Eshelby's Energy Momentum Research Articles

On dual conservation laws in planar elasticity

Dual conservation laws of linear planar elasticity theory have been systematically studied based on stress function formalism. By employing generalized symmetry transformation or Lie–Bäcklund transformation, a class of new dual conservation laws in planar elasticity have been discovered based on Noether theorem and its Bessel–Hagen generalization. These dual conservation laws represent variational symmetry properties of complementary potential energy, which stems from the symmetry properties of compatibility conditions––a biharmonic equation in two dimension. The physical implications of these dual conservation laws are discussed briefly. In particular, a dual-Eshelby tensor is constructed and compared with the Eshelby's energy momentum tensor.

Conserving Galerkin weak formulations for computational fracture mechanics

AbstractIn this paper, a notion of invariant Galerkin‐variational weak forms is proposed. Two specific invariant variational weak forms, the J‐invariant and the L‐invariant, are constructed based on the corresponding conservation laws in elasticity, one of which is the conservation of Eshelby's energy‐momentum (Eshelby. Philos. Trans. Roy. Soc. 1951; 87: 12; In Solid State Physics, Setitz F, Turnbull D (eds). Academic Press: New York, 1956; 331; Rice, J. Appl. Mech. 1968; 35: 379).It is shown that the finite element solution obtained from the invariant Galerkin weak formulations proposed here can conserve the value of J‐integral, or L‐integral exactly. In other words, the J and L integrals of the Galerkin finite element solutions are path independent in the discrete sense. It is argued that by using the J‐invariant Galerkin weak form to compute near crack‐tip field in an elastic solid, one may accurately calculate the crack extension energy release rate and subsequently the stress intensity factors in numerical computations, because the flux of the energy‐momentum is conserved in discrete computations. This may provide an alternative means to accurately simulate crack growth and propagation. Copyright © 2002 John Wiley & Sons, Ltd.

Material forces due to crack-inclusion interaction

The interaction between a crack and an elastically misfitting inclusion is investigated numerically. The microstructural change of the inclusion's morphology associated with this interaction is discussed using linear elasticity. The misfit is considered purely dilatational and no far field stresses are taken into account. After briefly discussing the theory of configurational forces using Eshelby's energy momentum tensor, a generalized balance equation is obtained to calculate material forces on a crack tip and on the interface of an inclusion. Utilizing a finite element formulation, the generalized balance equation is solved and results are presented for cracks interacting with both hard and soft inclusions.

Numerical assessment of T-stress computation using a p-version finite element method

Two path independent integrals for T-stress computations, one based on the Betti-Rayleigh reciprocal theorem and the other based on Eshelby's energy momentum tensor are studied. Analytical as well as numerical equivalence between the two integrals is found. To quantify and assess the accuracy of computed values, error analysis for the proposed numerical computation of the T-stress is presented. Specifically, it is found that the error of the computed T-stress is proportional to the ratio of the stress intensity factor divided by the square root of the characteristic dimension of the integration domain where the path independent integral is evaluated. Using a highly accurate hierarchical p-version finite element method, the convergence and accuracy of computed values are easily monitored, and it is shown for numerical examples that the error of the computed T-stress complies with the described error analysis. We conclude that path independent integrals, in conjunction with hierarchical p-version finite element methods, provide a powerful and robust tool to obtain highly accurate numerical results for the T-stress.

Generalized Griffith criterion for dynamic fracture and the stability of crack motion at high velocities.

We use Eshelby's energy momentum tensor of dynamic elasticity to compute the forces acting on a moving crack front in a three-dimensional elastic solid [Philos. Mag. 42, 1401 (1951)]. The crack front is allowed to be any curve in three dimensions, but its curvature is assumed small enough so that near the front the dynamics is locally governed by two-dimensional physics. In this case the component of the elastic force on the crack front that is tangent to the front vanishes. However, both the other components, parallel and perpendicular to the direction of motion, do not vanish. We propose that the dynamics of cracks that are allowed to deviate from straight line motion is governed by a vector equation that reflects a balance of elastic forces with dissipative forces at the crack tip, and a phenomenological model for those dissipative forces is advanced. Under certain assumptions for the parameters that characterize the model for the dissipative forces, we find a second order dynamic instability for the crack trajectory. This is signaled by the existence of a critical velocity V(c) such that for velocities V<V(c) the motion is governed by K(II)=0, while for V>V(c) it is governed by K(II) not equal to 0. This result provides a qualitative explanation for some experimental results associated with dynamic fracture instabilities in thin brittle plates. When deviations from straight line motion are suppressed, the usual equation of straight line crack motion based on a Griffiths-like criterion is recovered.

