resistance, it 132 is also used for structural components in industrial furnace applications [1]. Thus, it is important to understand the constitutive behavior of this material.
Several researchers in the past have investigated the behavior of Hastelloy X.
Lai [2] investigated changes in the hardness and impact toughness at room temperature of Hastelloy X after aging up to 1000h. The author found that it exhibits age hardening at 540°C and 650°C. The author also observed a slight increase in hardness at 871°C followed by overaging after 4000h. Yasuo Kondo et al. [3] presented experimental results on the changes in the tensile mechanical properties of Hastelloy X after being used in the liner tube of a HENDEL hot gas duct under high temperature helium gas for about 6000h. They observed that the 0.2% proof stress and total elongation were slightly decreased when Hastelloy X was exposed to high temperature helium. Static tensile stress-strain curves of Hastelloy X were reported in Mil Handbook 5h [4]. Swindeman and Brinkman [5] reviewed high temperature mechanical properties for materials including Hastelloy X, used in a pressure vessel using stress-strain curves obtained experimentally. They studied the effect of cold work, chemistry, and heat treatment variations in Hastelloy X. Aghaie Khafri and Golarzi [6] characterized the hot deformation behavior of Hastelloy X using hot compression tests in the temperature range of 900°C-1150°C and at varying strain rates between 0.001 and 0.5/s. They showed that softening mechanisms, dynamic recovery and dynamic recrystallization occurred during hot working. Zhao et al. [7] studied the phase precipitation in Hastelloy X heat-treated at 750°C, 850°C and 900°C for 26 and 100 hours. They also provided a TTT (time-temperature-transformation) diagram by combining the new experimental results with the existing literature data.
This TTT diagram depicted the presence of 6 23 C M and σ phases at temperatures < 900°C and µ phase between 800°C and 980°C. Hong et al. [8] performed low cycle fatigue tests of Hastelloy X in the temperature range of 650°C-870°C under various strain rates. They found that the Coffin-Manson (C-M) plot was different from those at other temperatures and also the fatigue life significantly decreased at the total strain range less than 0.6%. Krompholz and Ullrich [9] determined J-integral R curves for Hastelloy X upto 660°C using potential drop technique and showed that there is a decrease in J-integral value at 660°C. Rowley and Thornton [10] modeled visco plastic behavior of Hastelloy X using the Bodner Partom unified constitutive model.
Static compression experiments were conducted at different temperatures to get the parameters for the model. The model results of Hastelloy X for isothermal uniaxial tensile tests showed good agreement with the experimental stress-strain data.
Based upon the careful literature search, it was determined that there is no detailed study of the dynamic characterization of Hastelloy X under thermomechanical loading. Hence this study focuses mainly on the dynamic constitutive behavior of Hastelloy X at room and elevated temperatures.

Material and Specimen Geometry
The material used in this experimental study is Hastelloy X, supplied by Haynes International. The chemical composition of the material is listed in Table 1  Specimens, having a diameter of 7.62mm and thickness of 3.81mm were used for dynamic characterization at room temperature and specimens with a diameter of 6.35mm and thickness of 3.175mm were used at elevated temperatures.

Quasi-Static Characterization
The quasi-static compression tests were performed using Instron Materials Testing System-5585. ASTM standard E9-89a was followed for the specimen dimensions and testing procedure. Experiments were performed at a compressionextension rate of 1mm/min and the tests were continued up to a strain of 25%.
Molybdenum disulphide was used as a lubricant between the specimen and loading head. is propelled using an air-operated gun. A clay pulse shaper of thickness 2mm was placed at the impact end of the incident bar as shown in Fig. 1 to improve force equilibrium conditions at the specimen-bar interface. The theoretical details of SHPB can be obtained from Kolsky [11]. The specimen was sandwiched between the incident bar and the transmission bar. Molybdenum disulfide was used as a lubricant between the specimen and the contacting surfaces of the bars to minimize friction.

Dynamic Characterization
When the striker bar impacts the incident bar, an elastic compressive stress pulse, referred to as the incident pulse, is generated. The generated pulse deforms the Room temperature experiments were carried out at strain rates ranging from 1000-4000/s. Different strain rates are obtained by varying the velocities of the striker bar. Using one-dimensional wave theory, the true strain and true stress in the specimen 136 can be determined from the reflected and transmitted strain pulses respectively as given in Eqs. 1 and 2.
The expressions for the forces at the specimen incident bar interface and at the specimen transmission bar interface are given in equations Eqs. 3 and 4 respectively. For experiments at elevated temperatures, the SHPB apparatus in conjunction with the induction coil heating system was utilized as shown in Fig. 2. A special fixture was designed and fabricated to load the specimen. Two tungsten carbide inserts were placed between the two pressure bars and the specimen was sandwiched between the tungsten carbide inserts as shown in Fig. 2. The inserts were used to eliminate the temperature gradient in the bars and thus protect the strain gages mounted on them.
The impedance of the inserts was matched with the bars; hence they do not disturb the stress wave profiles in the bar. The impedance matching requires the diameter of these tungsten carbide inserts to be smaller than the main pressure bars. This is the reason 138 for the specimen diameter for high temperature testing being smaller than that for room temperature testing. The bars were kept apart initially, later the specimen and carbide inserts were heated in isolation to the desired temperature (usually about 20-50°C higher than the test temperature) and soon after the bars were brought manually into contact with the specimen. The temperature of the specimen was monitored using a 0.127mm chromel-alumel thermocouple, which was spot welded onto the specimen.
In most of the experiments, it takes less than two minutes to heat the specimen to the required temperature and it takes less than 10 seconds to bring the pressure bars into contact with the tungsten inserts and fire the gun.
High temperature experiments were carried out from room temperature to 1100°C under identical strain rate of about 5000/s. Molybdenum disulphide was used as lubricant in the initial experiments and subsequently this was replaced with Boron Nitride. Molybdenum disulphide works well till about 300°C and Boron nitride is supposed to work till about 1200°C. Both the lubricants show about same amount of barreling (1 to 2% differences in the center diameter to outside diameter).

