Analytical solutions for an infinite transversely isotropic functionally graded sectorial plate subjected to a concentrated force or couple at the tip

This paper considers the elastic responses of an infinite sectorial plate made of transversely isotropic functionally graded material (FGM), which is subjected to a concentrated force or couple at the tip. There is no load acting on the upper and lower surfaces, and the elastic coefficients can vary arbitrarily through the plate thickness. No constraint is required on the symmetry of the plate in the thickness direction. Based on the displacement assumption for the bending of an FGM plate and by using the complex variable method, this paper presents the general solutions to the basic equations governing transversely isotropic FGM plates, which are expressed in terms of four analytical functions (or complex potentials). The boundary conditions are a combination of those from the plane elasticity and those from the classical plate theory. For a particular boundary value problem, such as the ones considered here for a sectorial plate, with the specific conditions for determining solutions, the four analytic functions can be assumed in appropriate forms, which contain only some unknown constants. Once these constants are determined from the specific conditions, the complete solutions are readily derived too. Among the solutions presented here, the solutions for the infinite FGM sectorial plate under a concentrated couple are absolutely new to the literature, and they are also applicable to isotropic FGM sectorial plates. The solutions degenerate into the ones for a homogeneous sectorial plate, which coincide with the available solutions from the plane elasticity theory. There are three-dimensional correction terms in the mid-plane displacements.     

Size-dependent effective dynamic properties of unidirectional nanocomposites with interface energy effects

This article studies the size effect on wave propagation characteristics of plane longitudinal and transverse elastic waves in a two-phase nanocomposite consisting of transversely isotropic and unidirectionally oriented identical cylindrical nanofibers embedded in a transversely isotropic homogeneous matrix. The surface elasticity theory is employed to incorporate the interfacial stress effects. The effect of random distribution of nanofibers in the composite medium is taken into account via a generalized self-consistent multiple scattering model. The phase velocities and attenuations of longitudinal and shear waves along with the associated dynamic effective elastic constants are calculated for a wide range of frequencies and fiber concentrations. The numerical results reveal that interface elasticity at nanometer length scales can significantly alter the overall dynamic mechanical properties of nanofiber-reinforced composites. Limiting cases are considered and excellent agreements with solutions available in the literature have been obtained.     

The rotation of a rigid spheroidal inclusion embedded in a transversely isotropic medium

The exact three-dimensional elasticity solutions are given for two problems related to a rigid spheroidal inclusion embedded in bonded contact with an infinite transversely isotropic elastic medium. The first is of axisymmetric nature in which the inclusion is given a constant rotation about its axis of revolution which coincides with the axis of symmetry of the material. The second problem is asymmetric where the spheroidal inclusion is given a constant rotation about a direction that is perpendicular to the axis of elastic symmetry of the material. The displacement potential representation for the equilibrium of three-dimensional transversely isotropic bodies is used to solve the problem. In both cases, the moment-rotation relationship for the spheroidal inclusion and its limiting configurations are obtained in closed form. Numerical results are presented to show the effect of the aspect ratio of the spheroid on the rotational stiffness.     

The shear properties and deformation mechanisms of porous metal fiber sintered sheets

Porous metal fiber sintered sheets (MFSSs) are a type of layered transversely isotropic open cell materials with low relative density (i.e., volume fraction of fibers), high specific stiffness and strength, and controllable precision for functional and structural applications. Based on a non-contact optical full field strain measurement system, the in-plane and transverse shear properties of SMFFs with relative densities ranging from 15% to 34% are investigated. For the in-plane shear, the modulus and strength are found to depend linearly upon the relative density. The associated deformation is mainly due to fiber stretching, accompanied by the direction change of metal fibers. When the shear loading is applied in the transverse direction, the deformation of the material is mainly owing to fiber bending, followed by the separation failure of the fiber joints. Measured results show that the transverse shear modulus and strength have quartic and cubic dependence upon the relative density respectively and are much lower than their in-plane counterparts. Simple micromechanics models are proposed for the in-plane and transverse moduli and strengths of MFSSs in shear. The predicted relationships between the shear mechanical properties of MFSSs and their relative density are obtained and are in good agreement with the measured ones.     

