Linear Elastic Body Research Articles

Rolling/sliding of a linear elastic body on a deformable layered substrate

We analyze a three-dimensional frictional contact problem between a homogeneous and isotropic linear elastic body and a substrate consisting of a homogeneous and isotropic linear elastic half-space coated with a transversely isotropic elastic layer. The problem formulation includes the kinematics of the body, including its rolling/sliding on the substrate. Expressions relating the displacements of a point on the contact surface to the surface tractions acting there have been derived. We have also proposed a variational problem for the determination of surface tractions at points of the contact surface. An example problem involving the motion of a homogeneous spherical rigid ball on a substrate made of a homogeneous thin layer of a linear elastic material and bonded to a rigid base is studied. It is shown that there can be at most one adhesion zone followed by a slipping zone. Numerical results illustrating the dependence of the slip velocity upon the resultant frictional force are also presented graphically.

One-Dimensional Interaction between Shock Wave and Low-Porosity Foam. 2nd Report. Analysis and Comparison with Experiments.

Numerical simulations of the one dimensional unsteady motion of low-porosity foam, induced by the shock wave are studied. Interactive motions of the gas and the foam are analyzed using the Lagrangian coordinates. An elastic model for a low-porosity foam is assumed to be a simple non linear elastic body following the measured stress-strain relation of foam. One end of the foam in the axial x-direction is fixed to the end wall of the shock tube, and the other end interacts with the shock wave. In the case of uni-axial stress loading, the movement of the body is free to expand in both the y-and z-directions. In the case of uni-axial strain loading, there is no movement of the foam in the y-and z-directions. Results for the foam are compared with those for the rubber. Very good agreements are found between numerical and experimental results in the case of rubber, while fairly good agreements in the case of polyurethane foam.

Significance of specimen size for the KR-curve behavior of quasi-brittle materials

Abstract The influence of specimen size on the crack resistance ( K R -curve) behavior of a coarse-grained carbon is investigated using compact tension samples of varying size. The results are compared to previous measurements on asbestos cement. In order to analyse and estimate the toughening contribution appearing due to crack-wake related processes, the unloading compliance is measured during the K R curve test. Subsequent renotching of the extended crack and comparison with the behavior of an ideal linear-elastic body enables the crack-closure forces to be estimated.

On almost constant contact stress distributions by shape optimization

This paper addresses the problem of finding shapes of contacting bodies avoiding undesirable stress concentrations. It has previously been shown that designing the shape of a rigid body in contact with a fixed linear elastic body by minimizing the equilibrium potential energy under an isoparametric constraint results in a uniform contact pressure distribution. As an extension of this result, it is shown here that when the shape of an elastic body in contact with a flat rigid foundation is chosen on the same premises, the uniform pressure distribution is found only if displacement gradients can be considered small. From the point of view of applications, an important conclusion is that this smallness holds in a case when linear elasticity is physically valid.

Variation of elastic T-stresses along slightly wavy 3D crack fronts

The non-singular terms in the series expansion of the elastic crack-tip stress field, commonly referred to as the elastic T-stresses, play an important role in fracture mechanics in areas such as the stability of a crack path and the two-parameter characterization of elastic-plastic crack-tip deformation. In this paper, a first order perturbation analysis is performed to study some basic properties of the T-stress variation along a slightly wavy 3D crack front. The analysis employs important properties of Bueckner-Rice 3D weight function fields. Using the Boussinesq-Papkovitch potential representation for the mode I weight function field, it is shown that, for coplanar cracks in an infinite isotropic and homogeneous linear elastic body, the mean T-stress along an arbitrary crack front is independent of the shape and size of the crack. Further, a universal relation is discovered between the mean T-stress and the stress field at the same crack front location under the same loading but in the absence of a crack. First-order-accurate solutions are given for the T-stress variation along a slightly wavy crack front with nearly circular or straight confifurations. Specifically, cosine wave functions are adopted to describe smooth polygonal and slightly undulating planar crack shapes. The results indicate that T11, the 2D T-stress component acting normal to the crack front, increases with the curvature of the crack front as it bows out but T33, acting parallel to the crack front, decreases with the crack front curvature.

On the problem of optimizing contact force distributions

The problem of optimizing the distribution of contact forces between a rigid obstacle and a discretized linear elastic body is considered. The design variables are the initial gaps between the potential contact nodal points and the obstacle. Two different cost functionals are investigated: the first reflects the objective of minimizing the maximum contact force; the second is the equilibrium potential energy. Contrary to what has been claimed in the literature, it is shown that these cost functionals do not give, in general, the same optimal design. However, it is also shown that, if a certain frequently realized assumption is met by the system flexibility matrix, then this equality does hold.

Adhesive joints and interfaces of linear elastic bodies in loading and unloading. Semicoercive hemivariational inequalities

The present paper deals with the behaviour of adhesive joints and of interfaces in elastic bodies. Especially the possibility of debonding effects in loading conditions is considered. The joints and/or interfaces are simulated by nonmonotone, possibly multivalued stress-strain laws expressed by nonconvex superpotentials. The variational formulation of the problem is a hemivariational inequality which is studied concerning the existence and the approximation of the solution. Both the coercive and the semicoercive cases are studied.

