Orthotropic Half-plane Research Articles

Stress intensity factors around two co-planar Griffith cracks in an orthotropic layer sandwiched between two identical orthotropic half planes

An integral transform technique is employed to solve the elastodynamic problem of steady-state propagation of two collinear Griffith cracks located in an orthotropic elastic layer of finite thickness 2 h sandwiched between two identical orthotropic half planes. For large h analytical expressions for the stress intensity factors are determined up to the order h −4. Graphical plots of the numerical results are also presented.

Contact model of cracks in orthotropic materials

We consider a plane strain problem for an orthotropic half plane loaded at infinity and containing a crack along its fixed edge. To remove a singularity near the right crack tip, we introduce an artificial contact zone. The problem under consideration is reduced to the mixed Dirichlet-Riemann boundaryvalue problem. We present the exact solution of this problem and deduce formulas for stresses in the contact zone and on the continuation of the crack and for the stress intensity factors. By using both analytic and numerical methods, we prove that the energy-release rate is quasiinvariant in the process of crack propagation relative to the size of the contact zone. On the basis of these results, we propose an algorithm for the evaluation of the paramenters of fracture of composites of finite dimensions with interface cracks. As a special case, we develop a model of interface cracks with actual contact zone and establish the dependences of the length of this zone on the external load and elasticity moduli of the material.

Interaction of elastic waves by an interfacial Griffith crack

A study is made of the problem of finding the distribution of stress field in the vicinity of a Griffith crack located at the interface of two dissimilar orthotropic half-planes. The crack is opened by the interaction of a plane harmonic elastic wave incident normally on the crack. The Fredholm integral equation is derived for the determination of diffraction wave field. The problem is then reduced to a pair of simultaneous singular integral equations which are solved by using the Jacobi polynomials. An expression for the stress intensity factors is obtained in the end.

The Role of Bearing Plates in the Five-Point Bending Tests of Structural-Size Lumber

Abstract The five-point bending test is configured as a continuous two-span beam having point loads at each midspan. It was proposed by others as an approach to evaluating the shear capacity of structural size lumber and engineered wood products because earlier tests with wood-based structural panels yielded shear failure modes. In the tests with structural panels, the thickness/depth ratio was large relative to that in structural lumber, although the length/depth ratios were the same. In addition, the density of the structural panels was greater than that of structural lumber. At the load-bearing positions, rounded bearing surfaces were used for the tests of structural panels, but flat bearing plates have been the practice in tests of structural size lumber. In this study, finite-element analyses of the bearing plates and the resulting stress distributions on structural lumber were conducted, first on an orthotropic half-plane and then in the five-point test geometry for structural lumber. The analyses used linear elastic and nonlinear compressive stiffness perpendicular to the grain and nonlinear shear modulus. Our testing showed that elementary beam theory failed to accurately predict the stresses in the critical region of the specimens. In addition, it appears unlikely that sufficient bearing can be developed in smaller lumber sizes to make the five-point bending method reflective of the true shear capacity.

STRESS DECAY IN AN ORTHOTROPIC HALF-PLANE UNDER LOCAL SELF-EQUILIBRATED EDGE LOADING

An exact solution is presented for the stresses in a specially orthotropic half-plane under a localized, symmetric, self-equilibrated combination of uniform normal edge loadings. “Smeared” orthotropic material properties are used to represent a broad range of balanced angle-ply graphite-epoxy laminates. Expressions for the three stresses in the half-plane are presented for isotropic material and for two orthotropic material families that contain many of the practical laminates. Particular attention Is given to the diffusion of the edge loading into the half-space and to the shear stresses caused by the discontinuous edge loading, both of which are strongly influenced by material properties. Of special interest to designers of composite structures may be a simple approximate formula, in terms of material properties, which is proposed for a lower bound on the normalized 90%-decay distance for the normal stress. Also, the results serve as further illustration of the fact that the meaningful application of Salnt-Venant's principle to specially orthotropic materials requires knowledge of their elastic properties.

Stress intensity factors for an interfacial crack between an orthotropic half-plane bonded to a dissimilar orthotropic layer with a punch

The plane strain problem of determining Stress Intensity Factors (SIF) for a moving interfacial Griffith crack between an elastic orthotropic half-plane and a dissimilar orthotropic layer with a moving punch situated along the boundary of the layer have been considered. The problem is reduced to the solution of three simultaneous singular integral equations with Cauchy-type singularities. Expressions for SIF for the case of a general loading are obtained. Numerical results for some particular cases are also presented graphically.

