Cauchy Medium Research Articles

Non-uniqueness, stability and bifurcation analyses in elasto-viscoplastic boundary value problems with no inertia

The article focuses on non-uniqueness, bifurcation and stability conditions in elasto-viscoplastic boundary value problems when inertia terms are neglected. Analytical and numerical studies are presented to investigate the capability of an elasto-viscoplastic model to regularize the behavior in the occurrence of strain localization with respect to number of strain bands formed and mesh dependency. It is found that elasto-viscoplasticity in a Cauchy medium neither restores the uniqueness of the solution nor provides mesh independent results. A high value of the viscosity parameter can sometimes provide results that are mesh independent, up to a certain limit strain, as it actually modifies the response of the constitutive law by an ad-hoc increase of its hardening branch. On the contrary, coupling elasto-viscoplasticity with a second gradient model that introduces an internal length parameter reproduces realistically the rate dependent behavior and regularizes the results.

Self-consistent assessments for the effective properties of two-phase composites within strain gradient elasticity

Analytical method for the second-order homogenization of two-phase composites within Mindlin–Toupin strain gradient elasticity theory is proposed. Direct approach and self-consistent approximation are used to reduce the homogenization problem to the problem of determination of averaged Cauchy stresses, double stresses and static moments of Cauchy stresses inside the inclusions under prescribed quadratic boundary conditions. The ellipsoidal shape of inclusions and orthotropic properties of phases are assumed. Extended equivalent inclusion method with linear eigenstrain is proposed to derive the explicit relations between the Eshelby-like tensors and corresponding concentration tensors that are used to define the averaged field variables inside the inclusions. Obtained analytical solutions allow to evaluate the effective classical and gradient elastic moduli of composite materials accounting for the phases properties, volume fraction, shape and size of inclusions. Presented solution for the effective gradient moduli covers the full range of volume fraction and correctly predicts the absence of gradient effects for the homogeneous classical Cauchy medium. Examples of calculations for the composites with spherical inclusions are given. Micro-scale definition of the well-known simplified strain gradient elasticity theory is provided. Namely, it is shown that this theory is the phenomenological continuum model for the composites with isotropic matrix and with the small volume fraction of stiff isotropic spherical inclusions, which have the same Poisson’s ratio to those one of matrix phase.

Generalized macroscale model for Cosserat elasticity using Generalized Multiscale Finite Element Method

In this paper, we develop a multiscale computational approach for heterogeneous Cosserat media without scale separation and high contrast. Cosserat medium has been used in many applications to describe materials whose particles possess rotational degrees of freedom. Many previous findings investigate Cosserat medium in the presence of heterogeneities. Approaches that include homogenization and numerical homogenization have been developed for problems with scale separation, such as periodicity. In these approaches, macroscopic models are derived that show the resulting equations can be Cauchy medium or Cosserat medium depending on the Cosserat intrinsic length scale. In our paper, we consider computational macroscopic models for Cosserat medium without scale separation. We use Generalized Multiscale Finite Element Method (GMsFEM) to derive a macroscopic model on a coarse computational grid, that does not resolve the small scales and the contrast. In this approach, multiple macroscopic basis functions are used to describe the small scales. The resulting macroscopic model is similar to multicontinua model as it contains multiple macroscopic parameters at each macroscopic point (or grid). We present numerical results for different heterogeneous media types. Our numerical results show that using a few multiscale basis functions (i.e., macroscopic parameters), we can achieve a good accuracy.

A novel XFEM cohesive fracture framework for modeling nonlocal slip in randomly discrete fiber reinforced cementitious composites

We develop and implement a new nonlocal crack-bridging model for evaluating toughening and strengthening mechanisms of fiber reinforced cementitious composites. The quasibrittle fracture of cement matrix is characterized by XFEM-cohesive crack framework. To explore fiber-bridging mechanism, the horizontal fiber phases corresponding to different embedding length are superimposed on the near-crack region of the matrix phase, while an effective homogeneous Cauchy medium is assumed outside the multiphase region. The sliding interface for each fiber phase is described by a slip field, leading to a new nonlocal slip model being proposed which can be used to treat arbitrary fiber distributions. This model is formulated in a variationally consistent framework. In this study, it is verified that the toughening effect is induced by the interfacial shear stress rather than the fiber tensile stress. Prior to evaluate new problems, the capability of the proposed model is validated to reproduce consistent fracture behaviors with the experimentations. It is concluded from our systematic studies that a higher fiber modulus leads to a higher loading capacity, but not a higher composite toughness, while unaffects its residual strength. It is found that the enhancing interfacial strength has improved the composites toughness dramatically, but has limited contribution to the strengthening effect.

