Keywords: fiber-reinforced composites, metal-matrix composites, plasticity, finite elements, homogenization, generalized plane strain.

A numerical model is formulated to predict the macroscopic response of ductile-matrix composites reinforced by a regular array of long, parallel fibers, subjected to any 3D stress. A micromechanical approach is employed, based on homogenization theory for periodic media, which consists in submitting a Representative Volume Element (RVE) to any prescribed increasing macroscopic strain and finding the corresponding macroscopic stress (or vice-versa).

Special finite elements are formulated to analyze any cross-section of the RVE (a prism of unlimited length, embedding a single fiber) under `generalized plane strain conditions' (i.e., with strain fields invariant along the fiber axis ). Consider a finite element with nodes; the three components of the displacement of all the nodes are taken as degrees of freedom (d.o.f.s) and collected in an array . Each component of the 3-D displacement field is modeled so as to depend linearly on over any element. Concisely:

where is the in-plane shape matrix. gathers three additional d.o.f.s, common to all the elements by which the RVE was discretized, corresponding to three of the macroscopic strain components.

The six independent (infinitesimal) engineering strains associated with eq. (12) are collected into an array

where is a constant matrix and is the in-plane compatibility matrix.

Finite elements of this type, with 3 or 4 nodes each, are employed to discretize any cross-section of the RVE of a metal-matrix composite. Both the fiber and the matrix material are supposed to be perfectly plastic and to comply with -plasticity.

It is well known that finite element solutions dealing with incompressible materials are prone to `locking' effects when approaching the fully plastic range. According to [1], this problem is circumvented by introducing some `modified strains' ( ) and re-defining the in-plane compatibility matrix ( ) of the 4-noded elements:

can be split into 4 sub-matrices , each one given by

where is the shape matrix relevant to the th node. The compatibility matrix for 3-noded elements needs no modification.

Finally, particular kinematic conditions are enforced along the boundary of the model to accommodate the periodicity of the microscopic strain field over the RVE.

The effectiveness of the proposed model in predicting the macroscopic response of MMCs beyond the elasticity limit was assessed through a number of numerical applications, where the macroscopic strains were monotonically increased until a macroscopic yielding was detected. The numerical results obtained were compared both with theoretical approximations for the macroscopic strength domain of the composite, available in the literature [2,3], and with experimental results obtained by other authors [4]. All the comparisons were quite satisfactory.

Such a model turns out to be definitely advantageous in terms of allocated memory and number of kinematic constraints to be enforced, in comparison with fully 3D finite element models.

1

J.C. Nagtegaal, D.M. Parks, J.R. Rice, "On numerically accurate finite element solutions in the fully plastic range", Comp. Meth. Appl. Mech. Engng., 4, 153-177, 1974. doi:10.1016/0045-7825(74)90032-2

2

P. de Buhan, A. Taliercio, "A homogenization approach to the yield strength of composite materials", Eur. J. Mech. A/Solids, 10, 129-154, 1991.

3

A. Taliercio, "Lower and upper bounds to the macroscopic strength domain of a fiber reinforced composite material", Int. J. Plasticity, 8, 741-762, 1992. doi:10.1016/0749-6419(92)90026-9

4

G.J. Dvorak, Y.A. Bahei-El-Din, Y. Macheret, C.H. Liu, "An experimental study of the elastic-plastic behaviour of a fibrous boron-aluminium composite", J. Mech. Phys. Solids, 36, 655-687, 1988. doi:10.1016/0022-5096(88)90003-8

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