Showing papers in "Journal of The Mechanics and Physics of Solids in 2014"

The role of mechanics during brain development

Abstract: Convolutions are a classical hallmark of most mammalian brains. Brain surface morphology is often associated with intelligence and closely correlated with neurological dysfunction. Yet, we know surprisingly little about the underlying mechanisms of cortical folding. Here we identify the role of the key anatomic players during the folding process: cortical thickness, stiffness, and growth. To establish estimates for the critical time, pressure, and the wavelength at the onset of folding, we derive an analytical model using the Foppl–von Karman theory. Analytical modeling provides a quick first insight into the critical conditions at the onset of folding, yet it fails to predict the evolution of complex instability patterns in the post-critical regime. To predict realistic surface morphologies, we establish a computational model using the continuum theory of finite growth. Computational modeling not only confirms our analytical estimates, but is also capable of predicting the formation of complex surface morphologies with asymmetric patterns and secondary folds. Taken together, our analytical and computational models explain why larger mammalian brains tend to be more convoluted than smaller brains. Both models provide mechanistic interpretations of the classical malformations of lissencephaly and polymicrogyria. Understanding the process of cortical folding in the mammalian brain has direct implications on the diagnostics of neurological disorders including severe retardation, epilepsy, schizophrenia, and autism.

Phase field modeling of fracture in rubbery polymers. Part I: Finite elasticity coupled with brittle failure

Abstract: This work presents a new phase field model for rate-independent crack propagation in rubbery polymers at large strains and considers details of its numerical implementation. The approach accounts for micro-mechanically based features of both the elastic bulk response as well as the crack toughness of idealized polymer networks. The proposed diffusive crack modeling based on the introduction of a crack phase field overcomes difficulties associated with the computational realization of sharp crack discontinuities, in particular when it comes to complex crack topologies. The crack phase field governs a crack density function, which describes the macroscopic crack surface in the polymer per unit of the reference volume. It provides the basis for the constitutive modeling of a degrading free energy storage and a crack threshold function with a Griffith-type critical energy release rate, that governs the crack propagation in the polymer. Both the energy storage as well as the critical energy release due to fracture can be related to classical statistical network theories of polymers. The proposed framework of diffusive fracture in polymers is formulated in terms of a rate-type variational principle that determines the evolution of the coupled primary variable fields, i.e. the deformation field and the crack phase field. On the computational side, we outline a staggered solution procedure based on a one-pass operator split of the coupled equations, that successively updates in a typical time step the crack phase field and the displacement field. Such a solution algorithm is extremely robust, easy to implement and ideally suited for engineering problems. We finally demonstrate the performance of the phase field formulation of fracture at large strains by means of representative numerical examples.

A chemo-mechanical model of lithiation in silicon

Abstract: We present a chemo-mechanical model to investigate the lithiation-induced phase transformation, morphological evolution, stress generation and fracture in crystalline silicon nanowires ( c -SiNWs). The model couples lithium (Li) diffusion with elasto-plastic deformation in a three-dimensional (3D) setting. Several key features observed from recent transmission electron microscopy (TEM) studies are incorporated into the model, including the sharp interface between the lithiated amorphous shell and unlithiated crystalline core, crystallographic orientation dependent Li–Si reaction rate, and large-strain plasticity. Our simulation results demonstrate that the model faithfully predicts the anisotropic swelling of lithiated SiNWs observed from previous experimental studies. Stress analysis from the finite-deformation model reveals that the SiNWs are prone to surface fracture at the angular sites where two adjacent {110} facets intersect, consistent with previous experimental observations. The mechanistic understanding of the morphological evolution and stress generation sheds light on the design of failure-resistant nanostructured electrodes. Our model offers a framework for the study of the chemo-mechanical degradation in high-capacity electrode materials.

Continuum dislocation dynamics: Towards a physical theory of crystal plasticity

Abstract: The plastic deformation of metals is the result of the motion and interaction of dislocations, line defects of the crystalline structure. Continuum models of plasticity, however, remain largely phenomenological to date, usually do not consider dislocation motion, and fail when materials behavior becomes size dependent. In this work we present a novel plasticity theory based on systematic physical averages of the kinematics and dynamics of dislocation systems. We demonstrate that this theory can predict microstructure evolution and size effects in accordance with experiments and discrete dislocation simulations. The theory is based on only four internal variables per slip system and features physical boundary conditions, dislocation pile ups, dislocation curvature, dislocation multiplication and dislocation loss. The presented theory therefore marks a major step towards a physically based theory of crystal plasticity.

