Elastic study on singularities interacting with interfaces using alternating technique: Part I. Anisotropic trimaterial

Schwarz–Neumann's alternating technique is applied to singularity problems in an anisotropic `trimaterial', which denotes an infinite body composed of three dissimilar materials bonded along two parallel interfaces. Linear elastic materials under general plane deformations are assumed, in which the plane of deformation is perpendicular to the two parallel interface planes.It is well known that if the solution is known for singularities in a homogeneous anisotropic medium, the solution for the same singularities in an anisotropic bimaterial can be constructed by the method of analytic continuation. It is shown here that the solution for singularities in a homogeneous medium may also be used as a base of the solution for the same singularities in a trimaterial. The alternating technique is applied to derive the trimaterial solution in a series form, whose convergence is guaranteed. The solution procedure is universal in the sense that no specific information about the singularity is needed. The energetic forces exerted on a dislocation due to interfaces are also evaluated from the trimaterial solution. The trimaterial solution studied here can be applied to a variety of problems, e.g. a bimaterial (including a half-plane problem), a finite thin film on semi-infinite substrate, and a finite strip of thin film, etc. Some examples are presented to verify the usefulness of the obtained solutions.     

Corner stress singularities in an FGM thin plate

Describing the behaviors of stress singularities correctly is essential for obtaining accurate numerical solutions of complicated problems with stress singularities. This analysis derives asymptotic solutions for functionally graded material (FGM) thin plates with geometrically induced stress singularities. The classical thin plate theory is used to establish the equilibrium equations for FGM thin plates. It is assumed that the Young’s modulus varies along the thickness and Poisson’s ratio is constant. The eigenfunction expansion method is employed to the equilibrium equations in terms of displacement components for an asymptotic analysis in the vicinity of a sharp corner. The characteristic equations for determining the stress singularity order at the corner vertex and the corresponding corner functions are explicitly given for different combinations of boundary conditions along the radial edges forming the sharp corner. The non-homogeneous elasticity properties are present only in the characteristic equations corresponding to boundary conditions involving simple support. Finally, the effects of material non-homogeneity following a power law on the stress singularity orders are thoroughly examined by showing the minimum real values of the roots of the characteristic equations varying with the material properties and vertex angle.     

Strain-hardening extrusion—I. Construction of slip-line fields from experimental flow patterns

A possible method of solving problems in strain-hardening flows is by perturbation of known perfectly-plastic solutions. It is shown that vertex singularities, which are possessed by most such solutions, are not admissible in steady hardening flows. The structure of regular local solutions, both perfectly-plastic and strain-hardening, is investigated, and it is shown how vertex singularities can be replaced by regular local corner solutions.A scheme for constructing strain-hardening slip-line fields based on experimental flow patterns is described, which uses the maximum shear strain-rate directions calculated from the digitized and smoothed flow pattern to perturb local Hencky-Prandtl nets, which are then patched together in conformity with the topology of the solution to form a complete slip-line field. This method has been implemented in a computer program to construct slip-line fields from flow patterns for extrusion through wedge-shaped dies, and some fields computed by the program are presented.     

Elastic study on singularities interacting with interfaces using alternating technique: Part II. Isotropic trimaterial

Singularity problems in an isotropic trimaterial are analyzed by the same procedure as in an anisotropic trimaterial of Part I [Int. J. Solids Struct. 39, 943–957]. `Trimaterial' denotes an infinite body composed of three dissimilar materials bonded along two parallel interfaces. Linear elastic isotropic materials under plane deformations are assumed, in which the plane of deformation is perpendicular to the two parallel interface planes, and thus Muskhelishvili's complex potentials are used. The method of analytic continuation is alternatively applied across the two parallel interfaces in order to derive the trimaterial solution in a series form from the corresponding homogeneous solution. A variety of problems, e.g. a bimaterial (including a half-plane problem), a finite thin film on semi-infinite substrate, and a finite strip of thin film, etc, can be analyzed as special cases of the present study. A film/substrate structure with a dislocation is exemplified to verify the usefulness of the solutions obtained.     

Singularities of elastic stresses and of harmonic functions at conical notches or inclusions

The nature of singularities at the vertex of conical notches and inclusions is found for problems of potential theory and for elastostatic problems of torsion and of axisymmetric stress. A solution in terms of spherical harmonics and a general numerical solution based upon the field equations are used to determine the power dependence of the field quantities upon the distance from the apex of the cone. Eigenvalues representing the exponent are computed for various values of cone angle and for various Poisson ratios.     

On the influence of cohesive stress-separation laws on elastic stress singularities

The nature of the stress field occurring at the vertex of an angular elastic plate under in-plane loading is reconsidered. An additional boundary condition is introduced. This boundary condition reflects the action of cohesive stress-separation laws. Companion asymptotic analysis proceeds routinely on employing coupled eigenfunction expansions. Results show that a number of configurations that had previously contained stress singularities become singularity free.     

