Computational Physics Group

Coupled Multi-scale Cohesive Modeling of Failure in
Heterogeneous Adhesives

M.G. Kulkarni¹, K. Matous² and P.H. Geubelle<sup>1

1</sup>Department 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 modifycations. 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 verifieded 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.

Download the paper here