Antiplane Elasticity Research Articles

Interplay Between Nucleation and Kinetics in Dynamic Twinning

Abstract In this work, we apply a phase-field modeling framework to elucidate the interplay between nucleation and kinetics in the dynamic evolution of twinning interfaces. The key feature of this phase-field approach is the ability to transparently and explicitly specify nucleation and kinetic behavior in the model, in contrast to other regularized interface models. We use this to study two distinct problems where it is essential to explicitly specify the kinetic and nucleation behavior governing twin evolution. First, we study twinning interfaces in 2D. When these interfaces are driven to move, we find that significant levels of twin nucleation occur ahead of the moving interface. Essentially, the finite interface velocity and the relaxation time of the stresses ahead of the interface allow for nucleation to occur before the interface is able to propagate to that point. Second, we study the growth of needle twins in antiplane elasticity. We show that both nucleation and anisotropic kinetics are essential to obtain predictions of needle twins. While standard regularized interface approaches do not permit the transparent specification of anisotropic kinetics, this is readily possible with the phase-field approach that we have used here.

An isoparametric inclusion model for determining the thermo-elastic fields produced by varying Eigen-temperature gradients

The existence of inclusions or inhomogeneities may disrupt heat flow, resulting in increased stresses and temperature fluctuations at the material interface. Analyzing the thermo-elastic fields surrounding such microstructures is crucial for unveiling the intricate failure mechanisms within materials. Based on the method of Green's function and contour integral, this work presents a complete set of explicit analytical solutions for thermal and elastic fields of an arbitrary polygonal inclusion subjected to linearly varying eigen-temperature gradients or thermal eigenstrains. In contrast to the previous studies on potentials showing analogy with anti-plane elasticity, a mathematical analogy of Green's functions between steady-state heat conduction and thermal inclusion for plane elasticity is accordingly established. By employing the linear interpolation of the nodal eigen-temperature gradients or eigenstrains, the work further extends the traditional isoparametric element idea to the thermos-elastic inclusions, and proposes a novel isoparametric inclusion model to solve arbitrarily shaped inclusions with any distributed eigen-temperature gradients. Unlike the finite element method, which requires meshing for both the matrix and inclusion, the present model exhibits efficiency, flexibility, and versatility with the mesh discretization only conducted inside the inclusion domain. The inclusion based isoparametric method may have potential applications in mechanical engineering and material science for analyzing the thermo-structural behavior of various microstructures.

Determining the volume fraction in 2-phase composites and bodies using time varying applied fields

A body Θ containing two phases, which may form a periodic composite with microstructure much smaller that the body, or which may have structure on a length scale comparable to the body, is subjected to slowly time varying boundary conditions that would produce an approximate uniform field in Θ were it filled with homogeneous material. Here slowly time varying means that the wavelengths and attenuation lengths of waves at the frequencies associated with the time variation are much larger than the size of Θ, so that we can make a quasistatic approximation. At least one of the two phase does not have an instantaneous response but rather depends on fields at prior times. The fields may be those associated with electricity, magnetism, fluid flow in porous media, or antiplane elasticity. We find, subject to these approximations, that the time variation of the boundary conditions can be designed so boundary measurements at a specific time t=t0 exactly yield the volume fractions of the phases, independent of the detailed geometric configuration of the phases. Moreover, for specially tailored time variations, the volume fraction can be exactly determined from measurements at any time t, not just at the specific time t=t0. We also show how time varying boundary conditions, not oscillating at the single frequency ω0, can be designed to exactly retrieve the response at ω0.

A screw dislocation in a three-phase composite composed of an anisotropic elastic half-plane bonded to an isotropic elastic half-plane reinforced by a circular inhomogeneity

Using complex variable techniques, we solve the problem in anti-plane elasticity associated with a screw dislocation in a three-phase composite which is composed of an anisotropic elastic half-plane perfectly bonded to an isotropic elastic half-plane containing a circular elastic inhomogeneity. Analytical solutions are obtained for three typical cases: a screw dislocation in the isotropic half-plane in the exterior of the circular inhomogeneity, a screw dislocation inside the circular elastic inhomogeneity, a screw dislocation within the anisotropic half-plane. Explicit expressions of the image force acting on the screw dislocation are derived for each of the three cases. The numerical results indicate that at least one equilibrium position for the screw dislocation co-exists in each phase under certain conditions on the elastic constants.

