Stroh Formalism For Anisotropic Elasticity Research Articles

A semi-analytical matrix formalism for stress singularities in anisotropic multi-material corners with frictional boundary and interface conditions

A new general semi-analytic procedure for the characterisation of singular asymptotic elastic states in the vicinity of the apex of linearly elastic anisotropic multi-material corners, including frictional contact, is developed and tested. The corners can consist of any finite number of homogeneous wedges defined by polar sectors. The variability of configurations covered is enormous as frictional contact can be considered on one or more outer boundary surfaces or interfaces, in addition to a large variety of homogeneous boundary conditions and perfect bonding or frictionless sliding interface conditions between wedges in the corner. The Coulomb rate-independent and dry frictional contact law is assumed. One of the novelties is that, in addition to the singularity exponent λ, the angle ω of the friction tangential stress vector on each frictional contact surface is an a priory unknown to be determined by solving a nonlinear corner eigensystem. The procedure, which considers power-law stress singularities, is based on the Stroh formalism of anisotropic elasticity, assuming generalised plane strain (2.5D) conditions, and on the semi-analytic matrix formalism for wedge transfer-matrices and boundary and interface condition matrices. This makes it, firstly, very suitable for computational implementations, secondly, very efficient especially in cases with several perfectly bonded homogeneous wedges, and, thirdly, very accurate due to its fully semi-analytic nature. The code developed is tested by solving a large variety of examples, comparing the present results with those obtained by solving closed-form corner-eigenequations deduced by previous authors for specific useful practical configurations, confirming the extremely high accuracy of the present code in the computation of λ and ω. The differences observed in some cases with anisotropic materials are explained by the fact that some of the previous authors did not take the true 3D Coulomb friction law into account.

Transient thermal stress analysis of temperature-dependent anisotropic viscoelastic solids

Consider the transient thermal stress analysis for a material that is anisotropic (direction-dependent), viscoelastic (time-dependent), and thermorheologically simple (temperature-dependent). To solve this problem, three fundamental theories are employed: (1) complex variable Stroh formalism for anisotropic elasticity, (2) elastic-viscoelastic correspondence principle (EVCP) or time-stepping method (TSM) for viscoelasticity, and (3) time-temperature superposition principle (TTSP) for thermorheologically simple materials. By using the Stroh formalism, all different kinds of anisotropic elastic materials can be analyzed by using the same formulae. With EVCP and TSM, the problem of viscoelasticity can be solved via its corresponding problem of elasticity. By TTSP, the temperature-dependent constitutive relation can be expressed by using the reference temperature and the reduced time defined through shift factor. Thus, through the combined use of these theories, the complicated problems of transient anisotropic thermo-viscoelasticity can be solved by using the available solution tool for anisotropic elasticity such as the boundary element method (BEM). Based upon this concept, two novel BEMs are developed for the transient thermal stress analysis of temperature-dependent anisotropic viscoelastic solids. One is BEM-EVCP, and the other is BEM-TSM. The final results show that both of them can provide accurate solutions, and BEM-TSM is much more efficient than BEM-EVCP.

Proof of the Strong Eshelby Conjecture for Plane and Anti-plane Anisotropic Inclusion Problems

Based on the Stroh formalism for anisotropic elasticity and the complex variable function method, we prove in this paper that the strong Eshelby conjecture holds for simply-connected anisotropic inclusion problems under plane or anti-plane deformation. The interfaces can be either perfect or dislocation-like. For these inclusion problems, if the induced stress field inside the inclusion is uniform for a single uniform eigenstrain, the inclusion is of the elliptic shape. Thanks to the generality of the proof method, we obtain also alternative proofs of the strong Eshelby conjecture for isotropic inclusion problems, which are given in the Appendix.

Computing stress singularities in transversely isotropic multimaterial corners by means of explicit expressions of the orthonormalized Stroh-eigenvectors

Composite materials reinforced by unidirectional long fibers behave macroscopically as homogeneous transversely isotropic linear elastic materials. A general, accurate and computationally efficient procedure for the evaluation of singularity exponents and singular functions characterizing singular stress fields in multimaterial corners involving this kind of material is presented in this paper. To take full advantage of the sextic Stroh formalism of anisotropic elasticity applied to this particular problem, the complete set of explicit expressions of the eigenvalues and eigenvectors of the real 6 × 6 fundamental elasticity matrix N has been deduced for all the non-degenerate and degenerate (repeated roots of the sextic Stroh equation) cases. These expressions will also facilitate further applications of the Stroh formalism to these materials. Several numerical examples of singularity analysis of multimaterial corners appearing in adhesively bonded joints and damaged cross-ply laminates of composite materials are presented.

