Numerical Example

```An Isogeometric Layerwise Approach
for the Buckling Analysis of
Delaminated Laminates with Contact
Background
Layerwise theories of composite laminates provide accurate
predictions of the three-dimensional stress state which is in
sharp contrast to the class of equivalent single layer
theories that yield no or limited information about the
transverse stress components [1-2].
Delaminations may cause obvious reduction of compressive
The two ways to deal with a buckling analysis are:
(a) following the history of load-displacement variations in
a non-linear analysis or (b) solving an eigenvalue problem
for the critical buckling load and corresponding mode. The
buckling modes of composite laminates with delaminations
often accompanied with physically inadmissible mode
shapes as addressed here (cf Fig 6).
Isogeometric analysis [3], is a novel concept in computational mechanics, which employs the basis functions used to
describe the geometry also to approximate the physical
response in an isoparametric sense. An isogeometric
analysis specific refinement scheme (k-refinement) enables
the adjustment of continuity across element boundaries. In
the following we exploit the smoothness, the higher order
continuity and refinement properties of isogeometric
analysis for the buckling analysis of composites.
Aerospace Engineering
Isogeometric Layerwise Model Considering
Delaminations
The layerwise theory considers separate stiffness
contributions for each layer, thus avoiding a
homogenization of the elasticity properties through the
composite’s thickness. Correspondingly, the displacement
field for each layer is interpolated by a separate C0continuous NURBS (non-uniform rational B-spline) Ansatz.
The isogeometric layerwise refinement schemes of a two
layer laminate model are shown in Figure 1.
PhD Candidate: Yujie Guo
Department: ASM
Section:
Aerospace Structures and
Computational Mechanics
Supervisor: Martin Ruess
Promoter:
Zafer Gürdal
Start date: 18-9-2011
Funding:
CSC
In the isogeometric framework, discontinuities at the
delamination interfaces can be ensured by knot repetition
at the layer boundaries. Knots are a non-decreasing set of
points subdividing the parameter space in which the
NURBS Ansatz is specified into elements. The delaminated
and undelaminated regions are modeled as separate
patches and connected on the level of degrees of freedom.
Performance
The iteration history of the buckling load and the maximum
overlap for four different penalty factors are shown in
Figure 7.
Fig 3: Schematic representation of
a delamination model
Isogeometric Contact Iteration Strategy
non-physical overlaps of
laminate plys, (Fig 6,
red domain).
contact analysis (Fig 4)
to remove the critical
overlap and to regain
control over a reliable
buckling analysis. The
stiffness representation
of delaminated structure
is stepwise corrected by
a contact stiffness contribution based on a
penalty approach. The
free scalability of mode
shapes
requires
in
to remain in each iteration step consistent with
the original problem.
Fig 7: Iteration history
Revised buckling state results
The first three buckling modes of a [00/900] laminate with
α=0.3 and α=0.5 are considered. Various positions β of the
delamination zone are investigated and shown in Figures
8-9. The variation of the buckling load for the second and
third buckling mode is illustrated with respect to the
different delamination models.
Fig 4: Contact iteration loop
Numerical Example - Contact Study
Fig 1: Isogeometric refinement schemes of
a two layer laminate model
Using multiple NURBS-patches in isogeometric analysis
allows to model the delaminated domains separately and
independent of each other. In the following we summarize
the strong coupling concept for isogeometric multi-patch
models with respect to the multi-layer modeling in the
framework of the delamination analysis of composite
laminates. Exemplarily, the two NURBS-patches shown in
Figure 2 are chosen to illustrate the basic principles of the
coupling concept applying different basis functions through
the thickness of a laminate. The proposed concept even
allows the coupling of incompatible patches.
A two layered [00/900] laminate with a pre-existing
delamination at the ply interface is considered. The plate is
in a state of plane strain and has a slenderness s=L/h=10,
and a ratio of the delamination length α=0.4.
Fig 8: Effect of contact constraint on the buckling load
of [00/900] laminate with α=0.3
Fig 5: Delamination model for a two layer
composite plate
The mode shape history with regard to the contact analysis
is shown in Figure 6.
Original
Iteration 3
Final
Fig 2: Multi-patch connection
Fig 6: Changes of the buckling mode shapes
Fig 9: Effect of contact constraint on the buckling load
of [00/900] laminate with α=0.5
References
[1] Carrera E. (2002). Theories and finite elements for multilayered, anisotropic, composite plates and shells, Archives of Computational Methods in Engineering 9(2), pp 87-140
[2] Reddy J.N. (2004). Mechanics of laminated composite plates and shells, CRC Press LLC, Boca Raton
[3] Hughes T. J. R., Cottrell J.A., Bazilevs Y.,(2005). Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Computer Methods in Applied Mechanics and
Engineering 194, pp 4135-4195
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