Dilatational Misfit Strains Research Articles

Phase-Field Modeling of Equilibrium Precipitate Shapes Under the Influence of Coherency Stresses

Coherency misfit stresses and their related anisotropies are known to influence the equilibrium shapes of precipitates. Additionally, mechanical properties of the alloys are also dependent on the shapes of the precipitates. Therefore, in order to investigate the mechanical response of a material which undergoes precipitation during heat treatment, it is important to derive the range of precipitate shapes that evolve. In this regard, several studies have been conducted in the past using sharp interface approaches where the influence of elastic energy anisotropy on the precipitate shapes has been investigated. In this paper, we propose a diffuse interface approach which allows us to minimize grid-anisotropy related issues applicable in sharp-interface methods. In this context, we introduce a novel phase-field method where we minimize the functional consisting of the elastic and surface energy contributions while preserving the precipitate volume. Using this method we reproduce the shape-bifurcation diagrams for the cases of pure dilatational misfit that have been studied previously using sharp interface methods and then extend them to include interfacial energy anisotropy with different anisotropy strengths which has not been a part of previous sharp-interface models. While we restrict ourselves to cubic anisotropies in both elastic and interfacial energies in this study, the model is generic enough to handle any combination of anisotropies in both the bulk and interfacial terms. Further, we have examined the influence of asymmetry in dilatational misfit strains along with interfacial energy anisotropy on precipitate morphologies.

Shape evolution of fcc Co–Fe precipitate particles during coarsening in a Cu matrix

The shape evolution of fcc Co–Fe precipitate particles during coarsening in Cu-1.37 at.% Co-0.83 at.% Fe alloy single crystals has been examined by the transmission electron microscopy observation. Co–Fe particles change from sphere to cuboid and eventually flatten into a plate-like one with increasing particle size. The size-dependent equilibrium shapes of Co–Fe particles were discussed by considering the elastic strain energy of particles together with the interface energy between the particles and the matrix. To evaluate the elastic strain energy, elastically homogeneous but anisotropic particles with purely dilatational misfit strains are taken into account. The experimentally observed shape transition can be satisfactorily explained by the simple energy analysis.

弾性相互作用エネルギーと界面エネルギーが析出物の形状と分布に及ぼす影響：異方性弾性論による 2 次元解析

Elastic interaction energy between two super-circular inclusions with purely dilatational misfit strains is evaluated. For the calculation, a two dimensional model is used together with linear elasticity for the cubic anisotropy of copper. When two inclusions lie parallel to <100> of the copper crystal, attractive interaction is observed between the inclusions. The maximum of the attractive interaction is realized when the two inclusions exist close to each other. As the inclusion shape becomes circular to square-like, the maximum interaction energy decreases and the distance between the two inclusions to give the maximum increases. Furthermore, the two inclusions are found to be most stable when they have the same size. In addition to the elastic interaction energy, interface energy of the matrix/inclusion interface is also considered. The results show that the two inclusions tend to become the same size even for a situation when the interface energy is much more dominant compared with the elastic interaction energy.

Elastic strain energy due to misfit strains in a polyhedral precipitate composed of low-index planes

The dependence of precipitate shape on the elastic strain energy caused by anisotropy of the elastic moduli is discussed for polyhedral precipitates with purely dilatational misfit strains. By considering cubic structures for the precipitates and matrix phase, the elastic strain energy is calculated for the precipitate shapes of a {1 0 0} cube, a {1 1 0} rhombic-dodecahedron, a {1 1 1} octahedron, and the truncated shapes of all three polyhedra. The anisotropy of the elastic moduli of Cu is used for the numerical calculations. The polyhedral shapes of the {1 1 1} octahedron and {1 0 0} cube result in the highest and lowest values of the elastic strain energy per unit volume of precipitate. The calculated values of the elastic strain energy obtained for the truncated polyhedra are compared with values estimated using a geometrical method that considers the area changes of the {1 0 0}, {1 1 0} and {1 1 1} interfaces.