Model Description
Johnson and Cook proposed an empirical constitutive model [12,13] to predict the plastic behavior of metals that are subjected to large strains, high strain rates and high temperatures. It is one of the most widely used models due to its simplicity and yet effective form. According to the model, the flow stress can be written as 6) whereσ is the flow stress, A and B are the strain hardening parameters and typically A is referred as the yield stress at reference strain rate and reference temperature, C is a dimensionless strain hardening coefficient, p ε is the plastic strain, where melt T is the melting temperature, reference T is the reference temperature and T is the test temperature. The expression in the first set of parentheses in Eq. 6 represents stress as a function of strain, the expression in the second set of parentheses represents the effect of strain rate and the expression in the third set represents the effect of temperature.

Determination of Parameters of Model
As shown in Eq. 6, there is a set of five model parameters that need to be identified (A, B, C, n and m). A, B and n are obtained from a quasi-static test at room temperature. The parameter A is often defined as the yield stress of the reference strain rate [12]. Here, the quasi-static strain rate is considered as the reference strain rate.
Parameter A (yield stress) is determined by considering 0.2% strain offset from the quasi-static test. Once A is determined, parameters B and n are obtained by fitting the quasi-static test data to the model as given in Eq. 8.
Having identified A, B and n, the next step is to determine the value of C.
Dynamic SHPB experiments at reference temperature and different strain rates are used to estimate the value of C. The initial yield point and strain rate ( p ε ) from each dynamic test was used to determine C. In doing so, it was ensured that the heat generated from plastic work was not taken into account, thus ignoring the temperature effect. The parameter C for a given dynamic test can be calculated as given in Eq. 9, The calculated value of C at different strain rates may not be the same due to variations in the experimental data so an average value of C is obtained. Now the final task is to identify the value of m. A set of experiments at different temperatures but identical strain rate are used to determine the value of m. A Similar procedure is followed to that of parameter C, but the effect of temperature is included. As each of the parameters is identified from separate experiments, the final set of parameters may or may not predict the material behavior when all the terms vary at the same time. Therefore, the identified parameters may predict few experiments very well but not other experiments. So it is important to optimize the identified parameters, which can be done by minimizing the average relative error (Δ) between the experimental data and the predicted flow stress [14].  Table 2.

Quasi-Static Response
The true stress versus true stain curve for Hastelloy X under quasi-static conditions is shown in Fig. 3. It has two regions: the initial linear portion and the non-

Dynamic Constitutive Response at Room Temperature
The real time strain-pulses obtained for Hastelloy X at an average strain rate of 2500/s are shown in The dynamic true stress-true strain curve for different strain rates ranging from 1000/s to 4000/s are plotted along with the quasi-static true stress-true strain curve for Hastelloy X at room temperature in Fig 6. Hastelloy X shows rate dependency from quasi-static loading to dynamic loading. The dynamic yield strength is about 50% higher than the quasi-static yield strength, while the dynamic flow stress is about 200MPa higher than the quasi-static flow stress. The constituent elements in the material are responsible for the rate sensitivity of Hastelloy X. Under dynamic loading conditions, the material showed small increase in yield strength and flow stress as the strain rate is increased from 1000/s to 4000/s.

Dynamic Constitutive Response at Elevated Temperatures
A series of experiments were conducted to investigate the dynamic constitutive behavior of Hastelloy X at different temperatures under identical strain rate of about 5000/s. Fig. 7 shows the dynamic true stress-strain curves for Hastelloy X at different temperatures. All experiments were carried out with the same striker bar and at the same pressure. Fig. 7 clearly shows that as the temperature increases, the yield strength of the material decreases except at temperatures of 900°C and 1000°C. The true strain in the specimen increases and the flow stress decreases as the temperature increases. As the temperature increases, the material shows more ductile behavior and this result in a decrease in its yield strength and increase in the strain for a given applied stress. 700°C, then increases and shows a peak at 900°C and then again monotonically decreases as the temperature is further increased. The trend reported in this paper is in good agreement with the previous findings for Ni base superalloys [15][16][17][18][19][20] under quasi-static loadings. Bettge et al. [15] studied the temperature dependence of yield strength of Ni-base superalloy IN 738LC and concluded that the yield strength of the material decreases up to 450°C, then increases up to 750°C and finally decreases sharply. Sajjadi et al. [19] observed that the yield strength of GTD-111 under tensile loading decreases with increasing temperature and reaches a minimum value of 750MPa at 600°C. With a further increase in temperature to 750°C, the yield strength increased up to 1090MPa and then suddenly decreases. Three important factors were proposed [16,17,19] for the behavior of Ni base superalloys. These are (i) Grain boundary embrittlement, (ii) deformation mechanisms and (iii) ' γ coarsening. Grain boundary embrittlement by carbide particles and deformation mechanisms has almost 146 the same influence on the ductility of these alloys with temperature. The presence of transgranular cracks are more noticeable at higher temperatures but more intergranular cracks can be seen between 750°C to 950°C leading to increase in the yield strength of the material. Also the ratio of intergranular cracks to transgranular cracks decreases above 950°C. The transition of crack initiation on the surface from transgranular to intergranular is also noticed in other experiments of Hastelloy X [2,7,8,21]. It was also found that in the tensile properties of Hastelloy X above 950°C, there is a change in the fracture mode from transgranular dimple rupture to intergranular [3]. For Hastelloy X-280, the presence of ' 6 6 , C M C M and C M 23 phases in the 650°C to 930°C temperature range creates a ductility minimum at those temperatures [22].

Johnson Cook Constitutive Model Response
As the Johnson Cook model could only predict the stress in the plastic region, the elastic region was not considered for plotting. The comparison between the experimental data and the predicted data at various strain rates is shown in Fig. 9. A small deviation between the experimental data and the predicted data was observed at all three strain rates and the deviation was more at lower strain rates compared to higher strain rates. The predictability of the Johnson-Cook model for the dynamic plastic response of Hastelloy X was also quantified by calculating average relative error (Δ) and correlation coefficient(R). Average relative error was calculated using Eq. 10 and correlation coefficient (R) can be mathematically expressed as given in Eq.
11 [14].    Table 3. There is good correlation between the experimental data and the predicted data for all the three strain rates and the average relative error was low at higher strain rates compared to lower strain rates.
Higher value of 2 R (0.985) was observed for strain rate of 2500/s. Even though the value of 2 R (0.983) is high for a strain rate of 1700/s, it has a high relative error (7.24%) which indicates larger deviation between the experimental data and the predicted data. It should be noted that higher value of 2 R may not necessarily indicate better performance [23]. The model has the tendency to be biased towards higher or lower values, whereas Δ is calculated through a term by term comparison of relative error between the experimental data and the predicted data, therefore it produces unbiased statistics. Better correlation can be achieved if the value of Δ is low and this can noticed for a strain rate of 3900/s.  Table 3. Predictability of the model for the dynamic plastic response of Hastelloy X at room temperature under various strain rates.   Table 4. Predictability of the model for the dynamic plastic response of Hastelloy X at different temperatures.