Thermal weight function of cracked bodies subjected to thermal loading

A linear elastic body which contains a crack and is subjected to thermal loading is considered. The conventional weight function method for mechanical loading derived by Rice is extended to thermal loading. The thermal weight function is a universal function for a given cracked body and can be determined from any arbitrary loading system if the stress intensity factor and mean stress field of the system are known as functions of crack length. The thermal weight function may be thought of as Green's function for the stress intensity factor of thermal cracked bodies. Once the thermal weight function is determined, the thermally induced stress intensity factor can be simply and efficiently evaluated through the integration of the product of temperature and thermal weight function of the cracked body. Many complicated thermoelastic cracked problems which may cause difficulty in conventional study may be solved easily by using the thermal weight function method. As a demonstration, some examples of thermal loading problems are solved by the thermal weight function method, and the results are compared with available results in the published literature.

A new method for solving dynamically accelerating crack problems. I. The case of a semi-infinite mode 𝐼𝐼𝐼 crack in elastic material revisited

Presented here is a new method for constructing solutions to dynamically accelerating, semi-infinite crack problems. The problem of a semi-infinite, anti-plane shear (mode III) crack accelerating dynamically in an infinite, linear, homogeneous, and isotropic elastic body has been solved previously by Freund and Kostrov. However, their methods are based upon the construction of a certain Green’s function for the ordinary two-dimensional wave equation and do not generalize to either the opening mode problem in elastic material or viscoelastic material. What is presented here is a new approach based upon integral transforms and complex variable techniques that does, in principle, generalize to both the opening modes of deformation and viscoelastic material. Moreover, the method presented here produces directly, for the mode III crack, a simple closed form expression for the crack-face profile for arbitrary applied crack-face tractions. Generalizations to opening modes of deformation and viscoelastic material produce integral equations for the crack-face displacement profile that in some cases admit closed form solutions and otherwise can be solved numerically. In contrast, the method of Freund and Kostrov yields, in mode III, an expression for the stress in front of the crack, but for opening modes provides only the stress intensity factor.

Characterization of Acoustic Emission Signals from Mode I Crack

Separation of crack growth signals is of fundamental importance for detecting, locating, and determining the significance of an internal flaw using acoustic emission techniques. The methodology used to determine the acoustic emission signal in this paper incorporates a source model that is an actual crack propagation and arrest event. This source model includes the effects of the following physical parameters: location of the crack, crack orientation, crack tip velocity, duration of crack propagation, and mode of fracture. The integral equation method is used to examine a semi‐infinite crack in an infinite, linear elastic body loaded under the conditions of plane strain. The dynamic mode I stress caused by a crack propagating with a prescribed velocity is determined, and then the displacements at any point are calculated. These time‐dependent displacements are the analytical form of the acoustic emission waveforms. Numerical results examine the effect of crack tip velocity, duration of propagation, and di...

On debonding and delamination effects in adhesively bonded cracks —A boundary integral approach

The present paper studies the influence of adhesives on the behaviour of cracks in two-dimensional linear elastic bodies. Especially the delamination and debonding effects are studied. The adhesive material is assumed to introduce non-monotone, possibly multivalued laws which can be described via non-convex superpotentials. The direct boundary integral equation method is extended for this problem. It gives rise to two equivalent multivalued integral equations holding on each crack. Numerical examples concerning the resulting stress intensity factors illustrate the theory.

Effects of wall flexibility on the dynamic response of liquid storage tanks

A knowledge of the free vibration characteristics of tank — liquid systems is essential in order to predict dynamic responses. It has been demonstrated that in a dynamic situation impulsive forces are significant because of the elastic compliance of the tank wall. In this paper, the effects of wall flexibility on the dynamic response of liquid storage tanks is considered. Variational principles are used to obtain a functional describing coupled oscillations between a linear elastic body and a liquid of small wave heights. A complementary Rayleigh's quotient has been introduced to obtain coupled natural frequencies and the dynamic pressure distribution in the axisymmetric natural modes of vibration. Both low frequencies (liquid sloshing mode) and high frequencies (tank bulging mode) have been included in this study.

Integrated topology and boundary shape optimization of 2-D solids

This study is concerned with the development of an integrated procedure for the computation of the optimal topology as well as the optimal boundary shape of a two-dimensional, linear elastic body. The topology is computed by regarding the body as a domain of the plane with a high density of material and the objective is to maximize the overall stiffness, subject to a constraint on the material volume of the body. This optimal topology is then used as the basis for a shape optimal design method that regards the body as given by boundary curves. For this case the objective is to minimize the maximum value of the Von Mises equivalent stress in the body, subject to an isoperimetric constraint on the area as well as a constraint on the stiffness. The solution procedures for the shape design are based on variational formulations for the problems and the results of a variational analysis are implemented via finite element discretizations. The discretization grids are generated automatically by an elliptical method for general two-dimensional domains. Computational results are presented for the design of a fillet, a beam and a portal frame.

