Parametric constitutive model of plain-weave fabric reinforced composite ply

A computational model that allows to explicitly determine orthotropic elastic constants of plain-weave fabric-reinforced composite ply as functions of microstructure parameters has been developed in this study. These relationships are not given in the form of analytical formulae (as it is in the case of approximate analytical models) but in the form of an extensive database of numerically evaluated results for different microstructure instances and a numerical scheme that interpolates the results. To build the database, a standard finite-element-based homogenization technique of a periodic representative volume element is employed. As a result, a numerical algorithm is provided that may be easily employed in FE codes as a part of a regular constitutive subroutine. Sensitivity of the composite elastic constants with respect to the microstructure parameters is also directly available from the model.     

Boundary element analysis for effective stiffness tensors: effect of fabric tensor determination method

Second-rank fabric tensors have been extensively used to describe structural anisotropy and to predict orthotropic elastic constants. However, there are many different definitions of, and approaches to, determining the fabric tensor. Most commonly used is a fabric tensor based on mean intercept length measurements, but star volume distribution and star length distribution are commonly used, particularly in studies of trabecular bone. Here, we investigate the effect of the fabric tensor definition on elastic constant predictions using both synthetic, idealized microstructures as well as a micrograph of a porous ceramic. We use an efficient implantation of a symmetric Galerkin boundary element method to model the mechanical response of the various microstructures, and also use a boundary element approach to calculate the necessary volume averages of stress and strain to obtain the effective properties of the media.     

The effects of shear and near tip deformations on energy release rate and mode mixity of edge-cracked orthotropic layers

In this paper semi-analytical expressions are derived for the energy release rate and the stress intensity factors of edge-cracked homogeneous and orthotropic layers subject to arbitrary generalized end forces. The expressions are accurate for long and short cracks. Following the work of Li et al. [Li S, Wang J, Thouless MD. The effects of shear on delamination in layered materials. J Mech Phys Solids 2004;52(1):193-214] for isotropic bi-material layers, the derivation extends the method proposed by Suo [Suo ZG. Delamination specimens for orthotropic materials. J Appl Mech 1990;57(3):627-34] for axial forces and bending moments in order to include the contribution of the shear forces. The shear contribution to the fracture parameters depends on the shear deformations along the layer and the elastic near tip deformation of the material. Li et al. [Li S, Wang J, Thouless MD. The effects of shear on delamination in layered materials. J Mech Phys Solids 2004;52(1):193-214] derived semi-analytical expressions for the fracture parameters that depend on the crack tip stress resultants, the elastic constants and five numerically-determined constants globally describing the effect of shear. In this paper analogous constants are derived for orthotropic layers and defined by semi-analytical expressions that highlight their physical significance and allow separation of the different contributions. The derivation is based on the assumption that the near tip deformation can be described by means of relative rotations between the cross sections of the different sub-layers at the crack tip (root rotations). The root rotations depend linearly on the crack tip stress resultants through compliance coefficients that are derived numerically in the paper for a wide range of orthotropic materials. Applications to different mixed mode delamination and peeling problems, for which accurate two-dimensional finite element solutions can be found in the literature, highlight the accuracy of the proposed expressions.     

On the Relationship Between Microstructure of the Cortical Bone and its Overall Elastic Properties

The relationship between microstructure of the cortical bone and its effective elastic properties is discussed. We utilize results of Kachanov et al (1994) on materials with cracks/pores of diverse shapes. Bone's microstructure is modeled using available micrographs. The calculated anisotropic elastic constants for porous cortical bone are compared with available experimental data. For Young's moduli and shear moduli the agreement is good, whereas Poisson's ratios differ significantly. Possible reasons for this difference are discussed. This revised version was published online in August 2006 with corrections to the Cover Date.     

Dependence of elastic constants of an anisotropic porous material upon porosity and fabric

The elastic properties of an anisotropic porous material can be represented as functions of the material's solid volume fraction (or porosity) and the principal diameters of the material's fabric ellipsoid. The fabric ellipsoid is a measure of the anisotropy of the microstructure of a material. The definitions and measurement techniques for fabric ellipsoids in granular materials, foams, cancellous bone, and rocks are discussed. The principal results presented in this work are algebraic expressions for the dependence of the orthotropic elastic constants upon both solid volume fraction and the fabric ellipsoid.     

Identification of stiffness tensor components of wood cell walls by means of nanoindentation

A new procedure to characterize the full set of elastic constants of wood cell walls was developed. For the first time, not only the longitudinal modulus, but also the transverse- and the shear modulus were determined in one experimental setup at micron scale. For this purpose, nanoindentation experiments were performed at variable angles between the indentation direction and the direction of cellulose microfibrils in wood cell walls. Using an approach based on anisotropic indentation theory a relationship between the indentation moduli obtained experimentally and the elastic material constants of the cell wall was derived. Using an error minimization procedure, the values of the elastic material constants were finally calculated. As typically observed for natural materials, our experimental results are characterized by high variability. Particularly the elastic modulus in longitudinal cell direction is highly sensitive to small changes in the local orientation of cellulose microfibrils. Nonetheless, reasonable estimates of 26.3 GPa for the longitudinal elastic modulus of the secondary wood cell wall S2, 4.5 GPa for the transverse modulus, and – for the first time – a value of 4.8 GPa for the shear modulus of wood cell wall material were obtained.     

