Coaxiality of strain and stress in anisotropic linear elasticity

For a linearly elastic anisotropic body there are at least two rotations of the principal axes of strain such that the stress and strain tensors become coaxial. These rotations correspond to critical points for the stored energy, viewed as a function of the relative orientation between the body and the strain tensor.Supported by Gruppo Nazionale per la Fisica Matematica of C.N.R. (Italy).     

Strain energy density bounds for linear anisotropic elastic materials

Upper and lower bounds are presented for the magnitude of the strain energy density in linear anisotropic elastic materials. One set of bounds is given in terms of the magnitude of the stress field, another in terms of the magnitude of the strain field. Explicit algebraic formulas are given for the bounds in the case of cubic, transversely isotropic, hexagonal and tetragonal symmetry. In the case of orthotropic symmetry the explicit bounds depend upon the solution of a cubic equation, and in the case of the monoclinic and triclinic symmetries, on the solution of sixth order equations.     

Stationarity of the strain energy density for some classes of anisotropic solids

Homogeneous, anisotropic and linearly elastic solids, subjected to a given state of strain (or stress), are considered. The problem dealt with consists in finding the mutual orientations of the principal directions of strain to the material symmetry axes in order to make the strain energy density stationary. Such relative orientations are described through three Euler’s angles. When the stationarity problem is formulated for the generally anisotropic solid, it is shown that the necessary condition for stationarity demands for coaxiality of the stress and the strain tensors. From this feature, a procedure which leads to closed form solutions is proposed. To this end, tetragonal and cubic symmetry classes, together with transverse isotropy, are carefully dealt with, and for each case all the sets of Euler’s angles corresponding to critical points of the energy density are found and discussed. For these symmetries, three material parameters are then defined, which play a crucial role in ordering the energy values corresponding to each solution.     

Optimization of the stored energy and coaxiality of strain and stress in finite elasticity

The strain energy density of a hyperelastic anisotropic body which is rotated before being subjected to a given but arbitrary deformation is viewed as a smooth function defined on the group of rotations, parametrized by the deformation gradient. It is shown that the critical points of this function correspond to rotations which, when composed with the prescribed deformation, yield a total strain tensor which is coaxial with the corresponding stress. For any type of material symmetry, there are at least two such rotations. Coaxiality of stress and strain for all deformations is shown to be a sufficient condition for the isotropicity of hyperelastic materials.Research supported by GNFM of CNR (Italy).     

The combined energy variational principles in linear elasticity

In this paper, a new kind of mixed energy variational principles in linear elasticity—the combined energy variational principles is presented. First, we define the conjugate body of an elastic body, which is obtained by changing the boundary conditions of the elastic body. Next, we decompose the conjugate body into two component-states, construct functionals of potential energy and complementary energy, respectively, for the component-states and define the additional hybrid energy between the component-states. Thus the functionals of combined energy can be constructed. Three typical combined energy variational principles are demonstrated and the other forms of combined energy variational principles are given. The application of the proposed principles to the calculation of thin plates with complicated boundaries is shown.     

Conservation laws in elasticity. III. Planar linear anisotropic elastostatics

The first order conservation laws for an arbitrary homogeneous linear planar elastic material are completely classified. In all cases, both isotropic and anisotropic, besides the standard Betti reciprocity laws, there are two infinite-dimensional families of conservation laws, each depending on an arbitrary analytic function of two complex variables.     

The completeness and a new derivation of the Stroh formalism of anisotropic linear elasticity

In this paper we present a new, simpler and unified derivation of the Stroh formalism of anisotropic linear elasticity, for both nondegenerate and degenerate cases. It is based on the potential representation and Jordan canonical representation theorems. The completeness of the Stroh formalism is proved in the derivation process itself. This new approach is also extended to piezoelastic problems. Besides, we show that the eigenvalues of the fundamental elastic matrix in planar anisotropic elasticity are always distinct, except for the case of isotropy. The project supported by the National Natural Science Foundation of China (19525207 and 19891180).     

Integral bounds on the strain energy for the traction problem in finite elasticity

For the traction boundary value problem in finite elasticity, a bound is obtained for the total strain energy in terms of the L₂ integral norms of the surface tractions and body forces, under the assumptions that the unstressed body occupies a convex domain and the displacement gradients are sufficiently small.This is an extension of known results in linear (infinitesimal) elasticity into finite elasticity.     

Hamiltonian formulation of anisotropic elasticity

The general expression of Hamiltonian partial differential equation for 3-D anisotropic elasticity is derived in this paper. It is a two-variable mixed formulation based on which new analytic and semi-analytic methods can be introduced.     

