Stress concentration in incompressible multicomponent materials (original) (raw)
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Zeitschrift für angewandte Mathematik und Physik, 2006
We consider a transversal loading of a linearly elastic isotropic media containing the identical isotropic aligned circular fibers at non-dilute concentration c. By the use of solution obtained by the Kolosov-Muskhelishvili complex potential method for two interacting circles subjected to three different applied stresses at infinity, and exact integral representations for both the stress and strain distributions in a microinhomogeneous medium, one estimates the effective moduli of the composite accurately to order c 2 .
A multicontinuum theory for structural analysis of composite material systems
Composites Engineering, 1995
The success of modern continuum mechanics in modelling problems in solid mechanics is truly remarkable. For instance, the general theories of elasticity, plasticity, and viscoeleasticity all rely on the continuum hypothesis. However, while continuum mechanics has provided a powerful means of studying the physics of deformation of composite materials, there are situations when the continuum hypothesis is simply inadequate. These problems are generally associated with inelastic behavior and are mainly attributed to the necessity to homogenize two distinctly different materials into a single continuum. In this paper, we introduce a multicontinuum theory designed specifically for the analysis of composite material systems. The chief attribute of the theory is its ability to do structural analysis while allowing each constituent to retain its own identity. Major analytical and numerical advances in the theory originally developed by Hansen et al. [Hansen, A. C., Walker, J. L. and Donovan, R. P. (1994). A finite element formulation for composite structures based on a volume fraction mixture theory. ht. J. Engng Sci. 32, I-17.1 are presented. The utility of the theory is demonstrated by using constituent information to predict the yield surface of a unidirectional boron/aluminum composite in the course of an analysis carried out at the structural level. NOMENCLATURE Both direct tensor notation and contracted matrix notation are employed in this article. In direct notation, vectors are denoted by boldface characters, while second order tensors are underscored with a tilde. Fourth order tensors are both bold and underscored with a tilde. Microstructure tensor Contracted matrix form of the microstructure tensor Body force density for constituent (Y Composite material stiffness matrix Material matrix for constituent cy Young's modulus Uniform strain field applied to a unit cell. _ Shear modulus Unit normal to a surface Momentum supply for constituent cy Spatial domain Boundary of spatial domain R Stress traction Stress traction for constituent 01 Partial stress tensor for constituent o(Displacement for constituent (Y Volume Position vector Gradient operator Strain tensor for the composite Strain tensor for constituent cy Poisson's ratio Poisson's ratio for constituent (Y Dispersed density for constituent LY Volume fraction for constituent 01 Stress tensor for the composite Stress tensor for constituent 01 Uniform stress field applied to a unit cell Shear yield stress Subscripts p* Arbitrary constituent Arbitrary constituent m Composite material matrix constituent r Composite material reinforcement constituent 'On educational leave from Battelle-Pacific Northwest Laboratories.
Elastic properties of composite materials
Mathematical Models and Computer Simulations, 2010
A developed system is presented for computer aided calculation of the effective elastic properties of composite materials (CM) with various reinforcement structure (3D reinforced, 4D reinforced, textile reinforcement). The computation was based on the finite element method for the solution of the so called local problems L pq arising on applying the asymptotic homogenization method worked out by N.S. Bakhvalov and B.Ye. Pobedrya. The calculation results for effective elastic properties of CM obtained by the developed software system are presented as well as some character istics of the system application to the above listed types of reinforcement structures. CM (composites), which have been intensively developed since the 1960s, are still one of the leading classes of engineering materials due to their outstanding properties, i.e., low weight (even in comparison with aluminum alloys), high stiffness and strength, as well as their high chemical resistance, machinability, etc. The greatest disadvantage of these materials is their high price, which had earlier limited the scope of its application in civil engineering, but it has largely been overcome, since the price of the end product (e.g., aircraft or sea vessels) is now much less dependent on the prices of components and materials.
INTERFACE RESPONSE THEORY OF COMPOSITE ELASTIC MEDIA
Journal De Physique, 1989
Reçu le 2 février 1989, accepté sous forme définitive le 21 mars 1989) Résumé. 2014 Une théorie générale permettant d'étudier n'importe quel matériau élastique et composite est proposée. Son application aux composites lamellaires est ensuite développée. Ces résultats généraux sont illustrés par des exemples de nouveaux modes de vibration localisés dans des couches fluide et solide comprises entre deux autres solides semi-infinis.
