A recently introduced nonlocal peridynamic theory removes the obstacles present in classical continuum mechanics that limit the prediction of crack initiation and growth in materials. It is also applicable at different length scales. This study presents an alternative approach for the derivation of peridynamic equations of motion based on the principle of virtual work. It also presents solutions for the longitudinal vibration of a bar subjected to an initial stretch, propagation of a pre-existing crack in a plate subjected to velocity boundary conditions, and crack initiation and growth in a plate with a circular cutout. Furthermore, damage growth in composites involves complex and progressive failure modes. Current computational tools are incapable of predicting failure in composite materials mainly due to their mathematical structure. However, the peridynamic theory removes these obstacles by taking into account non-local interactions between material points. Hence, an application of the peridynamic theory to predict how damage propagates in fiber reinforced composite materials subjected to mechanical and thermal loading conditions is presented. Finally, an analysis approach based on a merger of the finite element method and the peridynamic theory is proposed. Its validity is established through qualitative and quantitative comparisons against the test results for a stiffened composite curved panel with a central slot under combined internal pressure and axial tension. The predicted initial and final failure loads, as well as the final failure modes, are in close agreement with the experimental observations. This proposed approach demonstrates the capability of the PD approach to assess the durability of complex composite structures.

This thesis explores the science of solid mechanics via the theory of peridynamics. Peridynamics has several key advantages over the classical theory of elasticity. The most notable of which is the ease with which fractures in the the material are handled. The goal here is to study the two theories and how they relate for problems in which the classical method is known to work well. While it is known that state-based peridynamic models agree with classical elasticity as the horizon radius vanishes, similar results for bond-based models have yet to be developed. In this study, we use numerical simulations to investigate the behavior of bond-based peridynamic models under this limit for a number of cases where analytic solutions of the classical elasticity problem are known. To carry out this study, the integral-based peridynamic model is solved using the finite element method in two dimensions and compared against solutions using the classical approach. / Master of Science

This study concerns the development of the peridynamic strain energy density function for a Neo-Hookean type membrane under equibiaxial, planar and uniaxial loading conditions. The material parameters for each loading case are determined by equating the peridynamic strain energy to those of the classical continuum mechanics. Therefore, the peridynamic equations of motion are derived based on the Neo-Hookean model under the assumption of incompressibility. Numerical results concern the deformation of a membrane without and with a defect in the form of a hole, an inclusion and a crack under equibiaxial, planar and uniaxial loading conditions. As part of the verification process, the peridynamic predictions are compared with those of finite element analysis. For all defect types and loading conditions, the comparisons indicate excellent agreement.

Doctor of Philosophy / Department of Mechanical and Nuclear Engineering / Kevin B. Lease / Xiao J. Xin / Peridynamics is a non-local continuum theory that formulates problems in terms of integration of interactions between the material points. Because the governing equation of motion in the peridynamic theory involves only integrals of displacements, rather than derivatives of displacements, this new theory offers great advantages in dealing with problems that contain discontinuities. Integration of the interaction force plays an important role in the formulation and numerical implementation of the peridynamic theory. In this study two enhanced methods of integration for peridynamics have been developed. In the first method, the continuum is discretized into cubic cells, and different geometric configurations over the cell and the horizon of interaction are categorized in detail. Integration of the peridynamic force over different intersection volumes are calculated accurately using an adaptive trapezoidal integration scheme with a combined relative-absolute error control. Numerical test examples are provided to demonstrate the accuracy of this new adaptive integration method. The bond-based peridynamic constitutive model is used in the calculation but this new method is also applicable to state-based peridynamics. In the second method, an integration method with fixed Gaussian points is employed to accurately calculate the integration of the peridynamic force. The moving least square approximation method is incorporated for interpolating the displacement field from the Gaussian points. A compensation factor is introduced to correct the soft boundary effect on the nodes near the boundaries. This work also uses linear viscous damping to minimize the dynamic effect in the solution process. Numerical results show the accuracy and effectiveness of this Gaussian integration method. Finally current research progress and prospective directions for several topics are discussed.

This study presents peridynamic field equations for mechanical deformation, thermal diffusion, moisture diffusion, electric potential distribution, porous flow and atomic diffusion in either an uncoupled or a coupled manner. It is a nonlocal theory with an internal length parameter. Therefore, it can capture physical phenomenon for the problems which include non-local effects and are not suitable for classical theories. Moreover, governing equations of peridynamics are based on integro-differential equations which permits the determination of the field variable in spite of discontinuities. Inherent with the nonlocal formulations, the imposition of the boundary conditions requires volume constraints. This study also describes the implementation of the essential and natural boundary conditions, and demonstrates the accuracy of their implementation. Solutions coupled field problems concerning plastic deformations, thermomechanics, hygrothermomechanics, hydraulic fracturing, thermal cracking of fuel pellet and electromigration are constructed. Their comparisons with the finite element predictions establish the validity of the PD field equations for coupled field analysis.

