Isotropic Matrix Research Articles

Elastic constants influence on the L4-L5-S1 annuli fibrosus behavior, a probabilistic finite element analysis.

A probabilistic finite element (FE) analysis of the L4-L5 and L5-S1 human annulus fibrosus (AF) was conducted to obtain a better understanding of the biomechanics of the AF and to quantify its influence on the range of motion (ROM) of the L4-L5 and L5-S1 segments. The FE models were composed of the AF and the upper and lower endplates. The AF was represented as a continuous material composed of a hyperelastic isotropic Yeoh matrix reinforced with two families of fibers described with an exponential energy function. The caudal endplate was fully restricted and 8 Nm pure moment was applied to the cranial endplate in flexion, extension, lateral flexion and axial rotation. The mechanical constants were determined randomly based on a normal distribution and average values reported. Results of the 576 models show that the ROM was more sensitive to the initial stiffness of the fibers rather than to the stiffening coefficient represented in the exponential function. The ROM was more sensitive to the input variables in extension, flexion, axial rotation and lateral bending. The analysis showed an increased probability for the L5-S1 ROM to be higher in flexion, extension and axial rotation, and smaller in lateral flexion, with respect to the L4-L5 ROM. An equation was proposed to obtain the ROM as a function of the elastic constants of the fibers and it may be used to facilitate the calibration process of the human spine segments and to understand the influence of each elastic constant on the ROM.

Surface tension and a self-consistent theory of soft composite solids with elastic inclusions.

The importance of surface tension effects is being recognized in the context of soft composite solids, where they are found to significantly affect the mechanical properties, such as the elastic response to an external stress. It has recently been discovered that Eshelby's inclusion theory breaks down when the inclusion size approaches the elastocapillary length L≡γ/E, where γ is the inclusion/host surface tension and E is the host Young's modulus. Extending our recent results for liquid inclusions, here we model the elastic behavior of a non-dilute distribution of isotropic elastic spherical inclusions in a soft isotropic elastic matrix, subject to a prescribed infinitesimal far-field loading. Within our framework, the composite stiffness is uniquely determined by the elastocapillary length L, the spherical inclusion radius R, and the stiffness contrast parameter C, which is the ratio of the inclusion to the matrix stiffness. We compare the results with those from the case of liquid inclusions, and we derive an analytical expression for elastic cloaking of the composite by the inclusions. Remarkably, we find that the composite stiffness is influenced significantly by surface tension even for inclusions two orders of magnitude more stiff than the host matrix. Finally, we show how to simultaneously determine the surface tension and the inclusion stiffness using two independent constraints provided by global and local measurements.

Constitutive modelling of fibre reinforced nonhomogenous hyperelastic materials

In the paper two classes of poly-convex stored energy functions for transversally isotropic hyper-elastic materials have been proposed. Within these classes, a simplified model of hyper-elastic materials with an isotropic matrix reinforced with fibre family was derived and implemented in finite element software ABAQUS/Standard. The two numerical examples proving correctness of the implementation and showing its positive physical features are presented.

Equivalent-inclusion approach for the conductivity of isotropic matrix composites with anisotropic inclusions

Many effective medium approximations for effective conductivity are elaborated for matrix composites made from isotropic continuous matrix and isotropic inclusions associated with simple shapes such as circles or spheres, ... In this paper, we focus specially on the effective conductivity of the isotropic composites containing the disorderly oriented anisotropic inclusions. We aim to replace those inhomogeneities by simple equivalent circular (spherical) isotropic inclusions with modified conductivities. Available simple approximations for the equivalent circular (spherical)-inclusion media then can be used to estimate the effective conductivity of the original composite. The equivalent-inclusion approach agrees well with numerical extended finite elements results.

A pressure-dependent phenomenological constitutive model for transversely isotropic foams

Based on the elongated Voronoi models, the yield behaviors of transversely isotropic foams have been explored by using finite element method. A pressure-dependent phenomenological constitutive model (stress potential function) has been established for transversely isotropic foams through the characteristic stress and strain. The stress potential function and its evolution can be determined in terms of the uniaxial and equi-triaxial tensile/compressive characteristic stress-strain curves. The stress potential function is expected to predict the elastoplastic mechanical responses of transversely isotropic foams. The material properties (i.e., relative density, transversely isotropic degree, matrix material and cell morphology) exhibit significant influence on the profile of initial yield surfaces. The pressure dependence of yield surfaces decreases as the relative density and transversely isotropic degree increase. The predictions of pressure-dependent constitutive model match excellently with the simulated yield data under different stress states and characteristic strain stages.

