Restrictions on nonlinear constitutive equations for elastic rods

Constitutive equations for the resultant forces and moments applied to a rod-like body necessarily couple the influences of the rod geometry and the constitutive nature of the three-dimensional material from which the rod was constructed. Consequently, even when the nonlinear constitutive equation of the three-dimensional material is known, the influence of the rod geometry on the constitutive response of the rod is not known. The main objective of this paper is to develop restrictions on the constitutive equations of nonlinear elastic rods which ensure that exact solutions of the rod equations are consistent with exact nonlinear solutions of the three-dimensional equations for all homogeneous deformations. Since these restrictions are nonlinear in nature they provide valuable general theoretical guidance for specific constitutive assumptions about the coupling of material and geometric properties of rods. Also, an example of a straight beam clamped at one end and subjected to a shear force at the other end is used to examine the validity of the proposed value for the transverse shear deformation coefficient.     

On minimal representations for constitutive equations of anisotropic elastic materials

The problem of determining minimal representations for anisotropic elastic constitutive equations is proposed and investigated. For elastic constitutive equations in any given case of anisotropy, it is shown that there exist generating sets consisting of six generators and such generating sets are minimal in all possible generating sets. This fact implies that most of the established results for representations of elastic constitutive equations are not minimal and remain to be sharpened. For elastic constitutive equations in some cases of anisotropy, including orthotropy, transverse isotropy, the trigonal crystal class S ₆, and the classes C _2mh , m=1, 2, 3,..., etc., representations in terms of minimal generating sets are presented for the first time.     

Remarks on ellipticity for the generalized Blatz-Ko constitutive model for a compressible nonlinearly elastic solid

One of the most widely used constitutive models for compressible isotropic nonlinearly elastic solids is the generalized Blatz-Ko material for foam-rubber and its various specializations. For this model, a unified derivation of necessary and sufficient conditions for ellipticity of the governing three-dimensional displacement equations of equilibrium is provided. When the parameterf occurring in the generalized Blatz-Ko model is in the range 0f<1, it is shown that ellipticity is always lost at sufficiently large stretches, while forf=1, the equilibrium equations are globally elliptic. The implications of these results for a variety of physical problems are discussed.     

Non-linear elastic constitutive equations

The non-linear elastic constitutive equations proposed by Kauderer, [1], are a particular case of a class of quasi-linear equations. They cannot give an account of some specific non-linear effects of the Poynting and the Kelvin types. It is more useful in two and three dimensional problems to start with general non-linear forms proposed initially by Brillouin, [2].     

The spreading of nonlinear elastic surface waves

A theory for the lateral spreading of a beam of nonlinear surface acoustic waves across the surface of an arbitrary, homogeneous, elastic half-space is developed. The resulting evolution equation generalizes that obtained for uni-directional waves by replacing an ordinary derivative by a diffusion operator of Schrödinger type. The coefficients arising in the evolution equation are related to partial derivatives of the dispersion relation for linearized surface waves on the half space. Details are given for isotropic materials and for two special cases of beams travelling along axes of high elastic symmetry.     

Non-linear modal equations for thin elastic shells

In this paper we derive non-linear modal equations for thin elastic shells of arbitrary geometry. Geometric non-linearities are accounted for by utilizing the strain-displacement relations of the Sanders-Koiter non-linear shell theory. Arbitrary initial imperfections are accounted for and the shell thickness is free to vary within the limits of thin shell theory. The derivation gives the coefficients of the modal equations as integral expressions over the surface of the shell. The resulting equations are well-suited for practical applications. Weighting factors are introduced to allow for reduction of our results to the Love shell theory and to the Donnell approximation. The equations are specialized for a finite simply supported circular cylinder and numerical results are compared to those previously published in the literature.     