150
The comparison between the experimental data and the predicted data at different temperatures is shown in Fig. 10. Model predictions for the dynamic plastic response of Hastelloy X at different temperatures are shown in Table 4. The model was not used to predict the dynamic plastic response of Hastelloy X above 700°C as the material showed different behavior after 700°C as discussed in the previous section. The model predicted very well for all the three temperatures chosen, with Δ ranging between 2.21% and 6.90%. Excellent correlation between the experimental data and the predicted data was achieved for the temperature of 700°C and this can be observed in Table 4 showing a low value of Δ (2.21%). Relatively high value of Δ (6.90%) was observed for a temperature of 500°C.

Conclusions
A series of experiments were conducted to investigate the dynamic constitutive behavior of Hastelloy X at room and elevated temperatures. Clay was used as a pulse shaper in all the experiments to improve force equilibrium conditions at the barspecimen interfaces. A Johnson-Cook model was used to predict the dynamic plastic response of Hastelloy X under varying strain rates and at different temperatures.
Following are the major conclusions of this study: • Quasi-static experiments showed yield strength of 380MPa and a Young's Modulus of 190GPa for Hastelloy X.
• The material showed rate dependency from quasi-static to dynamic loading.
An increase of 50% in yield stress and 200MPa in flow stress was observed from the quasi-static to dynamic loading. Under dynamic loading, the flow stress of the material decreases as the temperature increases.
• The yield strength of the specimen monotonically decreases as the temperature increases up to about 700°C. Between 700°C and 1000°C the yield stress increases before again decreasing as the temperature is further increased.
• The Johnson-Cook model predicted very well for all the experiments and the best correlation was observed for a strain rate of 3900/s (Δ=2.58%) in case of room temperatures experiments and for a temperature of 700°C (Δ=2.21%) in case of high temperature experiments. 156

Abstract
A comprehensive series of experiments were conducted to study the dynamic response of rectangular Hastelloy X plates at room and elevated temperatures when subjected to shock wave loading. A shock tube apparatus, capable of testing materials at temperatures up to 900 ºC, was developed. Propane gas was used as the heating source to effectively provide an extreme heating environment that is both robust and capable of providing uniform heating during shock loading. A cooling system was also implemented to prevent the shock tube from reaching high temperatures. High-speed photography coupled with the optical technique of Digital Image Correlation (DIC) was used to obtain the real-time 3D deformation of the specimen under shock wave loading. To eliminate the influence of thermal radiations at high temperatures, the DIC technique was used in conjunction with band pass optical filters and a high intensity light source to obtain the full-field deformation. In addition, a high speed camera was utilized to record the side-view deformation images and this information was used to validate the data obtained from the high temperature 3D DIC technique. The results showed that uniform heating of the specimen was consistently achieved with the designed heating system. For the same applied incident pressure, the highest impulse was imparted on the specimen at room temperature. As a consequence of temperature dependent material properties, the specimen demonstrated an increasing trend in back face (nozzles side) deflection, in-plane strain and out-of-plane velocity with increasing temperature.

Introduction
The objective of this study was to investigate the performance of Hastelloy X under a combination of extreme mechanical and thermal environments. For this purpose, a unique experimental set up was established to evaluate the performance of Hastelloy X at high temperatures when subjected to a controlled shock wave loading.
The present shock tube apparatus was modified for testing Hastelloy X at high temperatures (up to 900 °C). Propane gas was used as the heating source to effectively provide an extreme heating environment that is both robust and capable of providing uniform heating during shock loading. A 3D digital image correlation ( Hastelloy X is a nickel based superalloy that possesses excellent high temperature strength and oxidation resistance. It is widely used in gas turbine operations, petrochemical applications and in aircraft parts. Due to its high oxidation resistance, it is also used for structural components in industrial furnace applications . Several researchers have previously investigated the behavior of Hastelloy X.  investigated changes in the hardness and room temperature impact toughness of Hastelloy X after aging at 538 °C, 649 °C, 760 °C and 871 °C up to 10000 h. The author found that it exhibits age hardening at 649 °C and 760 °C. The results also indicated a slight increase in hardness at 871 °C followed by over aging after 4000 h. (Kondo et al., 1988) presented experimental results on the changes in the tensile mechanical properties of Hastelloy X after being used in the liner tube of a HENDEL hot gas duct under high temperature helium gas for about 6000 h. They observed that the 0.2 % proof stress and total elongation were slightly decreased when Hastelloy X was exposed to high temperature helium. plot was different from those at other temperatures and also the fatigue life significantly decreased at the total strain range less than 0.6 %. Recently,  studied the dynamic constitutive behavior of Hastelloy X at different temperatures ranging from 25 °C to 1100 °C and observed that the yield stress under dynamic loading decreases up to 700 °C, after which it shows a peak at 900 °C before beginning to decrease again as the temperature is further increased. They also observed that the amount of strain hardening and flow stress of Hastelloy X decreases with increase in temperature.
Also, there is a large volume of literature available on the response of monolithic plates and beams subjected to shock loading. (Menkes and Opat, 1973) characterized the response of aluminum beams under shock loading. They identified three major damage modes, which respectively are inelastic deformation, tensile tearing and transverse shear. Later, (Teeling-Smith and Nurick, 1991),  and  observed similar failure modes for circular, stiffened square and square plates, respectively. (Wierzbicki and Nurick, 1996) investigated thin clamped plates for different loading radii and applied impulse. They observed that the smaller loading radii led to a more localized damage towards the center of the plate, while the more distributed load produced tearing fracture at the clamped boundary. To date, there have been no studies reported on the damage behavior of aerospace materials under shock loading when exposed to extreme environments. Thus, the present study focuses on the dynamic response of Hastelloy X 160 when subjected to controlled shock loading under extreme environments. The results showed that uniform heating of the specimen was consistently achieved with the designed heating system. The high temperature flame proof paint used in the experiments withstood the temperatures and the intensive shock wave loading. For the same applied incident pressure, the highest impulse was imparted on the specimen at room temperature. However, due to the temperature dependent material properties, the specimen showed an increasing trend in back face deflection, in-plane strain and outof-plane velocity with increasing temperature.

