Effective Transverse Shear Modulus Research Articles

Effective mechanical behaviors of transverse isotropic materials containing compressible liquid inclusions with surface effects

Porous media with fluid-saturated microchannels, prevalent in biomaterials, tissue engineering materials and flexible electronic devices, can be modelled as transverse isotropic materials. Surface effects can influence significantly the macro-mechanical responses of such materials, particularly for soft materials with micro- or nano-scale channels. In this study, we first develop constitutive laws for fluid saturated porous materials with microchannels by integrating the top-down (homogenization) approach with the bottom-up (micromechanics) approach. We then explicitly establish a connection between surface effects and macro-mechanical responses. Employing the generalized self-consistent model (GSCM), we estimate the effective parameters (i.e., coefficients in constitutive equations governing macro-mechanical responses), with fluid compressibility and surface effects accounted for. Our findings reveal that, as surface energy increases, the effective transverse shear modulus is enlarged, the effective plane strain bulk modulus, the unit uniaxial straining modulus and the cross modulus are all reduced, but the axial shear modulus remains nearly unchanged. As an application, we characterize the mechanical behaviors of the transverse isotropic fluid-saturated porous material with surface effects by connecting the classic Mandel solution with the estimated effective parameters. The pore pressure exhibits the Mandel-Cryer effect. The insights gained from this study are valuable for comprehending and exploring the intricate ways in which surface effects influence the mechanical responses of transversely isotropic fluid-saturated porous materials featuring sufficiently small channels.

Harmonic viscoelastic response of 3D histology-informed white matter model

White matter (WM) consists of bundles of long axons embedded in a glial matrix, which lead to anisotropic mechanical properties of brain tissue, and this complicates direct numerical simulations of WM viscoelastic response. The detailed axonal geometry contains scales that range from axonal diameter (microscale) to many diameters (mesoscale) imposing an additional challenge to numerical simulations. Here we describe the development of a 3D homogenization model for the central nervous system (CNS) that accounts for the anisotropy introduced by the axon/neuroglia composite, the axonal trace curvature, and the tissue dynamic response in the frequency domain. Homogenized models that allow the incorporation of all the above factors are important for accurately simulating the tissue's mechanical behavior, and this in turn is essential in interpreting non-invasive elastography measurements.Geometric and material parameters affect the material properties and thus the response of the brain tissue. More complex, orthotropic, or anisotropic material properties are to be considered as necessitated by the 3D tissue structure. An assembly of micro-scale 3D representative elemental volumes (REVs) is constructed, leading to an integrated mesoscale WM finite element model. Assemblies of microscopic REVs, with orientations based on geometrical reconstructions driven by confocal microscopy data are employed to form the elements of the WM model. Each REV carries local material properties based on a finite element model of biphasic (axon-glial matrix) unidirectional composite. The viscoelastic response of the microscopic REVs is extracted based on geometric information and fiber volume fractions calculated from the relative distance between the local elements and global axonal trace. The response of the WM tissue model is homogenized by averaging the shear moduli over the total volume (thus deriving effective properties) under realistic external loading conditions. Under harmonic shear loading, it is proven that that the effective transverse shear moduli are higher than the axial moduli when the axon moduli are higher than the glial. Methodologically, the process of using micro-scale 3D REVs to describe more complex axon geometries avoids the partition process in traditional composite finite element methods (based on partition of finite element grids) and constitutes a robust algorithm to automatically build a WM model based on available axonal trace information. Analytically, the model provides unmatched simulation flexibility and computational power as the position, orientation, and the magnitude of each tissue building block is calculated using real tissue data, as are the training and testing processes at each level of the multiscale WM tissue.

On the elastic far-field response of a two-dimensional coated circular inhomogeneity: Analysis and applications

