Linear Anisotropic Elasticity Research Articles

Screw dislocation dipoles in niobium: combination of STM observations and atomistic simulations

We recently developed an experimental device that allows us to observe the slip traces under stress at the atomic scale. Here, we report experimental results obtained at the latter scale on Nb single crystals making it possible to observe dislocation dipoles (DD), which are evidenced by two slip traces formed by emerging moving dislocations of opposite Burgers vectors ending very close to each other. The geometry and stability of the DD were fully characterized in the framework of linear anisotropic elasticity theory and by atomistic simulations. This allows us to calculate a local opposite stress impeding dislocation motion of the dislocations of the dipole.

A new spectral representation of the strain energy function for linear anisotropic elasticity with a generalization for damage

A new spectral representation of the strain energy function for linear anisotropic elasticity is proposed in the form of a sum of 6 scalar eigen-stiffnesses times the squares of their associated scalar eigen-strains. Since this new representation is general, it is no less or more general than the standard representation. However, the spectral form reveals fundamental coupled eigen-modes of deformation and energy associated with general anisotropic response. Specifically, each eigen-strain is the inner product of the strain tensor with its symmetric eigen-tensor. The 6 eigen-strains are linearly independent and the 6 eigen-tensors are also linearly independent. Moreover, unlike the eigenvectors of a symmetric matrix, distinct pairs of eigen-tensors are not all orthogonal. The 15 constants that define the eigen-tensors together with the 6 eigen-stiffnesses form 21 independent material constants that characterize the fundamental physics of material anisotropy. These 21 constants are invariant to the orientation of the base vectors fixed in the material. This spectral representation also simplifies the restrictions on these material constants which ensure that the strain energy is a positive-definite function of strain. In addition, a simple damage model is proposed based on this spectral representation and a generalization for finite deformations is briefly discussed.

Determination of the Spherical Couple-Stress in Polar Linear Isotropic Elasticity

It is well known that the conventional couple-stress theory leaves the spherical part of the couple-stress indeterminate. This indeterminacy problem is recently resolved for fibrous composites subjected to either small or large deformations and containing a single family of fibres resistant in bending (Soldatos in Math. Mech. Solids, 2021, https://doi.org/10.1177/10812865211061595). However, the problem remains still unsolved in simpler cases where the implied preference material direction is not related to fibre bending resistance, and even in the simplest possible case where the polar material of interest is linearly elastic and isotropic. This communication aims (i) to show that a relevant virtual spin concept employed (Soldatos in Math. Mech. Solids, 2021, https://doi.org/10.1177/10812865211061595) is further applicable in the latter case of polar linear isotropic elasticity, (ii) to demonstrate the process in which that concept thus leads to determination of the spherical part of the couple-stress, (iii) to exemplify this process by providing a couple of simple illustrative examples, (iv) to specify and discuss the reason that the outlined method meets a hurdle in cases of linear anisotropic elasticity that is due to one or more preferential material directions, and, hence, (v) to further discuss the manner in which that newly identified difficulty is currently confronted, and may thus be handled successfully.

Planar Gliding and Vacancy Condensation: The Role of Dislocations in the Chemomechanical Degradation of Layered Transition-Metal Oxides

Stacking faults driven by dislocations have been observed in layered transition-metal oxide cathodes both in cycled and uncycled materials. The reversibility of stacking-sequence changes directly impacts the material performance. Irreversible glide due to lattice invariance or local compositional changes can initiate a catastrophic sequence of degradation mechanisms. In this study we compare the chemomechanical properties of LiCoO2 and LiNiO2 by combining density functional theory and anisotropic linear elasticity theory. We calculate stacking fault energies as a function of Li content and quantify the extent to which excess Ni hinders stacking-sequence changes. We then characterize screw dislocations, which mediate stacking-sequence changes, and find a peculiarly compliant behavior of LiNiO2 due to the interaction of Jahn–Teller distortions with the dislocation strain field. Finally, we analyze the tendency of vacancies to segregate along dislocation lines. This study represents the first instance of explicit ab initio atomistic dislocation models in layered oxides and paves the way for the understanding and optimization of the chemomechanical behavior of cathode active materials during battery operation.

