Fourth-order Elasticity Tensor Research Articles

A 3D RVE-based computational homogenization approach for predicting the effective fourth-order elasticity tensor of periodic porous materials

A 3D RVE-based computational homogenization approach for predicting the effective fourth-order elasticity tensor of periodic porous materials

New insight into large deformation analysis of stretch-based and invariant-based rubber-like hyperelastic elastomers

A nonlinear finite element formulation for large deformation analysis of hyperelastic membranes in Total Lagrangian framework is developed. In many cases, it is convenient to utilize the invariant-based strain energy density function to analyze the structures when employing the finite element method. However, in this contribution, both invariant-based and stretch-based formulations are developed. Formulations are completely general and not restricted to axisymmetric loading and geometry of membrane. Furthermore, initially flat and non-flat membrane can be investigated by this method. Incompressible as well as compressible materials can be modeled. A method for imposing the plane stress condition to the formulation for compressible materials is presented. Moreover, isotropic and anisotropic materials can be considered. Some problems are analyzed and results compared to the available ones in the literature. The comparison between solution based on invariant-based method and stretch-based method with exact fourth-order elasticity tensor obtained in this work demonstrates that the stretch-based method derived in this paper works very well. The presented examples demonstrate that the developed formulations have no restriction or simplification with respect to the modeled object geometry and inherent anisotropy.

Nonlinear Finite Element Calculations of Layered SAW Resonators.

In this work the nonlinear behavior of layered SAW resonators is studied with the help of Finite Element (FE) computations. The full calculations depend strongly on the availability of accurate tensor data. While there are accurate material data for linear computations, the complete sets of higher-order material constants, needed for nonlinear simulations, are still not available for relevant materials. To overcome this problem, scaling factors were used for each available nonlinear tensor. The approach here considers piezoelectricity, dielectricity, eletrostriction and elasticity constants up to fourth order. These factors act as a phenomenological estimate for incomplete tensor data. Since no set of fourth order material constants for LiTaO3 is available, an isotropic approximation for the fourth order elastic constants was applied. As a result, it was found that the fourth order elastic tensor is dominated by one fourth order Lamé constant. With the help of the FE model, derived in two different, but equivalent ways, we investigate the nonlinear behavior of a SAW resonator with a layered material stack. The focus was set to third order nonlinearity. Accordingly, the modeling approach is validated using measurements of third order effects in test resonators. In addition, the acoustic field distribution is analyzed.

Multiscale embedded models to determine effective mechanical properties of composite materials: Asymptotic Homogenization Method combined to Finite Element Method

A reliable way to determine the effective properties of heterogeneous media plays an important role in engineering design. As an experimental test to obtain effective properties may be tedious and expensive to perform, and as computational approaches, when all heterogeneities of the media are considered during the simulations, might become impractical due to computational effort, homogenization procedures are a good alternative to estimate the effective properties of the media, mainly considering damage and manufacturing defects in the material. Among several homogenization methods, the Asymptotic Homogenization Method (AHM), a well-established mathematically based method, is considered for deriving the relations for the effective properties of the media, and the Finite Element Method (FEM) is used to solve the equilibrium relations. In this work, a fiber-reinforced composite is considered, with three different regions depicted: intact, damaged fibers, and regions with resin/void pockets. The proposed AHM-FEM approach is used to simulate a tensile test in all scenarios with standard and multiscale embedded Unit Cells used to obtain the effective elastic modulus on the fiber direction. Discussions regarding computational efficiency are carried out, as well as numerical comparisons among the types of Unit Cells used to model the media. In a second scenario, the complete fourth-order elasticity tensor of the media is investigated, considering the fibers as both isotropic and transversely isotropic. The results for the transversely isotropic fibers approach literature data, while when isotropic fibers are considered, poor results are achieved. Finally, the approach is extrapolated to obtain the mechanical properties of metamaterials, defined here as a composite reinforced by two different kinds of fibers.

