Optimal Orthotropy for Minimum Elastic Energy by the Polar Method

The polar method is a minimal invariant representation in plane elasticity. A plane orthotropic elastic behaviour is expressed by five polar invariants related to the elastic symmetries. In this paper, considering the orthotropy orientation and the polar invariants as optimisation parameters, we discuss the problem of minimising the elastic energy for a given state of stress. The minimisation with respect to the orientation is solved in order to find the associated optimal elastic energy for given polar invariants. Then, this quantity is minimised with respect to the polar invariants which characterise the magnitude of the anisotropic components of the elastic stiffness tensor. Optimal uncoupled composite laminates corresponding to this optimum are presented for membrane and bending loadings.     

A novel internal dissipation inequality by isotropy and its implication for inelastic constitutive characterization

In the present work a novel inelastic deformation caused internal dissipation inequality by isotropy is revealed. This inequality has the most concise form among a variety of internal dissipation inequalities, including the one widely used in constitutive characterization of isotropic finite strain elastoplasticity and viscoelasticiy. Further, the evolution term describing the difference between the rate of deformation tensor and the “principal rate” of the elastic logarithmic strain tensor is set, according to the standard practice by isotropy, to equal a rank-two isotropic tensor function of the corresponding branch stress, with the tensor function having an eigenspace identical to the eigenspace of the branch stress tensor. Through that a general form of evolution equation for the elastic logarithmic strain is formulated and some interesting and important results are derived. Namely, by isotropy the evolution of the elastic logarithmic strain tensor is embodied separately by the evolutions of its eigenvalues and eigenprojections, with the evolution of the eigenprojections driven by the rate of deformation tensor and the evolution of the eigenvalues connected to specific material behavior. It can be proved that by isotropy the evolution term in the present dissipation inequality stands for the essential form of the evolution term in the extensively applied dissipation inequality.     

Invariant characteristic representations for classical and micropolar anisotropic elasticity tensors

By extending and developing the characteristic notion of the classical linear elasticity initiated by Lord Kelvin, a new type of representation for classical and micropolar anisotropic elasticity tensors is introduced. The new representation provides general expressions for characteristic forms of the two kinds of elasticity tensors under the material symmetry restriction and has many properties of physical and mathematical significance. For all types of material symmetries of interest, such new representations are constructed explicitly in terms of certain invariant constants and unit vectors in directions of material symmetry axes and hence they furnish invariants which can completely characterize the classical and micropolar linear elasticities. The results given are shown to be useful. In the case of classical elasticity, the spectral properties disclosed by our results are consistent with those given by similar established results.     

Anisotropy of plane complex elastic bodies

The problem of the search of the invariants of an anisotropic elastic tensor representing the mechanical response of a complex elastic body in a two dimensional space is addressed, in particular for a tensor that does not possess all the tensor symmetries typical of classical elasticity. The invariants of the stiffness tensor are found and all the possible types of orthotropy are discussed.     

Estimation of overall properties of random polycrystals with the use of invariant decompositions of Hooke’s tensor

In the paper the theoretical analysis of bounds and self-consistent estimates of overall properties of linear random polycrystals composed of arbitrarily anisotropic grains is presented. In the study two invariant decompositions of Hooke’s tensors are used. The applied method enables derivation of novel expressions for estimates of the bulk and shear moduli, which depend on invariants of local stiffness tensor. With use of these expressions the materials are considered for which at the local level constraints are imposed on deformation or some stresses are unsustained.     

A Representation Theorem for Material Tensors of?Weakly-Textured Polycrystals and?Its Applications in?Elasticity

Material tensors pertaining to polycrystalline aggregates should manifest also the influence of crystallographic texture on the material properties in question. In this paper we make use of tensors which form bases of irreducible representations of the rotation group and prove a representation theorem by which a given material tensor of a weakly-textured polycrystal is expressed as a linear combination of an orthonormal set of irreducible basis tensors, with the components given explicitly in terms of texture coefficients and a set of undetermined material parameters. Once the irreducible basis tensors that appear in the formula are determined, the representation formula, which is valid for all texture and crystal symmetries, will delineate quantitatively the effect of crystallographic texture on the material tensor in question. We present an integral formula and an orthonormalization process which serve as the basis for a procedure to determine explicitly the irreducible basis tensors required in the representation formula. For applications we determine a set of irreducible basis tensors for the elasticity tensor and a set for fourth-order tensors that define constitutive equations in incompressible elasticity and Hill’s quadratic yield functions in plasticity. We show that orientation averaging of a tensor can be done easily if we have in hand a set of irreducible basis tensors for the decomposition of the tensor in question. As illustration we derive a formula, which is valid for all texture and crystal symmetries, for the elasticity tensor under the Voigt model.     