Effects of octahedral to spherical precipitate-shape transition on elastic strain energy due to misfit precipitates in materials with cubic structures

When elastic moduli are anisotropic, the elastic strain energy of materials containing precipitates with purely dilatational misfit strains depends on the precipitate shape. This dependence is discussed for the octahedral to spherical shape transition. Precipitates modelled as superspherical shapes described by ( x 1 2/ a 2) p/2 +( x 2 2/ a 2) p/2 +( x 3 2/ a 2) p/2 ⩽1 with various values of p (1⩽ p⩽2) and embedded in matrices with cubic structure are treated. The micromechanics inclusion problem is solved with the assumption of the same elastic moduli for the precipitates and matrices to calculate the elastic strain energy. When the anisotropy ratio of the elastic moduli is larger than unity, octahedral precipitates cause larger elastic strain energy than spherical precipitates. However, the precipitate-shape dependence of the elastic strain energy is not monotonic and the elastic strain energy peaks for an intermediate shape between octahedral and spherical. The stability of misfit precipitates of octahedral shape is discussed.

Load transfer at imperfect interfaces––dislocation-like model

A dislocation-like model describing the boundary conditions of a partially debonded interface is verified experimentally. The effect of the imperfect interface on the load transfer is studied by photoelastically measuring the elastic deformation in bimaterials due to an inclusion with dilatational misfit strain. The boundary conditions to be modeled are that the radial and the tangential tractions are continuous across the interface and the displacements may be discontinuous from one solid to another. The discontinuity of displacement across the interface is assumed linearly proportional to the displacement at the interface of the constituent where the stress source is. The maximum shear stress distributions measured from the isochromatic fringe patterns are in good agreement with the theoretical calculations. The results show that an imperfect interface could be viewed as a continuum entity with interface rigidity as proposed by the dislocation-like model.

Two-dimensional analysis on elastic strain energy due to a uniformly eigenstrained supercircular inclusion in an elastically anisotropic material

In Ni-base alloys, the shape of γ ′ precipitates with purely dilatational misfit strains often becomes an intermediate shape between the sphere and cuboid. The present study provides a new two-dimensional model to understand the elastic strain energy caused by the intermediate precipitate shape. A two-dimensional inclusion problem is solved for an inclusion shape described by ( x 1 2/ a 2) p/2 +( x 2 2/ a 2) p/2 ⩽1 ( p⩾2). This shape that may be called a supercircle becomes a circle and a square when p=2 and p→∞, respectively. Intermediate shapes between the circle and square can be expressed by choosing appropriate values of p. Using the two-dimensional Green function of elasticity, the elastic states are examined for an infinitely extended material containing a uniformly eigenstrained supercircular inclusion. Anisotropic elastic moduli of cubic crystals are considered as those of the material containing the supercircular inclusion. The averaged Eshelby tensors of the supercircular inclusions are calculated as a function of p and the anisotropic elastic moduli. Using the averaged Eshelby tensors calculated, the elastic strain energy due to the supercircular inclusion with purely dilatational eigenstrains and its variation with the shape change from circular to square are discussed.

On the shape evolution of a two-dimensional coherent precipitate with a general misfit strain

For a coherent precipitate in two dimensions, any misfit strain, i.e. stress-free transformation strain is expressed with two principal terms of ε and η. Therefore, all the transformation problems can be divided into three cases, (a) a purely dilatational case with ε= η; (b) a tetragonal case of opposite signs with εη≤0; and (c) a tetragonal case of same signs with εη>0. For case (a) with purely dilatational misfit strain, if elastic anisotropy is uniform, a soft particle takes on a plate-like shape along an elastically soft direction. A hard particle, however, tends to take on a shape of high symmetry with edges parallel to elastically soft directions. A homogeneous particle tends to follow the hard particle case in the early stage of evolution, but eventually changes to the pattern of a soft particle. For case (b) with tetragonal misfit strain of opposite signs, there exists always a preferred orientation at an angle of tan −1(− η/ε) 1/2 with respect to a principal direction. This is true whether the particle is soft or hard, i.e. regardless of inhomogeneity. The third case (c), a misfit strain of same signs with εη>0, represents a combination of the previous two. A hard particle of this case tends to change its morphology from a plump shape to a somewhat elongated one with its major axis parallel to the preferred orientation due to the tetragonal part of opposite signs. A soft particle becomes a plate lying along a direction between an elastically soft direction and the preferred orientation.