Microstructure-related Stoneley waves and their effect on the scattering properties of a 2D Cauchy/relaxed-micromorphic interface

In this paper we set up the full two-dimensional plane wave solution for scattering from an interface separating a classical Cauchy medium from a relaxed micromorphic medium. Both media are assumed to be isotropic and semi-infinite to ease the semi-analytical implementation of the associated boundary value problem.Generalized macroscopic boundary conditions are presented (continuity of macroscopic displacement, continuity of generalized tractions and, eventually, additional conditions involving purely microstructural constraints), which allow for the effective description of the scattering properties of an interface between a homogeneous solid and a mechanical metamaterial. The associated “generalized energy flux” is introduced so as to quantify the energy which is transmitted at the interface via a simple scalar, macroscopic quantity.Two cases are considered in which the left homogeneous medium is “stiffer” and “softer” than the right metamaterial and the transmission coefficient is obtained as a function of the frequency and of the direction of propagation of the incident wave. We show that the contrast of the macroscopic stiffnesses of the two media, together with the type of boundary conditions, strongly influence the onset of Stoneley (or evanescent) waves at the interface. This allows for the tailoring of the scattering properties of the interface at both low and high frequencies, ranging from zones of complete transmission to zones of zero transmission well beyond the band-gap region.

Multiphase continuum models for fiber-reinforced materials

This contribution addresses the formulation of a generalized continuum model called multiphase model aimed at describing more accurately the mechanical behavior of fiber-reinforced materials. Improving on the classical macroscopic description of heterogeneous materials by an effective homogeneous Cauchy medium, such models rely on the superposition of several continua (or phases) possessing their own kinematics at the macroscopic level and being in mutual interaction (in the same spirit of deformable porous media). Up to now, they have only been formulated based on phenomenological assumptions and the identification of the corresponding constitutive parameters remained unclear. The aim of this paper is three-fold. First, a homogenization procedure is described, enabling to derive constitutive parameters from the resolution of a generalized auxiliary problem on a classical heterogeneous microstructure. Second, analytical and numerical derivation of these properties is performed in various cases. Finally, illustrative applications on boundary-value problems assess the validity of the homogenization procedure and illustrate the relevance of such generalized models which are able to capture scale effects and to model crack bridging and delaminated configurations at the macroscopic level. It is also shown to encompass results of shear-lag models for analyzing stress transfers in fiber/matrix composites.

Dynamical analysis of homogenized second gradient anisotropic media for textile composite structures and analysis of size effects

In order to predict the dispersion relation of 3D composite structures in the low frequency range, we construct effective first and second order grade continuum models. The effective properties of textile composites are obtained computationally by an equivalent strain energy method based on the response of the representative volume unit cell (RUC) under prescribed boundary conditions as described in Goda and Ganghoffer (2016). The expressions of the phase velocities for the three modes of wave propagation in a 3D context (longitudinal, horizontal shear and vertical shear) reveal that the second order continuum is dispersive, due to the presence of the second order elasticity constants. The shape change of the phase velocity when increasing the wave number shows the dispersive behavior of the second gradient medium, whereas Cauchy medium is non dispersive. Plots of the iso-frequency contour for the two investigated composites in the case of second gradient and Cauchy effective medium show that the second gradient contributions does not modify the anisotropic behavior of the considered composites. Important size effects on the dynamical behavior are shown, especially reflected by the dispersive behavior and the anisotropic dynamic responses, due to the significant overall increase of the second order rigidity matrix when increasing the RUC size.

Numerical homogenization of mesoscopic loss in poroelastic media

This contribution deals with the numerical homogenization of mesoscopic flow phenomena in fluid-saturated poroelastic media. Under compression, mesoscopic heterogeneities induce pore pressure gradients and consequently pressure diffusion of the pore fluid. Since this process takes place on a scale much smaller than the observable level, the dissipation mechanism is considered as a local phenomenon. The heterogeneous poroelastic medium is substituted by an overall homogeneous Cauchy medium accounting for viscoelastic properties. Applying volume averaging techniques we derive a consistent upscaling procedure based on an appropriate extension of the Hill-Mandel lemma. We introduce various sets of boundary conditions for the poroelastic problem and discuss the relation between size of the SVEm (Statistical Volume Element) and maximum diffusion length. Numerical examples for two- and three-dimensional problems are given.

Micromechanical approach for the micropolar modeling of heterogeneous periodic media

Computational homogenization is adopted to assess the homogenized two-dimensional response of periodic composite materials where the typical microstructural dimension is not negligible with respect to the structural sizes. A micropolar homogenization is, therefore, considered coupling a Cosserat medium at the macro-level with a Cauchy medium at the micro-level, where a repetitive Unit Cell (UC) is selected. A third order polynomial map is used to apply deformation modes on the repetitive UC consistent with the macro-level strain components. Hence, the perturbation displacement field arising in the heterogeneous medium is characterized. Thus, a newly defined micromechanical approach, based on the decomposition of the perturbation fields in terms of functions which depend on the macroscopic strain components, is adopted. Then, to estimate the effective micropolar constitutive response, the well known identification procedure based on the Hill-Mandel macro-homogeneity condition is exploited. Numerical examples for a specific composite with cubic symmetry are shown. The influence of the selection of the UC is analyzed and some critical issues are outlined.

