Effects of curvilinear anisotropy on radially symmetric stresses in anisotropic linearly elastic solids

It has been known for some time that certain radial anisotropies in some linear elasticity problems can give rise to stress singularities which are absent in the corresponding isotropic problems. Recently related issues were examined by other authors in the context of plane strain axisymmetric deformations of a hollow circular cylindrically anisotropic linearly elastic cylinder under uniform external pressure, an anisotropic analog of the classic isotropic Lamé problem. In the isotropic case, as the external radius increases, the stresses rapidly approach those for a traction-free cavity in an infinite medium under remotely applied uniform compression. However, it has been shown that this does not occur when the cylinder is even slightly anisotropic. In this paper, we provide further elaboration on these issues. For the externally pressurized hollow cylinder (or disk), it is shown that for radially orthotropic materials, the maximum hoop stress occurs always on the inner boundary (as in the isotropic case) but that the stress concentration factor is infinite. For circumferentially orthotropic materials, if the tube is sufficiently thin, the maximum hoop stress always occurs on the inner boundary whereas for sufficiently thick tubes, the maximum hoop stress occurs at the outer boundary. For the case of an internally pressurized tube, the anisotropic problem does not give rise to such radical differences in stress behavior from the isotropic problem. Such differences do, however, arise in the problem of an anisotropic disk, in plane stress, rotating at a constant angular velocity about its center, as well as in the three-dimensional problem governing radially symmetric deformations of anisotropic externally pressurized hollow spheres. The anisotropies of concern here do arise in technological applications such as the processing of fiber composites as well as the casting of metals.     

Rotating Variable-thickness Orthotropic Cylinder Containing a Solid Core of Uniform-thickness

This paper presents accurate elastic solutions for the rotating variable-thickness and/or uniform-thickness orthotropic circular cylinders. The present circular cylinder may contain a uniform-thickness solid core of rigid or homogeneously isotropic material. Different cases of rotating cylinders of various cores are investigated. These cylinders include completely isotropic solid cylinder, uniform-thickness orthotropic cylinder containing an isotropic core, variable-thickness orthotropic cylinder containing an isotropic core, uniform-thickness orthotropic cylinder containing a rigid core, and variable-thickness orthotropic cylinder containing a rigid core. For all cases studied, exact elastic solutions are obtained and numerical results are presented. The results include the radial, hoop, and axial stresses and radial displacement of the five cylinder configurations. The distributions of displacement and stresses through the radial direction of the rotating cylinder are obtained and comparisons between different cases are made at the same angular velocity.     

Plane strain problems in second-order elasticity theory

The equations of second-order elasticity are developed in polar coordinates R, θ for plane strain deformations of incompressible isotropic elastic materials. By considering a ‘displacement function’ the second-order problem is reduced to the solution of an equation of the form ⁴ψ = g(R, Θ) where ² is Laplace's differential operator and g(R, Θ) depends only on the first-order solution. The displacement function technique is then applied to obtain a second-order solution to the problem of an elastic body contained between two concentric rigid circular boundaries, when the outer boundary is held fixed and the inner is subjected to a rigid body translation.     

The effect of transverse normal strain in contact of an orthotropic beam pressed against a circular surface

The contact problem of a straight orthotropic beam pressed onto a rigid circular surface is considered using beam theories that account for transverse shear and transverse normal deformations. The circular nature of the rigid surface emphasizes the difference between Euler Bernoulli theory behavior, where point loads develop at the edge of contact, and the higher order theories that predict non-singular pressure distributions. While Timoshenko beam theory is the simplest theory that addresses this behavior, the prediction of a maximum value of pressure at the edge of contact contradicts the elasticity theory result that contact pressure must drop to zero. Transverse normal strain is therefore introduced, both to study this fundamental discrepancy and to include an important effect in many contact problems. To investigate this effect, higher order beam theories that account for both constant and linear transverse normal strain through the beam thickness are derived using the principle of virtual work. The resulting orthotropic beam theories depend on the bending stiffness (EI), shear stiffness (GA), axial stiffness (EA₁) and transverse normal stiffness (EA₂), which are independent stiffness parameters that can differ by orders of magnitude. The above mentioned contact problem is then solved analytically for these theories, along with the Timoshenko beam model which assumes zero transverse normal strain. The results for different orthotropic materials show that inclusion of transverse normal deformation has a significant effect on the contact pressure solution. Furthermore, the solution using higher order beam theories encompasses the two extremes of a Hertz-like contact pressure when the half contact length is smaller than the thickness of the beam, and the Timoshenko beam theory case when the half contact length is much larger than the thickness. Concerning the behavior of the pressure at the edge of contact, adherence to the boundary conditions required by the principle of virtual work, shows that while the pressure does tend to zero, it does not become zero unless artificially enforced. In this regard the solution for the case of linear strain is better than that for constant strain. All beam solutions are validated with plane elasticity solutions obtained using the commercial finite element software ABAQUS.     

