scroll identifier for mobile
main-content

This handbook covers all areas of nonlocal continuum mechanics including theoretical aspects,computational procedures, and experimental advances. The multidisciplinary scope of articles that comprise this reference are written by internationally recognized experts in the field and stand as the most-up-to-date, established knowledge base on using nonlocal continuum mechanics to characterize material behavior for advanced composites and nano-materials, as well as for engineering scale structures. The handbook is at once a comprehensive reference for academic researchers and engineers in industry concerned with nonlocal continuum mechanics for materials and structures as well as a supplement for graduate courses on a range of topics.

### 1. Size Effects and Material Length Scales in Nanoindentation for Metals

In nanoindentation experiments at submicron indentation depths, the hardness decreases with the increasing indentation depth. This phenomenon is termed as the indentation size effect. In order to predict the indentation size effect, the classical continuum needs to be enhanced with the strain gradient plasticity theory. The strain gradient plasticity theory provides a nonlocal term in addition to the classical theory. A material length scale parameter is required to be incorporated into the constitutive expression in order to characterize the size effects in different materials. By comparing the model of hardness as a function of the indentation depth with the nanoindentation experimental results, the length scale can be determined. Recent nanoindentation experiments on polycrystalline metals have shown an additional hardening segment in the hardness curves instead of the solely decreasing hardness as a function of the indentation depth. It is believed that the accumulation of dislocations near the grain boundaries during nanoindentation causes the additional increase in hardness. In order to isolate the influence of the grain boundary, bicrystal metals are tested near the grain boundary at different distances. The results show that the hardness increases with the decreasing distance between the indenter and the grain boundary, providing a new type of size effect. The length scales at different distances are determined using the modified model of hardness and the nanoindentation experimental results on bicrystal metals.

### 2. Size Effects During Nanoindentation: Molecular Dynamics Simulation

In this chapter, the molecular dynamics (MD) simulation of nanoindentation experiment is revisited. The MD simulation provides valuable insight into the atomistic process occurring during nanoindentation. First, the simulation details and methodology for MD analysis of nanoindentation are presented. The effects of boundary conditions on the nanoindentation response are studied in more detail. The dislocation evolution patterns are then studied using the information provided by atomistic simulation. Different characteristics of metallic sample during nanoindentation experiment, which have been predicted by theoretical models, are investigated. Next, the nature of size effects in samples with small length scales are studied during nanoindentation. The results indicate that the size effects at small indentation depths cannot be modeled using the forest hardening model, and the source exhaustion mechanism controls the size effects at the initial stages of nanoindentation. The total dislocation length increases by increasing the dislocation density which reduces the material strength according to the exhaustion hardening mechanisms. The dislocation interactions with each other become important as the dislocation content increases. Finally, the effects of grain boundary (GB) on the controlling mechanisms of size effects are studied using molecular dynamics.

### 3. Molecular Dynamics-Decorated Finite Element Method (MDeFEM): Application to the Gating Mechanism of Mechanosensitive Channels

Many fundamentally important biological processes rely on the mechanical responses of membrane proteins and their assemblies in the membrane environment, which are multiscale in nature and represent a significant challenge in modeling and simulation. For example, in mechanotransduction, mechanical stimuli can be introduced through macroscopic-scale contacts, which are transduced to mesoscopic-scale (micron) distances and can eventually lead to microscopic-scale (nanometer) conformational changes in membrane-bound protein or protein complexes. This is a fascinating process that spans a large range of length scales and time scales. The involvement of membrane environment and critical issues such as cooperativity calls for the need for an efficient multi-scale computational approach. The goal of the present research is to develop a hierarchical approach to study the mechanical behaviors of membrane proteins with a special emphasis on the gating mechanisms of mechanosensitive (MS) channels. This requires the formulation of modeling and numerical methods that can effectively bridge the disparate length and time scales. A top-down approach is proposed to achieve this by effectively treating biomolecules and their assemblies as integrated structures, in which the most important components of the biomolecule (e.g., MS channel) are modeled as continuum objects, yet their mechanical/physical properties, as well as their interactions, are derived from atomistic simulations. Molecular dynamics (MD) simulations at the nanoscale are used to obtain information on the physical properties and interactions among protein, lipid membrane, and solvent molecules, as well as relevant energetic and temporal characteristics. Effective continuum models are developed to incorporate these atomistic features, and the conformational response of macromolecule(s) to external mechanical perturbations is simulated using finite element (FEM) analyses with in situ mechanochemical coupling. Results from the continuum mechanics analysis provide further insights into the gating transition of MS channels at structural and physical levels, and specific predictions are proposed for further experimental investigations. It is anticipated that the hierarchical framework is uniquely suited for the analysis of many biomolecules and their assemblies under external mechanical stimuli.

Liangliang Zhu, Qiang Cui, Yilun Liu, Yuan Yan, Hang Xiao, Xi Chen

### 4. Spherical Indentation on a Prestressed Elastic Coating/Substrate System

While there have been many studies on the indentation test of thin film/substrate systems, the primary goal has been determining the film properties. However, there was very little effort to probe the properties of both the film and the substrate (the latter may be as important as the film properties). Moreover, a prestress usually exists in the film, typically resulted from mismatched deformation or material properties. In this study, we establish a spherical indentation framework to examine the material properties of both the film and substrate as well as determining the film prestress. An indentation test is performed at two prescribed depths, and functional forms are established between indentation parameters and material variables. An effective reverse analysis algorithm is established to deduce the desired material properties. The potential error sensitivity is also examined in a systematic way. The study on prestressed film/substrate systems has many potential applications in engineering.

James A. Mills, Xi Chen

### 5. Experimentation and Modeling of Mechanical Integrity and Instability at Metal/Ceramic Interfaces

Controlling the mechanical integrity of metal/ceramic interfaces is important for a wide range of technological applications. Achievement of such control requires a number of key elements, including establishing appropriate experimental protocols for quantifying mechanical response of metal/ceramic interfacial regions under well-defined loading conditions, understanding how interfacial compositional and structural characteristics impact such interfacial mechanical response, and elucidating unit interface physics and predicting interfacial mechanical response via development of multiscale physics-based models. Achieving this combined testing, understanding, and modeling will ultimately lead to effective control of mechanical integrity of metal/ceramic interfaces and true interfacial engineering through targeted modification of the interfacial composition and structure.Major breakthroughs in the improvement of interfacial mechanical integrity can be enabled by understanding and controlling key physical factors, including interfacial architectural and chemical features governing the mechanical response of metal/ceramic interfacial regions (MCIRs), thus leading to unprecedented interfacial mechanical performance that meets/exceeds the demands of future applications. Guided by a multiscale integrated computational materials engineering (ICME) framework, the mechanical integrity of MCIRs can be substantially improved by a variety of architectural and chemical enhancements/refinements. Recent research efforts by the authors aim to provide a fundamental, physics-based understanding of the failure mechanisms of MCIRs by constructing a novel, multiscale, computation-guided, and experiment-validated ICME framework. Interfacial refinements to be explored within this framework include addition of alloying impurities as well as geometrical features such as multilayered and stepped interfacial architectures. The findings can then be consolidated into a high fidelity, experiment-validated, micro- and mesoscale modeling tool to significantly accelerate the discovery-design-implementation cycle of advanced MCIRs. In this chapter, we summarize some preliminary results on shear failure and instability of various metal/ceramic interfacial regions, outline the theoretical background of this research thrust, and identify challenges and opportunities in this area.

