Orthotropic Half-plane Research Articles

The Flamant problem for an orthotropic half-plane

The Flamant problem for an orthotropic half-plane

RESEARCH OF THE STRESS-STRAIN STATE OF THE ORTHOTROPIC HALF-PLANE UNDER THE PLANAR DEFORMATION CONDITIONS

The first basic boundary problem of the elasticity theory connected with the determination of the stress-strain state of the orthotropic half-plane under the planar deformation conditions has been considered. The stresses at the boundary y = 0 are known. The stresses tend to zero at infinity. It is necessary to determine the stress and displacement at any point of the half-plane. A brief overview of the scientific works which highlight the methods and approaches to solving the problems of the theory of elasticity, the strength as for the determination of stresses and deformations in orthotropic bodies, in particular plates, slabs, and beams is provided. The solution of the given boundary problem for the orthotropic half-plane is sought in the transformant space of the one-dimensional integral Fourier transformation. At the same time, all the main equations of the problem and the boundary conditions are subjected to direct one-dimensional integral Fourier transformation. The solution of the formulated planar problem is based on the construction of the Fourier transform of the stress function, which satisfies the corresponding analogue of the biharmonic differential equation in the transformant space for the case of an orthotropic material. The form of the transformant of the stress function depends on the values of the orthotropic material elastic constants, namely on the values of the roots of the characteristic equation obtained in the transformant space. One of three possible cases have been considered. The relations between the transformant of the stress function and the transformants of stresses and displacements are established. The transformants of the stress function are expressed in terms of four auxiliary functions, which are related to the loads on the surface of the half-plane. We find two auxiliary functions from the boundary condition ( y = 0 ). The infinity conditions make it possible to establish the connection between two found auxiliary functions and the other two functions. After substituting the found expressions into transformants of stresses and displacements and applying the inverse integral Fourier transformation, we obtain the true values of stresses and displacements at the points of the orthotropic half-plane. The solutions for the specific cases have been obtained and the numerical results have been analyzed. The obtained calculations demonstrate the adequacy of the results and the logic of using the chosen method for the solution of the given problem.

Elastic and Thermoelastic Responses of Orthotropic Half-Planes.

A unified approach is presented for constructing explicit solutions to the plane elasticity and thermoelasticity problems for orthotropic half-planes. The solutions are constructed in forms which decrease the distance from the loaded segments of the boundary for any feasible relationship between the elastic moduli of orthotropic materials. For the construction, the direct integration method was employed to reduce the formulated problems to a governing equation for a key function. In turn, the governing equation was reduced to an integral equation of the second kind, whose explicit analytical solution was constructed by using the resolvent-kernel algorithm.

Stress analysis for a thin film bonded to an orthotropic substrate under thermal loading

In this paper, the interaction between an isotropic thin film and an orthotropic substrate is considered under the plane strain state. The perfectly bonded thin film is exposed to a predefined temperature gradient. The strain compatibility condition and the plane elasticity formulation are utilized to extract the governing integral equations. Employing the Gauss- Chebyshev discretization method, the interfacial shear as well as the flexural surface in-plane stresses are determined. The influences of the material stiffness and the orthotropy parameters on both surface and subsurface stresses are investigated. The numerical results confirm that the proper adjustment of the orthotropic substrate properties can alleviated the severe stress gradients. The results indicate that the surface in-plane stress tensile peak significantly decreases if the stiffness of the orthotropic half-plane along the film interface is lower than its corresponding value through the depth. Also, the maximum von Mises stress in the substrate decreases about 50% for an orthotropic material possessing a large shear parameter.

