Stability of an elastic cube under dead loading: Two equal forces

A cube of incompressible neo-Hookean material undergoes a pure homogeneous deformation and is held in equilibrium by three specified pairs of equal and opposite forces, two of which are the same, applied normally to its faces and uniformly distributed over them. The possible equilibrium states are determined and the stability of each is studied with respect to arbitrary superposed infinitesimal deformations. The stability limits are found to be different from those obtained when only infinitesimal deformations having the same principal directions as those of the basic equilibrium state are considered. The differences arise from rotational and shearing types of instabilities that may occur in the general case. A critical inference is drawn concerning the nature of the dead loading conditions employed.     

Bifurcation of rotating thick-walled elastic tubes

The deformation of a circular cylindrical elastic tube of finite wall thickness rotating about its axis is examined. A circular cylindrical deformed configuration is considered first, and the angular speed analysed as a function of an azimuthai deformation parameter at fixed axial extension for an arbitrary form of incompressible, isotropic elastic strain-energy function. This extends the analysis given previously (Haughton and Ogden, 1980) for membrane tubes.Bifurcation from a circular cylindrical configuration is then investigated. Prismatic, axisymmetric and asymmetric bifurcation modes are discussed separately. Their relative importance is assessed in relation to the wall thickness and length of the tube, the magnitude of the axial extension, and the angular speed turning-points. Numerical results are given for a specific form of strain-energy function.Amongst other results it is found that (i) for long tubes, asymmetric modes of bifurcation can occur at low values of the angular speed and before any possible axisymmetric or prismatic modes and (ii) for short tubes, there is a range of values of the axial extension (including zero) for which no bifurcation can occur during rotation.     

Shear-band bifurcation of an isotropic compressible hyperelastic solid

This paper concerns shear-band bifurcations from the homogeneous finite plane deformation of an isotropic compressible hyperelastic solid. The governing equations for the incremental plane deformation superposed on the initial finite deformation are derived and then the equilibrium equations in terms of incremental displacements are classified into the elliptic type, parabolic type, etc. From this classification follows a restriction which should be placed on the strain-energy function in order that the equilibrium equations may be either elliptic or parabolic for all principal stretches. For the hyperelastic solid complying with this restriction, the condition for the shear-band bifurcation is obtained. Finally, the incremental displacement field of an infinite series of shear bands in a slab sandwiched between slippery rigid layers is established.     

Simple shear is not so simple

For homogeneous, isotropic, non-linearly elastic materials, the form of the homogeneous deformation consistent with the application of a Cauchy shear stress is derived here for both compressible and incompressible materials. It is shown that this deformation is not simple shear, in contrast to the situation in linear elasticity. Instead, it consists of a triaxial stretch superposed on a classical simple shear deformation, for which the amount of shear cannot be greater than 1. In other words, the faces of a cubic block cannot be slanted by an angle greater than 45° by the application of a pure shear stress alone. The results are illustrated for those materials for which the strain-energy function does not depend on the principal second invariant of strain. For the case of a block deformed into a parallelepiped, the tractions on the inclined faces necessary to maintain the derived deformation are calculated.     

Non-axisymmetric warping of a heavy circular plate on a flexible ring support

In this paper, we study the deformation and stability of a circular plate under its own weight and supported by a flexible concentric ring. Both bilateral and unilateral supports are considered. Von Karman’s plate model is adopted to formulate the equations of motion. A nonlinear Galerkin’s method based on two sets of assumed functions is used to discretize and solve the governing equations. Vibration method is used to predict the stability of the deformations. The linear analysis conducted previously predicts that the deformation is always axisymmetric. The current nonlinear analysis, however, shows that the axisymmetric deformation may become unstable when the dimensionless load, i.e., a ratio between the weight per unit area and the flexural rigidity of the plate, reaches a critical value. At this critical load, a stable non-axisymmetric deformation of the form cos nθ emerges following a pitch-fork bifurcation, where the integer n depends on the stiffness and the radius of the ring support. When the load increases further, more than one stable non-axisymmetric deformation may coexist. In a stable non-axisymmetric deformation with bilateral support, tension on the ring support may develop when the load reaches another critical value. In this situation, the circular plate will separate from the supporting ring in part of the angular region if the bilateral support is replaced by a unilateral one. The deformation with unilateral support is in general larger than the one with bilateral support.     

