[Show abstract][Hide abstract] ABSTRACT: Many important biomaterials are composed of multiple layers of networked fibers. While there is a growing interest in modeling and simulation of the mechanical response of these biomaterials, a theoretical foundation for such simulations has yet to be firmly established. Moreover, correctly identifying and matching key geometric features is a critically important first step for performing reliable mechanical simulations. The present work addresses these issues in two ways. First, using methods of geometric probability we develop theoretical estimates for the mean linear and areal fiber intersection densities for two-dimensional fibrous networks. These densities are expressed in terms of the fiber density and the orientation distribution function, both of which are relatively easy-to-measure properties. Secondly, we develop a random walk algorithm for geometric simulation of two-dimensional fibrous networks which can accurately reproduce the prescribed fiber density and orientation distribution function. Furthermore, the linear and areal fiber intersection densities obtained with the algorithm are in agreement with the theoretical estimates. Both theoretical and computational results are compared with those obtained by post-processing of SEM images of actual scaffolds. These comparisons reveal difficulties inherent to resolving fine details of multilayered fibrous networks. The methods provided herein can provide a rational means to define and generate key geometric features from experimentally measured or prescribed scaffold structural data.
Full-text · Article · Oct 2014 · Acta Biomaterialia
[Show abstract][Hide abstract] ABSTRACT: Interactions in linear elastic solids containing inhomogeneities are examined using integral equations. Direct and reflected interactions are identified. Direct interactions occur simply because elastic fields emitted by inhomogeneities affect each other. Reflected interactions occur because elastic fields emitted by inhomogeneities are reflected by the specimen boundary back to the individual inhomogeneities. It is shown that the reflected interactions are of critical importance to analysis of representative volume elements. Further, the reflected interactions are expressed in simple terms, so that one can obtain explicit approximate expressions for the effective stiffness tensor for linear elastic solids containing ellipsoidal and non-ellipsoidal inhomogeneities. For ellipsoidal inhomogeneities, the new approximation is closely related to that of Mori and Tanaka. In general, the new approximation can be used to recover Ponte-Castañeda-Willis' and Kanaun-Levin's approximations. Connections with Maxwell's approximation are established.
Full-text · Article · Jul 2014 · Journal of the Mechanics and Physics of Solids
[Show abstract][Hide abstract] ABSTRACT: This paper presents three new coupling methods for interior penalty discontinuous Galerkin finite element methods and boundary element methods. The new methods allow one to use discontinuous basis functions on the interface between the subdomains represented by the finite element and boundary element methods. This feature is particularly important when discontinuous Galerkin finite element methods are used. Error and stability analysis is presented for one of the methods. Numerical examples suggest that all three methods exhibit very similar convergence properties, consistent with available theoretical results.
No preview · Article · Jan 2012 · SIAM Journal on Scientific Computing
[Show abstract][Hide abstract] ABSTRACT: It is demonstrated that for an isolated Mode I planar crack embedded in an infinite body, the stress intensity factor along the crack front is a function independent of the elastic constants.
No preview · Article · Dec 2011 · International Journal of Fracture
[Show abstract][Hide abstract] ABSTRACT: Two closely-related fictitious domain methods for solving problems involving multiple interfaces are introduced. Like other fictitious domain methods, the proposed methods simplify the task of finite element mesh generation and provide access to solvers that can take advantage of uniform structured grids. The proposed methods do not involve the Lagrange multipliers, which makes them quite different from existing fictitious domain methods. This difference leads to an advantageous form of the inf–sup condition, and allows one to avoid time-consuming integration over curvilinear surfaces. In principle, the proposed methods have the same rate of convergence as existing fictitious domain methods. Nevertheless it is shown that, at the cost of introducing additional unknowns, one can improve the quality of the solution near the interfaces. The methods are presented using a two-dimensional model problem formulated in the context of linearized theory of elasticity. The model problem is sufficient for presenting method details and mathematical foundations. Although the model problem is formulated in two dimensions and involves only one interface, there are no apparent conceptual difficulties to extending the methods to three dimensions and multiple interfaces. Further, it is possible to extend the methods to nonlinear problems involving multiple interfaces.
