A variational treatment of material configurations with application to interface motion and microstructural evolution
Abstract
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 physicallymore »
- Authors:
-
- Univ. of Michigan, Ann Arbor, MI (United States)
- Publication Date:
- Research Org.:
- Univ. of Michigan, Ann Arbor, MI (United States)
- Sponsoring Org.:
- USDOE Office of Science (SC), Basic Energy Sciences (BES)
- OSTI Identifier:
- 1332707
- Alternate Identifier(s):
- OSTI ID: 1421991
- Grant/Contract Number:
- SC0008637
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Journal of the Mechanics and Physics of Solids
- Additional Journal Information:
- Journal Name: Journal of the Mechanics and Physics of Solids; Journal ID: ISSN 0022-5096
- Publisher:
- Elsevier
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 36 MATERIALS SCIENCE; 97 MATHEMATICS AND COMPUTING; free energy; thermodynamic driving force; gradient regularization; configurational stress; spline basis functions
Citation Formats
Teichert, Gregory H., Rudraraju, Shiva, and Garikipati, Krishna. A variational treatment of material configurations with application to interface motion and microstructural evolution. United States: N. p., 2016.
Web. doi:10.1016/j.jmps.2016.11.008.
Teichert, Gregory H., Rudraraju, Shiva, & Garikipati, Krishna. A variational treatment of material configurations with application to interface motion and microstructural evolution. United States. https://doi.org/10.1016/j.jmps.2016.11.008
Teichert, Gregory H., Rudraraju, Shiva, and Garikipati, Krishna. Sun .
"A variational treatment of material configurations with application to interface motion and microstructural evolution". United States. https://doi.org/10.1016/j.jmps.2016.11.008. https://www.osti.gov/servlets/purl/1332707.
@article{osti_1332707,
title = {A variational treatment of material configurations with application to interface motion and microstructural evolution},
author = {Teichert, Gregory H. and Rudraraju, Shiva and Garikipati, Krishna},
abstractNote = {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.},
doi = {10.1016/j.jmps.2016.11.008},
journal = {Journal of the Mechanics and Physics of Solids},
number = ,
volume = ,
place = {United States},
year = {Sun Nov 20 00:00:00 EST 2016},
month = {Sun Nov 20 00:00:00 EST 2016}
}
Web of Science
Works referenced in this record:
Coupled phase transformations and plasticity as a field theory of deformation incompatibility
journal, December 2011
- Acharya, Amit; Fressengeas, Claude
- International Journal of Fracture, Vol. 174, Issue 1
Local minimizers and planar interfaces in a phase-transition model with interfacial energy
journal, July 2010
- Ball, J. M.; Crooks, E. C. M.
- Calculus of Variations and Partial Differential Equations, Vol. 40, Issue 3-4
Numerical Schemes for the Hamilton–Jacobi and Level Set Equations on Triangulated Domains
journal, September 1998
- Barth, Timothy J.; Sethian, James A.
- Journal of Computational Physics, Vol. 145, Issue 1
Elastic Energy Minimization and the Recoverable Strains of Polycrystalline Shape-Memory Materials
journal, October 1997
- Bhattacharya, Kaushik; Kohn, Robert V.
- Archive for Rational Mechanics and Analysis, Vol. 139, Issue 2
Crystal symmetry and the reversibility of martensitic transformations
journal, March 2004
- Bhattacharya, Kaushik; Conti, Sergio; Zanzotto, Giovanni
- Nature, Vol. 428, Issue 6978
Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations
journal, September 1982
- Brooks, Alexander N.; Hughes, Thomas J. R.
- Computer Methods in Applied Mechanics and Engineering, Vol. 32, Issue 1-3
XVIII. A theory concerning the constitution of matter
journal, February 1892
- Burton, Charles V.
- The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, Vol. 33, Issue 201
Configurational forces for quasi-incompressible large strain electro-viscoelasticity – Application to fracture mechanics
journal, November 2014
- Denzer, Ralf; Menzel, Andreas
- European Journal of Mechanics - A/Solids, Vol. 48
Recent Advances in Models for Thermal Oxidation of Silicon
journal, November 2001
- Garikipati, Krishna; Rao, Vinay S.