Measurement and modeling of the mechanical and electrochemical response of amorphous Si thin film electrodes during cyclic lithiation

Abstract: A combination of experimental measurements and numerical simulations are used to characterize the mechanical and electrochemical response of thin film amorphous Si electrodes during cyclic lithiation. Parameters extracted from the experiment include the variation of elastic modulus and the flow stress as functions of Li concentration; the strain rate sensitivity; the diffusion coefficient for Li transport in the electrode; the free energy of mixing as a function of Li concentration in the electrode; the exchange current density for the Lithium insertion reaction; as well as reaction rates and diffusion coefficients characterizing the rate of formation of solid-electrolyte interphase layer at the electrode surface. Model predictions are compared with experimental measurements; and the implications for practical Si based electrodes are discussed.

Tension, compression, and bending of superelastic shape memory alloy tubes

Abstract: While many uniaxial tension experiments of shape memory alloys (SMAs) have been published in the literature, relatively few experimental studies address their behavior in compression or bending, despite the prevalence of this latter deformation mode in applications. In this study, superelastic NiTi tubes from a single lot of material were characterized in tension, compression, and pure bending, which allowed us to make direct comparisons between the deformation modes for the first time. Custom built fixtures were used to overcome some long-standing experimental difficulties with performing well-controlled loading and accurate measurements during uniaxial compression (avoiding buckling) and large-rotation bending. In all experiments, the isothermal, global, mechanical responses were measured, and stereo digital image correlation (DIC) was used to measure the evolution of the strain fields on the tube's outer surface. As is characteristic of textured NiTi, our tubes exhibited significant tension–compression asymmetry in their uniaxial responses. Stress-induced transformations in tension exhibited flat force plateaus accompanied by strain localization and propagation. No such localization, however, was observed in compression, and the stress “plateaus” during compression always maintained a positive tangent modulus. While our uniaxial results are similar to the observations of previous researchers, the DIC strain measurements provided details of localized strain behavior with more clarity and allowed more quantitative measurements to be made. Consistent with the tension–compression asymmetry, our bending experiments showed a significant shift of the neutral axis towards the compression side. Furthermore, the tube exhibited strain localization on the tension side, but no localization on the compression side during bending. This is a new observation that has not been explored before. Detailed analysis of the strain distribution across the tube diameter revealed that the traditional assumption of elementary beam theory, that plane sections remain plane, does not hold. Yet when the strain was averaged over a few diameters of axial length, the tensile and compressive responses input into elementary beam theory predicted the global bending response with reasonable accuracy. While it is encouraging that a simple model could predict the moment–curvature response, we recommend that beam theory be used with caution. The averaged strain field can under/over predict local strains by as much as two-fold due to the localized deformation morphology.

Flexoelectricity in soft materials and biological membranes

Abstract: Flexoelectricity and the concomitant emergence of electromechanical size-effects at the nanoscale have been recently exploited to propose tantalizing concepts such as the creation of “apparently piezoelectric” materials without piezoelectric materials, e.g. graphene, emergence of “giant” piezoelectricity at the nanoscale, enhanced energy harvesting, among others. The aforementioned developments pertain primarily to hard ceramic crystals. In this work, we develop a nonlinear theoretical framework for flexoelectricity in soft materials. Using the concept of soft electret materials, we illustrate an interesting nonlinear interplay between the so-called Maxwell stress effect and flexoelectricity, and propose the design of a novel class of apparently piezoelectric materials whose constituents are intrinsically non-piezoelectric. In particular, we show that the electret-Maxwell stress based mechanism can be combined with flexoelectricity to achieve unprecedentedly high values of electromechanical coupling. Flexoelectricity is also important for a special class of soft materials: biological membranes. In this context, flexoelectricity manifests itself as the development of polarization upon changes in curvature. Flexoelectricity is found to be important in a number of biological functions including hearing, ion transport and in some situations where mechanotransduction is necessary. In this work, we present a simple linearized theory of flexoelectricity in biological membranes and some illustrative examples.

Effect of Lode parameter on plastic flow localization after proportional loading at low stress triaxialities

Abstract: The effect of the stress state on the localization of plastic flow in a Levy–von Mises material is investigated numerically. A unit cell model is built with a spherical central void that acts as a defect triggering the onset of flow localization along a narrow band. Periodic boundary conditions are defined along all boundaries of the unit cell. Shear and normal loading is applied such that the macroscopic stress triaxiality and Lode parameter remain constant throughout the entire loading history. Due to the initially orthogonal symmetry of the unit cell model the deformation-induced anisotropy associated with void shape changes, both co-rotational and radial loading paths are considered. The simulation results demonstrate that the macroscopic equivalent plastic strain at the onset of localization after monotonic proportional loading decreases in stress triaxiality and is a convex, non-symmetric function of the Lode parameter. In addition to predicting the onset of localization through unit cell analysis, an analytical criterion is proposed for monotonic proportional loading which defines an open convex envelope in terms of the shear and normal stresses acting on the plane of localization.

