Thermal and mechanical analysis of material response to non-steady ramp and steady shock wave loading

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Journal of the Mechanics and Physics of Solids 54 (2006) 237–265 www.elsevier.com/locate/jmps

Thermal and mechanical analysis of material response to non-steady ramp and steady shock wave loading J.L. Ding School of Mechanical and Materials Engineering and Institute for Shock Physics, Washington State University, Pullman, WA 99164-2920, USA Received 5 March 2005; received in revised form 9 September 2005; accepted 13 September 2005

Abstract Ramp wave experiments on the Sandia Z accelerator provide a new approach to study the rapid compression response of materials at pressures, temperatures and stress or strain rates not attainable in conventional shock experiments. Due to its shockless nature, the ramp wave experiment is often termed as an isentropic (or quasi-isentropic) compression experiment (ICE). However, in reality there is always some entropy produced when materials are subjected to large amplitude compression even under shockless loading. The entropy production mechanisms that cause deformation to deviate from the isentropic process can be attributed to mechanical and thermal dissipations. The former is due to inelasticity associated with various deformation mechanisms and the rate effect that is inherent in all the deformation processes and the latter is due to irreversible heat conduction. The main purpose of the current study is to gain insights into the effects of ramp and shock loading on the entropy production and thermomechanical responses of materials. Another purpose is to investigate the role of heat conduction in the material response to both the non-steady ramp wave and steady shock. Numerical simulations are used to address the aforementioned research objectives. The thermomechanical response associated with a steady shock wave is investigated first by solving a set of nonlinear ordinary differential equations. Using the steady wave solutions as the reference, the material responses under non-steady ramp waves are then studied with numerical wave propagation simulation. It is demonstrated that the material response to ramp and shock loading is essentially a manifestation of the interaction between the time scale associated with the loading and the intrinsic time scales associated with mechanical deformation and heat transfer. At lower loading rates as Tel.: +1 509 335 3226; fax: +1 509 335 4662.

E-mail address: [email protected]. 0022-5096/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jmps.2005.09.003

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encountered in ramp loading, the loading path is closer to an isentrope and results in lower entropy production. The reasonable ramp rate to obtain a quasi-isentropic state depends on the intrinsic time scales of the dissipation mechanisms which are strongly material dependent. Thus shockless loading does not necessarily produce an isentropic response. Between two equilibrium states, heat conduction was shown to have significant effect on the temperature history but it contributes little to the overall temperature change if the specific heat remains constant. It also affects the history of entropy, but only the irreversible part of heat conduction contributes to the net entropy change. The various types of thermomechanical responses of materials would manifest themselves more significantly in terms of the thermal history than the mechanical history. Thus temperature measurement appears to be an important experimental tool in distinguishing the various mechanisms for the thermomechancial responses of the materials. r 2005 Elsevier Ltd. All rights reserved. Keywords: Shock wave; Ramp wave; Isentrope; Thermal dissipation; Mechanical dissipation

1. Introduction Equation of State (EOS) is an essential piece of information for describing the material behavior under high pressure. A well established experimental method for developing EOS is the shock wave experiment. In this type of experiment, material is compressed through shock loading generated by plate impact, explosives, or energy deposition. The equilibrium state attained at the end of the steady shock loading path can be determined by the use of conservation laws for mass, momentum, and energy combined with experimental data associated with the equilibrium states, typically shock speed and particle velocity or stress. A locus of the equilibrium states attained through various steady shock loading paths constitutes the so-called principal Hugoniot, which is essentially an EOS constrained by the jump conditions for shock waves. Using the Hugoniot as a reference, a general EOS, e.g. the Mie-Gru¨neison EOS, may be established. The details about shock compression experiments and analysis, Hugoniot, and EOS can be found in Asay and Shahinpoor (1993). Besides shock wave experiments, another experimental technique that has been developed for EOS measurement is the ramp wave experiment. With the proper selection or design of materials for the impactor, ramp waves can be generated in a plate impact experiment. The materials that have been used for the impactor in plate impact ramp wave experiments include fused silica (Barker and Hollenbach, 1970), ceramics (Asay and Chhabildas, 1980), material with graded density (Barker, 1983), and layered material (Chhabildas and Barker, 1987). More recently, a significant development in the ramp wave experiment has been made at Sandia National Laboratories. Their experiment uses pulsed power as the energy source and is capable of driving very large amplitude (several Mbar) ramp wave. The details of this technique can be found in Asay (1999), Hall et al. (2001). Unlike weak shock, the wave profile of a strong or large amplitude shock is not discernable except for its terminal equilibrium state. In contrast, the large amplitude ramp wave experiment developed at Sandia provides a continuous wave profile that extends to the high pressure regime and has a finite rise time. Thus, the experiment provides a new and unique loading path to explore the material behavior under extreme pressure.

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Due to its shockless nature, the ramp wave experiment is often termed as an isentropic (or quasi-isentropic) compression experiment (ICE). For a deformation process to be isentropic or quasi-isentropic, the entropy change has to be close to zero. However, in reality there is always some entropy change associated with large amplitude compression even under shockless loading. The change could be due to heat transfer or entropy production associated with the irreversible processes. The entropy production can be generally attributed to mechanical and thermal dissipations. The former is due to inelasticity associated with various deformation mechanisms including phase transformation and the rate effect that is inherent in all the deformation processes, while the latter is due to irreversible heat conduction. Because of the inherent entropy change associated with high rate, large deformation processes, a pressing and critical question is how close the resultant stress strain curve obtained from the ramp wave experiment is to isentrope. This question also brings up some other fundamental issues in the material response to large amplitude ramp and shock wave loadings, such as the sources of entropy change and the relative roles between mechanical work and heat conduction and their interaction, etc. The quantitative answers to these questions are of course material specific. One major purpose of the current research is to gain some general insight and a qualitative understanding of these issues that are essential to the full utilization of this new experimental capability. Another purpose is to investigate the role of heat conduction in the material response to both non-steady ramp and steady shock wave loadings. The role of heat conduction in high strain rate deformation has been a long standing issue. It is usually neglected, i.e. the deformation is treated as adiabatic. However, for strong shock it is known that heat conduction can be significant (Wallace, 1980, 1981a; Drumheller, 1998). Thus it seems necessary that the role of heat conduction in large amplitude or strong ramp wave should at least be clarified. Since the relative roles of the mechanical work and heat conduction and their associated dissipation mechanisms would manifest themselves more significantly in terms of the temperature history, this part of the study is parallel to the analysis of the temperature profile associated with a large amplitude wave.

