Tarek I. Zohdi

An introduction to computational micromechanics

Some Basics of the Mechanics of Solid Continua.- Fundamental Weak Formulations.- Fundamental Micro-Macro Concepts.- A Basic Finite Element Implementation.- Computational/Statistical Testing Methods.- Various Extensions and Further Interpretations of Partitioning.- Domain Decomposition Analogies and Extensions.- Nonconvex-Nonderivative Genetic Material Design.- Modeling Coupled Multifield Processes.- Closing Comments.

Analysis and adaptive modeling of highly heterogeneous elastic structures

In this paper, we develop a theory and methodology for obtaining approximate solutions to boundary value problems describing the deformation of highly heterogeneous linearly-elastic structures. The method, which represents a significant departure from traditional homogenization methods, provides a systematic and rigorous approach towards resolving the effects of microstructure of different scales on the macroscopic response of complex heterogeneous structures. An early variant of this method was first introduced in [IO]. The method, referred to as HDPM (Homogenized Dirichlet Projection Method), proceeds by first solving an auxiliary homogenized boundary value problem, describing the deformation of a structure with the same exterior geometry, but with a selected set of uniform material properties. The adequacy of this homogenized solution is then determined using an a posteriori error estimate that provides a measure of the error of the homogenized solution in subdomains of the structure compared to the solution of the fine-scale heterogeneous problem, without specific knowledge of this fine-scale solution. In those subdomains where the homogenized solution is deemed acceptable, it is retained. However, in subdomains where the homogenized solution is inadequate, as is determined when the estimated error exceeds a preset tolerance, a local boundary value problem is constructed by projecting the homogenized displacements onto a partition designed to isolate these subdomains. These local boundary value problems are then solved in those subdomains where the homogenized solution is inaccurate using the exact microstructure with the approximate local boundary conditions. A posteriori error estimates are made to ascertain the quality of the resulting solution. If the quality of the solution remains inadequate, above a preset error tolerance, a two stage adaptive procedure is implemented. Stage-I (‘material adaptivity’) corresponds to modifying the homogenized structure’s material properties. If, after Stage I, the solution quality is still inadequate, the subdomains of local solution are enlarged, thereby modeling in greater detail the actual microstructure, and the local solution process is. repeated (Stage-II, ‘subdomain unrefinement’). The main feature of this method is that only in subdomains where the error in the usual homogenized solution is above a preset tolerance is the microstructure taken into account. The cost of this method is shown to be orders of magnitude cheaper than direct huge-scale computational simulations of micromechanical events. The results of several numerical experiments are provided to demonstrate the method and verify theoretical estimates. In this work, we present a method for the solution of boundary value problems modeling the deformation of structures composed of highly heterogeneous linearly-elastic materials. We concentrate on materials formed by a homogeneous matrix embedded with particulate matter of different properties (Fig. 1). However, the results presented are not limited to materials of this type, and can, in theory, be used for materials with virtually any microstructure. A straightforward calculation reveals that an accurate numerical approximation of fine-scale solutions of

Coffee-ring effect-based three dimensional patterning of micro/nanoparticle assembly with a single droplet.

We develop a novel patterning technique to create 3D patterns of micro and nanoparticle assembly via evaporative self-assembly based on the coffee-ring effect of an evaporating suspension. The principle of the technique is analyzed theoretically by the scaling analysis of main parameters of the process and the scaling effect, the effect of the volume, the concentration of the suspension, and the effect of surface treatment on the patterning are studied. On the basis of the presented technique, we demonstrate that the patterns of 3D assembly of various sizes of microparticles (Silica), metal oxide nanoparticles (TiO(2), ZnO), and metallic nanoparticles (Ag) can be successfully generated by low-concentrated particle suspension (1.25-5 wt %) without additional sintering steps, and we also show the geometries of the patterns can be finely controlled by adjusting the parameters of the process.

Genetic design of solids possessing a random–particulate microstructure

There exists a variety of difficulties in the computational design of macroscopic solid material properties formed by doping a homogeneous base matrix material with randomly distributed particles having different properties. Three primary problems are the wide array of free microdesign variables, such as particle topology, property phase contrasts and volume fraction, which render the associated objective functions to be highly non–convex; that the associated objective functions are not differentiable with respect to design variables, primarily due to prescribed constraints, such as prespecified restrictions on the microscale stress–field behaviour; and the effective responses of various finite–sized samples, of equal volume but of different random particle distributions, exhibit mutual fluctuations, leading to amplified noise in optimization strategies where objective function sensitivities or comparisons are needed. The focus of this paper is the development of a statistical genetic algorithm which can handle difficulties due to non–convexity, lack of regularity and size effects. Theoretical properties of the overall approach are investigated. Semi–analytical and large–scale numerical examples, involving finite–element type discretizations, are given to illustrate its practical application.

Computational micro-macro material testing

A key to the success of many modern structural components is the tailored behavior of the material. A relatively inexpensive way to obtain macroscopically desired responses is to enhance a base matrix properties by the addition of microscopic matter, i.e. to manipulate the microstructure. Accordingly, in many modern engineering designs, materials with highly complex microstructures are now in use. The macroscopic characteristics of modified base materials are the aggregate response of an assemblage of different “pure” components, for example several particles or fibers suspended in a binding matrix material (Figure 1.1). In the construction of such materials, the basic philosophy is to select material combinations to produce aggregate responses possessing desirable properties from each component. For example, in structural engineering applications, the classical choice is a harder particulate phase that serves as a stiffening agent for the base matrix material. Such inhomogeneities are encountered in metal matrix composites, concrete, etc. A variety of materials are characterized by particulate inhomogeneities as shown in Figures 1.2 and 1.3.