Wen Jin Meng, Shuai Shao

### 6. Uniqueness of Elastoplastic Properties Measured by Instrumented Indentation

Indentation is widely used to extract material elastoplastic properties from the measured load-displacement curves. One of the most well-established indentation technique utilizes dual (or plural) sharp indenters (which have different apex angles) to deduce key parameters such as the elastic modulus, yield stress, and work-hardening exponent for materials that obey the power-law constitutive relationship. Here we show the existence of “mystical materials,” which have distinct elastoplastic properties, yet they yield almost identical indentation behaviors, even when the indenter angle is varied in a large range. These mystical materials are, therefore, indistinguishable by many existing indentation analyses unless extreme (and often impractical) indenter angles are used. Explicit procedures of deriving these mystical materials are established, and the general characteristics of the mystical materials are discussed. In many cases, for a given indenter angle range, a material would have infinite numbers of mystical siblings, and the existence maps of the mystical materials are also obtained. Furthermore, we propose two alternative techniques to effectively distinguish these mystical materials. In addition, a critical strain is identified as the upper bound of the detectable range of indentation, and moderate tailoring of the constitutive behavior beyond this range cannot be effectively detected by the reverse analysis of the load-displacement curve. The topics in this chapter address the important question of the uniqueness of indentation test, as well as providing useful guidelines to properly use the indentation technique to measure material elastoplastic properties.

L. Liu, Xi Chen, N. Ogasawara, N. Chiba

### 7. Helical Buckling Behaviors of the Nanowire/Substrate System

When a nanowire is deposited on a compliant soft substrate or embedded in matrix, it may buckle into a helical coil form when the system is compressed. Using theoretical and finite element method (FEM) analyses, the detailed three-dimensional coil buckling mechanism for a silicon nanowire (SiNW) on a poly-dimethylsiloxne (PDMS) substrate is discussed. A continuum mechanics approach based on the minimization of the strain energy in the SiNW and elastomeric substrate is developed, and the helical buckling spacing and amplitude are deduced, taking into account the influences of the elastic properties and dimensions of SiNWs. These features are verified by systematic FEM simulations and parallel experiments. When the debonding of SiNW from the surface of the substrate is considered, the buckling profile of the nanowire can be divided into three regimes, i.e., the in-plane buckling, the disordered buckling in the out-of-plane direction, and the helical buckling, depending on the debonding density. For a nanowire embedded in matrix, the buckled profile is almost perfectly circular in the axial direction; with increasing compression, the buckling spacing decreases almost linearly, while the amplitude scales with the 1/2 power of the compressive strain; the transition strain from 2D mode to 3D helical mode decreases with the Young’s modulus of the wire and approaches to ~1.25% when the modulus is high enough, which is much smaller than nanowires on the surface of substrates. The study may shed useful insights on the design and optimization of high-performance stretchable electronics and 3D complex nanostructures.

Youlong Chen, Yilun Liu, Xi Chen

### 8. Hydrogen Embrittlement Cracking Produced by Indentation Test

Indentation is a convenient method to evaluate mechanical properties of materials as well as to simulate contact fracture with locally plastic deformation. Indentation experiment has been widely used for brittle solids, including ceramics and glass, for evaluating the fracture properties. With the aid of computational framework, simulation of crack propagation (for quasi-static and dynamic impact) is conducted to characterize “brittleness” of materials. In this review, we explore the applicability of indentation method for hydrogen embrittlement cracking (HEC). HEC is an important issue in the development of hydrogen-based energy systems. Especially high-strength steels tend to suffer from HE cracking, which leads to a significant decrease in the mechanical properties of the steels, including the critical stress for crack initiation and resistance to crack propagation. For such materials integrity for HEC, convenient material testing is necessary. In this review, the first part describes new indentation methodology to evaluate threshold stress intensity factor K ISCC, and the latter one is investigation into HEC morphology due to residual stress produced by indentation impression. Our findings will be useful for predicting K ISCC for HE instead of conventional long-term test with fracture mechanics testing. It will also indicate the stress criterion of HE cracking from an indentation impression crater, when the formed crater (for instance due to shot peening or foreign object contact) is exposed to a hydrogen environment.

Akio Yonezu, Xi Chen

### 9. Continuous Stiffness Measurement Nanoindentation Experiments on Polymeric Glasses: Strain Rate Alteration

In many studies using continuous stiffness measurement (CSM) nanoindentation technique, it is assumed that the strain rate remains constant during the whole experiment since the loading rate divided by the load ( Ṗ / P $$\dot{P}/P$$ ) is considered as a constant input parameter. Using the CSM method, the soundness of this assumption in nanoindentation of polymeric glasses is investigated by conducting a series of experiments on annealed poly(methyl methacrylate) (PMMA) and polycarbonate (PC) at different set Ṗ / P $$\dot{P}/P$$ values. Evaluating the variation of the actual Ṗ / P $$\dot{P}/P$$ value during the course of a single test shows that this parameter varies intensely at shallow indentation depths, and it reaches a stabilized value after a significant depth which is not material dependent. In addition, the strain rate variation is examined through two methods: first, using the definition of the strain rate as the descent rate of the indenter divided by its instantaneous depth ( ḣ / h $$\dot{h}/h$$ ) and second, considering the relationship between the strain rate and the load and hardness variations during the test. Based on the findings, the strain rate is greatly larger at shallow indentations, and the depth beyond which it attains the constant value depends on the material and the set Ṗ / P $$\dot{P}/P$$ ratio. Lastly, incorporating the relationship between the hardness and strain rate, it is revealed that although the strain rate variation changes the material hardness, its effect does not give a justification for the observed indentation size effect (ISE); therefore, other contributing parameters are discussed for their possible effects on this phenomenon.

### 10. Shear Transformation Zones in Amorphous Polymers: Geometrical and Micromechanical Properties

Glassy polymers are extensively used as high impact resistant, low density, and clear materials in industries. Due to the lack of the long-range order in the microstructures of glassy solids, plastic deformation is different from that in crystalline solids. Shear transformation zones (STZs) are believed to be the plasticity carriers in amorphous solids and defined as the localized atomic or molecular deformation patches induced by shear. Despite a great effort in characterizing these local disturbance regions in metallic glasses (MGs), there are still many unknowns relating to the microstructural and micromechanical characteristics of STZs in glassy polymers. This chapter is aimed at investigating the flow phenomenon in polycarbonate (PC) and poly(methyl methacrylate) (PMMA) as glassy polymers and obtaining the mechanical and geometrical characteristics of their STZs. To achieve this goal, the nanoindentation experiments are performed on samples with two different thermal histories: as-cast and annealed, and temperature and strain rate dependency of the yield stress of PC and PMMA are studied. Based on the experimental results, it is showed that the flow in PC and PMMA is a homogeneous phenomenon at tested temperatures and strain rates. The homogeneous flow theory is then applied to analyze the STZs quantitatively. The achieved results are discussed for their possible uniqueness or applicability to all glassy polymers in the context of amorphous plasticity.

