Richard M. Christensen

Solutions for effective shear properties in three phase sphere and cylinder models

Abstract S olutions are presented for the effective shear modulus of two types of composite material models. The first type is that of a macroscopically isotropic composite medium containing spherical inclusions. The corresponding model employed is that involving three phases: the spherical inclusion, a spherical annulus of matrix material and an outer region of equivalent homogeneous material of unlimited extent. The corresponding two-dimensional, polar model is used to represent a transversely isotropic, fiber reinforced medium. In the latter case only the transverse effective shear modulus is obtained. The relative volumes of the inclusion phase to the matrix annulus phase in the three phase models are taken to be the given volume fractions of the inclusion phases in the composite materials at large. The results are found to differ from those of the well-known Kerner and Hermans formulae for the same models. The latter works are now understood to violate a continuity condition at the matrix to equivalent homogeneous medium interface. The present results are compared extensively with results from other related models. Conditions of linear elasticity are assumed.

A High-Order Theory of Plate Deformation—Part 2: Laminated Plates

The high-order theory of plate deformation developed in Part I of this work is extended here to model the behavior of laminated plates. Through comparison with elasticity solutions, it is shown the present theory correctly models effects not attainable from the classical theory. /Author/

A HIGH-ORDER THEORY OF PLATE DEFORMATION—PART 1: HOMOGENEOUS PLATES

A theory of plate deformation is derived which accounts for the effects of transverse shear deformation, transverse normal strain, and a nonlinear distribution of the in-plane displacements with respect to the thickness coordinate. The theory is compared with lower-order plate theories through application to a particular problem involving a plate acted upon by a sinusoidal surface pressure. Comparison is also made with the exact elasticity solution of this problem. It is found that when the ratio of the characteristic length of the load pattern to the plate thickness is of the order of unity, lower-order theories are inadequate and the present high-order theory is required to give meaningful results. The present work treats homogeneous plates while Part 2 involves laminated plates. /Author/

Mechanics of cellular and other low-density materials

Both two-dimensional and three-dimensional low density materials are surveyed. The microstructure is usually in a cellular form with some characteristic dimension(s) being small compared to the cell size and with the density range approaching that at which the loss of material integrity occurs. The mechanical properties of stiffness and strength are considered, along with applications and future opportunities.

Mechanics of low density materials

Abstract Mechanics analyses are used to derive the effective elastic moduli for low density materials. Both open cell and closed cell geometric models are employed in the case of isotropic media. The five independent effective moduli are derived for a low density transversely isotropic medium. Compressive strength, as defined by elastic stability, is also derived for open cell and closed cell isotropic materials. The theoretical results are compared with some experimental results, and also are assessed with respect to previous work.

STRESS BASED YIELD/FAILURE CRITERIA FOR FIBER COMPOSITES

Abstract A macroscopic yield or failure criterion is derived for fiber composite materials. The derivation decomposes naturally into two modes of yield/failure, one being matrix dominated, the other being fiber dominated, thus there are two governing criteria. The resulting forms are quadratic in the components of the average stress tensor with two material parameters for each mode of yield/failure. The physical context of the formulation is that of aligned fiber systems with a polymeric matrix phase under high fiber concentration conditions.

Effective properties of composite materials containing voids

Recent results of theoretical and practical importance prove that the two-dimensional (in-plane) effective (average) Young’s modulus for an isotropic elastic material containing voids is independent of the Poisson’s ratio of the matrix material. This result is true regardless of the shape and morphology of the voids so long as isotropy is maintained. The present work uses this proof to obtain explicit analytical forms for the effective Young’s modulus property, forms which simplify greatly because of this characteristic. In some cases, the optimal morphology for the voids can be identified, giving the shapes of the voids, at fixed volume, that maximize the effective Young’s modulus in the two-dimensional situation. Recognizing that two-dimensional isotropy is a subset of three-dimensional transversely isotropic media, it is shown in this more general case that three of the five properties are independent of Poisson’s ratio, leaving only two that depend upon it. For three-dimensionally isotropic composite media containing voids, it is shown that a somewhat comparable situation exists whereby the three-dimensional Young’s modulus is insensitive to variations in Poisson’s ratio, vm, over the range 0 ≤ vm ≤ ½, although the same is not true for negative values of vm. This further extends the practical usefulness of the two-dimensional result to three-dimensional conditions for realistic values of vm.

Two Theoretical Elasticity Micromechanics Models

This study is concerned with the Composite Spheres (Cylinders) Model and the Generalized Self Consistent Method (GSCM). A detailed examination of the two models proves the two models are the same in the limited cases where they both give solutions. In this process of comparison between the two models, a new solution is found for the shear property of a closed cell foam type idealization.

Coefficient of Thermal Expansion for Composites with Randomly Oriented Fibers

The effective coefficient of thermal expansion is derived for fiber com posites in two dimensional quasi-isotropic form and three-dimensional isotropic form. These systems represent constructions that have fibers ran domly oriented in a plane and in three dimensions. The effective coeffi cients of thermal expansion are related to the thermal-mechanical proper ties of individual fiber and matrix phases. These results are put into asymp totic forms appropriate to very stiff fiber systems. The asymptotic predic tions are evaluated against the complete forms, and both results are com pared with an experimental result.

A Two-Property Yield, Failure (Fracture) Criterion for Homogeneous, Isotropic Materials

A critical review is given of the various historical attempts to formulate a general, three-dimensional theory of failure for broad classes of homogeneous, isotropic elastic materials. Following that, a recently developed two-parameter yield/failure criterion is compared with the historical efforts and it is further interpreted and extended. Specifically, the yield/failure criterion is combined with a fracture restriction that places limits on certain tensile stress states, without involving any additional parameters. An evaluation is conducted using available experimental data obtained from a variety of materials types. The two materials parameters are given a primary designation as yield type properties over a specified range of ductile behavior, and as failure or fracture type properties over the complementary brittle range.