Wu, C.H. (2003) A crystal structure-based eigentransformation and its work-conjugate material stress. EUROMECH Colloquium 445- Mechanics of Material Forces [ pdf ]

In the abstract of his 1970 paper, Eshelby stated: "The force on a dislocation or point defect, as understood in solid-state physics, and the crack extension force of fracture mechanics are examples of quantities which measure the rate at which the total energy of a physical system varies as some kind of departure from uniformity within it changes its configuration." He then went on to demonstrate that the elastic energy-momentum tensor proves to be a useful tool in calculating such forces. The 'forces' turn out to be the appropriate traction vectors associated with the energy-momentum (stress) tensor. It is therefore natural and perhaps even fundamental to look for the 'strain tensor' that can be paired with the 'stress tensor' to form work. The 'strain rate' would then be that some kind of departure from uniformity within a physical system. In this paper, we examine the configurational changes brought about by atomic diffusion in a nonuniform alloy crystal. The transformation from a reference, single-parameter simple cubic cell to a six-parameter alloy crystal cell, called the eigentransformation, is identified as the needed kinematic tensor.

The pairing of a chemical potential and its associated concentration rate in the thermodynamic identity is well known. The existence of an experimentally determinable molar volume as a function of molar concentrations is also widely used in chemical engineering. What is perhaps less known and infrequently used is the fact that a spatially nonuniform molar volume leads to a field of geometrically incompatible eigenstrain, or eigentransformation in finite deformation. This incompatibility forces the material environment to deform, and the result is a strain energy trapped inside the material body. The change of this energy with respect to the eigentransformation is a generalized configurational stress, which, in the limit as the eigentransformation tends to the identity transformation, tends to the classical energy momentum tensor of Eshelby, or the so-called configurational stress. It is shown that the generalized configurational stress is an integral part of the chemical potentials that are responsible for atomic diffusion. This cycle of cause and effect is demonstrated in an axially symmetric setting where the material configuration is taken to be cylindrically orthotropic.
Symmetry can be used in many occasions to bare the simple meaning of a complex mathematical expression hiding under the disguise of tensors and index notation. We used it to elucidate the "missing" term in surface chemical potential in 1996 and are now applying it to identify the chemical potential in the bulk. Both Professor Thomas C. T. Ting and I are civil engineering graduates of Tai-Da, the world-renowned National Taiwan University, but did not know each other until we joined UIC. It has been a wonderful friendship of many stimulating anisotropic discussions and numerous delicious potluck dinners. Happy birthday, Tom!

Keywords: Energy momentum tensor, configurational stress, chemical potential, and eigentransformation.

Electromigration (EM) in a metal line is the phenomenon of flow of the metal atoms along the line. The flow is driven by the current of electrons and is also affected by the variation of chemical potential along the line. The replacement of atoms leads to a change in eigenstrain, which, in turn, alters the chemical potential. Since the chemical potential is very often dominated by a stress term, this back flow is very often attributed to the presence of a stress gradient. In microelectronic applications metal lines are usually subjected to the actions of large thermally induced eigenstrain even before the current is turned on. This large eigenstrain, together with the EM-induced eigenstrain, may lead to a nonlinear back flow, which is the subject of investigation of this paper.

Keywords: Electromigration, chemical potential, and eigentransformation.

The molar volume of a mixture is used to define an eigentransformation as a function of the molar concentrations. It is shown that an appropriately defined energy momuntum tensor is work-conjugate to the rate of the eigentransformation.

The misfit deformation in a film-substrate system is mostly concentrated in the film when the stiffness of the system is dominated by that of the substrate. Such is generally the case in microelectronics applications. For many semiconductor materials with electronic properties suitable for device applications, the associated mismatch parameter routinely falls in the range say, from -5% to +5%. Elastic strains of such magnitude provide a source of free energy for configurational modifications. To examine such a possibility, the perturbation in elastic deformation is first obtained in terms of a perturbation in a configurational variable/parameter. This perturbation in elastic deformation and the misfit deformation are traditionally treated as two separate infinitesimal deformations. As a result, the resulting stability condition is either a function of the mismatch strain energy density, for the case of a morphological perturbation, or completely unrelated to the underlying mismatch, for the case of a compositional perturbation. In either situation, the sign of the mismatch is totally immaterial. In this paper, the perturbation in elastic deformation is taken as a small deformation superimposed on the large mismatch deformation in a nonlinear setting. The role of the mismatch is thus more fully explored and uncovered.

