On the theory of dislocation and generalized disclination fields and its application to straight and stepped symmetrical tilt boundaries

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Abstract

The theory of dislocation and generalized dislocation fields is developed within a second-order mechanical framework where the description of the internal state of the body and the balance equations involve the stress and hyper-stress tensors, work-conjugates to the strain and second-order distortion tensors. Consistently, the free energy density depends on the elastic strain and second-order distortion tensors. To obtain a continuous setting, the theory uses the duality between the discontinuity of the elastic displacement vector and distortion tensor fields and the incompatibility of the smooth second-order elastic distortion field. The conservation of these discontinuities across arbitrary patches provides transport relationships for the motion of dislocations and generalized disclinations serving as a kinematic basis for the description of plasticity and phase transformation. Closure of the theory derives from constitutive relationships for the mobility of dislocations and generalized disclinations compatible with the thermodynamic requirement of positive dissipation. In contrast with dislocations, the driving forces for generalized disclinations involve the hyper-stress tensor, not the stress tensor. The resulting theory is able to address boundary value problems for the elasto-plasticity of solids coupled with phase transformation along arbitrary loading paths. We provide examples showing generalized disclination distributions in plane state situations such as straight symmetrical tilt boundaries and symmetrical tilt boundaries involving steps and ledges.

Introduction

Engineering situations where the elastic displacement field presents discontinuities across bounded surfaces are commonplace in deforming solids. They include grain boundaries and triple lines, inclusions in a matrix of dissimilar material, shear bands, incoherent interfaces, twinning and phase transformation interfaces (interphases), etc. Crystal defect models such as dislocations and disclinations were simultaneously defined by Volterra (1907) early in the last century to account for bounded displacement discontinuities in solids. In the sole presence of a dislocation, the elastic displacement vector field encounters a constant discontinuity across some surface S bounded by the dislocation line. A translation takes place across S, while continuity of the elastic strain and rotation is maintained. The discontinuity of the elastic displacement induces an incompatibility of the elastic strain tensor, reflected in a non-vanishing tensor field of crystal defect densities α referred to as Nye’s tensor of dislocation densities in the elastic theory of continuously distributed dislocations (ECDD) (Bilby, Bullough, Smith, 1955, Kondo, 1953, Kröner, 1981, Nye, 1953). In ECDD, the elastic curvature tensor is defined as the gradient of the elastic rotation vector, and it is therefore compatible. Nevertheless, it contributes to the incompatibility reflected in Nye’s tensor α. When the elastic rotation vector/tensor fields additionally present a constant discontinuity across S, the elastic curvature tensor field also turns out to be incompatible, and the discontinuity of the displacement field has rigid-body character (Volterra, 1907, Weingarten, 1901). Such a situation is typically met in polycrystals, across grain and subgrain boundaries. The incompatibility of curvature is reflected in the disclination density tensor field θ (deWit, 1970), which complements the dislocation density tensor α in the description of lattice incompatibility. The (α, θ) defect densities and displacement fields are then described by the elastic field theory of dislocations and disclinations of deWit (1970), which reduces to ECDD when θ=0.

Beyond Volterra’s construct and in addition to discontinuities of the elastic displacement and rotation tensor (the skew-symmetric part of the elastic distortion tensor), the elastic strain tensor, i.e. the symmetric part of the elastic distortion tensor, may also encounter discontinuities across bounded surfaces in the body. Then, the discontinuity of the elastic displacement field is not that of a rigid body as in Weingarten’s theorem, but of a deforming body. As shown in Acharya and Fressengeas (2012), the discontinuity of the (first-order) elastic distortion translates into the incompatibility of the second-order elastic distortion (i.e. the second gradient of the elastic displacement in compatible elasticity), and into the existence of a non-vanishing third-order tensor field of crystal defect densities π, referred to in this paper as the generalized disclination (“g-disclination”) density tensor field. This tensor degenerates into the standard disclination density tensor θ when the elastic strain tensor retains continuity. It vanishes when the discontinuity of the elastic displacement is a mere translation.