The inplane orthotropic couple-stress elasticity constants of elliptical cell honeycombs

The macroscopic mechanical behavior of elliptical cell honeycombs are characterized using couple-stress elasticity theory. General expressions for the couple-stress elastic constants are obtained through a combination of non-dimensional analysis and numerical analysis. The expressions for the six in-plane orthotropic couple-stress compliances are derived in terms of the honeycomb cell size, cell wall thickness, cell ellipticity and the linear elastic properties of the cell wall material. The derived expressions are validated through numerical analyses.     

粘弹正交各向异性空心圆柱体周向SH波

基于Kelvin-Voight模型,推导了粘弹正交各向异性空心圆柱体周向SH波控制方程,得出了SH波特性受两个弹性常数与粘性系数的影响。通过第一类和第二类Bessel函数,求出频散方程。计算了四种不同径厚比下的相速度频散曲线和衰减曲线,分析了径厚比的变化对频散和衰减曲线的影响。为了与正交各向异性相比较,计算了粘弹各向同性空心圆柱体中周向SH波,并讨论了两个弹性常数和粘性系数对频散曲线和衰减曲线的影响。     

建筑膜材料双轴向拉伸弹性常数的估算方法

以机织物增强建筑膜材料为对象,研究了通过单轴向拉伸试验来获取双轴向拉伸弹性常数的方法。根据膜材料双轴向拉伸弹性模量的确定方法,以二维正交各向异性材料的本构关系为基础,以膜材料经纬向拉伸应力比为设定加载参数,导出了机织膜材料单轴向和双轴向拉伸弹性模量之间的关系,提出了采用单轴向拉伸弹性常数估算建筑膜材料双轴向拉伸弹性常数的方法。通过参考文献中对GF/PTFE和PVDF1202T膜材料拉伸试验的测试结果,验证研究所提出方法的适用性。结果表明,在工程实践中利用单轴向拉伸试验来获得膜材料双轴向拉伸弹性模量是可行的。     

Does fabric tensor exist for a fabric?

It is shown that the mean intercept length distribution for planar fibre networks or for materials composed of a set of plates is not in general elliptic and cannot be expressed analytically in terms of a second-order tensor. However, our numerical computations indicate that the polar plot of the mean intercept length at the angle of measurement may become nearly ellipsoidal as the microstructure (fibres or plates) become less discretely organized, but yet remain orthotropic. The equations presented in this study may be used to obtain fibre (plate) orientation density functions from the experimental data on mean intercept length distribution.     

A generalized method for characterizing elastic anisotropy in solid living tissues

The microstructure of cortical bone may exhibit either transverse isotropic or orthotropic symmetry, thus requiring either five or nine independent elastic stiffness coefficients (or compliances), respectively, to describe its elastic anisotropy. Our previous analysis to describe this anisotropy in terms of two scalar quantities for the transverse isotropic case is extended here to include orthotropic symmetry. The new results for orthotropic symmetry are compared with previous calculations using the transverse isotropic analysis on the same sets of anisotropic elastic constants for bone, determined either by mechanical or by ultrasonic experiments. In addition, the orthotropic calculation has been applied to full sets of orthotropic elastic stiffness coefficients of a large variety of wood species. Although having some resemblance to plexiform bone in microstructural organization, there is a dramatic difference in both the shear and the compressive elastic anisotropy between the two materials: wood is at least one order of magnitude more anisotropic than bone.     

Central crack in plane orthotropic rectangular sheet

The plane problem of a central crack in a rectangular sheet of orthotropic material is considered. The solution is found by an extension of the modified mapping-collocation technique, originally formulated for plane isotropic analysis. Application of the technique outlined in this paper for plane orthotropic problems to a wider class of geometries and loading is evident. The numerical results indicate a dependence of the orthotropic stress intensity factors on both geometric and elastic constants over a certain parameter range.     

Combined effects of relative density and material distribution on the mechanical properties of metallic honeycombs

This paper presents the combined effects of relative density and material distribution on the elastic constants and the yield strengths of metallic honeycombs. Periodic regular hexagonal cell is employed as the structural model. Cell wall bending, transverse shear and axial stretching/compression are taken as the deformation mechanisms in the analysis. Closed-form solutions for the yield strengths and all the five independent elastic constants are obtained for honeycombs with cell walls of uniform thickness. For honeycombs with cell walls of non-uniform thickness, the closed-form solutions would be too lengthy to use in practical applications. We instead provide numerical results to show the combined effects of relative density and material distribution on the initial and full yield strengths and all the five independent elastic constants of metallic honeycombs. The results can serve as a guide for the optimal design of metallic honeycombs.     