Small torsional vibration superposed on finitely deformed state of a circular cylindrical rod of transversely isotropic elastic material

Starting with a class of small deformations superposed on a finitely deformed state of a transversely isotropic elastic solid, we study a problem of small torsional vibration superposed on homogeneous finitely deformed state of a circular cylindrical rod made of transversely isotropic elastic material. It has been found that free vibration is possible and, due to anisotropy, the speed of propagation of waves of torsion along the cylinder is increased or decreased according as the initial stressed state is under tension or compression.     

Toughening by partial or full bridging of cracks in ceramics and fiber reinforced composites

A complete solution is given for a fully or partially bridged straight crack in transversely isotropic elastic materials which may correspond to unidirectionally fiber-reinforced ceramics or other brittle composites. The stiffness of the bridging materials may have an arbitrary variation along the crack, representing partially failed fibers or ligaments. The crack may have any orientation with respect to the axis of the material symmetry. The solution is explicit in terms of the Chebychev polynomials when the bridging-forces are linearly dependent on the crack-opening-displacement. In addition, uniformly valid asymptotic solutions are developed for fully or partially bridging cracks. For the case when the crack is short relative to a length scale which depends on the material properties, the method yields a complete asymptotic solution when the bridging forces are linearly or non-linearly dependent on the crack-opening-displacement (a square-root dependence, corresponding to continuous fibers, is used for illustration). For the case of long cracks, the proposed asymptotic is effective, but the results are not presented in this work.The mechanism of crack kinking is studied for an oblique partially or fully bridged, or unbridged crack in a macroscopically transversely isotropic elastic solid. The crack is assumed to grow in the matrix material (containing unbroken strong fibers) under local driving forces which are calculated on the basis of the overall anisotropic material response. The results of various fracture criteria are studied. It is illustrated that, under far-field tensile forces normal to the crack, the criterion of the maximum opening mode stress intensity factor in the homogenized anisotropic solid (i.e., the orientation for which the strength of the singularity associated with the tensile hoop stress is maximum) produces results which suggest crack growth more or less parallel to the fibers, whereas the results based on the maximum Mode I stress intensity factor in the isotropic matrix material and/or on the local symmetry criterion (again, for the isotropic matrix) predict crack extension more or less normal to the reinforcing fibers.     

Generalized stochastic approach for constitutive equation in linear elasticity: a random matrix model

This work is concerned with the construction of stochastic models for random elasticity matrices, allowing either for the generation of elasticity tensors exhibiting some material symmetry properties almost surely (integrating the statistical dependence between the random stiffness components) or for the modeling of random media that requires the mean of a stochastic anisotropy measure to be controlled apart from the level of statistical fluctuations. To this aim, we first introduced a decomposition of the stochastic elasticity tensor on a deterministic tensor basis and considered the probabilistic modeling of the random components, having recourse to the MaxEnt principle. Strategies for random generation and estimation were further reviewed, and the approach was exemplified in the case of a material that was transversely isotropic almost surely. In a second stage, we made use of such derivations to propose a generalized model for random elasticity matrices that took into account, almost separately, constraints on both the level of stochastic anisotropy and the level of statistical fluctuations. An example was finally provided and showed the efficiency of the approach. Copyright © 2011 John Wiley & Sons, Ltd.     

Thermomechanical analysis of elastic–plastic fibrous composites comprising an inhomogeneous interphase

An analytical approach is presented to investigate thermomechanical response of composites consisting of a transversely isotropic fiber, an inhomogeneous interphase and an elastic–plastic matrix. Using the existing cubic variation to describe the continuous change of the material properties of the interphase and dividing the interphase into a number of subdomains, the continuously varying material properties of the interphase are approximated by the constant ones of these subdomains, and the deformations and stresses of the interphase are described with the same formulae as those of transversely isotropic fibers. The analytical expressions of elastic–plastic deformations and stresses of the matrix are obtained from the basic equations of axisymmetric problems in elasticity, the assumption of generalized plane strain, the linear strain–hardening stress–plastic strain relation, Tresca’s yield condition, the associated flow rule and impressibility of plastic deformation. The boundary conditions of the composites and the continuities of the radial displacement and stress between different components are used to determine all the unknown constants and the obtained analytical solution is applied to thermomechanical analysis of the composites. The effects of the inhomogeneity of the interphase, and the plasticity and material properties of the matrix on the thermomechanical response of the composites are investigated.     