Finite Gradient Models with Enriched RBF-Based Interpolation

A finite strain gradient model for the 3D analysis of materials containing spherical voids is presented. A two-scale approach is proposed: a least-squares methodology for RVE analysis with quadratic displacements and a full high-order continuum with both fourth-order and sixth-order elasticity tensors. A meshless method is adopted using radial basis function interpolation with polynomial enrichment. Both the first and second derivatives of the resulting shape functions are described in detail. Complete expressions for the deformation gradient F and its gradient ∇F are derived and a consistent linearization is performed to ensure the Newton solution. A total of seven constitutive properties is required. The classical Lamé parameters corresponding to the pristine material are considered constant. From RVE homogenization, seven properties are obtained, two homogenized Lamé parameters plus five gradient-related properties. Two validation 3D numerical examples are presented. The first example exhibits the size effect (i.e., the stiffening of smaller specimens) and the second example shows the absence of stress singularity and hence the convergence of the discretization method.

A 3D SPH-based entirely Lagrangian meshfree hydroelastic FSI solver for anisotropic composite structures

This paper presents the first 3D entirely Lagrangian meshfree hydroelastic FSI (Fluid-Structure Interaction) solver for reproduction of incompressible fluid flows interacting with anisotropic/isotropic composite elastic structures as well as the first Hamiltonian SPH for anisotropic structures. To achieve this development, we have carefully (i) reformulated the HSPH (Hamiltonian Smoothed Particle Hydrodynamics) isotropic structure model with consideration of material anisotropy of structures, (ii) extended the 2D HSPH structure model and corresponding ISPH-HSPH FSI solver for 3D composite structures and their interactions with incompressible fluids. Regarding the advancement (i), the reformulation from isotropic to anisotropic structure model has been conducted through a careful revisit on the basis of stress-strain responses. The fourth-order elasticity tensor and transformation (rotation) of coordinate systems are considered for development of the anisotropic HSPH structure model. Then, the 3D HSPH structure model for anisotropic/composite structures is coupled with a refined projection-based Incompressible SPH (ISPH) fluid model. The proposed structure model and FSI solver are capable of handling large material anisotropies and discontinuities at material interfaces without use of any artificial stabilizers/smoothing schemes. Validations are conducted coherently. First, the newly proposed anisotropic HSPH structure model is verified through both 2D/3D classical benchmark tests with exact theoretical solutions. Then, followed by validations of HSPH for 3D composites, the corresponding coupled ISPH-HSPH FSI solver is applied to two hydroelastic FSI tests including slamming of an anisotropic composite hull.

Hyperelasticity models extending Hooke’s law from small to moderate strains and experimental verification of their scope of application

Lagrangian formulations of Hill’s linear isotropic hyperelastic material models based on the one-parameter (r) Itskov family of strain tensors (including the Hencky, Pelzer, and Mooney strain tensors generating the H, P, and M material models, respectively) were developed. Explicit basis-free expressions for fourth-order elasticity tensors were obtained for these material models. These expressions use the right Cauchy–Green deformation tensor. The developed formulations of isotropic hyperelastic material models were implemented in the commercial finite element MSC.Marc code. Simple tension stress–strain diagrams were obtained for dog-bone shaped flat specimens made of Duothane QA965 polyurethane. The experimental data were used to fit the parameters E and ν (having the meaning of Young’s modulus and Poisson’s ratio for specimens at small strains) for the H, P, and M material models and the parameters C1 and C2 for the Mooney–Rivlin (M–R) isotropic incompressible material model included in the library of standard material models of the MSC.Marc code. For the investigated material, the H and M material models were shown to adequately reproduce stress–strain diagrams up to 50% elongation, and the P and M–R models up to 150% elongation. In addition, cylindrical specimens were made of the same polyurethane and tested for simple torsion (with constrained edges) and generalized torsion (with edges free to move in the longitudinal direction). In both problems, the resultant moments obtained by computer simulations using the P and M–R material models agree well with the experimental data on rod torsion up to buckling; computer simulations using the H and M material models give less satisfactory agreement with the experimental data. Similar qualitative agreement is obtained between the experimental and simulated resultant axial force and axial elongation; however, in this case, the quantitative agreement is less satisfactory than for the resultant moment.

