An adaptive concurrent multilevel computational model of failure in a heterogeneous-material structure has been developed. As used here, "concurrent" is a term of art characterizing a class of structural/material models that (1) incorporate submodels representing material substructures at different spatial scales from macroscopic to microscopic, (2) the equations of the various models are solved simultaneously, and (3) the solutions at the various scales are coupled. The present model applies, more specifically, to a unidirectionalfiber/ matrix composite material structure. The model can be used to simulate and analyze the initiation and growth of damage, starting from microstructural damage in the form of debonding at fiber/matrix interfaces.

During the solution process, three levels of different spatial resolution adaptively evolve to increase the accuracy of solutions by reducing modeling and discretization errors. The top level, denoted level 0, is the level of pure macroscopic analysis by use of a continuum damage model that is solved by a conventional finite-element method. The middle level, denoted level 1, is that of asymptotic homogenizationbased macroscopic-microscopic representative- volume-element (RVE) modeling. The lowest level, denoted level 2, is that of pure micromechanical modeling by use of a Voronoi-cell finite-element model. The micromechanical modeling involves explicit computations of stresses and strains at the microscale, including those associated with debonding at fiber/matrix interfaces.

The continuum damage model homogenizes the damage incurred through initiation and growth of interfacial debonding in a microstructural RVE with nonuniform distribution of fibers. In addition, arbitrary loading conditions can be represented effectively in this model. The continuum damage model is then used in an adaptive concurrent multi-level computational model to analyze the multi-scale evolution of damage. Damage by fibermatrix interface debonding is explicitly modeled over extended microstructural regions at critical locations.

The modeling at level 1 is used to monitor the breakdown of continuum laws and signal the need for microscopic analyses: Physical criteria involving variables at the macroscopic and microstructural RVE levels are used to detect modeling errors and thus to trigger switching from level 0 to level 1 or from level 1 to level 2, as needed. A transition layer is placed between levels 1 and 2 to enable a smooth transition between the spatial scales of those levels.

The adaptation subprocess enables effective domain decomposition, maintaining a balance between computational efficiency and accuracy. Macroscopic analysis by means of the continuum damage model is done for high computational efficiency. Pure micromechanical analysis is computationally exhaustive, and the adaptation optimally reduces the region of micromechanical analysis to a minimum. The Voronoi-cell finite-element model is effectively utilized for efficient micromechanical analysis of extended microstructural regions.

Variants of the Voronoi-cell finite-element model (VCFEM) have been developed for solving different problems:

  • An extended Voronoi-cell model (XVCFEM) is meant to be used to simulate propagation of multiple cohesive cracks in a brittle material. The cracks are represented by a cohesive zone submodel and their incremental directions and growth lengths are determined in terms of the cohesive energy near the crack tips. Extension to the VCFEM is achieved through enhancements in stress functions in an assumed stress hybrid formulation. In addition to polynomial terms, the stress functions include branch functions in conjunction with level set methods, and multi-resolution wavelet functions in the vicinities of crack tips.
  • The (X-VCFEM) can be applied to interfacial debonding with arbitrary matrix cohesive cracking. For describing the onset and growth of damage along the fiber-matrix interface, normal and tangential cohesive zone models are coupled into the VCFEM.
  • Another version of the VCFEM is intended for use in analysis of transient elastodynamics in the time domain. In this version, the inertia field is approximated in terms of stresses so as to satisfy the equilibrium equation a priori. The weak forms of kinematics and traction reciprocity are obtained by minimization of a complementary variational submodel. A stress wave is treated as a local disturbance that propagates through the material, resulting in high stress gradients near the wavefront. Therefore, localization and multi-resolution properties of the wavelet functions are exploited to enhance computational efficiency by enriching the stress functions only locally near the wavefront. The enrichment is carried out adaptively by employing a posteriori local error estimators that determine the required translation and dilation of each wavelet function at each time step.

This work was done by Somnath Ghosh of Ohio State University Research Foundation for the Army Research Laboratory. For more information, download the Technical Support Package (free white paper) at under the Information Sciences category. ARL-0012

This Brief includes a Technical Support Package (TSP).
Multi-Scale Model of Failure in a Composite Material

(reference ARL-0012) is currently available for download from the TSP library.

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This article first appeared in the April, 2007 issue of Defense Tech Briefs Magazine.

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