In this paper, we present a linearized theory of the anelastic dynamic behavior of incompatible media containing dislocations and g-disclinations subjected to arbitrary loading paths. To describe the internal mechanical state of the body, the theory involves the stress and second-order stress (“double-stress” or “hyper-stress”) tensors as work-conjugates to the strain and second-order distortion tensors. Similar theories were presented in Acharya, Fressengeas, 2012, Acharya, Fressengeas, 2015, but the choice was made therein to use the couple-stress theory of Mindlin and Tiersten (1962) for this description, leading to some restrictions in the applied loading. In this presentation, the mechanical equilibrium equations involve both the symmetric and skew-symmetric parts of both the stress and hyper-stress tensors, in a setting proposed by Mindlin and Eshel (1968). The free energy density is assumed to be a second degree polynomial of the elastic strain and second-order distortion (i.e. strain-gradient and curvature) tensor fields. Thus, the elastic constitutive laws are linear, but non-local in the presence of generalized disclinations. As is customary in the thermodynamics of continuous media (Coleman and Gurtin, 1967), the elastic constitutive laws will be identified by demanding vanishing dissipation in all reversible processes, and the mobility laws restricted by further requiring that defects mobility warrant non-negative dissipation. As limiting cases, the theory should contain the earlier mechanical field theory of dislocations (Acharya, 2001) when the elastic second-order distortion is compatible, and the theory of dislocations/disclinations (Fressengeas et al., 2011) when only the symmetric part of the latter (i.e. the strain-gradient) is compatible. In the absence of both dislocations and disclinations, it should also reduce to the gradient theory of compatible elasticity of Mindlin and Eshel (1968).

This paper is primarily concerned with continuous modeling at nanoscale of crystal defects and crystal defect ensembles, particularly grain and phase boundaries. Models involving statiscally stored dislocations are unfit for such small-scale analyses, although the microstructure and dynamics of these crystal defect ensembles have far-reaching effects on the macroscopic properties of polycrystalline materials. Earlier continuum mechanics models have dealt with coherent or incoherent grain boundaries, using interfacial elements with phenomenological prescriptions deriving from atomistic or dislocation dynamics calculations (Warner, Sansoz, Molinari, 2006, Zbib, Overman, Akasheh, Bahr, 2011), or surface-dislocation distributions, see e.g. Berbenni et al. (2013); Vattré, 2017, Vattré, 2017. However, in contrast with the present approach, these authors did not use continuous dislocation and g-disclination fields in this description, nor did they account for the elastic/plastic distortion incompatibility taking place at grain boundaries. Consequently, while these models can reproduce some of the features of grain boundaries, they cannot account for their core structure and core mechanisms such as the absorption/emission of dislocations. By using the theory of dislocation fields (Acharya, 2001), the incompatibility of the (first-order) elastic/plastic distortion was accounted for at mesoscale along grain boundaries in Fressengeas (2019); Fressengeas and Upadhyay (2020), implying Burgers vector conservation and tangential continuity of the elastic/plastic distortion across static boundaries or boundaries propagating normally to themselves. The presence over a finite-width layer across the boundary of continuous dislocation distributions allowed for the description of boundary curvature (Fressengeas, 2019) and of slip transfer (Fressengeas and Upadhyay, 2020), but second-order distortion incompatibility is needed for the description of core properties at nanoscale. Continuous core description of symmetrical tilt grain boundaries were given in Cordier et al. (2014); Fressengeas et al. (2014); Taupin et al. (2013) using the elasto-plastic theory of dislocation and disclination fields (Fressengeas, Taupin, Capolungo, 2011, Fressengeas, Taupin, Upadhyay, Capolungo, 2012) and in Cleja-Tigoiu et al. (2019). In these papers, grain boundaries were seen as periodic arrays of disclination dipoles distributed over a thin layer, for example a few Å  thick in Cu (Fressengeas, Taupin, Capolungo, 2014, Taupin, Capolungo, Fressengeas, Das, Upadhyay, 2013). A continuous distribution of the elastic energy arising from the associated incompatible strains and curvatures was obtained in this area. Accounting for rotational incompatibility allowed for a stable defect core structure, and dislocation-grain boundary interactions could therefore be described (Taupin et al., 2017). Further, plasticity mechanisms such as grain boundary migration were explained in terms of coupled transport of dislocation and disclination densities. However, incoherent interfaces and phase boundaries, or grain boundaries with steps or ledges, were beyond the scope of these investigations, because they involve discontinuities of the elastic/plastic strain across the boundary. The present dislocation and generalized disclination theory is used in the following to extend this approach to non-planar interfaces showing steps and ledges, thus opening the way to dealing with curved interfaces and dilation-retraction mechanisms at interfaces. Symmetrical tilt boundaries are chosen for simplicity. The paper then proceeds in three steps. Firstly, we detail a plane restriction of the theory sufficient for symmetrical interfaces. We show the role of generalized disclinations in this context, and illustrate the differences between the “couple-stress” Mindlin-Tiersten and “hyper-stress” Mindlin-Eshel treatments. Secondly, we discuss the shear strain discontinuities revealed by Couillard et al. (2013) in straight symmetrical tilt boundaries. These features cannot easily be accounted for in dislocation/standard disclination theories, and they are interpreted here in terms of generalized disclination dipoles. Finally, the elastic structure of steps and ledges is detailed in terms of elastic stretch and rotation discontinuities, and described using the associated generalized disclination dipoles. In addition to providing a realistic description of the structure of tilt boundaries and showing the contributions of generalized disclinations to the latter, specific objectives of this presentation include evidencing the benefits offered by a continuous rendition of lattice incompatibility below inter-atomic spacing and by accounting for the non-locality of the elastic response in the core region of the boundaries through second-order distortions.

