Hypoelastic form of equations in nonlinear elasticity theory

It is shown that the complete system of equations of elasticity theory for an isotropic medium admits a unique representation in the hypoelastic form (the tensor of the rate of change of stresses is a linear function of the tensor of strain rates with coefficients depending on the invariants of the stress tensor). It is necessary to this end that the hypothesis be satisfied on the determination of strains by stresses which are unknown. Any arbitrariness in the choice of the coefficients of the hypoelastic relation may result in the thermodynamic identity being infringed.     

Determining the effective thermoelastic characteristics of regularly laminated composite in the asymmetric theory of elasticity

The asymmetric theory of elasticity is used to model a hybrid laminated composite of regular structure with all phases isotropic. The effective thermoelastic characteristics of the composite are determined. It is shown that the equations derived can be used to determine stress–strain state in all the phases of the composite using the average components of the tensors of force stresses, couple stresses, strains, and wryness in a layered material, which is of fundamental importance for the design of composites based on structural theories of failure     

A study of elasticity and plasticity equations under arbitrary displacements and strains

Equations of geometrically nonlinear theory of elasticity with finite displacements and strains are analyzed. The equations are composed using three versions of physical relations and applied to solve the problem of tension-compression of a straight bar. It is shown that the use of the classical relations between the components of the stress tensor and the Cauchy-Green strain tensor in the problem of compression of the bar results in the appearance of “spurious” static loss of stability such that the bar axis remains straight if the stresses are referred to unit areas before the deformation (conditional stresses). However, in the problem of tension, the classical relations do not permit one to describe the phenomenon of static instability (neck formation as the plastic instability occurs). These drawbacks disappear if one uses the third version of the physical equations, composed as relations between the true stresses referred to unit areas of the deformed faces on which they act and the true elongations and shears. The relations of the third version are most correct; they permit one to pass to self-consistent equations of elasticity and plasticity under small strains and finite displacements, and they should be recommended for practical use. As an example, such relations are composed for the flow theory.     

A Refined Shear Deformation Theory of Bending of Isotropic Plates*

ABSTRACT

A nonlinear, in-plane displacement assumption is proposed, based on an undetermined variation df/dz of transverse shear strains through the plate thickness. A second-order ordinary differential equation for f(z) and two surface conditions, as well as a set of eighth-order partial differential equations and four associated boundary conditions, are derived from the principle of minimum potential energy. Coupling exists between the partial and ordinary differential equations. In the homogeneous solutions for the former, in addition to an interior solution contribution, there exist two edge-zone solution contributions, one of which induces self-equilibrated (in the thickness direction) boundary stresses. Three examples are calculated using the present theory. The last gives the stress couple and maximum-stress concentration factors at the free edge of a circular hole in a large bent plate. Numerical results for the examples are compared with those given by three-dimensional elasticity theory and several two-dimensional theories. It is found that the present theory can accurately predict nonlinear variations of in-plane stresses through the thickness of a plate.     

A refined theory of elastic thick plates for extensional deformation

Based on elasticity theory, various two-dimensional (2D) equations and solutions for extensional deformation have been deduced systematically and directly from the three-dimensional (3D) theory of thick rectangular plates by using the Papkovich–Neuber solution and the Lur’e method without ad hoc assumptions. These equations and solutions can be used to construct a refined theory of thick plates for extensional deformation. It is shown that the displacements and stresses of the plate can be represented by the displacements and transverse normal strain of the midplane. In the case of homogeneous boundary conditions, the exact solutions for the plate are derived, and the exact equations consist of three governing differential equations: the biharmonic equation, the shear equation, and the transcendental equation. With the present theory a solution of these can satisfy all the fundamental equations of 3D elasticity. Moreover, the refined theory of thick plate for bending deformation constructed by Cheng is improved, and some physical or mathematical explanations and proof are provided to support our justification. It is important to note that the refined theory is consistent with the decomposition theorem by Gregory. In the case of nonhomogeneous boundary conditions, the approximate governing differential equations and solutions for the plate are accurate up to the second-order terms with respect to plate thickness. The correctness of the stress assumptions in the classic plane-stress problems is revised. In an example it is shown that the exact or accurate solutions may be obtained by applying the refined theory deduced herein.     