Effective toughness of heterogeneous media

Abstract: We propose a versatile approach to computing the effective toughness of heterogeneous media. This approach focusses on the material property independent of the details of the boundary condition. The key idea is what we call a surfing boundary condition, where a steadily propagating crack opening displacement is applied as a boundary condition to a large domain while the crack set is allowed to evolve as it chooses. The approach is verified and used to study examples in brittle fracture. We demonstrate that effective toughness is different from effective or weighted surface area of the crack set. Furthermore, we demonstrate that elastic heterogeneity can have a profound effect on fracture toughness: it can be a significant toughening mechanism and it can lead to toughness asymmetry wherein the toughness depends not only on the direction but also on the sense of propagation. The role of length-scale is also discussed.

Fully coupled peridynamic thermomechanics

Abstract: This chapter concerns the derivation of the coupled peridynamic (PD) thermomechanics equations based on thermodynamic considerations. The generalized peridynamic model for fully coupled thermomechanics is derived using the conservation of energy and the free-energy function. Subsequently, the bond-based coupled PD thermomechanics equations are obtained by reducing the generalized formulation. These equations are also cast into their nondimensional forms. After describing the numerical solution scheme, solutions to certain coupled thermomechanical problems with known previous solutions are presented.

Dynamic stress–strain states for metal foams using a 3D cellular model

Abstract: Dynamic uniaxial impact behaviour of metal foams using a 3D cell-based finite element model is examined. At sufficiently high loading rates, these materials respond by forming ‘shock or consolidation waves’ ( Tan et al., 2005a , Tan et al., 2005b ). However, the existing dynamic experimental methods have limitations in fully informing this behaviour, particularly for solving boundary/initial value problems. Recently, the problem of the shock-like response of an open-cell foam has been examined by Barnes et al. (2014) using the Hugoniot-curve representations. The present study is somewhat complementary to that approach and additionally aims to provide insight into the ‘rate sensitivity’ mechanism applicable to cellular materials. To assist our understanding of the ‘loading rate sensitivity’ behaviour of cellular materials, a virtual ‘test’ method based on the direct impact technique is explored. Following a continuum representation of the response, the strain field calculation method is employed to determine the local strains ahead of and behind the resulting ‘shock front’. The dynamic stress–strain states in the densification stage are found to be different from the quasi-static ones. It is evident that the constitutive behaviour of the cellular material is deformation-mode dependent. The nature of the ‘rate sensitivity’ revealed for cellular materials in this paper is different from the strain-rate sensitivity of dense metals. It is shown that the dynamic stress–strain states behind a shock front of the cellular material lie on a unique curve and each point on the curve corresponds to a particular ‘impact velocity’, referred as the velocity upstream of the shock in this study. The dynamic stress–strain curve is related to a layer-wise collapse mode, whilst the equivalent quasi-static curve is related to a random shear band collapse mode. The findings herein are aimed at improving the experimental test techniques used to characterise the rate-sensitivity behaviour of real cellular materials and providing data appropriate to solving dynamic loading problems in which cellular metals are utilised.

Design of materials with prescribed nonlinear properties

Abstract: We systematically design materials using topology optimization to achieve prescribed nonlinear properties under finite deformation. Instead of a formal homogenization procedure, a numerical experiment is proposed to evaluate the material performance in longitudinal and transverse tensile tests under finite deformation, i.e. stress–strain relations and Poisson׳s ratio. By minimizing errors between actual and prescribed properties, materials are tailored to achieve the target. Both two dimensional (2D) truss-based and continuum materials are designed with various prescribed nonlinear properties. The numerical examples illustrate optimized materials with rubber-like behavior and also optimized materials with extreme strain-independent Poisson׳s ratio for axial strain intervals of e i ∈ [ 0.00 , 0.30 ] .

A Cahn–Hilliard-type phase-field theory for species diffusion coupled with large elastic deformations: Application to phase-separating Li-ion electrode materials

Abstract: We formulate a unified framework of balance laws and thermodynamically-consistent constitutive equations which couple Cahn–Hilliard-type species diffusion with large elastic deformations of a body. The traditional Cahn–Hilliard theory, which is based on the species concentration c and its spatial gradient ∇ c , leads to a partial differential equation for the concentration which involves fourth-order spatial derivatives in c; this necessitates use of basis functions in finite-element solution procedures that are piecewise smooth and globally C 1 - continuous . In order to use standard C 0 - continuous finite-elements to implement our phase-field model, we use a split-method to reduce the fourth-order equation into two second-order partial differential equations (pdes). These two pdes, when taken together with the pde representing the balance of forces, represent the three governing pdes for chemo-mechanically coupled problems. These are amenable to finite-element solution methods which employ standard C 0 - continuous finite-element basis functions. We have numerically implemented our theory by writing a user-element subroutine for the widely used finite-element program Abaqus/Standard. We use this numerically implemented theory to first study the diffusion-only problem of spinodal decomposition in the absence of any mechanical deformation. Next, we use our fully coupled theory and numerical-implementation to study the combined effects of diffusion and stress on the lithiation of a representative spheroidal-shaped particle of a phase-separating electrode material.