International Journal of Engineering Science, 2011
The work is devoted to the calculation of static elastic fields in 3D-composite materials consisting of a homogeneous host medium (matrix) and an array of isolated heterogeneous inclusions. A self-consistent effective field method allows reducing this problem to the problem for a typical cell of the composite that contains a finite number of the inclusions. The volume integral equations for strain and stress fields in a heterogeneous medium are used. Discretization of these equations is performed by the radial Gaussian functions centered at a system of approximating nodes. Such functions allow calculating the elements of the matrix of the discretized problem in explicit analytical form. For a regular grid of approximating nodes, the matrix of the discretized problem has the Toeplitz properties, and matrix-vector products with such matrices may be calculated by the fast fourier transform technique. The latter accelerates significantly the iterative procedure. First, the method is applied to the calculation of elastic fields in a homogeneous medium with a spherical heterogeneous inclusion and then, to composites with periodic and random sets of spherical inclusions. Simple cubic and FCC lattices of the inclusions which material is stiffer or softer than the material of the matrix are considered. The calculations are performed for cells that contain various numbers of the inclusions, and the predicted effective constants of the composites are compared with the numerical solutions of other authors. Finally, a composite material with a random set of spherical inclusions is considered. It is shown that the consideration of a composite cell that contains a dozen of randomly distributed inclusions allows predicting the composite effective elastic constants with sufficient accuracy.
Nonlinear elasticity of composite materials
The European Physical Journal B, 2009
We investigate the elastic properties of model composites, consisting in a dispersion of nonlinear (spherical or cylindrical) inhomogeneities into a linear solid matrix. Both phases are considered isotropic. Under the simplifying hypotheses of small deformation for the material body and of small volume fraction of the embedded phase, we develop a homogenization procedure based on the Eshelby theory, aimed at describing nonlinear features. We obtain the bulk and shear moduli and Landau coefficients of the overall material in terms of the elastic behavior of the constituents and of their volume fractions. The mixing laws for the nonlinear properties describe a complex scenario where possible strong amplifications of the nonlinearities may arise in some given conditions.
Peridynamic Micromechanics of Random Structure Composites
Springer eBooks, 2012
In this chapter, we consider the solution methods of the GIE of peridynamic micromechanics. These methods are based on extraction from the material properties a constituent of the matrix properties. Effective moduli are expressed through the average local interface polarization tensor over the surface of the extended inclusion phase rather than over an entire space. Any spatial derivatives of displacement fields are not required. The basic hypotheses of locally elastic micromechanics are generalized to their peridynamic counterparts. In particular, in the generalized effective field method (EFM) proposed, the classical effective field hypothesis is relaxed, and the hypothesis of the ellipsoidal symmetry of the random structure of CMs is not used. One demonstrates some similarity and difference with respect to other methods (the dilute approximation and Mori-Tanaka approach) of micromechanics of peridynamic CMs. Estimation of macroscopic effective response of heterogeneous media with random structures in an averaged (or homogenized) meaning in terms of the mechanical and geometrical properties of constituents is a central focus of micromechanics denoted as micro-to-macro modeling. The general results establishing the links between the effective properties and the corresponding mechanical and transformation influence functions were inspired by Hill [640] for locally elastic composites. Some basic representations analogous to the mentioned above were generalized in Chap. 17 to the thermoperidynamics of CMs. The displacement field estimations in the constituents, in turn, are based on a substitution into the one or another micromechanical scheme of a solution (called basic problem) for one inclusion inside the infinite matrix subjected to some effective field. So, for locally elastic random structure CMs, a number of micromechanical models inspired by Eshelby [449] (see Chap. 3) were proposed in the literature for describing the thermoelastic behavior of composites with ellipsoidal inclusions (see Chaps. 8-12). Numerical solutions for the basic problem are considered in Sect. 18.1, whereas the different micromechanical models are generalized to their peridynamic counterparts in Sects. 18.2-18.4.
Overall dynamic constitutive relations of layered elastic composites
Journal of The Mechanics and Physics of Solids, 2011
A method for homogenization of a heterogeneous (finite or periodic) elastic composite is presented. It allows direct, consistent, and accurate evaluation of the averaged overall frequency-dependent dynamic material constitutive relations. It is shown that when the spatial variation of the field variables is restricted by a Bloch-form (Floquet-form) periodicity, then these relations together with the overall conservation and kinematical equations accurately yield the displacement or stress modeshapes and, necessarily, the dispersion relations. It also gives as a matter of course point-wise solution of the elasto-dynamic field equations, to any desired degree of accuracy. The resulting overall dynamic constitutive relations however, are general and need not be restricted by the Bloch-form periodicity. The formulation is based on micro-mechanical modeling of a representative unit cell of the composite proposed by Nemat-Nasser and coworkers; see, e.g., [1] and [2].