Electromechanical fracture analysis for corners and cracks in piezoelectric materials

Corners and cracks are usually studied separately in the literature. To build a bridge connecting these two different but similar topics, in this paper the solutions for piezoelectric multi-wedges, which cover corners and interface corners, are used to study the cracks and interface cracks in piezoelectric materials. Moreover, the stress/electric intensity factors defined for cracks, interface cracks and interface corners are also extended to the general corners. By taking the special feature of Stroh formalism for anisotropic elasticity, all the solutions presented in this paper for piezoelectric materials preserve the same matrix form as those of the corresponding anisotropic problems. To see more clearly about the piezoeffects on the corners and cracks, most of the complex matrix form solutions are expanded in real component form for two typical piezoelectric ceramics with different poling directions.

Micro-plasticity of surface steps under adhesive contact: Part I—Surface yielding controlled by single-dislocation nucleation

Mechanics of nano- and meso-scale contacts of rough surfaces is of fundamental importance in understanding deformation and failure mechanisms of a solid surface, and in engineering fabrication and reliability of small surface structures. We present a micro-mechanical dislocation model of contact-induced deformation of a surface step or ledge, as a unit process model to construct a meso-scale model of plastic deformations near and at a rough surface. This paper (Part I) considers onset of contact-induced surface yielding controlled by single-dislocation nucleation from a surface step. The Stroh formalism of anisotropic elasticity and conservation integrals are used to evaluate the driving force on the dislocation. The driving force together with a dislocation nucleation criterion is used to construct a contact-strength map of a surface step in terms of contact pressure, step height, surface adhesion and lattice resistance. Atomistic simulations of atomic surface-step indentation on a gold (1 0 0) surface have been also carried out with the embedded atom method. As predicted by the continuum dislocation model, the atomistic simulations also indicate that surface adhesion plays a significant role in dislocation nucleation processes. Instabilities due to adhesion and dislocation nucleation are evident. The atomistic simulation is used to calibrate the continuum dislocation nucleation criterion, while the continuum dislocation modeling captures the dislocation energetics in the inhomogeneous stress field of the surface-step under contact loading. Results show that dislocations in certain slip planes can be easily nucleated but will stay in equilibrium positions very close to the surface step, while dislocations in some other slip planes easily move away from the surface into the bulk. This phenomenon is called contact-induced near-surface dislocation segregation. As a consequence, we predict the existence of a thin tensile-stress sub-layer adjacent to the surface within the boundary layer of near-surface plastic deformation. In the companion paper (Part II), we analyze the surface hardening behavior caused by multiple dislocations.

Singularity analysis of anisotropic multimaterial corners

Singular stress states induced at the tip of linear elastic multimaterial corners are characterized in terms of the order of stress singularities and angular variation of stresses and displacements. Linear elastic materials of an arbitrary nature are considered, namely anisotropic, orthotropic, transversely isotropic, isotropic, etc. Thus, in terms of Stroh formalism of anisotropic elasticity, the scope of the present work includes mathematically non-degenerate and degenerate materials. Multimaterial corners composed of materials of different nature are typically present at any metal-composite, or composite-composite adhesive joint. Several works are available in the literature dealing with a singularity analysis of multimaterial corners but involving (in the vast majority) only materials of the same nature (e.g., either isotropic or orthotropic). Although many different corner configurations have been studied in literature, with almost any kind of boundary conditions, there is an obvious lack of a general procedure for the singularity characterization of multimaterial corners without any limitation in the nature of the materials. With the procedure developed here, and implemented in a computer code, multimaterial corners, with no limitation in the nature of the materials and any homogeneous orthogonal boundary conditions, could be analyzed. As a particular case, stress singularity orders in corners involving extraordinary degenerate materials are, to the authors’ knowledge, presented for the first time. The present work is based on an original idea by Ting (1997) in which an efficient procedure for a singularity analysis of anisotropic non-degenerate multimaterial corners is introduced by means of the use of a transfer matrix.

Analysis for Elastic Strips Under Concentrated Loads

The problem of a general anisotropic elastic strip under concentrated loads is analyzed. By using the Stroh formalism for anisotropic elasticity in conjunction with the Fourier transform, the elastic fields for a concentrated load are expressed as integrals which can be evaluated by residue theory to give eigenfunction expansions. The eigenfunction expansions can be divided into two parts. The first part arises from the singularities at the origin of the integrands of the Fourier integrals and corresponds to classical beam bending and stretching solutions. The second part consists of the other terms in the eigenfunction expansions and corresponds to exponentially decaying self-equilibrium solutions. The solution for a concentrated load is used to investigate the problems of a pair of collinear compressive loads and three-point bending.