The universal program of linear elasticity

Universal displacements are those displacements that can be maintained, in the absence of body forces, by applying only boundary tractions for any material in a given class of materials. Therefore, equilibrium equations must be satisfied for arbitrary elastic moduli for a given anisotropy class. These conditions can be expressed as a set of partial differential equations for the displacement field that we call universality constraints. The classification of universal displacements in homogeneous linear elasticity has been completed for all the eight anisotropy classes. Here, we extend our previous work by studying universal displacements in inhomogeneous anisotropic linear elasticity assuming that the directions of anisotropy are known. We show that universality constraints of inhomogeneous linear elasticity include those of homogeneous linear elasticity. For each class and for its known universal displacements, we find the most general inhomogeneous elastic moduli that are consistent with the universality constrains. It is known that the larger the symmetry group, the larger the space of universal displacements. We show that the larger the symmetry group, the more severe the universality constraints are on the inhomogeneities of the elastic moduli. In particular, we show that inhomogeneous isotropic and inhomogeneous cubic linear elastic solids do not admit universal displacements and we completely characterize the universal inhomogeneities for the other six anisotropy classes.

Stress partitioning and effective behavior of N-phase laminates in anisotropic elasticity from a fast explicit method

In this work, a fast explicit method, easy to implement numerically, is proposed in order to compute the effective behavior and the distribution of stresses in a general N-phase laminate made of parallel, planar and perfectly bonded interfaces. The solutions are exact for a homogeneous far-field loading and work for an arbitrary number of phases, a general linear anisotropic elasticity, as well as different uniform thermal and plastic strains in the phases. A simple direct analytical formula is also derived to compute the stress in a given phase once the effective behavior of the laminate is known. Moreover, the correctness of the proposed method is checked by comparisons with finite element simulation results on a same boundary value problem, showing excellent agreements. An application of the method is performed for a near-β titanium alloy with elongated grains, by comparing the level of internal stresses for different elastic loadings within a N-phase laminate made of 100,000 orientations and a 2-phase laminate of equal volume fraction with maximal elastic contrast. Interestingly, the maximum von Mises stress of the 2-phase laminate is always the lowest, which is explained by a volume fraction effect. Finally, comparisons with elastic self-consistent models considering oblate spheroidal grains of different aspect ratios are performed.

Stroh formalism for various types of materials and deformations

Abstract The Stroh formalism is a complex variable formulation developed originally for solving the problems of two-dimensional linear anisotropic elasticity. By separation of the third variable for the linear variation of displacements along the thickness direction, it was proved to be applicable for the problems with coupled stretching-bending deformation. By the Radon transform which maps a three-dimensional solid to a two-dimensional plane, it can be applied to the three-dimensional deformation. By the elastic-viscoelastic correspondence principle, it is also valid for the viscoelastic materials in the Laplace domain. By expansion of the matrix dimension, it can be generalized to the coupled-field materials such as piezoelectric, piezomagnetic and magneto-electro-elastic materials. By introducing a small perturbation on the material constants, it can also be applied to the degenerate materials such as isotropic materials. Thus, in this paper, the Stroh formalism for several different types of materials (anisotropic elastic, piezoelectric, piezomagnetic, magneto-electro-elastic, viscoelastic) and deformations (two-dimensional, coupled stretching-bending, three-dimensional) are organized together and presented in the same mathematical form.

Structure and properties of dislocations and the twin boundary on (101) in β-cyclotetramethylene tetranitramine

ABSTRACT Dispersion-corrected density functional theory calculations of the generalised stacking fault energy surface of the (101) plane in monoclinic β-cyclotetramethylene tetranitramine (β-HMX) indicate that a stacking fault with an energy of approximately 130 mJ/m at the translation vector is metastable. Using the energy of the metastable stacking fault and anisotropic linear elasticity theory, we have evaluated whether dislocations with total Burgers vectors , and will split into partial dislocations on . Our results suggest that it is not favourable for the screw dislocation to split into partials but that the edge dislocation will split into partials separated by a narrow, 19.8 Å, stacking fault. Both screw and edge dislocations are predicted to split into two partials with splitting distances of about 28.7 and 43.4 Å, respectively. The screw and edge dislocation are also predicted to split into partials, with equilibrium distances of 16.3 and 43.2 Å, respectively. These results are consistent with recent analyses of Knoop indentation experiments in which the slip system is inactive whereas dislocations are glissile. Since twinning is an important deformation mechanism in β-HMX, dispersion-corrected density functional theory calculations have also been used to compute the energy of the twin boundary, for which we report a value of 163 mJ/m.