复合材料圆柱曲板在轴压下的非线性弹性稳定问题

本文在文献[1, 2]的基础上, 考虑了纤维和基体本构关系的非线性, 讨论了正交铺层和±θ角铺层的情况, 采用小弹-塑性形变理论、复合材料微观力学的复合定律和线性稳定理论。得到了求解复合材料园柱曲板在轴压下非线性弹性失稳时临界载荷的算式。给出了算例和讨论及测定单向复合材料层片非线性弹性常数的方案。      

Winkler-Pasternak地基上四边固支层合板的振动

本文在文[1]的基础上,利用Bolotin方法,在考虑横向剪切变形的情况下,首次获得了Winkler-Pasternak地基上四边固支对称正交铺设层合板自由振动问题的渐近解析解.对于不同层数、不同长厚比的层合板以及在不同地基反力系数的情况下,文中进行了具体的数值计算,给出了相应的数值结果.作为本文一个简化算例,计算了弹性地基上正交各向异性厚板自由振动的固有频率,与其他文献所给结果非常一致.      

Composite materials whose phases have partially equal elastic moduli

Hill [J. Mech. Phys. Solids 11 (1963) 357, 12 (1964) 199] discovered that, regardless of its microstructure, a linearly elastic composite of two isotropic phases with identical shear moduli is isotropic and has the effective shear modulus equal to the phase ones. The present work generalizes this result to anisotropic phase composites by showing and exploiting the fact that uniform strain and stress fields exist in every composite whose phases have certain common elastic moduli. Precisely, a coordinate-free condition is given to characterize this specific class of elastic composites; an efficient algebraic method is elaborated to find the uniform strain and stress fields of such a composite and to obtain the structure of the effective elastic moduli in terms of the phase ones; sufficient microstructure-independent conditions are deduced for the orthogonal group symmetry of the effective elastic moduli. These results are applied to elastic composites consisting of isotropic, transversely isotropic and orthotropic phases.     

The inplane elastic properties of circular cell and elliptical cell honeycombs

Summary The inplane elastic properties of perfectly circular and elliptic cell honeycombs are derived through an analytical method and validated numerically. In the case of perfectly circular cell hexagonally packed honeycomb, the inplane elastic properties are shown to be isotropic. However, a departure from circularity of the cells leading to cell ellipticity results in the inplane properties becoming orthotropic. The orthotropic elastic constants are also derived analytically and validated numerically.     

Hierarchical approach to modelling of liquid-crystalline polymers

Summary Liquid-crystalline polymers exhibit fascinating structure at many size scales. This paper describes how computer models at different size scales can be linked together by a hierarchical approach to give a better understanding of the properties and behaviour of these materials. The atomic scale is considered first, where semi-empirical molecular orbital techniques are used to calculate the torsional energy functions associated with rotating the backbone bonds of the polymer. Secondly, entire chains are simulated using a Monte Carlo technique based on the torsional energy functions to deduce the persistence length of the polymer of interest. A theoretical relationship enables the Frank elastic constants to be determined from the persistence length. The elastic constants may then be used as input parameters for models both to predict microstructure and to help understand the role of defects in the shear flow of these materials. The hierarchical approach provides a meaningful framework within which data obtained from small-scale models are used to parameterise models at a larger scale.     

Interface crack problems in graded orthotropic media: Analytical and computational approaches

Interface crack problems in graded orthotropic media are considered using analytical and computational techniques. In the analytical formulation an interface crack between a graded orthotropic coating and a homogeneous orthotropic substrate is considered. The principal axes of orthotropy are assumed to be parallel and perpendicular to the crack plane. Mechanical properties of the medium are assumed to be continuous with discontinuous derivatives at the interface. The problem is formulated in terms of the averaged constants of plane orthotropic elasticity and reduced to a pair of singular integral equations which are solved numerically to compute the mixed mode stress intensity factors and the energy release rate. In the second part of the study, enriched finite elements are formulated and implemented for graded orthotropic materials. Comparisons of the finite element and analytical results show that enriched finite element technique is capable of producing highly accurate results for crack problems in graded orthotropic media. Finally, periodic interface cracking and the four point bending test for graded orthotropic solids are modeled using enriched finite elements and the results are briefly discussed.     

Ply cracking and stiffness degradation in cross-ply laminates under biaxial extension, bending and thermal loading

Transverse ply cracking often leads to the loss of stiffness and reduction in thermal expansion coefficients. This paper presents the thermoelastic degradation of general cross-ply laminates, containing transverse ply cracks, subjected to biaxial extension, bending and thermal loading. The stress and displacement fields are calculated by using the state space equation method [Zhang D, Ye JQ, Sheng HY. Free-edge and ply cracking effect in cross-ply laminated composites under uniform extension and thermal loading. Compos Struct [in press].]. By this approach, a laminated plate may be composed of an arbitrary number of orthotropic layers, each of which may have different material properties and thickness. The method takes into account all independent material constants and guarantees continuous fields of all interlaminar stresses across interfaces between material layers. After introducing the concept of the effective thermoelastic properties of a laminate, the degradations of axial elastic moduli, Poisson’s ratios, thermal expansion coefficients and flexural moduli are predicted and compared with numerical results from other methods or available test results. It is found that the theory provides good predictions of the stiffness degradation in both symmetric and antisymmetric cross-ply laminates. The predictions of stiffness reduction in nonsymmetric cross-ply laminates can be used as benchmark test for other methods.