A dislocation density based crystal plasticity finite element model: Application to a two-phase polycrystalline HCP/BCC composites

Abstract: We present a multiscale model for anisotropic, elasto-plastic, rate- and temperature-sensitive deformation of polycrystalline aggregates to large plastic strains. The model accounts for a dislocation-based hardening law for multiple slip modes and links a single-crystal to a polycrystalline response using a crystal plasticity finite element based homogenization. It is capable of predicting local stress and strain fields based on evolving microstructure including the explicit evolution of dislocation density and crystallographic grain reorientation. We apply the model to simulate monotonic mechanical response of a hexagonal close-packed metal, zirconium (Zr), and a body-centered cubic metal, niobium (Nb), and study the texture evolution and deformation mechanisms in a two-phase Zr/Nb layered composite under severe plastic deformation. The model predicts well the texture in both co-deforming phases to very large plastic strains. In addition, it offers insights into the active slip systems underlying texture evolution, indicating that the observed textures develop by a combination of prismatic, pyramidal, and anomalous basal slip in Zr and primarily {110}〈111〉 slip and secondly {112}〈111〉 slip in Nb.

Thermomechanics of monolayer graphene: Rippling, thermal expansion and elasticity

Abstract: Thermomechanical properties of monolayer graphene with thermal fluctuation are studied by both statistical mechanics analysis and molecular dynamics (MD) simulations. While the statistical mechanics analysis in the present study is limited by a harmonic approximation, significant anharmonic effects are revealed by MD simulations. The amplitude of out-ofplane thermal fluctuation is calculated for graphene membranes under both zero stress and zero strain conditions. It is found that the fluctuation amplitude follows a power-law scaling with respect to the linear dimension of the membrane, but the roughness exponents are different for the two conditions due to anharmonic interactions between bending and stretching modes. Such thermal fluctuation or rippling is found to be responsible for the effectively negative in-plane thermal expansion of graphene at relatively low temperatures, while a transition to positive thermal expansion is predicted as the anharmonic interactions suppress the rippling effect at high temperatures. Subject to equi-biaxial tension, the amplitude of thermal rippling decreases nonlinearly, and the in-plane stress-strain relation of graphene becomes nonlinear even at infinitesimal strain, in contrast with classical theory of linear elasticity. It is found that the tangent biaxial modulus of graphene depends on strain non-monotonically, decreases with increasing temperature, and depends on membrane size. Both statistical mechanics and MD simulations suggest considerable entropic contribution to the thermomechanical properties of graphene, and as a result thermal rippling is intricately coupled with thermal expansion and thermoelasticity for monolayer graphene membranes.

Towards optimal design of locally resonant acoustic metamaterials

Abstract: The paper presents an in-depth analysis of solid locally resonant acoustic metamaterials (LRAMs) consisting of rubber-coated inclusions. Dispersion properties of two-dimensional LRAMs are studied by means of finite-element modal analysis. For an incompressible rubber, only one practically important spectral band gap is found for in-plane modes in a low-frequency range. This result is in striking contrast with the compressible coating case, previously studied in the literature. For inclusions with a circular cross-section, the lower bound of the band gap can be evaluated exactly by means of the derived analytical solution, which is also valid for compressible coatings and can therefore be used to determine lower bounds of higher band gaps as well. The influence of geometric and material parameters, filling fraction and inclusion shape on the width of the lowest band gap is investigated in detail. Based on the results of this analysis, an optimal microstructure of LRAMs yielding the widest low-frequency band gap is proposed. To achieve the band gap at the lowest possible frequencies in LRAMs suitable for practical applications, the use of the tungsten core material is advised, as a safe and economically viable alternative to commonly considered lead and gold. Two configurations of LRAM with various sizes of coated tungsten cylindrical inclusions with circular cross-section are considered. The evolution of dispersion spectra due to the presence of different inclusions is investigated, and the parameters for optimal design of LRAMs are determined.