Periodic elliptic holes in anisotropic elasticity

The problem of collinear periodic elliptic holes in an anisotropic medium is examined in this paper. By means of Stroh formalism and the conformal mapping method, explicit full domain solutions for the periodic hole problems are presented. The solutions are valid not only for plane problems but also for antiplane problems and the problems whose implane and antiplane deformations are coupled. The stress concentration around the holes is analysed.     

An integral bound for the strain energy in nonlinear elasticity in terms of the boundary displacements

For the displacement boundary value problem in nonlinear elastostatics with zero body force, an integral bound for the strain energy is obtained in terms of theL ₂-norms of the given boundary displacements and their tangential derivatives (assumed sufficiently small). The constants involved depend upon the strain energy density function and upon the geometry of the domain.     

Application of the strain energy density criterion to ductile fracture

The strain energy density criterion due to Sih is used to predict fracture loads of two thin plates subjected to large elastic-plastic deformation. The prediction is achieved with a finite element analysis which is based on Hill's variational principle for incremental deformations capable of solving gross yielding problems involving arbitrary amounts of deformation. The computed results are in excellent agreement with those obtained in Sih's earlier analysis and with an experimental observation.     

Local strain energy density near sharp V-notches in plates

Conditions of fracture of the local strain energy density, which were first formulated by Sih for a sharp V-notch with an arbitrary tip angle, are proposed. The edges of the considered V-notch are free from loading. If loading schemes of type I and type II are used, it is shown that the known brittle fracture conditions proposed by Sih contradict one of the basic postulates in fracture mechanics: the greater the intensity of stresses or elastic energy near the V-notch tip, the greater the probability of crack propagation. The proposed new conditions of fracture (in a polar coordinate system) are obtained as a result of independent determination of the energy densities of changes in volume and shape. In this case, the above-mentioned contradiction is eliminated.     

Equations of the linear theory of elasticity of anisotropic materials,reduced to three independent wave equations

Novosibirsk. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, No. 6, pp. 143–150, November–December, 1994.     

Theory of elasticity of anisotropic bodies

International Applied Mechanics -     

On displacement methods in planar anisotropic elasticity

Two displacement formulation methods are presented for problems of planar anisotropic elasticity. The first displacement method is based on solving the two governing partial differential equations simultaneously/ This method is a recapitulation of the orignal work of Eshelby, Read and Shockley [7] on generalized plane deformations of anisotropic elastic materials in the context of planar anisotropic elasticity.The second displacement method is based on solving the two governing equations separately. This formulation introduces a displacement function, which satisfies a fourth-order partial differential equation that is identical in the form to the one given by Lekhnitskii [6] for monoclinic materials using a stress function. Moreover, this method parallels the traditional Airy stress function method and thus the Lekhnitskii method for pure plane problems. Both the new approach and the Airy stress function method start with the equilibrium equations and use the same extended version of Green's theorem (Chou and Pagano [13], p. 114; Gao [11]) to derive the expressions for stress or displacement components in terms of a potential (stress or displacement) function (see also Gao [10, 11]). It is therefore anticipated that the displacement function involved in this new method could also be evaluated from measured data, as was done by Lin and Rowlands [17] to determine the Airy stress function experimentally.The two different displacement methods lead to two general solutions for problems of planar anisotropic elasticity. Although the two solutions differ in expressions, both of the depend on the complex roots of the same characteristic equation. Furthermore, this characteristic equation is identical to that obtained by Lekhnitskii [6] using a stress formulation. It is therefore concluded that the two displacement methods and Lekhnitskii's stress method are all equivalent for problems of planar anisotropic elasticity (see Gao and Rowlands [8] for detailed discussions).     

Crack growth instability studied by the strain energy density theory

The strain energy density theory has successfully been used to address the problem of material damage and structural failure in problems of engineering interest. The theory makes use of the strain energy density function, dW/dV, and focuses attention in its stationary values. The directions of crack growth and yielding are determined from the minimum and maximum values of dW/dV, respectively, along the circumference of a circle centered at the point of failure initiation. Failure by crack growth or yielding takes place when these values of dW/dV become equal to their critical values which are material constants. In the present work the basic principles of the strain energy density theory were reviewed. Furthermore, this theory was used to study three problems of structural failure, namely the problem of slow stable growth of an inclined crack in a plate subjected to uniaxial tension, the problem of fracture instability of a plate with a central crack and two notches, and the problem of unstable crack growth in a circular disc subjected to two equal and opposite forces. The results of stress analysis were combined with the strain energy density theory to obtain the whole history of crack growth from initiation to instability. A length parameter was introduced to define the fracture instability of a mechanical system. Fracture trajectories were obtained for fast unstable crack propagation.