Some New Implications of Certain Constitutive Inequalities in Plane Isotropic Elasticity

In plane isotropic elasticity a strengthened form of the Ordered–Forces inequality is shown to imply that the restriction of the strain-energy function to the class of deformation gradients which share the same average of the principal stretches is bounded from below by the strain energy corresponding to the conformal deformations in this class. For boundary conditions of place, this property (together with a certain version of the Pressure–Compression inequality) is then used (i) to show that the plane radial conformal deformations are stable with respect to all radial variations of class C ¹ and (ii) to obtain explicit lower bounds for the total energy associated with arbitrary plane radial deformations. For the same type of boundary conditions and together with a different version of the Pressure–Compression inequality, an analogous property in plane isotropic elasticity (established in [3] under the assumption that the material satisfies a strengthened form of the Baker–Ericksen inequality and according to which the restriction of the strain-energy function to the class of deformation gradients which share the same determinant is bounded from below by the strain energy corresponding to the conformal deformations in that class) is used (i) to show that the plane radial conformal deformations are stable with respect to all variations of class C ¹ and (ii) to obtain explicit lower bounds for the total energy associated with any plane deformation.     

Asymmetric bifurcations in a pressurised circular thin plate under initial tension

It has been known for some time that under certain circumstances the axisymmetric solution describing the deformation experienced by a stretched circular thin plate or membrane under sufficiently strong normal pressure does not represent an energy-minimum configuration. By using the method of adjacent equilibrium a set of coordinate-free bifurcation equations is derived here by adopting the Föppl–von Kármán plate theory. A particular class of asymmetric bifurcation solutions is then investigated by reduction to a system of ordinary differential equations with variable coefficients. The localised character of the eigenmodes is confirmed numerically and we also look briefly at the role played by the background tension on this phenomenon.     

Finite deformation theory for soft ferromagnetic elastic solids

A general theory of finite deformation of soft ferromagnetic elastic solids is formulated following the linear theory developed earlier by Pao and Yeh. The constitutive equations, field equations, and the boundary conditions of this theory are applied to analyse the buckling of a plate under the action of a uniform magnetic field. A nontrivial equilibrium configuration for the deformed plate is shown to exist, and the critical value of the externally applied magnetic induction at which the plate buckles is determined. It is demonstrated that the non-linear deformation affects the critical magnetic induction considerably.     

Rapid Sliding Indentation with Friction of a Pre-Stressed Thermoelastic Material

A rigid indentor travels with a constant speed over the surface of an isotropic thermoelastic half-space. Friction exists between the indentor and half-space, and the latter is initially in equilibrium at a uniform temperature under a uniform normal pre-stress. This pre-stress, below but near yield, is assumed to produce deformations that dominate the additional deformations due to indentation. Thus, the process is treated as one of small deformations superposed upon (relatively) large. The governing equations for the superposed deformation are those of nonisotropic coupled thermoelasticity. A steady-state two-dimensional study uses robust asymptotic analytical solutions to reduce the associated mixed boundary value problem to a classical singular integral equation which can be solved analytically. The solutions show that the pre-stress-induced de facto nonisotropy alters the values of the rotational and dilatational wave and Rayleigh speeds in the half-space and, in the case of a compressive pre-stress, generates a second, lower, critical speed. In addition, pre-stress generates noncritical sliding speeds at which the friction-dependent integral equation eigenvalue changes sign. For purposes of illustration, expressions for the half-space surface temperature change and its average over the contact zone, the equations necessary to determine contact zone size and location, the resultant moment on the indentor, and the maximum compressive stress on the contact zone are presented for a parabolic indentor. This revised version was published online in August 2006 with corrections to the Cover Date.     

Postbuckling of FGM plates with piezoelectric actuators under thermo-electro-mechanical loadings

A postbuckling analysis is presented for a simply supported, shear deformable functionally graded plate with piezoelectric actuators subjected to the combined action of mechanical, electrical and thermal loads. The temperature field considered is assumed to be of uniform distribution over the plate surface and through the plate thickness and the electric field considered only has non-zero-valued component E_Z. The material properties of functionally graded materials (FGMs) are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents, and the material properties of both FGM and piezoelectric layers are assumed to be temperature-dependent. The governing equations are based on a higher order shear deformation plate theory that includes thermo-piezoelectric effects. The initial geometric imperfection of the plate is taken into account. Two cases of the in-plane boundary conditions are considered. A two step perturbation technique is employed to determine buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling behavior of perfect and imperfect, geometrically mid-plane symmetric FGM plates with fully covered or embedded piezoelectric actuators under different sets of thermal and electric loading conditions. The effects played by temperature rise, volume fraction distribution, applied voltage, the character of in-plane boundary conditions, as well as initial geometric imperfections are studied.     