### 11. Properties of Material Interfaces: Dynamic Local Versus Nonlocal

Interfaces in the materials are known entities since last century described as early as in the interfacial excess energy formulations by Gibbs (Boßelmann et al. 2007). The interface effect (or surface effect) is also widely referred to as the interface stress (or surface stress) that consists of two parts, both arise from the distorted atomic structure near the interface (or surface): the first part is the interface (or surface) residual stress which is independent of the deformation of solids, and the second part is the interface (or surface) elasticity which contributes to the stress field related to the deformation. Plastic deformation, in particular, the initial yielding point (i.e., the yield surface), is sensitive to the local stress (or local strain) of a heterogeneous material, which includes both the local (surface/interface) residual stress and local stress–strain relationship. The plastic deformation at the interfaces also considers the tension and compression along the interface and stress mismatch because of the material property differences. In the nanomaterials, the surface and interface stresses become even more important owing to the nanoscale size of the particles and interface areas.

Devendra Verma, Chandra Prakash, Vikas Tomar

### 12. Nanostructural Response to Plastic Deformation in Glassy Polymers

A closed form stress-strain relation is proposed for modeling the postyield behavior of amorphous polymers based on the shear transformation zones (STZs) dynamics and free volume evolution. Use is made of the classical free volume theory by Cohn and Turnbull (J Chem Phys 31:1164, 1959), and also STZ-mediated plasticity model for amorphous metals by Spaepen (Acta Metall 25:407, 1977) and Argon (Acta Metall 27:47, 1979) for developing a new homogenous plasticity framework for glassy polymers. The variations of free volume content and STZs activation energy during large deformation are parametrized considering the previous experimental measurements using positron annihilation lifetime spectroscopy (PALS) and thermal analysis with differential scanning calorimetry (DSC), respectively. The proposed model captures the softening-hardening behavior of glassy polymers at large strains with a single formula. This study shows that the postyield softening of the glassy polymers is a result of the reduction of the STZs nucleation energy as a consequence of increased free volume content during the plastic straining up to a steady-state point. Beyond the steady-state strain where the STZ nucleation energy reaches a plateau, the increased number density of STZs, which is required for finite strain, brings about the secondary hardening continuing up to the fracture point. This model also accurately predicts the effect of strain rate, temperature, and thermal history of the sample on its postyield behavior which is in consonance with experimental observations. Implication of the model for interpreting the localization and indentation size effect of polymers is also discussed.

### 13. Indentation Fatigue Mechanics

Instrumented indentation has been widely used in the determination of mechanical properties of materials due to its fast, simple, precise, and nondestructive merits over the past few years. In this chapter, we will present an emerging indentation technique, referred to as indentation fatigue, where a fatigue load is applied on a sample via a flat punch indenter, and establish the framework of mechanics of indentation fatigue to extract fatigue properties of materials. Through extensive experimental, theoretical, and computational investigations, we demonstrate a similarity between the indentation fatigue depth propagation and the fatigue crack growth, and propose an indentation fatigue depth propagation law and indentation fatigue strength law to describe indentation fatigue-induced deformation and failure of materials, respectively. This study provides an alternative approach for determining fatigue properties, as well as for studying the fatigue mechanisms of materials, especially for materials that are not available or feasible for conventional fatigue tests.

Baoxing Xu, Xi Chen, Zhufeng Yue

### 14. Crack Initiation and Propagation in Laminated Composite Materials

Jun Xu, Yanting Zheng

### 15. Micromorphic Approach to Gradient Plasticity and Damage

Eringen and Mindlin’s original micromorphic continuum model is presented and extended towards finite elastic-plastic deformations. The framework is generalized to any additional kinematic degrees of freedom related to plasticity and/or damage mechanisms. It provides a systematic method to develop size–dependent plasticity and damage models, closely related to phase field approaches, that can be applied to hardening and/or softening material behavior. The regularization power of the method is illustrated in the case of damage in single crystals. Special attention is given to the various possible finite deformation formulations enhancing existing frameworks for finite elastoplasticity and damage.

Samuel Forest

In this chapter, two cases of thermodynamic-based higher order gradient plasticity theories are presented and applied to the stretch-surface passivation problem for investigating the material behavior under the nonproportional loading condition. This chapter incorporates the thermal and mechanical responses of microsystems. It addresses phenomena such as size and boundary effects and in particular microscale heat transfer in fast-transient processes. The stored energy of cold work is considered in the development of the recoverable counterpart of the free energy. The main distinction between the two cases is the presence of the dissipative higher order microstress quantities ?? ijk dis $${\mathcal{S}}_{ijk}^{\mathrm{dis}}$$ . Fleck et al. (Soc. A-Math. Phys. 470:2170, 2014, ASME 82:7, 2015) noted that ?? ijk dis $${\mathcal{S}}_{ijk}^{\mathrm{dis}}$$ always gives rise to the stress jump phenomenon, which causes a controversial dispute in the field of strain gradient plasticity theory with respect to whether it is physically acceptable or not, under the nonproportional loading condition. The finite element solution for the stretch-surface passivation problem is also presented by using the commercial finite element package ABAQUS/standard (User’s Manual (Version 6.12). Dassault Systemes Simulia Corp., Providence, 2012) via the user-subroutine UEL. The model is validated by comparing with three sets of small-scale experiments. The numerical simulation part, which is largely composed of four subparts, is followed. In the first part, the occurrence of the stress jump phenomenon under the stretch-surface passivation condition is introduced in conjunction with the aforementioned three experiments. The second part is carried out in order to clearly show the results to be contrary to each other from the two classes of strain gradient plasticity models. An extensive parametric study is presented in the third part in terms of the effects of the various material parameters on the stress-strain response for the two cases of strain gradient plasticity models, respectively. The evolution of the free energy and dissipation potentials are also investigated at elevated temperatures. In the last part, the two-dimensional simulation is given to examine the gradient and grain boundary effect, the mesh sensitivity of the two-dimensional model, and the stress jump phenomenon. Finally, some significant conclusions are presented.

### 17. Micropolar Crystal Plasticity

This chapter considers advances over the past 15 years achieved by the authors and coworkers on generalized crystal plasticity to address size and configuration effects in dislocation plasticity at the micron scale. The specific approaches addressed here focus on micropolar and micromorphic theories rather than adopting strain gradient theory as the starting point, as motivated by the pioneering ideas of Eringen (Eringen and Suhubi 1964; Eringen and Claus Jr 1969; Eringen 1999). It is demonstrated with examples that for isotropic elasticity and specific sets of slip systems, a dislocation-based formulation of micropolar or micromorphic type provides results comparable to discrete dislocation dynamics and has much in common with the structure of Gurtin’s slip gradient theory (Gurtin 2002; Gurtin et al. 2007).