A peridynamic failure analysis of fiber-reinforced composite laminates using finite element discontinuous Galerkin approximations

This paper presents a discontinuous Galerkin weak form for bond-based peridynamic models to predict the damage of fiber-reinforced composite laminates. To represent the anisotropy of a laminate in a peridynamic model, a lamina is simplified as a transversely isotropic medium under a plane stress condition. The laminated structure is modeled by stacking the surface mesh layers along the thickness direction according to the laminate sequence. To avoid a mesh dependence on either the fiber orientation or the discretization, the spherical harmonic expansion theory is employed to construct a function for the micro-elastic modulus in terms of the bond-fiber angle. The laminate material is decomposed into an isotropic matrix material part and a transversely isotropic material part. The bond stiffness can be evaluated using the engineering material constants, based on the equivalence between the elastic energy density in the peridynamic theory and the elastic energy density in the classic continuum mechanics theory. Benchmark tests are conducted to verify the proposed model. Numerical results illustrate that the convergence of simulations with different horizon sizes and meshes can be achieved. In terms of damage analysis, the proposed model can capture the dynamic process of the complex coupling of the inner-layer and delamination damage modes.     

Static and dynamic analysis of transversely isotropic,moderately thick sandwich beams by analogy

Summary A flexural theory of elastic sandwich beams is derived which renders quite precise results within a wide range of ratios of dimensions, mass densities, and elastic constants of the core and faces. The assumptions of the Timoshenko theory of shear-deformable beams are applied to each of the homogeneous, linear elastic, transversely isotropic layers individually. Core and faces are perfectly bonded. The principle of virtual work is applied to derive the equations of motion of a symmetrically designed three-layer beam and its boundary conditions. By definition of an effective cross-sectional rotation the complex problem is reduced to a problem of a homogeneous beam with effective stiffnesses and with corresponding boundary conditions. Thus, methods of classical mechanics become directly applicable to the higher-order problem. Excellent agreement of the results of illustrative examples is observed when compared to solutions of other higher-order laminate theories as well as to exact solutions of the theory of elasticity.     

Computing stress singularities in transversely isotropic multimaterial corners by means of explicit expressions of the orthonormalized Stroh-eigenvectors

Composite materials reinforced by unidirectional long fibers behave macroscopically as homogeneous transversely isotropic linear elastic materials. A general, accurate and computationally efficient procedure for the evaluation of singularity exponents and singular functions characterizing singular stress fields in multimaterial corners involving this kind of material is presented in this paper. To take full advantage of the sextic Stroh formalism of anisotropic elasticity applied to this particular problem, the complete set of explicit expressions of the eigenvalues and eigenvectors of the real 6 × 6 fundamental elasticity matrix N has been deduced for all the non-degenerate and degenerate (repeated roots of the sextic Stroh equation) cases. These expressions will also facilitate further applications of the Stroh formalism to these materials. Several numerical examples of singularity analysis of multimaterial corners appearing in adhesively bonded joints and damaged cross-ply laminates of composite materials are presented.     

Modeling the effective elastic properties of nanocomposites with circular straight CNT fibers reinforced in the epoxy matrix

