Coupled Multi-scale Cohesive
Modeling of Failure in
M.G. Kulkarni1, K. Matous2 and P.H.
1Department of Aerospace Engineering
University of Illinois at Urbana-Champaign
Urbana, IL 61801, USA.
2Department of Aerospace and Mechanical Engineering
University of Notre Dame
Notre Dame, IN 46556, USA.
A multi-scale cohesive numerical framework is
proposed to simulate the failure of heterogeneous adhesively bonded
systems. This multi-scale scheme is based on Hill's variational
principle of energy equivalence between the higher and lower level
scales. It provides an easy way to obtain accurate homogenized
macroscopic properties while capturing the physics of failure processes
at the micro-scale in sufficient detail. We use an isotropic
rate-dependent damage model to mimic the failure response of the
constituents of heterogeneous adhesives. The finite element method is
used to solve the equilibrium equation at each scale. A nested
iterative scheme inspired by the return mapping algorithm used in
computational inelasticity is implemented. We propose a computationally
attractive technique to couple the macro- and micro-scales for
rate-dependent constitutive laws. We introduce an adhesive patch test
to study the numerical performance including spatial and temporal
convergence of the multi-scale scheme. We compare the solution of the
multi-scale cohesive scheme with a direct numerical simulation.
Finally, we solve mode I and mode II fracture problems to demonstrate
failure at the macro-scale.
We have presented the formulation and implementation
of a fully coupled multi-scale cohesive scheme for the simulation of
failure of structures bonded with heterogeneous adhesives. The emphasis
is placed on modeling of the failure simultaneously at macro- and
micro-scales through a computationally attractive nested scheme linking
the macro- and microscopic finite element models. Mode-mixity can be
naturally captured by our multi-scale scheme through a macro-micro load
coupling. The multi-scale scheme with an embedded rate dependent
constitutive model eeffectively captures the eeffect of disparate
loading rates at each macroscopic cohesive Gauss point. The loading
rates are conceivably high in front of the crack tip and reduce as one
moves away from the crack tip. Such eeffect cannot be easily accounted
for in the LEFM analytical solution and justify the necessity of the
multi-scale scheme. We note that in this work, we have limited
ourselves to two levels of the multi-scale strategy, although the
formulation and computational implementation presented are equally
extensible to more levels with appropriate modifycations. We have
proposed an adhesive patch test to assess the numerical characteristics
of the scheme including the spatial and temporal convergence. The
multi-scale cohesive solution has been verifieded by comparing it with
a direct numerical simulation performed at a single scale. The order of
convergence at both scales is presented. To reduce the computational
effort in the multi-scale simulations, we have proposed a spatial
adaptivity criterion, which relies on the adaptive introduction and
extraction of an adhesive unit cell in the path of the crack. The
multi-scale solution is compared to the analytical solution for the
mode I (DCB) and mode II failure problems.
The authors gratefully acknowledge the support from the
CMMI division of the NSF under the grant number 0527965.
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© 2010 Notre Dame and Dr.