Antiplane anisotropic elastodynamics: Babinet’s principle

Babinet’s principle is established for certain problems in antiplane anisotropic elasticity. For example, the scattered field behind a finite straight crack is exactly cancelled by the total field behind an aperture in an infinite flat rigid screen, where the crack and the aperture have the same length and the incident wave is the same for both problems. The method used involves Fourier integrals and dual integral equations.

Scaling in Anti-Plane Elasticity on Random Shear Modulus Fields with Fractal and Hurst Effects

The scale dependence of the effective anti-plane shear modulus response in microstructures with statistical ergodicity and spatial wide-sense stationarity is investigated. In particular, Cauchy and Dagum autocorrelation functions which can decouple the fractal and the Hurst effects are used to describe the random shear modulus fields. The resulting stochastic boundary value problems (BVPs) are set up in line with the Hill–Mandel condition of elastostatics for different sizes of statistical volume elements (SVEs). These BVPs are solved using a physics-based cellular automaton (CA) method that is applicable for anti-plane elasticity to study the scaling of SVEs towards a representative volume element (RVE). This progression from SVE to RVE is described through a scaling function, which is best approximated by the same form as the Cauchy and Dagum autocorrelation functions. The scaling function is obtained by fitting the scaling data from simulations conducted over a large number of random field realizations. The numerical simulation results show that the scaling function is strongly dependent on the fractal dimension D, the Hurst parameter H, and the mesoscale δ, and is weakly dependent on the autocorrelation function. Specifically, it is found that a larger D and a smaller H results in a higher rate of convergence towards an RVE with respect to δ.

A non-circular Eshelby inclusion near a non-parabolic inhomogeneity admitting internal uniform stresses

With the aid of conformal mapping and analytic continuation, we prove that within the framework of anti-plane elasticity, a non-parabolic open elastic inhomogeneity can still admit an internal uniform stress field despite the presence of a nearby non-circular Eshelby inclusion undergoing uniform anti-plane eigenstrains when the surrounding elastic matrix is subjected to uniform remote stresses. The non-circular inclusion can take the form of a Booth’s lemniscate inclusion, a generalized Booth’s lemniscate inclusion or a cardioid inclusion. Our analysis indicates that the uniform stress field within the non-parabolic inhomogeneity is independent of the specific open shape of the inhomogeneity and is also unaffected by the existence of the nearby non-circular inclusion. On the other hand, the non-parabolic shape of the inhomogeneity is caused solely by the presence of the non-circular inclusion.

Multicoated elastic inhomogeneities of arbitrary shape neutral to multiple fields

We rigorously establish the interesting result that in anti-plane elasticity an elastic epitrochoidal inhomogeneity can be made neutral to multiple uniform fields applied in the matrix via the insertion of two intermediate coatings. Using a two-term conformal mapping function, the simply connected domain occupied by the epitrochoidal inhomogeneity and its surrounding inner and outer coatings is mapped onto the interior of the unit circle in the image plane. The mismatch parameters are determined in an analytical manner by solving a set of two non-linear equations. An elastic inhomogeneity of arbitrary shape can be made neutral to multiple fields through the insertion of N coatings when the proposed mapping function for the simply connected domain occupied by the multicoated inhomogeneity is described in terms of a polynomial of finite degree containing N non-constant terms. In this case, the mismatch parameters are determined by iteratively solving a set of N non-linear equations.

Degenerate scale problem for some configurations with rigid line tips in antiplane elasticity or Laplace equation

This paper provides several solutions of the degenerate scale for some configurations with rigid line tips in antiplane elasticity. For solving the problem, it is necessary to investigate the conformal mapping function in advance. It is found that when a particular dimension reaches its critical value, a closed form solution for complex potential on the mapping plane can be found. This solution takes zero value for displacement along the boundary. However, it is not vanishing in the exterior region to the contour.

Solution for the degenerate scale for a rigid curve in antiplane elasticity by using a weakly singular integral equation

This paper provides a numerical solution for the degenerate scale for a rigid curve in antiplane elasticity. The degenerate scale problem for the rigid curve is formulated on the usage of the logarithmic potential. After assuming the displacement to be a vanishing value along the rigid curve, the boundary integral equation (BIE) is formulated. The problem can be first formulated in the degenerate scale. After making a coordinate transform, we can obtain the relevant BIE in the ordinary scale. Finally, a numerical solution is achieved. Several numerical examples are provided. In addition, the degenerate scale problem for the multiple rigid curves is also solved.

