Constitutive Tensors Research Articles

Thermodinamically-consistent dynamic continualization of block-lattice materials

Lattice-like materials featuring periodic planar tessellation of regular rigid blocks connected by linear elastic interfaces and chiral or achiral properties are considered. The chirality results from a uniform rotation of the blocks with respect to their centroidal joining line and leads to interesting auxetic and dispersive acoustic behaviors. The governing equations of the discrete Lagrangian model are properly manipulated via the novel enhanced continualization scheme in such a way to obtain equivalent non-local integral and gradient-type higher-order continua. Based on the formal Taylor series expansion of the integral kernels or the corresponding pseudo-differential functions accounting for shift operators and proper pseudo-differential downscaling laws, the proposed enhanced continualization technique allows formulating homogeneous non-local continuum models of increasing orders, analytically featured by characteristic non-local constitutive and inertial terms. The enhanced continualization shows thermodynamic consistency in the definition of the overall non-local constitutive tensors, as well as qualitative agreement and quantitative convergent matching of the actual complex frequency band structure. The theoretical findings are successfully verified though the study of wave dispersion and attenuation properties as referred to a representative tetrachiral geometry.

Symmetric and antisymmetric constitutive tensors for bi-isotropic and bi-anisotropic media

The Maxwell equations and the constitutive relations describe the classical propagation of electromagnetic waves in continuous matter. Here, we investigate the effects stemming from extended constitutive relations on the propagation of waves in bi-isotropic and bi-anisotropic media using a classical general approach based on the evaluation of dispersion relations and refractive indices. For the bi-anisotropic media, we specify two classes of magnetoelectric parameters represented by symmetric and antisymmetric tensors. The three cases examined have provided real and distinct refractive indices for two propagating modes, which implies birefringence. The propagating modes were also carried out in all cases. The anisotropy or birefringence effect, given by the rotatory power or phase difference, was evaluated in terms of the magnetoelectric parameters of the theory in each case. The propagation orthogonal to the vectors used to parametrize the symmetric and antisymmetric magnetoelectric tensors is described by distinct modes, representing a route to identify the kind of bi-anisotropic medium examined. The group velocity and Poynting vector were also evaluated for all the cases examined to discuss the energy propagation in these anisotropic media.

Time decay for several porous thermoviscoelastic systems of Moore–Gibson–Thompson type

In this paper, we consider several problems arising in the theory of thermoelastic bodies with voids. Four particular cases are considered depending on the choice of the constitutive tensors, assuming different dissipation mechanisms determined by Moore–Gibson–Thompson-type viscosity. For all of them, the existence and uniqueness of solutions are shown by using semigroup arguments. The energy decay of the solutions is also analyzed for each case.

Distance to plane elasticity orthotropy by Euler–Lagrange method

Constitutive tensors are of common use in mechanics of materials. To determine the relevant symmetry class of an experimental tensor is still a tedious problem. For instance, it requires numerical methods in three-dimensional elasticity. We address here the more affordable case of plane (bi-dimensional) elasticity, which has not been fully solved yet. We recall first Vianello's orthogonal projection method, valid for both the isotropic and the square symmetric (tetragonal) symmetry classes. We then solve in a closed-form the problem of the distance to plane elasticity orthotropy, thanks to the Euler-Lagrange method.

Study on dispersion and wave velocity in 2D elliptic granular crystals by a micromechanics-based micromorphic model

This study considers two-dimensional elliptic granular crystals respectively with different aspect ratios. Using a micromechanics-based micromorphic model, macroscopic micromorphic constitutive modulus tensors of elliptic granular crystals are obtained. Two modes of Taylor expansion of relative displacements are used to establish contact relation between particles. This study obtains micromorphic transverse-rotational and longitudinal waves. Obvious dispersions of micromorphic waves are given in 2D elliptic granular crystals. With increase of aspect ratio, width of frequency band gap increases when wave propagates along [Formula: see text] direction but decreases when wave propagates along [Formula: see text] direction. Two modes of Taylor expansion of relative displacements only have influence on dispersions and frequency band gap of transverse-rotational waves. Velocities of longitudinal and transverse-rotational waves first decrease to zero, keep zero for a range of frequency and then increase to a larger wave velocity with increase of frequency for different aspect ratios. The situation with a larger aspect ratio leads to larger transverse-rotational and longitudinal velocities, and ratio between transverse-rotational and longitudinal velocities decreases from 0.7 to 0.59, when wave propagates along [Formula: see text] direction. However, it leads to smaller transverse-rotational and longitudinal velocities, and ratio between two velocities decreases from 0.88 to 0.56, when wave propagates along [Formula: see text] direction.

