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.     

Realizable compressibility and conductivity in isotropic two-phase composites

It is shown in a very simple manner how any bulk modulus and any conductivity of isotropic two-phase composites located between the Hashin and Shtrikman bounds can be realized exactly by suitable microstructures. The latter are assemblages of two types of composite spheres.     

Locally-exact homogenization theory for transversely isotropic unidirectional composites

The locally-exact homogenization theory for unidirectional composites with square periodicity and isotropic phases proposed by Drago and Pindera [18] is extended to architectures with hexagonal symmetry and transversely isotropic phases. The theory employs Fourier series representation for the displacement fields in the fiber and matrix phases in the cylindrical coordinate system that satisfies exactly the equilibrium equations and continuity conditions in the unit cell's interior. The inseparable exterior problem involves satisfaction of periodicity conditions for the hexagonal unit cell geometry demonstrated herein to be readily achievable using the previously introduced balanced variational principle for square geometries. This variational principle plays a key role in the employed unit cell solution, ensuring rapid convergence of the Fourier series coefficients with relatively few harmonic terms, yielding converged homogenized moduli and local stress fields with little computational effort. The solution's stability is illustrated using the dilute case which is shown to reduce to the Eshelby solution regardless of the employed number of harmonic terms. Comparison with published results and predictions of a finite-volume based homogenization in a wide fiber volume range and different fiber/matrix modulus contrast validates the approach's accuracy, and its utility is demonstrated through rapid local stress recovery in a multi-scale application. This extension completes the development of the theory for three important classes of unidirectional reinforcement arrays, thereby providing an efficient alternative to finite-element based homogenization techniques or approximate micromechanical schemes, as well as an efficient standard against which other methods may be compared.     

Statistical continuum theory for inelastic behavior of a two-phase medium

A statistical continuum mechanics formulation is presented to predict the inelastic behavior of a medium consisting of two isotropic phases. The phase distribution and morphology are represented by a two-point probability function. The isotropic behavior of the single phase medium is represented by a power law relationship between the strain rate and the resolved local shear stress. It is assumed that the elastic contribution to deformation is negligible. A Green’s function solution to the equations of stress equilibrium is used to obtain the constitutive law for the heterogeneous medium. This relationship links the local velocity gradient to the macroscopic velocity gradient and local viscoplastic modulus. The statistical continuum theory is introduced into the localization relation to obtain a closed form solution. Using a Taylor series expansion an approximate solution is obtained and compared to the Taylor’s upper-bound for the inelastic effective modulus. The model is applied for the two classical cases of spherical and unidirectional discontinuous fiber-reinforced two-phase media with varying size and orientation.     

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 continuum micromechanical theory of overall plasticity for particulate composites including particle size effect

Classical continuum micromechanics cannot predict the particle size dependence of the overall plasticity for composite materials, a simple analytical micromechanical method is proposed in this paper to investigate this size dependence. The matrix material is idealized as a micropolar continuum, an average equivalent inclusion method is advanced and the Mori–Tanaka's method is extended to a micropolar medium to evaluate the effective elastic modulus tensor. The overall plasticity of composites is predicted by a new secant moduli method based on the second order moment of strain and torsion of the matrix in a framework of micropolar theory. The computed results show that the size dependence is more pronounced when the particle's size approaches to the matrix characteristic length, and for large particle sizes, the prediction coincides with that predicted by classical micromechanical models. The method is analytical in nature, and it can capture the particle size dependence on the overall plastic behavior for particulate composites, and the prediction agrees well with the experimental results presented in literature. The proposed model can be considered as a natural extension of the widely used secant moduli method from a heterogeneous Cauchy medium to a micropolar composite.     

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.     

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.     

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 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.