Contact stresses for a footing on an orthotropic elastic medium

AbstractThis paper deals with the contact stress analysis for a strip footing interacting with (i) a homogeneous orthotropic halfplane and (ii) multilayered orthotropic halfplane. The contact stresses have been obtained using a combined approach(finite element method‐analytical)in which the stiffness matrix for the footing has been obtained using the finite element method(F.E.M) while for the different cases of orthotropic medium(halfplane) considered here, suitable analytical solutions have been used. The halfplane needs to be discretised only along the contact length. Hence the solution can be obtained with a minimum input data and also the solution scheme is computationally efficient. Results have been presented for the contact stress distribution, displacement profile and bending stress distribution in the footing.

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.

Stress singularities in 2D orthotropic corners

A new technique for analysis of two-dimensional linear elastostatic solutions with stress singularities at orthotropic corners is developed. An explicit and general representation of the associated eigenequation given as ‘zero determinant condition’ of a matrix with half dimension in comparison with the current approach is derived by application of Stroh relations of anisotropic elasticity. The technique is directly applicable to anisotropic corners. Analytical formulae for stress singularity exponents, roots of the associated eigenequations, at orthotropic half-plane and semiinfinite crack problems for all combinations of basic homogeneous boundary conditions, including slip with friction, are derived. it is noteworthy that the singularity exponents are invariant with respect to the relative orientation of boundary edges and orthotropic material for nine of these combinations. Numerical analysis of singularity exponents for some configurations typical in the modelization of material tests of fiber-matrix composite materials is presented.

Moving Griffith crack at the interface of two dissimilar orthotropic half planes

The plane strain problem for determining the dynamic stress intensity factor in orthotropic medium when a Griffith crack is situated at the interface of two dissimilar half spaces is considered. By use of the Fourier transform technique, the mixed boundary value problem is reduced to solving a pair of simultaneous singular integral equations which finally have been solved by using Jacobi polynomials. Numerical results for the dynamic stress intensity factors are obtained for some particular cases and the results are presented graphically.

Transient response of an orthotropic elastic medium with a cavity

A class of traction boundary-value problems related to an orthotropic elastic half plane with an arbitrary shaped cavity is considered. The problems under consideration are formulated in the Laplace transform space by using a boundary integral equation method based on Green's functions for unit impulsive loads applied in the interior of an orthotropic half plane. Time domain solutions are obtained by solving the boundary integral equation numerically and applying a numerical scheme for Laplace inversion. It is found that the accurate numerical evaluation of half plane Green's functions is critical for both the accuracy and the numerical stability of boundary integral equation solutions. The accuracy of present results is confirmed by comparing with the existing solutions for transient pressurization of a circular cavity in an isotropic elastic full plane. Selected time histories of displacements and hoop stresses of an orthotropic half plane due to a pressurised circular cavity located at a finite depth below the free surface are presented. Surface displacement profiles of isotropic, and of orthotropic graphite-epoxy and cobalt half planes with circular cavities excited by a surface loading (transducer) pulse are also presented. Features of the response profiles are discussed for potential applications in nondestructive evaluation of composites, geomechanics and other disciplines. The boundary integral equation scheme presented in this study can also be extended to analyse displacement and mixed boundary-value problems realted to an orthotropic elastic half plane.

An Exact Stiffness Method for Elastodynamics of a Layered Orthotropic Half-Plane

A method is presented in this paper to compute displacements and stresses of a multilayered orthotropic elastic half-plane under time-harmonic excitations. The half-plane region under consideration consists of a number of layers with different thicknesses and material properties. Exact layer stiffness matrices describing the relationship between Fourier transforms of displacements and tractions at the upper and bottom surface of each layer are established explicitly by using the analytical general solutions for displacements and stresses of a homogeneous orthotropic elastic medium. The global stiff ness matrix which is also symmetric and banded is assembled by considering the traction continuity conditions at the interface between adjacent layers of the multilayered half-plane. The numerical solution of the global stiffness equation results in the solutions for Fourier transform of displacements at layer interfaces. Thereafter displacements and stresses of the multilayered plane can be obtained by the numerical integration of Fourier integrals. Only negative exponential terms of Fourier transform parameter are found to appear in the elements of layer stiffness matrices. This ensures the numerical stability in the solution of the global stiffness equation. In addition, the size of the final equation system is nearly onehalf of that corresponding to the conventional matrix approach for layered media based on the determination of layer arbitrary coefficients. The present method provides accurate solutions for both displacements and stresses over a wide range of frequencies and layer thicknesses. Selected numerical results are presented to portray the influence of layering, material orthotropy, and frequency of excitation on the response of five layered systems. Time-domain solutions are also presented to demonstrate the features of transient surface displacements due to a surface loading pulse applied to layered orthotropic half-planes.