Analytical method for calculating adhesive joint fracture parameters

By treating planar adhesive joints as being comprised of one or more generic adhesive sandwiches, beam theory and the J- integral have been used to develop analytical solutions for the generalized strain energy release rate of the adhesive layer. The calculations proceed from a knowledge of the reactions in the adherends at the ends of the adhesive sandwich, and allow for variations in stress through the thickness of the adhesive layer which is assumed to be in plane strain. Nonlinearities due to geometric and material response are accommodated, and a closedform solution is obtained. The method compares well with published finite element and compliance results for the strain energy release rate of joints under pure mode I, pure mode II and mixed mode conditions. The J-integral and the principle of superposition in a linear elastic body were then used to calculate the mode I and mode II stress intensity factors in a generic cracked adhesive sandwich. The method provides a closed-form solution for the stress intensity factors of linear elastic adhesive sandwiches having adherends with approximately equal ratios of thickness to flexural rigidity. In the case of dissimilar adherends, a system of linear differential equations is solved to estimate the required stresses in the adhesive layer.

Design sensitivity coefficients for linear elastic bodies with zones and corners by the derivative boundary element method

The subject of this paper is the efficient and accurate determination of design sensitivity coefficients (DSCs) in an elastic solid with zones and corners. Direct differentiation of the relevant derivative boundary element method (DBEM) formulation of the problem is carried out here. Corners and zones in a body are treated carefully with conforming elements. A numerical implementation of the method is carried out with isoparametric quadratic boundary elements. The power of this approach is demonstrated through several numerical examples. It is shown that the DSCs of the various mechanical quantities of interest are obtained accurately and efficiently. In one example, even the DSC of the stress at a corner, with respect to the corner angle, is obtained very accurately by this method.

Existence theorems for contact with friction of two linear elastic bodies

In this article, I give an iteration method for solving the Signorini problem by the idea of concentration cancellation. The nonlinear quantity in the mathematical formulation of the problem caused by contact-friction effect on the boundary is relaxed to a convex functional. Therefore, the relaxed problem can be solved by using convex minimization techniques which are numerically stable. Finally, good convergence property of the proposed method is proved.

Integral parameters for thermal fracture J′, θ and [formula omitted

This paper discusses the difference between parameters J', J θ and J ∗ in the case of linear elastic bodies under heat loading conditions. It shows for non-singular temperature at the crack tip that (i) J' ≈ J θ ; (ii) for temperature-dependent elastic parameters, J' and/or J θ must be evaluated in an incremental way; (iii) for constant elastic parameters J' and/or J θ are equal to the values obtained from the current state and they are equal to J ∗ . The physical meaning of the above mentioned parameters is discussed.

Initiation, Propagation, and Kinking of an In-Plane Crack

Consider an infinite linear elastic body containing a semi-infinite crack loaded by a longitudinal stress pulse parallel to the crack face. After the stress wave strikes the crack at time t = 0, the stress gradually intensifies at the crack tip. At some finite delay time tt, the crack begins to extend straight ahead with constant speed vo. At some later time tb, the crack suddenly stops, then kinks and propagates with constant speed vc, making an angle δ with the original crack. The exact full field solution of the propagating crack is constructed by using a superposition of fundamental solutions for a particular class of problems. When the crack suddenly stops, the stress field for a stationary crack with the new crack length is radiated out from the stopped crack tip. The dynamic stress intensity factor at the kinked crack tip can then be obtained by using a perturbation method.

Shape optimal design of elastic bodies using a mixed variational formulation

This study is concerned with the development of a variational formulation and a procedure for the computational solution for the shape optimal design of a two-dimensional linear elastic body. The objective is to minimize the maximum value of the Von Mises equivalent stress in the body, subject to an isoperimetric constraint on the area. The optimality conditions for this problem are derived using a mixed variational formulation, where the equations defining the elastostatics problem are dealt with as additional constraints for the optimization. The results of the analysis are implemented via a finite element discretization. The discretized model is tested in two numerical examples, the shape optimization of a hole in a biaxially loaded sheet, and of the design of a fillet.

Free vibration analysis of liquid storage tanks

To obtain the response of structures subjected to dynamic loading a knowledge of the free vibration characteristics of the structure is essential. The enclosed fluid in the tank exerts inertial loading on the vessel, and part of the fluid may be coupled to the motion of the vessel. for partially filled tanks/shells the free surface motions of the fluid may be coupled with the tank/shell motion, further complicating the analysis. In this paper, variational principles have been used to obtain a functional describing coupled oscillations between a linear elastic body and a liquid of small wave heights. A complementary Rayleigh's quotient has then been introduced to obtain the natural frequencies of circular cylindrical tanks partially filled with liquid and oscillating in an axisymmetric manner.