2. Formulation To facilitate the discussion, the differential equations associated with the conservation laws for thermomechanics are listed in the following. Details of these equations can be found in Malvern (1969). Conservation of mass: qr þ r
ðr uÞ ¼ 0, qt

(1)

where r is the current density and u the particle velocity; Conservation of linear momentum: sji;j þ rf i ¼ ru_ i ,

(2)

where sij is the Cauchy stress tensor and fi the body force per unit mass, u_ i is the material derivative of particle velocity (ui), i.e. acceleration;

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Conservation of energy: r_e ¼ sij e_ij  qi;i þ rr,

(3)

where e is the internal energy per unit mass, e_ij is the rate of deformation tensor or natural strain, qi is the outward heat flux vector and r is the internal heat source per unit mass. To address the entropy change, the second law of thermodynamics needs to be considered. The law can be written in the following form: r

ds r Xr  ðqi =yÞ;i , dt y

(4)

where s is the entropy per unit mass and y is the absolute temperature. In terms of entropy production rate ð_gÞ, it can also be expressed as   ds r 1
qi  g_ ¼   (5) ¼ g_ mech þ g_ con , dt y r y ;i where g_ mech

  ds r 1  q X0 ¼  dt y ry i;i

(6)

represents the entropy production due to mechanical dissipation, and g_ con ¼ 

1 qi y;i X0 ry2

(7)

represents the entropy production due to thermal dissipation associated with heat conduction. In the current study, the mechanical dissipation is assumed to be due to inelastic deformation and viscous stress svij . The latter can be written as svij ¼ Qdij þ s0vij where Q ¼ svkk =3 is the mean viscous stress and s0vij is the deviatoric viscous stress. The total strain ðeij Þ is equal to eeij þ epij where eeij and epij are the elastic and inelastic strain respectively, and the total stress ðsij Þ is seij þ svij where seij is the elastic or equilibrium stress. With the above decomposition of stress and strain, Equation (3) can then be written as r_e ¼ ðseij þ svij Þð_eeij þ e_pij Þ  qi;i þ rr ¼ seij e_eij þ Q_ekk þ s0vij e_ 0ij þ seij e_ pij  qi;i þ rr,

(8)

where e_0ij is the deviatoric strain rate. Using entropy (s) and ðeeij Þ as the state variables and excluding other possible internal state variables, the general equation of state can be written as r_e ¼ ry_s þ seij e_eij .

(9)

Combining Eqs. (8) and (9) together then yields ry_s ¼ Q_ekk þ s0vij e_0ij þ seij e_pij  qi;i þ rr.

(10)

In the current study, the viscous stress is assumed to be contributed by mean stress (Q) only, i.e. the deviatoric viscous stress ðs0vij Þ is zero and there is no internal heat source (r). Thus Eq. (10) can be simplified to ry_s ¼ Q_ekk þ seij e_ pij  qi;i .

(11)

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Without heat conduction, i.e. qi;j ¼ 0, all the entropy change is due to the entropy production associated with mechanical dissipation. Hence g_ mech ¼

1 ðQ_ekk þ seij e_pij Þ. ry

(12)

Based on the same state variables, the rate of change of seij and y can be written as follows (Wallace, 1981b): s_ eij ¼ Bijkl e_ ekl  rGij y_s ¼ Bijkl ð_ekl  e_ pkl Þ  rGij y_s,

(13)

y_ ¼ yGij e_eij þ ðy=C v Þ_s ¼ yGij ð_eij  e_pij Þ þ ðy=C v Þ_s,

(14)

where Bijkl

 qseij  ¼ e  qekl s

is the adiabatic stress strain coefficient,  1 qseij  Gij ¼  ry qs ee ij

is the Gru¨neisen parameter, and Cv is the specific heat at constant elastic configuration. 2.1. Constitutive law A general thermomechanical material model is used in this study. Without compromising the current research objectives, the deformation induced anisotropy is neglected and the material is treated as isotropic throughout the deformation process. The volumetric response of the material is described by the Mie–Gruneisen equation of state and the deviatoric response is described by an elastic–viscoplastic model. The Mie–Gruneisen equation of state is in the form of   Gm P ¼ PH 1  þ Gre, (15) 2 where PH is the Hugoniot pressure, m ¼ v0 =v  1 with v0 and v being the initial and current specific volume, respectively. G ¼ G0 ðv=v0 Þ with G0 being the initial Gruneisen ratio, and e is the internal energy per unit mass as described earlier. PH is expressed in terms of u  U 0 relation, i.e. U 0 ¼ c0 þ s0 u, where U0 is the shock speed, u the particle velocity as described earlier, and c0 and s0 are constants. Under the assumption of isotropy, the Gru¨neisen parameter is written as Gij ¼ Gdij . The viscous mean stress is a function of volumetric strain rate, i.e. Q ¼ f ð_ekk Þ.

(16)

The elastic part of the deviatoric response is described by se0ij ¼ 2Gee0ij ,

(17)

where G is the shear modulus and is treated as a constant in this study. The inelastic part of the deviatoric response is described by the viscoplastic model proposed by Swegle and

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Grady (1985), i.e. g_ p ¼ A0 t2v ¼ A0 ðt  t0 Þ2 , ð13s0ij s0ij Þ1=2 ,

p

(18)

ð13e_pij e_pij Þ1=2 ,

where t ¼ g_ ¼ and t0 is the yield strength which is assumed to be a constant, i.e. no hardening or softening. For the thermal response, the classical linear Fourier heat conduction law is used, i.e. qi ¼ ky;i ,

(19)

where k is the thermal conductivity. The specific heat (Cv) is given by C v ¼ 3NK þ ½g0 ðv=v0 Þg y,

(20)

where N is the Avogadro number, K is the Boltzmann constant, and g0 and g are material constants (Wallace, 1981a). 3. 1-D steady shock In this part of the study the characteristics of a steady wave are investigated first. This study provides an insight into mechanical and thermal contributions to the overall material response and also serves as a baseline for non-steady wave study. Here, the strength effect is neglected, i.e. the material is treated as a viscous fluid. The strength effect will be investigated by direct wave propagation simulation described in Section 5. Under uniaxial strain conditions as encountered in planar compression of isotropic material, e_22 ¼ e_33 ¼ 0,

(21)

e_11 ¼ dv=v ¼ dr=r ¼ e_kk ,

(22)

and

the tensorial governing equations illustrated in section II can be simplified as follows. s_ e ¼ K e_11 þ rGy_s,

(23)

y_ ¼ yG_e11 þ ðy=C v Þ_s,

(24)

ry_s ¼ Q_e11  J ;x ,

(25)

where J ¼ qx ¼ ky;x is the x component of the heat flux vector. K in Eq. (23) is the adiabatic bulk modulus which can be calculated by K ¼ vðqP=qvÞjs , where P is given in Eq. (15). The conservation of linear momentum for a steady wave leads to s_ ¼