Material and Specimen Geometry
The material used in this experimental study was Hastelloy X, supplied by Haynes International. The chemical composition of the material is listed in Table 1.
The table shows that Hastelloy X has a high content of Ni, Cr, Fe and Mo. The density of the material is 8220 kg/m 3 and the melting temperature ranges between 1260 °C and 1355 °C . The dynamic true compressive stress-strain curves of Hastelloy X for different temperatures at an average strain rate of about 5000 s -1 were obtained using a split Hopkinson pressure bar setup and are plotted in Fig.1 loading to dynamic loading. The dynamic yield strength was about 50 % higher than the quasi-static yield strength, while the dynamic flow stress was approximately 200 MPa higher than the quasi-static flow stress. At elevated temperatures, the yield strength of the material decreases as the temperature increases with an exception at 900 °C and 1000 °C. The true strain in the specimen increases and the flow stress decreases as the temperature increases . This behavior is due to an increase in material ductility at elevated temperatures. The Johnson-Cook parameters were determined for Hastelloy X from the experimental data. In the present study, rectangular plates 203 mm long, 51 mm wide and 3 mm thick were used in the experiments.

Experimental Procedures
For the present study, three experiments were conducted for each temperature in order to ensure repeatability as well as to better evaluate specimen performance and behavior under shock loading. Experiments were carried out for 25 °C, 360 °C, 700 °C and 900 °C temperatures.

Shock Tube
The shock tube apparatus was utilized to generate controlled blast loadings as shown in Fig. 2(a) Tekalur et al., 2009;). The shock tube has an overall length of 8 m, consisting of a driver, driven and muzzle section. The high-pressure driver section and the low pressure driven section are separated by a diaphragm. By pressurizing the high-pressure section, a pressure difference across the diaphragm is created. When this difference reaches a critical value, the diaphragm ruptures. This rapid release of gas creates a shock wave, which travels down the tube to impart dynamic loading on the specimen. A typical pressure profile obtained from the transducer closest to the specimen at room temperature is shown in Fig. 2(c). In the present study, a simply stacked diaphragm of 2 plies of 2.5 mil mylar sheets with a total thickness of 0.5 mm was utilized to generate a shock wave loading that impinged on the specimen with an incident peak pressure of approximately 0.25 MPa, a reflected peak pressure of approximately 0.72 MPa, an incident shock wave speed of 665 ms -1 and a reflected shock wave speed of 235 ms -1 . The shock wave generated had a short rise time (~80 µs) and showed an exponential decay period of approximately 4500 µs.

High Temperature Shock Tube Setup
The shock tube set up was modified for testing Hastelloy X at high temperatures. Initial efforts of using quartz lamps as the heating source were given up due to the fragile nature of the heaters. It was determined that the quartz lamps would not be suitable to withstand the intense and violent loads developed during shock loading since the quartz lamps would have to be placed very close to the specimen in order to direct most of the heat onto the specimen. Later, it was decided to use propane gas as the heating source and it was directed onto the specimen via four nozzles as shown in Fig. 3. Two nozzles were located on each side of the specimen. The nozzles on the left side were directed at the center of the specimen, while the nozzles on the right side were directed towards the top and bottom of the specimen, respectively. Different temperatures were obtained by adjusting the distance between the specimen and nozzles. For each temperature, the angle and intensity of all four flames were adjusted to obtain uniform heating of the specimen between the supports. The temperature on the front face (blast side) of the specimen was monitored at three different locations (between the supports) using chromel-alumel thermocouples to ensure uniform temperature distribution on the specimen. The temperatures recorded at these locations for 360 °C, 700 °C, and 900 °C are shown in Fig. 4. It can be seen that a uniform temperature was achieved between the supports for all the temperatures with a maximum gradient of +15 °C. Later, temperatures were also recorded on the front face (blast side) and back face (nozzles side) of the specimen using chromelalumel thermocouples and infrared thermometer respectively to ensure uniform temperature on both sides of the specimen. To protect the supports from the high temperatures, a high temperature flame resistant paint was applied and then wrapped with a heat shield. A cooling system was also implemented to prevent the muzzle section of the shock tube from reaching high temperatures. This was crucial in protecting the highly sensitive pressure sensors located towards the end of the muzzle.
The cooling system, as shown in Fig. 5, consists of a copper coil tightly wound around the muzzle end in which tap water is circulated. In addition, a distance of 5 mm was maintained between the specimen and the muzzle to protect the shock tube from reaching high temperatures. For a distance of 5 mm away from the specimen, a maximum temperature of 70 °C was observed at the location where the closest pressure sensor was mounted. Both heating and cooling systems together permit the specimen to be heated to temperatures up to 900 °C while protecting the surrounding equipment, thus allowing for precise pressure measurements during each experiment.
For each experiment, the temperature at the center of the specimen was monitored during the shock loading. Fig. 6 shows the temperature vs. time curves for 360 °C, 700 °C and 900 °C experiments. Fig. 6 clearly shows that the steady temperature was achieved in the specimen before the shock wave loading occurs. At  Real speckle pattern the instant the shock wave impinges on the specimen, the thermocouple disengages from the specimen and the recorded data shows a peak in the temperature-time curve.

High Speed Photography Systems
Two high-speed photography systems were utilized to capture the real-time 3D deformation data of the specimen during the blast experiment. The experimental setup, shown in Fig. 7, consisted of a back-view 3D Digital Image Correlation (DIC) system with two cameras facing the back side of the specimen to obtain the real-time full field data. Another high speed camera was placed perpendicular to the side surface of the specimen to capture the side-view deformation images. All cameras were Photron SA1 high-speed digital cameras and were used at a frame-rate of 30,000 fps with an image resolution of 256×512 pixels for a one second time duration. These cameras were synchronized to ensure that the images and data can be correlated and compared.