A circular Eshelby inclusion with linear eigenstrains interacting with a coated non-parabolic inhomogeneity with internal uniform anti-plane stresses

We establish criteria which guarantee the uniformity of stresses inside a coated non-parabolic open inhomogeneity located in the vicinity of a circular Eshelby inclusion undergoing linear anti-plane eigenstrains when the surrounding matrix is subjected to uniform remote anti-plane stresses. The associated inverse problem in anti-plane elasticity which identifies the required geometry of the constituents of the composite is successfully solved using conformal mapping techniques and analytic continuation. Three conditions on the remote loading in the matrix, uniform eigenstrains imposed on the non-parabolic inhomogeneity and linear eigenstrains imposed on the circular Eshelby inclusion for given geometric and material parameters of the composite are found in order to ensure the uniformity of internal stresses. The internal uniform stress field is found to be independent of the specific open shapes of the inhomogeneity-coating and coating-matrix interfaces, the shear modulus of the coating and also of the existence of the nearby circular Eshelby inclusion. However, the open shapes of the two interfaces are significantly influenced by the presence of the circular Eshelby inclusion. We also discuss the general case in which arbitrary polynomial eigenstrains are imposed on the circular Eshelby inclusion.

An extension of the Eshelby conjecture to domains of general shape in anti-plane elasticity

According to the Eshelby conjecture, an ellipse or ellipsoid is the only shape that induces an interior uniform strain under a uniform far-field loading. We extend the Eshelby conjecture to domains of general shape for anti-plane elasticity. Specifically, we show that for each positive integer N, an inclusion induces an interior uniform strain under a polynomial loading of degree N if and only if the exterior conformal map of the inclusion is a Laurent series of degree N. Furthermore, for the isotropic case, we characterize the shape of an inclusion by only using the first-degree polynomial loading and explicitly solve the interior potential of the inclusion in terms of the Grunsky coefficients.

Higher-order imperfect interface modeling via complex variables based asymptotic analysis

We present a new methodology to derive imperfect interface models for the problems with interphase layers. The test case is potential problems, e.g., thermal conductivity, antiplane elasticity, etc. The methodology combines classical asymptotic analysis with concepts from the theory of complex-valued functions. Its major advantage over existing asymptotic approaches is the straightforward derivation of jump conditions that involve surface differential operators of arbitrary order, resulting in a hierarchy of models that maintain arbitrary-order accuracy with respect to the layer thickness and its curvature. Unlike low-order models, the derived higher-order models can accurately represent layers that are significantly softer or stiffer than the adjacent bulk materials, exhibit varying curvature, or are of finite thickness with respect to the characteristic length scale of the adjacent bulk regions. The interface models obtained via our methodology are compared with existing models of different orders, their limiting behavior is validated with respect to known interface regimes, and the improved accuracy of higher-order variants is illustrated for a benchmark example. While here we limited ourselves to scalar problems in two dimensions, the extension to vector problems in two-dimensions is straightforward. We also discuss the pathway to extend our methodology to scalar and vector problems in three-dimensions.

Generalized Hill-Mendel lemma and equivalent inclusion method for determining the effective thermal conductivity of composites with imperfect interfaces

The present work aims at determining the effective thermal conductivity of two- or three-dimensional composites with imperfect interfaces between their constituent phases. These imperfect interfaces are described by the highly conducting, lowly conducting or general thermal imperfect model. To achieve the objective, the classical Hill-Mendel lemma is first extended to include the effects of imperfect interfaces and an equivalent inclusion method (EIM) is proposed. The basic idea of EIM is to replace an inclusion embedded in a matrix via an imperfect interface by an equivalent inclusion inserted in the same matrix via a perfect interface. Using EIM and applying the dilute distribution, Mori-Tanaka, self-consistent, generalized self-consistent and differential schemes, the effective thermal conductivities of layered composites and some particle-reinforced composites with imperfect interfaces are analytically and explicitly determined. These results are compared with the Voigt, Reuss and Hashin-Shtrikman bounds and checked against the numerical results provided by the fast Fourier transform (FFT) method. These comparisons and checks show that the methods proposed in this work are particularly efficient. The methods and results of the present work are directly transposable to other transport phenomena and anti-plane elasticity by their strict mathematical analogy with thermal conduction.