The paper studies the conditions under which the far-field response of a two-dimensional coated circular inhomogeneity embedded into an infinite matrix and subjected to uniform stresses at infinity is identical to that of a perfectly bonded inhomogeneity. The problem is considered in plane strain and antiplane settings. All constituents of the composite systems are assumed to be isotropic or transversely isotropic (with the axis Oz in longitudinal direction) and linearly elastic. For the plane strain problem and hydrostatic load or antiplane problem, it is shown that there always exists an equivalent inhomogeneity of the radius equal to the external radius of the coating that produces the elastic fields inside the matrix that are identical to those of the original coated inhomogeneity. For the plane strain and deviatoric load, the elastic fields in the matrix due to these two composite systems are always different, except for the equal shear moduli case. However, it is rigorously proved here that, for the deviatoric load and any combination of the material parameters, there exists the equivalent inhomogeneity of the radius equal to the external radius of the coating that induces the same dipole moments as those induced by the coated inhomogeneity. The existence of the equivalent inhomogeneities whose radius is different from the external radius of the coating is also investigated. The application of the proposed procedure to the homogenization problems leads to the new closed-form expression for the effective transverse shear modulus of transversely isotropic unidirectional composites. The findings presented here provide an insight on the influence of the interphases that could be useful in the analysis of some types of inverse problems.

Size-dependent effective behaviors of multiferroic fibrous composites with interface stress

The objective of this work is to study the size-dependent effective behaviors of multiferroic fibrous composites with interface stress subjected to generalized in-plane deformation and out-of-plane electromagnetic fields. Using the complex variable approach together with several micromechanical models, closed-form expressions of the effective moduli are obtained. In contrast to the composite with perfect bonding, the effective property formulations show the dependence on the size of the fibers. The derived solutions are applied to several examples to demonstrate the effect of interface stress, and to examine the results among different micromechanical models. Numerical results show that while most of the effective moduli increase as the radius of the fiber decreases, the magnetoelectric voltage coefficient, the figure of merit of the multiferroic material, decreases. Further, when considering the effective transverse shear modulus of composite with interface stress, a higher order approximation is required for the equivalency between an interphase and an imperfect interface if the interphase is not sufficiently thin or is very stiff.

Effect of residual interface stress on thermo-elastic properties of unidirectional fiber-reinforced nanocomposites

Surface/interface effect plays a significant role in the study of the mechanical properties of nanocomposites. Most previous papers in the literature only considered the surface/interface elasticity, whereas some papers only considered the residual surface/interface stress (surface/interface tension). In this paper, an energy-based surface/interface theory is applied to systematically study the effective thermo-elastic properties of unidirectional fiber-reinforced nanocomposites, in which both the surface/interface elasticity and the residual surface/interface stress are included. The emphasis is particularly placed on the influence of the residual interface stress on the effective thermo-elastic properties of such nanocomposites, since this influence was ignored by many previous authors. Analytical expressions of five effective transversely isotropic elastic constants are derived, in which a modified generalized self-consistent method is suggested to obtain an explicit expression of the size-dependent effective transverse shear modulus. Furthermore, with an introduced concept of ‘equivalent fiber’ (i.e., a fiber together with its interface), the effective thermal expansion coefficients and the effective specific heat at constant strain of the fiber-reinforced nanocomposite are obtained. Finally, numerical examples are illustrated, and the effect of residual interface stress on the effective thermo-elastic properties of the fibrous nanocomposite is discussed. It is shown that the residual interface stress has a significant effect on the overall thermo-elastic properties of the nanocomposites.

Effective stiffness and thermal expansion coefficients of unidirectional composites with fibers surrounded by cylindrically orthotropic matrix layers

An approach to predict the effective thermoelastic properties of unidirectional composites consisting of cylindrically orthotropic fibers surrounded by layers of cylindrically orthotropic matrix is developed and applied to carbon/carbon composites. Micromechanical modeling is based on elastic solutions for basic loadcases including prescribed axial tension, transverse hydrostatic loading, axial and in-plane shear, and unconstrained thermal expansion. The composite cylinder assemblage model (Hashin, 1990) is utilized to predict axial elastic properties, transverse bulk modulus and thermal expansion coefficients of the overall composite. The effective transverse shear modulus is evaluated using a self-consistent method. The approach is illustrated by considering a representative carbon/carbon material system containing pyrolytic carbon matrix of two different texture levels.

