A variational treatment of material configurations with application to interface motion and microstructural evolution

Teichert, Gregory H.; Rudraraju, Shiva; Garikipati, Krishna

doi:10.1016/j.jmps.2016.11.008

Title: A variational treatment of material configurations with application to interface motion and microstructural evolution

Journal Article · Sun Nov 20 00:00:00 EST 2016 · Journal of the Mechanics and Physics of Solids

DOI:https://doi.org/10.1016/j.jmps.2016.11.008· OSTI ID:1332707

Teichert, Gregory H. ^[1]; Rudraraju, Shiva ^[1]; Garikipati, Krishna ^[1]

Univ. of Michigan, Ann Arbor, MI (United States)

We present a unified variational treatment of evolving configurations in crystalline solids with microstructure. The crux of our treatment lies in the introduction of a vector configurational field. This field lies in the material, or configurational, manifold, in contrast with the traditional displacement field, which we regard as lying in the spatial manifold. We identify two distinct cases which describe (a) problems in which the configurational field's evolution is localized to a mathematically sharp interface, and (b) those in which the configurational field's evolution can extend throughout the volume. The first case is suitable for describing incoherent phase interfaces in polycrystalline solids, and the latter is useful for describing smooth changes in crystal structure and naturally incorporates coherent (diffuse) phase interfaces. These descriptions also lead to parameterizations of the free energies for the two cases, from which variational treatments can be developed and equilibrium conditions obtained. For sharp interfaces that are out-of-equilibrium, the second law of thermodynamics furnishes restrictions on the kinetic law for the interface velocity. The class of problems in which the material undergoes configurational changes between distinct, stable crystal structures are characterized by free energy density functions that are non-convex with respect to configurational strain. For physically meaningful solutions and mathematical well-posedness, it becomes necessary to incorporate interfacial energy. This we have done by introducing a configurational strain gradient dependence in the free energy density function following ideas laid out by Toupin (Arch. Rat. Mech. Anal., 11, 1962, 385-414). The variational treatment leads to a system of partial differential equations governing the configuration that is coupled with the traditional equations of nonlinear elasticity. The coupled system of equations governs the configurational change in crystal structure, and elastic deformation driven by elastic, Eshelbian, and configurational stresses. As a result, numerical examples are presented to demonstrate interface motion as well as evolving microstructures of crystal structures.

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Research Organization:: Univ. of Michigan, Ann Arbor, MI (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Basic Energy Sciences (BES)

Grant/Contract Number:: SC0008637

OSTI ID:: 1332707

Alternate ID(s):: OSTI ID: 1421991

Journal Information:: Journal of the Mechanics and Physics of Solids, Journal Name: Journal of the Mechanics and Physics of Solids; ISSN 0022-5096

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

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Cited by: 5 works

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References (27)

