Elsevier

Mechanics of Materials

Volume 148, September 2020, 103489
Mechanics of Materials

Research paper
Homogenized strength criterion for composite reinforced with orthogonal systems of fibers

https://doi.org/10.1016/j.mechmat.2020.103489Get rights and content

Abstract

It is demonstrated that having solved the periodicity cell problem for fiber reinforced composite, it is possible to obtain the homogenized (also referred to effective, overall, macroscopic) strength criterion for the composite (i.e. the criterion for the strength of the constitutive elements of composite in terms of homogenized stresses or strains). The method of construction of the homogenized strength criterion is adapted for the case when the periodicity cell problem is solved numerically. For composite reinforced with orthogonal systems of fibers, the homogenized strength criterion may be obtained in explicit form. It possible because the maximum local stresses in composites reinforced with orthogonal systems of fibers appear in certain areas and have special forms. An example is present for the case when Mises strength criterion is used for fibers and binder. Although the homogenized strength criterion is written in the explicit form, the numerical computations are still required to compute the constants in this.

Introduction

The problem of rigor construction of the strength criterion is an oldest problem in the theory of composite materials. Numerous publications are devoted to this problem, see, e.g. (Aboudi, 1991; Nemat-Nasser, 1991; Milton, 2002; Bakhvalov and Panasenko, 1989). The list is not completed even with the references in these books. In (Kalamkarov and Kolpakov, 1997) (see also (Annin et al., 1990; Kalamkarov and Kolpakov, 1996)) a theoretical scheme for construction of effective strength criterion of composite (a strength criterion of composite microstructure in terms of homogenized stresses or strains) was presented. The implementation of this scheme requires the knowledge of distribution of microscopic stresses over periodicity cell or representative volume of composite. The calculation of microscopic stresses is a non-trivial problem, since it requires solving the elasticity theory problem of a general form (the only exception is layered composites, for which solution can be obtained in explicit form (Kalamkarov and Kolpakov, 1997). In general case, solution can be obtained only numerically. This paper discusses the issues of numerical solution of periodicity cell problems for composites reinforced with orthogonal systems of fiber and practical construction of homogenized strength criterion for such composites.

Section snippets

Homogenization method as applied to composite reinforced with systems of fibers

We consider composite obtained by stocking layers of unidirectional fibers. The fibers in the layers are parallel to each other and the layers are stacked parallel to each other (for definiteness, the layers are perpendicular to the axis, Fig. 1). The direction of the fibers in the s-th layer is described by the guiding vector vs. The space between the fibers is filled with a binder forming the matrix of composite. In this paper, it is assumed that the fibers and the matrix are perfectly

The strength of the composite

Our goal is to obtain the homogenized strength criterion of composite, i.e. the strength criterion for constitutive components (fibers and binder) of composite written in terms of the homogenized stresses or strains.

The strength of composite at the micro level is determined by the local (microscopic) stresses arising in the constitutive components of composite when the composite is subjected to the homogenized (macroscopic) strains εmn(x). If macroscopic strains are known (calculated), then by

Numerical analysis of the microscopic stress-strain state of the composite material

For a cross-reinforced composite, the local stress-strain state in composite is of general form even it macroscopic loads of the simplest form are applied to the composite. We consider the basic types of macroscopic strains and calculate the corresponding local stresses in the composite. In our computations, we use the following characteristics of the components. Fibers: Young's modulus Ef=170GPa; Poisson's ratio νf=0.3. Binder: Young's modulus Eb=2GPa, Poisson's ratio νb=0.36. These material

Numerical construction of the composite strength criterion

According to (12), in order to verify whether or not the strength criterion is fulfilled for specific homogenized strains ɛmn(x), it is necessary to solve PCP for these homogenized strains. Due to the linearity of the problem, it suffices to have a set of solutions Zmn(y) to PCP, more precisely the corresponding stresses σijmn=aijkl(y)Zk,lmn(y), for the six basic homogenized strains εmn=δmn, where δmn is Kronecker symbol. Solution of these six problems takes a not long time (from ten minutes to

Construction of the homogenized strength criterion in an explicit form

In some special cases, part of the above computations can be performed explicitly. For example, it may be done for thin fiber reinforced plate subjected to the macroscopic in-plane deformations. Such plates are widely used in practice. The macroscopic in-plane deformations (i.e. deformations with no bending) are widely meet in practice.

We assume the fiber layers are parallel to the Oxz-plane, and the plate is subjected only in-plane macroscopic deformations ɛ11, ɛ33, ɛ13 (i.e., bending is

Tension-compression of an orthogonally reinforced composite under 45

Consider the case when the matrix is soft as compared to the binder. Let the material subjected to the macroscopic (homogenized) uniaxial tension along Ox-axis: ε11=e. Fig. 8 displays corresponding deformation of PC (Oy1-axis in the “fast” variables corresponds to Ox-axis in the “slow” variables).

Fig. 8 displays deformation of PC with the “jump” condition [Z11(y)]1=e[y1e1]1 in (5) on the faces perpendicular to the Oy1-axis. Due to symmetry, this solution can be guessed. Analysis of the solution

Conclusions

The theoretical scheme proposed in Kalamkarov and Kolpakov, 1997) for constructing the homogenized strength criterion of composite material can be practically realized for fiber reinforced composite numerically. In this paper, we do it by using programs written in APDL programming language of the ANSYS finite element complex. In some cases, the homogenized strength criterion can be obtained in an explicit form, although to determine the numerical values of the parameters in it, numerical

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