Effect of deformation frequency on temperature and stress oscillations in cyclic phase transition of NiTi shape memory alloy

Abstract: Distinctive temperature and stress oscillations can be observed in superelastic shape memory alloys (SMAs) when they subject to displacement-controlled cyclic phase transition. In this paper, we examine the effect of the deformation frequency on the thermal and mechanical responses of the polycrystalline superelastic NiTi rods under stress-induced cyclic phase transition. By synchronized measurement of the evolutions in overall temperature and stress–strain curve over the frequency range of 0.0004–1 Hz (corresponding average strain rate range of 4.8×10−5/s–1.2×10−1/s) in stagnant air, it was found that both the temperature evolution and the stress–strain curve vary significantly with the frequency and the number of cycles. For each frequency, steady-state cyclic thermal and mechanical responses of the specimen were reached after a transient stage, exhibiting stabilization. In the steady-state, the average temperature oscillated around a mean temperature plateau which increased monotonically with the frequency and rose rapidly in the high frequency range due to the rapid accumulation of hysteresis heat. The oscillation was mainly caused by the release and absorption of latent heat and increased with the frequency, eventually reaching a saturation value. The variations in the stress responses followed similar frequency dependence as the temperature. The steady-state stress–strain hysteresis loop area, as a measure of the material׳s damping capacity, first increased then decreased with the frequency in a non-monotonic manner. The experimental data were analyzed and discussed based on the simplified lumped heat transfer analysis and the Clausius–Clapeyron relationship, incorporating the inherent thermomechanical coupling in the material׳s response. We found that, for given material's properties and specimen geometries, all such frequency-dependent variations in temperature, stress and damping capacity were essentially determined by the competition between the time scale of the heat release (i.e. the phase transition frequency) and the time scale of the heat transfer to the ambient. The results emphasize that, the two time scales of loading and heat transfer must be clearly specified when characterizing and modeling the cyclic behavior of SMA materials.

Defects controlled wrinkling and topological design in graphene

Abstract: Due to its atomic scale thickness, the deformation energy in a free standing graphene sheet can be easily released through out-of-plane wrinkles which, if controllable, may be used to tune the electrical and mechanical properties of graphene. Here we adopt a generalized von Karman equation for a flexible solid membrane to describe graphene wrinkling induced by a prescribed distribution of topological defects such as disclinations (heptagons or pentagons) and dislocations (heptagon–pentagon dipoles). In this framework, a given distribution of topological defects in a graphene sheet is represented as an eigenstrain field which is determined from a Poisson equation and can be conveniently implemented in finite element (FEM) simulations. Comparison with atomistic simulations indicates that the proposed model, with only three parameters (i.e., bond length, stretching modulus and bending stiffness), is capable of accurately predicting the atomic scale wrinkles near disclination/dislocation cores while also capturing the large scale graphene configurations under specific defect distributions such as those leading to a sinusoidal surface ruga 2 or a catenoid funnel.

Size-dependence of the dielectric breakdown strength from nano- to millimeter scale

Abstract: Dielectric breakdown decisively determines the reliability of nano- to centimeter sized electronic devices and components. Nevertheless, a systematic investigation of this phenomenon over the relevant lengths scales and materials classes is still missing. Here, the thickness and permittivity-dependence of the dielectric breakdown strength of insulating crystalline and polymer materials from the millimeter down to the nanometer scale is investigated. While the dependence of breakdown strength on permittivity was found to be thickness-independent for materials in the nm–mm range, the magnitude of the breakdown strength was found to change from a thickness-independent, intrinsic regime, to a thickness-dependent, extrinsic regime. The transition-thickness is interpreted as the characteristic length of a breakdown-initiating conducting filament. The results are in agreement with a model, where the dielectric breakdown strength is defined in terms of breakdown toughness and length of a conducting filament.

3D printed, bio-inspired prototypes and analytical models for structured suture interfaces with geometrically-tuned deformation and failure behavior

