New S-Type Bounds of M-Eigenvalues for Elasticity Tensors with Applications

In this paper, based on the extreme eigenvalues of the matrices arisen from the given elasticity tensor, S-type upper bounds for the M-eigenvalues of elasticity tensors are established. Finally, S-type sufficient conditions are introduced for the strong ellipticity of elasticity tensors based on the S-type M-eigenvalue inclusion sets.

Analysis of unbalanced laminate composites with imperfect interphase: Effective properties via asymptotic homogenization method

This work addresses the Asymptotic Homogenization Method (AHM) to find all the non-zero independent constants of the fourth-order elasticity tensor of a theoretically infinite periodically laminated composite. The concept of Unit Cell describes the domain, comprised of two orthotropic composite plies separated by an isotropic interphase. A general case with an unbalanced composite is considered. Thus, the coupled components of the tensor are expected. Both analytical and numerical solutions are derived. In addition, an interphase degradation model is proposed to evaluate its effect on the effective properties of the media. Two different stacking sequences are considered with five degrees of interphase imperfection each. The results show good agreement between the analytical and numerical solutions. In addition, it is clear that the more imperfect the interphase is, the more affected the effective properties of the media are, especially those dependent on the stacking direction.

Determination of the anisotropic elasticity tensor by mechanical spectroscopy

A method is proposed to identify the fully anisotropic elasticity tensor by applying the impulse excitation technique. A specially designed batch of several differently oriented bar-shaped specimens with rectangular cross section is analyzed with respect to the eigenfrequencies of their fundamental flexural and torsional modes. Estimations based on the equations for the calculation of the isotropic Young’s modulus and the shear modulus from the ASTM standard allow a first approximation of the elasticity tensor from a selected subset of the measured eigenfrequencies. Subsequently, a more precise determination of the elasticity tensor is achieved by a numerical modal analysis using the finite element method. In this course, a Newton–Raphson optimization method is applied to solve the inverse problem. The proposed approach is demonstrated on a batch of specimen fabricated from the nickel-base alloy IN718 by selective laser melting.

On decompositions and approximations of conjugate partial-symmetric tensors

Hermitian matrices have played an important role in matrix theory and complex quadratic optimization. The high-order generalization of Hermitian matrices, conjugate partial-symmetric (CPS) tensors, have shown growing interest recently in tensor theory and computation, particularly in application-driven complex polynomial optimization problems. In this paper, we study CPS tensors with a focus on ranks, computing rank-one decompositions and approximations, as well as their applications. We prove constructively that any CPS tensor can be decomposed into a sum of rank-one CPS tensors, which provides an explicit method to compute such rank-one decompositions. Three types of ranks for CPS tensors are defined and shown to be different in general. This leads to the invalidity of the conjugate version of Comon’s conjecture. We then study rank-one approximations and matricizations of CPS tensors. By carefully unfolding CPS tensors to Hermitian matrices, rank-one equivalence can be preserved. This enables us to develop new convex optimization models and algorithms to compute best rank-one approximations of CPS tensors. Numerical experiments from data sets in radar wave form design, elasticity tensor, and quantum entanglement are performed to justify the capability of our methods.

Two-Scale Topology Optimization With Parameterized Cellular Structures

Advances in additive manufacturing enable the fabrication of complex structures with intricate geometric details. It also escalates the potential for high-resolution structure design. However, the increasingly finer design brings computational challenges for structural optimization approaches such as topology optimization (TO) since the number of variables to optimize increases with the resolutions. To address this issue, two-scale TO paves an avenue for high-resolution structural design. The design domain is first discretized to a coarse scale, and the material property distribution is optimized, then using micro-structures to fill each property field. In this paper, instead of finding optimal properties of two scales separately, we reformulate the two-scale TO problem and optimize the design variables concurrently in both scales. By introducing parameterized periodic cellular structures, the minimal surface level-parameter is defined as the material design parameter and is implemented directly in the optimization problem. A numerical homogenization method is employed to calculate the elasticity tensor of the cellular materials. The stiffness matrices of the cellular structures derived as a function of the level parameters, using the homogenization results. An additional constraint on the level parameter is introduced in the structural optimization framework to enhance adjacent cellulars interfaces’ compatibility. Based on the parameterized micro-structure, the optimization problem is solved concurrently with an iterative solver. The reliability of the proposed approach has been validated with different engineering design cases. Numerical results show a noticeable increase in structure stiffness using the level parameter directly in the optimization problem than the state-of-art mapping technique.

