On configurational compatibility and multiscale energy momentum tensors

Li, Shaofan; Linder, Christian; Foulk, James W.

doi:10.1016/j.jmps.2006.11.002

Cited by 11 publications

(10 citation statements)

References 75 publications

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“…with the J integrals [38] and Li [39] have used such an Eshelby stress tensor of dislocations without interaction with a force stress tensor. They called it compatibility momentum tensor.…”

Section: Static Casementioning

confidence: 99%

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The gauge theory of dislocations: conservation and balance laws

Lazar

Anastassiadis

2008

Philosophical Magazine

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We derive conservation and balance laws for the translational gauge theory of dislocations by applying Noether's theorem. We present an improved translational gauge theory of dislocations including the dislocation density tensor and the dislocation current tensor. The invariance of the variational principle under the continuous group of transformations is studied. Through Lie's-infinitesimal invariance criterion we obtain conserved translational and rotational currents for the total Lagrangian made up of an elastic and dislocation part. We calculate the broken scaling current. Looking only on one part of the whole system, the conservation laws are changed into balance laws. Because of the lack of translational, rotational and dilatation invariance for each part, a configurational force, moment and power appears. The corresponding J, L and M integrals are obtained. Only isotropic and homogeneous materials are considered and we restrict ourselves to a linear theory. We choose constitutive laws for the most general linear form of material isotropy. Also we give the conservation and balance laws corresponding to the gauge symmetry and the addition of solutions. From the addition of solutions we derive a reciprocity theorem for the gauge theory of dislocations. Also, we derive the conservation laws for stress-free states of dislocations.Comment: 28 page

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“…with the J integrals [38] and Li [39] have used such an Eshelby stress tensor of dislocations without interaction with a force stress tensor. They called it compatibility momentum tensor.…”

Section: Static Casementioning

confidence: 99%

“…This is the case for a pure dislocation theory without force stresses. If f j,i = constant, we recover the symmetry of constant pre-distortion used by Li et al [38] and Li [39].…”

Section: Dislocation Partmentioning

confidence: 99%

The gauge theory of dislocations: conservation and balance laws

Lazar

Anastassiadis

2008

Philosophical Magazine

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“…Similarly via the compatibility condition (6), it is easy to show that S f k,k ¼ 0, if there is no inelastic deformation in the solid. We may call S f k the compatibility-momentum tensor, because it is derived based on the symmetry condition of compatibility conditions [13]. Conceptually, configurational compatibility forms a duality pair with configurational force [14].…”

Section: Multiscale Energy-momentum Tensormentioning

confidence: 99%

A multiscale Griffith criterion

2007

Philosophical Magazine Letters

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In this letter, a compatibility-momentum tensor is proposed based on the strain compatibility condition of continuum mechanics. By rescaling the compatibilitymomentum tensor, we construct a so-called multiscale energy-momentum tensor and the corresponding L-integral, which is path-independent if the interior region of the integration contour is dislocation-free. By applying the invariant L-integral to the field of a mode III elastoplastic crack under small-scale yielding condition, we derive a multiscale criterion for macroscopically brittle fractures.

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“…Those applications exploit the two char acteristic material properties of piezoelectric ceramics in the form of the direct and the converse piezoelectric effect, where an elec tric field is produced when the material undergoes a mechanical deformation and where a deformation is induced when the speci men is subject to an electric field, respectively [1], When operated in a certain limited regime, a linear piezoelectric model can be used to describe these effects [2], Piezoelectric ceramics are generally classified as a brittle mate rial with the possible appearance of cracks, which after initiation start to propagate and may lead to the failure of the whole device. Concepts like the energy release rate, stress intensity factors, or the /-integral have been intro duced to better understand failure mechanisms in materials [4][5][6][7]. Fracture criteria for linear piezoelectricity were inspired by the results obtained for purely mechanical materials [3].…”

Section: Introductionmentioning

confidence: 99%

A Complex Variable Solution Based Analysis of Electric Displacement Saturation for a Cracked Piezoelectric Material

Linder

2014

Journal of Applied Mechanics

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A C o m p le x V a ria b le S o lu tio n B ased A n a ly s is of E le c tric D is p la c e m e n t S a tu ra tio n fo r a C racked P ie z o e le c tric M a te ria lThis paper presents an analysis of the effect of electric displacement saturation for a fail ing piezoelectric ceramic material based on a complex variable solution of a Mode III and a Mode I crack. This particular electric nonlinearity is caused by a reduction of the ionic movement in the material in the presence of high electric fields. Total and strain energy release rates are computed for varying far field stresses, electric displacements, and electric fields and compared for cases without and with full electric displacement saturation to further advance the understanding of failure initiation in piezoelectric ceramics. Basic Equations of Linear PiezoelectricityWithin the small strain range, the primary unknowns of a piezo electric solid 08, which occupies a configuration B C R , are the mechanical displacement field m,-: B -* R 3 and the electric potential

R 3xm 3 given aswith R 3x^ representing the space of symmetric tensors, and the electric field e, : B -> R 3 defined by e, = -q>,iHere, (•) , • = d(-)/dxj is the standard gradient operation with respect to the coordinate x,. For the quasi-static case considered Journal of Applied Mechanics

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On configurational compatibility and multiscale energy momentum tensors

Cited by 11 publications

References 75 publications

The gauge theory of dislocations: conservation and balance laws

The gauge theory of dislocations: conservation and balance laws

A multiscale Griffith criterion

A Complex Variable Solution Based Analysis of Electric Displacement Saturation for a Cracked Piezoelectric Material

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