ðr0 U 0 Þ2 e_ , r

(26)

where U0 is the shock speed defined earlier. The viscous mean stress is a function of e_11 , i.e. Q ¼ f ð_e11 Þ,

(27)

s ¼ se þ Q,

(28)

and

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where s (total longitudinal stress), se, and Q are considered as positive in compression. The functional form of the viscous mean stress is estimated with Eq. (18) by setting t0 to be zero and converting t and g into equivalent longitudinal stress and longitudinal strain. The converted form is given by Q ¼ bj_e11 j1=2 ,

(29)

where b is a constant. To gain more insight into the sources of entropy change, Eq. (25) is rewritten in the following form: s_ ¼

1 Q_e11 J ;x ðQ_e11  J ;x Þ ¼  ¼ g_ mech þ g_ con þ s_heat , ry ry ry

(30)

where g_ mech ¼ Q_e11 =ry is the entropy production rate due to mechanical dissipation (here the mechanical dissipation is due to the viscous mean stress), g_ con ¼ ð1=ry2 ÞJy;x as described in Eq. (5) is the entropy production rate associated with heat conduction, and   J ;x 1 J  g_ con ¼  s_heat ¼  r y ;x ry is the rate of entropy change associated with the reversible heat transfer. Similar to the decomposition of entropy rate, Eq. (24) can also be rewritten in the following form: y_ ¼ y_ comp þ y_ gmech þ y_ gcon þ y_ heat , (31) _ where ycomp ¼ yG_e11 represents the rate of temperature change associated with isentropic compression, y_ gmech ¼ y_gmech =C v is the rate of temperature change associated with mechanical dissipation, y_ gcon ¼ y_gcon =C v is associated with thermal dissipation, and y_ heat ¼ y_sheat =C v is associated with reversible heat conduction. For a material that follows linear Fourier law described in Eq. (19), g_ con ¼ 

ky2;x 1 Jy ¼ , ;x ry2 ry2

(32)

y_g a y_ gcon ¼ con ¼ y2;x , y Cv

(33)

J ;x y_ con ¼ y_ gcon þ y_ heat ¼  ¼ aðy;x Þ;x , rC v

(34)

where a ¼ k=rC v is the thermal diffusivity. For aluminum, a is around 84  106 m2/s (Holman, 2002). In summary, during ramp or shock loading, the material is subjected to both mechanical compression and heat transfer by conduction. Each of them has its reversible and irreversible or dissipative components. The decomposition described above quantifies each individual contribution. The material constants used in this study follow those used in Wallace (1981a) when applicable. The peak stress investigated is also the same as that studied in Wallace (1981a), i.e. 0.8 Mbar. The constant b used in Eq. (29) was estimated from the data obtained for 6061 aluminum (Swegle and Grady, 1985). The estimated value was 1  104 GPa s1/2. This value was used as a reference for the parametric study of mechanical dissipation due to viscous mean stress. Three different values for b, i.e. b1 ¼ 1  103 , b2 ¼ 1  104 , and

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Table 1 Material properties for the thermo-mechanical model used in the current study r0 (kg/m3) Initial density c0 (GPa) s0 (m/s) G0 G (GPa) A0 (GPa2 s)1 k (W/m K) g0 (104 cal/mol K2) g

2785 76.74 5.33  103 2.05 27.58 1  108 237 3.30 1.8

b3 ¼ 1  105 GPa s1=2 were investigated. For convenience and continuity of the paper, the relevant materials constants including those used in Wallace (1981a) are listed in Table 1. To investigate the steady shock response, we need to solve the coupled nonlinear ordinary differential equations, i.e. Eqs. (23), (26), (30) and (31). As shown in Wallace (1981a), under steady wave conditions, the spatial gradient of J can be replaced by its material time derivative as J ;x ¼

J_ r . U 0 r0

(35)

The numerical method for solving these type of equations is well developed and the solution software is available as part of the standard numerical library. However, caution needs to be exercised for the numerical stability. The stability issue was discussed in Drumheller (1998), where the heat conduction effect for a steady shock was examined using an ideal gas as an example. In contrast to the space-domain formulation used in Drumheller (1998), the current formulations are based in the time domain. Furthermore, the current study is much more comprehensive and covers more and broader issues. In the following, three different cases are considered, namely, viscous material, thermoelastic material, and thermoviscous material. This terminology follows that used in Drumheller (1998). The steady state responses of these three different types of materials are all shown in Fig. 1. 3.1. Viscous material This is the case corresponding to zero thermal conductivity, i.e. k ¼ 0. Thus, the only dissipation mechanism is the mechanical dissipation due to the viscous mean stress, Q. Accordingly, Eqs. (30) and (31) are simplified, respectively, to Q_e11 ¼ g_ mech , ry

(36)

y_ ¼ y_ comp þ y_ gmech .

(37)

s_ ¼ and

Fig. 1(a) depicts the three steady state velocity profiles corresponding to three different values b1 ¼ 1  103 , b2 ¼ 1  104 , and b3 ¼ 1  105 GPa s1=2 for the parameter b used

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Fig. 1. Steady shock response of viscous, thermoelastic, and thermoviscous materials with three different viscosities (b1 ¼ 1  103 , b2 ¼ 1  104 , and b3 ¼ 1  105 GPa s1=2 ): (a) velocity profiles for viscous material; (b) velocity versus volume compression relations; (c) stress versus volume compression relations, s ¼ se þ Q; (d) energy versus volume compression relations; (e) temperature versus volume compression relations; (f) entropy versus volume compression relations; (g) strain rate versus volume compression relations.

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in Eq. (29). As b or viscosity increases, both the shock thickness and the risetime increase. The relations between the volume compression and velocity, stress, and energy for various values of b or strain rate dependence are shown in Figs. 1(b), (c) and (d), respectively. For viscous material, these relations are independent of b because they result directly from conservation of mass, momentum, and energy. As energy and volume are uniquely determined, the temperature, entropy, and the equilibrium stress are also uniquely determined as a result of the equation of state. See Figs. 1(e), (f) and (c). Since both the total stress (s) and the equilibrium stress (se) are uniquely determined, the difference between them, i.e. Q, is also uniquely determined. However, due to different rate dependences as a result of different values for b, the same Q leads to different strain rates as shown in Fig. 1(g). Different strain rates then entail different risetimes or shock thicknesses as depicted in Fig. 1(a). 3.2. Thermoelastic material In this case, the mechanical viscosity or the parameter b is zero. Thus, the only dissipation source is heat conduction. Accordingly, Eqs. (30) and (31) are simplified, respectively, to s_ ¼

J_ ¼ g_ con þ s_heat , yU 0 r0

(38)

and y_ ¼ y_ comp þ y_ gloc .