J. R. Mayeur, D. L. McDowell, Samuel Forest

### 18. Micromorphic Crystal Plasticity

The micromorphic approach to crystal plasticity represents an extension of the micropolar (Cosserat) framework, which is presented in a separate chapter. Cosserat crystal plasticity is contained as a special constrained case in the same way as the Cosserat theory is a special restricted case of Eringen's micromorphic model, as explained also in a separate chapter. The micromorphic theory is presented along the lines of Aslan et al. (Int J Eng Sci 49:1311–1325, 2011) and Forest et al. (Micromorphic approach to crystal plasticity and phase transformation. In: Schroeder J, Hackl K (eds) Plasticity and beyond. CISM international centre for mechanical sciences, courses and lectures, vol 550, Springer, pp 131–198, 2014) and compared to the micropolar model in some applications. These extensions of conventional crystal plasticity aim at incorporating the dislocation density tensor introduced by Kröner (Initial studies of a plasticity theory based upon statistical mechanics. In: Kanninen M, Adler W, Rosenfield A, Jaffee R (eds) Inelastic behaviour of solids. McGraw-Hill, pp 137–147, 1969). and Cermelli and Gurtin (J Mech Phys Solids 49:1539–1568, 2001) into the constitutive framework. The concept of dislocation density tensor is equivalent to that of the so-called geometrically necessary dislocations (GND) introduced by Ashby (The deformation of plastically non-homogeneous alloys. In: Kelly A, Nicholson R (eds) Strengthening methods in crystals. Applied Science Publishers, London, pp 137–192, 1971). The applications presented in this chapter deal with pile-up formation in laminate microstructures and strain localization phenomena in polycrystals.

Samuel Forest, J. R. Mayeur, D. L. McDowell

### 19. Cosserat Approach to Localization in Geomaterials

A renewed interest toward Cosserat or micropolar continuum has driven researchers to the development of specific models for upscaling discrete media such as masonry, granular assemblies, fault gouges, porous media, and biomaterials. Cosserat continuum is a special case of what is called micromorphic, generalized or higher-order continua. Due to the presence of internal lengths in its formulation, Cosserat continuum is quite attractive for addressing problems involving strain localization. It enables modeling the shear band thickness evolution, tracking the postlocalization regime, and correctly dissipating the energy when using numerical schemes. In this chapter, we summarize the fundamental governing equations of a Cosserat continuum under multiphysical couplings. Several examples of the numerical advantages of Cosserat continuum are also presented regarding softening behavior, strain localization, finite element formulation, reduced integration, and hourglass control. The classically used constitutive models in Cosserat elastoplasticity are presented and some common approaches for upscaling and homogenization in Cosserat continuum are discussed. Finally, a simple illustrative example of the adiabatic shearing of a rock layer under constant shear stress is presented in order to juxtapose a rate-independent Cosserat with a rate-dependent Cauchy formulation as far as it concerns strain localization.

Ioannis Stefanou, Jean Sulem, Hadrien Rattez

### 20. Dispersion of Waves in Micromorphic Media and Metamaterials

In this contribution we discuss the interest of using enriched continuum models of the micromorphic type for the description of dispersive phenomena in metamaterials. Dispersion is defined as that phenomenon according to which the speed of propagation of elastic waves is not a constant, but depends on the wavelength of the traveling wave. In practice, all materials exhibit dispersion if one considers waves with sufficiently small wavelengths, since all materials have a discrete structure when going down at a suitably small scale. Given the discrete substructure of matter, it is easy to understand that the material properties vary when varying the scale at which the material itself is observed. It is hence not astonishing that the speed of propagation of waves changes as well when considering waves with smaller wavelengths.In an effort directed toward the modeling of dispersion in materials with architectured microstructures (metamaterials), different linear-elastic, isotropic, micromorphic models are introduced, and their peculiar dispersive behaviors are discussed by means of the analysis of the associated dispersion curves. The role of different micro-inertias related to both independent and constrained motions of the microstructure is also analyzed. A special focus is given to those metamaterials which have the unusual characteristic of being able to stop the propagation of mechanical waves and which are usually called band-gap metamaterials. We show that, in the considered linear-elastic, isotropic case, the relaxed micromorphic model, recently introduced by the authors, is the only enriched model simultaneously allowing for the description of non-localities and multiple band-gaps in mechanical metamaterials.

### 21. Implicit Nonlocality in the Framework of Viscoplasticity

The considerations are addressed to the notion of implicit nonlocality in mechanical models. The term implicit means that there is no direct measure of nonlocal action in a model (like classical or fractional gradients, etc. in explicit nonlocal models), but some phenomenological material parameters can be interpreted as one that maps some experimentally observed phenomena responsible for the scale effects.The overall discussion is conducted in the framework of the Perzyna Theory of Viscoplasticity where the role of the implicit length scale parameter plays the relaxation time of the mechanical disturbance. In this sense, in the viscoplastic range of the material behavior, the deformation at each material point contributes to the finite surrounding. The important consequence is that the solution of the IBVP described by Perzyna’s theory is unique – the relaxation time is the regularizing parameter.

Wojciech Sumelka, Tomasz Łodygowski

### 22. Finite Element Analysis of Thermodynamically Consistent Strain Gradient Plasticity Theory and Applications

In this chapter, a coupled thermomechanical gradient-enhanced continuum plasticity theory containing the flow rules of the grain interior and grain boundary areas is developed within the thermodynamically consistent framework. Two-dimensional finite element implementation for the proposed gradient plasticity theory is carried out to examine the micro-mechanical and thermal characteristics of small-scale metallic volumes. The proposed model is conceptually based on the dislocation interaction mechanisms and thermal activation energy. The thermodynamic conjugate microstresses are decomposed into dissipative and energetic components; correspondingly, the dissipative and energetic length scales for both the grain interior and grain boundary are incorporated in the proposed model, and an additional length scale related to the geometrically necessary dislocation-induced strengthening is also included. Not only the partial heat dissipation caused by the fast transient time but also the distribution of temperature caused by the transition from the plastic work to the heat is included into the coupled thermomechanical model by deriving a generalized heat equation. The derived constitutive framework and two-dimensional finite element model are validated through the comparison with the experimental observations conducted on microscale thin films. The proposed enhanced model is examined by solving the simple shear problem and the square plate problem to explore the thermomechanical characteristics of small-scale metallic materials. Finally, some significant conclusions are presented.

### 23. Fractional Nonlocal Continuum Mechanics and Microstructural Models

Models of physical lattices with long-range interactions for nonlocal continuum are suggested. The lattice long-range interactions are described by exact fractional-order difference operators. Continuous limit of suggested lattice operators gives continuum fractional derivatives of non-integer orders. The proposed approach gives a new microstructural basis to formulation of theory of nonlocal materials with power-law nonlocality. Moreover these lattice models, which is based on exact fractional differences, allow us to have a unified microscopic description of fractional nonlocal and standard local continuum.