A double-phase field model for multiple failures in composites

Fiber reinforced composites have very excellent mechanical properties due to their sophistic microstructures. At the same time these microstructures can induce complex failure mechanisms including intralaminar damage, interlaminar damage and the interactions between them. Nevertheless, high fidelity simulations for these complex failure mechanisms are still challenging. In this paper, we develop a double-phase field model for complex failure in fiber reinforced composite. In the model, two different phase fields, i.e., fiber phase field and matrix phase field, are adopted for characterizing fiber damage and matrix damage, respectively. The failure mechanisms of fiber damage as well as the other modes in matrix damage are identified through a new strain energy density form that contains four different effective strain variables as well as a new effective constitutive tensor. Then by coupling with redefined degradation functions corresponding to different failure mechanisms, specified damage initiation and evolution criteria can be embedded into the proposed model automatically. Moreover, the new model is implemented in an explicit manner. The proposed model is verified and validated through the comparison between the predicted results and that of the experiments on the failure in unidirectional block and composite laminate.

Multifield constitutive identification of non-conventional thermo-viscoelastic periodic Cauchy materials

The present work proposes a variational-asymptotic homogenization technique for non-conventional thermo-viscoelastic periodic microstructured materials. According to second sound theories, the heat flux vector linearly depends upon the history of temperature gradient and the heat conduction tensor represents the kernel of the relative hereditary integral, thus overcoming the paradox inferred from the usual Fourier’s law of thermal waves propagating at infinite speed. Down-scaling relations have been provided in the frequency domain, relating the transformed micro displacement and relative temperature fields to the corresponding macro variables and their gradients through frequency-dependent perturbation functions which convey the influence of the underlying microstructural heterogeneity. Average field equations of infinite order have been derived. Transformed field equations of the equivalent first-order medium have been obtained as Euler–Lagrange equations of a suitable functional whose first variation is properly truncated. They are characterized by frequency-dependent overall constitutive tensors, whose closed form has been provided. A benchmark test assesses the capabilities of the proposed homogenization method, where the solutions relative to a periodic, bi-phase, thermo-viscoelastic material and to the equivalent homogenized medium match under the hypothesis of periodic source terms.

On a system of independent arguments for constitutive tensor functions on the growing surface in micropolar continuum

The present study is devoted to the boundary value problems statements for the growing materials with microstructural features. The general form of tensor relations on the propagating growing surface is derived as a consequence of the conservation laws of momentum and angular momentum. The necessary system of independent arguments of constitutive differential constraints on the growing surface in micropolar continuum is determined. A complete set of joint rational invariants of the system of tensors and vectors that determine the thermodynamics of the production process of a woven 3D material is given and discussed. An invariant-complete formulation of the constitutive relations on the growing surface is obtained.

Setting the Free Material Design problem through the methods of optimal mass distribution

The paper deals with the Free Material Design {(text {FMD})} problem aimed at constructing the least compliant structures from an elastic material, the constitutive field of which plays the role of design variable in the form of a tensor valued measure lambda supported in the design domain. Point-wise the constitutive tensor is referred to a given anisotropy class mathscr {H}, while the integral of a cost c(lambda ) is bounded from above. The convex p-homogeneous elastic potential j is parameterized by the constitutive tensor. The work puts forward the existence result and shows that the original problem can be reduced to the Linear Constrained Problem (mathrm {LCP}) known from the theory of optimal mass distribution by G. Bouchitté and G. Buttazzo. A theorem linking solutions of {(text {FMD})} and (mathrm {LCP}) allows to effectively solve the original problem. The developed theory encompasses several optimal anisotropy design problems known in the literature as well as it unlocks new ones. By employing the derived optimality conditions we give several analytical examples of optimal designs.