Determination of boundary layers in the plane problem for three-layer strips. Part 2

In the first part of this paper, we considered the exact statement of the plane elasticity problem in displacements for strips made of various materials (problem A, an isotropic material; problem B, an orthotropic material with 2G ₁₂ < √E ₁ E ₂; problem C, an orthotropic material with 2G ₁₂ > √E ₁ E ₂). Further, we stated and solved the boundary layer problem (the problem on a solution decaying away from the boundary) for a sandwich strip of regular structure consisting of isotropic layers (problem AA). In the present paper, we use the solution of the plane problem to consider the problem for sandwich strips of regular structure with isotropic face layers and orthotropic filler (problem AB).     

Impact of a plane die on an elastic orthotropic half-plane

To study the process of impact of a rigid body on the surface of an elastic body made of a composite material, we consider a nonstationary dynamic contact problem about the impact of a plane rigid die on an elastic orthotropic half-plane. The problem is reduced to solving an integral equation of the first kind for the Laplace transform of the contact stresses under the die base. An approximate solution of the integral equation is constructed with the use of a special approximation to the symbol of the kernel of the integral equation in the complex plane. The inverse Laplace transform of the solution results in determining the scalar contact stress field on the die base, the force exerted by the die on the elastic medium, and the vertical displacement field of the free surface of the orthotropic medium out side the die. The solutions thus obtained permit studying specific features of the process of die penetration into an orthotropic medium and the strain properties of the medium.     

Elastic contact between a cylindrical surface and a flat surface - A non-Hertzian model of multi-asperity contact

Contact between curved rough bodies is an important engineering problem. The paper addresses the problem in its simplest form where a smooth rigid cylinder presses down an elastic half space bounded by a plane of uniformly spaced cylindrical asperities. Keeping the separation between the bodies unchanged the problem is inverted and solved using the method of complex variables. As the asperities deform as well as move as rigid bodies, contact lengths and positions develop non-symmetrically with respect to the initial axes of symmetry of the asperities. The resulting local contact pressures are non-Hertzian and the normal load for a given contact area is greater than that estimated using a priori Hertzian pressure profiles.     

Natural frequencies of thin,rectangular plates with holes or orthotropic “patches” carrying an elastically mounted mass

The approximate solution for the title problem is obtained in the case of simply supported and clamped rectangular plates made of isotropic or orthotropic materials. A variational approach (the well known Rayleigh–Ritz method) is used, where the displacement amplitude is expressed in terms of beam functions. This means that each coordinate function satisfies identically all the boundary conditions at the outer edge of the plate. Free vibration analysis has been performed on various different cases; solid isotropic and orthotropic plates, orthotropic plates with a hole and isotropic plates with an orthotropic inclusion or “patch”, carrying an elastically mounted concentrated mass. It is important to point out that the case of an orthotropic patch is interesting from a technological viewpoint since it constitutes a model of a repair implemented on the virgin structural element when it has suffered damage. This approach has been implemented by the aeronautical industry in some instances. The obtained results are in very good agreement with those of particular cases of simply supported plates available in the literature.     

Stability of frictional slipping at an anisotropic/isotropic interface

The stability of steady, quasi-static slip at a planar interface between an anisotropic elastic solid and an isotropic elastic solid is studied. The paper begins with an analysis of anti-plane sliding at an orthotropic/isotropic interface. Friction at the interface is assumed to follow a rate- and state-dependent law. The stability to spatial perturbations of the form exp(ikx₁), where k is the wavenumber and x₁ is the coordinate along which the interface is studied. An expression is derived for the critical wavenumber ∣k∣_cr above which there is stability. In-plane sliding at an anisotropic/isotropic interface is subsequently studied. In this case, slip couples with normal stress changes and a constitutive law for dynamic normal stress changes is adopted. Again a formula for ∣k∣_cr is derived and the results are specialized to the case of an orthotropic/isotropic interface. Numerical plots of the dependence of ∣k∣_cr on the orientation of the orthotropic solid, as well as on the material parameters are provided.     

Analytical solution of the contact problem of a rigid indenter and an anisotropic linear elastic layer

We use the Stroh formalism to study analytically generalized plane strain deformations of a linear elastic anisotropic layer bonded to a rigid substrate, and indented by a rigid cylindrical indenter. The mixed boundary-value problem is challenging since the a priori unknown deformed indented surface of the layer contacting the rigid cylinder is to be determined as a part of the solution of the problem. For a rigid parabolic prismatic indenter contacting either an isotropic layer or an orthotropic layer and a flat rigid punch indenting a half space, the computed solutions are found to agree well with those available in the literature. Parametric studies have been conducted to delimit the length and the thickness of the layer for which the derived relation between the axial load and the indentation depth caused by the rigid cylinder is valid. The indentation of a face centered cubic crystal with the plane of indentation oriented differently from the principal planes of symmetry has also been studied to illustrate the applicability of the technique to general layers made of anisotropic materials. Results presented herein can serve as benchmarks with which to compare solutions obtained by other methods.     