(39)

The conservation of mass and momentum leads to the same linear relation between volume compression and velocity and stress as shown in Figs. 1(b) and (c), respectively. The conservation of energy leads to the relation between energy and volume compression shown in Fig. 1(d) for the thermoelastic material. The unique relation between volume compression and energy also leads to the unique relations between volume compression

Fig. 2. Heat flux versus volume compression relation for thermoelastic and thermoviscous materials under steady shock condition. Heat conduction is negligible for b ¼ b1 .

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Fig. 3. (a) Contributions to entropy change for thermoelastic material under steady shock condition; (b) contributions to temperature change for thermoelastic material under steady shock condition.

and temperature and entropy shown in Figs. 1(e) and (f), respectively, as a result of the equation of state. The heat flux as a function of volume compression is shown in Fig. 2. As discussed by Wallace (1980), Drumheller (1998), and Zel’dovich and Raizer (1966), a thermoelastic shock generally does not exist in reality. One way to look at this issue is that for this case, conservation laws are sufficient to determine the heat flux that is needed to maintain a steady wave condition, which in turn leads to a unique relation for energy, temperature, and entropy. Thus Fourier heat conduction law is a redundant condition. _ Consequently, we have J40 at the places where yp0 or yx X0. This is not consistent with _ 0 Þðr=r Þ Fourier’s law. Notice that for a steady wave, y;x can be written as y;x ¼ ðy=U 0 similar to Eq. (35). More insight can also be gained from the individual contributions to entropy and temperature as shown in Figs. 3(a) and (b), respectively. The fact that g_ con o0 as shown in Fig. 3(a) is clearly a violation of the 2nd law of thermodynamics. Another interesting observation from Fig. 3(a) is that the net contribution of reversible heat conduction to the overall entropy change is zero. This can be easily proved by integrating s_heat and utilizing the condition that J is zero at both the initial and final states, i.e.      Z t Z t Z t dS heat 1 J 1 d J 1 J final dt ¼  dt ¼  ¼ 0. (40) dt ¼  r y ;x r0 U 0 dt y r0 U 0 y initial dt 0 0 0 Since there is no mechanical dissipation, the sources for temperature rise are isentropic compression and heat conduction. However as shown in Fig. 3(b), the net contribution to temperature rise from heat conduction is essentially zero, i.e. the net temperature change between the initial and final equilibrium states is due to isentropic compression. This can also be shown as follows. From Eqs. (34) and (35), the rate of change of temperature due to heat conduction is y_ con ¼

1 _ J. C v r0 U 0

(41)

Thus if Cv is constant, the integrals of both sides from the initial to the final state are zero again due to the fact that J is zero at these states. For the current temperature and volume compression ranges, the total change of the Cv’s between the initial and final states is about 5% of the initial Cv. It is important to mention that yheat and yg-cond are the ‘‘net change’’

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of the temperature due to reversible heat conduction and entropy production associated with heat conduction instead of physical temperature. In other words, yheat of 3000 K only means reversible heat conduction would introduce a net temperature change of 3293 K, i.e. the difference between 3000 K and the initial temperature, 293 K. This combined with other contributions, i.e. ycomp and yg-cond , result in the physical temperature of the material. It should also be mentioned that the strain rate in this case is not part of the formulation and is indeterminate. 3.3. Thermoviscous material In this case, neither k nor b is zero, i.e. both mechanical work and heat conduction are active. The conservation of mass and momentum again lead to a linear relation between volume compression and velocity and stress as shown in Figs. 1(a) and (b), respectively. However the relation between volume compression and energy varies with the parameter b as shown in Fig. 1(d). For b ¼ b1 or 103 GPa s1/2, the relation is essentially the same as that for the viscous material, i.e. the role of heat conduction is insignificant. As the value of b becomes smaller, the contribution from heat conduction becomes greater. The change of the partition between mechanical work and heat transfer results in different energy relations. The temperature and entropy relations also vary accordingly as shown in Figs. 1(e) and (f), respectively. The various individual contributions to entropy production are detailed in Figs. 4(a) and (b) for b ¼ b2 and b3, respectively. For b ¼ b1 , the contribution from heat conduction is negligible as mentioned earlier and all the entropy production is due to gmech. For b ¼ b2 , gmech is the major entropy producer. The contribution from heat conduction is relatively small. Notice again that the net change of entropy due to reversible heat conduction is zero. For b ¼ b3 , heat conduction becomes the dominant entropy producer. Unlike gmech and gcon which are both monotonically increasing functions of V/V0, Sheat increases rapidly in the initial loading stage and then starts to decrease. For this case, the decrease also leads to the decrease of the total entropy at the later loading stage. The temperature relations for b ¼ b1 , b2, and b3 shown in Fig. 5

Fig. 4. (a) Contributions to entropy change for thermoviscous material with b ¼ b2 under steady shock condition; (b) contributions to entropy change for thermoviscous material with b ¼ b3 under steady shock condition.

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Fig. 5. (a) Contributions to temperature change for thermoviscous material with b ¼ b1 under steady shock condition; (b) contributions to temperature change for thermoviscous material with b ¼ b2 under steady shock condition; (c) contributions to temperature change for thermoviscous material with b ¼ b3 under steady shock condition.

follow similar patterns to those for entropy except that the overall temperature is a monotonically increasing function of volume compression. Similar to what was observed for the thermoelastic material, for b ¼ b2 and b3, although there are net changes for yg-cond and yheat, their combined contribution, i.e. the net temperature change due to heat conduction, is essentially zero. Thus heat conduction has significant effect on the temperature history between the initial and final equilibrium states, but its contribution to the net temperature change is very limited if any. In other words, the net temperature change is mainly due to the mechanical work. The heat flux versus volume compression is shown in Fig. 2. There is no appreciable heat flux for b ¼ b1 . The corresponding heat or energy transfer is shown in Fig. 6. Notice that the material absorbs heat at the earlier part of the loading and releases it at the later stage. The decrease of Sheat shown in Fig. 4 is due to this heat release. The correlation between Figs. 4–6 can be better appreciated with the aid of the time histories of the temperature and entropy as shown in Fig. 7 for the case of b ¼ b3 . Fig. 7(a) is a comparison of the temperature histories between the thermoviscous and viscous cases. The histories of the various contributions to the temperature and entropy change are shown in Figs. 7(b) and (c), respectively. Fig. (a) shows that heat conduction produces a more spreading and

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Fig. 6. Heat transfer rate versus volume compression relation for thermoviscous material with b ¼ b2 , and b3 under steady shock condition.

