Plastic flow in solids with interfaces
A non-equilibrium theory of isothermal and diffusionless evolution of incoherent interfaces within a plastically deforming solid is developed. The irreversible dynamics of the interface are driven by its normal motion, incoherency (slip and misorientation), and an intrinsic plastic flow; and purely by plastic deformation in the bulk away from the interface. Using the continuum theory for defect distribution (in bulk and over the interface) we formulate a general kinematical framework, derive relevant balance laws and jump conditions, and prescribe a thermodynamically consistent constitutive/kinetic structure for interface evolution.
keywords: Continuous distribution of dislocations, Finite strain elasto-plasticity, Incoherent interfaces, Interface evolution.
The motivation for the current work is derived from the processes in material evolution where a moving interface plastically deforms the bulk material region, for example recrystallization and impact induced plasticity [10, 25, 34]. The interface is taken to be a sharp surface separating two distinct regions such as different phases (during phase transition), different crystals (in poly-crystalline materials), and differently oriented single crystals (grain boundary), or a wave front during dynamic deformation. Even after we assume the processes to be isothermal and diffusionless and make simplifying assumptions about the bulk and the interface, the rich dynamics of an interface offers a wide gamut of problems to the material scientist  and the mathematician  alike. The challenge on one hand is to construct physical models which are amenable to experimental verification and numerical implementation, and on the other hand to analyze the resulting partial differential equations for their well-posedness and properties of the solutions.
The structural nature of the interface is characterized on the basis of its behavior upon relaxation of local stresses. We call an interface incoherent if, upon relaxation, it is locally mapped into two disjoint configurations. Otherwise, we call it coherent. An incoherent interface in an otherwise defect-free solid, after stress relaxation, will result into two separate solids [6, 33]. Incoherency is expressed in terms of the incompatibility of the distortion field and leads to interface dislocation density as a smeared-out defect distribution (cf. Bilby and coworkers [2, 3, 5] and Ch. of ). The interfacial dislocation density along with its bulk counterpart contributes to the Burgers vector for arbitrary closed curves (crossing the interface) in the body. If the interface is coherent then its defect distribution, and consequently its contribution to the net Burgers vector, vanishes identically.
We consider plasticity to be a purely dissipative phenomenon driven by irreversible changes in the microstructure. Even in the bulk, away from the interface, the evolution of plastic flow is a complicated non-linear problem coupled with elasticity and non-local microstructural interactions. Many of the underlying mechanisms remain poorly understood  and it is becoming increasingly necessary to develop the theory at a microstructural level. One related concern is to understand the plastic behavior at internal boundaries separating different phases or grains . The plastic flow behavior at such boundaries will depend on interface motion, relative distortion of the neighboring grains, and the local shape (for example orientation and curvature). It is clear that a theory for plastic flow at the interface cannot be, in general, modeled along the same lines as the theory associated with the bulk.
Interfaces in solids, with an associated energy density, have been well studied in the context of continuum thermodynamics. We note, in particular, the earlier work done to obtain equilibrium conditions for coherent/incoherent interfaces within elastically deforming solids [26, 31, 32, 33]. These conditions were obtained by minimizing the total energy (bulk and interfacial) under appropriate variations in the domain (see Remark 2.3 for further discussion). Gurtin and coworkers [6, 18, 20, 22] extended these results to dynamic interfaces and demonstrated the validity of equilibrium interfacial conditions in wider settings than were previously considered. Their methodology relies upon a version of the virtual work principle where contributions from configurational forces were considered in addition to those from classical forces. All these theories, however, assume the bulk surrounding the interface to be defect-free and thus neglect any possibility of interaction between interfacial and bulk defect densities. They therefore fall short of modeling the behavior of interfaces in a plastically deforming medium. On the other hand, some recent strain gradient plasticity models with interface energies dependent on (infinitesimal) plastic strains [1, 14, 15] incorporate interfacial flow rules along the same principles as those in the bulk. These relations furnish boundary data for the bulk equations. While restricting themselves to infinitesimal strains, these models also neglect any coupling with other processes (for example the motion and the relative distortion of the interface).