Contact problems for several transversely isotropic elastic layers on a smooth elastic half-space

The idea of generalized images, first used by the author for the case of crack problems, is applied here to solve a contact problem for n transversely isotropic elastic layers, with smooth interfaces, resting on a smooth elastic half-space, made of a different transversely isotropic material. A rigid punch of arbitrary shape is pressed against the top layer’s free surface. The governing integral equation is derived for the case of two layers; it is mathematically equivalent to that of an electrostatic problem of an infinite row of coaxial charged disks in the shape of the domain of contact. This result is then generalized for an arbitrary number of layers. As a comparison, the method of integral transforms is also used to solve the problem. The main difference of our integral transform approach with the existing ones is in separating of our half-space solution from the integral transform terms. It is shown that both methods lead to the same results, thus giving a new interpretation to the integral transform as a sum of an infinite series of generalized images.     

Finite element modeling of elasto-plastic contact between rough surfaces

This paper presents a finite element calculation of frictionless, non-adhesive, contact between a rigid plane and an elasto-plastic solid with a self-affine fractal surface. The calculations are conducted within an explicit dynamic Lagrangian framework. The elasto-plastic response of the material is described by a J₂ isotropic plasticity law. Parametric studies are used to establish general relations between contact properties and key material parameters. In all cases, the contact area A rises linearly with the applied load. The rate of increase grows as the yield stress σ_y decreases, scaling as a power of σ_y over the range typical of real materials. Results for A from different plasticity laws and surface morphologies can all be described by a simple scaling formula. Plasticity produces qualitative changes in the distributions of local pressures in the contact and of the size of connected contact regions. The probability of large local pressures is decreased, while large clusters become more likely. Loading-unloading cycles are considered and the total plastic work is found to be nearly constant over a wide range of yield stresses.     

The contact problem of a rigid cylinder on an elastic layer

Summary This paper deals with the contact problem of a rigid cylinder pressed on an elastic layer connected rigidly to a rigid base. It is assumed that there is no friction between cylinder and layer and that the cylinder is long enough to ensure a plane deformation. Asymptotic solutions are presented when the ratio of the half width c of the contact area to the thickness b of the layer is small and also when c/b is large. The breakdown of the asymptotic solution for large values of c/b when the material is incompressible, discussed by Koiter [6], is overcome by considering a more general solution of the Wiener-Hopf integral equation encountered. The results of both asymptotic solutions match so well that a satisfactory solution is obtained for all values c/b and for 00.5.     

Two-dimensional transient dynamic response of orthotropic layered media

In this study, two-dimensional transient dynamic response of orthotropic plane layered media is investigated. The plane multilayered media consist of N different generally orthotropic, homogeneous and linearly elastic layers with different ply angles. In the generally orthotropic layer, representing a ply reinforced by unidirectional fibers with an arbitrary orientation angle, the principal material directions do not coincide with body coordinate axes. The solution is obtained by employing a numerical technique which combines the use of Fourier transform with the method of characteristics. The numerical results are displayed in curves denoting the variations of stress and displacement components with time at different locations. These curves clearly reveal, in wave profiles, the scattering effects caused by the reflections and refractions of waves at the boundaries and at the interfaces of the layers, and also the effects of anisotropy caused by fiber orientation angle. The curves properly predict the sharp variations in the response at the neighborhood of the wave fronts, which shows the power of the numerical technique employed in the study. By suitably adjusting the elastic constants, the results for multilayered media with transversely isotropic layers, or layers with cubic symmetry, or isotropic layers can easily be obtained from the general formulation. Furthermore, solutions for some special cases, including Lamb’s problem for an elastic half-space, are obtained and compared with the available solutions in the literature and very good agreement is found. Preliminary version presented at the Second International Congress on Mechatronics (MECH2K3), Graz, Austria, July 14-17, 2003.     