Crack field analysis by optical DIC of short cracks in Zircaloy-4

The prediction of fatigue life is most difficult for short cracks, as their local conditions may differ from what is predicted using the remote applied loading and crack geometry. Digital image correlation (DIC) can be utilized to analyze images from an optical microscope (OM), which facilitates the local characterization of the crack field. This paper presents a Finite-Element-based approach that uses DIC-obtained displacement data to retrieve the crack field and quantify the local crack driving force. With the assumption of linear anisotropic elasticity, the change in both the Mode I and Mode II crack intensity factors over a fatigue cycle can be extracted using the interaction integral method. This allows the determination of the local driving force for short crack propagation. The application of this method is demonstrated by the full-field analysis of short fatigue cracks in blocky alpha Zircaloy-4. The sensitivity of the analysis is investigated to factors that include DIC subset size, uncertainties in crack tip position and lower quality data in the crack vicinity. This technique requires no prior knowledge of theoretical solutions or far-field boundary conditions, and it can be applied to the tip of a tortuous crack by defining an appropriate local frame of reference. The analysis is applied here to cracks under similar mode I loading that propagate on the prism plane at different rates in directions that are parallel and perpendicular to the c-axis of the hexagonal unit cell.

The lattice dislocation trapping mechanism at the ferrite/cementite interface in the Isaichev orientation relationship

We analyze the lattice dislocation trapping mechanism at the ferrite/cementite interface of the Isaichev orientation relationship by atomistic simulations combined with the anisotropic linear elasticity theory and disregistry analysis. We find that the lattice dislocation trapping ability is varied by initial position of the lattice dislocation. The lattice dislocation near the interface is attracted to the interface by the image force generated by the interface shear, while the lattice dislocation located far is either attracted to or repelled from the interface, or even oscillates around the introduced position, depending on the combination of the stress field induced by the misfit dislocation array and the image stress field induced by the lattice dislocation.

Subtraction singularity technique applied to the regularization of singular and hypersingular integrals in high-order curved boundary elements in plane anisotropic elasticity

The numerical solutions of boundary integral equations by the Boundary Element Method (BEM) have been applied in several areas of computational engineering and science such as elasticity and fracture mechanics. The BEM formulations often require the evaluation of complex singular and hypersingular integrals. Therefore, BEM requires special integration schemes for singular elements. The Subtraction Singularity Technique (SST) is a general procedure for evaluating such integrals, which allows for the integral over high-order curved boundary elements. The SST regularises these kernels through the Taylor expansion of the integral kernels around the source point. This study presents the expressions from Taylor expansions, which are required for the regularization of singular and hypersingular boundary integral equations of plane linear anisotropic elasticity. These expressions have been implemented into an academic BEM code, which enables the integration over high-order straight and curved boundary elements. The proposed scheme leads to excellent performance. The results obtained by the proposed scheme are in excellent agreement with reference responses available in the literature.

On the characterisation of polar fibrous composites when fibres resist bending – Part III: The spherical part of the couple-stress

Part II (Soldatos, 2018b) identified some theoretical disagreement between the generally anisotropic polar linear elasticity of Mindlin and Tiersten (1962) and its counterpart developed in (Spencer and Soldatos, 2007; Soldatos, 2014) for fibrous composites with embedded fibres resistant in bending. The present communication shows that this disagreement is essentially due to inherent features of fibre-splay types of deformation and, consequently, generalises the Cosserat and Cosserat (1909) couple-stress theory in a manner that creates room for newly emerged fibre-splay type of kinematic variables to enter and be accounted for. Relevant fundamental theorems, associated in Part II with the Mindlin and Tiersten model, are generalised accordingly to meet the needs and requirements of the proposed new formulation. Interestingly and importantly, the outlined generalisation enables formation of a convincing answer to a long-standing question regarding the indeterminacy of the spherical part of the couple-stress tensor, at least as far as polar elasticity of fibre-reinforced materials is concerned. The manner thus is demonstrated in which the spherical part of the couple-stress can be determined in polar linear elasticity of fibrous composites that exhibit transverse isotropy due to an embedded family of fibres resistant in bending.

