Irreducible expressions for isotropic functions of two tensors

Polynomial expressions are given for scalar-valued, skew-symmetric second-order-tensor-valued and symmetric second-order-tensor-valued isotropic functions of two skew-symmetric second-order tensors, of one skew-symmetric and one symmetric second-order tensor and of two symmetric second-order tensors. It is shown that all terms of degree α, β in two tensors R, S (say) appearing in a given expression are linearly independent and that this is the case for all values of α and β.     

Families of continuous spin tensors and applications in continuum mechanics

A new method is proposed for generating families of continuous spin tensors associated with families of corotational rates of second-order tensors using isotropic tensor functions of the same tensor arguments and different forms of continuous antisymmetric scalar spin functions of scalar arguments. Tensor functions are represented in terms of eigenprojections of a symmetric tensor S, which is one of the arguments of these functions. Each member of the generated family is represented as the sum of some basic spin tensor associated with the basic corotational tensor rate and the above-mentioned tensor function, whose structure is matched to the structure of the tensor function required to construct the twirl tensor of the triad of orthonormal eigenvectors of the tensor S (but this twirl tensor itself does not belong to the family of continuous spin tensors). The developed method is used in continuum mechanics to generate two families of continuous spin tensors associated with two families of objective corotational rates: Lagrangian and Eulerian. In these families, isotropic tensor functions are constructed using Lagrangian and Eulerian tensor arguments of the kinematic type, respectively. It is shown that if the same scalar spin function is used in deriving tensor functions of Lagrangian and Eulerian tensor arguments, then the corotational tensor rates associated with the generated spin tensors are objective (Lagrangian and Eulerian) counterparts of each other. It is shown that the spin tensors associated with the classical Eulerian corotational tensor rates (Zaremba–Jaumann, Green–Naghdi, d-rate) and their Lagrangian counterparts (including material rate) belong to the generated families of continuous spin tensors. It is also shown that both of these families of continuous spin tensors are subfamilies of the families of material spin tensors derived by Xiao et al. (J Elast 52:1–41, 1998). It is noted that the twirl tensors of the Lagrangian and Eulerian triads associated with the Gurtin–Spear corotational rates of tensors belong to the families of material spin tensors but do not belong to the families of continuous spin tensors. The final section gives expressions of continuous spin tensors from families associated with the families of Lagrangian and Eulerian corotational tensor rates which are appropriate for applications.     

Apparent stiffness tensors for porous solids using symmetric Galerkin boundary elements

Representative volume elements (RVEs) from porous or cellular solids can often be too large for numerical or experimental determination of effective elastic constants. Volume elements which are smaller than the RVE can be useful in extracting apparent elastic stiffness tensors which provide bounds on the homogenized elastic stiffness tensor. Here, we make efficient use of boundary element analysis to compute the volume averages of stress and strain needed for such an analysis. For boundary conditions which satisfy the Hill criterion, we demonstrate the extraction of apparent elastic stiffness tensors using a symmetric Galerkin boundary element method. We apply the analysis method to two examples of a porous ceramic. Finally, we extract the eigenvalues of the fabric tensor for the example problem and provide predictions on the apparent elastic stiffnesses as a function of solid volume fraction.     

Large strain elastic-plastic theory and nonlinear finite element analysis based on metric transformation tensors