Abstract: Geometrically structured interfaces in nature possess enhanced, and often surprising, mechanical properties, and provide inspiration for materials design. This paper investigates the mechanics of deformation and failure mechanisms of suture interface designs through analytical models and experiments on 3D printed polymer physical prototypes. Suture waveforms with generalized trapezoidal geometries (trapezoidal, rectangular, anti-trapezoidal, and triangular) are studied and characterized by several important geometric parameters: the presence or absence of a bonded tip region, the tip angle, and the geometry. It is shown that a wide range (in some cases as great as an order of magnitude) in stiffness, strength, and toughness is achievable dependent on tip bonding, tip angle, and geometry. Suture interfaces with a bonded tip region exhibit a higher initial stiffness due to the greater load bearing by the skeletal teeth, a double peak in the stress–strain curve corresponding to the failure of the bonded tip and the failure of the slanted interface region or tooth, respectively, and an additional failure and toughening mechanism due to the failure of the bonded tip. Anti-trapezoidal geometries promote the greatest amplification of properties for suture interfaces with a bonded tip due the large tip interface area. The tip angle and geometry govern the stress distributions in the teeth and the ratio of normal to shear stresses in the interfacial layers, which together determine the failure mechanism of the interface and/or the teeth. Rectangular suture interfaces fail by simple shearing of the interfaces. Trapezoidal and triangular suture interfaces fail by a combination of shear and tensile normal stresses in the interface, leading to plastic deformation, cavitation events, and subsequent stretching of interface ligaments with mostly elastic deformation in the teeth. Anti-trapezoidal suture interfaces with small tip angles have high stress concentrations in the teeth and fail catastrophically by tooth failure, whereas larger tip angles exhibit a shear failure of the interfaces. Therefore, larger tip angles and trapezoidal or triangular geometries promote graceful failure, and smaller tip angles and anti-trapezoidal geometries promote more brittle-like failure. This dependence is reminiscent of biological systems, which exhibit a range of failure behaviors with limited materials and varied geometry. Triangular geometries uniquely exhibit uniform stress distributions in its teeth and promote the greatest amplification of mechanical properties. In both the bonded and unbonded cases, the predictions from the presented analytical models and experimental results on 3D printed prototypes show excellent agreement. This validates the analytical models and allows for the models to be used as a tool for the design of new materials and interfaces with tailored mechanical behavior.

Designing nacre-like materials for simultaneous stiffness, strength and toughness: Optimum materials, composition, microstructure and size

Abstract: Nacre, bone and spider silk are staggered composites where inclusions of high aspect ratio reinforce a softer matrix. Such staggered composites have emerged through natural selection as the best configuration to produce stiffness, strength and toughness simultaneously. As a result, these remarkable materials are increasingly serving as model for synthetic composites with unusual and attractive performance. While several models have been developed to predict basic properties for biological and bio-inspired staggered composites, the designer is still left to struggle with finding optimum parameters. Unresolved issues include choosing optimum properties for inclusions and matrix, and resolving the contradictory effects of certain design variables. Here we overcome these difficulties with a multi-objective optimization for simultaneous high stiffness, strength and energy absorption in staggered composites. Our optimization scheme includes material properties for inclusions and matrix as design variables. This process reveals new guidelines, for example the staggered microstructure is only advantageous if the tablets are at least five times stronger than the interfaces, and only if high volume concentrations of tablets are used. We finally compile the results into a step-by-step optimization procedure which can be applied for the design of any type of high-performance staggered composite and at any length scale. The procedure produces optimum designs which are consistent with the materials and microstructure of natural nacre, confirming that this natural material is indeed optimized for mechanical performance.

An extended constitutive correspondence formulation of peridynamics based on nonlinear bond-strain measures

Abstract: An extension of the constitutive correspondence framework of peridynamics is proposed. The main motivation is to address unphysical deformation modes which are shown to be permitted in the original constitutive formulation. The specific problem of matter interpenetration observed in numerical discretizations of peridynamics has usually been treated by adding short-range forces between neighboring particles in the discretization. Here, we propose a solution that is rooted directly within the nonlocal theory. The basic approach is to introduce generalized nonlocal peridynamic strain tensors based on corresponding bond-level Seth–Hill strain measures which inherently avoid violations of the matter interpenetration constraint. Several analytic examples are used to show that the modified theory avoids issues of matter interpenetration in cases where the original theory fails. The resulting extended constitutive correspondence framework supports general classic constitutive laws as originally intended and is also shown to be ordinary.

A hierarchical computational model for stretchable interconnects with fractal-inspired designs

Abstract: Stretchable electronics that require functional components with high areal coverages, antennas with small sizes and/or electrodes with invisibility under magnetic resonance imaging can benefit from the use of electrical wiring constructs that adopt fractal inspired layouts. Due to the complex and diverse microstructures inherent in high order interconnects/electrodes/antennas with such designs, traditional non-linear postbuckling analyses based on conventional finite element analyses (FEA) can be cumbersome and time-consuming. Here, we introduce a hierarchical computational model (HCM) based on the mechanism of ordered unraveling for postbuckling analysis of fractal inspired interconnects, in designs previously referred to as ‘self-similar’, under stretching. The model reduces the computational efforts of traditional approaches by many orders of magnitude, but with accurate predictions, as validated by experiments and FEA. As the fractal order increases from 1 to 4, the elastic stretchability can be enhanced by � 200 times, clearly illustrating the advantage of simple concepts in fractal design. These results, and the model in general, can be exploited in the development of optimal designs in wide ranging classes of stretchable electronics systems.