2D ELASTICITY TENSOR INVARIANTS, INVARIANTS DEFINITE POSITIVE CRITERIA

In this paper, we constructed relationships with the differents 2D elasticity tensor invariants. Indeed, let ${\bf A}$ be a 2D elasticity tensor. Rotation group action leads to a pair of Lax in linear elasticity. This pair of Lax leads to five independent invariants chosen among six. The definite positive criteria are established with the determined invariants. We believe that this approach finds interesting applications, as in the one of elastic material classification or approaches in orbit space description.

Classification of first strain-gradient elasticity tensors by symmetry planes

First strain-gradient elasticity is a generalized continuum theory capable of modelling size effects in materials. This extended capability comes from the inclusion in the mechanical energy density of terms related to the strain-gradient. In its linear formulation, the constitutive law is defined by three elasticity tensors whose orders range from four to six. In the present contribution, the symmetry properties of the sixth-order elasticity tensors involved in this model are investigated. If their classification with respect to the orthogonal symmetry group is known, their classification with respect to symmetry planes is still missing. This last classification is important since it is deeply connected with some identification procedures. The classification of sixth-order elasticity tensors in terms of invariance properties with respect to symmetry planes is given in the present contribution. Precisely, it is demonstrated that there exist 11 reflection symmetry classes. This classification is distinct from the one obtained with respect to the orthogonal group, according to which there exist 17 different symmetry classes. These results for the sixth-order elasticity tensor are very different from those obtained for the classical fourth-order elasticity tensor, since in the latter case the two classifications coincide. A few numerical examples are provided to illustrate how some different orthogonal classes merge into one reflection class.

Sensitivity of the Second Order Homogenized Elasticity Tensor to Topological Microstructural Changes

The multiscale elasticity model of solids with singular geometrical perturbations of microstructure is considered for the purposes, e.g., of optimum design. The homogenized linear elasticity tensors of first and second orders are considered in the framework of periodic Sobolev spaces. In particular, the sensitivity analysis of second order homogenized elasticity tensor to topological microstructural changes is performed. The derivation of the proposed sensitivities relies on the concept of topological derivative applied within a multiscale constitutive model. The microstructure is topologically perturbed by the nucleation of a small circular inclusion that allows for deriving the sensitivity in its closed form with the help of appropriate adjoint states. The resulting topological derivative is given by a sixth order tensor field over the microstructural domain, which measures how the second order homogenized elasticity tensor changes when a small circular inclusion is introduced at the microscopic level. As a result, the topological derivatives of functionals for multiscale models can be obtained and used in numerical methods of shape and topology optimization of microstructures, including synthesis and optimal design of metamaterials by taking into account the second order mechanical effects. The analysis is performed in two spatial dimensions however the results are valid in three spatial dimensions as well.

Characterization of the symmetry class of an elasticity tensor using polynomial covariants

We formulate effective necessary and sufficient conditions to identify the symmetry class of an elasticity tensor, a fourth-order tensor which is the cornerstone of the theory of elasticity and a toy model for linear constitutive laws in physics. The novelty is that these conditions are written using polynomial covariants. As a corollary, we deduce that the symmetry classes are affine algebraic sets, a result which seems to be new. Meanwhile, we have been lead to produce a minimal set of 70 generators for the covariant algebra of a fourth-order harmonic tensor and introduce an original generalized cross-product on totally symmetric tensors. Finally, using these tensorial covariants, we produce a new minimal set of 294 generators for the invariant algebra of the elasticity tensor.