Singularity of dynamic stress fields around an interface crack between viscoelastic bodies

The singular nature of the dynamic stress fields around an interface crack located between two dissimilar isotropic linearly viscoelastic bodies is studied. A harmonic load is imposed on the surfaces of the interface crack. The dynamic stress fields around the crack are obtained by solving a set of simultaneous singular integral equations in terms of the normal and tangent crack dislocation densities. The singularity of the dynamic stress fields near the crack tips is embodied in the fundamental solutions of the singular integral equations. The investigation of the fundamental solutions indicates that the singularity and oscillation indices of the stress fields are both dependent upon the material constants and the frequency of the harmonic load. This observation is different from the well-known −1/2 oscillating singularity for elastic bi-materials. The explanation for the differences between viscoelastic and elastic bi-materials can be given by the additional viscosity mismatch in the case of viscoelastic bi-materials. As an example, the standard linear solid model of a viscoelastic material is used. The effects of the frequency and the material constants (short-term modulus, long-term modulus and relaxation time) on the singularity and the oscillation indices are studied numerically.     

Stress singularities in bimaterial bodies of revolution

An eigenfunction expansion solution is first developed for determining the stress singularities of bimaterial bodies of revolution by directly solving the equilibrium equations of three-dimensional elasticity in terms of displacement functions. The characteristic equations are explicitly given for determining the stress singularities in the vicinity of the interface corner of two intersecting bodies of revolution having a sharp corner with free boundary conditions along the corner. The characteristic equations are found to be equivalent to a combination of the characteristic equations for plane elasticity problems and St. Venant torsion problems. The strength of stress singularities varying with the interface angles is also investigated. The first known asymptotic solutions for the displacement and stress fields are also explicitly shown for some interface angles. The present results will be useful not only for understanding the singularity behaviors of stresses in the vicinity of a revolution interface corner, but also for developing accurate numerical solutions with fast convergence for stress or vibration analysis of a body of revolution having an interface corner.     

Analytical solution for the singular stress distribution due to V-notch in an orthotropic plate material

On the basis of general solutions of two-dimensional linear elasticity, displacement and singular stress fields near the singular point in orthotropic materials are derived in closed form expressions. According to the presented expressions, analysis formulas of displacement and singular stress fields near the tip of a V-notch under the symmetric and the anti-symmetric modes are obtained subsequently. The open literatures devoted to developing stress singularity near the tip of the V-notch in anisotropic or orthotropic materials. In this study, however, not only direct eigenequations were derived, but also the explicit solutions of displacement and singular stress fields were obtained. At the end, the correctness of the formulas of the singular stress field near the tip of the V-notch has been verified by FEM analysis.     

Plane Elastic Problem of a Crack Normal to and Terminating at Bimaterial Interface of Isotropic and Orthotropic Half Plates

In this paper, the displacement and stress fields for a crack normal to and terminating at a bimaterial interface of isotropic and orthotropic half planes are studied as a plane problem. The eigenequation, by which the order of stress singularity is determined, is given in an explicit form. A discriminant function is presented to judge whether the stress singularity at the crack tip is greater than -1/2 or not. An explicit closed form expression is derived for the displacement and stress distribution near the crack tip.     

Elastodynamic analysis of the Finite punch and finite crack problems in orthotropic materials

The transient elastodynamic response of the finite punch and finite crack problems in orthotropic materials is examined. Solution for the stress intensity factor history around the punch corner and crack tip is found. Laplace and Fourier transforms together with the Wiener–Hopf technique are employed to solve the equations of motion in terms of displacements. A detailed analysis is made in the simplified case when a flat rigid punch indents an elastic orthotropic half-plane, the punch approaches with a constant velocity normally to the boundary of the half-plane. An asymptotic expression for the singular stress near the punch corner is analyzed leading to an explicit expression for the dynamic stress intensity factor which is valid for the time the dilatational wave takes to travel twice the punch width. In the crack problem, a finite crack is considered in an infinite orthotropic plane. The crack faces are loaded by impact uniform pressure in mode I. An expression for the dynamic stress intensity factor is found which is valid while the dilatational wave travels the crack length twice. Results for orthotropic materials are shown to converge to known solutions for isotropic materials derived independently.     

Stress intensities at interface corners in anisotropic bimaterials

Asymptotic stress and displacement fields near the tip of a sharp anisotropic bimaterial interface corner are computed using a combination of the Stroh formalism and the Williams eigenfunction expansion method. From the asymptotic fields, the path independent H-integral is developed and implemented to calculate anisotropic interface corner stress intensities. The calculation procedure is demonstrated for two glass–silicon interface corner configurations that commonly arise in practice in the microsensor industry. In each case, the bimaterial interface corner experiences mixed mode I and II loading, and accurate estimates of both stress intensities are obtained.     