Modelling crystal plasticity by 3D dislocation dynamics and the finite element method: The Discrete-Continuous Model revisited

Abstract: A unified model coupling 3D dislocation dynamics (DD) simulations with the finite element (FE) method is revisited. The so-called Discrete-Continuous Model (DCM) aims to predict plastic flow at the (sub-)micron length scale of materials with complex boundary conditions. The evolution of the dislocation microstructure and the short-range dislocation–dislocation interactions are calculated with a DD code. The long-range mechanical fields due to the dislocations are calculated by a FE code, taking into account the boundary conditions. The coupling procedure is based on eigenstrain theory, and the precise manner in which the plastic slip, i.e. the dislocation glide as calculated by the DD code, is transferred to the integration points of the FE mesh is described in full detail. Several test cases are presented, and the DCM is applied to plastic flow in a single-crystal Nickel-based superalloy.

Relating pore shape to the non-linear response of periodic elastomeric structures

Abstract: By introducing a periodic array of pores in an elastic matrix, instabilities with wavelengths that are of the order of the size of the microstructure can be triggered. Interestingly, these instabilities can be utilized to design a novel class of responsive materials. Possible applications include materials with unusual properties such as negative Poisson's ratio, phononic and photonic switches and colorful and reconfigurable displays. Although shape plays an important role in the design and performance of periodic materials, so far only the non-linear response of structures with circular and elliptical pores has been investigated and the effect of the pore shape on the structural response has not yet been explored. Here, we numerically explore the effect of pore shape on the non-linear response of a square array of pores in an elastomeric matrix. Our results show that pore shape can be used effectively to design material with desired properties and to control attractive features of soft porous systems, such as their stiffness, critical strain and negative Poisson's ratio.

Phase field approach to martensitic phase transformations with large strains and interface stresses

Abstract: Thermodynamically consistent phase field theory for multivariant martensitic transformations, which includes large strains and interface stresses, is developed. Theory is formulated in a way that some geometrically nonlinear terms do not disappear in the geometrically linear limit, which in particular allowed us to introduce the expression for the interface stresses consistent with the sharp interface approach. Namely, for the propagating nonequilibrium interface, a structural part of the interface Cauchy stresses reduces to a biaxial tension with the magnitude equal to the temperature-dependent interface energy. Additional elastic and viscous contributions to the interface stresses do not require separate constitutive equations and are determined by solution of the coupled system of phase field and mechanics equations. Ginzburg–Landau equations are derived for the evolution of the order parameters and temperature evolution equation. Boundary conditions for the order parameters include variation of the surface energy during phase transformation. Because elastic energy is defined per unit volume of unloaded (intermediate) configuration, additional contributions to the Ginzburg–Landau equations and the expression for entropy appear, which are important even for small strains. A complete system of equations for fifth- and sixth-degree polynomials in terms of the order parameters is presented in the reference and actual configurations. An analytical solution for the propagating interface and critical martensitic nucleus which includes distribution of components of interface stresses has been found for the sixth-degree polynomial. This required resolving a fundamental problem in the interface and surface science: how to define the Gibbsian dividing surface, i.e., the sharp interface equivalent to the finite-width interface. An unexpected, simple solution was found utilizing the principle of static equivalence. In fact, even two equations for determination of the dividing surface follow from the equivalence of the resultant force and zero-moment condition. For the obtained analytical solution for the propagating interface, both conditions determine the same dividing surface, i.e., the theory is noncontradictory. A similar formalism can be developed for the phase field approach to diffusive phase transformations described by the Cahn–Hilliard equation, twinning, dislocations, fracture, and their interaction.

Metaconcrete: designed aggregates to enhance dynamic performance

Abstract: We propose a new type of concrete for the attenuation of elastic waves induced by dynamic excitation In this metamaterial, which we call metaconcrete, the stone, sand, and gravel aggregates of standard concrete are replaced with spherical inclusions consisting of a heavy metal core coated with a soft outer layer These engineered aggregates can be tuned so that particular frequencies of a propagating blast wave will activate resonant oscillations of the heavy mass within the inclusions The resonant behavior causes the system to exhibit negative effective mass, and this interaction between the wave motion and the resonant aggregates results in the attenuation of the applied dynamic loading We introduce the concept of negative mass by deriving the effective momentum mass for the system and we define the geometrical and material parameters for the design of resonant aggregates We develop finite element models for the analysis of metaconcrete behavior, defining a section of slab containing a periodic arrangement of inclusions By computing the energy histories for the system when subject to a blast load, we show that there is a transfer of energy between the inclusions and the surrounding mortar The inclusions are able to absorb a significant portion of the applied energy, resulting in a reduction in the amount of stress carried by the mortar phase and greatly improving the ability of the material to resist damage under explosive dynamic loading