Computational methods for inverse finite elastostatics

In the inverse motion problem in finite hyper-elasticity, the classical formulation relies on conservation laws based on Eshelby's energy-momentum tensor. This formulation is shown to be lacking in several regards for a particular class of inverse motion problems where the deformed configuration and Cauchy traction are given and the undeformed configuration must be calculated. It is shown that for finite element calculations a simple re-examination of the equilibrium equations provides a more suitable finite element formulation. This formulation is also shown to involve only minor changes to existing elements designed for forward motion calculations. Examples illustrating the method in simple and complex situations involving a Neo-Hookean material are presented.

On Interface Crack Growth in Composite Plates

Analysis of fracture growth, and in particular at interfaces, is pertinent not only to load-carrying members in composite structures but also as regards, e.g., adhesive joints, thin films, and coatings. Ordinarily linear fracture mechanics then constitutes the common tool to solve two-dimensional problems occasionally based on beam theory. In the present more general effort, an analysis is first carried out for determination of the energy release rate at general loading of multilayered plates with local crack advance either at interfaces or parallel to such. The procedure is accomplished for arbitrary hyperelastic material properties within von Karman plate theory and the results are expressed by aid of an Eshelby energy momentum tensor. By a feasible superposition it is then shown that the original nonlinear plate problem may be reduced to that of an equivalent beam in case of linear material properties. As a consequence of the so-established principle, the magnitude of mode-dependent singular stress amplitude factors is then directly determinable from earlier two-dimensional linear beam solutions for isotropic and anisotropic bimaterials and relevant at determination of cohesive and adhesive fracture. The procedure is illustrated by an analysis of combined buckling and crack growth of a delaminated plate having a nontrivial crack contour.

On interface equilibrium and inclusion problems

The condition for interface equilibrium, in which the Eshelby's energymomentum tensor is recognized as a generalization of the chemical potential in Gibbs' classical results, is used to evaluate the stability of the shape of inclusions in an infinite body with Eshelby's results for infinitesimal transformation strains.

Pseudomomentum and material forces in inhomogeneous materials: Application to the fracture of electromagnetic materials in electromagnetoelastic fields

The balance of pseudomomentum (covariant material, canonical momentum) is established in different ways for both pure finite-strain anisotropic elasticity and electromagnetoelasticity in the Galilean approximation for materially inhomogeneous solids. This balance law relates pseudomomentum. Eshelby's energy-momentum tensor, and the material inhomogeneity force. The relationship with the Hamiltonian canonical formulation of finite-strain elasticity is outlined and consequences for the evaluation of energy release rate via path-independent integrals in electro- or magnetoelasticity are drawn. This work, of a fairly general nature, builds on the pioneering results of the Stanford group around G. Herrmann.

Energy-momentum tensors in elastostatics: Some reflections on the general theory

The analysis has to do with a Green-elastic continuum at finite strain, and with the changes in total potential energy due to notional variations of any kind of material heterogeneity. The role of Eshelby's energy-momentum tensors is demonstrated for a much wider class of variations than hitherto, and by a new self-contained approach. Stress jumps at phase boundaries and interfacial dislocations are taken into account. Within this same framework it is shown further how integral identities and conservation laws can be generated straightforwardly for homogeneous media.

A comparison of methods for calculating energy release rates

We compare two methods for calculating the energy released during quasi-static crack advance. One method is based on a surface integral (line integral in two dimensions) expression for the energy release rate, whereas the other method is based on a volume (area) integral representation. A concise derivation of the volume (area) integral expression is given using a virtual-work-type identity for Eshelby's energy momentum tensor. The finite-element implementation of the volume (area) integral formulation corresponds to the virtual crack extension method. Within the context of this formulation, we outline a procedure for calculating pointwise values of the energy release rate along a three-dimensional crack front. For illustrative purposes, numerical examples are presented for a fully plastic, plane strain, edge-cracked panel subject to combined tension and bending.

Three-dimensional J-integral calculations of part-through surface crack problems

In the field of nonlinear fracture mechanics, the J-integral is well known to play a significant role in predicting crack propagation and fracture strength. In this paper, we develop the three dimensional form of the J-integral using Eshelby's energy momentum tensor on any closed surface surrounding a crack front. Twenty-node isoparametric hexahedra elements in ADINA are used to calculate stress and strain distributions and Gaussian surface integration is used to evaluate J-integral values. Using the above method, we calculate the stress intensity factors for part-through surface cracks and verify the accuracy of these results.