Abstract: Abstract In small arterial vessels, fluid mechanics involving linear viscous fluid does not reproduce experimental results that correspond to non-parabolic profiles of velocity across the vessel diameter. In this paper, an alternative approach is pursued introducing long-range interactions that describe the interactions of non-adjacent fluid volume elements due to the presence of red blood cells and other dispersed cells in plasma. These non-local forces are defined as linearly dependent on the product of the volumes of the considered elements and on their relative velocity. Moreover, as the distance between two volume elements increases, the non-local forces decay with a material distance-decaying function. Assuming that decaying function belongs to a power-law functional class of real order, a fractional operator of the relative velocity appears in the resulting governing equation. It is shown that the mesoscale approach involving Hagen-Poiseuille law is able to reproduce experimentally measured profiles of velocity with a great accuracy. Additionally as the dimension of the vessel increases, non-local forces become negligible and the proposed model reverts to the classical Hagen-Poiseuille model. PubDate: 2019-10-01
Abstract: Abstract This paper reviews several existing peridynamic models for frictional contact (previously documented only in the gray literature), and extends them to remedy various shortcomings. In particular, we introduce a state-based nonlocal friction formulation that corrects for loss of angular momentum balance and objectivity in a widely used frictional extension of short-range contact forces. We demonstrate the properties of various peridynamic contact models by applying them in finite element and meshfree peridynamic analyses of benchmark problems and an impact/penetration test. PubDate: 2019-10-01
Abstract: Abstract A mechanical model is introduced for predicting the initiation and evolution of complex fracture patterns without the need for a damage variable or law. The model, a continuum variant of Newton’s second law, uses integral rather than partial differential operators where the region of integration is over finite domain. The force interaction is derived from a novel nonconvex strain energy density function, resulting in a nonmonotonic material model. The resulting equation of motion is proved to be mathematically well-posed. The model has the capacity to simulate nucleation and growth of multiple, mutually interacting dynamic fractures. In the limit of zero region of integration, the model reproduces the classic Griffith model of brittle fracture. The simplicity of the formulation avoids the need for supplemental kinetic relations that dictate crack growth or the need for an explicit damage evolution law. PubDate: 2019-10-01
Abstract: Abstract The primary mechanism of damage evolution of brittle materials in peridynamic theory is based on individual pair-wise bond strain. A critical value for bond strain is derived based on linear elastic fracture mechanic theory. It is a function of the horizon, the radius of non-locality of the material. The horizon must be larger than the material point spacing but not too large which would significantly slow the simulation. Typical sizes used for the horizon range between three and six times the material point spacing, if a constant multiple is used it could be said that the critical strain is a function of the model’s material point grid resolution. This critical strain function ensures that the fracture toughness is independent of the horizon and grid resolution. This works well when modeled materials have pre-existing cracks. However, since the peridynamic fracture toughness manifests as a critical strain it imposes an artificial strength onto the modeled material that may not represent the real strength of the material. When the material strength is less than the model’s critical strain certain flaw insertion methods can be used to capture the strength behavior. However, if the material strength is larger than the critical strain then the flaw size that represents that strength is too small to model. To address this discrepancy, multi-resolution models have been previously used, having differently sized horizons, such that the region that is expected to nucleate failure is represented correctly while the bulk material is modeled with a coarser grid and larger horizon. Such a multiscale approach could be designed from the beginning of the simulation or exhibit adaptive refinement during crack propagation. Such multiscale approaches add significant complexity to the simulation framework and fundamental model descriptions. This paper introduces a method that is able to decouple the model strength from the horizon and grid resolution by using a refinement overlay technique. This overlay is virtual and does not change the explicit material resolution as adaptive techniques do. PubDate: 2019-10-01
Abstract: Abstract A basic problem of micromechanics is analysis of one inclusion in the infinite matrix subjected to a homogeneous remote loading. A heterogeneous medium with the bond-based peridynamic properties (see Silling, J Mech Phys Solids 48:175–209, 2000) of constituents is considered. At first, volumetric boundary conditions are set up at the external boundary of a final domain obtained from the original infinite domain by truncation. One also considers a periodization approach (for a dilute concentration of inclusion). At last, a group of the iteration methods is considered where the displacement field is decomposed as linear displacement corresponding to the homogeneous loading of the infinite homogeneous medium and a perturbation field introduced by one inclusion. This perturbation field is found by the iteration method for entirely infinite sample with an initial approximation given by a driving term which has a compact support. The form of the mentioned solutions is adapted for subsequent incorporation into one or another micromechanical approach for peristatic composite materials. The methods are demonstrated by numerical examples for 1D case. A convergence of numerical results for the peristatic composite bar to the corresponding exact evaluation for the local elastic theory is shown. PubDate: 2019-10-01