ABSTRACT

The chemical potential used in interdiffusion analysis was derived by Li, Oriani and Darken (1966), and Larche and Cahn (1982). It contains the trace of the stress tensor as the essential elasticity contribution to the configurational force conjugate to the material composition. As a result, the underlying diffusion equation is totally independent of any accompanying elastic field. In particular, when an alloy epifilm is annealed, the theory implies that the rather large lattice mismatch has no effect on the ensuing diffusion process. However, it is perhaps intuitively clear by now - almost fifty years since Eshelby published his first paper on energy momentum tensor in 1951 - that the trace of the (canonical) Eshelby stress tensor should be the total elasticity contribution to the desired configurational force. This conjecture is formally established in this paper for an n-component substitutional solid. Since the elastic energy is now a part of the chemical potential, the interplay between a composition-generated deformation and another elastic field may become important via the interaction energy. As an example, the effect of this interaction is calculated for the spinodal decomposition of a binary alloy solid/epifilm. The modification of the critical temperature is such that it is now a function of mismatch.

Solid surface stresses are known to behave like the prestress in a prestressed membrane that is perfectly fitted on the bounding surface of a bulk material. The inclusion of such a surface stress in an otherwise traction-free crack surface leads to additional loads for the bulk material: a pair of point forces, one at each crack-tip, a uniformly distributed compressive load on the convex side of the crack, and a uniformly distributed tensile load on the concave side. As a result, the values of the stress intensity factors are altered, and the crack-tip stress fields become 1/r singular, in addition to being one over square root r singular. The severity of the added singularity does not carry any particular physical significance in that the configurational equilibrium is always an energy condition and never a stress criterion. Indeed, the new configurational equilibrium condition is now also dependent on the surface stress as well as the curvature of the crack. The dependence on curvature becomes more and more pronounced as the radius of curvature becomes smaller and smaller.

ABSTRACT

The placement of a point load at a crack tip does not contribute to the magnitude of the stress intensity factors (SIF) at that tip. It nevertheless can couple with other SIF-producing loads and lead to a change in the potential energy of the system. As a result, the configurational equilibrium condition or the fracture criterion of linear elastic fracture mechanics is altered. The inclusion of surface stress leads to yet another point load with similar effect. In the context of linear elasticity, however, the change produced by surface stress is of second order, while that affected by a point force is of first order in that the surface tension is actually reduced by an amount equal to the applied force. This is another one of those situations where the applied mechanical force actually combines with the so-called fictitious force in bringing about configurational changes. The splitting of a double-cantilever-beam by a wedging force, which is not the same as a crack-tip point load, is used to demonstrate that the wedging force at fracture is of the order of magnitude of the surface tension.

The title problem is studied for a linear surface energy density with two surface stress coefficients. In terms of a Youngs modulus E and an applied traction T, the perturbation of a smooth configuration affected by a surface stress sigma is proportional to (sigma)T/E, which is small but depends on the sign of T. It is shown that, through the existence of an interaction energy between the applied traction and the surface stress, a Griffith crack can actually be strengthened (weakened) by a crack-parallel tension (compression) via surface stress, an intuitively reasonable conclusion that does not follow from linear elasticity. In general, surface stress can effectively reduce an applied stress-intensity factor to a lower effective stress-intensity factor.

When a crack is curvilinearly propagating in an inhomogeneous stress field, the incremental crack-extension trajectory in the neighborhood of the crack tip x = 0 may be defined by  y=k x x / 2, where the y-axis is perpendicular to the crack path at x = 0. The total energy of the system is obtained in this paper as a quadratic function of k and x.