Our aim is to generalize the above mentioned works by developing a continuum theory for interface evolution in a plastically deforming solid under isothermal and diffusionless conditions. Both the bulk and the interface are assumed to possess a continuous distribution of defects, whose density is related to the local elastic and plastic distortion maps. The role of a relaxed manifold is emphasized in the multiplicative decomposition of deformation gradient in the bulk and at the interface. We restrict our developments to the point of positing specific kinetic laws and therefore stop short of formulating complete boundary-initial-value problems. We however provide a detailed description of the associated kinematics, derive all the necessary balance laws and jump conditions, and use physical and material symmetries to restrict the form of constitutive/kinetic relations. In particular, we derive local dissipation inequalities and highlight the interplay between various dissipative mechanisms and the associated driving forces. The bulk behavior in this paper is modeled after our recent work [16, 17] on bulk plasticity.
The central results in this paper are:
The multiplicative decomposition of the interface deformation gradient is equivalently given in terms of two sets of (interfacial) elastic and plastic distortions, cf. (2.87). Both of these coincide for coherent interfaces.
The relationship between bulk and interface dislocation densities given in (3.44), which also highlights the fact that interface dislocation density, unlike the bulk dislocation density, does not have a vanishing divergence.
The dissipation inequality (4.36) arising due to interface motion, plastic flow at the interface, and change in relative distortion across the interface. This inequality demonstrates the underlying coupling between the interface motion, the tangential plastic distortion of the neighboring grains, and the relative tangential distortion of the grains. It provides a starting point for developing kinetic laws governing the out of equilibrium thermodynamic process. Otherwise, in thermodynamic equilibrium, it furnishes additional balance laws to be satisfied at the interface.
Our work furnishes the pre-requisite information about kinematics, dissipation, and the basic requirements for constitutive equations needed for the formulation of complete boundary-initial-value problems in the study of dynamic incoherent interfaces within plastically deforming solids.
We have divided this work into three parts. In the first, we prepare the necessary background for studying the thermodynamics of energetic interfaces within a bulk medium. The second part is concerned with the interface dislocation density as a measure of defect distribution over the interface and its relation with the bulk dislocation density. The final part deals with the energetics and kinetics of incoherent interfaces moving within plastically deforming solids. We make certain constitutive assumptions about the nature of interfacial energies and use them to evaluate the net dissipation at the interface. Motivated by the dissipation inequality, and exploiting various symmetries of the physical space and the material, we formulate restrictions on kinetic laws at the interface.
2 Preliminaries for the theory
In the following we prepare the ground work for the next two sections. Our discussion on the kinematics and thermodynamics of surfaces, in Subsections 2.2 and 2.3, is largely based upon the work of Gurtin and coworkers [6, 18, 20, 22] and Šilhavý [39, 40, 41]. Our derivation of the interface dissipation inequality, cf. (2.70) or (2.76), is however different and appears to be new. Similar relations were obtained in [6, 18, 22] within the framework of configurational mechanics .
2.1 Three-dimensional continuum
The translation space of a real three-dimensional Euclidean point space is denoted by . Let be the space of linear transformations from to (second order tensors). The groups of invertible tensors, orthogonal tensors, and rotations are denoted by , , and , respectively. The spaces of symmetric, symmetric positive definite, and skew tensors are represented by , , and , respectively. The determinant and the cofactor of are denoted by and , respectively, where if (superscripts and denote the transpose and the inverse, respectively, and ). The space is equipped with the Euclidean inner product and norm defined by () and , respectively, where is the trace operator.
We use both indicial as well as bold notation to represent vector and tensor fields. The components in the indicial notation are written with respect to the three-dimensional Cartesian coordinate system. Indices are denoted with roman alphabets appearing as subscripts. Summation is assumed for repeated indices unless stated otherwise. Let be the three-dimensional permutation symbol; it is if is an even permutation of , if it is an odd permutation, and if any index is repeated.