Oblique surface cracking and crack closure in an orthotropic medium under contact loading

This article presents analytical methods for analysis of an inclined surface crack in an orthotropic medium subjected to contact loading. Governing partial differential equations are derived in accordance with plane theory of orthotropic elasticity. Contact and crack problems are formulated separately utilizing Fourier transformation techniques. The contact problem consists of a rigid flat punch pressing upon an orthotropic half-plane, and is reduced to a singular integral equation of the second kind. The surface crack is assumed to be tilted at an arbitrary angle with respect to the normal of the half-plane surface. Three different crack configurations are considered, which are completely closed, partially-closed, and fully-open cracks. Frictional contact between the crack faces is implemented in the cases of completely closed, and partially-closed cracks. A single singular integral equation is derived for the completely closed crack, whereas each of the problems of partially- and fully-open cracks requires derivation of a set of two singular integral equations. The contact problem and each of the crack problems are solved numerically by means of the expansion-collocation technique. A flowchart is developed and implemented to detect the type of crack closure configuration under a given contact loading. The parametric analyses carried out illustrate the influences of factors such as crack inclination angle, crack face and contact surface friction coefficients, material orthotropy, and relative punch location on crack face contact stresses, degree of crack closure, and mode I and II stress intensity factors.

Rayleigh wave motions in an orthotropic half-space under time-harmonic loadings: A theoretical study

Rayleigh surface wave motion in a homogeneous orthotropic elastic solid half-space under time-harmonic sources is theoretically investigated. In this article, closed-form expressions of the Rayleigh wave field as a result of a time-harmonic concentrated line load are simply determined by using reciprocity theorems. For the verification purpose, analytical predictions are also obtained through Fourier transform technique with the aid of the residue theorem. The solutions found by the two approaches are shown to be mathematically the same. As an example of calculation, surface waves due to a set of buried loadings in the orthotropic half-plane are then examined in detail. A discussion of these results is presented to explore the influence of the loading types on the displacement amplitudes of the generated wave fields.

The fundamental problems on orthotropic half-plane with holes

The fundamental problems on orthotropic half-plane with holes

Interfacial debonding of an orthotropic half-plane bonded to a rigid foundation

A partial debonding of the interface of an orthotropic elastic half-plane bonded to a rigid foundation is studied when shear and compressive loads are applied. A debonding/crack face smoothly contacts with a rigid base under applied compressive load, but relatively slides. The Fourier transform method is used to reduce a mixed boundary value problem to dual integral equations. Full elastic fields at any position are obtained in terms of preliminary functions with the aid of the well-known integrals of Bessel functions for odd and even shear loads. Asymptotic elastic stresses near the debonding tips together with stress intensity factors are determined explicitly. Only inverse square-root singularity near the debonding tips appears. The influence of material orthotropy parameters on energy release rate and stress distribution is discussed.

Mixed-Mode Stress Intensity Factors in a Homogeneous Orthotropic Medium Loaded by a Frictional Sliding Rigid Flat Stamp

In this study, the crack problem for a homogeneous orthotropic medium loaded by a sliding rigid flat punch is considered. The homogeneous orthotropic medium is assumed to be a half-plane and is subjected to both normal and tangential forces through the sliding action of the punch. The crack on the homogeneous orthotropic medium is supposed to a depth of and is parallel to the direction of the normal force. The effect of the geometrical parameters and coefficient of friction on the mixed-mode stress intensity factors (mode I and mode II) is investigated using a computational approach using the finite element method. Augmented Lagrange method is used for the contact algorithm between the rigid flat punch and homogeneous orthotropic half-plane. This study may provide insight to the engineers in understanding the crack mechanisms in orthotropic materials in a comprehensive way and to identify early crack propagations under frictional loadings accurately.

Thermal stress intensity factor for an edge crack in orthotropic composite media

The purpose of this article is to determine the thermal stress intensity factor (SIF) of an edge crack situated in an orthotropic strip of finite thickness h bonded to an orthotropic half plane, under thermal and mechanical loadings. The problem is reduced to a singular integral equation of the first kind. The integral equation is converted to a pair of the singular integral equations using asymptotic expansion method. The analytical expression of the stress intensity factor at the edge crack tip is found for concentrated point loading. The numerical values of SIF are computed for different point loading for various crack lengths and different ratios of thermal conductivities for different particular cases when the composite orthotropic material is the mixture of Steel-Myler and E-Glass epoxy. The significant feature of the article is the pictorial presentations of the variations of SIF due to the effect of ratios of thermal conductivities of both the materials.