Peridynamics is a reformulation of classical continuum mechanics that utilizes integral equations in place of partial differential equations to remove the difficulty in handling discontinuities, such as cracks or interfaces, within a body. Damage is included within the constitutive model; initiation and propagation can occur without resorting to special crack growth criteria necessary in other commonly utilized approaches. Predicting damage and residual strengths of composite materials involves capturing complex, distinct and progressive failure modes. The peridynamic laminate theory correctly predicts the load redistribution in general laminate layups in the presence of complex failure modes through the use of multiple interaction types.This study presents two approaches to obtain the critical peridynamic failure parameters necessary to capture the residual strength of a composite structure. The validity of both approaches is first demonstrated by considering the residual strength of isotropic materials. The peridynamic theory is used to predict the crack growth and final failure load in both a diagonally loaded square plate with a center crack, as well as a four-point shear specimen subjected to asymmetric loading.This study also establishes the validity of each approach by considering composite laminate specimens in which each failure mode is isolated. Finally, the failure loads and final failure modes are predicted in a laminate with various hole diameters subjected to tensile and compressive loads.

Geopolymer is a novel cementitious material which can be a potential alternative to ordinary Portland cement (OPC) for all practical applications. However, until now research on this revolutionary material is limited mainly to experimental studies, which have the limitations in considering the details of the atomic- and meso-scale structure and atomic scale mechanisms that govern the properties at the macro-scale. Most experimental studies on geopolymer have been conducted focusing only on the macroscopic properties and considering it as a single-phase material. However, research has shown that geopolymer is a composite material consisting of geopolymer binder (GB), unreacted source material, and, in the presence of Ca in the source material, calcium silicate hydrate (CSH). Therefore, in this research, a multiscale/multiphysics modeling approach has been taken to understand geopolymer structure and mechanical properties under varying conditions and at different length scales. First, GB was prepared at the atomic scale using molecular dynamics (MD) simulations with varying Si/Al ratios and water contents within the nano voids. The MD simulated geopolymer structure was validated based on comparison with experiments using X-ray pair distribution function (PDF), infra-red (IR) spectra, coordination of atoms, and density. The results indicate that the highest strength occurs at a Si/Al ratio of 2-3 and the presence of molecular water negatively affects the mechanical properties of GB. The loss of strength for GB with increased water content is linked to the diffusion of Na atoms and subsequent weakening of Al tetrahedra. The GB was also subjected to nanoindentation using MD and the effect of indenter size and loading rate was investigated at an atomic scale. A clear correlation between the indenter size and observed hardness of GB was observed which proves indentation size effects (ISE). Realizing the composite nature of geopolymer, the presence of unreacted and secondary phases such as quartz and CSH in geopolymer was also investigated. To do that, the mechanical properties of GB, the secondary phases and their interfaces was first determined from MD simulations. Using the MD generated properties, a meso-scale model of geopolymer composite was prepared in Peridynamics (PD) framework which considered large particles of GB and secondary phases of nanometers in size which cannot be easily modeled in MD. The meso-scale model provides a larger platform to study geopolymer in the presence of large nano-voids and multiple phases. Results from the PD simulations were directly comparable to experimentally observed mechanical properties. Findings of this study can be directly used in future to construct more advanced and sophisticated models of geopolymer and will be instrumental in designing the synthesis condition for geopolymer with superior mechanical properties.