The present paper is concerned with an efficient framework for a nonlinear finite element procedure for the rate-independent finite strain analysis of solids undergoing large elastic-plastic deformations. The formulation relies on the introduction of a mixed-variant metric transformation tensor which will be multiplicatively decomposed into a plastic and an elastic part. This leads to the definition of an appropriate logarithmic strain measure whose rate is shown to be additively decomposed into elastic and plastic strain rate tensors. The mixed-variant logarithmic elastic strain tensor provides a basis for the definition of a local isotropic hyperelastic stress response in the elastic-plastic solid. Additionally, the plastic material behavior is assumed to be governed by a generalized J ₂ yield criterion and rate-independent isochoric plastic strain rates are computed using an associated flow rule. On the numerical side, the computation of the logarithmic strain tensors is based on 1st and higher order Padé approximations. Estimates of the stress and strain histories are obtained via a highly stable and accurate explicit scalar integration procedure which employs a plastic predictor followed by an elastic corrector step. The development of a consistent elastic-plastic tangent operator as well as its implementation into a nonlinear finite element program will also be discussed. Finally, the numerical solution of finite strain elastic-plastic problems is presented to demonstrate the efficiency of the algorithm. Received: 17 May 1998     

An alternative model for anisotropic elasticity based on fabric tensors

Motivated by the mechanical analysis of multiphase or damaged materials, a general approach relating fabric tensors characterizing microstructure to the fourth rank elasticity tensor is proposed. Using a Fourier expansion in spherical harmonics, the orientation distribution function of a positive, radially symmetric microstructural property is approximated by a scalar and a symmetric, traceless second rank tensor. Following this approximation, a general expression of the elastic free energy potential is derived from representation theorems for anisotropic scalar functions. Based on a homogeneity assumption for the elastic constitutive law with respect to the microstructural property, a particular elasticity model is developed that involves three independent constants beside the fabric tensors. Strict positive definiteness of the corresponding elasticity tensor is ensured under explicit conditions on the independent constants for arbitrary fabric tensors.     

Free vibration of symmetric angle-ply thick laminated composite cylindrical shells

Analytical solutions for the vibrations of thick symmetric angle-ply laminated composite cylindrical shells are established using the first-order shear deformation theory. The complex method is developed to deal with the partial differential governing equations of thick symmetric angle-ply laminated composite cylindrical shells. Numerical results are provided for comparison, and coincide with existing results in the literature. The frequency characteristics for thick symmetric angle-ply laminated composite cylindrical shells with different H/R and L/R ratios are studied in comparison with those of symmetric cross-ply laminates. Also, the influence of lamination angle and number of lamination layers on frequency is investigated in details.     

Notes on a paper by C. Chan and D. E. Carlson, “second-order incompressible elastic torsion”

It is shown that the procedure for the determination of second-order effects of isotropic incompressible hyperelastic materials as given by Chan and Carlson[1] takes on complete generality whenever the deformation or the material is such that the strain-energy function W(I₁, I₂) is a symmetric function of the principal strain invariants I₁ and I₂. Explicit formulae are given for the second-order solutions for the two major situations when W(I₁, I₂) is a symmetric function of its arguments.     

Twist,tilt, and symmetric grain boundaries in hexagonal materials

Grain boundaries of characteristic geometry, e.g., twist, tilt, and symmetric boundaries are often used as reference boundaries in analyses of boundary networks in polycrystalline materials. This article deals with the issue of proper identification of characteristic boundaries in the case of materials with hexagonal D _6h symmetry. To identify all boundaries of characteristic types, both analytical calculations and numerical searches are used. The first approach provides exact parameters of the characteristic boundaries, whereas the second one gives boundaries which can be classified as characteristic if some tolerance is allowed. In both methods, all symmetrically equivalent boundary representations are taken into consideration. The obtained sets of twist, tilt, symmetric, and 180°-tilt boundaries are presented in the form of two-dimensional maps containing stereographic projections of the corresponding boundary plane normals for selected grain misorientations. These diagrams facilitate interpretation of experimental distributions of grain boundaries; with the representation used, they can be directly linked to experimental distributions. Examples of such diagrams for lattice parameter ratios c/a of $\sqrt{5/2}$ and $\sqrt{20/21}$ are presented. They are compared to example boundary distributions in Ti alloy and distributions of WC/WC boundaries in WC–Co composites available in the literature.     

