Showing papers in "Journal of The Mechanics and Physics of Solids in 2008"

A microstructure-dependent Timoshenko beam model based on a modified couple stress theory

Abstract: A microstructure-dependent Timoshenko beam model is developed using a variational formulation. It is based on a modified couple stress theory and Hamilton's principle. The new model contains a material length scale parameter and can capture the size effect, unlike the classical Timoshenko beam theory. Moreover, both bending and axial deformations are considered, and the Poisson effect is incorporated in the current model, which differ from existing Timoshenko beam models. The newly developed non-classical beam model recovers the classical Timoshenko beam model when the material length scale parameter and Poisson's ratio are both set to be zero. In addition, the current Timoshenko beam model reduces to a microstructure-dependent Bernoulli–Euler beam model when the normality assumption is reinstated, which also incorporates the Poisson effect and can be further reduced to the classical Bernoulli–Euler beam model. To illustrate the new Timoshenko beam model, the static bending and free vibration problems of a simply supported beam are solved by directly applying the formulas derived. The numerical results for the static bending problem reveal that both the deflection and rotation of the simply supported beam predicted by the new model are smaller than those predicted by the classical Timoshenko beam model. Also, the differences in both the deflection and rotation predicted by the two models are very large when the beam thickness is small, but they are diminishing with the increase of the beam thickness. Similar trends are observed for the free vibration problem, where it is shown that the natural frequency predicted by the new model is higher than that by the classical model, with the difference between them being significantly large only for very thin beams. These predicted trends of the size effect in beam bending at the micron scale agree with those observed experimentally. Finally, the Poisson effect on the beam deflection, rotation and natural frequency is found to be significant, which is especially true when the classical Timoshenko beam model is used. This indicates that the assumption of Poisson's effect being negligible, which is commonly used in existing beam theories, is inadequate and should be individually verified or simply abandoned in order to obtain more accurate and reliable results.

A theory of coupled diffusion and large deformation in polymeric gels

Abstract: A large quantity of small molecules may migrate into a network of long polymers, causing the network to swell, forming an aggregate known as a polymeric gel. This paper formulates a theory of the coupled mass transport and large deformation. The free energy of the gel results from two molecular processes: stretching the network and mixing the network with the small molecules. Both the small molecules and the long polymers are taken to be incompressible, a constraint that we enforce by using a Lagrange multiplier, which coincides with the osmosis pressure or the swelling stress. The gel can undergo large deformation of two modes. The first mode results from the fast process of local rearrangement of molecules, allowing the gel to change shape but not volume. The second mode results from the slow process of long-range migration of the small molecules, allowing the gel to change both shape and volume. We assume that the local rearrangement is instantaneous, and model the long-range migration by assuming that the small molecules diffuse inside the gel. The theory is illustrated with a layer of a gel constrained in its plane and subject to a weight in the normal direction. We also predict the scaling behavior of a gel under a conical indenter.

A nonlinear field theory of deformable dielectrics

Abstract: Two difficulties have long troubled the field theory of dielectric solids. First, when two electric charges are placed inside a dielectric solid, the force between them is not a measurable quantity. Second, when a dielectric solid deforms, the true electric field and true electric displacement are not work conjugates. These difficulties are circumvented in a new formulation of the theory in this paper. Imagine that each material particle in a dielectric is attached with a weight and a battery, and prescribe a field of virtual displacement and a field of virtual voltage. Associated with the virtual work done by the weights and inertia, define the nominal stress as the conjugate to the gradient of the virtual displacement. Associated with the virtual work done by the batteries, define the nominal electric displacement as the conjugate to the gradient of virtual voltage. The approach does not start with Newton's laws of mechanics and Maxwell–Faraday theory of electrostatics, but produces them as consequences. The definitions lead to familiar and decoupled field equations. Electromechanical coupling enters the theory through material laws. In the limiting case of a fluid dielectric, the theory recovers the Maxwell stress. The approach is developed for finite deformation, and is applicable to both elastic and inelastic dielectrics. As applications of the theory, we discuss material laws for elastic dielectrics, and study infinitesimal fields superimposed upon a given field, including phenomena such as vibration, wave propagation, and bifurcation.

Nonlocal shell model for elastic wave propagation in single- and double-walled carbon nanotubes

Abstract: This paper investigates the transverse and torsional wave in single- and double-walled carbon nanotubes (SWCNTs and DWCNTs), focusing on the effect of carbon nanotube microstructure on wave dispersion. The SWCNTs and DWCNTs are modeled as nonlocal single and double elastic cylindrical shells. Molecular dynamics (MD) simulations indicate that the wave dispersion predicted by the nonlocal elastic cylindrical shell theory shows good agreement with that of the MD simulations in a wide frequency range up to the terahertz region. The nonlocal elastic shell theory provides a better prediction of the dispersion relationships than the classical shell theory when the wavenumber is large enough for the carbon nanotube microstructure to have a significant influence on the wave dispersion. The nonlocal shell models are required when the wavelengths are approximately less than 2.36×10 −9 and 0.95×10 −9 m for transverse wave in armchair (15,15) SWCNT and torsional wave in armchair (10,10) SWCNT, respectively. Moreover, an MD-based estimation of the scale coefficient e 0 for the nonlocal elastic cylindrical shell model is suggested. Due to the small-scale effects of SWCNTs and the interlayer van der Waals interaction of DWCNTs, the phase difference of the transverse wave in the inner and outer tube can be observed in MD simulations in wave propagation at high frequency. However, the van der Waals interaction has little effect on the phase difference of transverse wave.