Debonding of FRP and thin films from an elastic half-plane using a coupled FE-BIE model

A Finite Element-Boundary Integral Equation (FE-BIE) coupling method is proposed to investigate a flexible bar weakly attached to an elastic orthotropic half-plane. Firstly, the analysis focused on the case of a bar subjected to horizontal forces and thermal loads considering interfacial displacements linearly proportional to the tangential traction. Secondly, the debonding behaviour of a composite reinforcement glued to a substrate has been modelled. Using an incremental nonlinear analysis, a bilinear elastic-softening interfacial traction-slip law has been implemented simulating the delamination of pure mode II. Finally, the influence of the anchorage length on the ultimate bearing capacity of the adhesive joint has been investigated.

Propagation of a Dugdale crack between two orthotropic half-planes

Abstract The aim of this work is to study the propagation of a Dugdale crack between two orthotropic half-planes. The two mediums are of the same unidirectional composite material. The fibers are oriented symmetrically with respect to the interface. The applied propagation criteria are deduced from the revisited Griffith theory (Ferdjani, H. and J.-J. Marigo, European Journal of Mechanics - A/Solids, 2015. 53: p. 1–9). To resolve the problem, the Lekhnitskii-Eshelby-Stroh representation is used (Suo, Z. Proc. R. Soc. Lond. A 427, 331–358 (1990)). The fracture threshold and the evolution of the applied load are determined for different cases of fiber orientation and crack length. A comparison with the Griffith's model is also presented.

Contact problem for a composite material with nacre inspired microstructure

Bi-material composites with nacre inspired brick and mortar microstructures, characterized by stiff elements of one phase with high aspect ratio separated by thin layers of the second one, are considered. Such microstructure is proved to provide an efficient solution for the problem of a crack arrest. However, contrary to the case of a homogeneous material, an external pressure, applied to a part of the composite boundary, can cause significant tensile stresses which increase the danger of crack nucleation. Investigation of the influence of microstructure parameters on the magnitude of tensile stresses is performed by means of the classical Flamant-like problem of an orthotropic half-plane subjected to a normal external distributed loading. Adequate analysis of this problem represents a serious computational task due to the geometry of the considered layout and the high contrast between the composite constituents. This difficulty is presently circumvented by deriving a micro-to-macro analysis in the framework of which an analytical solution of the auxiliary elasticity problem, followed by the discrete Fourier transform and the higher-order theory are employed. As a result, full scale continuum modeling of both composite constituents without employing any simplifying assumptions is presented. In the framework of the present proposed modeling, the influence of stiff elements aspect ratio on the overall stress distribution is demonstrated.

Interaction of Three Interfacial Cracks between an Orthotropic Half-Plane Bonded to a Dissimilar Orthotropic Layer with Punch

Abstract This article deals with the interactions between a central crack and a pair of outer cracks situated at the interface of an orthotropic elastic half-plane bonded to a dissimilar orthotropic layer with a punch. The problem is reduced to the solution of three simultaneous singular integral equations that are finally solved using Jacobi polynomials. The phenomena of crack shielding and crack amplification have been depicted through graphs for different particular cases.

Response of an orthotropic half-plane subjected to transient anti-plane loading with multiple edge cracks

This study deals with the problem of multiple edge cracks in an elastic orthotropic half-plane under transient loading. The dislocation solution is utilized to derive integral equations for multiple interacting edge cracks in an orthotropic half-plane. These equations are solved numerically thereby obtaining the dislocation density function on the crack faces and stress intensity factors at crack tips. Numerical results are obtained to illustrate the variation of the dynamic stress intensity factors as a function of crack length and material properties.

