JEC COMPOSITES MAGAZINE - Issue #90 - June/July 2014 - 49

The VCE method

In LMS Samtech Samcef , VCE is not used to simulate the crack
propagation, mainly for two reasons. The first one is that VCE
(just like VCCT) is a fracture mechanics approach. It is well
known, and this is demonstrated even on simple configurations
(see e.g. [5,6]), that a very fine mesh is needed in the vicinity of
the crack front to obtain accurate results. The second reason is
linked to the simulation of the crack propagation itself. In order
to keep a model with a reasonable size, the mesh refinement
should follow the crack front during propagation. For composites, re-meshing is a difficult task, because the crack fronts
don't remain straight during propagation and only hexahedral
(or prismatic) solid elements can be used when 3D laminates
are modelled [7]. Moreover, the strategy for the crack propagation itself is either based on a geometric parameterization
- which is very difficult to implement when several cracks must
be managed - or based on nodes release [8]. In both cases, a
general and robust solution procedure is difficult to implement
and only simple cracked configurations have been successfully
studied.

In this fracture mechanics approach, which is very similar to
VCCT (virtual crack closure technique), the energy release
rates by mode GI, GII and GIII (opening, shearing and tearing
modes, respectively, see Figure 3) are computed at each node
modelling the crack fronts, as illustrated in Figure 4.

Fig. 3: The 3 modes of a crack propagation

The cohesive elements approach
In the cohesive elements approach, zero thickness interface elements are defined between the hexahedral elements representing the plies (Figure 5).
Fig. 4: Computation of the energy release rate at the nodes modelling the
crack front

First, the total energy release rate is given by a finite difference
on the potential energy p with respect to a local virtual crack
propagation DA, where pinit and ppert correspond to the energy
associated to the initial and perturbed crack size, respectively:

Then the contributions to the different modes are evaluated
based on GT and on the relative displacements Ui and reaction
forces Ri in the local coordinate system attached to each node
modelling the crack front:

Once computed, these values are inserted in a failure criterion,
for instance (1), in order to check whether or not the crack will
propagate:

Fig. 5: Interface element

A specific material model including damage is assigned to these
interface elements. In LMS Samtech Samcef , the approach described in [9] is implemented and extended in [10,5] in order to
propose exponential, bi-triangular and polynomial constitutive
laws for the interface (Figure 6). A potential ed including the
damage variables dI, dII and dIII is defined in order to represent
the 3 modes of crack propagation of Figure 3. Only the components of the strain tensor relevant in the interface are considered (e33, g31 and g32). As usual in continuum damage mechanics,
the damage di impacts the initial material stiffness, here noted
k0. The damage does not appear when the crack tends to be
i
closed, that is for _.

(1)
In (1), GIc, GIIc and GIIIc are the fracture toughness values,
which are determined based on specific (sometimes standardized) tests such as DCB (double cantilever beam) and ENF
(end notched flexure). See [5] for more details.

The damage variables take their values between 0 (initial value;
no damage) and 1 (final value; completely damaged interface).
The value of the damage increases with the loading, which is
associated to the so-called thermodynamic forces. In order to
manage mixed mode behaviours, an equivalent thermodynamic
No90 June - July 2014 /

jec composites magazine 49

JEC COMPOSITES MAGAZINE - Issue #90 - June/July 2014

Table of Contents for the Digital Edition of JEC COMPOSITES MAGAZINE - Issue #90 - June/July 2014