A thermoviscoelastic model for amorphous shape memory polymers: Incorporating structural and stress relaxation

Abstract: A thermoviscoelastic constitutive model is developed for amorphous shape memory polymers (SMP) based on the hypothesis that structural and stress relaxation are the primary molecular mechanisms of the shape memory effect and its time-dependence. This work represents a new and fundamentally different approach to modeling amorphous SMPs. A principal feature of the constitutive model is the incorporation of the nonlinear Adam–Gibbs model of structural relaxation and a modified Eyring model of viscous flow into a continuum finite–deformation thermoviscoelastic framework. Comparisons with experiments show that the model can reproduce the strain–temperature response, the temperature and strain-rate dependent stress–strain response, and important features of the temperature dependence of the shape memory response. Because the model includes structural relaxation, the shape memory response also exhibits a dependence on the cooling and heating rates.

Finite deformation thermo-mechanical behavior of thermally induced shape memory polymers

Abstract: Shape memory polymers (SMPs) are polymers that can demonstrate programmable shape memory effects. Typically, an SMP is pre-deformed from an initial shape to a deformed shape by applying a mechanical load at the temperature TH>Tg. It will maintain this deformed shape after subsequently lowering the temperature to TL Tg, where the initial shape is recovered. In this paper, the finite deformation thermo-mechanical behaviors of amorphous SMPs are experimentally investigated. Based on the experimental observations and an understanding of the underlying physical mechanism of the shape memory behavior, a three-dimensional (3D) constitutive model is developed to describe the finite deformation thermo-mechanical response of SMPs. The model in this paper has been implemented into an ABAQUS user material subroutine (UMAT) for finite element analysis, and numerical simulations of the thermo-mechanical experiments verify the efficiency of the model. This model will serve as a modeling tool for the design of more complicated SMP-based structures and devices.

Mechanics of deformation-triggered pattern transformations and superelastic behavior in periodic elastomeric structures

Abstract: Recently, novel and uniform deformation-induced pattern transformations have been found in periodic elastomeric cellular solids upon reaching a critical value of applied load [ Mullin, T., Deschanel, S., Bertoldi, K., Boyce, M.C., 2007. Pattern transformation triggered by deformation. Phys. Rev. Lett. 99, 084301 ; Boyce, M.C., Prange, S.M., Bertoldi, K., Deschanel, S., Mullin, T., 2008. Mechanics of periodic elastomeric structures. In: Boukamel, Laiarinandrasana, Meo, Verron (Eds.), Constitutive Models for Rubber, vol. V. Taylor & Francis Group, London, pp. 3–7 ]. Here, the mechanics of the deformation behavior of several periodically patterned two-dimensional elastomeric sheets are investigated experimentally and through numerical simulation. Square and oblique lattices of circular voids and rectangular lattices of elliptical voids are studied. The numerical results clearly show the mechanism of the pattern switch for each microstructure to be a form of local elastic instability, giving reversible and repeatable transformation events as confirmed by experiments. Post-deformation transformation is observed to accentuate the new pattern and is found to be elastic and to occur at nearly constant stress, resulting in a superelastic behavior. The deformation-induced transformations have been physically realized on structures constructed at the millimeter length scale. This behavior should also persist at the micro and nano length scales, providing opportunities for transformative photonic and phononic crystals which can switch in a controlled manner and also exploiting the phenomenon to imprint complex patterns.

An interacting micro-crack damage model for failure of brittle materials under compression

Abstract: A model is developed for brittle failure under compressive loading with an explicit accounting of micro-crack interactions. The model incorporates a pre-existing flaw distribution in the material. The macroscopic inelastic deformation is assumed to be due to the nucleation and growth of tensile “wing” micro-cracks associated with frictional sliding on these flaws. Interactions among the cracks are modeled by means of a crack-matrix-effective-medium approach in which each crack experiences a stress field different from that acting on isolated cracks. This yields an effective stress intensity factor at the crack tips which is utilized in the formulation of the crack growth dynamics. Load-induced damage in the material is defined in terms of a scalar crack density parameter, the evolution of which is a function of the existing flaw distribution and the crack growth dynamics. This methodology is applied for the case of uniaxial compression under constant strain rate loading. The model provides a natural prediction of a peak stress (defined as the compressive strength of the material) and also of a transition strain rate, beyond which the compressive strength increases dramatically with the imposed strain rate. The influences of the crack growth dynamics, the initial flaw distribution, and the imposed strain rate on the constitutive response and the damage evolution are studied. It is shown that different characteristics of the flaw distribution are dominant at different imposed strain rates: at low rates the spread of the distribution is critical, while at high strain rates the total flaw density is critical.