Computational contact mechanics analysis of laterally graded orthotropic half-planes

PurposeFrictional sliding contact problems between laterally graded orthotropic half-planes and a flat rigid stamp are investigated. The presented study aims at guiding engineering applications in the prediction of the contact response of orthotropic laterally graded members.Design/methodology/approachThe solution procedure is based on a finite element (FE) approach which is conducted with an efficient FE analysis software ANSYS. The spatial gradations of the orthotropic stiffness constants through the horizontal axis are enabled utilizing the homogeneous FE approach. The Augmented Lagrangian contact algorithm is used as an iterative non-linear solution method in the contact analysis.FindingsThe accuracy of the proposed FE solution method is approved by using the comparisons of the results with those computed using an analytical technique. The prominent results indicate that the surface contact stresses can be mitigated upon increasing the degree of orthotropy and positive lateral gradations.Originality/valueOne can infer from the literature survey that, the contact mechanics analysis of orthotropic laterally graded materials has not been investigated so far. In this study, an FE method-based computational solution procedure for the aforementioned problem is addressed. The presented study aims at guiding engineering applications in the prediction of the contact response of orthotropic laterally graded members. Additionally, this study provides some useful points related to computational contact mechanics analysis of orthotropic structures.

Rolling contact problem for an orthotropic medium

In this paper, the two-dimensional tractive rolling contact problem between a rigid cylinder and an orthotropic half plane is considered. A pair of singular integral equations is derived for the contact tractions using the Fourier transform technique. The governing integral equations are decoupled using the Goodman approximation and then solved numerically. Finally, the distribution of surface tractions is achieved in the contact region. This study emphasizes the significant effect of the orthotropic material parameters and the coefficient of friction on the contact tractions as well as the rolling contact characteristics for an orthotropic medium.

Stresses produced in an orthotropic half-plane under a moving line load

The stresses and displacements of an orthotropic half-plane were obtained under a line load moving at a constant speed over its surface. The Fourier transform was applied to solve the problem at all speeds. It was verified that the stresses and displacements can be readily classified into three regimes according to the speed of the moving load. The expressions for the stresses and displacements have shapes similar to those of the solutions to the corresponding problem for an isotropic half-plane.

Interface crack between different orthotropic media under uniform heat flow.

In this paper, plane thermo-elastic solutions are presented for the problem of a crack in two bonded homogeneous orthotropic media with a graded interfacial zone. The graded interfacial zone is treated as a nonhomogeneous interlayer having spatially varying thermo-elastic moduli between dissimilar, homogeneous orthotropic half-planes, which is assumed to vary exponentially in the direction perpendicular to the crack surface. Using singular integral equation method, the mixed boundary value conditions with respect to the temperature field and those with respect to the stress field are reduced to a system of singular integral equations and solved numerically. Numerical results are obtained to show the influence of non-homogeneity parameters of the material thermo-elastic properties, the orthotropy parameters and the dimensionless thermal resistance on the temperature distribution and the thermal stress intensity factors.

A coupled FE-BIE model for the static analysis of Timoshenko beams bonded to an orthotropic elastic half-plane

Interface displacements, surface tractions and stresses of a flexible beam bonded to an elastic orthotropic half-plane are analysed by means of a Finite Element-Boundary Integral Equation (FE-BIE) method. Numerical results are obtained by using locking-free shear deformable beams and piecewise constant interfacial reactions. Making use of the generalised Green's function for the half-plane, the mechanical behaviour of fully bonded or detached beams subjected to force, couple or thermal load is investigated. The special cases of a beam in bilateral frictionless contact with the half-plane and a beam having a vanishing bending rigidity (thin film) are considered also. In particular, the maximum bending moment of beams subjected to a vertical point force are compared with some closed-form solutions of the contact problem of a rigid indenter and with the solution of an infinite Euler–Bernoulli beam in bilateral frictionless contact with an isotropic substrate.