Anti-plane Shear Cracks Approaching a Bi-material Interface

A novel procedure is proposed for evaluation of stress intensity factors of planar Mode III shear cracks perpendicular to a nearby interface between two isotropic elastic solids. Shear cracks traversing a flat layer bonded to two different elastic solids are also analyzed. The method is based on superposition of singular near tip stress and displacement fields generated by both the main crack and certain image cracks. Both the main and the image cracks are loaded by self-equilibrating shear tractions of different magnitude, such that matching parts of the said fields are made to satisfy traction and displacement continuity conditions at the interface. Selected comparisons with results obtained by different methods show good agreement. Applications of the method to other crack problems are discussed.     

Analysis of anisotropic in-plane corners

Singular behavior of the mechanical in-plane fields occurs at a laminate reinforcement patch corner due to the geometry and different material properties in the reinforced and non-reinforced domain, respectively. Adopting Lekhnitskii’s approach of the complex potential method, an asymptotic analysis of the mechanical fields is performed near laminate reinforcement patch corners. The mechanical in-plane fields at the two-dimensionally modeled interface corner can be determined in closed-form manner. Various configurations of interface corners are examined and their effect on the singular characteristics of the cross-sectional force field is studied. It is found that for a characterization of the singular behaviour of the in-plane forces each singular in-plane force term has to be considered and that the corresponding displacement modes are useful for understanding this behaviour.     

Three-dimensional singularities in double lap joints

A 3-dimensional numerical analysis of a double lap joint specimen is presented. It is focused on the stress field on the interface plane where delamination would probably start. Combinations of isotropic and orthotropic layers are studied. It is shown that the most favorable point for delamination onset according to the linear fracture mechanics (LFM) approach lies near the 3-D corner, where all three modes of fracture are present, for all cases checked. The shear energy, which represents the tendency for yielding, is at the corner itself. The singular region for common combinations of adhesives-adherents is so small that regular continuum mechanics tools may be limited in failure predictions. The special features of the 3-D singularity at the comer point is discussed. Limitations of LFM for delamination onset prediction may lead to an extended type of failure criteria which is based on the shear energy state of the adhesive as well as the regular stress intensity factors. Two-dimensional solutions would show higher critical loads for the same problem.     

Screw dislocations in a three-phase anisotropic medium

The displacement and stress fields of a screw dislocation in a three-phase anisotropic medium with orthotropic symmetry are analyzed. Exact expressions are given in series form. In the limiting case where the two outer phases are rigid, the series expressions converge to simple functions which are identical (apart from the elastic constants) to those for an isotropic medium. This equivalence relation facilitates the analysis of dislocation pileups. The inhomogeneity effect on pileups is examined.     

A unified and integrated numerical‐analytical approach for evaluation of Jk‐integrals of an interfacial crack in orthotropic and isotropic bimaterials

A new unified and integrated method for the numerical‐analytical calculation of J_k‐integrals of an in‐plane traction free interfacial crack in homogeneous orthotropic and isotropic bimaterials is presented. The numerical algorithm, based on the boundary element crack shape sensitivities, is generic and flexible. It applies to both straight and curved interfacial cracks in anisotropic and/or isotropic bimaterials. The shape functions of semidiscontinuous quadratic quarter point crack tip elements are correctly scaled to adapt the singular oscillatory near tip field of tractions. The length of crack is designated as the design variable to compute the strain energy release rate precisely. Although an analytical equation relating J₁ and stress intensity factors (SIFs) exists, a similar relation for J₂ in debonded anisotropic solids for decoupling SIFs is not available. An analytical expression was recently derived by this author for J₂ in aligned orthotropic/orthotropic bimaterials with a straight interface crack. Using this new relation and the present computed J_k values, the SIFs can be decoupled without the need for an auxiliary equation. Here, the aforementioned analytical relation is reconstructed for cubic symmetry/isotropic bimaterials and used with the present numerical algorithm. An example with known analytical SIFs is presented. The numerical and analytical magnitudes of J_k for an interface crack in orthotropic/orthotropic and cubic symmetry/isotropic bimaterials show an excellent agreement.     

Finite element evaluation of free-edge singular stress fields in anisotropic materials

A finite element formulation based on the work of Yamada and Okumura¹⁴ is presented to determine the order of singularity and angular variation of the stress and displacement fields surrounding a singular point on a free edge of anisotropic materials. Emphasis is placed on the computational aspects of this method when applied to configurations including fully bonded multi-material junctions intersecting a free edge as well as materials containing cracks intersecting a free edge. The study shows that the singularity of the three-dimensional stress field may be accurately determined with a relatively small number of elements only when a proper level of numerical integration is used. The method is applied to isotropic and orthotropic materials with a crack intersecting a free edge and an anisotropic three-material junction intersecting a free edge. The efficiency and accuracy of the method indicates it could be used to develop a numerical solution for the singular field that could in turn be used to create free-edge enriched finite elements.     