The chemical potential associated with surfaces of stressed solids is found to consist of four terms, instead of the widely accepted two. The boundary conditions are also affected by the presence of a surface deformation-dependent surface stress. The stability of the surfaces of stressed solids is reinvestigated, using the revised equations. It is found that the stability condition is sensitive to the sign of the applied stress.

This paper concerns itself with the determination of the nonlinear stress-strain relation of a solid-filled hydroxyl-terminated polybutadiene (HTPB) composite, where solid particles are homogeneously dispersed in the rubber matrix. A stress-stretch master curve was obtained for the glass bead-filled Mooney-Rivlin rubber.

Stress-driven morphological instability in solids is governed by a surface-diffusion mechanism that is dependent upon the chemical potential that is merely the sum of the surface energy and the surface value of the bulk strain energy. This version of the chemical potential is shown to be incomplete and the exact expression is rigorously derived in the context of finite deformation. The role of the surface stress in the new and complete formulation for the surface-roughening phenomenon is demonstrated.

This paper is dedicated to Professor C. C. Yu of National Taiwan University on the occasion of his eightieth birthday. Professor Yu's teaching always stressed the importance of physical intuition and he was particularly fond of the expression "the simpler the better" in his instruction. Our interest in chemical potential began with the now well-known stress-driven morphological instability in solids. The phenomenon is governed by a surface-diffusion mechanism that is dependent on the also well-accepted chemical potential that is merely the sum of the surface energy and the surface value of the bulk strain energy. The same potential also forms the basis of analysis for many other phenomena involving mass transfer along strained surfaces. The results of such theoretical investigations reveal a common, nonphysical conclusion that the associated physical phenomena are insensitive to the sign of the surface straining. Professor Yu's teaching has led us to the discovery of the incompleteness of the two-term chemical potential that has been accepted as the exact expression since 1972. This discovery has been received with a certain amount of skepticism. Our objective in this paper is to use a simpler situation to reveal in a more transparent, if not indeed better way the full expression of the potential. The axial deformation of an annulus is undoubtedly the simplest candidate one can find. Thank you dear Professor Yu, and happy birthday!

The one-dimensional continuum modeling of stress-induced phase transformations in solids presumes the existence of a double-well Helmholtz free energy. The co-existence of two uniformly strained phases is characterized by a step jump in axial strain and there is no characteristic thickness associated with the interface. Following the original concept of Cahn and Hilliard in their dealings with nonuniform systems, a "gradient energy" is introduced to account for the transition. The phenomenological length parameter associated with the gradient energy, as well as the energy barrier associated with the double-well, is then expressed in terms of the interfacial energy and a suitably defined thickness for the interface. These latter quantities can be experimentally measured. An estimate for the interfacial energy is also obtained from the self-equilibrating solution of two butt-joined semi-infinite strips.

A regular perturbation procedure is applied to obtain the stress intensity factors at the tips of regularly perturbed cracks. It is shown that the second term of a two-term expansion is not always of the order of the crack slenderness. The notch-tip singularity associated with a singularly perturbed crack is obtained by the method of matched asymptotic expansions.

ABSTRACT

The presence of a smooth depression on the surface of a stretched half-plane magnifies the surface stress at the trough of the depression. Similarly, the stress at the foot of a smooth protrusion is also magnified. The associated stress concentration factors are determined, via a regular perturbation procedure, in terms of a small aspect ratio that measures the slight unevenness of the surface. When the depressions protrusion profile has corners, the method of matched asymptotic expansions is used to obtain the desired analytic solution. The case of a shallow notch, as well as that of a triangular protrusion, is solved explicitly.

Cohesive elasticity is the grade-3 theory of elasticity developed by Mindlin in 1965. It has a modulus of cohesion that gives rise to surface tension. The concept of adhesion is introduced, and interfacial energies and energy of adhesion are defined. The interfacial-energy solution may also be used to define a grain boundary energy. Also presented are the thin film energy and the concept of an interface-phase. The stretching of a thin film is analyzed in detail; and it is found that the apparent Young's modulus obtained from a film is higher than that obtained from a plate.