High Temperature DIC System under Dynamic Loading
The 3D DIC technique is one of the most recent non-contact methods for analyzing full-field shape, motion and deformation (Gardner et al., 2012;Luo et al., 1993;Sutton et al., 2009). Two cameras capture two images from different angles simultaneously. By correlating these two images, one can obtain the three dimensional shape of the surface. Correlating this deformed shape to a reference (zero-load) shape gives full-field in-plane and out-of-plane deformations. However, there were limitations in using DIC for high temperature applications. It was observed that when a material exceeds a certain temperature, the self radiation of the heated material leads to decorrelation effects in the images. This phenomenon occurs because the radiated light of the heated material will dramatically intensify the brightness while decreasing the contrast of the captured image . Therefore, most of the high temperature DIC applications were limited to 600 °C (Bing et al., 2009;De Strycker et al., 2010;. Recently, (Grant et al., 2009)  When an object is heated, it emits electromagnetic waves of all wavelengths ranging from infrared to ultraviolet depending on the temperature of the object. As the temperature increases, the peak (primary) wavelength slowly shifts towards the shorter wavelength and the amount of thermal radiation increases very fast. To extract accurate and reliable information using DIC, the specimen is required to have either a natural or artificial speckle pattern with random variations of gray scale intensity which serves as a carrier of deformation information. For high temperature experiments, commercially available flame proof paint (VHT) which can sustain temperatures up to 1200 °C was used to speckle the specimen. Fig. 9 shows the proposed experimental set up with bandpass filters, high speed cameras and the high energy flash unit. The image recorded using this proposed set up for Hastelloy X at 900 °C without any loading is shown in Fig. 10. When the image at 900 °C (Fig. 10b) is compared to the reference image (Fig. 10a), there was no significant change in the  intensities of these images. This indicates that the bandpass filters eliminated the blackbody radiation and helped acquire high quality image. Also, Fig. 10b ensures that the flash unit provided sufficient light for the specimen. The captured images using the proposed set up can be used to generate the full field deformation data.

Pressure profile & Impulse
The pressure profiles measured for Hastelloy X at different temperatures under shock loading are shown in Fig. 11a. Three important observations can be made from these pressure profiles. (i) The incident and reflected peak pressures are the same for different temperatures. This is consistent with the previous findings  which showed that the reflected peak pressure for different materials only period of the reflected pressure decreased from 9 ms at 25 °C to 6.5 ms at 900 °C. Due to thermal softening at higher temperatures, higher deflections were observed in the specimens which caused the pressure to drop relatively faster than at room temperature. Thus, the rigid plate will experience reflected pressure for a longer duration and obtain more impulse from the blast loading. (iii) Humps are observed during the decay of the reflected pressure profile for different temperatures. These humps are an artifact of the specimen reverberating after reaching its peak deflection as shown in Fig. 11b. As the temperature increases, the specimen takes a longer time to reach its maximum deflection and therefore the humps for high temperature experiments are observed at later times. Fig. 12 shows the impulse imparted on the specimen for 25 °C, 360 °C, 700 °C and 900 °C temperatures. The impulse imparted on the specimen is calculated by integrating the force-time data of the reflected pressure profile. As discussed above, at higher temperatures, the time period of the reflected pressure profile decreased which reduced the impulse imparted on the specimen. Since the overall time period of the reflected pressure profile was longer at room temperature and the reflected peak pressure was the same for different temperatures, the specimen at room temperature obtained a maximum impulse of 2.9 Ns from the blast loading. When the temperature was increased from 25 °C to 900 °C, the maximum impulse achieved in the specimen decreased by approximately 41% (2.9 Ns to 1.7 Ns).

Real-time Deformation Images
The real-time observations of the transient behavior of Hastelloy X under shock loading for different temperatures are shown in Fig. 13. The shock wave is propagating from the right side of the image to the left side causing deformation in the specimens. Time t=0 µs represents the beginning of the event where the shock wave impinged on the specimen. The side-view images were shown at different times during the event. The final image for each temperature represents the maximum defection of °C reached its maximum deflections at 2300 µs, 2900 µs and 3600 µs respectively.

Digital Image Correlation Analysis
The deflection, in-plane strain (ε yy ) and velocity of the back face for each temperature were obtained using the 3D Digital Image Correlation (DIC) technique.
For higher temperatures, as discussed in section 3.3.1, the DIC technique was used in conjunction with an optical blue pass filter and high intensity light source. Since the thermal radiation greatly influences the image quality above 600 °C, the results obtained using the proposed DIC method for 700 °C and 900 °C need to be validated.
To verify the performance of the proposed DIC method, the back face mid-point deflections obtained using the proposed DIC technique were compared with the deflections measured from the high speed side-view images (Fig. 13). The mid-point back face deflections obtained using DIC for different temperatures is shown in Fig. 17. These deflections were also in agreement with the deflections measured using the high speed images (as shown in Fig. 13). All of the curves exhibit a maximum plastic deformation in its first peak, followed by elastic The average in-plane strain-rate observed in these experiments was 10 s -1 .
Using the Johnson-Cook parameters as listed in

