A plane theory of inextensible transversely isotropic elastic composites

A plane strain or plane stress configuration of an inextensible transversely isotropic linear elastic material, with the axis of symmetry in the plane, leads to a harmonic lateral displacement field in stretched coordinates. Various displacement and traction conditions lead to standard and nonstandard boundary value problems of potential theory. Examples for a rectangular plane, half-plane and infinite plate with elliptic hole, are presented in illustration.     

A continuum theory for fibre-reinforced composites

The effective stiffness theory for fibre reinforced materials with a hexagonal layout of fibres is presented. The theory is illustrated by the dispersion curves of plane steadystate time-harmonic waves. The limiting phase velocities at vanishing wave numbers serve in the determination of the elastic moduli of the equivalent homogeneous transversely isotropic medium.     

A continuum theory of elastic material surfaces

A mathematical framework is developed to study the mechanical behavior of material surfaces. The tensorial nature of surface stress is established using the force and moment balance laws. Bodies whose boundaries are material surfaces are discussed and the relation between surface and body stress examined. Elastic surfaces are defined and a linear theory with non-vanishing residual stress derived. The free-surface problem is posed within the linear theory and uniqueness of solution demonstrated. Predictions of the linear theory are noted and compared with the corresponding classical results. A note on frame-indifference and symmetry for material surfaces is appended.     

Physical constants of a nonlinear microstructural theory of two-phase elastic mixtures for several granular structural composites

Translated from Prikladnaya Mekhanika, Vol. 32, No. 12, pp. 80–89, December, 1996.     

Bounds and perturbation series for incompressible elastic composites with transverse isotropic symmetry

The effective elastic behavior of a transversely isotropic composite made from two incompressible elastic materials is examined. The set of all effective elasticity tensors for transversely isotropic finite rank laminar microstructures is described. The extremal property of this class of microstructures is used to derive a new more precise characterization of the set of effective shear moduli.The perturbation series for the effective elasticity tensor is considered. An explicit formula for the second order perturbation tensor is derived. We describe precisely the set of tensors that correspond to all second order perturbations consistent with transverse isotropy. We apply analytic methods [cf. 27] to show that all second order perturbation tensors are realized by finite rank laminar microstructures.Supported by NSF through Grant DMS-3907658.     

Generalization of strain-gradient theory to finite elastic deformation for isotropic materials

This paper concerns finite deformation in the strain-gradient continuum. In order to take account of the geometric nonlinearity, the original strain-gradient theory which is based on the infinitesimal strain tensor is rewritten given the Green–Lagrange strain tensor. Following introducing the generalized isotropic Saint Venant–Kirchhoff material model for the strain-gradient elasticity, the boundary value problem is investigated in not only the material configuration but also the spatial configuration building upon the principle of virtual work for a three-dimensional solid. By presenting one example, the convergence of the strain-gradient and classical theories is studied.     

A note on isotropic elastic response

For an important class of incompressible isotropic elastic solids, the response function for the extra stress is a (tensor-valued) function of scalar type. It is shown here that the stress response for compressible isotropic elastic solids cannot be of scalar type.     

Effective conductivity in isotropic heterogeneous media using a strong-contrast statistical continuum theory

The strong-contrast formulation is used to predict the effective conductivity of a porous material. The distribution, shape and orientation of the two phases are taken into account using two- and three-point probability distribution functions. A new approximation for the three-point probability function appropriate for two-phase media is proposed and discussed. Computed results for the effective conductivity using the strong-contrast formulation are compared to the Voigt and the Hashin-Shtrikman upper-bound estimates. These results show that the predicted effective conductivity is lower than both Voigt and Hashin-Shtrikman bounds. Compared to previous results using the weak-contrast formulation, the strong-contrast formulation seems to provide a better estimate for the effect of the microstructure on the conductivity.     

Magnetooptical, electrooptical and photoelastic effects in an elastic polarizable and magnetizable isotropic continuum

The magnetooptical, electrooptical and photoelastic behaviour of an elastic polarizable and magnetizable isotropic continuum are investigated from a dynamical point of view, starting from balance equations and constitutive relations. The most original result of the theory is the fact that the continuum exhibits the Cotton-Mouton effect, together with linear birefringence of transverse sound waves. This is compared with experimental data and quantum theory results.As expected, the continuum does not exhibit Faraday rotation.     