BR2 discontinuous Galerkin methods for finite hyperelastic deformations

In this work we introduce a DG framework for nonlinear elasticity based on a Bassi-Rebay (BR2) formulation. The framework encompasses compressible and incompressible hyperelastic materials and is capable of dealing with large deformations. In order to achieve stability, we combine higher-order lifting operators for the BR2 stabilisation term with an adaptive stabilisation strategy which relies on the BR2 Laplace operator stabilisation and a penalty parameter based on the spectrum of the fourth-order elasticity tensor. Dirichlet boundary conditions for the displacement can be imposed by means of Lagrange multipliers and Nitsche method. Efficiency of the solution strategy is achieved by means of state-of-the-art agglomeration based h-multigrid preconditioners and the code implementation supports distributed memory execution on modern parallel architectures. Several benchmark test cases are proposed in order to investigate some relevant computational aspects, namely the performance of the h-multigrid iterative solver varying the stabilisation parameters and the influence of Dirichlet boundary conditions on Newton's method globalisation strategy.

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.

Effective elasticity of a medium with many parallel fractures

Summary We consider an alternative way of obtaining the effective elastic properties of a cracked medium. Similarly, to the popular linear-slip model, we assume flat, parallel fractures, and long wavelengths. However, we do not treat fractures as weakness planes of displacement discontinuity. In contrast to the classical models, we represent fractures by a thin layer embedded in the background medium. In other words, we follow the Schoenberg-Douma matrix formalism for Backus averaging, but we relax the assumptions of infinite weakness and marginal thickness of a layer so that it does not correspond to the linear-slip plane. To represent the properties of a fracture, we need a fourth order elasticity tensor and a thickness parameter. The effective tensor becomes more complicated, but it may describe a higher concentration of parallel cracks more accurately. Apart from the derivations of the effective elasticity tensors, we perform numerical experiments in which we compare the performance of our approach with a linear-slip model in the context of highly fractured media. Our model becomes pertinent if filled-in or empty cracks occupy more than one per cent of the effective medium.

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.

Symmetry classes in piezoelectricity from second-order symmetries

The piezoelectricity law is a constitutive model that describes how mechanical andelectric fields are coupled within a material. In its linear formulation this law comprises threeconstitutive tensors of increasing order: the second order permittivity tensor S, the third orderpiezoelectricity tensor P and the fourth-order elasticity tensor C. In a first part of the paper,the symmetry classes of the piezoelectricity tensor alone are investigated. Using a new approachbased on the use of the so-called clips operations, we establish the 16 symmetry classes of thistensor and provide their associated normal forms. Second order orthogonal transformations(plane symmetries and $\pi$-angle rotations) are then used to characterize and classify directly 11out of the 16 symmetry classes of the piezoelectricity tensor. An additional step to distinguishthe remaining classes is proposed

Bounds of M-eigenvalues and strong ellipticity conditions for elasticity tensors

ABSTRACT Novel upper and lower bounds are derived for the M-eigenvalues of a fourth-order elasticity tensor by using the extreme eigenvalues of the symmetric matrices extracted from the given elasticity tensor. The new upper bound of the M-spectral radius is tighter than that given by S. Li et al. in [J Comput Appl Math. 2019;356]. Based on the new lower bound of the M-eigenvalues, a stronger sufficient condition is presented for the strong ellipticity of an elasticity tensor. Numerical examples are also given to demonstrate the results.

A lattice method for seismic wave propagation in nonlinear viscoelastic media

SUMMARYNonlinear wave propagation is an important consideration in several geophysical problems as heterogeneities within the subsurface give rise to nonlinear stress–strain relationships in rocks. To address these problems, where the inclusion of subsurface complexity is accounted for, numerical solutions are required. Here, we outline a discrete particle or lattice numerical method that can simulate dynamic and static deformation in an isotropic viscoelastic nonlinear medium where the nonlinearity includes the fourth-order elastic tensor coefficients. The relationship between the local interaction constants in the lattice method and the macroscopic nonlinear coefficients is derived and the relationships presented. The theoretical dispersion properties of the system is determined from a nonlinear perturbation method and are validated against some numerical results. There is an excellent agreement between the theory and numerical results and the method reproduces the common features of nonlinear wave propagation, for example, harmonics, waveform distortion and spectra shifts. As expected the system is shown to be dependent on the wave amplitudes and an example is presented which shows the complexity that can arise in both dynamic and static deformation in a complex model where nonlinearity is included.