Minimization of thermal expansion of symmetric, balanced, angle ply laminates by optimization of fiber path configurations

Optimal fiber path configurations that minimize the sum of the coefficients of thermal expansion (CTE) values along the principal material directions for a class of laminates are presented. Previous studies suggest that balanced, symmetric, angle ply laminates exhibit negative CTE values along the principal directions. Using the sum of the CTE values along the principal material directions as an effective measure of the coefficient of thermal expansion (CTE_eff), we have shown and provided a proof that the smallest value of CTE_eff is rendered by straight fiber path configurations. The laminates considered are sufficiently thin so as to neglect the thermal stresses induced through the thickness of the laminate. It is found that the minimal CTE_eff values occur for [+45/−45]_ns lay-ups. This result is supported by numerical studies that consider curvilinear fiber paths. The possibility of obtaining zero CTE values along both principal material directions and the conditions that render this situation are also examined.     

Static Josephson effects in circularly symmetric annular junctions

Static Josephson effects in circularly symmetric annular junctions are discussed. When the annular width W of a junction is smaller than _J, analytic solutions can be obtained. For annular width W of a junction larger than _J, the self-field effects must be considered; the solutions can be obtained numerically. For these two cases, called "small junction" and "large junction," respectively, the magnetic field dependence of the critical current I _m of a circularly symmetric annular junction is obtained. Further, for a width W larger than _J, the current density distribution is also given for several magnetic fields. In addition, for a given external magnetic field, when the feeding current I is smaller than the critical current I _m of the circular symmetric annular junction, there are multisolutions under such physical conditions. Finally, the stability of these solutions is also discussed.     

A discrete constitutive model for transverse and shear damage of symmetric laminates with arbitrary stacking sequence

A damage constitutive model in conjunction with a 2-D finite element discretization is presented for predicting onset and evolution of matrix cracking and subsequent stiffness reduction of symmetric composite laminates with arbitrary stacking sequence subjected to membrane loads. The formulation uses laminae crack densities as the only state variables, with crack growth driven by both mechanical stress and residual stress due to thermal expansion. The formulation is based on fracture mechanics in terms of basic materials properties, lamina moduli, and critical strain energy release rates G_IC and G_IIC, only. No additional adjustable parameters are needed to predict the damage evolution. Spurious strain localization and mesh size dependence are intrinsically absent in this formulation. Thus, there is no need to define a characteristic length. Comparison of model results to experimental data is presented for various laminate stacking sequences. Prediction of crack initiation, evolution, and stiffness degradation compare very well to experimental data.     

Computational aspects of tangent moduli tensors in rate-independent crystal elastoplasticity

The computational efficiencies of the continuum and consistent (algorithmic) tangent moduli tensors in rate-independent crystal elastoplasticity are examined in conjunction with the available implicit state update algorithms. It is, in this context, shown that the consistent tangent moduli associated with the state update algorithm with the exponential mapping coincide with the continuum tangent moduli. After verifying the reported performance of the exponential mapping algorithm in preserving the incompressibility of plastic deformation in a single crystal grain, we carry out numerical experiments to understand the convergence trends of the global Newton–Raphson iterative procedure with different kinds of tangent moduli tensors. Having done this, we are concerned with the performance of those tangent moduli tensors for the micro-scale analysis of a polycrystalline aggregate, which is regarded as a representative volume element, subjected to macro-scale uniform deformation in the context of the two-scale homogenization method.     

On the cross-spectral tensors for black-body emission into space

Abstract

The cross-spectral tensors for black-body emission into space are derived, and the results expressed in terms of Debye integrals which have recently been found to be important in the scalar theory of partial coherence. For the case of an infinite black-body surface, closed-form expressions can be obtained for the cross-spectral tensors which are exact within the constraints of the local thermodynamic equilibrium assumption and the idealization of a perfectly absorbing surface. The trace of the electric cross-spectral tensor agrees with the recent results in the scalar theory of emission from uniform planar Lambertian sources.     