Stress intensity factor analysis at an interfacial corner between anisotropic bimaterials under thermal stress

A numerical method using a path-independent H-integral based on the Betti reciprocal principle was developed to analyze the stress intensity factors of an interfacial corner between anisotropic bimaterials under thermal stress. According to the theory of linear elasticity, asymptotic stress near the tip of a sharp interfacial corner is generally singular as a result of a mismatch of the materials’ elastic constants. The eigenvalues and the eigenfunctions are obtained using the Williams eigenfunction method, which depends on the materials’ properties and the geometry of an interfacial corner. The order of the singularity related to the eigenvalue is real, complex or power-logarithmic. The amplitudes of the singular stress terms can be calculated using the H-integral. The stress and displacement fields around an interfacial corner for the H-integral are obtained using finite element analysis. A proposed definition of the stress intensity factors of an interfacial corner involves a smooth expansion of the stress intensity factors of an interfacial crack between dissimilar materials. The asymptotic solutions of stress and displacement around an interfacial corner are uniquely obtained using these stress intensity factors.     

Interfacial crack tip field in anisotropic/isotropic bimaterials

The co-cured joint is more efficient than the adhesively bonded joint for composite structures because of its several advantages. However, failure analysis of the co-cured joint has just a little been reported since the co-cured joining method was introduced. Observing based on the experimental results, failure starts at the interface corner of the co-cured joint. Therefore, it is important to consider stress intensities at the interface corner of the co-cured joint.

In this paper, the eigenvalue problem was used to determine asymptotic stress and displacement fields near the interface corner between composite and steel adherends. For obtaining stress intensities at the interface corner a path independent conservative line integral was used.     

Edge effect of a carbon fiber meeting a surface

This paper investigates the free edge effect on the stress field of a carbon fiber, which is embedded into an epoxy matrix. The fiber is assumed to possess cylindrical symmetry and to be transversely isotropic. The matrix is assumed to be of an isotropic material. The stress field is induced by a uniform tension applied on the matrix at points far away from the fiber surface.

The displacement and stress fields are explicitly derived and a stress singularity is shown to prevail. The singularity strength is shown to be a function of the material constants of the fiber as well as those of the matrix. Finally, the displacement and stress profiles are plotted as a function of the angle φ which is measured from the free surface.     

Analytical solution for a functionally graded beam with arbitrary graded material properties

The plane stress problem of an orthotropic functionally graded beam with arbitrary graded material properties along the thickness direction is investigated by the displacement function approach for the first time. A general two-dimensional solution is obtained for a functionally graded beam subjected to normal and shear tractions of arbitrary form on the top and bottom surfaces and under various end boundary conditions. For isotropic case explicit solutions are given to some specific through-the-thickness variations of Young’s modulus such as exponential model, linear model and reciprocal model. The influence of different grade models on the stress and displacement fields are illustrated in numerical examples. These analytical solutions can serve as a basis for establishing simplified theories and evaluating numerical solutions of functionally graded beams.     

T-shaped crack problem for bonded orthotropic layers

The plane elasticity problem of two perfectly bonded orthotropic layers containing cracks perpendicular to and along the interface is considered. Cracks are extended to intersect the boundaries and each other in such a way that a crack configuration suitable to study the T-shaped crack problem is obtained. The problem is reduced to the solution of a system of singular integral equations with Cauchy type singularities. Numerical results for stress intensity factors and energy release rates are presented for various loading conditions and for isotropic as well as orthotropic material pairs. These results indicate that elementary strength of material type calculations for energy release rates provide a good approximation to the actual elasticity solution even for relatively short cracks, as long as the layer thicknesses are not very different.     

Matrix form near tip solutions of interface corners

In this paper, the stresses near the tip of interface corners are expressed in matrix power function form. To make this matrix power function form valid for all possible cases of interface corners, all kinds of singularities including oscillatory and logarithmic singularity are discussed in this paper. The coefficient vector of this matrix power function form solution is then defined as a vector of stress intensity factors along the radial direction. Since the stress intensity factors are functions of radial direction, the maximum stress intensity factor of a certain radial direction may be useful for the fracture prediction of interface corners. This new definition of stress intensity factors is applicable for all possible singular orders??repeated or distinct, real or complex, and keeps the same unit for all possible interface corners. Therefore, it will be helpful for bridging the problems of cracks, corners, interface cracks and interface corners. Finally, the matrix form solutions are reduced to the scalar form solutions for two special cases. One is a special case of corner angles??cracks, and the other is a special case of anisotropic materials??isotropic materials. Through this reduction, new scalar form analytical solutions are obtained and specialized further to compare with the existing analytical solutions.     

The order of singularity at a multi-wedge corner of a composite plate

The plane problem of a composite body consisting of many dissimilar isotropic, homogeneous and elastic wedges, perfectly bonded along their common interfaces, so that the plate forms a composite full plane with a number of corners, where a number of interfaces coalesce, is considered. The loading of the plate is due to prescribed surface tractions at the outside boundary of the plate. The particular behavior of the stress and displacement fields at the close vicinity of each interface corner is studied and the dependence of the order of singularity there was established in relation with the mechanical properties of the wedges coalescing at the particular corner considered. Applications of this general theory to simple cases of a single wedge and a bimaterial wedge gave results coinciding with already existing simple relations.     

Finite element applications of viscoplasticity with singular yield surfaces

The side and corner flow rules for viscoplastic materials with singular yield surfaces are discussed and applied in numerical solution of several boundary-value problems. The differences in predictions of displacement and stress fields for different flow rules are demonstrated.