On computing the evolution of temperature for materials under dynamic loading

Abstract Modeling and simulation of the dynamic response of materials is important to many applications including the development of armor systems, understanding the safety of explosives, and assessing the crashworthiness of vehicles. Within such applications it is often critical to accurately compute the evolution of the temperature because it is a state variable that affects the kinetics of competing active processes within the material (e.g., dislocation motion, phase transformation, decomposition). Depending on the selection of an independent state variable, e.g. temperature or entropy, the approach for computing temperature is well understood based on the thermodynamic framework attributed to Coleman and Noll. However, different computational codes used for modeling the dynamic response of materials adopt different independent state variables. In this work, two thermodynamically consistent strategies for computing the temperature of a coupled thermodynamic state are compared and implemented into two different Lagrangian computational codes. The equivalence of these two approaches is established through the numerical solutions of several test problems. Finally, the implication of various approximations made to each of these approaches within the literature are assessed in the context of uniaxial stress conditions (split Hopkinson pressure bar experiments) and uniaxial strain conditions (plate impact experiments). It is shown that the temperature rate or energy partition approaches are equivalent when implemented in their complete forms, but that several common simplifying assumptions, that are warranted in the case of uniaxial stress, lead to significant errors in the resulting Hugoniot state for plate impact.

[1]  Marc J. Cawkwell,et al.  A dislocation density-based continuum model of the anisotropic shock response of single crystal α-cyclotrimethylene trinitramine , 2017 .

[2]  Walter Noll,et al.  The thermodynamics of elastic materials with heat conduction and viscosity , 1963 .

[3]  B. Banerjee,et al.  AN EXTENDED MECHANICAL THRESHOLD STRESS PLASTICITY MODEL: MODELING 6061-T6 ALUMINUM ALLOY , 2008 .

[4]  J. Langer.  Statistical thermodynamics of strain hardening in polycrystalline solids. , 2015, Physical review. E, Statistical, nonlinear, and soft matter physics.

[5]  David L. McDowell,et al.  Plane wave simulation of elastic-viscoplastic single crystals , 2014 .

[6]  Rahul,et al.  A fully anisotropic single crystal model for high strain rate loading conditions with an application to α-RDX , 2014 .

[7]  A. Hunter,et al.  Implementation of a dislocation-density based single-crystal model into a continuum shock hydrodynamics code , 2018 .

[8]  G. Ravichandran,et al.  Thermomechanical behavior of single crystalline tantalum in the static and dynamic regime , 2009 .

[9]  J. Rodríguez-Martínez,et al.  On the Taylor–Quinney coefficient in dynamically phase transforming materials. Application to 304 stainless steel , 2013 .

[10]  Gordon R. Johnson,et al.  Further Development of the EPIC-3 Computer Program for Three-Dimensional Analysis of Intense Impulsive Loading , 1978 .

[11]  Dean L. Preston,et al.  Model of plastic deformation for extreme loading conditions , 2003 .

[12]  T. Vogler.  On measuring the strength of metals at ultrahigh strain rates , 2009 .

[13]  R. Austin.  Elastic precursor wave decay in shock-compressed aluminum over a wide range of temperature , 2017 .

[14]  Laurent Stainier,et al.  Study and validation of a variational theory of thermo-mechanical coupling in finite visco-plasticity , 2010 .

[15]  Thermodynamically consistent relations involving plasticity, internal energy and thermal effects , 2005, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[16]  T. A. Mason,et al.  An experimental and numerical study of the localization behavior of tantalum and stainless steel , 2006 .

[17]  T. Wright,et al.  A continuum framework for finite viscoplasticity , 2001 .

[18]  Mark A. Kenamond,et al.  Coupling continuum dislocation transport with crystal plasticity for application to shock loading conditions , 2016 .

[19]  Y. Gupta,et al.  Nonlinear anisotropic description for the thermomechanical response of shocked single crystals: Inelastic deformation , 2006 .

[20]  Geoffrey Ingram Taylor,et al.  The Latent Energy Remaining in a Metal after Cold Working , 1934 .

[21]  A. Delpuech,et al.  RAMAN SCATTERING TEMPERATURE MEASUREMENT BEHIND A SHOCK WAVE , 1984 .

[22]  A. Arias,et al.  A hyperelastic-thermoviscoplastic constitutive model for semi-crystalline polymers: Application to PEEK under dynamic loading conditions , 2017 .

[23]  Y. Gupta,et al.  TEMPERATURE DETERMINATION IN SHOCKED CONDENSED MATERIALS USING RAMAN SCATTERING , 1997 .

[24]  T. Aslam,et al.  Temperature of shocked plastic bonded explosive PBX 9502 measured with spontaneous Stokes/anti-Stokes Raman , 2018 .

[25]  M. Cawkwell,et al.  Equations of state for the α and γ polymorphs of cyclotrimethylene trinitramine , 2016 .

[26]  E. Prince,et al.  The crystal structure of cyclotrimethylenetrinitramine , 1972 .

[27]  John D. Clayton,et al.  Dynamic plasticity and fracture in high density polycrystals: constitutive modeling and numerical simulation , 2005 .