Anisotropic peridynamics for homogenized microstructured materials

An anisotropic continuum-molecular model based on pair potentials for describing the linear static and dynamic behavior of periodic heterogeneous Cauchy materials is derived in the framework of peridynamic theory. Non-central pairwise actions are assumed as work-conjugates to specific pairwise deformation variables. A pairwise constitutive law, depending trigonometrically on the material fiber direction and preserving the elastic symmetries of the material, is properly defined. Based on this theoretical ground, a bond-based type continuum model with oriented material points, governed by six independent material micro-moduli, is obtained. Considering microstructured materials featuring periodic rigid massive blocks connected by soft elastic interfaces, the elastic potential functions of the continuum-molecular model are consistently identified in terms of geometrical and constitutive parameters of the microstructure. The identification is based on the formal analogy with the overall constitutive tensors of a homogeneous continuum obtained through continualization schemes and equivalent to the microstructured material. Consequently, the macroscopic elastic properties of the continuum-molecular model are described in terms of pairwise material properties, but its micro-moduli are also explicitly related to the specific properties of the linear elastic interfaces of the microstructured material. The accuracy of the proposed model in the static and dynamic response is successfully verified by comparison with high-fidelity finite element models of an auxetic tetrachiral block lattice considering different periodic cell orientations and elastic interface properties, as well as by comparison with analytical solutions for fully anisotropic elasticity.

Large strain Flory’s decomposition for Lagrangian modeling of viscoleastic solids and compressive fluids

The fundamental difference in the solution of solids and fluids relies on the respective constitutive laws. Based on the Rivlin-Saunders-Düster-Hartmann hyperelastic model and using the Flory’s strain decomposition, we present a new total Lagrangian viscoelastic constitutive model for both Kelvin-Voigt viscoelastic solids and free-surface compressive viscous isothermal fluids. A dissipative viscous virtual work is written as a function of the time rate of isochoric invariants and its relation with the viscous stress is derived. Local time derivatives are solved by backward finite difference, allowing a consistent tangent viscoelastic constitutive tensor. The virtual work principle is used to write the weak equilibrium equation and its position-based finite element counterpart. Dynamic time integration is carried out by the Newmark β method and the Newton-Raphson procedure is used to solve time steps. The formulation is validated against experimental and numerical literature results revealing good precision. Additional examples are shown in order to demonstrate the applicability and future possibilities of the technique.

Asymptotic homogenization approach for anisotropic micropolar modeling of periodic Cauchy materials

A micropolar-based asymptotic homogenization approach for the analysis of composite materials with periodic microstructure is proposed. The upscaling relations, conceived to determine the macro-descriptors (macro displacement and the micropolar rotation fields) as a function of the micro displacement field, are consistently derived in the asymptotic framework. In particular, the micropolar rotation field is expressed in terms of the microscopical infinitesimal rotation tensor and perturbation functions. The micro displacement field is, in turn, obtained by choosing a third order approximation of the asymptotic expansion, in which the macroscopic fields are expressed as a third order polynomial expansion. It follows that the macro descriptors are directly related to both perturbation functions and micropolar two-dimensional deformation modes. Furthermore, a properly conceived energy equivalence between the macroscopic point and a microscopic representative portion of the periodic composite material is introduced to derive the consistent overall micropolar constitutive tensors. It is pointed out that these constitutive tensors are not affected by the choice of the periodic cell. Moreover, in the case of vanishing microstructure the internal-length-scale-dependent constitutive tensors tend to zero, as expected. Finally, the capabilities of the proposed approach are shown through some illustrative examples.

Inverse homogenization using the topological derivative

PurposeThe purpose of this study is to solve the inverse homogenization problem, or so-called material design problem, using the topological derivative concept.Design/methodology/approachThe optimal topology is obtained through a relaxed formulation of the problem by replacing the characteristic function with a continuous design variable, so-called density variable. The constitutive tensor is then parametrized with the density variable through an analytical interpolation scheme that is based on the topological derivative concept. The intermediate values that may appear in the optimal topologies are removed by penalizing the perimeter functional.FindingsThe optimization process benefits from the intermediate values that provide the proposed method reaching to solutions that the topological derivative had not been able to find before. In addition, the presented theory opens the path to propose a new framework of research where the topological derivative uses classical optimization algorithms.Originality/valueThe proposed methodology allows us to use the topological derivative concept for solving the inverse homogenization problem and to fulfil the optimality conditions of the problem with the use of classical optimization algorithms. The authors solved several material design examples through a projected gradient algorithm to show the advantages of the proposed method.