Let and be the reference configuration and the spatial (or current) configuration with translation spaces and , respectively. There exists a bijective map between and ; therefore for every and time we have a unique given by
We assume to be continuous but piecewise differentiable over and continuously differentiable with respect to .
The derivative of a scalar valued differentiable function of tensors (where is the set of all real numbers) is a tensor defined by
where as . Similar definitions can be made for vector and tensor valued differentiable functions (of scalars, vectors, and tensors). In particular, if the domain of a function is we denote the derivative by ; and if it is then we write for the derivative. Such functions are called fields. The divergence and the curl of fields, on , are defined by (for )
for any fixed . Similar definitions hold for fields on ; in this case we denote divergence and curl by and , respectively. The material time derivative is the derivative of a function with respect to time for fixed ; we denote it by a superimposed dot.
The particle velocity is defined as . If is differentiable at , then the deformation gradient exists at and is given by . We assume and to be piecewise continuously differentiable over ; they (and their derivatives) are allowed to be discontinuous only across the singular surface.
2.2 Singular surface
A singular surface (or interface) is a two dimensional manifold in the interior of (or ) across which various fields (and their derivatives) may be discontinuous, which otherwise are continuous in the body. A singular surface in is given by
where is a continuously differentiable function. The unit normal to the surface and the normal velocity are defined by
respectively; the derivatives being evaluated at the surface. The projection tensor which map vectors in to vectors in , where is the tangent space at such that , is given by
where is the identity tensor in . Note that and .
The jump in a discontinuous field (say ) is defined on the singular surface and is denoted by
where and are the limit values of as one approaches the singular surface from either side. The side is the one into which the normal to the surface points. Let be another piecewise continuous field. The following relation can be verified by direct substitution using (2.8):
Derivatives on the surface
We first introduce the general idea of derivatives on manifolds embedded in a higher dimensional space (see for example [39, 41]). Let be a manifold in the space of tensors. The derivative of a scalar valued differentiable function of tensors is a tensor defined by (for )
where as , and is the orthogonal projection onto the tangent space of at . Similar definitions can be made for vector and tensor valued functions on manifolds.
Let , , and denote a scalar, vector, and tensor valued field, respectively, on . They are differentiable at if they have extensions , , and to a neighborhood of in which are differentiable at . The surface gradients of , , and at are defined by
In the rest of the paper we will use the same symbol for both the surface field and its extension. We define the surface divergence of as a scalar field ; and of as a vector field given by
for a fixed . Moreover, we call (or ) tangential if () and superficial if .
Define the curvature tensor by
It is straightforward to verify that (use (2.6)) and . Therefore, is an eigenvector of with zero eigenvalue. Since is symmetric, the spectral theorem implies that it has three real eigenvalues with mutually orthogonal eigenvectors. Let the two nontrivial eigenvalues be and with eigenvectors in . The mean and the Gaussian curvature associated with the surface are defined as
A function , is said to be a normal curve through at time if for each , and
It represents the rate of change in with respect to an observer sitting on and moving with the normal velocity of the interface. As an example, on differentiating (2.6) and using the definitions for surface divergence and normal time derivative, we obtain
Therefore, evolving surfaces are parallel if and only if is constant over at any fixed time.
where is arbitrary. For these relations can be combined to eliminate ,
Singular surface in the current configuration
The image of the singular surface in the current configuration is given by
The scalar function is continuous but, in general, only piecewise differentiable with respect to its arguments. The derivatives of can suffer jump discontinuities at . Differentiate (away from ) with respect to (at fixed ) and (at fixed ), and then restrict the result to the surface, to obtain respectively,
where indicates that either of or limit of the field can be used to satisfy the equation (due to smoothness of across the singular surface), and indicates the partial derivative of with respect to at fixed . Substitute (2.6) into (2.25) to get
The compatibility relations (2.21) and (2.22) yield the and value of the expressions on the right hand sides above identical, cf. (2.33) below. This leads us to define the normal to the surface and the spatial normal velocity by, cf. (2.6),
respectively; we obtain
The projection tensor which map vectors in to vectors in , where is the tangent space at such that , is given by
Surface deformation gradient and normal velocity
It is then easy to check that
Therefore, and . The cofactor of is defined by for arbitrary vectors . Let be two unit vectors such that forms a positively oriented orthogonal basis at . Then
where in the third equality we have used . On the other hand, employ (2.32) to conclude that () and hence . Therefore, remains non-zero because does not vanish. According to (2.33), is equal to the ratio of the infinitesimal areas (on the singular surface) in the current and the reference configuration. Use to write
Following Penrose  we define a unique tensor , the pseudoinverse (or the generalized inverse) of , such that
which also satisfies
as can be checked by direct substitution.