Abstract: Abstract In this work, we introduce convolutional neural networks designed to predict and analyze damage patterns on a disk resulting from molecular dynamic (MD) collision simulations. The simulations under consideration are specifically designed to produce cracks on the disk and, accordingly, numerical methods which require partial derivative information, such as finite element analysis, are not applicable. These simulations can, however, be carried out using peridynamics, a nonlocal extension of classical continuum mechanics based on integral equations which overcome the difficulties in modeling deformation discontinuities. Although this nonlocal extension provides a highly accurate model for the MD simulations, the computational complexity and corresponding run times increase greatly as the simulations grow larger. We propose the use of neural network approximations to complement peridynamic simulations by providing quick estimates which maintain much of the accuracy of the full simulations while reducing simulation times by a factor of 1500. We propose two distinct convolutional neural networks: one trained to perform the forward problem of predicting the damage pattern on a disk provided the location of a colliding object’s impact, and another trained to solve the inverse problem of identifying the collision location, angle, velocity, and size given the resulting damage pattern. PubDate: 2019-10-01
Abstract: Abstract This work introduces the nonlocal integral theory into the finite-deformation viscoplastic Gurson-type model for glassy polymer with plastic softening under strain rate loading. In order to improve computational efficiency and numerical robustness, nonlocal averaging on the void volume fraction is coupled with the update of stress and strain using ABAQUS dynamic subtoutines. From two numerical examples using nonlocal FEA on the unnotched square plate and the dumbbell-shaped specimen under tension respectively, the introduced length scale is demonstrated to regularize the dynamic initial-value problem well that remains hyperbolic by predicting the nonlocal void growth, the objective load responses, and the contours of localization bands accurately. In addition, the relationship between the width of localization band and the length scale is also studied. PubDate: 2019-04-01
Abstract: Abstract This study combines atomic force microscope (AFM) nanoindentation tests and peridynamic (PD) simulations to extract the elastic moduli of polystyrene (PS) films with varying thicknesses. AFM nanoindentation tests are applied to relatively hard PS thin films deposited on soft polymer (polydimethylsiloxane (PDMS)) substrates. Linear force versus deformation response was observed in nanoindentation experiments and numerical simulations since the soft PDMS substrate under the stiff PS films allowed bending of thin PS films instead of penetration of AFM tip towards the PS films. The elastic moduli of PS thin films are found to be increasing with increasing film thickness. The validity of both the simulation and experimental results is established by comparison against those previously published in the literature. PubDate: 2019-04-01
Abstract: Abstract Typical implementations of peridynamics use a constant or tapered micromodulus (or influence) function, the choice of which has been shown to have a large impact on the dispersion relation. In this work, a method for computing micromodulus function values at discretized points within a node’s horizon is presented for linearized peridynamics. The technique involves constructing a system of equations representing the desired dispersion relation and solving for the micromodulus function coefficients at discretized node locations. Both 1D and 2D formulations are presented. A straightforward implementation of the method results in negative coefficients, which improve wave propagation accuracy, but results in unstable solutions of fracture problems using a bond-breakage scheme. Two methods for addressing this issue are discussed: A hybrid method that uses a constant micromodulus function after damage has occurred at a node, and a constrained solution that results in only positive coefficients. The dispersion properties of the method are examined in detail, including the numerical anisotropy in 2D. Finally, results for wave propagation in 1D and 2D, static fracture, and dynamic fracture are given. PubDate: 2019-04-01
Abstract: Abstract Peridynamics (PD), a non-local generalization of classical continuum mechanics (CCM) allowing for discontinuous displacement fields, provides an attractive framework for the modeling and simulation of fracture mechanics applications. However, PD introduces new model parameters, such as the so-called horizon parameter. The length scale of the horizon is a priori unknown and need to be identified. Moreover, the treatment of the boundary conditions is also problematic due to the non-local nature of PD models. It has thus become crucial to calibrate the new PD parameters and assess the model adequacy based on experimental observations. The objective of the present paper is to review and catalog available experimental setups that have been used to date for the calibration and validation of peridynamics. We have identified and analyzed a total of 39 publications that compare PD-based simulation results with experimental data. In particular, we have systematically reported, whenever possible, either the relative error or the R-squared coefficient. The best correlations were obtained in the case of experiments involving aluminum and steel materials. Experiments based on imaging techniques were also considered. However, images provide large amounts of information and their comparison with simulations is in that case far from trivial. A total of six publications have been identified and summarized that introduce numerical techniques for extracting additional attributes from peridynamics simulations in order to facilitate the comparison against image-based data. PubDate: 2019-04-01
Abstract: Abstract This work presents a novel procedure to calibrate bond micromoduli in a discrete bond-based peridynamics setting. Conventionally, an analytical expression has been used based on strain energy equivalence. In the present work, the micromoduli for individual bonds in a neighborhood are calibrated by equating the effective peridynamic stiffness to the reference material stiffness and posed as a set of linear system of equations. The solution to this is found by a least squares approach which aims to reduce the error in the components of the stiffness matrix. Results for an isotropic case in a local neighborhood of particles are shown to demonstrate the feasibility of this approach. PubDate: 2019-04-01
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