Evolution of Strain Energy During Recrystallization of Plated Cu Films

Microstructural evolution within thin films is dictated by energy minimization that can arise due to different mechanisms. Conventional treatments that estimate the driving force associated with elastic strain energy often employ a fiber-textured microstructure to arrive at an analytical solution. By approximating the case of a recrystallized grain in a randomly oriented film as an elastically anisotropic inclusion in an elastically isotropic matrix, we can apply Eshelby's inclusion method to calculate both the interaction strains and the strain energy density generated by this elastic incompatibility. A comparison of these two approaches for grains with cubic symmetry reveals that a lower amount of elastic strain energy is generated in the case of an elastically anisotropic grain in an isotropic matrix, suggesting that it is less energetically favorable for Cu (111) grains to recrystallize in films possessing strong (111) texture than in randomly textured films. High-resolution X-ray diffraction measurements are used to extract the difference in the elastic response between recrystallized grains and the untransformed matrix, providing experimental quantification of the induced interaction strains. In addition, the corresponding elastic strain energy values are compared with the appearance of recrystallized (100) grains as a function of film thickness, allowing for a threshold energy density to be extracted. This threshold energy density can be correlated to the difference in surface energy values between (100)- and (111)-oriented grains.

Polymer-dispersed liquid crystal elastomers.

The need for mechanical manipulation during the curing of conventional liquid crystal elastomers diminishes their applicability in the field of shape-programmable soft materials and future applications in additive manufacturing. Here we report on polymer-dispersed liquid crystal elastomers, novel composite materials that eliminate this difficulty. Their thermal shape memory anisotropy is imprinted by curing in external magnetic field, providing for conventional moulding of macroscopically sized soft, thermomechanically active elastic objects of general shapes. The binary soft-soft composition of isotropic elastomer matrix, filled with freeze-fracture-fabricated, oriented liquid crystal elastomer microparticles as colloidal inclusions, allows for fine-tuning of thermal morphing behaviour. This is accomplished by adjusting the concentration, spatial distribution and orientation of microparticles or using blends of microparticles with different thermomechanical characteristics. We demonstrate that any Gaussian thermomechanical deformation mode (bend, cup, saddle, left and right twist) of a planar sample, as well as beat-like actuation, is attainable with bilayer microparticle configurations.

Analytical treatment on the effective material properties of a composite material with spheroidal and ellipsoidal inhomogeneities in an isotropic matrix

Effective material properties of a composite with spheroidal and ellipsoidal inhomogeneities in an isotropic matrix are investigated analytically using the dilute approximation and the Mori–Tanaka approximation together with the Eshelby's equivalent inclusion method. Both uniaxially aligned and uniformly randomly oriented spheroidal and ellipsoidal inhomogeneities are treated. For a spheroid, both oblate and prolate spheroidal inhomogeneities are considered. It is analytically shown that a composite with uniaxially aligned anisotropic ellipsoidal inhomogeneities in an isotropic matrix is anisotropic in general in thermal conductivity. It is also analytically shown that a composite with uniformly randomly oriented anisotropic ellipsoidal inhomogeneities in an isotropic matrix is exactly isotropic in thermal conductivity. Various special cases are also treated for the effective thermal conductivity of a composite with ellipsoidal and spheroidal inhomogeneities. Similar results are also obtained for the effective elastic moduli. Newly obtained expressions for the effective elastic moduli of a composite with isotropic spheroidal inhomogeneities are rather involved. Conversely, an effective thermal conductivity of a composite with anisotropic ellipsoidal inhomogeneities is relatively simple. An effective thermal conductivity of a composite with isotropic spheroidal inhomogeneities reduces to a known result (Kerner, E. H. [1956] “The electrical conductivity of composite media,” Proceedings of the Physical Society London Section B 69, 802–807; Hashin, Z. and Shtrikman, S. [1962] “A variational approach to the theory of the effective magnetic permeability of multiphase materials,” Journal of Applied Physics 33, 3125–3131.) as the spheroid aspect ratio approaches 1 (i.e., a sphere). The effective thermal conductivity of a composite with uniformly randomly oriented isotropic spheroidal inhomogeneities in an isotropic matrix obtained in this paper as a special case is similar to the one obtained by Hatta and Taya (Hatta, H. and Taya, M. [1985] “Effective thermal conductivity of a misoriented short fiber composite,” Journal Applied Physics 58, 2478–2486.) in some respects, but is different. Numerical results are shown for a composite with oblate spheroidal voids in an isotropic matrix.