Fig. 7. (a) Comparison of the temperature profiles for viscous and thermoviscous materials with b ¼ b3 under steady shock condition; (b) time histories of the contributions to temperature change for the thermoviscous material with b ¼ b3 under steady chock condition; (c) time histories of the contributions to entropy change for the thermoviscous material with b ¼ b3 under steady shock condition.

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smoothed temperature profile. Because of the steady wave condition, the slope of the temperature profile can be directly related to the temperature gradient. This figure also shows that the temperature gradient increases from zero to some maximum value and then decreases to zero. This change of temperature gradient results in heat absorption and release at different loading stage as illustrated by Eq. (34). For the viscous case, a in Eq. (34) is zero because k is zero. Zero or small thermal diffusivity leads to a large change of the temperature gradient, or ðy;x Þ;x which also results in a large temperature gradient according to Eq. (34). This is what is shown in Fig. 7(a) for the viscous case. Figs. (b) and (c) show that the heat conduction precedes the mechanical compression. Hence the temperature rise at the initial stage results mainly from the heat conduction. Furthermore, the heat conduction at the very early stage does not produce much entropy. In other words, temperature rise is mostly due to reversible heat transfer, yheat. As indicated by Eq. (32), the entropy production rate due to irreversible heat conduction is proportional to the square of the temperature gradient. At slightly later stages, additional heating due to irreversible heat conduction quickly surpasses reversible heat transfer resulting in large increases of yg-cond and gcond. Once the mechanical dissipation starts to set in, the material begins to release heat. The compensation of heating by mechanical dissipation and cooling by heat conduction significantly reduces the temperature rise rate and the temperature gradient. Lower temperature gradient combined with the higher temperature again results in lower entropy production associated with heat conduction at the later loading stage or the heat release phase as shown in Fig. 7(c). Furthermore, the magnitude of yheat during heat release is much larger than that during the heating. The large yheat during heat release wipes out the temperature rise contributed by both yheat and yg-cond during the heating stage and brings down the total contribution of heat conduction to temperature rise to essentially zero. It should also be noted that unlike thermoviscous material, yg-cond or gcond continues to increase even during most of the heat release stage for the thermoelastic material as shown in Fig. 3. Consequently, g_ con and y_ g-con have to decrease in order to bring the net change of temperature due to heat transfer to zero, but this violates the second law of thermodynamics.

Fig. 8. (a) Comparison of the strain rate versus volume compression relations in linear scale for the thermoviscous and viscous materials with b ¼ b3 under steady shock condition; (b) comparison of the velocity profiles for the thermoviscous and viscous materials with b ¼ b3 under steady shock condition.

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The strain rate relations for thermoviscous material are shown in Fig. 1(g) which depicts that the participation of heat conduction results in an overall lower strain rate compared to the corresponding viscous material. A comparison of the strain rate in linear scale for b ¼ b3 is shown in Fig. 8(a) and the corresponding velocity profiles are shown in Fig. 8(b). The relatively lower strain rate at the early stage, as shown in Fig. 8(a), is associated mainly with heating due to conduction instead of mechanical compression. The same mechanism also leads to a more spreading velocity profile. 4. 1-D nonsteady ramp wave and ramp–shock transition To study the thermomechanical material response associated with a propagating ramp wave, direct numerical simulation of wave propagation is necessary. In the current work, the simulation was carried out with a one dimensional hydrocode COPS (Gupta, 1978) which uses finite difference method to solve one dimensional nonlinear wave equations. The background of this well-developed numerical method and its application to shock dynamics can be found in Wilkins (1964), Walsh (1973) among others. To address the specific issues outlined in the current work, the thermodynamics framework discussed above and heat conduction had to be implemented in the code. The former was added to the code as a material module. The latter was implemented as follows. In each finite difference cell, the temperature was calculated using Eq. (24). The heat flux (J) was then calculated at the nodes using Eq. (19) and center difference method. From the heat fluxes at the nodes, J ;x can be calculated in the cell. As shown in Eqs. (24) and (25), J was used to calculate y_ and s_. However, J itself was calculated using temperature y. Since y and y_ are not centered at the same time, the terms in Eqs. (24) or (25) are not time-centered in an exactly consistent manner. However, with the very small time steps used, this did not seem to cause any appreciable error as will be shown later. The time step was evaluated based on the stability criteria for the respective hyperbolic (wave) (Walsh, 1973) and parabolic (heat conduction) (Holman, 2002) equations. A fraction of the smaller one between the two time steps evaluated from the aforementioned criteria was used in the calculation. To resolve the material response within the shock, the cell size has to be smaller than the shock thickness which can be obtained from the analyses described in the previous section. In a typical hydrocode, a cell-size dependent artificial viscosity was used to dampen out the oscillation (Walsh, 1973). This viscosity produces additional artificial entropy and temperature rise which would obscure the analysis. Hence it had to be removed in all the analyses. All the entropy production results precisely from the dissipation mechanisms included in the thermomechanical framework. The heat conduction part of the coding was evaluated by comparing the numerical solution with the analytical solution for a semi-infinite strip subjected to a prescribed temperature at one end (Holman, 2002). The two results are compared very well as shown in Fig. 9. The combined wave propagation and heat conduction coding was evaluated with the steady wave solutions obtained in the previous section as will be demonstrated later. In all the following calculations, the input is a prescribed ramp velocity history. Again no strength was considered in this part of the study and the material is treated as a thermoviscous material. Case 1: b ¼ b1 The results for this case are shown in Fig. 10. (a) shows the evolution of the velocity profile. As the ramp wave gets steeper, the corresponding strain rates get higher as shown

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Fig. 9. Comparison of the analytical (dash lines) and numerical (dotted lines) solutions of the temperature profiles at two different times for a semi-infinite plate subjected to a prescribed temperature at one end.

Fig. 10. (a) Evolution of the velocity profiles for a thermoviscous material with b ¼ b1 subjected to a prescribed ramp velocity input; (b) the strain rate versus volume compression relations for the corresponding velocity profiles; (c) the entropy versus volume compression relations for the corresponding velocity profiles; (d) the temperature versus volume compression relations corresponding to the velocity profiles 1 and 3.