No preview · Article · Nov 2011 · Mathematical Models and Methods in Applied Sciences
[Show abstract][Hide abstract] ABSTRACT: Self-assembly of a binary monolayer of charged particles is modeled using molecular dynamics and statistical mechanics. The equilibrium phase diagram for the system has three distinct phases: an ionic crystal; a geometrically ordered crystal with disordered charges; and a fluid. We show that self-assembly occurs near the phase transition between the ionic crystal and the fluid, and that the rate of ordering is sensitive to the applied pressure. By assuming an Arrhenius form for the rate of ordering, an optimality condition for the temperature and pressure is derived that maximizes the rate. Using the Clausius-Clapeyron equation, the optimal point on the phase boundary is expressed in terms of the thermodynamic changes in state variables across the boundary. The predicted optimal temperature and pressure conditions are in good agreement with numerical simulations and result in self-organization rates five times that of a simulation without applied pressure.
Full-text · Article · Oct 2011 · The Journal of Chemical Physics
[Show abstract][Hide abstract] ABSTRACT: Reduced-dimensionality, coarse-grained models are commonly employed to describe the structure and dynamics of large molecular systems. In those models, the dynamics is often described by Langevin equations of motion with phenomenological parameters. This paper presents a rigorous coarse-graining method for the dynamics of linear systems. In this method, as usual, the conformational space of the original atomistic system is divided into master and slave degrees of freedom. Under the assumption that the characteristic timescales of the masters are slower than those of the slaves, the method results in Langevin-type equations of motion governed by an effective potential of mean force. In addition, coarse-graining introduces hydrodynamic-like coupling among the masters as well as non-trivial inertial effects. Application of our method to the long-timescale part of the relaxation spectra of proteins shows that such dynamic coupling is essential for reproducing their relaxation rates and modes.
No preview · Article · Aug 2011 · The Journal of Chemical Physics
[Show abstract][Hide abstract] ABSTRACT: Thermodynamic analysis of brittle fracture specimens near the threshold developed by Rice (Thermodynamics of quasi-static
growth of Griffith cracks, J Mech Phys Solid 26:61–78, 1978) is extended to specimens undergoing microstructural changes.
The proposed extension gives rise to a generalization of the threshold concept that mirrors the way the resistance curve generalizes
the fracture toughness. In the absence of experimental data, the resistance curve near the threshold is constructed using
a basic lattice model.
KeywordsSub-critical crack growth–Threshold–Resistance curve
No preview · Article · Feb 2011 · International Journal of Fracture
[Show abstract][Hide abstract] ABSTRACT: This study is concerned with extending the use of boundary algebraic equations (BAEs) to problems involving irregular rather than regular lattices. Such an extension would be indispensable for solving multiscale problems defined on irregular lattices, as BAEs provide seamless bridging between discrete and continuum models. BAEs share many features with boundary integral equations and are particularly effective for solving problems involving infinite domains. However, it is shown that, BAEs for irregular lattices containing certain terms may require the same amount of computational effort as the original problem for which the equations are formulated. In this paper, we formulate a BAE for a model problem and expose the fundamental obstacle that prevents us from using that BAE as an effective tool. It is shown that, in contrast to regular lattices, BAEs for irregular lattices require a statistical rather than deterministic treatment. This is a very interesting direction for future research.
[Show abstract][Hide abstract] ABSTRACT: In this work, we are concerned that transmission of various boundary conditions through irregular lattices. The boundary conditions are parameterized using trigonometric Fourier series, and it is shown that, under certain conditions, transmission through irregular lattices can be well approximated by that through classical continuum. It is determined that such transmission must involve the wavelength of at least 12 lattice spacings; for smaller wavelength classical continuum approximations become increasingly inaccurate.