For there exist unique tensors and such that . For a non-invertible tensor there exists a unique positive semidefinite tensor and a (non-unique) orthogonal tensor such that . These statements follow from the polar decomposition theorem for invertible and non-invertible tensors. Recall (2.31) to write . Tensor thus satisfies . Define , where is the pseudoinverse of such that . Tensor is unique and satisfies
Moreover, tensor in the polar decomposition for is related to as . The expression
provides a decomposition for into unique tensors.
The surface gradient of normal velocity can be calculated from (2.31)
to rewrite (2.39) as
Consequently it is only for a flat interface () that we have (compare with ).
Let be a scalar function on the interface given by . The arguments of satisfy and and therefore form a submanifold, say , of . The partial derivatives and (with respect to and , respectively) are evaluated using an extension of and restricting the result to . Extension of is any smooth function defined over such that it is equal to on . These partial derivatives lie in the tangent space of and hence satisfy (cf. (2.11); for a proof see Appendix B of )
In the rest of the paper we will use same notation for the function and its extension.
where and are functions of only ; i.e., they are arbitrary orthonormal vectors orthogonal to . To find partial derivative fix in (2.43) and differentiate it on a one-parameter curve in the space of all unit vectors satisfying . Apply the definition of cofactor and use (2.32) to get . Therefore, by (2.42), . On the other hand, differentiating for fix yields
Hence the normal time derivative of is given by (compare with )
2.3 Balance laws and dissipation inequality
Assuming a purely mechanical environment and isothermal heat flow we obtain balance laws for mass and momentum, and the dissipation inequalities both for material points on the interface and away from it. We do not state the balance of energy since it is used, under isothermal conditions, only to calculate the net heat flux during the dissipative process.
Surface divergence theorem and surface transport theorem
In addition to divergence and transport theorems for piecewise smooth fields on (see for example Ch. of ) we will repeatedly use the following theorems for fields defined on . For a vector field continuously differentiable on
where is the outer unit normal to the closed curve bounding such that forms a positively oriented orthonormal basis on with as the tangent vector along . Moreover if is tangential, i.e. , then and (2.46) reduces to
The surface transport theorem for an evolving surface within a fixed region such that is given by 
where and is the outward unit normal on . If is arbitrary then we can always choose with such that at all i.e., orient in such a way that it is orthogonal to at all points on (cf. Figure 2). With this choice (2.48) reduces to
Similar theorems hold for scalar and tensor fields on .
Conservation of mass
Assume no net mass transfer in an arbitrary volume of . Also assume that there is no additional mass density associated with . The statement of conservation of mass then reduces to 
where is the referential mass density of the bulk, and
i.e., either the normal velocity vanishes or the referential mass density is continuous across .