181
The full field out-of-plane deflection (W) images for different temperatures, with a scale of 0 mm to 13.5 mm, are shown in Fig. 18. For a better comparison, the time scales were kept constant for all temperatures. These images ensure that the fullfield deformation data can be obtained from the modified DIC technique under shock wave loading at high temperatures. At time t=0 µs, the shock wave impinges on the specimen. For all temperatures, local deflection contours were observed during the early time of the event and the specimens exhibited very minimal out-of-plane deflections up to t=300 µs . For the specimen at room temperature, between t=300 µs and t=1900 µs, a significant amount of deflection was observed. At this time (t=1900 µs ), the specimens at 360 °C, 700 °C and 900 °C had shown significant deflections of 6.6 mm, 7.6 mm and 8 mm respectively. After t=1900 µs, as the specimen at room temperature began to reverberate, the specimens at higher temperatures continued to deform. The specimen at 360 °C deformed further for 400 µs before reaching its maximum deflection of 7.0 mm at 2300 µs. In comparison to the room temperature experiment, the specimens at 700 °C and 900 °C continued to deform further for 1000 µs and 1600 µs respectively before reaching their maximum deflections. The specimen at 900 °C deformed for a longer time of 3600 µs.
The in-plane strain (ε yy ) on the back face of the specimen is plotted in Fig. 19.
It can be observed from the figure that all the specimens exhibit very minimal in-plane strain of 0.05 % up to t=400 µs. After t=400 µs, the specimens showed significant bending and resulted in higher in-plane strain values. At room temperature, a maximum in-plane strain of 0.4 % was observed at t=1865 µs. As the temperature increases, the specimens exhibited higher deflections (more bending) causing an and 212 % respectively. Since the strains exceeded the elastic limit, the increase in yield stress at 900 °C (as explained in section 2) had no effect on the performance of Hastelloy X under shock loading.
The out-of-plane velocity on the back face of the specimen is shown in Fig. 20.
All experiments at different temperatures showed similar velocity profiles up to t=750 µs. At t=750 µs, all the specimens showed an approximate velocity of 4.25 ms -1 . The specimen at room temperature reached a maximum out-of-plane velocity of 4.5 ms -1 at t=875 µs and began to decelerate after that. In comparison to room temperature, the specimen at 360 °C reached same maximum velocity of 4.5 ms -1 (at t=975 µs) but was delayed by approximately 105 µs. A maximum out-of-plane velocity of 5.0 ms -1 and 5.6 ms -1 was observed at t=1100 µs and 1400 µs for the specimens at 700 °C and 900 °C respectively. As observed at room temperature, the specimen at higher temperatures also begins to decelerate at a later time.

Deformation Energy Evaluation
After the incident shock wave impinges upon the specimen, a reflected shock wave is generated and the reflected pressure is applied on the panel to deform the specimen. The total work done by the gas to deform the specimen is defined as the deformation energy. The deformation energy is calculated by evaluating the deflection-time data from the high speed images (Fig. 13) and the force-time data from the reflected pressure profile. Since the force was applied on the front face of the specimen, the deflection of the front face of the specimen was determined. By using curve fitting method, such as cubic spline curve fitting, the shape of the deformed front face was matched and the deflection of every point inside the shock loading area was calculated along the front face. Combining the deflection-time data and the force-time data will result in force-deflection data. Now the deformation energy can be obtained by integrating the force-deflection data  temperature. This indicates that the specimens at higher temperatures consume more energy during the shock loading process.

Post-mortem Analysis
The post-mortem images of the specimens tested at 25

Conclusions
A series of experiments were conducted to investigate the dynamic behavior of Hastelloy X at room and elevated temperatures under shock wave loading. A 3D DIC technique was used in conjunction with an optical blue pass filter and high intensity light source to record the transient behavior of Hastelloy X at elevated temperatures.
Following are the major conclusions of this study: • A unique experimental set up was built for studying materials at high temperatures (up to 900 °C) under shock loading.
• The DIC system has been modified and validated for obtaining the full-field data at elevated temperatures under dynamic loading. The mid-point deflections obtained from the DIC technique for 700 °C and 900 °C match very well with the deflections obtained using the high speed side-view images.
• The characteristic fluid structure time is determined to be ~100 µs for all the temperatures studied.
• The impulse imparted on the specimen decreased as the temperature increased from 25 °C to 900 °C.
• The maximum back face deflection of the specimen at 900 °C is approximately 160 % higher than the specimen at room temperature.
• The maximum in-plane strain showed an increase of 212 % as the temperature increased from room temperature (0.4 %) to 900 °C (1.25 %).

Conclusions
The main objective of this investigation is to advance the development of (1) The thermo-mechanical stress fields around the crack-tip are significantly affected by the curvature, temperature and non-homogeneity parameters. As the curvature at the crack-tip increases, the peak value of the stress components also increases.
(2) Crack-tip velocity plays a significant role in the crack extension angle. For a mixed-mode curving crack under thermo-mechanical loading, the crack extension angle increases with the increase in crack-tip velocity. Similar behavior was reported earlier for the case of homogeneous and FGMs without the curvature parameters. ).
(4) The thermo-mechanical stress fields are not significantly affected by transient parameters. As the crack-tip acceleration increases, the size of stress contours ahead of the crack-tip decreases slightly. There is not much significant effect on the rate of change of stress intensity factor.
The methods used to evaluate the energy as described by  were implemented and the results analyzed. The energy needed to deform the specimen increased from 10J to 30J as the temperature increased from 25°C to 900°C.

Future Work
The current research is a step forward in understanding the dynamic response of aerospace materials under extreme environments. It elucidates a more comprehensive understanding on the dynamic behavior and mechanisms of failure of materials subjected to high-intensity loadings. Different materials, ranging from functionally graded materials, Hastelloy X and Ti 2 AlC were studied. The proposed future projects are as follows, 1. The specimen dimensions should follow the below relation: where L is the length, D is the diameter and υ is the Poisson's ratio of the specimen.
2. Make sure the faces of the specimen are parallel and flat (use step collet while machining metal specimens to get parallel faces). The diameter of the specimen after the test should be smaller than the diameter of the pressure bars.