[Show abstract][Hide abstract] ABSTRACT: Recently, self-assembly has emerged as an efficient method to create scale-independent regular structures. A problem commonly encountered in experiment is little or no control over the kinetics of structure formation. In this work, an optimal choice of macroscopic control parameters is identified. Using Molecular Dynamics simulations, kinetics of self-assembly is evaluated in systems proposed by Whitesides et. al.. An idealized model is put forward to explain the results observed in simulations.
[Show abstract][Hide abstract] ABSTRACT: A simple macroscopic model is used to determine an optimal annealing schedule for self-assembly of binary monolayers of spherical particles. The model assumes that a single rate-controlling mechanism is responsible for the formation of spatially ordered structures and that its rate follows an Arrhenius form. The optimal schedule is derived in an analytical form using classical optimization methods. Molecular dynamics simulations of the self-assembly demonstrate that the proposed schedule outperforms other schedules commonly used for simulated annealing.
Full-text · Article · May 2009 · New Journal of Physics
[Show abstract][Hide abstract] ABSTRACT: In recent years, elastic network models (ENM) have been widely used to describe low-frequency collective motions in proteins. These models are often validated and calibrated by fitting mean-square atomic displacements estimated from x-ray crystallography (B-factors). We show that a proper calibration procedure must account for the rigid-body motion and constraints imposed by the crystalline environment on the protein. These fundamental aspects of protein dynamics in crystals are often ignored in currently used ENMs, leading to potentially erroneous network parameters. Here we develop an ENM that properly takes the rigid-body motion and crystalline constraints into account. Its application to the crystallographic B-factors reveals that they are dominated by rigid-body motion and thus are poorly suited for the calibration of models for internal protein dynamics. Furthermore, the translation libration screw (TLS) model that treats proteins as rigid bodies is considerably more successful in interpreting the experimental B-factors than ENMs. This conclusion is reached on the basis of a comparative study of various models of protein dynamics. To evaluate their performance, we used a data set of 330 protein structures that combined the sets previously used in the literature to test and validate different models. We further propose an extended TLS model that treats the bulk of the protein as a rigid body while allowing for flexibility of chain ends. This model outperforms other simple models of protein dynamics in interpreting the crystallographic B-factors.
No preview · Article · Feb 2008 · Physical Biology
[Show abstract][Hide abstract] ABSTRACT: Two micromechanical models are used to calculate the statistical distributions of the stress intensity factor of a crack in a polycrystalline plate containing stiff grains and soft grain boundaries. The first is a finite-element method based Monte Carlo procedure where the microstructure is represented by a Poisson-Voronoi tessellation. The effective elastic moduli of the uncracked plate and the stress intensity factor of the cracked plate are calculated for selected values of the parameters that quantify the level of elastic mismatch between the grains and grain boundaries. It is shown that the stress intensity factor is independent of the expected number of grains, and that it can be estimated using an analytical model involving a long crack whose tip is contained within a circular inhomogeneity surrounded by an infinitely extended homogenized material. The stress intensity factor distributions of this auxiliary problem, obtained using the method of continuously distributed dislocations, are in excellent agreement with those corresponding to the polycrystalline microstructure, and are very sensitive to the position within the inhomogeneity of the crack tip. These results suggest that fracture toughness experiments on polycrystalline plates can be considered experiments on the single grain containing the crack tip, and in turn reflect the a/w effects typical of finite-geometry specimens.
Full-text · Article · Jan 2008 · Journal of Engineering Mechanics
[Show abstract][Hide abstract] ABSTRACT: A method,for analyzing problems involving defects in lattices, is presented. Special attention is paid to problems in which the lattice containing the defect is infinite, and the response in a finite zone adjacent to the defect is nonlinear. It is shown that lattice Green's functions allow one to reduce such problems to algebraic problems whose size is comparable to that of the nonlinear zone. The proposed method is similar to a hybrid finite-boundary element method in which the interior nonlinear region is treated with a finite element method and the exterior linear region is treated with a boundary element method. Method details are explained using an anti-plane deformation model problem involving a cylindrical vacancy.