Development of an ABAQUS™ plug-in to evaluate the fourth-order elasticity tensor of a periodic material via homogenization by the asymptotic expansion method

This paper presents the development of an ABAQUS™ plug-in for estimation of the effective properties of heterogeneous three-dimensional periodic media using the asymptotic homogenization method (AHM). The developed plug-in is user-friendly and requires only basic skills in ABAQUS™. All aspects of the plug-in development are presented, including the solutions found to implement this approach within the finite element package in ABAQUS™. The application of the load cases is performed by using a Uexpan subroutine written in the Fortran language. The application of periodic boundary conditions, as well as the post-processing, are made in Python script. All codes are connected by an ABAQUS™ plug-in and can be called directly by the graphical user interface (GUI) of the software. In order to use the plug-in, the user must only generate the geometry, define the properties of the materials, and mesh the model. The remaining calculations are done automatically. Several case studies are presented and compared with the results found in the literature. Additionally, several element types were evaluated in order to test the versatility of the implemented method. The results show an excellent agreement with the results found in the literature. All of the implemented codes are provided as supplementary material to this paper.

Rheological model for rock-type materials under large deformations

In this paper the behaviour of rock-type materials is described with respect to the current configuration through rate-type constitutive equations, within the finite deformation framework, using the relative motion. The viscoplastic part of the constitutive equations concerns the long-time effects, i.e. the deformation by creep, while the instantaneous elastic response characterizes the short-time effect. The instantaneous elastic response expresses an appropriate objective time derivative of Cauchy stress tensor via the rate of strain, with the aid of fourth-order elastic stiffness tensor. Using mathematical arguments it follows that our new model extends, to finite strains, Cristescu’s rheological model Cristescu (1989)[2]. To solve boundary value problems for rock-type materials described by the proposed model, the appropriate variational equality for the velocity field, at a time t, is derived starting from the weak formulation of the incremental equilibrium problem. The rate form of the constitutive model is coupled with the variational equality, and serves to update the current state of the material.

Irreducible matrix resolution for symmetry classes of elasticity tensors

In linear elasticity, a fourth-order elasticity (stiffness) tensor of 21 independent components completely describes deformation properties elastic constants of a material. The main goal of the current work is to derive a compact matrix representation of the elasticity tensor that correlates with its intrinsic algebraic properties. Such representation can be useful in design of artificial materials. Owing to Voigt, the elasticity tensor is conventionally represented by a (6 × 6) symmetric matrix. In this paper, we construct two alternative matrix representations that conform with the irreducible decomposition of the elasticity tensor. The 3 × 7 matrix representation is in correspondence with the permutation transformations of indices and with the general linear transformation of the basis. An additional representation of the elasticity tensor by two scalars and three 3 × 3 matrices is suitable to describe the irreducible decomposition under the rotation transformations. We present the elasticity tensor of all crystal systems in these compact matrix forms and construct the hierarchy diagrams based on this representation.

Probing depth and lateral variations of upper-mantle seismic anisotropy from full-waveform inversion of teleseismic body-waves