The state of stress of an anisotropic half-plane with an elliptic inclusion

One solves the problem of the determination of the state of stress in an anisotropic half-plane in the presence of an elliptic inclusion of the same material, subjected to a preliminary strain. This problem is equivalent to the problem of the determination of the state of stress in a half-plane, due to a thermal “patch, ” coinciding with the inclusion domain. The solution of the problem is given in complex potentials. The case of an orthotropic half-plane with a circular inclusion is considered in detail. The results are illustrated by graphs.

Multiply Loaded Timoshenko Beam on a Stressed Orthotropic Half-Plane via a Thin Elastic Layer

The problem of bonded contact between a uniform finite Timoshenko beam and an orthotropic half-plane via a thin elastic layer is considered in this paper. The beam is loaded by distributions of normal and tangential forces, and a uniaxial stress load is applied to the half-plane. The Timoshenko beam theory is extended in such a way that the tangential load is included when the shear contribution to the beam central line deflection is calculated. The layer is formulated as a generalized Winkler cushion including also shear stresses and strains. Governing singular integral equations are stated and numerically solved for the unknown interface stresses. A comparison with a corresponding FE-model is also performed.

The crack problem for an orthotropic half-plane stiffened by elastic films

In this paper, various contact and crack problems for an orthotropic substrate, stiffened by elastic films, are considered. The film is modeled as a membrane and the substrate as an orthotropic half-plane with the principal axes of orthotropy parallel and perpendicular to the boundary. The problem is formulated in terms of a system of singular integral equations. Various asymptotic analyses are carried out in order to determine the nature of the stresses singularities. The special cases for which the solution is obtained and results are provided include the stiffened half-plane without the crack, the cracked half-plane without any stiffeners, the orthotropic half-plane with a single stiffener and a crack emanating from the endpoint of the stiffener, and a broken stiffener on a cracked half-plane. In each case the influence of the relative crack/stiffener dimensions, the film/substrate stiffness ratios, and the material orthotropy on the stress intensity factors is studied and some sample results are given.

Determination of the shape of a cavity in an orthotropic half-plane given the wave field at the boundary

The shape of a cavity in an elastic orthotropic half-plane, in which vibrations are excited by a point source situated on the surface, is determined from the vertical field of displacements on the surface. The problem is reduced to a system of three non-linear operator equations, and a linearized formulation is proposed for convex cavities. In the final analysis the problem is reduced to solving (1) a system of two boundary integral equations over the boundary of a known cavity, which is done by the boundary element method, and (2) a linear integral equation of the first kind with a smooth kernel. As the problem thus obtained is ill-posed, it is treated by regularization in Tikhonov's sense [1]. The effect of the initial approximation and the vibration frequency on the efficiency of the proposed algorithm is also investigated.

Vibrations of an orthotropic half-plane with a cavity

Vibrations of an orthotropic half-plane with a cavity

Multiply Loaded Rigid Punch on a Stressed Orthotopic Half-Plane via a Thin Elastic Layer

A uniaxially stressed orthotropic half-plane indented on the free edge by a multiply loaded rigid punch via a thin elastic layer is considered. The layer is formulated as a generalized Winkler cushion including also shear stresses and strains. Governing singular integral equations are stated for the unknown interface stresses between the cushion and the half-plane. Two kinds of friction conditions between the cushion and half-plane are treated, viz. completely adhesive and frictionless contact. An analytical solution for contact with a rigid cushion and a numerical solution with an elastic cushion are presented. Also, a comparison with a corresponding FEM model is performed. For frictionless contact, some analytical results concerning optimum design of the elastic cushion are given.

Combined periodic boundary-value problems and their applications in the theory of elasticity

The method described in [1–4] for solving the Hilbert-Riemann boundary value problems posed for a finite number of contours in a complex plane is extended to periodic problems. Mixed periodic problems of the static and stationary dynamic theory of elasticity for an isotropic and orthotropic half-plane and a composite plane that can be reduced to Hilbert-Riemann problems and solved in quadratures, using the method in question, are considered. In particular, problems concerned with a periodic system of punches with sections of complete adhesion and detachment pressing on an elastic half-plane are solved.

Erratum: “Rigid Punch on Stressed Orthotropic Half-Plane With Partial Slip” (Journal of Applied Mechanics, 1991, 58, pp. 128–133)

Erratum: “Rigid Punch on Stressed Orthotropic Half-Plane With Partial Slip” (Journal of Applied Mechanics, 1991, 58, pp. 128–133)