No preview · Article · Jul 2007 · Journal of Applied Mechanics
[Show abstract][Hide abstract] ABSTRACT: We present generalizations of Hill's classical results concerned with the macroscopic strain and stress measures. Generalizations involve polynomial boundary conditions and polynomial moments of the microscopic fields. It is shown that for higher-order polynomials certain boundary conditions and moments should be excluded from considerations in order to guarantee unique relationships between boundary data and macroscopic measures. Particularly simple relationships are obtained for spherical specimens, for which higher-order macroscopic measures are defined in terms of spherical harmonics. Also it is demonstrated that higher-order macroscopic measures and constitutive equations can be useful in multi-scale analysis of problems formulated in terms of integral equations.
No preview · Article · Jun 2007 · Journal of the Mechanics and Physics of Solids
[Show abstract][Hide abstract] ABSTRACT: Anti-plane deformation of square lattices containing interphases is analyzed. It is assumed that lattices are linear elastic but not necessarily isotropic, whereas interphases exhibit non-linear elastic behavior. It is demonstrated that such problems can be treated effectively using Green’s functions, which allow to eliminate the degrees of freedom outside of the interphase. Illustrative numerical examples focus on the determination of applied stresses leading to lattice instability.
Preview · Article · Aug 2006 · Acta Mechanica Sinica
[Show abstract][Hide abstract] ABSTRACT: We have used kinetic Monte Carlo simulations to study the kinetics of unfolding of cross-linked polymer chains under mechanical loading. As the ends of a chain are pulled apart, the force transmitted by each cross-link increases until it ruptures. The stochastic cross-link rupture process is assumed to be governed by first order kinetics with a rate that depends exponentially on the transmitted force. We have performed random searches to identify optimal cross-link configurations whose unfolding requires a large applied force (measure of strength) and/or large dissipated energy (measure of toughness). We found that such optimal chains always involve cross-links arranged to form parallel strands. The location of those optimal strands generally depends on the loading rate. Optimal chains with a small number of cross-links were found to be almost as strong and tough as optimal chains with a large number of cross-links. Furthermore, optimality of chains with a small number of cross-links can be easily destroyed by adding cross-links at random. The present findings are relevant for the interpretation of single molecule force probe spectroscopy studies of the mechanical unfolding of "load-bearing" proteins, whose native topology often involves parallel strand arrangements similar to the optimal configurations identified in the study.
Full-text · Article · Mar 2005 · Physical Review E
[Show abstract][Hide abstract] ABSTRACT: This paper presents a version of the fast multipole method (FMM) for integral equations describing conduction through three-dimensional periodic heterogeneous media. The proposed method is based on the standard rather than periodic fundamental solution, and therefore it is very close to the original FMM. In deriving the method, particular attention is paid to convergence of arising integral equations and lattice sums. It is shown that convergence can be achieved without introducing artificial compensatory sources or boundary conditions.
Preview · Article · Oct 2004 · Proceedings of The Royal Society A
[Show abstract][Hide abstract] ABSTRACT: Procedures for constructing boundary integral equations equivalent to linear boundary-value problems governed by partial differential equations are well established. In this paper, it is demonstrated how these procedures can be extended to linear boundary-value problems defined on lattices and governed by algebraic (‘difference’) equations. The boundary equations that arise are then themselves algebraic equations. Such ‘boundary algebraic equations’ (BAEs) are derived for fundamental boundary-value problems defined on both perfect lattices and lattices with defects. It is demonstrated that key advantages of representing a continuum boundary-value problem as an equation on the boundary, such as favourable spectral properties and minimal problem size, are preserved in the lattice environment. Certain spectral properties of BAEs are established rigorously, whereas others are supported by numerical experiments.