Modified equations of finite-size layered plates made of orthotropic material. Comparison of the results of numerical calculations with analytical solutions

This paper describes the modified bending equations of layered orthotropic plates in the first approximation. The approximation of the solution of the equation of the three-dimensional theory of elasticity by the Legendre polynomial segments is used to obtain differential equations of the elastic layer. For the approximation of equilibrium equations and boundary conditions of three-dimensional theory of elasticity, several approximations of each desired function (stresses and displacements) are used. The stresses at the internal points of the plate are determined from the defining equations for the orthotropic material, averaged with respect to the plate thickness. The construction of the bending equations of layered plates for each layer is carried out with the help of the elastic layer equations and the conjugation conditions on the boundaries between layers, which are conditions for the continuity of normal stresses and displacements. The numerical solution of the problem of bending of the rectangular layered plate obtained with the help of modified equations is compared with an analytical solution. It is determined that the maximum error in determining the stresses does not exceed 3 %.     

General solutions of weakened equations in terms of stresses in the theory of elasticity

The general solutions of some weakened systems of equations expressed in terms of stresses in the isotropic theory of elasticity are analyzed. These systems are not equivalent to the classical one and involve the equilibrium equations and only three of the six equations of compatibility (either diagonal or off-diagonal ones). In the framework of elasticity theory, an equivalence of the formulations of quasistatic boundary value problems based on such systems and expressed in terms of stresses is discussed.     

Near field behavior of in-plane crack extension in nonlinear incompressible material

The nonlinear theory of finite elasticity is applied to obtain the in-plane displacement and stresses in the immediate vicinity of the crack tip. Incompressibility, homogeneity, elasticity and isotropy are assumed for the material while the resultant shear stress and shear strain are assumed to follow a nonlinear hardening/softening behavior. The system of governing differential equations becomes nonelliptical when the strains are sufficiently large.     

Coordinates of principal stresses for elastic plane problem

I.IntroductionInelasticmechanics,thereisakindofproblemsthatcouldbesolveddirectlybyequilibriumequations,i.e.,whenal1oftheboundaryconditionsaretheknownstressesorforcessuchasthestressfunction.Becausestressfunctionsmustsatisfyharmonicequationorbi-harmonicequa…     

An asymmetric theory of nonlocal elasticity—Part 2. Continuum field

In this paper, an asymmetric theory of nonlocal elasticity with nonlocal body couple is developed on the basis of the axiom system in nonlocal continuum field theory. The Galileo invariance is used for determining the explicit form of the constitutive equations. It is shown that both continuum field theory and quasicontinuum theory give the same constitutive equations and field equations for the general theory of nonlocal elasticity. Finally, the relations among nonlocal theory, couple stress theory, and higher gradient theory are investigated.     

Stress distribution instability in the plane problem of the theory of elasticity of heterogeneous bodies

A possibility of degeneration of relationships between stresses and their derivatives with respect to the coordinates in the plane problem of the elasticity theory is considered. For particular dependences of the parameters of elasticity on coordinates, curves are given, for which degeneration conditions are satisfied. It is shown that even a small inhomogeneity of the medium causes stress instability.     

Waves in Constrained Linear Elastic Materials

This paper deals with the propagation of acceleration waves in constrained linear elastic materials, within the framework of the so-called linearized finite theory of elasticity, as defined by Hoger and Johnson in [12, 13]. In this theory, the constitutive equations are obtained by linearization of the corresponding finite constitutive equations with respect to the displacement gradient and significantly differ from those of the classical linear theory of elasticity. First, following the same procedure used for the constitutive equations, the amplitude condition for a general constraint is obtained. Explicit results for the amplitude condition for incompressible and inextensible materials are also given and compared with those of the classical linear theory of elasticity. In particular, it is shown that for the constraint of incompressibility the classical linear elasticity provides an amplitude condition that, coincidently, is correct, while for the constraint of inextensibility the disagreement is first order in the displacement gradient. Then, the propagation condition for the constraints of incompressibility and inextensibility is studied. For incompressible materials the propagation condition is solved and explicit values for the squares of the speeds of propagation are obtained. For inextensible materials the propagation condition is solved for plane acceleration waves propagating into a homogeneously strained material. For both constraints, it is shown that the squares of the speeds of propagation depend by terms that are first order in the displacement gradient, while in classical linear elasticity they are constant. This revised version was published online in July 2006 with corrections to the Cover Date.     