Selecting the bar:
1. Determine the impedance ( cA ρ ) of the specimen.
where ρ is the density, is the wave speed and A is the area.
2. Then select the pressure bars (steel or Aluminum) closer to the impedance of the specimen. We also have different diameters for the pressure bars.
Note: The basic thumb rule is that we use steel bars for the harder materials the lower the strain rate. The higher the pressure, the higher the strain rate).
(k) Lubricate both faces of the test specimen with Molybdenum disulfide lubricant and sandwich the specimen between the bars and align the specimen with respect to bar center.
(l) Place the pulse shaper at the impact end of the incident bar with a thin layer of KY jelly grease (if you are using lead pulse shaper) and align it with respect to bar center. We generally use clay and lead pulse shapers. These give us very good results for harder materials, but for the softer materials, you can try different pulse shapers. These include paper, copper etc.
(m) Release the nitrogen gas from the gas tank into the gas gun chamber until the required pressure level is achieved.
(n) Arm the oscillation to capture the strain gage voltage signals and make sure the arm holds until you release the projectile. If the arm is not holding, adjust trigger levels. (Note: if you are getting high noise in your signals more than 20mv, turn off the tube ights before the experiment).
(o) Once again, ensure that the specimen is well aligned between the bars and verify the status of the trigger hold before pressing the solenoid valve release button.
(p) Press solenoid valve control box button to release the projectile.
(q) Save captured voltage pulses onto a USB drive for further analysis of the data.
(r) A MATLAB program is written to read the data from the pulses and analyze the pulses using the one-dimensional wave theory stress and strain equations. After the experiments are performed, the pulses are used along with the MATLAB program to determine the equilibrium and true stressstrain plots of the specimen.
(s) After the experiment is completed, turn off the cylinder and make sure all the left over nitrogen gas in the gas chamber is released.
(t) After the data is transferred from the oscilloscope to USB drive, verify that in your computer and turn off the amplifier and oscilloscope.
Analyzing the results: 1. There are two MATLAB codes to analyze the data. 1. Verify_Equilibrium and 2. Steel/aluminum_SHPB. Use the appropriate codes to analyze the data.
Depending on the bars you used, the respective code has to be used.

199
2. Make sure the code has the right properties and dimensions of the pressure bars you used. These include diameter, wave speed, and diameter.
3. If we use hollow tubes, make sure you have the right dimensions in the code.
For solid bars, dimensions for the hollow tube should be zero.
4. First run the verify equilibrium code. Make sure the data you get from the oscilloscope has the following names for the four channels. TEK00000, TEK00001, TEK00002,and TEK00003. The code recognizes these names.
Make sure the codes and the data are in the same folder. 8. When you run the code, you get two figures. Figure 1 gives the incident and reflected pulses. Figure 2 gives the transmitted pulse. pulses by pressing 'zoom in' button at the top to get the right times. Then go to MATLAB main window and press 'ENTER'. 10. Input the values you found out and press 'ENTER'. 11. Now you will get 3 more figures. Figure 3 shows the incident, reflected and transmitted pulses. Figure 4 shows the incident, reflected and transmitted pulses you picked on before. Figure 5 shows the force ratio. Front face represents the forces calculated on the incident and reflected pulses. Back face represents the force calculated on the transmitted pulse. Ideally, these two fronts and back face should match perfectly.
12. Various factors decide the equilibrium. These include type of material tested, strain rate etc..
13. Make sure the incident and reflected pulses start at the same time on 16. Now open the SHPB code and make sure you have the same value for filter as in the verify_equilibrium code.
17. Enter the specimen thickness and diameter in inches.
18. Again, you get two figures. Figure 1 gives the incident and reflected pulses. 24. Go to MATLAB main window and you can see the strain rate. Note down this value. Next you will end up with final figure (Figure 6). This is eng. strain rate vs. time. 25. Be careful when you pick up the strain rate points. Consider the following points a. Make sure the region you pick is in the equilibrium. 2. Selecting the bar is same as explained before.
3. Experimental procedure is also similar to the above. Here, you place the pulse shaper on the flange. You can use paper, clay or lead.
4. Different striker bars can be used to perform experiments at different strain rates. Make sure the striker bar slides freely on the bars. 5. The specimen will be threaded at both ends to the pressure bars. There is no need to use the lubricant. 6. The connections remain the same as explained before. You can use the same amplifier and oscilloscope, and same settings. 7. The MATLAB codes have been modified and use the appropriate code to perform your analysis. The steps to run the code is same as explained for compression SHPB. 1. The tungsten carbide inserts will be used. The specimen will be sandwiched between these inserts.
2. The diameter of the specimen should be smaller than the inserts. The below figure shows the set up.

For experiments at elevated temperatures, the SHPB apparatus in conjunction
with the induction coil heating system will be utilized as shown in Fig. 2. 4. A special fixture is used to load the specimen. 5. The inserts were used to eliminate the temperature gradient in the bars and thus protect the strain gages mounted on them.
6. The impedance of the inserts was matched with the bars; hence they do not disturb the stress wave profiles in the bar. The impedance matching requires the diameter of these tungsten carbide inserts to be smaller than the main pressure bars. This is the reason for the specimen diameter for high temperature testing being smaller than that for room temperature testing. 7. By varying the power, higher temperatures can be achieved. 10. Make sure the wheel on the cooling unit is pinning smoothly and fast. If not, do not do the experiment. Increase the power supply, to heat the specimen.
11. When the regulator is turned ON, it should give a click sound after around 30s.
If it does not, turn it off and try again. If the problem persists, turn off the regulator and the problem can be determined.
12. Turning ON the power supply regulator, it will read 'cycle continuous' on the remote (smallest one), which is desired. 13. The system should already be set to manual power output again, which will allow to control the power. If it is not set, you can do by using the switch located to the immediate right of the dial on the regulator.
14. Make sure the dial on the regulator is zero, so there will be no immediate power output.
15. Now press 'start' button on the remote (small one that reads the diaplay). 16. The bars were kept apart initially, later the specimen and carbide inserts were heated in isolation to the desired temperature (usually about 20-50°C higher than the test temperature) and soon after the bars were brought manually into contact with the specimen. The temperature of the specimen was monitored using a 0.127mm chromel-alumel thermocouple, which was spot welded onto the specimen. 17. In most of the experiments, it takes less than two minutes to heat the specimen to the required temperature and it takes less than 10 seconds to bring the pressure bars into contact with the tungsten inserts and fire the gun.
18. Once the temperature is reached, hit 'stop' on the display and turn off the regulator and the induction heater. Now trigger the oscilloscope. If you trigger the oscilloscope before, due to magnetic fields from the induction heater, you will see lot of noise.

205
19. Allow the cooling unit to run for some time son that it reaches room temperature.
20. All other experimental procedure, data capturing, and analyzing the results remain the same as explained in compression SHPB section.

Note:
1. Always make sure the yield strength of the material you are testing is never beyond the yield strength pressure bars.
2. For testing ceramics of high strength, we need to use inserts so as to protect the bars from plastic deformation.