Effects of higher-order interface stresses on the elastic states of two-dimensional composites

We propose a simple model to simulate higher-order interface stresses along the interface between two neighboring media in two dimensions. The interface behavior is modeled from a thin interphase of constant thickness by taking a proper limit process. In the formulation the deformation of the thin interphase is approximated by the Kirchhoff–Love assumption of thin shell. To incorporate the higher-order interface stresses, we consider the bending effects resulting from the non-uniform surface stress across the layer thickness. The stress equilibrium conditions is fulfilled by consideration of balance for forces as well as stress couples. Depending on the difference in stiffness and length scales of the interphase, we show that the interfaces can be classified into four different types. This findings, upon suitable definitions of material parameters, agree with a rigorous asymptotic analysis proposed by Benveniste and Miloh [Benveniste, Y., Miloh, T., 2001. Imperfect soft and stiff interfaces in two-dimensional elasticity. Mech. Mater. 33 309–323]. To illustrate the higher-order effects, we derive analytically the stress concentration factor of an infinite plate containing a circular cavity with interface stresses of different orders subjected to a remote transverse shear loading. The closed-form expressions show how the orders of interface stresses influence the concentration factor in a successive manner. In addition, we examine the effective shear modulus of composites with circular inclusions with higher-order interface effects. The effective transverse shear modulus is derived based on the generalized self-consistent method.

Transverse shear stiffness of a chevron folded core used in sandwich construction

Using Kelsey et al. (1958) unit load method, upper and lower bounds for the effective transverse shear moduli of a chevron folded core used in sandwich construction are analytically derived and compared to finite element computations. We found that these bounds are generally loose and that in some cases chevron folded cores are 40% stiffer than honeycomb-like cores.

Micro-Mechanical Behavior of a Unidirectional Composite Subjected to Transverse Shear Loading

The effects of interphase between fibers and matrix on the micro-and macro-mechanical behaviors of fiber-reinforced composite lamina subjected to transverse shear load at remote distance have been studied. The interphase has been modeled by the compliant spring-layers that are linearly related to the normal and tangential tractions. Numerical analyses on composite basic cells have been carried out using the boundary element method. For undamaged composites the micro-level stresses at the matrix side of the interphase and effective shear modulus have been calculated as a function of the fiber volume fraction and the interphase stiffness. Results are presented for various interphase stiffnesses from perfect bonding to total debonding. For a square array composite results show that for a high interphase stiffness k > 10, an increase in a fiber volume fraction results in a higher effective transverse shear modulus. For a relatively low interphase stiffness k < 1, it is shown that an increase in the fiber volume fraction causes a decrease in the effective transverse shear modulus. For perfect bonding, the effective shear modulus for a hexagonal array composite is slightly larger than that for a square array composite. Also for the damaged composite with partially debonded interphase, local stress fields and effective shear moduli are calculated and a decrease in the effective shear modulus has been observed.

Optimization of transverse shear moduli for composite honeycomb cores

Effective transverse shear moduli of composite honeycomb cores are important material properties in analysis and design of sandwich structures. Based on a combined multi-objective optimization and homogenization approach, effective transverse shear moduli for composite honeycomb cores are optimized in this paper. An analytical approach using a two-scale homogenization technique is adopted to predict the effective transverse shear stiffnesses of thin-walled composite honeycomb cores with general configurations, and explicit formulas are given for three typical honeycomb cores consisting of sinusoidal, tubular, and hexagonal geometries. To maximize the performance of transverse shear behavior, a multi-objective method is employed to obtain optimal designs of periodic cellular cores. By assigning equal weighting (or penalty) factors to the transverse shear stiffnesses, the overall shear stiffness parameter is optimized. The optimization problem is solved using a sequential quadratic programming algorithm. The geometric and weighting factor effects on the optimal designs for the three cellular cores considered in this study are evaluated and discussed. An optimization procedure using finite element analysis and a response surface algorithm is also conducted to verify the proposed approach. The present combined homogenization and multi-objective optimization approach can be used to maximize transverse shear material properties of composite honeycomb structures.

Thermoelastic composites with columnar microstructure and cylindrically anisotropic phases. Part II: One-parameter generalized self-consistent estimates

The effective transverse shear modulus of a transversely isotropic composite with columnar microstructure and with two cylindrically orthotropic phases is estimated by a one-parameter generalized self-consistent model. The equation characterizing the effective transverse shear modulus is a fourth-order polynomial equation instead of the quadratic one determined by the classical generalized self-consistent model. The result obtained in Part II of this work extends the relevant one of Hashin [Z. Hashin, Thermoelastic properties and conductivity of carbon/carbon fiber composites, Mech. Mater. 8 (1990) 293–308].

