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145
Convergence of peridynamics to classical elasticity theory
 Journal of Elasticity
"... Abstract The peridynamic model of solid mechanics is a nonlocal theory containing a length scale. It is based on direct interactions between points in a continuum separated from each other by a finite distance. The maximum interaction distance provides a length scale for the material model. This pap ..."
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Cited by 29 (5 self)
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Abstract The peridynamic model of solid mechanics is a nonlocal theory containing a length scale. It is based on direct interactions between points in a continuum separated from each other by a finite distance. The maximum interaction distance provides a length scale for the material model. This paper addresses the question of whether the peridynamic model for an elastic material reproduces the classical local model as this length scale goes to zero. We show that if the motion, constitutive model, and any nonhomogeneities are sufficiently smooth, then the peridynamic stress tensor converges in this limit to a PiolaKirchhoff stress tensor that is a function only of the local deformation gradient tensor, as in the classical theory. This limiting PiolaKirchhoff stress tensor field is differentiable, and its divergence represents the force density due to internal forces. The limiting, or collapsed, stressstrain model satisfies the conditions in the classical theory for angular momentum balance, isotropy, objectivity, and hyperelasticity, provided the original peridynamic consti
Steadystate crack growth and work of fracture for solids characterized by strain gradient plasticity
 J. Mech. Phys. Solids
, 1997
"... Mode I steadystate crack growth is analysed under plane strain conditions in small scale yielding. The elasticplastic solid is characterized by a generalization of JZ flow theory which accounts for the influence of the gradients of plastic strains on hardening. The constitutive model involves one ..."
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Cited by 27 (3 self)
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Mode I steadystate crack growth is analysed under plane strain conditions in small scale yielding. The elasticplastic solid is characterized by a generalization of JZ flow theory which accounts for the influence of the gradients of plastic strains on hardening. The constitutive model involves one new parameter, a material ength 1, specifying the scale of nonuniform deformation at which hardening elevation owing to strain gradients becomes important. Gradients of plastic strain at a sharp crack tip result in a substantial increase in tractions ahead of the tip. This has important consequences for crack growth in materials that fail by decohesion or cleavage at the atomic scale. The new constitutive law is used in conjunction with a model which represents the fracture process by an embedded tractionseparation relation applied on the plane ahead of the crack tip. The ratio of the macroscopic work of fracture to the work of the fracture process is calculated as a function of the parameters characterizing the fracture process and the solid, with particular emphasis on the role of 1. 0 1997 Elsevier Science Ltd
Theories of elasticity with couplestress
 Arch. Rational Mech. Anal
, 1964
"... The concept of couplestress i familiar from the theory of elastic shells. It is customary to represent the action of one part of a shell upon another by a line distribution of forces and couples along a curve which divides the shell into two parts. If suitable other assumptions be made, the force p ..."
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Cited by 25 (0 self)
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The concept of couplestress i familiar from the theory of elastic shells. It is customary to represent the action of one part of a shell upon another by a line distribution of forces and couples along a curve which divides the shell into two parts. If suitable other assumptions be made, the force per unit
Élie Cartan’s torsion in geometry and in field theory, an essay
 ANNALES DE LA FONDATION LOUIS DE BROGLIE, MANUSCRIT
, 2007
"... We review the application of torsion in field theory. First we show how the notion of torsion emerges in differential geometry. In the context of a Cartan circuit, torsion is related to translations similar as curvature to rotations. Cartan’s investigations started by analyzing Einsteins general rel ..."