Balance of momentum
The balance laws for linear and angular momentum can be either stated as Euler’s postulates or can be deduced from the first law of thermodynamics . Let be a three-dimensional open subset of with boundary such that is nonempty and . Let and be unit vectors normal to and , respectively. Let be the bulk Piola stress and the specific body force vector. We assume the existence of a contact force between two subsets of along the curve of contact, which can be expressed in terms of a linear map (given by interface Piola stress ) acting on the normal to the contact curve . If there are no body forces associated with the singular surface then the balance of linear momentum for is given by (see Figure 1 where all the forces are shown)
Let be superficial, i.e. . This is motivated from the last term of the above equation where does not contribute to the net force (since is orthogonal to ) and therefore can be assumed to vanish without loss of generality. Integral equation (2.52) can be localized, using the transport and divergence theorems, to [18, 22]
The balance of angular momentum is given by
The interface Cauchy stress is a superficial tensor () which satisfies
for (with normal ) and (with normal ) as curves on the referential and spatial singular surface, respectively. Let be a positively oriented orthonormal basis on . Define by . The triad then forms a positively oriented orthonormal basis on , where is given by (2.28). Hence , which on repeated use of the definition of cofactor simplifies to
Stresses and are therefore related as (compare with , where is the bulk Cauchy Stress)
Let and be the free energy densities per unit volume of and per unit area of , respectively. Assume that has zero body force and kinetic energy density. For an arbitrary volume , with nonempty and , the mechanical version of second law of thermodynamics (under isothermal conditions) yields
A comment is in order for the term representing the power due to interfacial stress. At every point on the curve the contact force (between the surfaces divided by the curve) is given by and the rate of change in displacement, with respect to an observer sitting on (at the considered point) and moving with velocity , is given by . The change in displacement apparent to the observer sitting on but moving tangentially to the interface will depend on the chosen parametrization and so will the resulting power. This is undesirable and therefore we use only to calculate the power expended at the interface. Gurtin and coauthors [6, 20, 22] have imposed invariance with respect to tangential velocities in their formulation of configurational balance laws. This is equivalent to the requirement of invariance under re-parameterizations of the interface. Our viewpoint is different: We require (a priori) the mechanical power balance to be invariant under re-parametrization and write it in a form that satisfies this invariance automatically. Thus this requirement is automatically satisfied in the present formulation and accordingly yields no non-trivial information.
Before we proceed let us clarify the nature of interfacial stresses. The interface stress , in contrast to the bulk stress, does not act on a fixed set of material points but rather on material points momentarily occupying the surface . This is in accord with the mechanism responsible for surface tension in liquids. As the surface area increases, interstices are generated which are filled by molecules from the bulk liquid. In this way the surface tension remains sensibly constant while the surface area expands. Thus the matter occupying the surface does not actually stretch. Instead, the surface changes its area due to the continuous addition of mass. This physical situation stands in contrast to the treatment of surface tension in conventional continuum mechanics, in which the surface is regarded as a material surface if the motion of the liquid, regarded as a closed set, is continuous. In the conventional interpretation, surface tension is then a conventional force system acting on a persistent set of material points. However, in the actual physical situation, the surface is not material in the usual sense. Our framework accommodates such mechanisms while retaining the conventional interpretation of force. The contribution to mechanical power from interface stresses (as in (2.63) above) is consequently obtained not by its action on material velocities but on .
Using the transport, divergence, and localization theorems, (2.63) reduces to
Here (2.66) and (2.67) can be verified with the help of (2.9), (2.22), and (2.31) while (2.68) follows from the chain rule of differentiation. These identities, in addition to (2.51) and (2.54), reduce (2.65) to
or equivalently (on substituting from (2.41) and )
are bulk and interface Eshelby tensors defined over and , respectively. Dissipation inequalities (2.64) and (2.70), in addition to balance laws for mass and momentum, should be satisfied for every process.
The present setting differs from that of Gurtin [6, 22, 20] as we do not consider any explicit contribution from configurational forces in the global dissipation inequality (2.63) (compare with Equations (21-6) and (21-19) in ); the final results however coincide. We demonstrate this by assuming, for now, the interface energy density to be of the form . Such energies have been well studied in the contexts of phase equilibrium with interfacial energy [22, 33, 41]. The surface stress is given by . We can then obtain , where is tangential. Substituting from (2.20) and using the chain rule of differentiation yields
Substituting it in (2.70) we get
where is the exterior unit normal to . Since is arbitrary and is tangential, we can choose such that (i.e., orient such that is parallel to the tangent at every point on , cf. Figure 2). Upon localization of the resulting integral inequality we are finally led to
as a necessary condition for (2.63). The coefficient of is the net driving force for the motion of a coherent interface between two bulks phases. This coincides with the result obtained by Gurtin, cf. Equations (21-10a) and (21-26) in . The configurational shear appearing in those equations from  is equal to (see of  in this regard). At thermodynamic equilibrium the driving force vanishes thereby furnishing a balance relation to be satisfied at the interface. Such relations were also obtained via energy minimization [26, 31, 32, 33]. Ours is a dynamical theory, whereas results coming from energy minimization are really only relevant at equilibrium, and even then only for stable equilibria.
