Eshelby's Energy Momentum Tensor Research Articles

Irreversible thermodynamics of surfaces and interfaces: Special reference to the strained thin solid films on the substrates: Theory and practice

The realization of nanoscale devices largely depends on our ability to control and manipulate interfacial interactions and, thus, understanding of the mechanisms of surface/interface instabilities. In this work, theoretically as well as technologically important and distinct two thermodynamic systems, which are exposed to (isobaric) and isolated from (isochoric) external body forces and surface tractions, are formulated by using irreversible thermodynamics in combination with the generalized variational method. The starting point for the present formulation closely follows up the Fowler and Guggenheim [Statistical Thermodynamics (University Press, Cambridge, 1952)] interpretation of the Planck inequality [Über Prinzip Vermehrung Entropie: Ann. Phys. Series 2(32), 462 (1887)] for isothermal reversible and irreversible (natural) infinitesimal changes in heterogeneous systems (multi-phase and multi-component). By combining this fundamental principle with the interlink between the dissipation function and global internal entropy production postulates, two distinct sets of governing equations for the surface drift-diffusion flux as well as the rate of evaporation/condensation and/or the growth/recrystallization of amorphous solid thin films are obtained for isochoric and isobaric systems. The role of Eshelby's energy-momentum tensor in the generalized potential for the interface displacement is found to differ (opposite in sign) for isochoric and isobaric systems. To demonstrate the importance of these sign conflicts, two sets of computer experiments are performed on isochoric and isobaric systems. They showed us that the elastic strain energy density contribution to the generalized driving force for surface drift-diffusion alone favoring flat and smooth surfaces in isobaric systems regardless of the sign of the uniaxial stress (healing), rather than causing the surface roughness and even catastrophic crack initiation as the case in internally strained isochoric systems. Computer simulations allowed us to track down the dynamical behavior of test modules by furnishing surface and strain energy variations, combined with the Global Helmholtz free change, which indicates the existence of two regimes: initial smooth surface undulations followed up by the rather chaotic crack formation and propagation stage at the middle of the thin film supported by the stiff substrate. In this study, we mainly focused on the development kinetics of “Stranski–Krastanow” island-type morphology, initiated by the nucleation route rather than the surface roughening scheme. The physicomathematical model, which is based on the irreversible thermodynamics treatment of surfaces and interfaces with singularities [T. O. Ogurtani, J. Chem. Phys. 124, 144706 (2006)], furnishes us to have autocontrol on otherwise free-motion of the triple junction contour line between the substrate and the droplet without presuming any equilibrium dihedral contact (wetting) angles at edges. We have also demonstrated the formation of the Stranski–Krastanow (SK)-type doublet islanding (quantum dots) as a stationary nonequilibrium state in an epitaxially strained thin flat droplet on a rigid substrate by introducing the wetting potential—invoked by the quantum confinement—into the scenario and carefully selecting the system parameters (size and shape) for the isochoric system represented by [Ge/Si (100)]. It has been also shown that on the contrary to common perceptions, the Stranski–Krastanow islands are in genuine stationary nonequilibrium states in the sense of Prigogine if one invokes proper free-moving boundary conditions at triple junctions deduced from the irreversible thermodynamics rather than ad hoc periodic or reflecting constrains at the edges.

On Spatial Covariance, Second Law of Thermodynamics and Configurational Forces in Continua

This paper studies the transformation properties of the spatial balance of energy equation for a dissipative material, under the superposition of arbitrary spatial diffeomorphisms. The study reveals that for a dissipative material the transformed energy balance equation has some non-standard terms in it. These terms are related to a system of microforces with its own balance equation. These microforces act during the superposition of the spatial diffeomorphism, because of the dissipative properties of the material. Moreover, it is shown that for the case in question the stress tensor is additively decomposed into a conventional part given by the standard Doyle-Ericksen formula and a non-conventional one which is related to changes in the material internal structure in the course of deformation. On the basis of the second law of thermodynamics and the integrability condition of a Pfaffian form it is shown that the non-conventional part of the stress tensor can be related not only to dissipative but also to conservative response. A further insight to this conservative response is provided by exploiting the invariance properties of the balance of energy equation within the context of the material intrinsic “physical” metric concept. In this case, it is shown that the assumption of spatial covariance yields the standard conservation and balance laws of classical mechanics but it does not yield the standard Doyle-Ericksen formula. In fact, the Doyle-Ericksen formula has an additional term in it, which is related directly to the evolution of the material internal structure, as it is determined by the (time) evolution of the material metric in the spatial configuration. A formal connection between this term and the Eshelby energy-momentum tensor is derived as well.

A finite-strain elastic–plastic Cosserat theory for polycrystals with grain rotations

We investigate geometrically exact generalized continua of Cosserat micropolar type. A variational form of these models is recalled and extended to finite-strain elasto-plasticity based on the multiplicative decomposition of the deformation gradient. The stress driving the plastic evolution is the Eshelby energy momentum tensor. No plastic Cosserat rotation is introduced and the plastic spin is set to zero. It is argued that the traditional Cosserat couple modulus μ c should be set to zero for polycrystal specimens liable to fracture in shear, still leading to a complete Cosserat theory with independent rotations in the geometrically exact case in contrast to the infinitesimal, linearized model. A geometrical linearization of the presented finite-strain plasticity model is already shown to be well posed.

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.

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.

The thermodynamic driving force for rafting in superalloys

Eshelby's energy-momentum tensor is used to provide an analytical expression for the driving force for rafting in the elastic regime in a superalloy with a high volume fraction of γ′. The structure is modelled as a simple cubic array of γ′ cubes separated by thin sheets of γ. During rafting, the γ′ particles are constrained to remain tetragonal prisms. For tension along a cube axis, the driving force is proportional to the product of the tension σ, the fractional difference δ of lattice parameters of γ′ and γ and the fractional differencem of their elastic constantsc11 – c12. As in the calculation of Pineau for an isolated spheroid, needles are formed when this productσδm is positive. Two- and three-dimensional systems behave similarly. The initial plastic strain in γ is anelastic and in principle reversible. When the plastic strain exceedsmδ, platelets perpendicular to the stress axis are formed if the product σδ is negative.

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.

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.