On the adjacent equilibrium method in the stability theory of conservative elastic continua

The relationship of the adjacent equilibrium method, the regular perturbation method and the energy method for neutral equilibrium is studied. It is shown that unlike the adjacent equilibrium method, the regular perturbation method yields, for the problems under consideration, non-homogeneous perturbation equations and that adjacent states of equilibrium do not exist at the bifurcation point. These results are then compared with the result of the energy criterion for neutral equilibrium V₂[u] = 0. It is found that although the physical arguments are different in the three methods, the resulting stability equations are identical; thus showing why the adjacent equilibrium argument, even for cases when it is incorrect, yields correct critical loads. This is followed by a discussion of an incorrect derivation of a stability condition and a notion about a load type introduced in the stability literature, which are consequences of the assumption of the general existence of adjacent equilibrium states at bifurcation points.     

A comprehensive analysis of functionally graded sandwich plates: Part 1—Deflection and stresses

A two-dimensional solution is presented for bending analysis of simply supported functionally graded ceramic–metal sandwich plates. The sandwich plate faces are assumed to have isotropic, two-constituent material distribution through the thickness, and the modulus of elasticity and Poisson’s ratio of the faces are assumed to vary according to a power-law distribution in terms of the volume fractions of the constituents. The core layer is still homogeneous and made of an isotropic ceramic material. Several kinds of sandwich plates are used taking into account the symmetry of the plate and the thickness of each layer. We derive field equations for functionally graded sandwich plates whose deformations are governed by either the shear deformation theories or the classical theory. Displacement functions that identically satisfy boundary conditions are used to reduce the governing equations to a set of coupled ordinary differential equations with variable coefficients. Numerical results of the sinusoidal, third-order, first-order and classical theories are presented to show the effect of material distribution on the deflections and stresses.     

Finite deformations of fibre-reinforced elastic solids with fibre bending stiffness

In the conventional theory of finite deformations of fibre-reinforced elastic solids it is assumed that the strain-energy is an isotropic invariant function of the deformation and a unit vector A that defines the fibre direction and is convected with the material. This leads to a constitutive equation that involves no natural length. To incorporate fibre bending stiffness into a continuum theory, we make the more general assumption that the strain-energy depends on deformation, fibre direction, and the gradients of the fibre direction in the deformed configuration. The resulting extended theory requires, in general, a non-symmetric stress and the couple-stress. The constitutive equations for stress and couple-stress are formulated in a general way, and specialized to the case in which dependence on the fibre direction gradients is restricted to dependence on their directional derivatives in the fibre direction. This is further specialized to the case of plane strain, and finite pure bending of a thick plate is solved as an example. We also formulate and develop the linearized theory in which the stress and couple-stress are linear functions of the first and second spacial derivatives of the displacement. In this case for the symmetric part of the stress we recover the standard equations of transversely isotropic linear elasticity, with five elastic moduli, and find that, in the most general case, a further seven moduli are required to characterize the couple-stress.     

The Valanis-Landel Strain-Energy Function

A constitutive equation is derived for the Cauchy stress matrix for arbitrary deformations of an isotropic elastic solid characterized by a Valanis-Landel strain-energy function. A simple example is given of the way in which results for controllable deformations of an incompressible elastic solid, with a Valanis-Landel strain-energy function, can be obtained from the known results for the more general strain-energy function employed by Rivlin.     

On SH waves in a pre-stressed layered half-space for an incompressible elastic material

The propagation of Love waves along the boundary between a half-space and a layer of different pre-stressed material is examined for incompressible isotropic elastic materials. The secular equation is obtained for a general strain-energy function and analysed for particular deformations and materials. For the neo-Hookean strain-energy function, numerical results are obtained to illustrate the dependence of the wavespeed on the wave number and on the deformation.     