2.4 Elastic plastic deformation
The idea of stress-free local configurations is central to our theory. We assume both the bulk and the interface stress to be purely elastic in origin, wherein the deformation is measured with respect to the stress-free configuration. In a recent paper  we demonstrated, using the mean-stress theorem, that it is always possible to obtain a locally stress-free state (under equilibrium and in the absence of external forces) by cutting into parts with arbitrarily small volume. Moreover, if these sub-bodies cannot be made congruent in absence of any distortion then they do not form a connected set in a Euclidean space. The material is then said to be dislocated with no global differentiable map from to the disjoint set of sub-bodies [4, 28, 29, 30, 36]. The union of these unstressed sub-bodies is a three-dimensional non-Euclidean smooth manifold, say . A local configuration in is identified with the local tangent space, denoted by . The local map from to is represented by . The absence of a global differentiable map renders incompatible and therefore, unlike , it cannot be written as gradient of a differentiable map. The incompatibility of implies the existence of a continuous distribution of dislocations over (see the next section for details).
The argument used for the existence of stress-free local configurations in  assumes smoothness of bulk stress. If the stress field is non-smooth only over a set of measure zero, the stresses can still be relaxed on neighborhoods arbitrarily close to the singular region and therefore everywhere except over the set of zero measure. If singular regions have stresses associated with them, for example the surface stresses discussed above, then they also need to be relaxed. In the following we show that this can be done under equilibrium and vanishing external forces if the surface is cut into infinitesimal areas.
To this end consider an arbitrary subsurface and assume to be continuously differentiable over . A simple calculation (using (2.62) without the inertial term) then yields
where is the mean interface Cauchy stress and is the area of . The mean stress therefore vanishes if there are no external forces on . According to the mean value theorem, there exists such that (). Let the area become arbitrarily close to zero. Then, by continuity of , the surface stress can be brought arbitrarily close to zero.
While cutting , care is needed with surfaces where the bulk stress is singular. The neighborhood of a point on such surfaces is to be cut such that the length dimension parallel to the normal (of the surface) is arbitrarily small compare to other length dimensions. This way we will be left essentially with areas to be relaxed from stress, if any. The resulting stress-free configurations at the singular interface are of dimension one less than those obtained from the bulk. Their union forms a two dimensional smooth manifold . A local configuration in is identified with the local tangent space of . If the tangent space is mapped (locally) into two disjoint local configurations in , for reasons that will become clear below, then we call the singular interface incoherent (at ). We denote the two local configurations by and (in rest of the paper, a superscript will represent an association with configuration and with ; they are not to be confused as indices). Otherwise, if the mapping is injective then we call the singular interface coherent and denote the local configuration by . Incoherency of the interface implies a continuous distribution of dislocations over the interface; we postpone the discussion on this aspect till the next Section. The process of relaxation is illustrated through a cartoon in Figure 3.
Let be the local map from tangent space to at . Both and are assumed to be continuously differentiable except on the singular surface. The following decomposition
is admitted (conventional plasticity theories usually represent tensors and by and , respectively). Since we demand unloading to be elastic in nature, we call the elastic distortion. We identify with plastic distortion, for reasons that will become apparent when we discuss dissipation in Subsection 4.1. Define distortion maps on the surface
where superscripts and denote the association with the two local configurations in at a fixed material point. The projection tensors and are given by