Strain gradient plasticity analysis of elasto-plastic contact between rough surfaces

From a microscopic point of view, the real contact area between two rough surfaces is the sum of the areas of contact between facing asperities. Since the real contact area is a fraction of the nominal contact area, the real contact pressure is much higher than the nominal contact pressure, which results in plastic deformation of asperities. As plasticity is size dependent at size scales below tens of micrometers, with the general trend of smaller being harder, macroscopic plasticity is not suitable to describe plastic deformation of small asperities and thus fails to capture the real contact area and pressure accurately. Here we adopt conventional mechanism-based strain gradient plasticity (CMSGP) to analyze the contact between a rigid platen and an elasto-plastic solid with a rough surface. Flattening of a single sinusoidal asperity is analyzed first to highlight the difference between CMSGP and J₂ isotropic plasticity. For the rough surface contact, besides CMSGP, pure elastic and J₂ isotropic plasticity analysis is also carried out for comparison. In all cases, the contact area A rises linearly with the applied load, but with a different slope which implies that the mean contact pressures are different. CMSGP produces qualitative changes in the distributions of local contact pressures compared with pure elastic and J₂ isotropic plasticity analysis, furthermore, bounded by the two.     

Nonlinear Axisymmetric Static and Transient Response of Orthotropic Annular Shallow Spherical Caps

ABSTRACT

This investigation deals with the nonlinear axisymmetric static and transient response of orthotropic annular shallow spherical caps with a free edge hole and with a rigid central mass, subjected to uniformly distributed load and a central load. The dynamic analogue of Marguerre equations, in terms of normal displacement w and stress function Ψ, are employed. An orthogonal point collocation method is used for spatial discretization and a Newmark-β scheme is used for time integration. Static load, step function load, and sinusoidal pulse load are considered. The influence of annular ratio on the response is studied for isotropic (β = 1) and orthotropic (β = 3) clamped caps, with a rise to thickness ratio of 1.5.     

Indentation of a Rigid Sphere into an Elastic Half-Space in the Direction Orthogonal to the Axis of Material Symmetry

In this work a contact problem for a transversely isotropic half-space indented by a rigid sphere is considered. The axis of symmetry of the half-space is orthogonal to the axis of the applied contact load, which makes the problem fully three-dimensional. An exact solution to this contact problem is constructed. Stresses and strains are obtained in the form of contour integrals in the complex plane with explicitly determined dimensionless integrands. As an illustrative application of the constructed solution, failure beneath the indenter in a unidirectional composite is analyzed.     

Elastodynamic solution of a non-homogeneous orthotropic hollow cylinder

A theoretical method for analyzing the axisymmetric plane strain elastodynamic problem of a non-homogeneous orthotropic hollow cylinder is developed. Firstly, a new dependent variable is introduced to rewrite the governing equation, the boundary conditions and the initial conditions. Secondly, a special function is introduced to transform the inhomogeneous boundary conditions to homogeneous ones. By virtue of the orthogonal expansion technique, the equation with respect to the time variable is derived, of which the solution can be obtained. The displacement solution is finally obtained, which can be degenerated in a rather straightforward way into the solution for a homogeneous orthotropic hollow cylinder and isotropic solid cylinder as well as that for a non-homogeneous isotropic hollow cylinder. Using the present method, integral transform can be avoided and it can be used for hollow cylinders with arbitrary thickness and subjected to arbitrary dynamic loads. Numerical results are presented for a non-homogeneous orthotropic hollow cylinder subjected to dynamic internal pressure. The project supported by the National Natural Science Foundation of China (10172075 and 10002016)     

On interface crack models with contact zones situated in an anisotropic bimaterial

Summary A boundary value problem for two semi-infinite anisotropic spaces with mixed boundary conditions at the interface is considered. Assuming that the displacements are independent of the coordinate x ₃, stresses and derivatives of displacement jumps are expressed via a sectionally holomorphic vector function. By means of these relations the problem for an interface crack with an artificial contact zone in an orthotropic bimaterial is reduced to a combined Dirichlet-Riemann problem which is solved analytically. As a particular case of this solution, the contact zone model (in Comninou's sense) is derived. A simple transcendental equation and an asymptotic formula for the determination of the real contact zone length are obtained. The classical interface crack model with oscillating singularities at the crack tips is derived from the obtained solution as well. Analytical relations between fracture mechanical parameters of different models are found, and recommendations concerning their implementation are given. The dependencies of the contact zone lengths on material properties and external load coefficients are illustrated in graphical form. The practical applicability of the obtained results is demonstrated by means of a FEM analysis of a finite-sized orthotropic bimaterial with an interface crack. Received 19 October 1998; accepted for publication 13 November 1998     

Two Nonlinear Models of Brittle Fracture for Solids

This paper considers isotropic and orthotropic nonlinear constitutive relations for brittle materials in the case of plane stresses. Numerical solution algorithms based on the finite-element method are developed. The resulting material models are incorporated in the PIONER software. The correctness of crack path determination is examined by solving a test problem of crack propagation. The isotropic model gives mesh-dependent results, whereas the orthotropic model provides an adequate solution. It is shown that solutions obtained for the isotropic model are close to those obtained by eliminating failed elements.