In the present study, the consistent effective elastic properties of straight, circular carbon nanotube epoxy composites are derived using the micromechanics theory. The CNT composites are known to provide high stiffness and elastic properties when the shape of the fibers is cylindrical and straight. Accordingly, in the present work, the effective elastic moduli of composite are newly obtained for straight, circular CNTs aligned in the specified direction as well as distributed randomly in the matrix. In this direction, novel analytical expressions are proposed for four cases of fiber property. First, aligned, and straight CNTs are considered with transverse isotropy in fiber coordinates, and the composite properties are also transversely isotropic in global coordinates. The short comings in the earlier developments are effectively addressed by deriving the consistent form of the strain tensor and the stiffness tensor of the CNT nanocomposite. Subsequently, effective relations for composites reinforced with aligned, straight CNTs but fibers isotropic in local coordinates are newly developed under hydrostatic loading. The effect of the unsymmetric Eshelby tensor for cylindrical fibers on the overall properties of the nanocomposite is included by deriving the strain concentration tensors. Next, the random distribution of CNT fibers in the matrix is studied with fibers being transversely isotropic as well as isotropic when CNT nanocomposites are subjected to uniform loading. The corresponding relations for the effective elastic properties are newly derived. The modeling technique is validated with results reported, and the variations in the effective properties for different CNT volume fractions are presented.     

Computation of sensitivity of periodically excited dynamical systems

An analytical treatment is presented for bonded contact of a rigid disk inclusion embedded in a penny-shaped crack in a transversely isotropic full-space. Theoretical analysis is carried out using a complete potential function method, and with the aid of Hankel transforms. Boundary conditions propel the problem toward a set of triple integral equations, which are solved analytically and then reduced to a pair of Fredholm integral equations of the second kind. Furthermore, the results are evaluated and presented graphically using numerical schemes, and comparison is made with well-known classical solutions in transversely isotropic and isotropic media. This can be obtained as special cases for the problem to reveal the efficacy of the proposed method. Eventually, it can be seen that not only the presence of the crack around the disk inclusion decreases the axial stiffness, but also the extension of its length also reduces the fracture parameter, stress intensity factor, for different degrees of material anisotropy.     

A transverse shear deformation theory for homogeneous monoclinic plates

Summary A general two-dimensional theory suitable for the static and/or dynamic analysis of a transverse shear deformable plate, constructed of a homogeneous, monoclinic, linearly elastic material and subjected to any type of shear tractions at its lateral planes, is presented. Developed on the basis of Hamilton's principle, in conjunction with the method of Lagrange multipliers, this new theory accounts for an unlimited number of choices of through-thickness displacement distributions, while, starting with the smallest possible number of independent displacement components (five, for a shear deformation theory), it is capable of further operating with as many degrees of freedom as desired. For the particular case of a theory operating with five degrees of freedom, special attention is given to displacement expansions producing symmetric, through thicknes, distributions of transverse shear strain. For the cylindrical bending problem of a specially orthotropic plate, the governing equations of that five-degrees-of-freedom theory are solved and for three different choices of symmetric, through tickness, transverse shear deformation, numerical results are obtained and compared with corresponding results based on the exact three-dimensional solution existing in the literature. The comparisons made show clearly, that the multiple options offered by the new theory, by either suitably altering the displacement expansions or gradually increasing the degrees of freedom involved, will be found useful in future studies dealing with the static and/or dynamic analysis of homogeneous plates.     

Circular crested waves in anisotropic thermoelastic plates bordered with inviscid liquid

The propagation of circularly crested waves in a homogeneous, transversely isotropic, thermally conducting plate bordered with layers of inviscid liquid or half space of inviscid liquid on both sides is investigated in the context of conventional coupled thermoelasticity, Lord-Shulman and Green-Lindsay theories of thermoelasticity. Secular equations for circular homogeneous transversely isotropic plate in closed form and isolated mathematical conditions for symmetric and antisymmetric wave modes in completely separate terms are derived. The results for isotropic materials and uncoupled theories of thermoelasticity have been obtained as particular cases. The special cases such as short wavelength waves, thin plate waves and leaky Lamb waves of the secular equation are also deduced and discussed. The amplitudes of displacement components and temperature change have also been computed and studied. Finally, the numerical solution is carried out for transversely isotropic circular plate of cobalt material bordered with water. The dispersion curves for symmetric and antisymmetric wave modes, attenuation coefficient and amplitudes of displacement and temperature change in case of fundamental symmetric (S₀) and skew symmetric (A₀) modes are presented in order to illustrate and compare the theoretical results. The analytical and numerical results are found to be in close agreement.     