Coupled phase transformations and plasticity as a field theory of deformation incompatibility Acharya, Amit; Fressengeas, Claude International Journal of Fracture, Vol. 174, Issue 1 https://doi.org/10.1007/s10704-011-9656-0	journal	December 2011
Local minimizers and planar interfaces in a phase-transition model with interfacial energy Ball, J. M.; Crooks, E. C. M. Calculus of Variations and Partial Differential Equations, Vol. 40, Issue 3-4 https://doi.org/10.1007/s00526-010-0349-8	journal	July 2010
Numerical Schemes for the Hamilton–Jacobi and Level Set Equations on Triangulated Domains Barth, Timothy J.; Sethian, James A. Journal of Computational Physics, Vol. 145, Issue 1 https://doi.org/10.1006/jcph.1998.6007	journal	September 1998
Elastic Energy Minimization and the Recoverable Strains of Polycrystalline Shape-Memory Materials Bhattacharya, Kaushik; Kohn, Robert V. Archive for Rational Mechanics and Analysis, Vol. 139, Issue 2 https://doi.org/10.1007/s002050050049	journal	October 1997
Crystal symmetry and the reversibility of martensitic transformations Bhattacharya, Kaushik; Conti, Sergio; Zanzotto, Giovanni Nature, Vol. 428, Issue 6978 https://doi.org/10.1038/nature02378	journal	March 2004
Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations Brooks, Alexander N.; Hughes, Thomas J. R. Computer Methods in Applied Mechanics and Engineering, Vol. 32, Issue 1-3 https://doi.org/10.1016/0045-7825(82)90071-8	journal	September 1982
XVIII. A theory concerning the constitution of matter Burton, Charles V. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, Vol. 33, Issue 201 https://doi.org/10.1080/14786449208621454	journal	February 1892
Configurational forces for quasi-incompressible large strain electro-viscoelasticity – Application to fracture mechanics Denzer, Ralf; Menzel, Andreas European Journal of Mechanics - A/Solids, Vol. 48 https://doi.org/10.1016/j.euromechsol.2014.05.012	journal	November 2014
Recent Advances in Models for Thermal Oxidation of Silicon Garikipati, Krishna; Rao, Vinay S. Journal of Computational Physics, Vol. 174, Issue 1 https://doi.org/10.1006/jcph.2001.6884	journal	November 2001
Biological remodelling: Stationary energy, configurational change, internal variables and dissipation Garikipati, K.; Olberding, J.; Narayanan, H. Journal of the Mechanics and Physics of Solids, Vol. 54, Issue 7 https://doi.org/10.1016/j.jmps.2005.11.011	journal	July 2006
Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement Hughes, T. J. R.; Cottrell, J. A.; Bazilevs, Y. Computer Methods in Applied Mechanics and Engineering, Vol. 194, Issue 39-41 https://doi.org/10.1016/j.cma.2004.10.008	journal	October 2005
Configurational Forces in Continuous Theories of Elastic Ferroelectrics Kalpakides, Vassilios K.; Arvanitakis, Antonios I. IUTAM Symposium on Progress in the Theory and Numerics of Configurational Mechanics https://doi.org/10.1007/978-90-481-3447-2_21	book	January 2009
On the properties of the Eshelly tensor Kienzler, R.; Herrmann, G. Acta Mechanica, Vol. 125, Issue 1-4 https://doi.org/10.1007/BF01177300	journal	March 1997
A Dynamical Theory of the Electric and Luminiferous Medium. Part III. Relations with Material Media Larmor, J. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 190, Issue 0 https://doi.org/10.1098/rsta.1897.0020	journal	January 1897
An improved geometry-aware curvature discretization for level set methods: Application to tumor growth Macklin, Paul; Lowengrub, John Journal of Computational Physics, Vol. 215, Issue 2 https://doi.org/10.1016/j.jcp.2005.11.016	journal	July 2006
Material Forces: Concepts and Applications Maugin, Ge´rard A. Applied Mechanics Reviews, Vol. 48, Issue 5 https://doi.org/10.1115/1.3005101	journal	May 1995
On material forces and finite element discretizations Mueller, R.; Maugin, G. A. Computational Mechanics, Vol. 29, Issue 1 https://doi.org/10.1007/s00466-002-0322-2	journal	July 2002
Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations Osher, Stanley; Sethian, James A. Journal of Computational Physics, Vol. 79, Issue 1 https://doi.org/10.1016/0021-9991(88)90002-2	journal	November 1988
Configurational forces: are they needed? Podio-Guidugli, P. Mechanics Research Communications, Vol. 29, Issue 6 https://doi.org/10.1016/S0093-6413(02)00295-1	journal	November 2002
On modelling thermal oxidation of Silicon I: theory Rao, Vinay S.; Hughes, Thomas J. R. International Journal for Numerical Methods in Engineering, Vol. 47, Issue 1-3 https://doi.org/10.1002/(SICI)1097-0207(20000110/30)47:1/3<341::AID-NME774>3.0.CO;2-Z	journal	January 2000
On modelling thermal oxidation of Silicon II: numerical aspects Rao, Vinay S.; Hughes, Thomas J. R.; Garikipati, Krishna International Journal for Numerical Methods in Engineering, Vol. 47, Issue 1-3 https://doi.org/10.1002/(SICI)1097-0207(20000110/30)47:1/3<359::AID-NME775>3.0.CO;2-7	journal	January 2000
Three-dimensional isogeometric solutions to general boundary value problems of Toupin’s gradient elasticity theory at finite strains Rudraraju, S.; Van der Ven, A.; Garikipati, K. Computer Methods in Applied Mechanics and Engineering, Vol. 278 https://doi.org/10.1016/j.cma.2014.06.015	journal	August 2014
A Remark on Computing Distance Functions Russo, Giovanni; Smereka, Peter Journal of Computational Physics, Vol. 163, Issue 1 https://doi.org/10.1006/jcph.2000.6553	journal	September 2000
On spatial and material settings of hyperelastostatic crystal defects Steinmann, P. Journal of the Mechanics and Physics of Solids, Vol. 50, Issue 8 https://doi.org/10.1016/S0022-5096(01)00135-1	journal	August 2002
Elastic materials with couple-stresses Toupin, R. A. Archive for Rational Mechanics and Analysis, Vol. 11, Issue 1 https://doi.org/10.1007/BF00253945	journal	January 1962
On the spatial and material motion problems in nonlinear electro-elastostatics with consideration of free space Vu, Dk; Steinmann, P. Mathematics and Mechanics of Solids, Vol. 17, Issue 8 https://doi.org/10.1177/1081286511430161	journal	January 2012
On spatial and material covariant balance laws in elasticity Yavari, Arash; Marsden, Jerrold E.; Ortiz, Michael Journal of Mathematical Physics, Vol. 47, Issue 4 https://doi.org/10.1063/1.2190827	journal	April 2006

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