[28]  Y. Gupta,et al.  Nonlinear anisotropic description for shocked single crystals: Thermoelastic response and pure mode wave propagation , 2004 .

[29]  D. Dlott,et al.  Multichannel emission spectrometer for high dynamic range optical pyrometry of shock-driven materials. , 2016, The Review of scientific instruments.

[30]  Carl Eckart,et al.  The Thermodynamics of Irreversible Processes. IV. The Theory of Elasticity and Anelasticity , 1948 .

[31]  Wing Kam Liu,et al.  Nonlinear Finite Elements for Continua and Structures , 2000 .

[32]  Richard A. Regueiro,et al.  Finite strain elastoplasticity considering the Eshelby stress for materials undergoing plastic volume change , 2016 .

[33]  D. McDowell,et al.  A dislocation-based constitutive model for viscoplastic deformation of fcc metals at very high strain rates , 2011 .

[34]  E. Cerreta,et al.  Modeling the microstructural evolution of metallic polycrystalline materials under localization conditions , 2007 .

[35]  M. Zikry,et al.  The effects of microstructural defects on hot spot formation in cyclotrimethylenetrinitramine-polychlorotrifluoroethylene energetic aggregates , 2013 .

[36]  J. Clayton,et al.  Heterogeneous deformation and spall of an extruded tungsten alloy : plate impact experiments and crystal plasticity modeling , 2008 .

[37]  Kiran Solanki,et al.  On kinematic, thermodynamic, and kinetic coupling of a damage theory for polycrystalline material , 2010 .

[38]  Hussein M. Zbib,et al.  Multiscale dislocation dynamics simulations of shock compression in copper single crystal , 2005 .

[39]  R. Menikoff.  Complete Mie-Gruneisen Equation of State (update) , 2016 .

[40]  J. Asay,et al.  Numerical study of rate-dependent strength behavior under ramp and shock wave loading , 2009 .

[41]  V. Lubarda.  On thermodynamic potentials in linear thermoelasticity , 2004 .

[42]  M. Gurtin,et al.  Thermodynamics with Internal State Variables , 1967 .

[43]  M. Cawkwell,et al.  A single-crystal model for the high-strain rate deformation of cyclotrimethylene trinitramine including phase transformations and plastic slip , 2017 .

[44]  D. Benson.  Computational methods in Lagrangian and Eulerian hydrocodes , 1992 .

[45]  C. Bronkhorst,et al.  A model for finite-deformation nonlinear thermomechanical response of single crystal copper under shock conditions , 2013 .

[46]  L. Veeser,et al.  Thermal transport in shock wave-compressed solids using pulsed laser heating. , 2014, The Review of scientific instruments.

[47]  D. Kittell,et al.  A physically-based Mie–Grüneisen equation of state to determine hot spot temperature distributions , 2016 .

[48]  U. F. Kocks.  Thermodynamics and kinetics of slip , 1975 .

[49]  D. Burton.  Consistent finite-volume discretization of hydrodynamic conservation laws for unstructured grids , 1994 .

[50]  C. Greeff,et al.  The Cold Equation of State of Tantalum , 2009 .

[51]  M. Shashkov,et al.  The Construction of Compatible Hydrodynamics Algorithms Utilizing Conservation of Total Energy , 1998 .

[52]  J. Rice.  Inelastic constitutive relations for solids: An internal-variable theory and its application to metal plasticity , 1971 .

[53]  D. Rittel,et al.  Dynamic large strain characterization of tantalum using shear-compression and shear-tension testing , 2017 .

[54]  Dimitris Drikakis,et al.  On entropy generation and dissipation of kinetic energy in high-resolution shock-capturing schemes , 2008, J. Comput. Phys..

[55]  C. Truesdell,et al.  The Non-Linear Field Theories Of Mechanics , 1992 .

[56]  Donald E. Burton,et al.  Exact conservation of energy and momentum in staggered-grid hydrodynamics with arbitrary connectivity , 1991 .

[57]  David L. McDowell,et al.  Internal State Variable Theory , 2005 .

[58]  Michael B. Prime,et al.  Starck Ta PTW strength model recommendation for use with SESAME 93524 EoS , 2017 .

[59]  Somnath Ghosh,et al.  Evaluating the effects of loading parameters on single-crystal slip in tantalum using molecular mechanics , 2014 .

[60]  G. Ravichandran,et al.  Thermomechanical characterization of pure polycrystalline tantalum , 2007 .

[61]  A. Rosakis,et al.  A thermodynamic internal variable model for the partition of plastic work into heat and stored energy in metals , 2000 .

[62]  D. Eakins,et al.  On the formation of adiabatic shear bands in textured HCP polycrystals , 2016 .

[63]  P. Withers,et al.  Response and representation of ductile damage under varying shock loading conditions in tantalum , 2016 .

[64]  J. Boettger,et al.  Global Equation of State for Copper , 2011 .

[65]  D. Wallace,et al.  A model of the shear modulus , 1992 .

[66]  Pratheek Shanthraj,et al.  Dislocation density evolution and interactions in crystalline materials , 2011 .