An approach based on distributed dislocations and disclinations for crack problems in couple-stress elasticity

The technique of distributed dislocations proved to be in the past an effective approach in studying crack problems within classical elasticity. The present work is intended to extend this technique in studying crack problems within couple-stress elasticity, i.e. within a theory accounting for effects of microstructure. This extension is not an obvious one since rotations and couple-stresses are involved in the theory employed to analyze the crack problems. Here, the technique is introduced to study the case of a mode I crack. Due to the nature of the boundary conditions that arise in couple-stress elasticity, the crack is modeled by a continuous distribution of climb dislocations and constrained wedge disclinations (the concept of ‘constrained wedge disclination’ is first introduced in the present work). These distributions create both standard stresses and couple stresses in the body. In particular, it is shown that the mode-I case is governed by a system of coupled singular integral equations with both Cauchy-type and logarithmic kernels. The numerical solution of this system shows that a cracked solid governed by couple-stress elasticity behaves in a more rigid way (having increased stiffness) as compared to a solid governed by classical elasticity. Also, the stress level at the crack-tip region is appreciably higher than the one predicted by classical elasticity.     

Elastoplastic Model of Rocks with a Linear Structural Parameter

A closed mathematical model is formulated, which takes into account elastoplastic strains and the medium capability of accumulating the energy of internal self-balanced stresses. Satisfaction of the diffeomorphism postulate (assumption of displacement field smoothness) is not required; as a result, the strains depend on the stresses and second derivatives of the stresses with respect to the coordinates. The model involves a linear structural parameter. Relations that take into account local bending of the elementary volumes of the medium are derived.     

Automated Stress Separation Along Stress Trajectories

A procedure for the separation of principal stresses in automated photoelasticity is presented. It is based on the integration of indefinite equations of equilibrium along stress trajectories, also known as Lamè–Maxwell equations. A new algorithm for precise and reliable stress trajectory calculation, which is an essential feature of the procedure, has also been developed. Automated stress separation is carried out along stress trajectories starting from free boundaries. Experimental tests were performed on a disc in diametral compression and on a ring with internally applied pressure. Full-field principal stress values were obtained and results were compared with those from the theory of elasticity and with those obtained from the classical shear difference method. It was shown that the proposed method is more accurate and less affected by the presence of residual stresses or experimental errors at the boundaries than the shear difference method. In addition, the method requires little human interaction and is therefore well-suited for automated photoelasticity.     

Local and Global Injective Solution of the Rotationally Symmetric Sphere Problem

There are problems in the classical linear theory of elasticity whose closed form solutions, while satisfying the governing equations of equilibrium together with well-posed boundary conditions, predict the existence of regions, often quite small, inside the body where material overlaps. Of course, material overlapping is not physically realistic, and one possible way to prevent it combines linear theory with the requirement that the deformation field be injective. A formulation of minimization problems in classical linear elasticity proposed by Fosdick and Royer [3] imposes this requirement through a Lagrange multiplier technique. An existence theorem for minimizers of plane problems is also presented. In general, however, it is not certain that such minimizers exist. Here, the Euler–Lagrange equations corresponding to a family of three-dimensional problems is investigated. In classical linear elasticity, these problems do not have bounded solutions inside a body of anisotropic material for a range of material parameters. For another range of parameters, bounded solutions do exist but yield stresses that are infinite at a point inside the body. In addition, these solutions are not injective in a region surrounding this point, yielding unrealistic behavior such as overlapping of material. Applying the formulation of Fosdick and Royer on this family of problems, it is shown that both the displacements and the constitutive part of the stresses are bounded for all values of the material parameters and that the injectivity constraint is preserved. In addition, a penalty functional formulation of the constrained elastic problems is proposed, which allows to devise a numerical approach to compute the solutions of these problems. The approach consists of finding the displacement field that minimizes an augmented potential energy functional. This augmented functional is composed of the potential energy of linear elasticity theory and of a penalty functional divided by a penalty parameter. A sequence of solutions is then constructed, parameterized by the penalty parameter, that converges to a function that satisfies the first variation conditions for a minimizer of the constrained minimization problem when this parameter tends to infinity. This approach has the advantages of being mathematically appealling and computationally simple to implement.     