Laminations in Linearized Elasticity: The Isotropic Non-very Strongly Elliptic Case

The explicit computation of the effective elasticity tensor of the material produced by laminating two homogeneous elastic media is used to show that, in 2-dimensional and 3-dimensional linear elasticity, for any isotropic material a whose elasticity tensor is strongly elliptic, but not semipositive definite, we can select very strongly elliptic materials, so that through laminations between these with material a, we can create a nonstrongly elliptic media, whose existence contradicts properties concerning the propagation of elastic waves. This revised version was published online in August 2006 with corrections to the Cover Date.     

Theory of creep deformation with kinematic hardening for materials with different properties in tension and compression

A constitutive model for creep deformation that describes the loading-history-dependent behavior of initially isotropic materials with different properties in tension and compression under stress vector rotations limited by 50–60° is presented within a thermodynamic framework. In the proposed constitutive model a kinematic hardening rule is adopted. This model also introduces an effective equivalent stress in the creep potential that is based on the first and second invariants of the effective stress tensor, and on the joint invariant of the effective stress tensor and eigenvector associated with the maximum principal Cauchy stress. The formulation of the kinematic hardening rule is presented and discussed. All the material parameters in the model have been obtained from a series of proposed basic experiments with constant stresses. These model parameters are then used to predict the creep deformation of the aluminum alloy under multiaxial loading with constant stresses, and under non-proportional uniaxial and non-proportional multiaxial loadings for both isothermal and nonisothermal processes.     

General Theory of Coupled Thermally Stable Anisotropic Laminates

A general theory of thermally stable anisotropic laminates with bending-extension coupling is presented in the framework of linear thermo-elasticity. It is based upon the use of tensor invariants, found by the polar formalism for bi-dimensional anisotropic elasticity. The extension of the results to moisture absorption effects is also proved.     

Bootstrap Elasticity: From Linear to Nonlinear Constitutive Representations

The normal modes of the 6×6 symmetric matrix of elastic moduli for linear anisotropic elasticity, also called Kelvin modes, provide orthogonal basis sets for the six dimensional space of symmetric, second order tensors in three dimensional Euclidean space. In turn the partitioning of the six space, induced by these bases and the multiplicity of each eigenvalue, provides the means for constructing six term minimal representations of nonlinear constitutive equations for materials of any symmetry from triclinic to cubic. The constructions also for the first time show clear connections to the linear elastic moduli, which through the eigenvalues set the scale for most, but not all, of the tensor generators. This approach also provides an alternate way to construct the well-known three term Rivlin-Ericksen representation for nonlinear isotropic materials.     

Asymptotic solutions by the successive disordering method

We develop the periodic componentmethod [1] and represent the solution of a stochastic boundary value elasticity problem for a random quasiperiodic structure with a given disordering degree of inclusions in the matrix via the deviations from the corresponding solution for a random structure with a smaller disordering degree. An example in which the tensor of elastic properties of a composite is calculated is used to illustrate the asymptotic and differential approaches of the successive disordering method. The asymptotic approach permits representing the quasiperiodic structure with a given chaos coefficient and the desired tensor of effective elastic properties as a result of small successive disordering of an originally ideally periodic structure of a composite with known tensor of elastic properties. In the differential approach, a differential equation is obtained for the tensor of effective elastic properties as a function of the chaos coefficient. Its solution coincides with the solution provided by the asymptotic approach. The solution is generalized to the case of piezoactive composites, and a numerical analysis of the effective properties is performed for a PVF (polyvinylidene fluoride) piezoelectric with various quasiperiodic structures on the basis of the cubic structure with spherical inclusions of a high-module elastic material.     

The evolution of Hooke's law due to texture development in FCC polycrystals

Initially isotropic aggregates of crystalline grains show a texture-induced anisotropy of both their inelastic and elastic behavior when submitted to large inelastic deformations. The latter, however, is normally neglected, although experiments as well as numerical simulations clearly show a strong alteration of the elastic properties for certain materials. The main purpose of the work is to formulate a phenomenological model for the evolution of the elastic properties of cubic crystal aggregates. The effective elastic properties are determined by orientation averages of the local elasticity tensors. Arithmetic, geometric, and harmonic averages are compared. It can be shown that for cubic crystal aggregates all of these averages depend on the same irreducible fourth-order tensor, which represents the purely anisotropic portion of the effective elasticity tensor. Coupled equations for the flow rule and the evolution of the anisotropic part of the elasticity tensor are formulated. The flow rule is based on an anisotropic norm of the stress deviator defined by means of the elastic anisotropy. In the evolution equation for the anisotropic part of the elasticity tensor the direction of the rate of change depends only on the inelastic rate of deformation. The evolution equation is derived according to the theory of isotropic tensor functions. The transition from an elastically isotropic initial state to a (path-dependent) final anisotropic state is discussed for polycrystalline copper. The predictions of the model are compared with micro–macro simulations based on the Taylor–Lin model and experimental data.     