Mathematical modeling of the elastic properties of cubic crystals at small scales based on the Toupin\u2013Mindlin anisotropic first strain gradient elasticity

In this work, a mathematical modeling of the elastic properties of cubic crystals with centrosymmetry at small scales by means of the Toupin–Mindlin anisotropic first strain gradient elasticity theory is presented. In this framework, two constitutive tensors are involved, a constitutive tensor of fourth-rank of the elastic constants and a constitutive tensor of sixth-rank of the gradient-elastic constants. First, 3+11 material parameters (3 elastic and 11 gradient-elastic constants), 3 characteristic lengths and 1+6 isotropy conditions are derived. The 11 gradient-elastic constants are given in terms of the 11 gradient-elastic constants in Voigt notation. Second, the numerical values of the obtained quantities are computed for four representative cubic materials, namely aluminum (Al), copper (Cu), iron (Fe) and tungsten (W) using an interatomic potential (MEAM). The positive definiteness of the strain energy density is examined leading to 3 necessary and sufficient conditions for the elastic constants and 7 ones for the gradient-elastic constants in Voigt notation. Moreover, 5 lattice relations as well as 8 generalized Cauchy relations for the gradient-elastic constants are derived. Furthermore, using the normalized Voigt notation, a tensor equivalent matrix representation of the two constitutive tensors is given. A generalization of the Voigt average toward the sixth-rank constitutive tensor of the gradient-elastic constants is given in order to determine isotropic gradient-elastic constants. In addition, Mindlin’s isotropic first strain gradient elasticity theory is also considered offering through comparisons a deeper understanding of the influence of the anisotropy in a crystal as well as the increased complexity of the mathematical modeling.

Overall dynamic properties of locally resonant viscoelastic layered media based on consistent field integration for oblique anti-plane shear waves

In this work, the analysis of oblique anti-plane shear waves propagation and scattering in low frequency resonant micro-structured layered media with viscoelastic constituent layers is presented. The band structure of the infinitely periodic systems and scattering off a finite thickness slab of such media are determined using the transfer matrix method. A consistent dynamic field homogenization approach is applied, in which the micro-scale field equations are integrated and the overall macro-scale quantities are defined to be compatible with these integral forms. A reduced set of constitutive tensors is presented for general asymmetric repeating unit cells (RUC), utilizing the proposed homogenized macro-scale quantities combined with Onsager’s principle and presumed material form of elastodynamic reciprocity. This set can be further restricted by studying the form of the dispersion equation leading to a unique constitutive tensor. It is shown that for an asymmetric RUC, the full constitutive tensor, including off-diagonal elastic moduli and Willis coupling terms are required in order to match the scattering and band structure of the micro-structured media, but all the off-diagonal parameters vanish for a symmetric RUC. Therefore, it is possible to create an equivalent homogenized representation with a uniquely determined diagonal constitutive tensor for a symmetric RUC, though, as is the case also with asymmetric RUCs, all non-zero components will be wave-vector dependent. Numerical examples are presented to demonstrate the application and consistency of the proposed method. All the diagonal terms are converging to their appropriate Voigt or Reuss averages at the long-wavelength limit for all wave directions. The wave-vector dependent nature of the off-diagonal coupling constants can still be observed even in this limit. The conditions for lossy (passive) or lossless (fully elastic) systems are presented and are shown to impose weak requirements on overall constitutive tensors.