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in Fig. (b). (c) shows the relations between the entropy and volume compression as the ramp wave evolves. Because of the relatively high viscosity, the risetimes remain substantial and the role of heat conduction is negligible throughout the evolution. All of the entropy production was due to the mechanical dissipation. In the beginning, the entropy production is smaller and the material response is closer to an isentrope. As the ramp wave gets steeper, more entropy is produced by mechanical dissipation and the material response is farther away from isentrope. The evolution of the temperature profiles and the contributions of individual mechanisms for temperature rise are demonstrated in (d) through wave profiles 1 and 3. The temperature profiles eventually reach the steady state depicted in Fig. 5(a). Again heat conduction contributes little to the temperature rise throughout the whole process. At the early stage of ramp wave evolution such as that associated with the first wave profile, the majority of the temperature rise is due to isentropic compression. As the ramp wave evolves, the temperature rise due to mechanical dissipation becomes more significant and eventually dominates the temperature rise. Case 2: b ¼ b2 The results for this case are shown in Fig. 11. Because of the relatively low viscosity, the risetime for the corresponding steady shock is shorter than that with b ¼ b1 . This means that a ramp wave can also have a shorter risetime before it evolves into a shock. Accordingly the risetime for the input ramp wave and the total time duration used in this simulation are shorter than those used in the previous case. The shorter time span reduces the total distance a wave travels and the minimum sample size required for the numerical simulation, and provides better time and spatial resolutions that are required for simulating thinner ramp and shock waves. (a) shows the evolution of the velocity profile. The risetime for the first wave profile is about 20 ns, which is actually shorter than that for the steady shock in the previous case. Again, as the ramp wave gets steeper, the corresponding strain rate gets higher as shown in (b). The relation between the entropy and volume compression as the ramp wave evolves is shown in (c), where the solid lines represent the total entropy production and the dash lines represent the contribution from heat conduction. The difference is the contribution from mechanical dissipation. Because of the relatively low viscosity and short risetime associated with both ramp wave and evolving shocks, the role of heat conduction becomes more appreciable starting from the fourth wave profile. The evolution of the temperature profiles and the contributions of individual mechanisms for temperature rise are demonstrated in (d) through wave profiles 1 and 4. The steady state response was shown in Fig. 5(b). At the early stages as those associated with wave profiles 1 and 2, the temperature rise is mainly due to isentropic compression. There is some contribution from the mechanical dissipation, but none from the heat transfer. Recall that for b ¼ b1 , the mechanical dissipation is quite significant for a ramp wave with much longer risetime. When the wave evolves to stage (profile) 3, the contribution from mechanical dissipation starts to become significant, but there is still little contribution from heat conduction as demonstrated by the entropy relations shown in (c). From the wave profile 4, the role of heat conduction becomes more apparent as a result of shock formation. However, mechanical dissipation remains to be the dominant mechanism for temperature rise throughout the wave evolution. (e) shows the amount of heat delivered to or removed from the material through heat transfer as the wave passes through. The pattern is similar to that for the steady wave discussed in the previous section. As expected, the energy transferred by heat conduction increases as shock grows.

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Fig. 11. (a) Evolution of the velocity profiles for a thermoviscous material with b ¼ b2 subjected to a prescribed ramp velocity input; (b) the strain rate versus volume compression relations for the corresponding velocity profiles; (c) the entropy versus volume compression relations for the corresponding velocity profiles; (d) the temperature versus volume compression relations corresponding to the velocity profiles 1 and 4. (e) the heat transfer rate versus volume compression relation for the corresponding velocity profiles.

Case 3: b ¼ b3 The results for this case are shown in Fig. 12. Due to the even lower viscosity compared to the two previous cases, the time span used in this simulation is much shorter, but still larger than the risetime of the corresponding shock. (a) shows the evolution of the ramp velocity profile. Similar to the two previous cases, as the ramp wave evolves, the

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corresponding strain rate gets higher as shown in (b). The relation between the entropy and volume compression as the ramp waves evolve is shown in (c), where again the solid lines represent the total entropy production and the dashed lines represent the contribution

Fig. 12. (a) Evolution of the velocity profiles for a thermoviscous material with b ¼ b3 subjected to a prescribed ramp velocity input; (b) the strain rate versus volume compression relations for the corresponding velocity profiles; (c) the entropy versus volume compression relations for the corresponding velocity profiles; (d) the temperature versus volume compression relations corresponding to the velocity profiles 1 and 3; (e) comparison of the steady state temperature responses corresponding to wave profiles 4 and 5 calculated from wave simulation and those from steady state analysis presented in Fig. 5(c) (thin solid line); (f) the heat transfer rate versus volume compression relation for the corresponding velocity profiles.

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from heat conduction. Because of the short ramp wave risetime and low mechanical dissipation, heat conduction is appreciable from the very beginning. The evolution of the temperature profiles and the contributions of individual mechanisms for temperature rise are demonstrated in (d) through wave profiles 1 and 3. A comparison of the steady state temperature profiles calculated from wave simulation and steady state analysis presented in Fig. 5(c) is shown in (e). In general the overall consistency between the two results obtained with different approaches is very respectable and validates the results from the wave propagation simulation. As demonstrated by (c) and (d), the contribution from mechanical dissipation is smaller than that from isentropic compression throughout the wave evolution. Heat conduction is relatively insignificant in the beginning, but rapidly becomes the dominant mechanism for temperature rise. As expected, the amount of heat transfer is also much more significant than in the previous cases, as shown in (f). Wave profile 3 exemplifies an interesting temperature path. In the early stage of the loading, temperature rises rapidly with respect to the volume compression due to the heat transferred to the material by conduction. Subsequently, the heating by mechanical dissipation is compensated by the heat release by conduction resulting in much lower temperature rise rate with respect to volume compression. At the ramp shock transition, heat release by conduction diminishes and the temperature rise rate starts to increase again. At the end of the loading, the temperature and entropy associated with profile 3 continue to decrease slightly, while for wave profile 2 the entropy (and temperature) continues to increase. This is an evidence for continued heat transfer at the end of the loading, which leads to the continued changes of entropy and temperature. This is due to the much smaller sample size and smaller time span covered in this simulation than the previous ones. The continued heat transfer is attributed to the backward heat conduction from the evolving shock during this very short time period. 5. Effect of strength To study the strength effect, another set of simulations was performed. In the following, the thermoelastic-plastic case is analyzed first. A more general thermoviscous-elasticviscoplastic case is investigated afterwards. 5.1. Thermoelastic– plastic case In this case, the viscous mean stress is set to zero. Thus the mechanical dissipation is due to inelastic deformation only. Furthermore, a very large number, 1.7  1015 (GPa2 s)1, is assumed for the parameter A0 in Eq. (18). A large A0 implies a quasi rate-independent inelasticity model. The strength of the material is assumed to be 500 MPa. Fig. 13(a) shows the evolution of the velocity profiles. A two wave structure is formed in the early stage and it eventually evolves into an overdriven shock. Unfortunately, due to the lack of damping, oscillations seem to be inevitable at the ramp-shock transition as observed for profiles 3 and 4. To overcome this problem, a smoothing procedure is used to smooth out the oscillations. An example of the smoothed velocity profile compared to the original one is shown in Fig. 13(b). As will be shown in the next section, the insights gained from this smoothed responses are very reasonable. The entropy versus volume compression relation is shown in Fig. (c). The temperature evolution is demonstrated in Fig. (d) using the temperature profiles associated with the first and the last wave profiles. Unlike the