- Journal of Computational Physics, Vol. 174, Issue 1
Biological remodelling: Stationary energy, configurational change, internal variables and dissipation
journal, July 2006
- Garikipati, K.; Olberding, J.; Narayanan, H.
- Journal of the Mechanics and Physics of Solids, Vol. 54, Issue 7
Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement
journal, October 2005
- Hughes, T. J. R.; Cottrell, J. A.; Bazilevs, Y.
- Computer Methods in Applied Mechanics and Engineering, Vol. 194, Issue 39-41
Configurational Forces in Continuous Theories of Elastic Ferroelectrics
book, January 2009
- Kalpakides, Vassilios K.; Arvanitakis, Antonios I.
- IUTAM Symposium on Progress in the Theory and Numerics of Configurational Mechanics
On the properties of the Eshelly tensor
journal, March 1997
- Kienzler, R.; Herrmann, G.
- Acta Mechanica, Vol. 125, Issue 1-4
A Dynamical Theory of the Electric and Luminiferous Medium. Part III. Relations with Material Media
journal, January 1897
- Larmor, J.
- Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 190, Issue 0
An improved geometry-aware curvature discretization for level set methods: Application to tumor growth
journal, July 2006
- Macklin, Paul; Lowengrub, John
- Journal of Computational Physics, Vol. 215, Issue 2
Material Forces: Concepts and Applications
journal, May 1995
- Maugin, Ge´rard A.
- Applied Mechanics Reviews, Vol. 48, Issue 5
On material forces and finite element discretizations
journal, July 2002
- Mueller, R.; Maugin, G. A.
- Computational Mechanics, Vol. 29, Issue 1
Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations
journal, November 1988
- Osher, Stanley; Sethian, James A.
- Journal of Computational Physics, Vol. 79, Issue 1
Configurational forces: are they needed?
journal, November 2002
- Podio-Guidugli, P.
- Mechanics Research Communications, Vol. 29, Issue 6
On modelling thermal oxidation of Silicon I: theory
journal, January 2000
- Rao, Vinay S.; Hughes, Thomas J. R.
- International Journal for Numerical Methods in Engineering, Vol. 47, Issue 1-3
On modelling thermal oxidation of Silicon II: numerical aspects
journal, January 2000
- Rao, Vinay S.; Hughes, Thomas J. R.; Garikipati, Krishna
- International Journal for Numerical Methods in Engineering, Vol. 47, Issue 1-3
Three-dimensional isogeometric solutions to general boundary value problems of Toupin’s gradient elasticity theory at finite strains
journal, August 2014
- Rudraraju, S.; Van der Ven, A.; Garikipati, K.
- Computer Methods in Applied Mechanics and Engineering, Vol. 278
A Remark on Computing Distance Functions
journal, September 2000
- Russo, Giovanni; Smereka, Peter
- Journal of Computational Physics, Vol. 163, Issue 1
On spatial and material settings of hyperelastostatic crystal defects
journal, August 2002
- Steinmann, P.
- Journal of the Mechanics and Physics of Solids, Vol. 50, Issue 8
Elastic materials with couple-stresses
journal, January 1962
- Toupin, R. A.
- Archive for Rational Mechanics and Analysis, Vol. 11, Issue 1
On the spatial and material motion problems in nonlinear electro-elastostatics with consideration of free space
journal, January 2012
- Vu, Dk; Steinmann, P.
- Mathematics and Mechanics of Solids, Vol. 17, Issue 8
On spatial and material covariant balance laws in elasticity
journal, April 2006
- Yavari, Arash; Marsden, Jerrold E.; Ortiz, Michael
- Journal of Mathematical Physics, Vol. 47, Issue 4
A Remark on Computing Distance Functions
journal, September 2000
- Russo, Giovanni; Smereka, Peter
- Journal of Computational Physics, Vol. 163, Issue 1
Three-dimensional isogeometric solutions to general boundary value problems of Toupin’s gradient elasticity theory at finite strains
journal, August 2014
- Rudraraju, S.; Van der Ven, A.; Garikipati, K.
- Computer Methods in Applied Mechanics and Engineering, Vol. 278
Works referencing / citing this record:
PRISMS: An Integrated, Open-Source Framework for Accelerating Predictive Structural Materials Science
journal, August 2018
- Aagesen, L. K.; Adams, J. F.; Allison, J. E.
- JOM, Vol. 70, Issue 10