A new experimental method to characterize the mechanical properties of metallic nanowires is introduced. An accurate and fast mechanical characterization of nanowires requires simultaneous imaging and testing of nanowires. However, there exists no practical experimental procedure in the literature that provides a quantitative mechanical analysis and imaging of the nanowire specimens during mechanical testing. In this study, a customized atomic force microscope (AFM) is placed inside a scanning electron microscope (SEM) in order to locate the position of the nanowires. The tip of the atomic force microscope cantilever is utilized to bend and break the nanowires. The nanowires are prepared by electroplating of nickel ions into the nanoscale pores of the alumina membranes. Force versus bending displacement responses of these nanowires are measured experimentally and then compared against those of the finite element analysis and peridynamic simulations to extract their mechanical properties through an inverse approach.The average elastic modulus of nickel nanowires, which are extracted using finite element analysis and peridynamic simulations, varies between 220 GPa and 225 GPa. The elastic modulus of bulk nickel published in the literature is comparable to that of nickel nanowires. This observation agrees well with the previous findings on nanowires stating that the elastic modulus of nanowires with diameters over 100nm is similar to that of bulk counterparts. The average yield stress of nickel nanowires, which are extracted using finite element analysis and peridynamic simulations, is found to be between 3.6 GPa to 4.1 GPa. The average value of yield stress of nickel nanowires with 250nm diameter is significantly higher than that of bulk nickel. Higher yield stress of nickel nanowires observed in this study can be explained by the lower defect density of nickel nanowires when compared to their bulk counterparts.Deviation in the extracted mechanical properties is investigated by analyzing the major sources of uncertainty in the experimental procedure. The effects of the nanowire orientation, the loading position and the nanowire diameter on the mechanical test results are quantified using ANSYS simulations. Among all of these three sources of uncertainty investigated, the nanowire diameter has been found to have the most significant effect on the extracted mechanical properties.

Master of Science / Department of Mechanical and Nuclear Engineering / Daniel V. Swenson / This thesis provides a description of Visual EMU, a graphical user interface for the peridynamic EMU code. The peridynamic model is a fundamental method for computational mechanical analysis that makes no assumption of continuous or small deformation behavior and has no requirement for the concepts of stress and strain. The model does not require spatial derivatives and instead uses integral equations. A force density function, called the pairwise force function, is postulated to act between each pair of infinitesimally small particles if the particles are closer together than some finite distance. A spatial integration process is employed to determine the total force acting upon each particle and a time integration process is employed to track the positions of the particles due to the applied body forces and applied displacements. EMU is a computer code developed by Sandia National Laboratories that implements the peridynamic model. Visual EMU is a pre-processor for the EMU code that allows any user to enter all parameters and visualize the resulting material regions, peridynamic grid, and a preview of resulting nodes. Visual EMU can be used before starting a lengthy solution with potential errors. The language, visual layout, and code design of Visual EMU are described along with two examples and their results.

Peridynamics, presented by Silling in 2000 [1], is a reformulation of the elastic theory from differential equations to integral equations, which are more equipped to handle discontinuities, such as crack initiation and propagation. Because of this, peridynamics is an effective tool to address many of the problems relevant to the aerospace and defense industries. For example, airborne sand particles and raindrops cause local damage to aircraft in flight. This damage manifests itself as radial and subsurface lateral cracking, as well as increased surface roughness. All of these damage morphologies may result in undesired degradation of mechanical and optical properties. This dissertation aims to address the question of how peridynamics (PD) can be used as a tool to help understand impact problems and resultant damage. Three main types of problems will be discussed: (1) modeling of quasi-static nano- and micro-indentation in PD; (2) solid impact experiments and simulations involving glass micro-spheres impacting coated and uncoated advanced ceramics, and sand particles impacting optical glasses; and (3) the implementation of a new, fully three-dimensional hyperelastic material model in state-based PD to simulate nylon bead impact and capture the damage patterns relevant to raindrop impact. In the first portion, a new method for modeling indentation in PD is presented using the principle of viscous damping and automatic convergence checking. In these simulations, depth-controlled indentation is performed by splitting up the total indentation depth into multiple stages, and applying damping at each stage to ensure the system reaches equilibrium before allowing for failure. PD results show good agreement to experimental data, in terms of crack lengths and force-displacement curves. In a chapter about solid particle impact, two studies are presented. In the first, glass spheres with diameters ranging from 200 to 700 um impact multi-spectral zinc sulfide (MS-ZnS) with various coating systems. It was found that samples containing the REP coating had better resistance to damage than those without. This resistance was evident in all three damage metrics used: impact pit diameter, radial crack length, and lateral crack size. Simulations were carried out in bond-based PD, with good agreement to experiments regarding damage metrics and rebound velocity. The second solid particle impact study involved sand particles impacting four different types of optical glasses: BK7, alumino-boro-silicate, fused silica, and Pyrex. First, data from experiments was analyzed, and a multi-variable power law regression was performed to show that sand particle shape plays a significant role in resultant damage. This was confirmed via bond-based PD simulations, with damage quantities agreeing well with experimental values. Finally, the problem of how to model raindrop impact using nylon beads was examined. Due to the large amounts of elastic strain experienced by the nylon beads during impact experiments, it was determined that a hyperelastic material model could be a good fit. Based on elastic theory and classical continuum mechanics, a new, fully three-dimensional Neo-Hookean material model was implemented in nonordinary state-based peridynamics. This model was verified against results and finite element analysis, with very good agreement. Preliminary simulations including damage show good results, consistent with experiments.