Elastic waves interacting with a thin, prestressed, fiber-reinforced surface film

Elastic surface waves propagating at the interface between an isotropic substrate and a thin, transversely isotropic film are analyzed. The transverse isotropy is conferred by fibers lying parallel to the interface. A rigorous leading-order model of the thin-film/substrate interface is derived from the equations of three-dimensional elasticity for prestressed, transversely isotropic films having non- uniform properties. This is used to study Love waves.     

An elasticity solution for transversely isotropic,functionally graded circular plates

This article presents a new elasticity solution for transversely isotropic, functionally graded circular plates subject to axisymmetric loads. It is assumed that the material properties vary along the thickness of a circular plate according to an exponential form. By extending the displacement function presented by Plevako to the case of transversely isotropic material, we derived the governing equation of the problem studied. The displacement function was assumed as the sum of the Bessel function and polynomial function to obtain the analytical solution of a transversely isotropic, functionally graded circular plate under different boundary conditions. As a numerical example, the influence of the graded variations of the material properties on the displacements and stresses was studied. The results demonstrate that the graded variations have a significant effect on the mechanical behavior of a circular plate.     

Elastostatic Green’s functions for an arbitrary internal load in a transversely isotropic bi-material full-space

The problem of a full-space which is composed of two half-spaces with different transversely isotropic materials with an internal load at an arbitrary distance from the interface is considered. By virtue of Hu-Nowacki-Lekhnitskii potentials, the equations of equilibrium are uncoupled and solved with the aid of Hankel transform and Fourier decompositions. With the use of the transformed displacement- and stress-potential relations, all responses of the bi-material medium are derived in the form of line integrals. By appropriate limit processes, the solution can be shown to encompass the cases of (i) a homogeneous transversely isotropic full-space, and (ii) a homogeneous transversely isotropic half-space under arbitrary surface load. As the integrals for the displacement- and stress-Green’s functions, for an arbitrary point load can be evaluated explicitly, illustrative results are presented for the fundamental solution under different material anisotropy and relative moduli of the half-spaces and compared with existing solutions.     

Numerical simulation of injection/compression liquid composite molding. Part 2: preform compression

In the injection/compression liquid composite molding process (I/C-LCM), a liquid polymer resin is injected into a partially open mold, which contains a preform of reinforcing fibers. After some or all of the resin has been injected, the mold is closed, compressing the preform and causing additional resin flow. This paper addresses compression of the preform, with particular emphasis on modeling three-dimensional mold geometries and multi-layer preforms in which the layers have different mechanical responses. First, a new constitutive relation is developed to model the mechanical response of fiber mats during compression. We introduce a new form of nonlinear elasticity for transversely isotropic materials. A special case of this form is chosen that includes the compressive stress generated by changes in mat thickness, but suppresses all other responses. This avoids the need to model slip of the preform along the mold surface. Second, a finite element method, based on the principle of virtual displacement, is developed to solve for the deformation of the preform at any stage of mold closing. The formulation includes both geometric and material nonlinearities, and uses a full Newton–Raphson iteration in the solution. An open gap above the preform can be incorporated by treating the gap as a distinct material layer with a very small stiffness. Examples show that this approach successfully predicts compression in dry preforms for three-dimensional I/C-LCM molds.     

Nonlinear Viscoelastic Response of Unidirectional Polymeric Laminated Composite Plates Under Bending Loads

Nonlinear bending analysis of polymeric laminated composite plate is examined considering material nonlinearity for viscoelastic matrix material through a Micro–macro approach. The micromechanical Simplified Unit Cell Method (SUCM) in three-dimensional closed-form solution is used for the overall behavior of the unidirectional composite in any combination of loading conditions. The elastic fibers are transversely isotropic where Schapery single integral equation in multiaxial stress state describes the matrix material by recursive-iterative formulation. The finite difference Dynamic Relaxation (DR) method is utilized to study the bending behavior of Mindlin annular sector plate including geometric nonlinearity under uniform lateral pressure with clamped and hinged edge constraints. The unsymmetrical laminated plate deflection is predicted for different thicknesses and also various pressures in different time steps and they are compared with elastic finite element results. As a main objective, the deflection results of viscoelastic laminated sector plate are obtained for various fiber volume fractions in the composite system.