Some basic questions on mechanosensing in cell–substrate interaction

Abstract: Cells constantly probe their surrounding microenvironment by pushing and pulling on the extracellular matrix (ECM). While it is widely accepted that cell induced traction forces at the cell–matrix interface play essential roles in cell signaling, cell migration and tissue morphogenesis, a number of puzzling questions remain with respect to mechanosensing in cell–substrate interactions. Here we show that these open questions can be addressed by modeling the cell–substrate system as a pre-strained elastic disk attached to an elastic substrate via molecular bonds at the interface. Based on this model, we establish analytical and numerical solutions for the displacement and stress fields in both cell and substrate, as well as traction forces at the cell–substrate interface. We show that the cell traction generally increases with distance away from the cell center and that the traction-distance relationship changes from linear on soft substrates to exponential on stiff substrates. These results indicate that cell adhesion and migration behaviors can be regulated by cell shape and substrate stiffness. Our analysis also reveals that the cell traction increases linearly with substrate stiffness on soft substrates but then levels off to a constant value on stiff substrates. This biphasic behavior in the dependence of cell traction on substrate stiffness immediately sheds light on an existing debate on whether cells sense mechanical force or deformation when interacting with their surroundings. Finally, it is shown that the cell induced deformation field decays exponentially with distance away from the cell. The characteristic length of this decay is comparable to the cell size and provides a quantitative measure of how far cells feel into the ECM.

Singularity-free dislocation dynamics with strain gradient elasticity

Abstract: The singular nature of the elastic fields produced by dislocations presents conceptual challenges and computational difficulties in the implementation of discrete dislocation-based models of plasticity. In the context of classical elasticity, attempts to regularize the elastic fields of discrete dislocations encounter intrinsic difficulties. On the other hand, in gradient elasticity, the issue of singularity can be removed at the outset and smooth elastic fields of dislocations are available. In this work we consider theoretical and numerical aspects of the non-singular theory of discrete dislocation loops in gradient elasticity of Helmholtz type, with interest in its applications to three dimensional dislocation dynamics (DD) simulations. The gradient solution is developed and compared to its singular and non-singular counterparts in classical elasticity using the unified framework of eigenstrain theory. The fundamental equations of curved dislocation theory are given as non-singular line integrals suitable for numerical implementation using fast one-dimensional quadrature. These include expressions for the interaction energy between two dislocation loops and the line integral form of the generalized solid angle associated with dislocations having a spread core. The single characteristic length scale of Helmholtz elasticity is determined from independent molecular statics (MS) calculations. The gradient solution is implemented numerically within our variational formulation of DD, with several examples illustrating the viability of the non-singular solution. The displacement field around a dislocation loop is shown to be smooth, and the loop self-energy non-divergent, as expected from atomic configurations of crystalline materials. The loop nucleation energy barrier and its dependence on the applied shear stress are computed and shown to be in good agreement with atomistic calculations. DD simulations of Lomer–Cottrell junctions in Al show that the strength of the junction and its configuration are easily obtained, without ad-hoc regularization of the singular fields. Numerical convergence studies related to the implementation of the non-singular theory in DD are presented.

Effect of inclusion density on ductile fracture toughness and roughness

Abstract: Three dimensional calculations of ductile fracture under mode I plane strain, small scale yielding conditions are carried out using an elastic-viscoplastic constitutive relation for a progressively cavitating solid with two populations of void nucleating second phase particles Larger inclusions that result in void nucleation at an early stage are modeled discretely while smaller particles that require large strains to nucleate voids are homogeneously distributed Full field solutions are obtained for eight volume fractions, ranging from 1% to 19%, of randomly distributed larger inclusions For each volume fraction calculations are carried out for seven random distributions of inclusion centers Crack growth resistance curves and fracture surface roughness statistics are calculated using standard procedures The crack growth resistance is characterized in terms of both J IC and the tearing modulus TR For all volume fractions considered, the computed fracture surfaces are self-affine over a size range of nearly two orders of magnitude with a microstructure independent roughness exponent of 053 with a standard error of 00023 The cut-off length of the scale invariant regime is found to depend on the inclusion volume fraction Consideration of the full statistics of the fracture surface roughness revealed other parameters that vary with inclusion volume fraction For smaller values of the discretely modeled inclusion volume fraction (r7%), there is a linear correlation between several measures of fracture surface roughness and both JIC and TR I n this regime crack growth is dominated by a void-by-void process For greater values of the discretely modeled inclusion volume fraction, crack growth mainly involves multiple void interactions and no such correlation is found