Micromechanical analysis of heterogeneous materials subjected to overall Cosserat strains

In the framework of the computational homogenization procedures, the problem of coupling a Cosserat continuum at the macroscopic level and a Cauchy medium at the microscopic level, where a heterogeneous periodic material is considered, is addressed. In particular, non-homogeneous higher-order boundary conditions are defined on the basis of a kinematic map, properly formulated for taking into account all the Cosserat deformation components and for satisfying all the governing equations at the micro-level in the case of a homogenized elastic material. Furthermore, the distribution of the perturbation fields, arising when the actual heterogeneous nature of the material is taken into account, is investigated. Contrary to the case of the first-order homogenization where periodic fluctuations arise, in the analyzed problem more complex distributions emerge.

Application of Clifford algebra $${C \ell_3(\mathbb{C})}$$ to continuum and engineering mechanics

Multivectorial algebra is of both academic and technological interest. Its application, however, is not always easy. A distinction must be made between polar and axial vectors and between scalars and pseudo-scalars. Eight element types are often considered even if they are not always identified as multivectors. In some cases, for simplicity’s sake, only vectorial algebra or quaternion algebra is explicitly used for physical and mechanical applications. It would, however, be more convenient to use more complex algebra directly in order to have a wider range of mechanical applications. The aim of this paper is to examine one particular type of Clifford algebra that could solve this problem. The present study focusses on showing how these quantities can be used to model mechanical and engineering problems. First, continuum mechanics in a Cauchy medium is investigated for elastic transformations. Second, a specific type of shot-peening application is studied. Applications are then used to illustrate the scope and efficiency of this type of modeling based on geometric algebra.

A multi-scale enriched model for the analysis of masonry panels

A multi-scale model for the structural analysis of the in-plane response of masonry panels, characterized by periodic arrangement of bricks and mortar, is presented. The model is based on the use of two scales: at the macroscopic level the Cosserat micropolar continuum is adopted, while at the microscopic scale the classical Cauchy medium is employed. A nonlinear constitutive law is introduced at the microscopic level, which includes damage, friction, crushing and unilateral contact effects for the mortar joints. The nonlinear homogenization is performed employing the Transformation Field Analysis (TFA) technique, properly extended to the macroscopic Cosserat continuum. A numerical procedure is developed and implemented in a Finite Element (FE) code in order to analyze some interesting structural problems. In particular, four numerical applications are presented: the first one analyzes the response of the masonry Representative Volume Element (RVE) subjected to a cyclic loading history; in the other three applications, a comparison between the numerically evaluated response and the micromechanical or experimental one is performed for some masonry panels.

A variational method for non-linear micropolar composites

Built upon Ponte Castañeda’s method for a Cauchy medium, a variational method for evaluating the effective non-linear behavior of micropolar composites is proposed. The same as for a Cauchy medium, it is shown that the proposed variational method can be interpreted as the secant moduli method based on second-order stress and couple stress moments. With simple examples, the interplay between material length parameters of a higher-order medium and its geometrical dimensions and/or material constants is highlighted. By using the new variational method, the influence of reinforcement size on the yielding and strain hardening of particulate composites is examined in a simple and analytical manner. The predictions agree well with existing experimental data for selected particulate metal matrix composite systems. The particle size effect is found to be more pronounced for shear loading and hard particles.

A continuum micromechanical theory of overall plasticity for particulate composites including particle size effect

Classical continuum micromechanics cannot predict the particle size dependence of the overall plasticity for composite materials, a simple analytical micromechanical method is proposed in this paper to investigate this size dependence. The matrix material is idealized as a micropolar continuum, an average equivalent inclusion method is advanced and the Mori–Tanaka's method is extended to a micropolar medium to evaluate the effective elastic modulus tensor. The overall plasticity of composites is predicted by a new secant moduli method based on the second order moment of strain and torsion of the matrix in a framework of micropolar theory. The computed results show that the size dependence is more pronounced when the particle's size approaches to the matrix characteristic length, and for large particle sizes, the prediction coincides with that predicted by classical micromechanical models. The method is analytical in nature, and it can capture the particle size dependence on the overall plastic behavior for particulate composites, and the prediction agrees well with the experimental results presented in literature. The proposed model can be considered as a natural extension of the widely used secant moduli method from a heterogeneous Cauchy medium to a micropolar composite.