SUMMARY While seismic anisotropy can potentially provide crucial insights into mantle dynamics, 3-D imaging of seismic anisotropy is still a challenging problem. Here, we present an extension of our regional full-waveform inversion method to image seismic anisotropy in the lithosphere and asthenosphere from teleseismic P and S waveforms. The models are parametrized in terms of density and the 21 elastic coefficients of the fourth-order elasticity tensor. The inversion method makes no a priori assumptions on the symmetry class or on the orientation of the symmetry axes. Instead, the elasticity tensors in the final models are decomposed with the projection method. This method allows us to determine the orientation of the symmetry axes and to extract the contributions of each symmetry class. From simple synthetic experiments, we demonstrate that our full-waveform inversion method is able to image complex 3-D anisotropic structures. In particular, the method is able to almost perfectly recover the general orientation of the symmetry axis or complex layered anisotropic models, which are both extremely challenging problems. We attribute this success to the joint exploitation of both P and S teleseismic waves, which constrain different parts of the elasticity tensor. Another key ingredient is the pre-conditioning of the gradient with an approximate inverse Hessian computed with scattering integrals. The inverse Hessian is crucial for mitigating the artefacts resulting from the uneven (mostly vertical) illumination of teleseismic acquisitions.

Full wave sensitivity of SK(K)S phases to arbitrary anisotropy in the upper and lower mantle

SUMMARY Core-refracted phases such as SKS and SKKS are commonly used to probe seismic anisotropy in the upper and lowermost portions of the Earth’s mantle. Measurements of SK(K)S splitting are often interpreted in the context of ray theory, and their frequency dependent sensitivity to anisotropy remains imperfectly understood, particularly for anisotropy in the lowermost mantle. The goal of this work is to obtain constraints on the frequency dependent sensitivity of SK(K)S phases to mantle anisotropy, particularly at the base of the mantle, through global wavefield simulations. We present results from a new numerical approach to modelling the effects of seismic anisotropy of arbitrary geometry on seismic wave propagation in global 3-D earth models using the spectral element solver AxiSEM3D. While previous versions of AxiSEM3D were capable of handling radially anisotropic input models, here we take advantage of the ability of the solver to handle the full fourth-order elasticity tensor, with 21 independent coefficients. We take advantage of the computational efficiency of the method to compute wavefields at the relatively short periods (5 s) that are needed to simulate SK(K)S phases. We benchmark the code for simple, single-layer anisotropic models by measuring the splitting (via both the splitting intensity and the traditional splitting parameters ϕ and δt) of synthetic waveforms and comparing them to well-understood analytical solutions. We then carry out a series of numerical experiments for laterally homogeneous upper mantle anisotropic models with different symmetry classes, and compare the splitting of synthetic waveforms to predictions from ray theory. We next investigate the full wave sensitivity of SK(K)S phases to lowermost mantle anisotropy, using elasticity models based on crystallographic preferred orientation of bridgmanite and post-perovskite. We find that SK(K)S phases have significant sensitivity to anisotropy at the base of the mantle, and while ray theoretical approximations capture the first-order aspects of the splitting behaviour, full wavefield simulations will allow for more accurate modelling of SK(K)S splitting data, particularly in the presence of lateral heterogeneity. Lastly, we present a cross-verification test of AxiSEM3D against the SPECFEM3D_GLOBE spectral element solver for global seismic waves in an anisotropic earth model that includes both radial and azimuthal anisotropy. A nearly perfect agreement is achieved, with a significantly lower computational cost for AxiSEM3D. Our results highlight the capability of AxiSEM3D to handle arbitrary anisotropy geometries and its potential for future studies aimed at unraveling the details of anisotropy at the base of the mantle.

Elasticity [formula omitted]-tensors and the strong ellipticity condition

In this paper, we establish two sufficient conditions for the strong ellipticity of any fourth-order elasticity tensor and investigate a class of tensors satisfying the strong ellipticity condition, the elasticity M-tensor. The first sufficient condition is that the strong ellipticity holds if the unfolding matrix of this fourth-order elasticity tensor can be modified into a positive definite one by preserving the summations of some corresponding entries. Second, an alternating projection algorithm is proposed to verify whether an elasticity tensor satisfies the first condition or not. Besides, the elasticity M-tensor is defined with respect to the M-eigenvalues of elasticity tensors. We prove that any nonsingular elasticity M-tensor satisfies the strong ellipticity condition by employing a Perron-Frobenius-type theorem for M-spectral radii of nonnegative elasticity tensors. Other equivalent definitions of nonsingular elasticity M-tensors are also established.