Dynamic interaction between a coherent precipitate and an edge dislocation

Dynamic interaction between a coherent precipitate and an edge dislocation is analyzed by means of a discrete atom method, which is based on classical statistical mechanics and linear elasticity. Precipitates having a dilatational misfit strain and elastic constants different from those of the matrix phase are treated in anisotropic elastic systems under a plane strain condition. A coherent interface transforms into a semicoherent one by nucleating dipolar dislocations at a stress concentration in a coherent precipitate. One of the dipolar dislocations glides along the precipitate-matrix interface to become a misfit dislocation, and the other slips into the matrix phase to become a lattice dislocation. In accordance with continuum elasticity, a coherent particle with a positive misfit strain migrates to the tension side of an edge dislocation, whereas a particle with a negative misfit diffuses to the region of compression. Morphological change is, however, caused by the dislocation as the particle tries to capitalize on the dislocation stress field, and the particle shape depends on its stiffness and elastic anisotropy. Under an applied shear, a hard coherent particle with a positive misfit strain is sheared along the shear direction, but a soft particle responds in the opposite direction. Elastic interaction between a coherent particle and an edge dislocation can be so strong that the particle-dislocation complex remains coupled even at a high shear strain applied to the system. Some composite applied stresses can cause an edge dislocation to split into two partials. One of the partials, a glissile component, is found to engage actively in the morphological evolution of a particle during a diffusional relaxation.

Morphology of coherent precipitates via a discrete atom method

Morphological evolution of coherent precipitates is analyzed by means of a discrete atom method, which is based on classical statistical mechanics and linear elasticity. Precipitates having a dilatational misfit strain and elastic constants different from those of the matrix phase are treated in dislocation-free, anisotropic elastic systems. In the first part, precipitate shapes in a three-phase system are studied under a plane strain condition. It is found that envelopment of one phase by another phase can have its origin entirely in strain energy reduction, even if the competing interfacial energies are similar to each other. If a hard particle has a misfit strain and a soft particle is misfit-free, the soft particle envelopes the hard particle, producing a concentric double-ring equilibrium shape. This is also true when the signs of the misfit strains are opposite. If a soft particle has a misfit strain and a hard one is misfit-free, the soft particle migrates away from the hard particle due to the coherency-induced image force. If both phases have misfit strains of the same sign, the soft particle wets partially the hard particle, but its other end stretches away from the hard particle. Employing a face centered cubic as a basis lattice, the discrete atom method is then extended to the three-dimensional case. The features predicted by the previous, two-dimensional model are reproduced. In an Isotropic system with a purely dilatational strain, a hard coherent particle takes on a spherical equilibrium shape, while a soft particle tends to have a plate-like equilibrium shape. In an anisotropic system with Zener's ratio, A = 2.33, a soft precipitate shows a series of shape transitions from a radial to four-fold, then to two-fold symmetry. The four-fold shape is concave cuboidal, and is of transient nature, as it is later replaced with a convex plate. The hard particle, however, maintains a convex, cuboidal equilibrium shape. Under an applied tensile stress, a soft particle with a positive misfit strain tends to become a plate perpendicular to the applied stress axis, while a hard particle elongates along the stress direction. If the elastic interaction between the applied stress and the coherency strain is strong enough, precipitates often split into smaller particles and then follow coarsening.