Vasily E. Tarasov

### 24. Fractional Differential Calculus and Continuum Mechanics

The present essay is an attempt to present a meaningful continuum mechanics formulation into the context of fractional calculus. The task is not easy, since people working on various fields using fractional calculus take for granted that a fractional physical problem is set up by simple substitution of the conventional derivatives to any kind of the plethora of fractional derivatives. However, that procedure is meaningless, although popular, since laws in science are derived through differentials and not through derivatives. One source of that mistake is that the fractional derivative of a variable with respect to itself is different from one. The other source of the same mistake is that the well-known derivatives are not able to form differentials. This leads to erroneous and meaningless quantities like fractional velocity and fractional strain. In reality those quantities, that nobody understands what physically represent, alter the dimensions of the physical quantities. In fact the dimension of the fractional velocity is L/Tα, contrary to the conventional L/T. Likewise, the dimension of the fractional strain is L−α, contrary to the conventional L0. That fact cannot be justified. Imagine that even in relativity theory, where everything is changed, like time, lengths, velocities, momentums, etc., the dimensions remain constant. Fractional calculus is allowed up to now to change the dimensions and to accept derivatives that are not able to form differentials, according to differential topology laws. Those handicaps have been pointed out in two recent conferences dedicated to fractional calculus by the authors, (K.A. Lazopoulos, in Fractional Vector Calculus and Fractional Continuum Mechanics, Conference “Mechanics though Mathematical Modelling”, celebrating the 70th birthday of Prof. T. Atanackovic, Novi Sad, 6–11 Sept, Abstract, p. 40, 2015; K.A. Lazopoulos, A.K. Lazopoulos, Fractional vector calculus and fractional continuum mechanics. Prog. Fract. Diff. Appl. 2(1), 67–86, 2016a) and were accepted by the fractional calculus community. The authors in their lectures (K.A. Lazopoulos, in Fractional Vector Calculus and Fractional Continuum Mechanics, Conference “Mechanics though Mathematical Modelling”, celebrating the 70th birthday of Prof. T. Atanackovic, Novi Sad, 6–11 Sept, Abstract, p. 40, 2015; K.A. Lazopoulos, in Fractional Differential Geometry of Curves and Surfaces, International Conference on Fractional Differentiation and Its Applications (ICFDA 2016), Novi Sad, 2016b; A.K. Lazopoulos, On Fractional Peridynamic Deformations, International Conference on Fractional Differentiation and Its Applications, Proceedings ICFDA 2016, Novi Sad, 2016c) and in the two recently published papers concerning fractional differential geometry of curves and surfaces (K.A. Lazopoulos, A.K. Lazopoulos, On the fractional differential geometry of curves and surfaces. Prog. Fract. Diff. Appl., No 2(3), 169–186, 2016b) and fractional continuum mechanics (K.A. Lazopoulos, A.K. Lazopoulos, Fractional vector calculus and fractional continuum mechanics. Prog. Fract. Diff. Appl. 2(1), 67–86, 2016a) added in the plethora of fractional derivatives one more, that called Leibniz L-fractional derivative. That derivative is able to yield differential and formulate fractional differential geometry. Using that derivative the dimensions of the various quantities remain constant and are equal to the dimensions of the conventional quantities. Since the establishment of fractional differential geometry is necessary for dealing with continuum mechanics, fractional differential geometry of curves and surfaces with the fractional field theory will be discussed first. Then the quantities and principles concerning fractional continuum mechanics will be derived. Finally, fractional viscoelasticity Zener model will be presented as application of the proposed theory, since it is of first priority for the fractional calculus people. Hence the present essay will be divided into two major chapters, the chapter of fractional differential geometry, and the chapter of the fractional continuum mechanics. It is pointed out that the well-known historical events concerning the evolution of the fractional calculus will be circumvented, since the goal of the authors is the presentation of the fractional analysis with derivatives able to form differentials, formulating not only fractional differential geometry but also establishing the fractional continuum mechanics principles. For instance, following the concepts of fractional differential and Leibniz’s L-fractional derivatives, proposed by the author (K.A. Lazopoulos, A.K. Lazopoulos, Fractional vector calculus and fractional continuum mechanics. Prog. Fract. Diff. Appl. 2(1), 67–86, 2016a), the L-fractional chain rule is introduced. Furthermore, the theory of curves and surfaces is revisited, into the context of fractional calculus. The fractional tangents, normals, curvature vectors, and radii of curvature of curves are defined. Further, the Serret-Frenet equations are revisited, into the context of fractional calculus. The proposed theory is implemented into a parabola and the curve configured by the Weierstrass function as well. The fractional bending problem of an inhomogeneous beam is also presented, as implementation of the proposed theory. In addition, the theory is extended on manifolds, defining the fractional first differential (tangent) spaces, along with the revisiting first and second fundamental forms for the surfaces. Yet, revisited operators like fractional gradient, divergence, and rotation are introduced, outlining revision of the vector field theorems. Finally, the viscoelastic mechanical Zener system is modelled with the help of Leibniz fractional derivative. The compliance and relaxation behavior of the viscoelastic systems is revisited and comparison with the conventional systems and the existing fractional viscoelastic systems are presented.

K. A. Lazopoulos, A. K. Lazopoulos

### 25. Continuum Homogenization of Fractal Media

This chapter reviews the modeling of fractal materials by homogenized continuum mechanics using calculus in non-integer dimensional spaces. The approach relies on expressing the global balance laws in terms of fractional integrals and, then, converting them to integer-order integrals in conventional (Euclidean) space. Via localization, this allows development of local balance laws of fractal media (continuity, linear and angular momenta, energy, and second law) and, in case of elastic responses, formulation of wave equations in several settings (1D and 3D wave motions, fractal Timoshenko beam, and elastodynamics under finite strains). Next, follows an account of extremum and variational principles, and fracture mechanics. In all the cases, the derived equations for fractal media depend explicitly on fractal dimensions and reduce to conventional forms for continuous media with Euclidean geometries upon setting the dimensions to integers.

Martin Ostoja-Starzewski, Jun Li, Paul N. Demmie

### 26. Modeling High-Speed Impact Failure of Metallic Materials: Nonlocal Approaches

Development and application of advanced, computationally intensive multiscale (macro-, meso-, and micro-mechanically) physically based models to describe physical phenomena associated with friction and wear in heterogeneous solids, particularly under high velocity impact loading conditions. Emphasis will be placed on the development of fundamental, thermodynamically consistent theories to describe high-velocity material wear failure processes in combinations of ductile and brittle materials for wear damage-related problems. The wear failure criterion will be based on dissipated energies due to plastic strains at elevated temperatures. Frictional coefficients will be identified for the contact surfaces based on temperature, strain rates, and roughness of the surfaces. In addition failure models for microstructural effects, such as shear bands and localized deformations, will be studied.The computations will be carried with Abaqus Explicit as a dynamic temperature-displacement analysis. The contact between sliding against each other’s surfaces is specified as surface-to-surface contact on the master-slave basis. The tangential behavior is defined as kinematic contact with finite sliding. The validation of computations utilizing the novel approach presented in this work is going to be conducted on the continuum level while comparing the obtained numerical results with the experimental results obtained in the laboratories in Metz, France. Reaction forces due to friction between the two specimens and temperature resulting from the dissipated energy during the friction experiment are going to be compared and discussed in detail. Additionally the indentation response at the macroscale, for decreasing the size of the indenter, will be used to critically assess and evaluate the length scale parameters.

