Post-Damage Constitutive Model Based on the Concept of Smeared Crack

Upon increasing the applied load, micro-cracks will develop in the matrix. These cracks cause reduction in stiffness of the laminate. In contrast to a predefined single dominant crack in isotropic materials, the orientation and location or even numbers of cracks in a laminate is unknown, thus this makes it difficult to deal with such cracks through the classical fracture mechanics approach. Instead, the so-called ‘smeared crack’ approach will be used. In this approach the reduction of load bearing capacity induced by a crack is described by stress-strain softening relationship. In this manner the discontinuity caused by a crack is smeared out, and this procedure can be implemented into a FEM code.

Once a crack is formed, it is assumed that it cannot transfer normal and shear stresses across the crack surfaces, i. e.Cj, Oj2and Oj3 ^ 0. The subscript 1 denotes the Cartesian axis perpendicular to the crack plane while 2 and 3 are in the crack plane. However, the ability to transfer the other stress components is not affected by the crack formation. Let the stress and strain vectors in the local (crack)

coordinate system be designated by {о} ande}cr, respectively. Thus, the post-damage constitutive model in the crack coordinate system is

{дстГ _ e, Ид£Г-х[вЫ

or written in its full form:

1 -и z _ и z _ 1

(1 + и)(1 – 2и), 2 _ (1 + и)(1 – 2и), 3 _ 2(1 + и)

Et is the modulus of the epoxy under uniaxial tensile loading at the instant of damage initiation, в is a small number which represents the loss of the stiffness in these three particular stress directions and the constant х allows the three stress components to decrease to a near zero value in a sufficiently short time duration.