A micropolar continuum model of diffusion creep

ABSTRACT Solid polycrystalline materials undergoing diffusion creep are usually described by Cauchy continuum models with a Newtonian viscous rheology dependent on the grain size. Such a continuum lacks the rotational degrees of freedom needed to describe grain rotation. Here we provide a more general continuum description of diffusion creep that includes grain rotation, and identifies the deformation of the material with that of a micropolar (Cosserat) fluid. We derive expressions for the micropolar constitutive tensors by a homogenisation of the physics describing a discrete collection of rigid grains, demanding an equivalent dissipation between the discrete and continuum descriptions. General constitutive laws are derived for both Coble (grain-boundary diffusion) and Nabarro-Herring (volume diffusion) creep. Detailed calculations are performed for a two-dimensional tiling of irregular hexagonal grains, which illustrates a potential coupling between the rotational and translational degrees of freedom. If only the plating out or removal of material at grain boundaries is considered, the constitutive laws are degenerate: modes of deformation that involve pure tangential motion at the grain boundaries are not resisted. This degeneracy can be removed by including the resistance to grain-boundary sliding, or by imposing additional constraints on the deformation.

Fourth‐order phase field model with spectral decomposition for simulating fracture in hyperelastic material

AbstractWe present a fourth‐order phase field model for fracture behavior simulations of hyperelastic material undergoing finite deformation. Governing equations of the fourth‐order phase field model consist of the biharmonic operator of the phase field, which requires the second‐order derivatives of shape function. Therefore, a 5 × 5 Jacobian matrix of isoparametric transformation is constructed. Neo‐Hooken model and Hencky model are adopted as the material constitutive models. The spectral decomposition of stored strain energy is used to distinguish the contributions of tension and compression, and the corresponding stress tensor and constitutive tensors are derived, and subsequently, the numerical framework of modeling fracture with the fourth‐order phase field model is implemented in details. Several typical numerical examples are conducted to demonstrate the robustness and effectiveness of the fourth‐order phase field model in simulating the fracture phenomenon of rubber‐like materials.

Decomposition of third-order constitutive tensors

Third-order tensors are widely used as a mathematical tool for modeling the physical properties of media in solid-state physics. In most cases, they arise as constitutive tensors of proportionality between basic physical quantities. The constitutive tensor can be considered the complete set of physical parameters of a medium. The algebraic features of the constitutive tensor can be used as a tool for proper identification of natural materials, such as crystals, and for designing artificial nanomaterials with prescribed properties. In this paper, we study the algebraic properties of a general asymmetric third-order tensor relative to its invariant decomposition. In correspondence with different groups acting on the basic vector space, we present the hierarchy of different types of tensor decomposition into invariant subtensors. In particular, we discuss the problem of non-uniqueness and reducibility of high-order tensor decomposition. For a general asymmetric third-order tensor, these features are described explicitly. In the case of special tensors with a prescribed symmetry, the decomposition is demonstrated to be irreducible and unique. We present the explicit results for two physically interesting models: the piezoelectric tensor as an example of pair symmetry and the Hall tensor as an example of pair skew-symmetry.

Oblique Scattering From Radially Inhomogeneous Bianisotropic Cylindrical Structures Analyzed Using a Highly Accurate Volterra Integral Equation Approach

We study the oblique scattering of arbitrarily polarized plane waves from radially inhomogeneous and bianisotropic cylinders in the general case where all the elements of the four constitutive tensors are nonzero. The analysis relies on an exact reformulation of Maxwell’s equations as a second-kind linear Volterra matrix integral equation. The discretization scheme uses a high order, entire-domain, Nystrom method that employs the Gauss–Legendre–Radau quadrature. The entries of the Nystrom matrix have simple closed-form expressions. The proposed algorithms have exponential convergence and can handle both single-layer and multilayer cylinders with either continuous or sectionally continuous inhomogeneity profiles.

The electromomentum coupling in generalized Willis media

By developing and applying a homogenization scheme for elastodynamics, Willis discovered that the momentum of composite materials is macroscopically coupled with their strain through a constitutive tensor. This now-termed Willis tensor not only enlarges the design space of metamaterials, but is also necessary for obtaining a meaningful effective description that respects basic physical laws. In this talk, I will show how additional tensors of Willis type emerge by generalizing the homogenization theory of Willis to thermoelastic-, piezomagnetic- or piezoelectric media. I will provide examples for the latter case that exhibit an electromomentum coupling. I will show that this coupling is necessary for describing the effective properties of asymmetric piezoelectric media using a homogenized description that respects reciprocity and energy conservation. Finally, I will demonstrate how this coupling can be used to realize a device that actively control the phase of elastic waves.