A numerical damage model for initially anisotropic materials

Significant progresses have been realized during the last decades on both macroscopic and micro-mechanical modeling of induced damage in brittle materials. Most damage models developed so far were devoted to initially isotropic materials. This work is devoted to modeling of induced damage in an initially anisotropic material. A numerical micro-mechanical damage model is proposed, using an Eshelby inclusion solution based homogenization method. Based on the numerical integration of the exact Green’s function and using an appropriate coordinate frame rotation method, an efficient numerical algorithm is proposed to determine the Hill tensor for an arbitrarily oriented family of cracks embedded in a transversely isotropic elastic matrix. Based on this, the effective elastic properties of cracked materials are determined through a rigorous up-scaling procedure using three different homogenization schemes, and taking into account interactions between the initial material anisotropy and induced cracks. A specific damage criterion is then defined in the framework of irreversible thermodynamics to describe the progressive growth of damage. The proposed model is finally implemented in a computer code and applied to study mechanical responses of cracked materials in different loading paths. Again, effects of the initial anisotropy and homogenization schemes are investigated.

The criterion for determining the direction of crack propagation in a random pattern composites

Presented paper contains results of fracture analysis of brittle composite materials with a random distribution of grains. The composite structure has been modelled as an isotropic matrix that surrounds circular grains with random diameters and space position. Analyses were preformed for the rectangular “numerical sample” by finite element method. FE mesh for the examples were generated using the authors’ computer program RandomGrain. Fracture analyses were accomplished with the authors’ computer program CrackPath3 executing the “fine mesh window” technique. Calculations were preformed in 2D space assuming the plane stress state. Current efforts focus on brittle materials such as rocks or concrete.

A micromechanical approach for homogenization of elastic metamaterials with dynamic microstructure.

An approximate homogenization technique is presented for generally anisotropic elastic metamaterials consisting of an elastic host material containing randomly distributed heterogeneities displaying frequency-dependent material properties. The dynamic response may arise from relaxation processes such as viscoelasticity or from dynamic microstructure. A Green's function approach is used to model elastic inhomogeneities embedded within a uniform elastic matrix as force sources that are excited by a time-varying, spatially uniform displacement field. Assuming dynamic subwavelength inhomogeneities only interact through their volume-averaged fields implies the macroscopic stress and momentum density fields are functions of both the microscopic strain and velocity fields, and may be related to the macroscopic strain and velocity fields through localization tensors. The macroscopic and microscopic fields are combined to yield a homogenization scheme that predicts the local effective stiffness, density and coupling tensors for an effective Willis-type constitutive equation. It is shown that when internal degrees of freedom of the inhomogeneities are present, Willis-type coupling becomes necessary on the macroscale. To demonstrate the utility of the homogenization technique, the effective properties of an isotropic elastic matrix material containing isotropic and anisotropic spherical inhomogeneities, isotropic spheroidal inhomogeneities and isotropic dynamic spherical inhomogeneities are presented and discussed.

Conduction of the three-dimensional model of a composite with structural anisotropy

The electric conduction of the model of a structurally anisotropic composite that represents an isotropic matrix with disk-shaped inclusions (oblate spheroids) is analytically studied. The disks have the same orientation and their centers are randomly distributed in the composite volume. The cases of dielectric and ideally conducting inclusions are considered. The conduction of the model in a broad concentration range is described by the approximate effective medium method. Conduction in the critical region (in the vicinity of the metal–dielectric phase transition point) is studied using the similarity hypothesis.

The research on thermoplastic properties cement composites impregnated with the waste of sulfur

The process of creating special polymerized sulfur composites through application of industrial waste material is a thermal treatment process in the temperature of about 150–155 ^{circ }hbox {C}. The result of such treatment is a special polymerized sulfur composite in a liquid state. This paper presents the effective plastic constants and coefficients of thermal expansion of stochastic special polymer sulfur composites, with isotropic porous matrix, reinforced by disoriented ellipsoidal inclusions with orthotropic symmetry of the thermoplastic properties. The investigations are based on the stochastic differential equations of solid mechanics. A model and algorithm for calculating the effective characteristics of special polymer sulfur composites are suggested. The effective thermoplastic characteristics of special polymer sulfur composites, with disoriented ellipsoidal inclusions, are calculated in two stages: First, the properties of materials with oriented inclusions are determined, and then, effective constants of a composite with disoriented inclusions are determined on the basis of the Voigt or Rice scheme. In this paper, the review of the literature has been done, among other things, about selecting and using special polymerized sulfur coating in buildings, about the methods of surface protection against corrosion of concretes, and about the methods of the evaluation of the protection layer on these materials. Special polymerized sulfur as polymer coating impregnation, which has received little attention in recent years, currently has some very interesting applications. An innovative overlay for use in areas where freezing and thawing are prevalent has been developed.