Effect of uniaxial stress on coarsening of precipitate clusters

The influence of moderate applied uniaxial stresses (σapp/C 44 ≈ 10−3) on the coarsening behavior of misfitting coherent precipitates in binary alloys has been studied. Three-dimensional (3-D) computer simulations of the coarsening have been performed for elastically homogeneous systems with tetragonal misfit strain and elastically heterogeneous systems with dilatational misfit strain. Precipitate shapes are restricted to spheres. Results depend on the sign of the misfit strain, the sign of the applied field, and the character of the elastic heterogeneity: precipitates softer than the matrix phase with positive (negative) misfit strain align along the direction of the applied stress for compressive (tensile) fields and arrange in planes perpendicular to it for tensile (compressive) fields. Precipitates harder than the matrix behave in the opposite way.

The elastic strain energy of a coherent inclusion with deviatoric misfit strains

The elastic strain energy of a misfitted coherent inclusion is discussed using the Eshelby method for ellipsoidal inclusions. Deviatoric and tetragonal misfit strains in an elastically inhomogeneous spheroidal inclusion are particularly considered. The variation of the elastic strain energy is evaluated as a function of the shape and orientation of the inclusion. Results obtained are compared with those for a coherent inclusion with purely dilatational misfit strains. Under certain deviatoric misfit strains and elastic moduli of the inclusion, the least elastic strain energy is achieved by a spheroid with an intermediate shape between a plate and a sphere or between a sphere and a needle.

On the Shape Evolution of Coherent Precipitates: Discrete Atom Method

ABSTRACTMorphological evolution of coherent precipitates with arbitrary transformation strain is studied via a discrete atom method. With a purely dilatational misfit strain, a soft precipitate tends to have a plate-like equilibrium shape, whereas a hard precipitate takes up a shape of high symmetry such as a circle. If the stiffness is comparable between the matrix and precipitate, however, the equilibrium shape depends on the degree of anisotropy, misfit strain, size, and interfacial energy. Shape bifurcation phenomena from circular to rectangular shapes are also revealed during clustering. With either a tetragonal misfit strain of mixed signs or a pure shear misfit strain, a precipitate takes up a plate-like shape whose major axis lies along the direction containing an invariant line in accordance with the continuum elasticity prediction. Both elastic anisotropy and elastic inhomo-geneity exert little influence on the preferred orientation relationship. With a misfit strain of combined shear and dilatation, a precipitate follows an orientation and shape dictated by both components of the misfit strain.

Thin Inclusions and Cracks in Anisotropic Media

By using the solution of eigenstrain problems in anisotropic media, flat ellipsoidal inclusions and crack problems are investigated. When a dilatational misfit strain is defined in a flat ellipsoidal inclusion (i.e., disk-shaped precipitate), the elastic strain energy associated with the misfit becomes minimum when the plane of the inclusion coincides with one of the crystalline planes, where the matrix and the inclusion are assumed as two different cubic crystals having the same crystalline directions. The minimum value, however surprisingly, depends only on the elastic moduli of the inclusion. When crystalline directions of the inclusion are parallel to the principal axes of the ellipsoid, the elastic strain energy is independent of the orientation of the inclusion with respect to the crystalline directions of the matrix and the constant value is equal to the minimum value in the first case. If an inclusion is simply defined by giving a uniform dilatational misfit strain in an ellipsoidal domain in a homogeneous material (i.e., matrix and inclusion are same material), the condition for the minimum elastic strain energy is the same as that of the first case, namely the minimum occurs when the plane of the inclusion is parallel to one of the crystalline directions. The method employed here can be applied equally to generally anisotropic materials. The method is also applicable to fracture problems by taking the elastic moduli of the inclusion as zero. As examples, an elliptical crack is considered for a simple tension and a pure shear. The interaction energy between the applied stress and a crack is calculated. By using this result, the Griffith criterion for fracture is derived for the penny-shaped crack. Some numerical results for cubic crystals are shown.