Computational Methods for Debonding in Composites

44 R. Rolfes et al.
Fig. 2.14 Radial return
algorithm
Hence, the stresses can be calculated through
and the internal variable is updated by
σn+1 =σ trial
n+1 −∆λn+1C el :mn+1
III
II
n
n+1
trial
n+1
(2.25)
¯ε pl
n+1 = ¯εpl
n +∆λn+1 �mn+1� (2.26)
Inserting in the active yield surface Eq. (2.21) leads formally to a nonlinear equation
in ∆λn+1 which is solved by the Newton-Raphson method. Figure 2.14 shows an
illustration of the applied integration algorithm.
2.3.2 Transversely Isotropic Elastic-Plastic Material Model
for Fiber Bundles
Fiber bundles, as used in the mesomechanical unit cell, exhibit a transversely
isotropic characteristic. Although plasticity is lower and the overall behavior is more
brittle than pure resin, plasticity also occurs. Especially under transverse and inplane
shear stress states considerable plastic deformations can be observed, whereas
under tensile and compressive loadings in fiber direction the fiber bundle exhibits an
elastic-brittle behavior. That is, the material behavior can be approximated as nearly
linear elastic until failure occurs and plasticity in fiber direction is neglected. Further,
yielding behavior and material failure are dependent on hydrostatic pressure.
Therefore, a transversely isotropic elastic-plastic material model with a pressure
dependent C1-continuous yield surface and a transversely isotropic damage formulation
is developed. For the infinitesimal strain tensor, an additive decomposition is
assumed:
ε = ε el + ε pl
(2.27)
For both the elastic and the plastic part of the transversally isotropic material
model, the representation of the constitutive equations is carried out in the format
of isotropic tensor functions by means of structural tensors. The structural tensor A,
reflecting the materials intrinsic characteristic, is defined as the dyadic product of