Drilling couples and refined constitutive equations in the resultant geometrically non-linear theory of elastic shells

It is well known that distribution of displacements through the shell thickness is non-linear, in general. We introduce a modified polar decomposition of shell deformation gradient and a vector of deviation from the linear displacement distribution. When strains are assumed to be small, this allows one to propose an explicit definition of the drilling couples which is proportional to tangential components of the deviation vector. The consistent second approximation to the complementary energy density of the geometrically non-linear theory of isotropic elastic shells is constructed. From differentiation of the density we obtain the consistently refined constitutive equations for 2D surface stretch and bending measures. These equations are then inverted for 2D stress resultants and stress couples. The second-order terms in these constitutive equations take consistent account of influence of undeformed midsurface curvatures. The drilling couples are explicitly expressed by the stress couples, undeformed midsurface curvatures, and amplitudes of quadratic part of displacement distribution through the thickness. The drilling couples are shown to be much smaller than the stress couples, and their influence on the stress and strain state of the shell is negligible. However, such very small drilling couples have to be admitted in non-linear analyses of irregular multi-shell structures, e.g. shells with branches, intersections, or technological junctions. In such shell problems six 2D couple resultants are required to preserve the structure of the resultant shell theory at the junctions during entire deformation process.     

A general nonlinear theory of elastic shells

This paper presents a general nonlinear theory of elastic shells for large deflections and finite strains in reference to a certain natural state. By expanding the displacement components into power series in the coordinate θ³ normal to the undeformed middle surface of shells, the expansions of the Cauchy-Green strain tensors are expressed in terms of these expanded displacement components. Through the modified Hellinger-Reissner variational principle for a three-dimensional elastic continuum, a set of the fundamental shell equations is derived in terms of the expanded Cauchy-Green strain tensors and Kirchhoff stress resultants. The Love-Kirchhoff hypothesis is not assumed and higher order stretching and bending are taken into consideration. For elastic shells of isotropic materials, assuming the strain-energy to be an analytic function of the strain measures, general nonlinear constitutive equations are then derived. Thus, a complete and consistent two-dimensional shell theory incorporating the geometrical and physical nonlinearities is established. The classical theories of shells are directly derivable from the present results by proper truncations of the series.     

The number of elastic moduli for anisotropic polynomial constitutive equations

This paper shows how to compute the number of elastic moduli for polynomial anisotropy constitutive equations. We investigate the algebraic structure of constitutive equations and we show that the knowledge of the number of elasticities for five types of anisotropy (described by cyclic groups) permits to determine the number of elasticities for all other symmetry groups relevant for elasticity theory. We compute the numbers of elastic moduli for polynomial constitutive relations of degree ≤6.     

Evaluation of a simple nonlinear elastic constitutive form

A nonlinear, two constant stress-deformation form is deduced for elastic materials. At very large stretch ratios of greater than about 3 or 4, the model exhibits the strain stiffening behavior common to many elastomers. The constitutive form is very simple since the two material constants enter it as multiplying constants times certain nonlinear deformation terms. The model is evaluated with respect to data upon natural rubber under both uniaxial and bi-axial stress conditions. The model is also used to evaluate data obtained from a nonlinear membrane inflation experiment. The latter experimental capability and corresponding data are new.Work performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore National Laboratory under Contract W-7405-Eng-48.     

General irreducible representations for constitutive equations of elastic crystals and transversely isotropic elastic solids

By means of the combined invariance restrictions due to material frame-indifference and material symmetry, the present paper provides general reduced forms for non-polynomial elastic constitutive equations of all 32 classes of crystals and transversely isotropic solids.Project supported by National Natural Science Foundation and National Postdoctoral Science Foundation of China.     

Cavitation for incompressible anisotropic nonlinearly elastic spheres

In this paper, the effect ofmaterial anisotropy on void nucleation and growth inincompressible nonlinearly elastic solids is examined. A bifurcation problem is considered for a solid sphere composed of an incompressible homogeneous nonlinearly elastic material which is transversely isotropic about the radial direction. Under a uniform radial tensile dead-load, a branch of radially symmetric configurations involving a traction-free internal cavity bifurcates from the undeformed configuration at sufficiently large loads. Closed form analytic solutions are obtained for a specific material model, which may be viewed as a generalization of the classic neo-Hookean model to anisotropic materials. In contrast to the situation for a neo-Hookean sphere, bifurcation here may occur locally either to the right (supercritical) or to the left (subcritical), depending on the degree of anisotropy. In the latter case, the cavity has finite radius on first appearance. Such a discontinuous change in stable equilibrium configurations is reminiscent of the snap-through buckling phenomenon of structural mechanics. Such dramatic cavitational instabilities were previously encountered by Antman and Negrón-Marrero [3] for anisotropiccompressible solids and by Horgan and Pence [17] forcomposite incompressible spheres.     

