Survey of Existence Results in Nonlinear Peridynamics in Comparison with Local Elastodynamics

Abstract We give an overview on existence results in nonlinear peridynamics, a nonlocal elasticity theory, and compare these results with those known for classical nonlinear elastodynamics. In peridynamics, it is possible to obtain stronger results under equal or even less restrictive assumptions on the stress.

[1]  I. Fonseca,et al.  Modern Methods in the Calculus of Variations: L^p Spaces , 2007 .

[2]  Athanasios E. Tzavaras,et al.  Construction of entropy solutions for one dimensional elastodynamics via time discretisation , 2000 .

[3]  Nicolas Sau,et al.  Peridynamic modeling of concrete structures , 2007 .

[4]  Q. Du,et al.  MATHEMATICAL ANALYSIS FOR THE PERIDYNAMIC NONLOCAL CONTINUUM THEORY , 2011 .

[5]  Peixiong Lin,et al.  Young measures and an application of compensated compactness to one-dimensional nonlinear elastodynamics , 1992 .

[6]  Burak Aksoylu,et al.  Results on Nonlocal Boundary Value Problems , 2010 .

[7]  Hans Engler,et al.  Global regular solutions for the dynamic antiplane shear problem in nonlinear viscoelasticity , 1989 .

[8]  Constantine M. Dafermos,et al.  Energy methods for quasilinear hyperbolic initial-boundary value problems. Applications to elastodynamics , 1985 .

[9]  J. Glimm.  Solutions in the large for nonlinear hyperbolic systems of equations , 1965 .

[10]  Hartmut Pecher.  On global regular solutions of third order partial differential equations , 1980 .

[11]  Carsten Carstensen,et al.  Time-Space Discretization of the Nonlinear Hyperbolic System \protect\mbox\boldmath\lowercase$u_tt = \operatornamediv (\sigma(\mbox\boldmath\uppercase$D$u)+ \mbox\boldmath\uppercase$D$u_t)$ , 2004 .

[12]  Wang Jun,et al.  On some theorems in the nonlocal theory of micropolar elasticity , 1993 .

[13]  Avner Friedman,et al.  Systems of nonlinear wave equations with nonlinear viscosity. , 1988 .

[14]  F. Bobaru,et al.  Studies of dynamic crack propagation and crack branching with peridynamics , 2010 .

[15]  T. Roubíček.  Nonlinear partial differential equations with applications , 2005 .

[16]  Kun Zhou,et al.  Mathematical and Numerical Analysis of Linear Peridynamic Models with Nonlocal Boundary Conditions , 2010, SIAM J. Numer. Anal..

[17]  Peter D. Lax,et al.  Development of Singularities of Solutions of Nonlinear Hyperbolic Partial Differential Equations , 1964 .

[18]  S. Altan,et al.  Existence in nonlocal elasticity , 1989 .

[19]  S. Altan,et al.  Uniqueness of initial-boundary value problems in nonlocal elasticity , 1989 .

[20]  A. Eringen,et al.  On nonlocal elasticity , 1972 .

[21]  A. Eringen,et al.  Theory of Nonlocal Elasticity and Some Applications , 1984 .

[22]  S. Silling.  Reformulation of Elasticity Theory for Discontinuities and Long-Range Forces , 2000 .

[23]  Etienne Emmrich,et al.  Evolution equations of second order with nonconvex potential and linear damping: existence via convergence of a full discretization , 2013 .

[24]  Sophia Demoulini,et al.  A Variational Approximation Scheme for¶Three-Dimensional Elastodynamics¶with Polyconvex Energy , 2001 .

[25]  Nicole Propst,et al.  Mathematical Foundations Of Elasticity , 2016 .

[26]  A. Erkip,et al.  The cauchy problem for a one dimensional nonlinear elastic peridynamic model , 2010, 1010.5689.

[27]  Silvia Sastre Gómez,et al.  Nonlocal diffusion problems , 2014 .

[28]  Etienne Emmrich,et al.  Well-posedness of the peridynamic model with Lipschitz continuous pairwise force function , 2013 .

[29]  Tomáš Roubíček,et al.  Relaxation in Optimization Theory and Variational Calculus , 1997 .

[30]  W. S. Kimball.  XI. New basic critical equations and methods in the calculus of variations , 1937 .

[31]  Etienne Emmrich,et al.  Measure-valued and weak solutions to the nonlinear peridynamic model in nonlocal elastodynamics , 2014 .

[32]  P. Pedregal.  Parametrized measures and variational principles , 1997 .

[33]  Sophia Demoulini,et al.  Young measure solutions for nonlinear evolutionary systems of mixed type , 1997 .