Stress singularities resulting from various boundary conditions in angular corners of plates of arbitrary thickness in extension

The stress singularities in angular corners of plates of arbitrary thickness with various boundary conditions subjected to in-plane loading are studied within the first-order plate theory. By adapting an eigenfunction expansion approach a set of characteristic equations for determining the structure and orders of singularities of the stress resultants in the vicinity of the vertex is developed. The characteristic equations derived in this paper incorporate that obtained within the classical plane theory of elasticity (M.L. Williams’ solution) and also describe the possible singular behaviour of the out-of-plane shear stress resultants induced by various boundary conditions.     

Stress analysis of an elastic cracked layer bonded to a viscoelastic substrate

The effect of a viscoelastic substrate on an elastic cracked layer under an in-plane concentrated load is solved and discussed in this study. Based on a correspondence principle, the viscoelastic solution is directly obtained from the corresponding elastic one. The elastic solution in an anisotropic trimaterial is solved as a rapidly convergent series in terms of complex potentials via the successive iterations of the alternating technique in order to satisfy the continuity condition along the interfaces between dissimilar media. This trimaterial solution is then applied to a problem of a thin layer bonded to a half-plane substrate. Using the standard solid model to formulate the viscoelastic constitutive equation, the real-time stress intensity factors can be directly obtained by performing the numerical calculations. The results obtained in this paper are useful in studying the problem with bone defects where a crack is assumed to exist in an elastic body made of the cortical bone that is bonded to a viscoelastic substrate made of the cancellous bone.     

Stress fields and Eshelby forces on half-plane inhomogeneities with eigenstrains and strip inclusions meeting a free surface

The stress field due to a half-plane inhomogeneity with plane eigenstrain is obtained by a limiting procedure from the one of a circular Eshelby inhomogeneity/inclusion. This field, which requires tractions to be applied at infinity to be sustained, has minimum strain energy versus any other superposed homogeneous one, and is the Eshelby solution inside plus the Hill jump conditions. By superposition, the stresses due to an infinite strip (Eshelby property domain) inhomogeneity with eigenstrain are obtained, and, by superposition periodic strips or laminates can be obtained. By cancelling the stresses on a free-surface, strips of inclusions meeting a free surface are solved. They exhibit tensile stresses under the free surface, and logarithmic singularities in the tensile stress at the vertex, which may initiate cracking. The Eshelby self-forces on the boundary of circular and half-plane inhomogeneities are computed.     

Localization,delocalization and compression fracture in externally pressurized thick cross-ply (very) long cylindrical shells with material non-linearity: A multi-scale and multi-physics analysis

Compression fracture in carbon fiber reinforced plastics (CFRP) involves multiple physical mechanisms operating at multiple scales ranging from angströms to cms and beyond. First, at the macro/meso-scale, combined effects of modal imperfections, transverse shear/normal deformation along with the non-linear hypo-elastic transverse shear (G_TT) material property on the emergence of interlaminar shear crippling type instability modes, related to the localization (onset of deformation softening), delocalization (onset of deformation hardening) and propagation of mode II compression fracture/damage, in thick imperfect cross-ply very long cylindrical shells under applied hydrostatic pressure, are investigated. The primary accomplishment is the (hitherto unavailable) computation of the layer-wise mode II stress intensity factor, energy release rate and kink crack band-width, under hydrostatic compression, from a non-linear finite element analysis (FEA), using Maxwell’s construction and Griffith׳s energy balance approach. Numerical results include the effects of hypoelastic (G_TT only) material property, on localization and delocalization leading to compression fracture.At the micro-scale, a novel three-dimensional eigenfunction expansion technique, based in part on separation of the cylinder length-variable and partly utilizing a modified Frobenius type series expansion in conjunction with an affine transformation to compute the local stress singularity, in the vicinity of a kinked-fiber/matrix trimaterial junction front. Such computed stress singularities represent a measure of the degree of inherent flaw sensitivity of unidirectional CFRP under compression. Finally, dislocation glide in graphite crystallites plays a dominant role in kink band nucleation and propagation at the nano-meter scale.     

A Mathematical Analogy and a Unified Asymptotic Formulation for Singular Elastic and Electromagnetic Fields at Multimaterial Wedges

In the present contribution, the mathematical analogy existing between the singular stress field in elasticity due to antiplane loading and the singular electromagnetic fields in electromagnetism is derived with reference to the problem of isotropic multimaterial wedges. These configurations, where dissimilar sectors converge to the same vertex, are commonly observed in composite materials and may lead to singularities. The proposed analogy permits to extend several elastic solutions for the power of the stress-singularity already available in the elasticity literature to the analogous electromagnetic problems and viceversa. Finally, electromagnetic structures that cannot be treated according to the proposed analogy, such as those related to bi-isotropic multimaterial wedges, are discussed.     