### 27. Strain Gradient Plasticity: Deformation Patterning, Localization, and Fracture

In this chapter, two different strain gradient plasticity models based on non-convex plastic energies are presented and compared through analytical estimates and numerical experiments. The models are formulated in the simple one-dimensional setting, and their ability to reproduce heterogeneous plastic strain processes is analyzed, focusing on strain localization phenomena observed in metallic materials at different length scales. In a geometrically linear context, both models are based on the additive decomposition of the strain into elastic and plastic parts. Moreover, they share the same non-convex plastic energy, and they are both characterized by the same nonlocal plastic energy as well, i.e., a quadratic form of the plastic strain gradient. In the first model, proposed in Yalçinkaya et al. (Int J Solids Struct 49:2625–2636, 2012) and Yalcinkaya (Microstructure evolution in crystal plasticity: strain path effects and dislocation slip patterning. Ph.D. thesis, Eindhoven University of Technology, 2011), the plastic energy is assumed to be conservative, and plastic dissipation is introduced through a viscous term, which makes the formulation rate-dependent. In the second model, developed in Del Piero et al. (J Mech Mater Struct 8(2–4):109–151, 2013), the plastic term is supposed to be totally dissipative. As a result, plastic deformations are not recoverable, and the resulting framework is rate-independent, contrary to the first model. First, the evolution problems resulting from the two theories are analytically solved in a special simplified case, and correlations between the shape of the plastic potential and the modeling predictions are established. Then, the models are numerically implemented by finite elements, and numerical solutions of two different one-dimensional problems, associated with different plastic energies, are determined. In the first problem, a double-well plastic energy is considered, and the evolution of plastic slip patterning observed in crystals at the mesoscale is reproduced. In the second problem, a convex-concave plastic energy is used to simulate the macroscopic response of a tensile steel bar, which experiences the so-called necking process, with plastic strains localization and final coalescing into fracture. Numerical results provided by the two models are analyzed and compared.

Giovanni Lancioni, Tuncay Yalçinkaya

### 28. Strain Gradient Crystal Plasticity: Thermodynamics and Implementation

This chapter studies the thermodynamical consistency and the finite element implementation aspects of a rate-dependent nonlocal (strain gradient) crystal plasticity model, which is used to address the modeling of the size-dependent behavior of polycrystalline metallic materials. The possibilities and required updates for the simulation of dislocation microstructure evolution, grain boundary-dislocation interaction mechanisms, and localization leading to necking and fracture phenomena are shortly discussed as well. The development of the model is conducted in terms of the displacement and the plastic slip, where the coupled fields are updated incrementally through finite element method. Numerical examples illustrate the size effect predictions in polycrystalline materials through Voronoi tessellation.

Tuncay Yalçinkaya

### 29. Strain Gradient Crystal Plasticity: Intergranular Microstructure Formation

This chapter addresses the formation and evolution of inhomogeneous plastic deformation field between grains in polycrystalline metals by focusing on continuum scale modeling of dislocation-grain boundary interactions within a strain gradient crystal plasticity (SGCP) framework. Thermodynamically consistent extension of a particular strain gradient plasticity model, addressed previously (see also, e.g., Yalcinkaya et al, J Mech Phys Solids 59:1–17, 2011), is presented which incorporates the effect of grain boundaries on plastic slip evolution explicitly. Among various choices, a potential-type non-dissipative grain boundary description in terms of grain boundary Burgers tensor (see, e.g., Gurtin, J Mech Phys Solids 56:640–662, 2008) is preferred since this is the essential descriptor to capture both the misorientation and grain boundary orientation effects. A mixed finite element formulation is used to discretize the problem in which both displacements and plastic slips are considered as primary variables. For the treatment of grain boundaries within the solution algorithm, an interface element is formulated. The capabilities of the framework is demonstrated through 3D bi-crystal and polycrystal examples, and potential extensions and currently pursued multi-scale modeling efforts are briefly discussed in the closure.

İzzet Özdemir, Tuncay Yalçinkaya

### 30. Microplane Models for Elasticity and Inelasticity of Engineering Materials

In the traditional approach to the modeling of mechanical behavior of engineering materials, the stress tensor is calculated directly from the prescribed strain tensor either by a closed form tensorial relation as in elasticity or by incremental analysis as in classical plasticity formulations in which the formulation is developed in terms of tensor invariants and their combinations. However, to model the general three-dimensional constitutive behavior of the so-called geomaterials at arbitrary nonproportional load paths that frequently arise in dynamic loadings, such direct approaches do not yield models with desired accuracy. Instead, microplane approach prescribes the constitutive behavior on planes of various orientations of the material microstructure independently, and the second-order stress tensor is obtained by imposing the equilibrium of second-order stress tensor with the microplane stress vectors. In this work, particular attention is devoted to the milestone microplane models for plain concrete, namely, the model M4 and the model M7. Furthermore, a novel autocalibrating version of the model M7 called the model M7Auto is presented as an alternative to both differential and integral type nonlocal formulations since the model M7Auto does not suffer from the shortcomings of these classical nonlocal approaches. Examples of the performance of the models M7 and M7Auto are shown by simulating well-known benchmark test data like three-point bending size effect test data of plain concrete beams using finite element meshes of the same element size and Nooru-Mohamed test data obtained at different load paths using finite element meshes having different element sizes, respectively.

Ferhun C. Caner, Valentín de Carlos Blasco, Mercè Ginjaume Egido

### 31. Modeling Temperature-Driven Ductile-to-Brittle Transition Fracture in Ferritic Steels

The most catastrophic brittle failure in ferritic steels is observed as their tendency of losing almost all of their toughness when the temperature drops below their ductile-to-brittle transition (DBT) temperature. There have been put large efforts in experimental and theoretical studies to clarify the controlling mechanism of this transition; however, it still remains unclear how to model accurately the coupled ductile∕brittle fracture behavior of ferritic steels in the region of ductile-to-brittle transition.Therefore, in this study, an important attempt is made to model coupled ductile∕brittle fracture by means of blended micro-void and micro-cracks. To this end, a thermomechanical finite strain-coupled plasticity and continuum damage mechanics models which incorporate the blended effects of micro-heterogeneities in the form of micro-cracks and micro-voids are proposed.In order to determine the proposed model material constant, a set of finite element model, where the proposed unified framework, which characterizes ductile-to-brittle fracture behavior of ferritic steels, is implemented as a VUMAT, is performed by modeling the benchmark experiment given in the experimental research published by Turba et al., then, using these models as a departure point, the fracture response of the small punch fracture testing is investigated numerically at 22∘C and − 196∘C and at which the fracture is characterized as ductile and brittle, respectively.

