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.

Interface modeling is a critical aspect in any multi-material system modeling. Even a small change in the interface model may lead to significant changes in material behavior of the microscale, and these changes may transfer up to higher scales influencing the strain and stress fields, and damaging behavior in the macroscale material. This work focuses on the effects of different interface models in nanocomposites composed of carbon nanotubes in polymer matrix materials and their applications as nanocomposite binders in energetic materials. These material systems include materials that span multiple scales from nano to macroscale, and thus require a detailed multiscale analysis. A hierarchical multiscale framework is employed here, where the effective material properties from subscales are obtained by solving the subscale boundary value problem. The information obtained from the subscale simulations are transferred up to higher scales to be used as input properties. A nonlocal continuum mechanics framework known as peridynamics is used to perform the computational simulations. Peridynamics uses integro-differential equations for conservation laws instead of partial differential equations as in the classical continuum mechanics. This makes it possible for peridynamics to inherently account for nonlocal effects such as damage initiation, crack growth, and crack branching without any modifications such as element deletion, adaptive mesh refinement, using enrichment functions and so on, which are commonly used in other numerical methods. Peridynamics is a particle-based method where the particles are allowed to interact with other particles within their horizon which serves as a cut-off distance for forming particle-to-particle bonds and therefore defines the extent of nonlocality. Peridynamics has different formulations regarding the bond interactions. A bond-based peridynamics framework is used here. A verified and validated in-house code is used for the simulations. The simulations for the carbon nanotube and nanofiber-based nanocomposites, and for nanocomposite bonded energetic materials start from the microscale and range up to the macroscale. For only the carbon nanotube-polymer nanocomposites, the interfaces include the CNT-polymer interfaces. For the energetic materials, the interfaces consider the CNT-polymer interfaces in the microscale and the grain-nanocomposite binder interfaces in the mesoscale. Peridynamics, being a nonlocal continuum mechanics method, by default will have nonlocal interfaces. The material systems investigated in this work first use different nonlocal interfaces in peridynamics which consider the bond between two particles at the interface to be connected in series or in parallel. The nonlocal interface model in peridynamics makes it challenging to control the interface properties and leads to fuzzy interfaces, i.e. interfaces of finite thickness. In this work, a local cohesive interface model is implemented in the peridynamics framework. Cohesive zones were originally used for modeling the growth of cracks by introducing cohesive forces that hold the crack surfaces together, thereby removing the stress singularity problem in linear elastic fracture mechanics. The idea of cohesive zones are applied to peridynamics interfaces, which introduces locality into the nonlocal framework. This interface model does not only remove the nonlocality at the peridynamics interfaces, but it leads to a higher fidelity interface model that is controllable by the user. The differences between the nonlocal and local interfaces are studied in detail in different scales and for different material systems. Implementing a local model into a nonlocal framework brings some challenges, namely obtaining and calibrating the cohesive interface properties for the materials used, the numerical problems with material interpenetration in extreme compression, and very small time steps that are required to resolve the material response. Some remedies are proposed for the problems encountered. The cohesive zone model used in this work can have different functional forms in normal and tangential direction to reflect differences in opening mode and frictional sliding behaviors. / Doctor of Philosophy / Multi-material systems have interface regions where a transition from one material to another occurs. How the interface region is modeled can change the response of a material to external loads even if the interface model is slightly different. This work focuses on the effect of different interface models in nanocomposites based on carbon nanotubes and in nanocomposite bonded energetic materials. These material systems include materials that span multiple scales from nano to the macroscale, and thus require a detailed multiscale analysis. Multiscale analysis of a material means analyzing the material at each scale that is involved for the given material system separately and passing relevant information between the scales. A hierarchical multiscale framework is employed here which is based on a bottom-up approach, where the material properties are obtained at the smaller scales and passed up to the larger scales to be used as the input properties. A nonlocal continuum mechanics in the form of peridynamics is used to perform the computational simulations. The nonlocality stems from the fact that the particles can interact not only with their closest neighbors, but with other particles within their horizon, which is the cut-off distance that dictates how far a material particle can make bonds with other particles. The main advantage of peridynamics is to be able to model cracks without any a priori knowledge about crack growth directions or patterns. Peridynamics has different formulations for representing the bond interactions. A bond-based peridynamics framework is used here. A verified and validated in-house code is used for the simulations. The simulations for the carbon nanotube-polymer nanocomposites and nanocomposite bonded energetic materials take place starting from the microscale up to the macroscale. For the carbon nanotube nanocomposite scale, the interfaces include the fiber-matrix interfaces. For the nanocomposite bonded energetic materials, the interfaces considered include the fiber-matrix interfaces in the microscale and the grain-binder interfaces in the mesoscale. Peridynamics, being a nonlocal continuum mechanics method, nominally includes nonlocal interfaces. The material systems investigated in this work first use different nonlocal interfaces in peridynamics which consider the bond between two particles at the interface to be connected in series or in parallel. The nonlocal interface model in peridynamics makes it challenging to control the interface properties and leads to fuzzy, or finite thickness interfaces. A local cohesive interface model is implemented in the peridynamics framework. Cohesive zones are originally used for modeling cracks by introducing cohesive forces that hold the crack surfaces together to remove the stress singularity at the crack in classical linear elastic fracture mechanics. The idea of cohesive zones are applied to peridynamics interfaces which introduces locality into the nonlocal framework. This interface model does not only remove the nonlocality at the peridynamics interfaces, but it leads to a higher fidelity interface model that is controllable by the user. The differences between the nonlocal and local interfaces are studied in detail in different scales and for different material systems. Implementing a local model into a nonlocal framework brings some challenges, namely obtaining and calibrating the cohesive interface properties for the materials used, the numerical problems with material interpenetration in extreme compression, and very small time steps that are required to resolve the material response. Some remedies are proposed to address these issues. The cohesive zone model used in this work have different mathematical models in normal and tangential directions. It is therefore capable of modeling mechanical and thermal problems including frictional heating. The mechanical results obtained by using cohesive interfaces show potential for developing similar local interface models for thermal and electrical conduction allowing for the expanded application of the approach to multiphysics problems in multiscale composite materials.

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.