Computation of the Dimensions of Symmetry Classes of Tensors Associated with the Finite two Dimensional Projective Special Linear Group

The dimensions of the symmetry classes of tensors associated with the projective special linear group of degree 2 over a field with q elements, PSL ₂(q), are found. Of course we will assume PSL ₂(q) as a subgroup of the symmetric group S _{q +1} because this group has a faithful action on the points of the underlying projective space. We also discuss the non-triviality of the symmetry classes of tensors associated with each irreducible character of PSL ₂(q). Received: October 14, 1998; revised version: November 30, 1999     

Adaptive cross approximation of tensors arising in the discretization of boundary integral operator shape derivatives

The shape derivative of a dense N×N BEM matrix is a sparse three-way tensor with O(N²) non-zero entries, to which standard BEM acceleration techniques such as the adaptive cross approximation (ACA) and FMM cannot be directly applied. The tensor can be used to compute shape sensitivities, or via adjoint equations, the gradient of an objective function. Although for many PDEs, calculation of the tensor can be avoided by expressing the shape derivative of the solution as the solution of a related PDE, this approach is not always easily amenable to BEM. Therefore, the computation of shape derivatives via the sparse three-way tensor is a valuable alternative, provided that efficient acceleration techniques exist. We propose a new algorithm for the approximation of BEM shape derivative tensors based on ACA that achieves the same complexity and error bounds as ACA for the BEM matrix itself. Numerical examples show that despite the much larger amount of data involved, the tensor approximation is only moderately slower than the matrix approximation. We also demonstrate the method on a shape optimization problem from the literature.     

A three-segment anisomorphic constant life diagram for the fatigue of symmetric angle-ply carbon/epoxy laminates at room temperature

The anisomorphic constant fatigue life (CFL) diagram approach that was developed in an earlier study is further tested for applicability to the matrix-dominated fatigue failure in symmetric angle-ply carbon/epoxy laminates. An extension of the CFL diagram approach is also attempted to improve the accuracy of fatigue life prediction. The original anisomorphic CFL diagram approach can be used for approximately predicting the CFL diagrams for the [±30]_3S and [±45]_3S laminates, while it fails to accurately predict the CFL diagram for the [±60]_3S laminate due to its significant local distortion. For accommodating the anisomorphic CFL diagram approach to the local distortion in CFL curves due to a significant change in mean stress sensitivity in fatigue, a transitional segment is inserted between the tension–tension and compression–compression dominated segments. It is demonstrated that the three-segment anisomorphic CFL diagram approach allows improved predictions of the CFL diagrams and S–N relationships for the angle-ply laminates.     

Creep and relaxation contribution tensors for spheroidal pores in hereditary solids: fraction-exponential operators approach

In this paper, we introduce creep and relaxation contribution tensors that describe the effect ofindividual pores on the overall viscoelastic properties of a porous material. Explicit analytical expressions for these tensors are obtained using the elastic–viscoelastic correspondence principle and the Laplace transform. This becomes possible when viscoelastic properties are expressed in terms of fraction-exponential operators of Rabotnov (J Appl Math Mech (PMM) 12:53–62, 1948). Creep and relaxation contribution tensors can serve as the basic building blocks for the calculation of overall viscoelastic properties of porous hereditary materials.     

Current simulation of symmetric contacts on CdTe

This article presents the calculated current-voltage characteristics of symmetric Metal-Semiconductor-Metal configurations for Schottky, Ohmic, and injecting-Ohmic contacts on high resistivity CdTe. The results clearly demonstrate that in the wide band-gap, semi-insulating semiconductors, such as high resistivity CdTe, the linearity of the I-V curves cannot be considered a proof of the ohmicity of the contacts. It is shown that the linear I-V curves are expected for a wide range of contact barriers. Furthermore, the slope of these linear curves is governed by the barrier height, rather than the bulk doping concentration. Therefore the deduction of bulk's resistivity from the I-V curves may be false.