Babür Deliktaş, Ismail Cem Turtuk, George Z. Voyiadjis

### 32. Size-Dependent Transverse Vibration of Microbeams

In this chapter, a new microstructure-dependent higher-order shear deformation beam model is introduced to investigate the vibrational characteristics of microbeams. This model captures both the size and shear deformation effects without the need for any shear correction factors. The governing differential equations and related boundary conditions are derived by implementing Hamilton’s principle on the basis of modified strain gradient theory in conjunction with trigonometric shear deformation beam theory. The free vibration problem for simply supported microbeams is analytically solved by employing the Navier solution procedure. Moreover, a new modified shear correction factor is firstly proposed for Timoshenko (first-order shear deformation) microbeam model. Several comparative results are presented to indicate the effects of material length-scale parameter ratio, slenderness ratio, and shear correction factor on the natural frequencies of microbeams. It is observed that effect of shear deformation becomes more considerable for both smaller slenderness ratios and higher modes.

Ömer Civalek, Bekir Akgöz

### 33. Axial Vibration of Strain Gradient Micro-rods

In this chapter, size-dependent axial vibration response of micro-sized rods is investigated on the basis of modified strain gradient elasticity theory. On the contrary to the classical rod model, the developed nonclassical micro-rod model includes additional material length scale parameters and can capture the size effect. If the additional material length scale parameters are equal to zero, the current model reduces to the classical one. The equation of motion together with initial conditions, classical and nonclassical corresponding boundary conditions, for micro-rods is derived by implementing Hamilton’s principle. The resulting higher-order equation is analytically solved for clamped-free and clamped-clamped boundary conditions. Finally, some illustrative examples are presented to indicate the influences of the additional material length scale parameters, size dependency, boundary conditions, and mode numbers on the natural frequencies. It is found that size effect is more significant when the micro-rod diameter is closer to the additional material length scale parameter. In addition, it is observed that the difference between natural frequencies evaluated by the present and classical models becomes more considerable for both lower values of slenderness ratio and higher modes.

Ömer Civalek, Bekir Akgöz, Babür Deliktaş

### 34. Peridynamics: Introduction

The peridynamic theory is a nonlocal extension of continuum mechanics that is compatible with the physical nature of cracks as discontinuities. It avoids the need to evaluate the partial derivatives of the deformation with respect to the spatial coordinates, instead using an integro-differential equation for the linear momentum balance. This chapter summarizes the peridynamic theory, emphasizing the continuum mechanical and thermodynamic aspects. Formulation of material models is discussed, including details on the statement of models using mathematical objects called peridynamic states that are nonlocal and nonlinear generalizations of second-order tensors. Damage evolution is treated within a nonlocal thermodynamic framework making use of the dependence of free energy on damage. Continuous, stable growth of damage can suddenly become unstable, leading to dynamic fracture. Peridynamics treats fracture and long-range forces on the same mathematical basis as continuous deformation and contact forces, extending the applicability of continuum mechanics to new classes of problems.

S. A. Silling

### 35. Recent Progress in Mathematical and Computational Aspects of Peridynamics

Recent developments in the mathematical and computational aspects of the nonlocal peridynamic model for material mechanics are provided. Based on a recently developed vector calculus for nonlocal operators, a mathematical framework is constructed that has proved useful for the mathematical analyses of peridynamic models and for the development of finite element discretizations of those models. A specific class of discretization algorithms referred to as asymptotically compatible schemes is discussed; this class consists of methods that converge to the proper limits as grid sizes and nonlocal effects tend to zero. Then, the multiscale nature of peridynamics is discussed including how, as a single model, it can account for phenomena occurring over a wide range of scales. The use of this feature of the model is shown to result in efficient finite element implementations. In addition, the mathematical and computational frameworks developed for peridynamic simulation problems are shown to extend to control, coefficient identification, and obstacle problems.

Marta D’Elia, Qiang Du, Max Gunzburger

### 36. Optimization-Based Coupling of Local and Nonlocal Models: Applications to Peridynamics

Nonlocal continuum theories for mechanics can capture strong nonlocal effects due to long-range forces in their governing equations. When these effects cannot be neglected, nonlocal models are more accurate than partial differential equations (PDEs); however, the accuracy comes at the price of a prohibitive computational cost, making local-to-nonlocal (LtN) coupling strategies mandatory.In this chapter, we review the state of the art of LtN methods where the efficiency of PDEs is combined with the accuracy of nonlocal models. Then, we focus on optimization-based coupling strategies that couch the coupling of the models into a control problem where the states are the solutions of the nonlocal and local equations, the objective is to minimize their mismatch on the overlap of the local and nonlocal problem domains, and the virtual controls are the nonlocal volume constraint and the local boundary condition. The strategy is described in the context of nonlocal and local elasticity and illustrated by numerical tests on three-dimensional realistic geometries. Additional numerical tests also prove the consistency of the method via patch tests.

Marta D’Elia, Pavel Bochev, David J. Littlewood, Mauro Perego

### 37. Bridging Local and Nonlocal Models: Convergence and Regularity

As nonlocal models become more widespread in applications, we focus on their connections with their classical counterparts and also on some theoretical aspects which impact their implementation. In this context we survey recent developments by the authors and prove some new results on regularity of solutions to nonlinear systems in the nonlocal framework. In particular, we focus on semilinear problems and also on higher-order problems with applications in the theory of plate deformations.

### 38. Dynamic Brittle Fracture from Nonlocal Double-Well Potentials: A State-Based Model

We introduce a regularized model for free fracture propagation based on nonlocal potentials. We work within the small deformation setting, and the model is developed within a state-based peridynamic formulation. At each instant of the evolution, we identify the softening zone where strains lie above the strength of the material. We show that deformation discontinuities associated with flaws larger than the length scale of nonlocality δ can become unstable and grow. An explicit inequality is found that shows that the volume of the softening zone goes to zero linearly with the length scale of nonlocal interaction. This scaling is consistent with the notion that a softening zone of width proportional to δ converges to a sharp fracture set as the length scale of nonlocal interaction goes to zero. Here the softening zone is interpreted as a regularization of the crack network. Inside quiescent regions with no cracks or softening, the nonlocal operator converges to the local elastic operator at a rate proportional to the radius of nonlocal interaction. This model is designed to be calibrated to measured values of critical energy release rate, shear modulus, and bulk modulus of material samples. For this model one is not restricted to Poisson ratios of 1∕4 and can choose the potentials so that small strain behavior is specified by the isotropic elasticity tensor for any material with prescribed shear and Lamé moduli.

