Coupled Multi-scale
Cohesive Modeling of Failure in
Heterogeneous Adhesives
M.G. Kulkarni1, K. Matous2
and P.H. Geubelle1
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.
Abstract
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.
Conclusions
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.
Acknowledgment
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.
Karel Matous