Thermal stresses in an isotropic trimaterial interacted with a pair of point heat source and heat sink

A general analytical solution for an isotropic trimaterial interacted with a point heat source is provided in this paper. Based on the method of analytical continuation in conjunction with the alternating technique, the solutions to heat conduction and thermoelasticity problems for three dissimilar media are first derived. A rapidly convergent series solution for both the temperature and stress functions, which is expressed in terms of an explicit general term of the complex potential of the corresponding homogeneous problem, is obtained in an elegant form. As a numerical illustration, the distributions of thermal stresses along the interface are presented for various material combinations and for different positions of the applied heat source and heat sink.     

Three-dimensional Green's functions in anisotropic trimaterials

Three-dimensional elastostatic Green's functions in anisotropic trimaterials are derived, for the first time, by applying the generalized Stroh's formalism and Fourier transforms. The Green's functions are expressed as a series summation with the first term corresponding to the full-space solution and other terms to the image solutions due to the interfaces. The most remarkable feature of the present solution is that the image solutions can be expressed by a simple line integral over a finite interval [0,2π]. By partitioning the trimaterial Green's function into a full-space solution and a complementary part, the line integral involves only regular functions if the singularity is within one of the three materials, being treated analytically owning to the explicit expression of the full-space solution. When the singularity is on the interface, which occurs if the field and source points are both on the same interface, the involved singularity is handled with the interfacial Green's functions.A numerical example is presented for a trimaterial system made of two anisotropic half spaces bonded perfectly by an isotropic adhesive layer, showing clearly the effect of material layering on the Green's displacements and stresses. Furthermore, by comparing the present Green's solution to the direct (two-dimensional) 2D integral expression which is also derived in this paper, it is shown that, the computational time for the calculation of the Green's function can be substantially reduced using the present solution, instead of the direct 2D integral method.     

纤维端部的界面裂纹分析

基于弹性力学空间轴对称问题的通解,研究了短纤维增强复合材料中纤维端部的轴对称币形和柱形界面裂纹尖端的应力奇异性,得到了裂纹尖端附近的奇异应力场．研究结果表明,这两种轴对称界面裂纹尖端的应力奇异性相同,并且与平面应变状态下相应模型的应力奇异性一致,材料性能对裂纹尖端附近奇异应力场的影响可用三个组合参数描述     

The Physical Fundamentals of the Ultrasonic Nondestructive Stress Analysis of Solids

Theoretical and experimental works on acoustoelasticity are briefly generalized. Studies conducted and scientific results obtained at the S. P. Timoshenko Institute of Mechanics and E. O. Paton Institute of Electric Welding of the National Academy of Sciences of Ukraine are highlighted. Special features of these works and their difference from those of other authors are pointed out. The basic principles and laws governing the propagation of longitudinal, shear, and surface waves in bi- and triaxially stressed bodies are briefly stated with regard for the orthotropy and nonlinear properties of the material. The experimentally proven principles and laws for elastic waves propagating in initially stressed bodies are formulated. The physical fundamentals of the ultrasonic nondestructive technique for determining bi- and triaxial stresses in solids are described. The determination of bi- and triaxial residual stresses in specimens and structural members is demonstrated by examples. The basic principles of the related (dielectric and electromagnetic) methods for stress analysis of polymeric materials are stated. The application of the electromagnetic method to the stress analysis of some polymeric materials is considered     

Hamiltonian principle based stress singularity analysis near crack corners of multi-material junctions

This paper presents a new method for the stress singularity analysis near the crack corners of a multi-material junctions. The stress singularities near the crack corners of multi-dissimilar isotropic elastic material junctions are studied analytically in terms of the methods developed in Hamiltonian system. The governing equations of plane elasticity in a sectorial domain are derived in Hamiltonian form via variable substitution and variational principle respectively. Both of the methods of global state variable separation and symplectic eigenfunction expansion are used to find the analytical solution of the problem. The relationships among the state vectors in different material spaces are obtained by means of coordinate transformation and consistent conditions between the two adjacent domains. The expression of the original problem is thus changed into a new form where the solutions of symplectic generalized eigenvalues and eigenvectors are needed. The closed form of expressions is established for the stress singularity analysis near the corner with arbitrary vertex angles. Numerical results are presented with several chosen angles and multi-material constants. To show the potential of the new method proposed, a semi-analytical finite element is furthermore developed for the numerical analysis of crack problems.     

Analysis of the stress singularity field at a vertex in 3D-bonded structures having a slanted side surface

The stress singularity that occurs at a vertex in a joint with a slanted side surface is investigated. The orders of stress singularity at a vertex and at a point on stress singularity lines for various material properties are determined using eigenanalysis. The stress distribution on an interface and the intensity of stress singularity at the vertex are investigated using BEM. It is shown that the order of stress singularity at the vertex in the joints can be reduced by slanting a side surface so as to decrease the angle between the interface and the side surface. The results of BEM analysis reveal that the distribution of stress on the interface is influenced by the slanted side surface. Finally, the 3D intensities of the singularity for stress components which are continuous at the interface are newly defined and determined for various material combinations.