Uniform stresses inside two hyperbolic inhomogeneities

We prove that the internal stresses inside two hyperbolic elastic inhomogeneities can indeed remain uniform when the matrix is subjected to uniform remote anti-plane and in-plane stresses. Conditions leading to internal uniform stresses are established rigorously for both anti-plane and plane elasticity. Once these conditions are satisfied, the internal uniform stresses inside the two hyperbolic inhomogeneities are determined explicitly.

Uniform stresses inside a non-elliptical inhomogeneity and a nearby half-plane with locally wavy interface

We develop a two-parameter conformal mapping function for a doubly connected domain to solve the inverse problem in anti-plane and plane elasticity associated with a non-elliptical inhomogeneity with internal uniform stresses embedded in a half-plane bonded to another half-plane also with internal uniform stresses via a locally wavy interface. The internal uniform stresses are found to be independent of the specific shapes of the locally wavy and non-elliptical interfaces. The permissible range of the two parameters in the mapping function for a one-to-one mapping is obtained. Typical geometries corresponding to the three-phase composite are illustrated.

Three-phase parabolic inhomogeneities with internal uniform stresses in plane and anti-plane elasticity

We examine the in-plane and anti-plane stress states inside a parabolic inhomogeneity which is bonded to an infinite matrix through an intermediate coating. The interfaces of the three-phase parabolic inhomogeneity are two confocal parabolas. The corresponding boundary value problems are studied in the physical plane rather than in the image plane. A simple condition is found that ensures that the internal stress state inside the parabolic inhomogeneity is uniform and hydrostatic. Furthermore, this condition is independent of the elastic properties of the coating and the two geometric parameters of the composite: in fact, the condition depends only on the elastic constants of the inhomogeneity and the matrix and the ratio between the two remote principal stresses. Once this condition is met, the mean stress in the coating is constant and the hoop stress on the coating side is also uniform along the entire inhomogeneity-coating interface. The unconditional uniformity of stresses inside a three-phase parabolic inhomogeneity is achieved when the matrix is subjected to uniform remote anti-plane shear stresses. The internal uniform anti-plane shear stresses inside the inhomogeneity are independent of the shear modulus of the coating and the two geometric parameters of the composite.

Uniformity of stresses inside a parabolic inhomogeneity

Within the context of anti-plane elasticity and plane elasticity, we prove that the stress field inside a parabolic elastic inhomogeneity still remains uniform when the surrounding matrix is subjected to uniform remote stresses. Explicit expressions are presented for the uniform stresses and rigid-body rotation inside the inhomogeneity as well as for the non-uniform stresses in the matrix.

The Degenerate Scales for Plane Elasticity Problems in Piecewise Homogeneous Media Under General Boundary Conditions

The degenerate scale issue for D boundary integral equations and boundary element methods has been already investigated for Laplace equation, antiplane and plane elasticity, bending plate for Dirichlet boundary condition. Recently, the problem of Robin and mixed boundary conditions and of piecewise heterogeneous domains have been considered for the case of Laplace equation. We investigate similar questions for plane elasticity for more general boundary conditions. For interior problems, it is shown that the degenerate scales do not depend on the boundary condition. For exterior problems, the two degenerate scales (homogeneous medium) or two of them (heterogeneous medium) are tightly linked with the behavior at infinity of the solutions. The dependence of this behavior on the boundary conditions is investigated. We give sufficient conditions for the uniqueness of the solution. Numerical applications are provided and validate the set of theoretical results.

Uniform fields inside two interacting non-parabolic and non-elliptical inhomogeneities

With the aid of conformal mapping for a doubly connected domain, we prove the existence of internal uniform stress fields inside two interacting elastic inhomogeneities: one of non-parabolic open shape and the other of non-elliptical closed shape, when the matrix is subjected to uniform remote anti-plane and in-plane stresses. The uniformity property is unconditional for anti-plane elasticity but conditional for plane elasticity. The internal uniform stress fields are independent of the specific open and closed shapes of the two inhomogeneities. Typical numerical examples are presented to demonstrate the feasibility and effectiveness of the proposed theory.