Robert Lipton, Eyad Said, Prashant K. Jha

### 39. Nonlocal Operators with Local Boundary Conditions: An Overview

We present novel governing operators in arbitrary dimension for nonlocal diffusion in homogeneous media. The operators are inspired by the theory of peridynamics (PD). They agree with the original PD operator in the bulk of the domain and simultaneously enforce local boundary conditions (BC). The main ingredients are periodic, antiperiodic, and mixed extensions of kernel functions together with even and odd parts of bivariate functions. We present different types of BC in 2D which include pure and mixed combinations of Neumann and Dirichlet BC. Our construction is systematic and easy to follow. We provide numerical experiments that validate our theoretical findings. When our novel operators are extended to vector-valued functions, they will allow the extension of PD to applications that require local BC. Furthermore, we hope that the ability to enforce local BC provides a remedy for surface effects seen in PD.We recently proved that the nonlocal diffusion operator is a function of the classical operator. This observation opened a gateway to incorporate local BC to nonlocal problems on bounded domains. The main tool we use to define the novel governing operators is functional calculus, in which we replace the classical governing operator by a suitable function of it. We present how to apply functional calculus to general nonlocal problems in a methodical way.

Burak Aksoylu, Fatih Celiker, Orsan Kilicer

### 40. Peridynamics and Nonlocal Diffusion Models: Fast Numerical Methods

We outline the recent developments of fast numerical methods for linear nonlocal diffusion and peridynamic models in one and two space dimensions. We show how the analysis was carried out to take full advantage of the structure of the stiffness matrices of the numerical methods in its storage, evaluation, and assembly and in the efficient solution of the corresponding numerical schemes. This significantly reduces the computational complexity and storage of the numerical methods over conventional ones, without using any lossy compression. For instance, we would use the same numerical quadratures for conventional methods to evaluate the singular integrals in the stiffness matrices, except that we only need to evaluate O(N) of them instead of O(N 2) of them. Numerical results are presented to show the utility of these fast methods.

Hong Wang

### 41. Peridynamic Functionally Graded and Porous Materials: Modeling Fracture and Damage

In this chapter, we present two peridynamic models for composite materials: a locally homogenized model (FH-PD model, based on results reported in Cheng et al. (Compos Struct 133: 529–546, 2015)) and an intermediately homogenized model (IH-PD model). We use these models to simulate fracture in functionally graded materials (FGMs) and in porous elastic materials. We analyze dynamic fracture, by eccentric impact, of a functionally graded plate with monotonically varying volume fraction of reinforcements. We study the influence of material gradients, elastic waves, and of contact time and magnitude of impact loading on the crack growth from a pre-notch in terms of crack path geometry and crack propagation speed. The results from FH-PD and IH-PD models show the same cracking behavior and final crack patterns. The simulations agree very well, through full failure, with experiments. We discuss advantages offered by the peridynamic models in dynamic fracture of FGMs compared with, for example, FEM-based models. The models lead to a better understanding of how cracks propagate in FGMs and of the factors that control crack path and its velocity in these materials. The IH-PD model has important advantages when compared with the FH-PD model when applied to composite materials with phases of disparate mechanical properties. An application to fracture of porous and elastic materials (following Chen et al. (Peridynamic model for damage and fracture in porous materials, 2017)) shows the major effect local heterogeneities have on fracture behavior and the importance of intermediate homogenization as a modeling approach of crack initiation and growth.

Ziguang Chen, Sina Niazi, Guanfeng Zhang, Florin Bobaru

### 42. Numerical Tools for Improved Convergence of Meshfree Peridynamic Discretizations

Peridynamic models have been employed to simulate a broad range of engineering applications concerning material failure and damage, with the majority of these simulations using a meshfree discretization. This chapter reviews that meshfree discretization, related issues present in peridynamic convergence studies, and possible remedies proposed in the literature. In particular, we discuss two numerical tools, partial-volume algorithms and influence functions, to improve the convergence behavior of numerical solutions in peridynamics. Numerical studies in this chapter involve static and dynamic simulations for linear elastic state-based peridynamic problems.

Pablo Seleson, David J. Littlewood

### 43. Well-Posed Nonlinear Nonlocal Fracture Models Associated with Double-Well Potentials

In this chapter, we consider a generic class of bond-based nonlocal nonlinear potentials and formulate the evolution over suitable function spaces. The peridynamic potential considered in this work is a differentiable version of the original bond-based model introduced in Silling (J Mech Phys Solids 48(1):175–209, 2000). The potential associated with the model has two wells where one well corresponds to linear elastic behavior and the other corresponds to brittle fracture (see Lipton (J Elast 117(1):21–50, 2014; 124(2):143–191, 2016)). The parameters in the potential can be directly related to the elastic tensor and fracture toughness. In this chapter we show that well-posed formulations of the model can be developed over different function spaces. Here we will consider formulations posed over Hölder spaces and Sobolev spaces. The motivation for the Hölder space formulation is to show a priori convergence for the discrete finite difference method. The motivation for the Sobolev formulation is to show a priori convergence for the finite element method. In the following chapter we will show that the discrete approximations converge to well-posed evolutions. The associated convergence rates are given explicitly in terms of time step and the size of the spatial mesh.

Prashant K. Jha, Robert Lipton

### 44. Finite Differences and Finite Elements in Nonlocal Fracture Modeling: A Priori Convergence Rates

In this chapter we present a rigorous convergence analysis of finite difference and finite element approximation of nonlinear nonlocal models. In the previous chapter, we considered a differentiable version of the original bond-based model introduced in Silling (J Mech Phys Solids 48(1):175–209, 2000). There we showed, for a fixed horizon of nonlocal interaction ??, that well-posed formulations of the model can be developed over Hölder spaces and Sobolev spaces. In this chapter we apply these formulations to show a priori convergence for the discrete finite difference and finite element methods. We show that the error made using the forward Euler in time and a finite difference (i.e., piecewise constant) discretization in space with time step Δt and spatial discretization h is of the order of O( Δt + h∕?? 2). For a central difference approximation in time and piecewise linear finite element approximation in space, the approximation error is of the order of O( Δt + h 2∕?? 2). We point out these are the first such error estimates for nonlinear nonlocal fracture formulations and are reported in Jha and Lipton (2017b Numerical analysis of nonlocal fracture models models in holder space. arXiv preprint arXiv:1701.02818. To appear in SIAM Journal on Numerical Analysis 2018) and Jha and Lipton (2017a, Finite element approximation of nonlocal fracture models. arXiv preprint arXiv:1710.07661). We then go on to prove the stability of the semi-discrete approximation and show that the energy of the discrete approximation is bounded in terms of work done by the body force and initial energy put into the system. We look forward to improvements and development of a posteriori error estimation in the coming years.

Prashant K. Jha, Robert Lipton

### 45. Dynamic Damage Propagation with Memory: A State-Based Model

A model for dynamic damage propagation is developed using nonlocal potentials. The model is posed using a state-based peridynamic formulation. The resulting evolution is seen to be well posed. At each instant of the evolution, we identify a damage set. On this set, the local strain has exceeded critical values either for tensile or hydrostatic strain, and damage has occurred. The damage set is nondecreasing with time and is associated with damage state variables defined at each point in the body. We show that a rate form of energy balance holds at each time during the evolution. Away from the damage set, we show that the nonlocal model converges to the linear elastic model in the limit of vanishing nonlocal interaction.

Robert Lipton, Eyad Said, Prashant K. Jha