The paper is organized as follows. Notations are settled in Section 2. Incompatibility in the theories of dislocations and g-disclinations is reviewed in Section 3. In Section 4, transport properties are presented. Section 5 deals with second-order equilibrium equations and boundary conditions. The elastic constitutive relations appropriate for second-order incompatible media are detailed in Section 6, where thermodynamic guidance following from dissipation arguments is also used to derive driving forces and admissible mobility laws for dislocation and g-disclination motion. Section 7 provides algorithms for the solution of boundary value problems in the elasto-static and anelastic dynamic configurations. In Section 8, a plane dislocation/g-disclination model adequate for the description of symmetrical tilt boundaries is presented, and the relevance in this description of both the Mindlin-Eshel mechanical framework and the g-disclination concept is shown from examples of planar and non-planar symmetrical tilt boundaries. A summary and concluding remarks follow.

Section snippets

Notations

A bold symbol denotes a tensor, as in: A. When there may be ambiguity, an arrow is superposed to represent a vector: V. All tensor subscripts refer to the basis (ei,i=1,2,3) of a rectangular Cartesian coordinate system. Vertical arrays of one, two or three dots represent contraction of the respective number of “adjacent” subscripts on two immediately neighboring tensors, in standard fashion. For example, the tensor A.B with components AikBkj results from the dot product of tensors A and B, and

Review of compatibility theory

In a simply-connected body D undergoing a continuous elasto-plastic deformation process, the displacement vector field u and the rotation vector fieldω=12curluderiving from this displacement are single-valued at any point and possibly defined between the atoms, below inter-atomic distances. Therefore, the conventional compatibility conditions on distortion, rotation, strain and curvature should be satisfied. If I denotes the identity tensor, X and x the reference (Lagrange) and current (Euler)

Transport

The dislocation, standard disclination and generalized disclination density fields (α, θ) and ξ are aeral densities of defect lines carrying a topological content, respectively (b, Ω) and Ξ. We now assume that they have associated velocity fields, (Vα, Vθ) and Vξ with respect to the material. At sufficiently small resolution length scales, (α, θ, ξ) represent individual defects, and the corresponding velocity fields reflect the motions of the defects’ cores. At larger length scales, they may

Equilibrium equations and boundary conditions

In the second-order theory being developed here, the virtual power of internal forces Pi is assumed to be a function of the strain rate and second-order distortion rate tensor fields (ϵ˙,G˙) in order to account for the kinematics of the dislocations and g-disclinations, and of their work-conjugate fields (Tsym, M), respectively the symmetric stress tensor and hyper-stress tensor fields:Pi=V(ϵ˙:Tsym+G˙M)dv.Because it is symmetric, the strain rate tensor extracts the symmetric part of the

Constitutive relations

The mechanical power dissipation is defined as the difference between the power of external loads applied to the body and the rate of free energy stored in the body:D=V(u˙.t+U˙:Λ)dSVψ˙dv.According to the second law of thermodynamics, D is non-negative in all dynamic processes, and vanishes only in purely elastic reversible processes. From the previous Section, it is readily seen that D can also be expressed as the volumetric integral:D=V(ϵ˙:Tsym+G˙M)dvVψ˙dv.or, in terms of the

Solution algorithms

In this Section, we detail algorithms for the solution of the boundary value problem in a body containing a prescribed distribution of dislocations and g-disclinations and submitted to external loading. In a first step, we consider the conservative elasto-static case where the crystal defects are motionless and, in a second step, the dissipative dynamic-anelastic case where the crystal defects are moving with respect to the material.

Case study of symmetrical tilt boundaries

With the aim of discussing below the distribution of dislocations and generalized disclinations in symmetrical tilt boundaries, either straight or presenting ledges, we now present a restriction of the above equations to a plane model with all defect lines normal to the plane.

Conclusion

The paper presented a second-order linearized field theory of crystal defects: dislocations and generalized disclinations, able to accommodate arbitrary discontinuities of the elastic displacement vector and distortion (strain and rotation) tensor fields, and to describe continuously the anelasticity (plasticity and phase transformation) of crystalline solids. It is a “linearized” theory because it uses the additive decomposition of the distortion into strain and rotation, and a “second-order”

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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