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Fig. 13. (a) Evolution of the velocity profiles for a thermoelastic–plastic material with A0 ¼ 1:7  1015 ðGPa2 sÞ1 and strength of 0.5 GPa subjected to a prescribed ramp velocity input; (b) comparison of the simulated and smoothed velocity profile 4; (c) the entropy versus volume compression relations for the corresponding velocity profiles; (d) the temperature versus volume compression relations corresponding to the velocity profiles 1 and 4.

dissipation associated with the mean viscous stress, the plastic deformation and dissipation can take place only when the applied stress exceeds a threshold imposed by the material strength. Furthermore, enough time must also be given for plastic deformation to develop except for the idealized rate independent case. The less rate sensitive (the larger A0 ) it is, then the shorter the time that is required for the inelastic deformation to develop. During the ramp loading, the loading rate is low or the loading time is long. Hence, there is plenty of time for the inelastic deformation to develop and the deformation is basically driven by an equivalent shear stress (seij in Eq. (12) or t in Eq. (18)) that is very close to t0 . In other words, the overstress is small and the amount of dissipation generated is very limited. Although there is some heat transfer associated with the plastic wave due to its short rise time in this case, the entropy production associated with the heat transfer is also insignificant. Thus the overall material response is close to isentrope. However, once the shock is formed, both the temperature and entropy rise sharply. In the early stage of shock loading, because of the strength and the very high loading rate or short loading time, there is little inelastic deformation or dissipation. As a result the material behaves essentially like a thermoelastic material and the initial temperature rise and entropy production are

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primarily due to heat conduction. The high overstress accumulated due to delayed inelastic deformation is relaxed at the end of the shock. This relaxation brings about a large amount of inelastic dissipation and a temperature rise at the end of the loading. 5.2. Thermoviscous– elastic– viscoplastic material In this case both viscous mean stress and inelastic deformation, i.e. both terms on the right hand side of Eq. (12), are active. The main purpose of this part of the study is to study the effect of rate sensitivity and strength associated with the inelastic deformation on the thermomechanical response of the material. Inclusion of viscous mean stress here is mainly to eliminate the oscillations observed in the previous case. For all the cases in this part of the study, b in Eq. (29) is set to be equal to b3 ¼ 1  105 GPa s1=2 . 5.2.1. Thermoviscous– elastic– plastic material In this case, the inelasticity model is the same as that used in case 5.1. This case serves as a baseline model for addressing the rate sensitivity and strength effects. The results are shown in Fig. 14. Because of the longer rise time induced by the viscous mean stress compared to that for the case 5.1, there is enough time for the inelastic deformation to develop during the loading. Hence the inelasticity is essentially driven by t0 during both ramp and shock loading and the inelastic dissipation is distributed throughout the loading process instead of being accumulated and released at the end of the shock as observed in Fig. 13. Compared to Fig. 12, the additional mechanical dissipation due to inelastic deformation reduces slightly the peak overall strain rate as shown in Fig. 14(b). This implies slightly increased risetimes for both ramp waves and shock and slightly reduced role of heat conduction. However, because the inelastic dissipation induced by t0 is small, the differences between what are observed in Figs. 12 and 14 are trivial and the overall material response is essentially dominated by the mean viscous stress and heat transfer. 5.2.2. Effect of rate sensitivity Rate sensitivity of the inelastic deformation can be increased by decreasing the value of A0 in Eq. (18). Shown in Fig. 15 are the results for A0 ¼ 1:7  1011 ðGPa2 sÞ1 . The general trend of the material response is similar to that shown in Fig. 14. With higher rate sensitivity, there is more contribution to entropy production and temperature rise from the inelastic deformation, although viscous mean stress is still the dominant mechanism for mechanical dissipation in this case. As mentioned earlier, inelastic deformation with higher rate sensitivity requires more time to develop. Thus the same ramp input as used in the previous case reflects a shorter loading time or higher loading rate with respect to the inelastic deformation. This leads to a larger overstress and inelastic dissipation which in turn leads to further reduction of the peak overall strain rate and the role of heat conduction. Shown in Fig. 16 are the results for even higher rate sensitivity with A0 ¼ 1:7  109 ðGPa2 sÞ1 . In this case, the inelastic deformation becomes the dominant mechanism for mechanical dissipation. Furthermore, the same ramp input reflects a much higher loading rate with respect to the inelastic deformation. At stages 3–5, the loading rate associated with the shock is too high for the inelastic deformation to develop completely during the shock loading. The delayed inelastic deformation and high overstress are relaxed at the end of the shock. This leads to a significant increase of entropy

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Fig. 14. (a) Evolution of the velocity profiles for a thermoviscous–elastic–plastic material with b ¼ b3 , A0 ¼ 1:7  1015 ðGPa2 sÞ1 , and strength of 0.5 GPa subjected to a prescribed ramp velocity input; (b) the strain rate versus volume compression relations for the corresponding velocity profiles. Solid lines represent total strain rate and dash lines represent equivalent plastic strain rate defined by ð23e_pij e_pij Þ1=2 ; (c) the entropy versus volume compression relations for the corresponding velocity profiles; (d) the temperature versus volume compression relations corresponding to the velocity profiles 1 and 5.

production and temperature rise and results in temperature and entropy profiles that are similar to those observed in Fig. 13. 5.2.3. Effect of strength Shown in Fig. 17 are the simulated results for the case in which the strength was increased from 0.5 GPa used in previous cases to 5 GPa but A0 is the same as that used in the quasi rate-independent model as shown in Figs. 13 and 14. Increased strength results in increased threshold and the driving force for inelastic deformation. Compared to Fig. 14, Fig. 17 shows delayed inelastic deformation due to the higher threshold, but larger inelastic dissipation due to the higher driving force. The increased inelastic dissipation reduces the role of the dissipation due to viscous mean stress and results in slightly lower strain rate. In this regard, the effect of strength is similar to that of rate sensitivity. The interaction of heat conduction with inelasticity and viscous mean stress is more complicated because the latter two would affect both the rate and duration of heat transfer. In general, the reduced strain rate would lead to reduced heat transfer rate, but delayed mechanical dissipation would also prolong the duration of heat absorption due to conduction.

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Fig. 15. (a) Evolution of the velocity profiles for a thermoviscous–elastic–plastic material with b ¼ b3 , A0 ¼ 1:7  1011 ðGPa2 sÞ1 , and strength of 0.5 GPa subjected to a prescribed ramp velocity input; (b) the strain rate versus volume compression relations for the corresponding velocity profiles. Solid lines represent total strain rate and dash lines represent equivalent plastic strain rate defined by ð23e_pij e_pij Þ1=2 ; (c) the entropy versus volume compression relations for the corresponding velocity profiles; (d) the temperature versus volume compression relations corresponding to the velocity profiles 1 and 5.