Invariants for Rari- and Multi-Constant Theories with Generalization to Anisotropy in Biological Tissues

The rari-constant theory of linear elasticity is based on the assumption that elasticity in solids is caused by only pair potentials with coaxial forces acting between atoms. The strain energy of each pair potential depends on the square of the strain between the atoms in the pair. This strain can be determined by taking the inner product of the strain tensor with a structural tensor that is the tensor product of a unit vector with itself. It is shown that the 15 independent constants in the rari-constant theory can be generated by a complete set of 15 structural tensors. It is also shown that the 6 additional independent constants in the multi-constant theory can be generated by taking the inner product of 6 of these structural tensors with the square of the strain tensor. A generalization of these results for nonlinear elasticity is discussed with reference to recent work which compares the structural and generalized structural tensor approaches to modeling fibrous tissues.     

Nonlinear viscoelasticity of polymer melts in different types of flow

The nonlinear viscoelastic properties of a fairly large class of polymeric fluids can be described with the factorable single integral constitutive equation. For this class of fluids, a connection between the rheological behaviour in different flow geometries can be defined if the strain tensor (or the damping function) is expressed as a function of the invariants of a tensor which describes the macroscopic strain, such as the Finger tensor. A number of these expressions, proposed in the literature, are tested on the basis of the measuring data for a low-density polyethylene melt. In the factorable BKZ constitutive equation the strain-energy function must be expressed as a function of the invariants of the Finger tensor. The paper demonstrates that the strain-energy function can be calculated from the simple shear and simple elongation strain measures, if it is assumed to be of the shape proposed by Valanis and Landel. The measuring data for the LDPE melt indicate that the Valanis-Landel hypothesis concerning the shape of the strainenergy function is probably not valid for polymer melts.     

单轴压缩下断续节理岩体动态损伤本构模型

断续节理将对工程岩体的强度及变形等力学特性产生显著影响,损伤力学中视节理为岩体的一种宏观损伤,因而采用损伤张量来刻画其对岩体的影响。目前学术界提出了用节理的几何、强度及变形等3类参数来描述节理的物理力学性质,而目前的岩体损伤张量计算方法都只涉及前2类参数,均没有涉及其变形参数即法向及切向刚度。为此,在前人研究的基础上,基于断裂及损伤理论提出了考虑节理法向及切向刚度的单轴压缩下单条断续节理引起的损伤张量计算公式,进而通过考虑节理间相互作用给出了单组单排或多排节理岩体损伤张量计算公式。其次,以岩石细观动态损伤模型为基础,结合宏细观损伤耦合观点提出了一个能够同时考虑节理几何、强度及变形参数的断续节理岩体动态损伤本构模型。最后,利用该模型讨论了节理参数及载荷应变率等对岩体动态力学特性的影响,认为节理长度减小及摩擦角增大将导致岩体动态峰值强度及弹性模量增大;岩体动态峰值强度及弹性模量则随着节理法向及切向刚度的增大分别减小或增大;而当节理法向及切向刚度按照同一比例增大时,岩体动态峰值强度及弹性模量则是增大的。岩体动态峰值强度与载荷应变率呈正相关。     

Periodic Homogenization and Material Symmetry in Linear Elasticity

Here homogenization theory is used to establish a connection between the symmetries of a periodic elastic structure associated with the microscopic properties of an elastic material and the material symmetries of the effective, macroscopic elasticity tensor. Previous results of this type exist but here more general symmetries on the microscale are considered. Using an explicit example, we show that it is possible for a material to be fully anisotropic on the microscale and yet the symmetry group on the macroscale can contain elements other than plus or minus the identity. Another example demonstrates that not all material symmetries of the macroscopic elastic tensor are generated by symmetries of the periodic elastic structure.     

Homogenization of Polycrystalline¶Aggregates

The effective elastic properties of a polycrystalline material depend on the single crystal elastic constants of the crystallites comprising the polycrystal and on the manner in which the crystallites are arranged. In this paper we apply the techniques of homogenization to put the problem of determining effective elastic constants in a precise mathematical framework that permits us to derive an expression for the effective elasticity tensor. We also study how the homogenized elasticity tensor changes as the probability characterizing the ensemble changes. Under the assumption that the field of orientations of the crystallographic axes of the crystallites is an independent random field, we show that our theory is compatible with the formulation used in texture analysis. In particular, we are able to prove that the physical assumption made by [10] in his study of weakly-textured polycrystals holds true. In addition, we establish some elementary bounds on the material constants that characterize the effective elasticity tensor of weakly-textured orthorhombic aggregates of cubic crystallites. Accepted: (June 15, 1999)