The analysis of thermoplastic characteristics of special polymer sulfur composite

Specific chemical environments step out in the industry objects. Portland cement composites (concrete and mortar) were impregnated by using the special polymerized sulfur and technical soot as a filler (polymer sulfur composite). Sulfur and technical soot was applied as the industrial waste. Portland cement composites were made of the same aggregate, cement and water. The process of special polymer sulfur composite applied as the industrial waste is a thermal treatment process in the temperature of about 150–155 ^{circ }hbox {C}. The result of such treatment is special polymer sulfur composite in a liquid state. This paper presents the plastic constants and coefficients of thermal expansion of special polymer sulfur composites, with isotropic porous matrix, reinforced by disoriented ellipsoidal inclusions with orthotropic symmetry of the thermoplastic properties. The investigations are based on the stochastic differential equations of solid mechanics. A model and algorithm for calculating the effective characteristics of special polymer sulfur composites are suggested. The effective thermoplastic characteristics of special polymer sulfur composites, with disoriented ellipsoidal inclusions, are calculated in two stages: First, the properties of materials with oriented inclusions are determined, and then effective constants of a composite with disoriented inclusions are determined on the basis of the Voigt or Rice scheme. A brief summary of new products related to special polymer sulfur composites is given as follows: Impregnation, repair, overlays and precast polymer concrete will be presented. Special polymer sulfur as polymer coating impregnation, which has received little attention in recent years, currently has some very interesting applications.

An anisotropic hyperelastic constitutive model for short glass fiber-reinforced polyamide

The present paper describes the development of a unified hyperelastic constitutive model based on anisotropic hyperelastic material with families of fibers approach and integrating both; pure isotropic matrix of polyamide-6.6 and anisotropic materials induced by the short glass fiber reinforcement (SGFR), into one framework. The material model is implemented in a four-node shell element. The model is used to predict the mechanical behavior of SGFR polyamide-6.6. Material parameters are identified using experimental data from tensile tests on unreinforced and 10, 20, and 30%wt reinforced, polyamide-6.6. The proposed model is validated with two points bending experimental results. Experimental and finite elements results are compared. Numerical results prove the good performances of the developed model to predict the anisotropic behavior of SGFR polyamide-6.6.

Intergranular mechanical equilibrium during the rolling deformation of polycrystalline metals based on Taylor principles

Taylor principles indicate that the Taylor strain tensor is identical to the macroscopic strain tensor during plastic deformation and prevails everywhere inside polycrystalline aggregates, in which real grain behaviors generally differ. These principles have been modified in many deformation models while considering strain and stress equilibria in local areas, e.g., grains, grain pairs or grain clusters. However, the Taylor strain tensor is still valid in the surrounding matrix of local areas. In this paper, a reaction stress model based on intergranular mechanical interactions is proposed for rolling deformation caused by penetrating slips and additional local slips while keeping reaction shear stresses below certain top limits. Both stress and strain equilibria are reached in entire rolling sheets in the model, and the same Taylor texture is predicted without the Taylor strain tensor anywhere inside the polycrystalline matrix, regardless if the isotropic matrix is rigid or elastic. Rolling-texture formation in experimental polycrystalline metals could be simulated based on the model if the relaxation effects of additional slips on reducing the top limits of reaction shear stresses are included.

How does tissue preparation affect skeletal muscle transverse isotropy?

The passive tensile properties of skeletal muscle play a key role in its physiological function. Previous research has identified conflicting reports of muscle transverse isotropy, with some data suggesting the longitudinal direction is stiffest, while others show the transverse direction is stiffest. Accurate constitutive models of skeletal muscle must be employed to provide correct recommendations for and observations of clinical methods. The goal of this work was to identify transversely isotropic tensile muscle properties as a function of post mortem handling. Six pairs of tibialis anterior muscles were harvested from Giant Flemish rabbits and split into two groups: fresh testing (within four hours post mortem), and non-fresh testing (subject to delayed testing and a freeze/thaw cycle). Longitudinal and transverse samples were removed from each muscle and tested to identify tensile modulus and relaxation behavior. Longitudinal non-fresh samples exhibited a higher initial modulus value and faster relaxation than longitudinal fresh, transverse fresh, and transverse rigor samples (p<0.05), while longitudinal fresh samples were less stiff at lower strain levels than longitudinal non-fresh, transverse fresh, and transverse non-fresh samples (p<0.05), but exhibited more nonlinear behavior. While fresh skeletal muscle exhibits a higher transverse modulus than longitudinal modulus, discrepancies in previously published data may be the result of a number of differences in experimental protocol. Constitutive modeling of fresh muscle should reflect these data by identifying the material as truly transversely isotropic and not as an isotropic matrix reinforced with fibers.