Ideal nonsymmetric 4D-medium as a model of invertible dynamic thermoelasticity

The space-time continuum (4D-medium) is considered, and a generalized model of reversible dynamic thermoelasticity is constructed as a theory of elasticity of an ideal (defect-free) nonsymmetric 4D-medium that is transversally-isotropic with respect to the time coordinate. The definitions of stresses and strains for the space-time continuum are introduced. The constitutive equations of the medium model relating the components of nonsymmetric stress and distortion 4D-tensors are stated. Physical interpretations of all tensor components of the thermomechanical properties are given. The Lagrangian of the generalized model of coupled dynamic thermoelasticity is presented, and the Euler equations are analyzed. It is shown that the three Euler equations are generalized equations of motion of the dynamic classical thermoelasticity, and the last, fourth, equation is a generalized heat equation which allows one to predict the wave properties of heat. An energy-consistent version of thermoelasticity is constructed where the Duhamel-Neumann and Maxwell-Cattaneo laws (a nonclassical generalization of the Fourier law for the heat flow) are direct consequences of the constitutive equations.     

Reduction of dimensions in random,elastic soil medium

The paper deals with an application of the plane strain analysis in a stochastic three-dimensional soil medium. In a framework of random elasticity theory, the geostatical state of stresses and the problem of a unit force acting in a statistically homogeneous half-space are considered. Only the modulus of elasticity is considered to be random and is modelled as a three-dimensional (3-D) homogeneous random field. As the result of imposed constrains due to the plane strain assumption the additional body and surface forces are induced. In order to determine them, additional equations must be introduced. The equations in a form of constrain relations are proposed in this paper. These equations are also valid for a case of uniformly distributed external loading.First, the two-dimensional (2-D) problem and its reduction to the uni-axial strain state, for the gravity forces and uniform, unlimited surface loading is considered. Then, it is generalised into a 2-D schematization of the 3-D state. Next, the problem of a unit force acting in a statistically homogeneous half-space is considered. For a 3-D state of stress and strain the resulting stresses are compared with those for a 2-D state. These stresses for the multidimensional state of strain and stress are presented as a sum of two components. The first one reflects plane strain state stresses and is given in a form of a 3-D random field. This term allows for incorporating a spatial, 3-D soil variability into a two-dimensional analysis. The second component can be treated as a correction term and it represents the longitudinal influence of a 3-D analysis.Some numerical results are presented in this paper. The proposed method can be regarded as a framework for further research aiming at application to a variety of geotechnical problems, for which the plane strain state is assumed.     

Exact three-dimensional analysis for static torsion of piezoelectric rods

Based on the theory of elasticity, exact analytical and numerical solutions of piezoelectric rods under static torsion are studied. In this paper, direct solution method is used. The main scope is to check the extension of validity of assumptions in previous papers that had been made based on linear distribution of electric potential through the cross section and their influences on deflection and the angle of rotation. Stress and electric induction functions are employed to obtain the exact solution of the static and electrostatic equilibrium equations under torsional loading. It is shown that previous assumptions are valid only in some types of piezoelectric materials, while in other types these assumptions lead to considerable deviations from accurate modeling. The present analytical solutions are compared with three-dimensional finite element analysis results and absolute agreements are found. At the end of this article, torsional rigidity, shape-effects on induced piezoelectric deformation and the range of valid region for linear distribution of electric potential assumption have been studied.