Integral kernels in the 2D Somigliana displacement and stress identities for anisotropic materials

The theory of the direct and indirect formulations of the Somigliana displacement and stress identities for anisotropic materials subjected to generalized plane strain deformations is revised and rounded off. By making use of the powerful Stroh formalism of anisotropic elasticity, explicit and compact formulae of all integral kernels appearing in these identities are derived and relations between these kernels are analysed. Somigliana stress identity is derived directly from the Betti theorem of reciprocity of work using the singularity source of a dislocation dipole (i.e. a kind of eigenstrain nucleus or concentrated initial strain). Explicit formula of the hypersingular integral kernel in the Somigliana stress identity reveals that this kernel is fully symmetrical with respect to index permutation when only in-plane components are considered. Finally, some comments on the applications of the results presented in the stress analysis of anisotropic materials by collocation BEM and Symmetric Galerkin BEM are presented.

Conservation laws and canonical forms in the Stroh formalism of anisotropic elasticity

We find and classify all first-order conservation laws in the Stroh formalism. All possible non-semisimple degeneracies are considered. The laws are found to depend on three arbitrary analytic functions. In some instances, there is an “extra” law which is quadratic in ∇ u \nabla u . Separable and inseparable canonical forms for the stored energy function are given for each type of degeneracy and they are used to compute the conservation laws. The existence of a real Stroh eigenvector is found to be a necessary and sufficient condition for separability. The laws themselves are stated in terms of the Stroh eigenvectors.

The elastic fields of a dislocation in ananisotropic strip

The elastic fields of a dislocation in a general anisotropic strip are analysed. The analysis is made by using the Stroh formalism for anisotropic elasticity in conjunction with the Fourier transform. The surfaces of the strip are considered either free or fixed. The image forces on an edge dislocation in tungsten and potassium are computed as functions of the position of the dislocation and the inclination of the slip planes. The interactive forces between two like dislocations on the same slip plane are also calculated.

Delamination cracks originating from transverse cracking in cross-ply laminates under various loadings

Delamination cracks originating from transverse cracking in two types of cross-ply laminates [90/0]s and [0/90]s are analysed for three types of loadings: plane strain extension, plane strain bending and antiplane shear. The asymptotic solution for stress and displacement field is constructed based upon the Stroh formalism for anisotropic elasticity and the method of eigenfunction expansion. Its structure is determined from appropriate near-field conditions leading to the eigenvalue equation, and then the singular hybrid finite element method in conjunction with a quadratic programming procedure for treating the partial contact of crack faces is employed to complete the field solution. The stress intensity factor properly defined for anisotropic interfacial cracks and the energy release rate are computed from the complete solution. The stability of crack growth and the effect of the relative ply thickness upon the crack growth stability are examined in terms of the energy release rate and the phase angle.

Generalized stroh formalism for anisotropic elasticity for general boundary conditions

The Stroh formalism is most elegant when the boundary conditions are simple, namely, they are prescribed in terms of traction or displacement. For mixed boundary conditions such as there for a slippery boundary, the concise matrix expressions of the Stroh formalism are destroyed. We present a generalized Stroh formalism which is applicable to a class of general boundary conditions. The general boundary conditions include the simple and slippery boundary conditions as special cases. For Green's functions for the half space, the general solution is applicable to the case when the surface of the half-space is a fixed, a free, a slippery, or other more general boundary. For the Griffith crack in the infinite space, the crack can be a slit-like crack with free surfaces, a rigid line inclusion (which is sometimes called an anticrack), or a rigid line with slippery surface or with other general surface conditions. It is worth mention that the modifications required on the Stroh formalism are minor. The generalized formalism and the final solutions look very similar to those of unmodified version. Yet the results are applicable to a rather wide range of boundary conditions.

Some identities and the structure of 𝑁ᵢ in the Stroh formalism of anisotropic elasticity

The Stroh formalism of anisotropic elasticity leads to a 6 × 6 6 \times 6 real matrix N that can be composed from three 3 × 3 3 \times 3 real matrices N i ( i = 1 , 2 , 3 ) {N_i} \left ( {i = 1, 2, 3} \right ) . The eigenvalues and eigenvectors of N are all complex. New identities are derived that express certain combinations of the eigenvalues and eigenvectors in terms of the real matrices N i {N_i} and the three real matrices H, S, L introduced by Barnett and Lothe. It is shown that the elements of N 1 {N_1} and N 3 {N_3} have simple expressions in terms of the reduced elastic compliances. We prove that − N 3 - {N_3} is positive semidefinite and, with this property, we present a direct proof that L is positive definite.