Eshelby instability pressure for nucleation of a phase change defect

The existence of a nucleation instability is demonstrated at the vanishing of the path-independent M integral in linear anisotropic elasticity so that a region that is arbitrarily small can grow at constant potential energy. It yields a quadratic equation for the critical pressure to balance the Eshelby dissipation and nucleate an inclusion undergoing both change in density and change in bulk modulus, and is independent of the radius of the defect. Regarding the shape of nucleation, the expression from Noether's theorem allows also a symmetry breaking mode as a penny-shape “pancake”, and by comparison of the M integrals for the pancake and the sphere, it would require a greater loss of potential energy of the system to nucleate a pancake, but less to grow it larger.

High-performance dune modules for solving large-scale, strongly anisotropic elliptic problems with applications to aerospace composites

The key innovation in this paper is an open-source, high-performance iterative solver for high contrast, strongly anisotropic elliptic partial differential equations implemented within dune-pdelab. The iterative solver exploits a robust, scalable two-level additive Schwarz preconditioner, GenEO (Spillane et al., 2014). The development of this solver has been motivated by the need to overcome the limitations of commercially available modelling tools for solving structural analysis simulations in aerospace composite applications. Our software toolbox dune-composites encapsulates the mathematical complexities of the underlying packages within an efficient C++ framework, providing an application interface to our new high-performance solver. We illustrate its use on a range of industrially motivated examples, which should enable other scientists to build on and extend dune-composites and the GenEO preconditioner for use in their own applications. We demonstrate the scalability of the solver on more than 15,000 cores of the UK national supercomputer Archer, solving an aerospace composite problem with over 200 million degrees of freedom in a few minutes. This scale of computation brings composites problems that would otherwise be unthinkable into the feasible range. To demonstrate the wider applicability of the new solver, we also confirm the robustness and scalability of the solver on SPE10, a challenging benchmark in subsurface flow/reservoir simulation. Program summaryProgram Title:dune-compositesProgram Files doi:http://dx.doi.org/10.17632/96mtdcmjsb.1Licensing provisions: BSD 3-clauseProgramming language: C++Nature of problem:dune-composites is designed to solve anisotropic linear elasticity equations for anisotropic, heterogeneous materials, e.g. composite materials. To achieve this, our contribution also implements a new preconditioner in dune-pdelab.Solution method: The anisotropic elliptic partial differential equations are solved via the finite element method. The resulting linear system is solved via an iterative solver with a robust, scalable two-level overlapping Schwarz preconditioner: GenEO.

Splitting of Initial Boundary Value Problems in Anisotropic Linear Elasticity Theory

The splitting of initial boundary value problems in the theories of elasticity is considered for some anisotropic media. In particular, the initial boundary value problems of the micropolar classical theory of elasticity are represented using the tensor-block matrix operators (or tensor operators). In the case of isotropic micropolar elastic media known also as isotropic or transversally isotropic classical media, we propose the tensor-block matrix operators of algebraic cofactors corresponding to the tensor-block matrix operators of the initial boundary value problems, which allows us to split these problems.

Cell-centered finite-volume method for elastic deformation of heterogeneous media with full-tensor properties

We propose a cell-centered finite-volume method for the heterogeneous anisotropic linear elasticity problem. The internal traction vector is represented as a discrete flux on the interface. This ‘elastic’ flux is decomposed into a discrete two-point flux approximation and a semi-discrete transversal part. We introduce an interpolation method in the presence of discontinuity in the full-tensor material properties. The scheme yields an accurate reconstruction of the gradient of the displacement in each cell. The formulation is based on the assumption of linearity of the displacement, and we enforce continuity of the internal traction vector and the displacement at the interface. The treatment of the boundary conditions can be complicated in finite-volume methods. Here, we describe a general treatment of boundary conditions that does not entail the introduction of additional degrees of freedom. The finite-volume method is tested for a series of challenging elasticity problems.

Distance of a stiffness tetrad to the symmetry classes of linear elasticity

In the scope of linear anisotropic elasticity, the fourth-order elasticity tensor or tetrad has to be identified. This can be done either by measurements or by numerical simulations. An important task is then to identify a given tetrad, probably with some experimental or numerical scattering, with one of the symmetry classes. For this purpose one needs a distance function between a given tetrad and the class of all tetrads with a particular symmetry, which is zero if the tetrad obeys this symmetry, or non-zero otherwise. In this paper we present a fast method to solve these problems. We firstly introduce the 8th order projectors that map any stiffness tetrad into the part that is invariant under the action of a specific fixed symmetry group. For this purpose we consider the seven out of the eight symmetry classes that are distinguishable in linear elasticity. Secondly, since the symmetry axes of the specific stiffness tensor under consideration is generally not aligned with the used tensorial basis of the projector, we need to rotate the sample stiffness. The optimal orientation is obtained when the distance between the rotated stiffness and the rotated and projected stiffness is minimal. Thus, we need to apply only linear mappings and minimize over three Euler angles. The latter is quite simple, as the domain of the Euler angles is periodic, and the number of local minima is limited. This procedure has the advantage that it is applicable in an algorithmic manner, and does not require an a priori identification of symmetry planes, symmetry axes or component symmetries, which are only apparent under special choices for the tensorial basis.

