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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 83
PROCEEDINGS OF THE EIGHTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY
Edited by: B.H.V. Topping, G. Montero and R. Montenegro
Paper 265

Solution of an Elastostatic Problem with Imperfect Bonding Using a Two Scale Finite Element Method

G. Mejak

University of Ljubljana, Slovenia

Full Bibliographic Reference for this paper
G. Mejak, "Solution of an Elastostatic Problem with Imperfect Bonding Using a Two Scale Finite Element Method", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Proceedings of the Eighth International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 265, 2006. doi:10.4203/ccp.83.265
Keywords: composite material, imperfect bonding, fracture, homogenization, multi scale computation, large scale quadratic programming.

Summary
Many modern materials have two length scales, a macro scale which is proportional to the physical size of the object and a micro scale which is proportional to the fine structure of the material. For linear problems with a periodic or statistically homogeneous structure homogenization theory gives the effective material properties in a closed form. In a general case however homogenization theory gives only existing results and thus numerical methods must be used. Since discretization must capture the small scale, direct application of standard numerical methods is ineffective. Thus to solve the problem several alternative methods were developed, one of them being the two scale finite element method [1]. In the present work we extend this method to a nonlinear problem of imperfect bonding.

Composite material made of two phases is considered. It is assumed that the boundaries between the phases are piecewise smooth and are contained strictly in the domain of the material. One phase then represents the matrix and the other inclusions. Both phases are made using a linear elastic material. Then the stress is given by the generalized Hooke's law with the elasticity tensor being a piecewise constant. Equilibrium equations are given as stationary equations of the energy functional which is the sum of the elastic and interfacial bonding energy. Interfacial energy has a density which is a function of the displacement jumps across the interface. From among several possible interface models a rate independent Needelman model [2] is applied. Evolution of the debonding state of the interface is governed by the finite interfacial bond strength model [3]. In the model the bonding is perfect until the interfacial stress exceeds the interfacial bond strength. At that moment the interface debonds and remains debonded afterwards. At the given debonding the state variation of the energy is restricted to the admissible set which is given by the prescribed boundary conditions and the non penetration conditions. It is presence of the the non penetration conditions and nonlinear interfacial energy that makes the problem nonlinear.

As the number of inclusions is large a direct application of the finite element method (FEM) is from the computational point of view very expensive. For example, in a two dimensional case with 256 inclusions with a periodic structure and with 32 nodes along each inclusion boundary, direct application of the FEM results in a sequence of the quadratic programming problems with 345,088 variables and 16,384 constraints. This is a rather large problem. However, noting that the problem has a cell structure the size of the problem is considerable reduced by applying the two scale finite element method. The reduced problem has 40,962 variables and the same number of constraints. This is a significant reduction of the problem which permits the solution of the problem even on a desktop computer. The method can be interpreted as a static condensation method where the finite element nodes that do not lay along the interface are eliminated. The minimization problem is thus reduced to the nonlinear programming problem with variables only along the interfaces and cell boundaries. The resulting mid size nonlinear programming problem is solved by the successive quadratic programming approach. The quadratic programming problems are solved using a working set method. In particular, we implemented a QPA module of the GALAHAD package. The choice of the working set method instead of the interior point method is done as at each iteration step a good estimate of the optimal active set is known from the previous iteration step.

The theoretical model has several material parameters, elasticity constants of the matrix and inclusions as well as parameters of the interfacial energy. Initial elastic response and the final totally debonding mode are determined by elastic constants while the shape of the strain stress curve depends upon the parameters of the interfacial energy. Geometry and the number of inclusions also affects the material response. It is observed that the material with a larger number of inclusions that is a composite with a finer mixture of phases, is less affected by the interfacial failure as a material with fewer inclusions. For the interfacial energy with a very narrow range where interfacial stresses sharply fall after debonding the strain stress curve zigzaggs. On the other hand it is a well shaped almost smooth curve if the interfacial energy has a long range.

Using this state of the art approach a problem with important practical applications is solved. It is shown that the two scale finite element method can also be applied to nonlinear problems.

References
1
G. Mejak, "Direct numerical computation of effective material properties", in "Multi-physics and multi-scale computer models in nonlinear analysis and optimal design of engineering structures under extreme conditions", A. Ibrahimbegovi and B. Brank (Editors), University of Ljubljana, FGG, Ljubljana, 571-574, 2004.
2
A. Needelman, "A continuum model for void nucleation by inclusion debonding", Journal of Applied Mechanics, 54, 525-531, 1987.
3
S.M. Arnold and B.A. Bednarcyk, "A new local debonding model with application to the transverse tensile and creep behaviour of continuously reinforced titanium composites", NASA/TM 2000-210029, 2000.

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