Transverse Shear Stiffness of Composite Honeycomb Cores and Efficiency of Material

Effective transverse shear moduli of composite honeycomb cores are important material properties in analysis and design of sandwich structures. An analytical approach using a two-scale homogenization technique is presented to predict the effective transverse shear stiffnesses of thin-walled composite honeycomb cores with general configurations. To improve the performance of core transverse shear behavior, a nondimensional index, the so-called efficiency of material (EOM), is introduced to evaluate the optimal design of periodic cellular cores. Explicit formulas of effective transverse shear stiffnesses and the corresponding efficiencies of material are provided for three typical honeycomb cores consisting of reinforced sinusoidal, elliptical, and reinforced hexagonal geometries. It has been shown that the efficiency of material is only determined by nondimensional core geometric ratios. The effects of these ratios on the EOM of effective transverse shear stiffnesses, which are examined in detail in this study, offer insight and guidelines for optimal design and selection of honeycomb cores. The explicit formulas for the effective transverse shear stiffness properties of thin-walled composite honeycomb cores and their related EOMs can be used effectively to predict and optimize the core transverse shear behavior in sandwich structures.

On the micromechanics theory of Reissner-Mindlin plates

A micromechanics model is developed for the Reissner-Mindlin plate. A generalized eigenstrain formulation, i.e., an eigencurvature/eigen-rotation formulation, is proposed, which is the analogue or counterpart of the eigenstrain formulation in linear elasticity. The micromechanics model of the Reissner-Mindlin plate is useful in the study of mechanical behavior of composite plates that contain randomly distributed inhomogeneities, whose sizes are close to the order of thickness of the plate; under those circumstances, the use of micromechanics of linear elasticity is not justified, and moreover, it is inconsistent with structural theories, such as the Reissner-Mindlin plate theory, that are actually used in engineering design. In this paper, the analytical solution of an elliptical inclusion embedded in an infinite thick plate is sought. In particular, the first order asymptotic (or approximated) solution of the elliptical inclusion problem is obtained in explicit form. Accordingly, the Eshelby tensors of the Reissner-Mindlin plate are derived, which relate eigencurvature and eigen-rotation to the induced curvature and shear deformation fields. Several variational inequalities of the Reissner-Mindlin plate are discussed and derived, including the comparison variational principles of Hashin-Shtrikman/Talbot-Willis, type. As an application, variational bounds are derived to estimate the effective elastic stiffness of Reissner-Mindlin plates, specifically, the flexural rigidity and transverse shear modulus. The newly derived bounds are congruous with the Reissner-Mindlin plate theory, and they provide an optimal estimation on effective rigidity as well as effective transverse shear modulus for unstructured composite thick plates.

Evaluation of transverse shear stiffness of structural core sandwich plates

Effective transverse shear moduli in the principal material directions of corrugated board were examined experimentally for five corrugated board types using the ASTM block shear test and the three-point bend test. It was found that shear moduli determined by the three-point bend test are significantly lower than those obtained from the block shear test. The difference is probably caused by local indentation of the board at the supports and contribution from bending deformation of the facings in the three-point bend test. Experimental results were compared to shear moduli obtained by finite-element analysis (FEA) and analytical predictions. Shear modulus along the corrugations, determined with the three-point bend test, was about half of the predicted value using FEA, while the shear modulus transverse to the corrugations was substantially below that prediction from FEA, apparently due to delamination damage of the core material inflicted during the corrugation process.

Micromechanical behavior of unidirectional composites under a transverse shear loading

Effects of fiber-matrix interphases on the micro-and macro-mechanical behaviors of unidirectionally fiber-reinforced composites subjected to transverse shear loading at remote distance have been studied. The interphases between fibers and matrix have been modeled by the spring-layer which accounts for continuity of tractions, but allows radial and circumferential displacement jumps across the interphase that are linearly related to the normal and tangential tractions. Numerical calculations for basic cells of the composites have been carried out using the boundary element method. For an undamaged composite the micro-level stresses at the matrix side of the interphase and effective shear stiffness have been computed as functions of fiber volume ratio and interphase stiffness k. Results are presented for various interphase stiffnesses from the perfect bonding to the case of total debonding. For a square array composite the results show that for a high interphase stiffness k>10, an increase of increases the effective transverse shear modulus G over bar of the composite. For a relatively low interphase stiffness k slightly decreases the effective transverse shear modulus. For the perfect bonding case, G over bar for a hexagonal array composite is slightly larger than that for a square array composite. Also for a damaged composite partially debonded at the interphase, local stress fields and effective shear modulus are calculated and a decrease in G over bar has been observed.