6. Discussion In this work, a comprehensive study was carried out to examine the effects of ramp and shock loading on the entropy production and thermomechanical responses of materials. As mentioned earlier, the source of entropy production can be generally attributed to mechanical and thermal dissipations. The various material responses observed are essentially a manifestation of the interaction between the loading rate or the time scale of the external loading and the intrinsic time scales associated with various dissipation mechanisms. For the mechanical dissipation, the intrinsic time scales are characterized by the parameters b in Eq. (29) and A0 in Eq. (18) for the viscous mean stress and inelastic deformation, respectively. These time scales are directly related to the rate sensitivity of the dissipation mechanisms. The higher the rate sensitivity, the larger the intrinsic time scales. When the loading time is long (or the loading rate is low) relative to the intrinsic time scales, both mean viscous stress (Q) and the overstress ðt  t0 Þ and their corresponding

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Fig. 16. (a) Evolution of the velocity profiles for a thermoviscous–elastic–plastic material with b ¼ b3 , A0 ¼ 1:7  109 ðGPa2 sÞ1 , and strength of 0.5 GPa subjected to a prescribed ramp velocity input; (b) the strain rate versus volume compression relations for the corresponding velocity profiles. Solid lines represent total strain rate and dash lines represent equivalent plastic strain rate defined by ð23e_pij e_pij Þ1=2 ; (c) the entropy versus volume compression relations for the corresponding velocity profiles; (d) the temperature versus volume compression relations corresponding to the velocity profiles 1 and 5.

dissipations are small. In this case, the loading essentially follows a quasi-equilibrium mechanical loading path that is close to an isentrope. The shorter the loading time (or the larger the loading rate) relative to the intrinsic time scales, the larger the viscous stress (Q) and the overstress ðt  t0 Þ. Larger viscous stress or overstress deviates the loading away from the equilibrium path and produces a larger amount of mechanical dissipation. The time scale associated with heat conduction is the thermal diffusivity a. Similar to the parameter A0 in Eq. (18), the larger a, the smaller the time scale. As demonstrated in Fig. 8, large a reduces the temperature gradient and the thermal dissipation associated with heat transfer. For material such as aluminum which has a very large a, heat conduction becomes significant when the imposed ðy;x Þ;x is extremely high as encountered in a very thin shock. This is analogous to the mechanical dissipation associated with a quasi rateindependent inelasticity model which becomes significant only when the loading rate is very high. For a modestly imposed ðy;x Þ;x , large a maintains a low temperature gradient throughout the loading process and eliminates heat conduction. Furthermore, for aluminum, the time scale for heat conduction is much smaller than those of mechanical dissipation. Thus a very slight increase of the risetime may completely eliminate heat

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Fig. 17. (a) Evolution of the velocity profiles for a thermoviscous–elastic–plastic material with b ¼ b3 , A0 ¼ 1:7  1015 ðGPa2 sÞ1 and strength of 5 GPa subjected to a prescribed ramp velocity input; (b) the strain rate versus volume compression relations for the corresponding velocity profiles. Solid lines represent total strain rate and dash lines represent equivalent plastic strain rate defined by ð23e_pij e_pij Þ1=2 ; (c) the entropy versus volume compression relations for the corresponding velocity profiles; (d) the temperature versus volume compression relations corresponding to the velocity profiles 1 and 5.

conduction, but is not sufficient to reduce mechanical dissipation substantially. An example can be found for the thermoviscous case with b ¼ b1 in which the shock risetime is short relative to the time scale of mechanical dissipation, but long with respect to the time scale of heat conduction. Thus heat conduction has little effect in this case. For materials with small a or a large time scale for heat conduction, heat conduction may not occur even under a large temperature gradient, such as the case for a viscous material with a very small b value. As also demonstrated in this study, the various types of thermomechanical responses of materials result in much more significant differences in the temperature path than the mechanical path. Thus temperature measurement, although it is a significant challenge, appears to be an important experimental tool to distinguish the various mechanisms responsible for the thermomechancial response of the materials. As also mentioned earlier, the role of heat conduction in high strain rate deformation has been a long standing issue. This is true even for steady shock, let alone non-steady waves. To the author’s knowledge,

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the current work is the first to deal with this issue for a non-steady and large amplitude wave. In a recent work by Sano and Sano (2001), the authors attempted to calculate the temperature profile associated with a 0.8 Mbar steady shock for 2024 aluminum. They included heat conduction in their analysis, but also made some strong assumptions about the wave profile. The temperature profile that they obtained is similar to that for viscous material or thermoviscous material with a large b value. It is clear from the current study that the temperature profile presented in Sano and Sano (2001) is only a very special case at best. It should also be reminded that the current thermal analysis is based on the linear Fourier heat conduction law. As the shock gets stronger, the validity of this liner relation may become more dubious. A discussion of the linear and nonlinear heat conduction and the temperature dependence of thermal conductivity, and their correlation with the structure of a shock can be found in Zel’dovich and Raizer (1966).

7. Conclusion In this study, the effects of high rate ramp and shock loading on the mechanical and thermal responses of materials are investigated comprehensively. It is demonstrated that the material responses to ramp and shock loading are essentially a manifestation of the interaction between the time scale associated with the loading and the intrinsic time scales associated with mechanical deformation and heat transfer. At lower loading rates as encountered in ramp loading, the loading path is closer to an isentrope and results in lower entropy production. The reasonable ramp rate to obtain quasi-isentropic response depends on the intrinsic time scales of the dissipation mechanisms which are strongly material dependent. Thus shockless loading does not necessarily produce an isentropic response. Between two equilibrium states, heat conduction was shown to have a significant effect on the temperature history, but it contributes little to the overall temperature change if the specific heat does not change significantly. It also affects the history of entropy, but only the irreversible part of heat conduction contributes to the net entropy change. The various types of thermomechanical responses of materials would manifest themselves more significantly in terms of the thermal history than the mechanical history. Thus temperature measurement appears to be an important experimental tool in distinguishing the various mechanisms for the thermomechancial responses of materials.

Acknowledgement The author would like to sincerely thank Sandia National Laboratories whose support has made this work possible. The interaction with the members of the Shock & Z-Pinch Department of Sandia, especially Drs. James Asay (now at Institute of Shock Physics, Washington State University), Christopher Deeney, and Marcus Knudson has been very rewarding and productive. The author would also like to express his deep appreciation to Dr. Y.M. Gupta, Director of the Institute for Shock Physics of Washington State University, for introducing the author to the field of shock physics and providing support throughout the author’s learning process.

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