On the characterisation of polar fibrous composites when fibres resist bending – Part II: Connection with anisotropic polar linear elasticity

This continuation of Part I (Soldatos, 2018) aims to make a connection between the polar linear elasticity for fibre-reinforce materials due to (Spencer and Soldatos, 2007; Soldatos, 2014, 2015) with the anisotropic version and the principal postulates of its counterpart due to (Mindlin and Tiersten, 1963). The outlined analysis, comparison and discussions are purely theoretical, and aim to collect and classify valuable information regarding the nature of continuous as well as weak discontinuity solutions of relevant well-posed boundary value problems. Emphasis is given on the fact that the compared pair of theoretical models has a common theoretical background (Cosserat and Cosserat, 1909) but different kinds of origin. Some new concepts and features, introduced in Part I in association with linear elastic behaviour of materials having embedded fibres resistant in bending, are thus shown relevant to more general linearly elastic, anisotropic, Cosserat-type material behaviour. The different routes followed for the origination of the compared pair of models is known to produce identical results in the case of conventional (non-polar) linear elasticity. The same is here found generally not true in the polar elasticity case, although considerable similarities are also observed. No definite answers are provided regarding the manner in which existing differences might be bridged or, if at all possible, eliminated. These are matters that require further study and thorough investigation.

Stress-controlled carbon diffusion channeling in bct-iron: A mean-field theory

Diffusion of interstitial carbon atoms in iron is the rate-limiting phenomenon of a number of phase transitions in body-centered (bcc) and body-centered tetragonal (bct) phases such as ferrite and martensite. These phases being rarely stress-free and undeformed, the influence of stress/strain on the diffusivity of carbon is essential, although scarcely documented. We developed a model of carbon elastodiffusion in bct-iron. We combined anisotropic linear elasticity theory of point defects, the dilute approximation of regular solutions and the multisite model of random walk into a coherent mean-field theory. The model allows predicting the effects of composition, temperature and mechanical loading on the anisotropy of carbon diffusion. Density functional theory calculations have provided most of the materials parameters. The predictions were successfully tested against kinetic Monte Carlo simulations. Our results show that compression of the crystal increases carbon diffusivity, while tension has the opposite effect. Axial straining is accompanied by a large anisotropy of diffusion. This effect could be exploited to produce stress-controlled diffusion channeling for the engineering of anisotropic microstructures during thermal ageing of martensitic Fe-C alloys.

Elastic Interaction between Dislocation and Interface in Elastically Anisotropic Ceramic Bimaterials Al<sub>2</sub>O<sub>3</sub>-AlN: Image Force Effect

The Al2O3 and AlN materials are often used as electrical insulators (electronic substrates) also the case of pressure sensors where the aluminum nitride (AlN) is selected as the piezoelectric layer and the alumina (Al2O3) as a solid substrate insulating. This study aims to investigate the mobility of dislocations near the heterophase interface of bimaterials based alumina (Al2O3) under the effect of the image force, without the effect of temperature, deformation and external stress These dislocations having a Burgers vector b = 1/3 , they are located in Al2O3. The interface is defined by its plane parallel to the dislocation line and disorientation varies between 0 and 180。 around the axis . Usually, the image force calculated in the context of the anisotropic linear elasticity using Barnett and Lothe theory with Stroh formalism, Fi = ΔE / d, where ΔE is the elastic interaction energy. The results show that dislocation motion under the image force effect depends on the elastic and crystallographic properties of the materials constituting the bicrystals and even disorientation of the interface which has an effect on the intensity of the elastic interaction energy. The dislocations are repelled to the interface if the difference in shear modulus between the two materials is positive (Δμ= μ2 − μ1>0), they are attracted to the interface in the opposite case (Δμ= μ2 − μ1<0).