Constitutive branching analysis of cylindrical bodies under in-plane equibiaxial dead-load tractions

Finite homogeneous deformations of hyperelastic cylindrical bodies subjected to in-plane equibiaxial dead-load tractions are analyzed. Four basic equilibrium problems are formulated considering incompressible and compressible isotropic bodies under plane stress and plane deformation condition. Depending on the form of the stored energy function, these plane problems, in addition to the obvious symmetric solutions, may admit asymmetric solutions. In other words, the body may assume an equilibrium configuration characterized by two unequal in-plane principal stretches corresponding to equal external forces. In this paper, a mathematical condition, in terms of the principal invariants, governing the global development of the asymmetric deformation branches is obtained and examined in detail with regard to different choices of the stored energy function. Moreover, explicit expressions for evaluating critical loads and bifurcation points are derived. With reference to neo-Hookean, Mooney-Rivlin and Ogden-Ball materials, a broad numerical analysis is performed and the qualitatively more interesting asymmetric equilibrium branches are shown. Finally, using the energy criterion, a number of considerations are put forward about the stability of the computed solutions.     

弹性地基加肋功能梯度板自由振动分析的无网格法

基于一阶剪切变形理论和移动最小二乘近似研究Winkler弹性地基上加肋功能梯度板的固有频率。假设功能梯度板的材料性质沿厚度方向按幂函数连续变化,基于物理中面和移动最小二乘近似分别推导功能梯度板和肋条的动能和势能,再通过引入位移协调条件,建立板和肋条节点参数转换关系,叠加两者的总能量,然后利用Hamilton原理推导加肋功能梯度板自由振动控制方程。采用完全转换法施加边界条件。通过将本文的计算结果与有限元以及文献的结果对比,验证方法的收敛性以及准确性。     

Thermal buckling and post-buckling response of imperfect temperature-dependent sandwich FGM plates resting on elastic foundation

Post-buckling behaviour of sandwich plates with functionally graded material (FGM) face sheets under uniform temperature rise loading is considered. It is assumed that the plate is in contact with a Pasternak-type elastic foundation during deformation, which acts in both compression and tension. The derivation of equations is based on the first-order shear deformation plate theory. Thermomechanical non-homogeneous properties of FGM layers vary smoothly by the distribution of power law across the thickness, and temperature dependency of material constituents is taken into account. Using the non-linear von-Karman strain-displacement relations, the equilibrium and compatibility equations of imperfect sandwich plates with FGM face sheets are derived. The boundary conditions for the plate are assumed to be simply supported in all edges. The governing equations are reduced to two coupled equation in terms of stress function and lateral deflection. Employing the single mode approach combined with Galerkin technique, an approximate closed-form solution is presented to calculate the critical buckling temperature and post-buckling equilibrium path of the plate. Presented numerical examples contain the influences of power law index, sandwich plate geometry, geometrical imperfection, temperature dependency, and the elastic foundation coefficients.     

On the General Structure of Small on Large Problems for Elastic Deformations of Varga Materials II: Axially Symmetric Deformations

In Part I of this article, we have formulated the general structure of the equations governing small plane strain deformations which are superimposed upon a known large plane strain deformation for the perfectly elastic incompressible 'modified' Varga material, and assuming only that the initial large plane deformation is a known solution of one of three first integrals previously derived by the authors. For axially summetric deformations there are only two such first integrals, one of which applies only to the single term Varga strain-energy function, and we give here the corresponding general equations for small superimposed deformations. As an illustration, a partial analysis for the case of small deformations superimposed upon the eversion of a thick spherical shell is examined. The Varga strain-energy functions are known to apply to both natural and synthetic rubber, provided the magnitude of the deformation is restricted. Their behaviour in both simple tension and equibiaxial tension, and in comparison to experimental data, is shown graphically in Part I of this paper [1]. This revised version was published online in August 2006 with corrections to the Cover Date.     

The plane elasticity problem of an isotropic wedge under normal and shear distributed loading––application in the case of a multi-material problem

The problem of a multi-material composite wedge under a normal and shear loading at its external faces is considered with a variable separable solution. The stress and displacement fields are determined using the equilibrium conditions for forces and moments and the appropriate Airy stress function. The infinite isotropic wedge under shear and normal distributed loading along its external faces is examined for different values of the order n of the radial coordinate r. The proposed solution is applied to the elastostatic problem of a composite isotropic k-materials infinite wedge under distributed loading along its external faces. Applications are made in the case of the two-materials composite wedge under linearly distributed loading along its external faces and in the case of a three-materials composite wedge under a parabolically distributed loading along its external faces.