Tensor-linear determinative equations for a nonlinear elastic anisotropic medium

International Applied Mechanics -     

Two exact micromechanics-based nonlocal constitutive equations for random linear elastic composite materials

A Hashin-Shtrikman-Willis variational principle is employed to derive two exact micromechanics-based nonlocal constitutive equations relating ensemble averages of stress and strain for two-phase, and also many types of multi-phase, random linear elastic composite materials. By exact is meant that the constitutive equations employ the complete spatially-varying ensemble-average strain field, not gradient approximations to it as were employed in the previous, related work of Drugan and Willis (J. Mech. Phys. Solids 44 (1996) 497) and Drugan (J. Mech. Phys. Solids 48 (2000) 1359) (and in other, more phenomenological works). Thus, the nonlocal constitutive equations obtained here are valid for arbitrary ensemble-average strain fields, not restricted to slowly-varying ones as is the case for gradient-approximate nonlocal constitutive equations. One approach presented shows how to solve the integral equations arising from the variational principle directly and exactly, for a special, physically reasonable choice of the homogeneous comparison material. The resulting nonlocal constitutive equation is applicable to composites of arbitrary anisotropy, and arbitrary phase contrast and volume fraction. One exact nonlocal constitutive equation derived using this approach is valid for two-phase composites having any statistically uniform distribution of phases, accounting for up through two-point statistics and arbitrary phase shape. It is also shown that the same approach can be used to derive exact nonlocal constitutive equations for a large class of composites comprised of more than two phases, still permitting arbitrary elastic anisotropy. The second approach presented employs three-dimensional Fourier transforms, resulting in a nonlocal constitutive equation valid for arbitrary choices of the comparison modulus for isotropic composites. This approach is based on use of the general representation of an isotropic fourth-rank tensor function of a vector variable, and its inverse. The exact nonlocal constitutive equations derived from these two approaches are applied to some example cases, directly rationalizing some recently-obtained numerical simulation results and assessing the accuracy of previous results based on gradient-approximate nonlocal constitutive equations.     

A note on incremental equations for a new class of constitutive relations for elastic bodies

Some new classes of constitutive relations for elastic bodies have been proposed in the literature, wherein the stresses and strains are obtained from implicit constitutive relations. A special case of the above relations corresponds to a class of constitutive equations where the linearized strain tensor is given as a nonlinear function of the stresses. For such constitutive equations we consider the problem of decomposing the stresses into two parts: one corresponds to a time-independent solution of the boundary value problem, plus a small (in comparison with the above) time-dependent stress tensor. The effect of this initial time-independent stress in the propagation of a small wave motion is studied for an infinite medium.     

General invariant representations of the constitutive equations for isotropic nonlinearly elastic materials

This paper develops general invariant representations of the constitutive equations for isotropic nonlinearly elastic materials. Different sets of mutually orthogonal unit tensor bases are constructed from the strain argument tensor by using the representation theorem and corresponding irreducible invariants are defined. Their relations and geometrical interpretations are established in three dimensional principal space. It is shown that the constitutive law linking the stress and strain tensors is revealed to be a simple relationship between two vectors in the principal space. Relative to two different sets of the basis tensors, the constitutive equations are transformed according to the transformation rule of vectors. When a potential function is assumed to exist, the vector associated with the stress tensor is expressed in terms of its gradient with respect to the vector associated with the strain tensor. The Hill’s stability condition is shown to be that the scalar product of the increment of those two vectors must be positive. When potential function exists, it becomes to be that the 3 × 3 constitutive matrix derived from its second order derivative with respect to the vector associated with the strain must be positive definite. By decomposing the second order symmetric tensor space into the direct sum of a coaxial tensor subspace and another one orthogonal to it, the closed form representations for the fourth order tangent operator and its inversion are derived in an extremely simple way.