[34]  P. Lax,et al.  Systems of conservation equations with a convex extension. , 1971, Proceedings of the National Academy of Sciences of the United States of America.

[35]  A. Cemal Eringen,et al.  Vistas of nonlocal continuum physics , 1992 .

[36]  Pablo Pedregal,et al.  Addendum to the paper: Hyperelasticity as a $$\Gamma $$Γ-limit of peridynamics when the horizon goes to zero , 2015 .

[37]  A. Erkip,et al.  Global existence and blow-up for a class of nonlocal nonlinear Cauchy problems arising in elasticity , 2009 .

[38]  George E. Andrews,et al.  On the existence of solutions to the equation , 1980 .

[39]  Erdogan Madenci,et al.  Combined finite element and peridynamic analyses for predicting failure in a stiffened composite curved panel with a central slot , 2012 .

[40]  Castrenze Polizzotto,et al.  Nonlocal elasticity and related variational principles , 2001 .

[41]  J. Ball,et al.  Asymptotic behaviour and changes of phase in one-dimensional nonlinear viscoelasticity , 1982 .

[42]  Sophia Demoulini.  Weak Solutions for a Class of Nonlinear Systems of Viscoelasticity , 2000 .

[43]  P. Ciarlet,et al.  Mathematical elasticity, volume I: Three-dimensional elasticity , 1989 .

[44]  Jerrold E. Marsden,et al.  Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity , 1977 .

[45]  Gero Friesecke,et al.  Implicit time discretization and global existence for a quasi-linear evolution equation with nonconvex energy , 1997 .

[46]  Erdogan Madenci,et al.  Predicting crack propagation with peridynamics: a comparative study , 2011 .

[47]  José M. Mazón,et al.  Nonlocal Diffusion Problems , 2010 .

[48]  Jun Wang,et al.  UNIQUENESS IN GENERALIZED NONLOCAL THERMOELASTICITY , 1993 .

[49]  J. Málek.  Weak and Measure-valued Solutions to Evolutionary PDEs , 1996 .

[50]  J. Ball.  Convexity conditions and existence theorems in nonlinear elasticity , 1976 .

[51]  A. C. Eringen,et al.  Nonlocal polar elastic continua , 1972 .

[52]  A. Erkip,et al.  Blow-up and global existence for a general class of nonlocal nonlinear coupled wave equations , 2010, 1009.1263.

[53]  Milan Jirásek,et al.  Nonlocal integral formulations of plasticity and damage : Survey of progress , 2002 .

[54]  A. Eringen,et al.  Nonlocal Continuum Field Theories , 2002 .

[55]  S. Antman.  Nonlinear problems of elasticity , 1994 .

[56]  P. LeFloch.  Hyperbolic Systems of Conservation Laws , 2002 .

[57]  Marc Oliver Rieger,et al.  Young Measure Solutions for Nonconvex Elastodynamics , 2003, SIAM J. Math. Anal..

[58]  J. Lions,et al.  Non-homogeneous boundary value problems and applications , 1972 .

[59]  A. Cemal Eringen,et al.  Linear theory of nonlocal elasticity and dispersion of plane waves , 1972 .

[60]  Carsten Carstensen,et al.  Esaim: Mathematical Modelling and Numerical Analysis Young-measure Approximations for Elastodynamics with Non-monotone Stress-strain Relations , 2022 .

[61]  R. J. DiPerna.  Convergence of approximate solutions to conservation laws , 1983 .

[62]  Andreas Prohl,et al.  Convergence of a Finite Element-Based Space-Time Discretization in Elastodynamics , 2008, SIAM J. Numer. Anal..

[63]  B. Altan,et al.  UNIQUENESS IN NONLOCAL THERMOELASTICITY , 1991 .

[64]  E. Kröner,et al.  Elasticity theory of materials with long range cohesive forces , 1967 .

[65]  José Carlos Bellido,et al.  Existence for Nonlocal Variational Problems in Peridynamics , 2014, SIAM J. Math. Anal..

[66]  Shirley Dex,et al.  JR 旅客販売総合システム（マルス）における運用及び管理について , 1991 .

[67]  Etienne Emmrich,et al.  Peridynamics: a nonlocal continuum theory. , 2013 .

[68]  E. Emmrich,et al.  On the well-posedness of the linear peridynamic model and its convergence towards the Navier equation of linear elasticity , 2007 .

[69]  J. Clements,et al.  EXISTENCE THEOREMS FOR A QUASILINEAR EVOLUTION EQUATION , 1974 .

[70]  R. Lipton.  Dynamic Brittle Fracture as a Small Horizon Limit of Peridynamics , 2013, 1305.4531.