Uncertain Loading and Quantifying Maximum Energy Concentration within Composite Structures

We introduce a systematic method for identifying the worst case load among all boundary loads of fixed energy. Here the worst case load is defined to be the one that delivers the largest fraction of input energy to a prescribed subdomain of interest. The worst case load is identified with the first eigenfunction of a suitably defined eigenvalue problem. The first eigenvalue for this problem is the maximum fraction of boundary energy that can be delivered to the subdomain. We compute worst case boundary loads and associated energy contained inside a prescribed subdomain through the numerical solution of the eigenvalue problem. We apply this computational method to bound the worst case load associated with an ensemble of random boundary loads given by a second order random process. Several examples are carried out on heterogeneous structures to illustrate the method

An inverse homogenization design method for stress control in composites

This thesis addresses the problem of optimal design of microstructure in composite materials. The work involves new developments in homogenization theory and numerical analysis. A computational design method for grading the microstructure in composite materials for the control of local stress in the vicinity of stress concentrations is developed. The method is based upon new rigorous multiscale stress criteria connecting the macroscopic or homogenized stress to local stress fluctuations at the scale of the microstructure. These methods are applied to three different types of design problems. The first treats the problem of optimal distribution of fibers with circular cross section inside a long shaft subject to torsion loading. The second treats the same problem but now the shaft cross section is filled with locally layered material. The third one treats the problem of composite design for a flange fixed at one end and loaded at the other end

Optimization of composite structures subject to local stress constraints

An extension of current methodologies is introduced for optimization of graded microstructure subject to local stress criteria. The method is based on new multiscale stress criteria given by macrostress modulation functions. The modulation functions quantify the intensity of local stress fluctuations at the scale of the microstructure due to the imposed macroscopic stress. The methodology is illustrated for long cylindrical shafts reinforced with stiff cylindrical elastic fibers with generators parallel to the shaft. Examples are presented for shaft cross sections that possess reentrant corners typically seen in lap joints and junctions of struts. It is shown that the computational methodology delivers graded fiber microgeometries that provide overall structural rigidity while at the same time tempering the influence of stress concentrations near reentrant corners. © 2006

Multiscale-Spectral GFEM and optimal oversampling

In this work we address the Multiscale Spectral Generalized Finite Element Method (MS-GFEM) developed in Babuška and Lipton (2011). We outline the numerical implementation of this method and present simulations that demonstrate contrast independent exponential convergence of MS-GFEM solutions. We introduce strategies to reduce the computational cost of generating the optimal oversampled local approximating spaces used here. These strategies retain accuracy while reducing the computational work necessary to generate local bases. Motivated by oversampling we develop a nearly optimal local basis based on a partition of unity on the boundary and the associated A-harmonic extensions

Design of fiber reinforced shafts subject to local stress constraints through inverse homogenization: A preliminary study on fiber size effect

A new inverse homogenization procedure is applied to design graded fiber reinforced shafts subject to local stress criteria. The method is based on new multiscale stress criteria given by macrostress modulation functions. The modulation functions quantify the intensity of local stress fluctuations at the scale of the microstructure due to the imposed macroscopic stress. The method is carried out for long cylindrical shafts reinforced with stiff cylindrical elastic fibers with generators parallel to the shaft. Benchmark examples are presented for shaft cross sections that possess reentrant corners typically seen in lap joints and junctions of struts. © 2006 Springer

Optimal design of composite structures for strength and stiffness: An inverse homogenization approach

We introduce a rigorously based numerical method for compliance minimization problems in the presence of pointwise stress constraints. The method is based on new multiscale quantities that measure the amplification of the local stress due to the microstructure. The design method is illustrated for two different kinds of problems. The first identifies suitably graded distributions of fibers inside shaft cross sections that impart sufficient overall stiffness while at the same time adequately control the amplitude of the local stress at each point. The second set of problems are carried out in the context of plane strain. In this study, we recover a novel class of designs made from locally layered media for minimum compliance subject to pointwise stress constraints. The stress-constrained designs place the more compliant material in the neighborhood of stress concentrators associated with abrupt changes in boundary loading and reentrant corners. © Springer-Verlag Berlin Heidelberg 2007

The penetration function and its application to microscale problems

The penetration function measures the effect of the boundary data on the energy of the solution of a second order linear elliptic PDE taken over an interior subdomain. Here the coefficients of the PDE are functions of position and often represent the material properties of non homogeneous media with microstructure. The penetration function is used to assess the accuracy of global-local approaches for recovering local solution features from coarse grained solutions such as those delivered by homogenization theory. © 2008 Springer Science + Business Media B.V

Multi-scale quasistatic damage evolution for polycrystalline materials

We present a new multi-scale model for linking higher order microstructure descriptions to failure initiation and damage propagation in polycrystalline media. The model gives an accurate local field description for predicting damage nucleation at the length scale of the polycrystalline texture. The new method allows the recovery of the local damage microstructure inside domains of microtexture and has the capability to capture the conditions for component failure through the propagation of damage across macroscopic length scales. Computational examples for damage evolutions for different load cases demonstrate the potential of this model. In the simulations component level failure is seen in the form of damage appearing along ligaments with length scales comparable to the structural component. © 2012 Elsevier Ltd. All rights reserved