Rigorous link between the conductivity and elastic moduli of fibre-reinforced composite materials

We derive rigorous cross-property relations linking the effective transverse electrical conductivity cr* and the effective transverse elastic moduli of any transversely isotropic, two-phase ‘fibre-reinforced’ composite whose phase boundaries are cylindrical surfaces with generators parallel to one axis. Specifically, upper and lower bounds are derived on the effective transverse bulk modulus k* in terms of cr* and on the effective transverse shear modulus //* in terms of cr*. These bounds enclose certain regions in the ct*-ac* and cr*-/r* planes, portions of which are attainable by certain microgeometries and thus optimal. Our bounds connecting the effective conductivity cr* to the effective bulk modulus ft* apply as well to anisotropic composites with square symmetry. The implications and utility of the bounds are explored for some general situations, as well as for specific microgeometries, including regular and random arrays of circular cylinders, hierarchical geometries corresponding to effective-medium theories, and checkerboard models. It is shown that knowledge of the effective conductivity can yield sharp estimates of the effective elastic moduli (and vice versa), even for infinite phase contrast.

The effective transverse moduli of a composite with degraded fiber-matrix interfaces

A Generalized Self Consistent Model (GSCM) is applied to a composite containing multiple-coated fibers and the general solution for this model subject to a constant transverse shear loading at infinity is derived. The effective transverse shear modulus of the composite is determined through the use of the equality of the strain energy in the inhomogeneous composite and the equivalent homogeneous medium. The stress field at the interface of a metal matrix composite under the same loading condition is calculated. In addition, the reduction in the transverse bulk and shear stiffnesses of the fiber-matrix interface in a ceramic composite subject to materials degradation is determined through inversion of the known or measured values of its effective elastic constants. The stability of the inversion process is examined as well.

Elastic Moduli of Thickly Coated Particle and Fiber-Reinforced Composites

Based on the models of Hashin (1962) and Hashin and Rosen (1964), the effective elastic moduli of thickly coated particle and fiber-reinforced composites are derived. The microgeometry of the composite is that of a progressively filled composite sphere or cylinder element model. The exact solutions of the effective bulk modulus κ of the particle-reinforced composite and those of the plain-strain bulk modulus κ23, axial shear modulus μ12, longitudinal Young’s modulus E11, major Poisson ratio ν12, of the fiber-reinforced one are derived by the replacement method. The bounds for the effective shear modulus μ and the effective transverse shear modulus μ23 of these two kinds of composite, respectively, are solved with the aid of Christensen and Lo’s (1979) formulations. By considering the six possible geometrical arrangements of the three constituent phases, the values of κ, and of κ23, μ12, E11, and ν12 are found to always lie within the Hashin-Shtrikman (1963) bounds, and the Hashin (1965), Hill (1964), and Walpole (1969) bounds, respectively, but unlike the two-phase composites, none coincides with their bounds. The bounds of μ and μ23 derived here are consistently tighter than their bounds but, as for the two-phase composites, one of the bounds sometimes may fall slightly below or above theirs and therefore it is suggested that these two sets of bounds be used in combination, always choosing the higher for the lower bound and the lower for the upper one.

Improved bounds on elastic and transport properties of fiber-reinforced composites: Effect of polydispersivity in fiber radius

Improved rigorous bounds on the effective elastic and transport properties of a transversely isotropic fiber-reinforced material composed of oriented, infinitely long, multisized circular cylinders distributed throughout a matrix are computed. Specifically, we evaluate such bounds on the effective axial shear modulus (which includes, by mathematical analogy, the transverse conductivity), effective transverse bulk modulus, and the effective transverse shear modulus. These are generally demonstrated to provide significant improvement over the Hill–Hashin bounds which incorporate only volume-fraction information. Although the upper bounds diverge from the lower bounds when the cylinders are much stiffer than the matrix, the improved lower bounds still yield relatively accurate estimates of the effective properties. Generally, increasing the degree of polydispersivity in cylinder size increases the effective transverse conductivity (or axial shear modulus) and effective transverse bulk modulus, and decreases (slightly) the effective transverse shear modulus for cases in which the fibers are more conducting or stiffer than the matrix.