A two-dimensional continuum theory for angle-ply laminates

A two-dimensional continuum theory of microstructure is developed for stress analysis of angle-ply laminates under in-plane loading. An example problem is used to evaluate the results of the theory against a reference solution obtained by the finite element method. The results are in satisfactory agreement; they also show that the in-plane stresses reach somewhat higher peak values than reported in previous literature.The theory is also presented in a simplified version, which is found to be adequate for predicting interlaminar stresses and in-plane stress resultants, but does not give acceptable results for the variation of in-plane stresses through the thickness of the laminations.     

A continuum theory for viscoelastic composite beams

Summary A dynamical continuum theory is developed for laminated composite beams. Starting with an assumed displacement- and temperature field, the one-dimensional approximate theory is consistently constructed within the frame of the three-dimensional theory of linear, nonisothermal, anisotropic, coupled viscoelasticity. Each constituent of the beam may possess different constant thickness and mechanical properties. All dynamic interactions between the adjacent constituents are included. Further, the effects of transverse shear and normal strains and rotatory inertia as well as those of cross-sectional distortion are all taken into account. The resulting equations consist of the macroscopic beam equations of motion and heat conduction, the kinematical relations, the initial and boundary conditions and the constitutive equations, and they govern the extensional, flexural and torsional motions of laminated composite beams. The special cases of constituents which made of either isotropic thermoviscoelastic or anisotropic thermoelastic materials are discussed briefly.Supported by the Office of Naval Research.With 1 figure     

A generalized continuum theory for granular materials

A generalized continuum theory for granular media is formulated by allowing for the possibility of rotation of granules. The basic balance laws are presented and based on thermodynamical consideration a set of constitutive equations are derived. The theory naturally gives rise to the generation of antisymmetric stress tensor and existence of couple stresses. The basic equations of motion are derived and it is shown that the theory contains Mohr-Coulomb criterion of limiting equilibrium as a special case. The problem of coupled porosity and microrotational wave propagation is investigated and the rectilinear shear flow of granular materials is discussed.     

A one-parameter generalized self-consistent model for isotropic multiphase composites

The classical generalized self-consistent model (GSCM) is recognized to be suitable and efficient for estimating the effective moduli of an isotropic composite consisting of an isotropic host matrix and an isotropic inclusion phase. The present work aims to enlarge the scope of the GSCM so that it becomes applicable to a good number of important situations where the phases cannot be differentiated as the host matrix and inclusions. This objective is achieved first by inserting into the unknown effective medium a coated composite sphere whose core is made of the unknown effective medium and whose coatings are formed of the constituent phases and then by imposing an energy equivalency condition. The equations thus obtained to characterize the effective bulk and shear moduli involve a microstructural parameter which turns out to be capable of describing in some sense how far a microstructure is from the host matrix/inclusion morphology. The important case of two-phase composites is studied in detail to illustrate the salient features of the proposed model.     

A continuum damage model for delaminations in laminated composites

Delamination, a typical mode of interfacial damage in laminated composites, has been considered in the context of continuum damage mechanics in this paper. Interfaces where delaminations could occur are introduced between the constituent layers. A simple but appropriate continuum damage representation is proposed. A single scalar damage parameter is employed and the degradation of the interface stiffness is established. Use has been made of the concept of a damage surface to derive the damage evolution law. The damage surface is constructed so that it combines the conventional stress-based and fracture-mechanics-based failure criteria which take account of mode interaction in mixed-mode delamination problems. The damage surface shrinks as damage develops and leads to a softening interfacial constitutive law. By adjusting the shrinkage rate of the damage surface, various interfacial constitutive laws found in the literature can be reproduced. An incremental interfacial constitutive law is also derived for use in damage analysis of laminated composites, which is a non-linear problem in nature. Numerical predictions for problems involving a DCB specimen under pure mode I delamination and mixed-mode delamination in a split beam are in good agreement with available experimental data or analytical solutions. The model has also been applied to the prediction of the failure strength of overlap ply-blocking specimens. The results have been compared with available experimental and alternative theoretical ones and discussed fully.