IMACON200A
Set camera facing desired event 1. Connect control i/o cable, which runs from camera to computer, to camera (should already be attached to computer) 2. Connect a BNC cable from the monitor output of the camera (clearly marked) to the stereo converter box used to connect trigger to photogenic flash light.
3. Connect a BNC cable to trigger 1 (clearly marked) on the camera, and utilizing a T joint, connect the trigger to the oscilloscope, with a second BnC branching to the make circuit used to trigger the apparatus. 4. When selecting which Photogenic to use, be sure that one is the flash with the bulb capable of producing persistent light -this will be identifiable as the bulb with an additional, smaller bulb, situated in the center of the flash ring of the rest of the bulb.  11. FINAL NOTE: The Imacon 200 was the first of its kind, and is worth $500,000. Treat it with respect -handle it and its components with care

ENERGY ANALYSIS
This manual is designed for the steps to use the energy and impulse analysis code.
(1) Obtain the original data The original data of shock tube experiments are from the Tektronix oscilloscope (TDS3014 or 3014C). The data must have following name: First channel: TEK00000.csv Second channel: TEK00001.csv Normally, there are two columns in these files. The unit of the first column is second (s). The unit of the second column is voltage (v).Please copy these files into the folder named "experimental data backup".
(2) Analyze the original data. This step is to analyze the original data to obtain the shock wave velocity, the peak pressure and the modified pressure profiles.
This step is carried out in the folder named "original data analysis". In this folder, the m file named profile_analysis.m is necessary. Other files can be deleted or replaced.
You must copy the original data files into this folder and then run the code. The running process is as follow, (1) The code will first ask you how many plys you use in the experiment. This information is only for your record. It does not matter the analysis process.
(2) The code will ask you the sensitivity of the sensors. This value is given in the box of the sensors. This value means how many milli-voltage related to 1 psi.
(3) Then, a figure with two plots of the pressure profiles will be given. Look at

217
(4) Then the code will inform you as follow, "The peak and velocity data have been saved into the file, which isnamed peak&velocity.txt and in the same folder of this code.
There are two more pressure data files in this folder: inc_sp.dat ref_sp.dat They can be used for energy and impulse evaluation.
The code will give some plots to verify your data.
Please double check them very carefully.
press any key to continue" (5) After pressing any key, the code will give four images:  The first two files will be used to analyze the energy and impulse. There are two column data in these two files. The first column is time with unit second (s). The second column is pressure data with unit psi.
The last file records the physical parameters, which needs to be input in the energy and impulse analysis.
(7) Please cut these three data files into the folder named "experimental data backup" and delete all of these files in the current folder.

(3) Analyze the incident and remaining energy and impulse
This step is to use the data obtained in step 2 to analyze the incident and remaining energy and impulse in a shock tube experiment.
This step is carried out in the folder named "gas energy and impulse analysis". In this folder, seven m files are necessary. They are, Main code: energy_impulse_analysis.m Other files can be deleted or replaced. You must copy the data files: inc_sp.dat and ref_sp.dat, into this folder and the correlated blank test data file, inc.dat, from the folder named "blank test data". The code running process is as follow, (1) The code will first show the format and unit of the data. Please be sure that the data should be the exact format.
(2) Then the code will give the total number of the data and ask you how many points you want to skip. This means if the original data is (3) Then the code will ask you to input physical parameters obtained in step 2 (saved in peak&velocity.txt).
(4) The code will calculate the physical parameter profiles. This process is automatic.
(5) Then the code will integrate the parameter profiles to obtain the energy components. This process is automatic.
(6) Finally, the code will ask you to input the name of the file which you want to save data into. Then all of the data will be saved into filename_gas_energy.mat. The energy and impulse components are saved into ten files as follow, Please copy and save these data into a safe folder.
(4) Analyze the deformation energy of the gas, momentum and kinetic energy of the specimen This step is to use the data obtained in step 2 and the high-speed side-view images to analyze the deformation energy of the gas, momentum and kinetic energy of the specimen in a shock tube experiment.
This step is carried out in the folder named "specimen momentum and energy". In this folder, only one m files, Deformation_momentum_kinetic_photron.m, are necessary.
Other files can be deleted or replaced. You must copy the data files: ref_sp.dat, and a series of high-speed images into this folder. The code running process is as follow, (1) The code will first show the format and unit of the data. Please be sure that the data should be the exact format.
(2) load the time series of the images You will have three ways to load the time series of the images.
(i) The time between two frames is same. You can input total number of frames and time between two frames. Then the code will generate the time series automatically.
(ii) The time between two frames is not same. You can input total number of frames and input time between two frames frame by frame.
(iii)The time between two frames is not same. The time between two frames is not same. Then you can just load that time series data file.
You can choose anyone and following the instruction.
(3) Length calibration. You can choose any image for length calibration. You will need to choose two points on this image and the vertical distance between these two points will be used to calibrate the length. Therefore, you need to know one real vertical scale between two points on the image.
For example: (i) the span of the supports is 6 inches (ii) the outer diameter of the shock tube is 5 inches The process will repeat three times. Thus, totally you will pick six times.
Please follow the instruction.
(4) Then you can do real measurement. Normally, you need to measure the deformation shape of the front face for every image. For each image, you 221 need to choose seven points on the front face. The code will let you pick up the top point of the front face first. Then it will let you pick up the bottom point of the front face. After this, the code will base on these two points to draw seven lines with equivalent distance. Please pick up the points from the top to the bottom.
(5) The code will ask you to input the mass of the specimen.
(6) The code will ask you to input the name of the file which you want to save you data into.

Dynamic Constitutive Response at Room Temperature
The real time strain-pulses obtained for Ti 2 AlC at an average strain rate of 2500/s are shown in Fig 1. The clay pulse shaper used in all these experiments helped to reduce high frequency oscillations in the incident stress wave and improved the force equilibrium conditions at the specimen-bar interface.
The dynamic true stress-true strain curve for different strain rates ranging from 1500/s to 4500/s are plotted for Ti 2 AlC at room temperature in  By t=100 μs, more cracks are clearly visible and they have completely propagated through the specimen. After this time, the damage begins to grow and the outer edge of the specimen breaks off, while the center part of the specimen stays intact.