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Cited by 16 (2 self)
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We review the application of torsion in field theory. First we show how the notion of torsion emerges in differential geometry. In the context of a Cartan circuit, torsion is related to translations similar as curvature to rotations. Cartan’s investigations started by analyzing Einsteins general relativity theory and by taking recourse to the theory of Cosserat continua. In these continua, the points of which carry independent translational and rotational degrees of freedom, there occur, besides ordinary (force) stresses, additionally spin moment stresses. In a 3dimensional continuized crystal with dislocation lines, a linear connection can be introduced that takes the crystal lattice structure as a basis for parallelism. Such a continuum has similar properties as a Cosserat continuum, and the dislocation density is equal to the torsion of this connection. Subsequently, these ideas are applied to 4dimensional spacetime. A translational gauge theory of gravity is displayed (in a Weitzenböck or teleparallel spacetime) as well as the viable EinsteinCartan theory (in a RiemannCartan spacetime). In both theories, the notion of torsion is contained in an essential
Nonlinear microstrain theories.
 Int. J. Solids Struct.,
, 2006
"... Abstract A hierarchy of higher order continua is presented that introduces additional degrees of freedom accounting for volume changes, rotation and straining of an underlying microstructure. An increase in the number of degrees of freedom represents a refinement of the material description. In add ..."
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Cited by 14 (2 self)
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Abstract A hierarchy of higher order continua is presented that introduces additional degrees of freedom accounting for volume changes, rotation and straining of an underlying microstructure. An increase in the number of degrees of freedom represents a refinement of the material description. In addition to available nonlinear Cosserat and micromorphic theories, general formulations of elastoviscoplastic behaviour are proposed for microdilatation and microstretch continua. A microstrain theory is introduced that is based on six additional degrees of freedom describing the pure straining of the microstructural element. In each case, balance equations and boundary conditions are derived, decompositions of the finite strain measures into elastic and plastic parts are provided. The formulation of finite deformation elastoviscoplastic constitutive equations relies on the introduction of the free energy and dissipation potentials, thus complying with requirements of continuum thermodynamics. Some guidelines for the selection of a suitable higher order model for a given material close the discussion.
GROUND STATES IN COMPLEX BODIES
, 2008
"... A unified framework for analyzing the existence of ground states in wide classes of elastic complex bodies is presented here. The approach makes use of classical semicontinuity results, Sobolev mappinngs and Cartesian currents. Weak diffeomorphisms are used to represent macroscopic deformations. So ..."
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Cited by 11 (2 self)
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A unified framework for analyzing the existence of ground states in wide classes of elastic complex bodies is presented here. The approach makes use of classical semicontinuity results, Sobolev mappinngs and Cartesian currents. Weak diffeomorphisms are used to represent macroscopic deformations. Sobolev maps and Cartesian currents describe the inner substructure of the material elements. Balance equations for irregular minimizers are derived. A contribution to the debate about the role of the balance of configurational actions follows. After describing a list of possible applications of the general results collected here, a concrete discussion of the existence of ground states in thermodynamically stable quasicrystals is presented at the end.
A physically based gradient plasticity theory’,
 Int. J. Plasticity,
, 2006
"... Abstract The intent of this work is to derive a physically motivated mathematical form for the gradient plasticity that can be used to interpret the size effects observed experimentally. The step of translating from the dislocationbased mechanics to a continuum formulation is explored. This paper ..."
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Cited by 10 (0 self)
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Abstract The intent of this work is to derive a physically motivated mathematical form for the gradient plasticity that can be used to interpret the size effects observed experimentally. The step of translating from the dislocationbased mechanics to a continuum formulation is explored. This paper addresses a possible, yet simple, link between the TaylorÕs model of dislocation hardening and the strain gradient plasticity. Evolution equations for the densities of statistically stored dislocations and geometrically necessary dislocations are used to establish this linkage. The dislocation processes of generation, motion, immobilization, recovery, and annihilation are considered in which the geometric obstacles contribute to the storage of statistical dislocations. As a result, a physically sound relation for the material length scale parameter is obtained as a function of the course of plastic deformation, grain size, and a set of macroscopic and microscopic physical parameters. Comparisons are made of this theory with experiments on microtorsion, microbending, and microindentation size effects.
Mixed finite element formulations of straingradient elasticity problems
, 2002
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