Constrained inverse formulations in random material design

The focus of this work is on the development of a computational strategy to design materials composed of randomly dispersed particulates suspended in a homogeneous binding matrix. The design objectives are to deliver prescribed macroscopic effective responses while simultaneously obeying constraints that reflect the distortion of the microscale stress fields, as well as the likelihood for fatigue damage. A nonderivative statistical genetic algorithm is developed which can handle difficulties in designing materials with random particulate microstructure due to objective function nonconvexity and lack of objective function regularity. Theoretical aspects are investigated and three-dimensional numerical examples are given.

A method of substructuring large-scale computational micromechanical problems

In this work, a method is developed to decompose or substructure large-scale micromechanical simulations into a set of computationally smaller problems. In the approach the global domain is partitioned into nonoverlapping subdomains. On the interior subdomain partitions an approximate globally kinematically admissible solution is projected. This allows the subdomains to be mutually decoupled, and therefore separately solvable. The subdomain boundary value problems are solved with the exact microstructural representation contained within their respective boundaries, but with approximate displacement boundary data. The resulting microstructural solution is the assembly of the subdomain solutions, each restricted to its corresponding subdomain. The approximate solution is far more inexpensive to compute than the direct problem. A posteriori error bounds are developed to quantify the quality of the approximate solution. Numerical simulations are presented to illustrate the essential concepts.

A computational study of interfacial debonding damage in fibrous composite materials

In this paper the effect of finite interface strength, and possible debonding, on the macroscopic response of a sample of fiber-reinforced composite material is computationally investigated via the finite element method. The sample consists of several fibers embedded in a homogeneous matrix, aligned in the longitudinal direction, and randomly distributed in the transverse direction. Plane strain conditions are enforced. Both the matrix and the fibers are assumed to behave in a linearly elastic manner. The approach is to employ unilateral constraints to model interface strength limits. However, because the debonded surfaces are unknown a priori, and depend on the internal fields, the originally linear elastic problem becomes nonlinear, and hence it must be solved in an iterative manner. Accordingly, a nested contact algorithm scheme is developed, based on an active set strategy, to efficiently simulate multiple interacting unilateral constraints. The nesting allows the nonlinear problem within a Newton step to be transformed into a sequence of linear sub-problems. Using the algorithm, numerical tests are performed on a widely used Aluminum/Boron fiber-reinforced composite combination to determine the effects of debonding on changes in macroscopic responses as a function of interface strength and loading. It is shown that the amount of debonded surface area correlates perfectly with the loss in the macroscopic stiffness of the material. This result lends credence to damage evolution laws, for homogenized material models, which employ interface separation surface area as the primary internal damage variable.

Fast, High-Throughput Creation of Size-Tunable Micro/Nanoparticle Clusters via Evaporative Self-Assembly in Picoliter-Scale Droplets of Particle Suspension

We report a fast, high-throughput method to create size-tunable micro/nanoparticle clusters via evaporative assembly in picoliter-scale droplets of particle suspension. Mediated by gravity force and surface tension force of a contacting surface, picoliter-scale droplets of the suspension are generated from a nanofabricated printing head. Rapid evaporative self-assembly of the particles on a hydrophobic surface leads to fast clustering of micro/nanoparticles and forms particle clusters of tunable sizes and controlled spacing. The evaporating behavior of the droplet is observed in real-time, and the clustering characteristics of the particles are understood based on the physics of evaporative-assembly. With this method, multiplex printing of various particle clusters with accurate positioning and alignment are demonstrated. Also, size-unifomity of the cluster arrays is thoroughly analyzed by examining the metallic nanoparticle cluster-arrays based on surface-enhanced Raman spectroscopy (SERS).

A direct particle-based computational framework for electrically enhanced thermo-mechanical sintering of powdered materials

As a method for bonding powdered materials, sintering has distinct advantages, such as the production of a near final-shape of the desired product, without the need for significant post-processing. However, sintering has certain deficiencies, such as incomplete or weak bonding. Research is ongoing to improve the process. One approach to improve sintering processes of powdered materials is via electrically enhanced bonding, whereby electricity is pumped through the material, while it is compressed in a press, in order to induce Joule-heating. This paper develops a computationally based model for the direct simulation of electrically enhanced sintering of powdered materials using particle-based methods. The overall approach is to construct three coupled sub-models which primarily involve: (a) particle-to-particle mechanical contact, (b) particle-to-particle thermal exchange and (c) particle-to-particle electrical current flow. These physical processes are strongly coupled, since the dynamics dictates which particles are in contact and the contacts determine the electrical flow. The flow of electricity controls the Joule-heating and the induced thermal fields, which soften the material, leading to enhanced particle binding. The strong multiphysics-coupled sub-models are solved iteratively within each time-step using a recursive staggering scheme, which employs temporal adaptivity to control the error. If the process does not converge (to within an error tolerance) within a preset number of iterations, the time-step is adapted (reduced) by utilizing an estimate of the spectral radius of the coupled system. The modular approach allows for easy replacement of submodels, if needed. Numerical examples are provided to illustrate the model and numerical solution scheme.