Frequency dependence of anisotropy in fluid saturated rocks – Part I: aligned cracks case

ABSTRACTA major cause of attenuation in fluid‐saturated media is the local fluid flow (or squirt flow) induced by a passing wave between pores of different shapes and sizes. Several squirt flow models have been derived for isotropic media. For anisotropic media however, most of the existing squirt flow models only provide the low‐ and high‐frequency limits of the saturated elastic properties. We develop a new squirt flow model to account for the frequency dependence of elastic properties and thus gain some insight into velocity dispersion and attenuation in anisotropic media. In this paper, we focus on media containing aligned compliant pores embedded in an isotropic background matrix. The low‐ and high‐frequency limits of the predicted fluid‐saturated elastic properties are respectively consistent with Gassmann theory and Mukerji–Mavko squirt flow model. Results are also expressed in terms of Thomsen anisotropy parameters. It turns out that the P‐wave anisotropy parameter ε tends to zero in the high‐frequency limit, whereas the δ parameter remains the only indicator of P‐S⊥ anisotropy. The S‐wave anisotropy parameter γ is not affected by the presence of fluid and remains the same for all frequency ranges. A new definition for attenuation anisotropy parameters is also proposed to quantify the attenuation anisotropy. In the most important case of liquid saturation, analytical expressions are derived for elastic properties, velocity anisotropy parameters, quality factors, and attenuation anisotropy parameters. A companion paper considers the case of cracks with an ellipsoidal distribution of orientations resulting from the application of anisotropic stress.

The influence of crack-orientation distribution on the mechanical properties of pre-cracked brittle media

Cracks in kerogen-rich shales and other brittle rock-like materials have a tremendous impact on their elastic properties and strength. In this paper, we investigate the effective mechanical properties of shale plates with pre-existing cracks. We employ the extended finite element method (XFEM) to investigate a pre-cracked medium with an elastic, isotropic and brittle shale matrix. We show how the mechanical properties of the orthotropic shale plates are dependent on the crack density and the standard deviation of crack angles. Both the Young's modulus and the Poisson's ratio of the cracked media exhibit a linear dependence on the standard deviation of crack angles, in contrast to the nonlinear dependence of the strength on the angle deviation. Finally, we propose mechanical models to capture the relationship between the mechanical properties and the distribution characteristics of pre-existing cracks in shales. These phenomenological models could be applied to estimate the fracking behavior of shales in engineering practice.

Frequency dependence of anisotropy in fluid saturated rocks – Part II: Stress‐induced anisotropy case

ABSTRACTA major cause of attenuation in fluid‐saturated media is the local fluid flow (or squirt flow) induced by a passing wave between pores of different shapes and sizes. Several squirt flow models have been derived for isotropic media. For anisotropic media, however, most of the existing squirt flow models only provide the low‐ and high‐frequency limits of the saturated elastic properties. We develop a new squirt flow model to account for the frequency dependence of elastic properties and thus gain some insight into velocity dispersion and attenuation in anisotropic media. In a companion paper, we focused on media containing aligned compliant pores embedded in an isotropic background matrix. In this paper, we investigate the case for which anisotropy results from the presence of cracks with an ellipsoidal distribution of orientations due to the application of anisotropic stress. The low‐ and high‐frequency limits of the predicted fluid‐saturated elastic properties are respectively consistent with Gassmann theory and Mukerji–Mavko squirt flow model. In the most important case of liquid saturation, analytical expressions are derived for elastic properties and Thomsen anisotropy parameters. The main observations drawn from this model are as follows. Crack closure perpendicular to the applied stress leads to an increase in seismic velocities as a function of stress in the direction of applied stress and a decrease in squirt‐flow‐induced dispersion and attenuation in this direction. The anisotropy of squirt flow dispersion engenders a decrease in the degree of anisotropy with frequency. The stress‐induced anisotropy remains elliptical, even in saturated media, for all frequency ranges.