Application Examples

The response of three types of glass fiber/epoxy polymer matrix laminates subject to a tensile loading are presented herein, see Ellyin and Kujawski (1995), Zhang et al. (2005). These are: (1) a unidirectional laminate under 45° off-axis loading; (2) a [0°/90°]ns cross-ply laminate under transverse loading, and (3) a [±45°]ns angle-ply laminate under tensile loading, see Fig. 3(a), (b), (c), respectively. The 45° off-axis tension on the unidirectional laminate (Fig. 3a) and the tensile load on

the [±45°]ns angle-ply laminate (Fig 3c) result in combined normal and shear traction boundary conditions as shown in figures with

Ox = Oy = Txy = °-5^

Fig. 3 (a) a unidirectional laminate under 45° off-axis loading; (b) a cross-ply laminate under
transverse loading; (c) a [±45°]ns angle-ply laminate under tensile loading.

The RUC for the unidirectional laminate is a unit cube containing a cylindrical fiber and the RUCs for both [0°/90°]ns cross-ply laminate and [±45°]ns angle-ply laminate are the same consisting of two cubes with the two cylindrical fibers at an angle of 90° (Fig. 4a, b ).

The finite element code ADINA was used to conduct the numerical analysis. The nonlinear viscoelastic constitutive model of the epoxy matrix and the post-damage constitutive model were implemented into the code through its user-defined subroutine. It is to be noted that in Eqn. (24) the multiaxial loads are the global stress components applied to the RUCs. The applied periodic boundary conditions, Eqn. (12), are in the form of global strain components. An iterative procedure is required to ensure proper proportion of the increments of the global strain components are applied so that Eqn. (24) is satisfied at each time step. The iteration procedure is as follows.

(i) For each time step, At, we apply a set of trial global strain increments, Aeij

(ii) The solution gives the stress distribution in the RUC, so the global stress components are the average values over the volume V of the RUC (see Eqn. 8.)

(iii) Equation (24) is checked and, if it is satisfied (within a certain error limit), then one proceeds to the next time step. If not, new values of Аёц are obtained and the steps (i) to (iii) are repeated. For a

small time step, it could be assumed that the increments of Аёц are proportional to the corresponding increment of average stress components, then the new values of Аёц can be estimated.

Two damage mechanisms were considered in the analyses, i. e. matrix cracking and fibre breaking. For the epoxy polymer matrix, a maximum principal strain criterion was adopted, i. e. if£1 > £cr = 4.8%,

the matrix element was assumed to be damaged and subsequently the post-damage constitutive relation would be used. (The manufacturer did not provide the failure strain for the neat resin, the value assumed here was taken from another resin system). For the glass fiber, the average axial strain of the fibre was monitored and if it exceeded a prescribed maximum value, i. e. £fa > єf max = 2.3%

then the fiber was assumed to have fractured and the calculation was then terminated. The above critical strain values were taken from experimental data and the fiber volume fraction of the laminates was assumed to be 52.5%.