Asperity contact theories: Do they predict linearity between contact area and load?

Abstract: During the last few years, the scientific community has been debating about which theory of contact between rough surfaces can be considered as the most accurate. The authors have been attracted by such a discussion and in this paper try to give their personal thought and contribution to this debate. We present a critical analysis of the principal contact theories of rough surfaces. We focus on the multiasperity contact models (which are all based on the original idea of Greenwood and Williamson (GW) [1966. Proc. R. Soc. London A 295, 300]), and also briefly discuss a relatively recent contact theory developed by Persson [2001. J. Chem. Phys. 115, 3840]. For small loads both asperity contact models and Persson's theory predict a linear relation between the area of true contact and the applied external load, but the two theories differ for the constant of proportionality. However, this is not the only difference between the two approaches. Indeed, we show that the fully calculated predictions of asperity contact models very rapidly deviates from the predicted linear relation already for very small and in many cases unrealistic vanishing applied loads and contact areas. Moreover, this deviation becomes more and more important as the PSD breadth parameter α (as defined by Nayak) increases. Therefore, the asymptotic linear relation of multiasperity contact theories turns out to be only an academic result. On the contrary, Persson's theory is not affected by α and shows a linear behavior between contact area and load up to 10–15% of the nominal contact area, i.e. for physical reasonable loads. The authors also prove that, at high separation, all multiasperity contact models, which take into account the influence of summit curvature variation as a function of summit height, necessarily converge to a (slightly) improved version of the GW model, which, therefore, remains one of the most important milestones in the field of contact mechanics of rough surfaces.

Multiple interacting circular nano-inhomogeneities with surface/interface effects

Abstract: A two-dimensional problem of multiple interacting circular nano-inhomogeneities or/and nano-pores is considered. The analysis is based on the Gurtin and Murdoch model [Gurtin, M.E., Murdoch, A.I., 1975. A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57, 291–323.] in which the interfaces between the nano-inhomogeneities and the matrix are regarded as material surfaces that possess their own mechanical properties and surface tension. The precise component forms of Gurtin and Murdoch's three-dimensional equations are derived for interfaces of arbitrary shape to provide a basis for critical review of various modifications used in the literature. The two-dimensional specification of these equations is considered and their representation in terms of complex variables is provided. A semi-analytical method is proposed to solve the problem. Solutions to several example problems are presented to: (i) examine the difference between the results obtained with the original and modified Gurtin and Murdoch's equations, (ii) compare the results obtained using Gurtin and Murdoch's model and those for a problem of nano-inhomogeneities with thin membrane-type interphase layers, and (iii) demonstrate the effectiveness of the approach in solving problems with multiple nano-inhomogeneities.

Strain gradient plasticity modelling of high-pressure torsion

Abstract: Gradient plasticity modelling combining a micro-structure-related constitutive description of the local material behaviour with a particular gradient plasticity frame is presented. The constitutive formulation is based on a phase mixture model in which the dislocation cell walls and the cell interiors are considered as separate ‘phases’, the respective dislocation densities entering as internal variables. Two distinct physical mechanisms, which give rise to gradient plasticity, are considered. The first one is associated with the occurrence of geometrically necessary dislocations leading to first-order strain gradients; the second one is associated with the reaction stresses due to plastic strain incompatibilities between neighbouring grains, which lead to second-order strain gradients. These two separate variants of gradient plasticity were applied to the case of high-pressure torsion: a process known to result in a fairly uniform, ultrafine grained structure of metals. It is shown that the two complementary variants of gradient plasticity can both account for the experimental results, thus resolving a controversial issue of the occurrence of a uniform micro-structure as a result of an inherently non-uniform process.

Buckling of a stiff film bound to a compliant substrate—Part I:: Formulation, linear stability of cylindrical patterns, secondary bifurcations

Abstract: The buckling of a thin elastic film bound to a compliant substrate is studied: we analyze the different patterns that arise as a function of the biaxial residual compressive stress in the film. We first clarify the boundary conditions to be used at the interface between film and substrate. We carry out the linear stability analysis of the classical pattern made of straight stripes, and point out secondary instabilities leading to the formation of undulating stripes, varicose, checkerboard or hexagonal patterns. Straight stripes are found to be stable in a narrow window of load parameters only. We present a weakly nonlinear post-buckling analysis of these patterns: for equi-biaxial residual compression, straight wrinkles are never stable and square checkerboard patterns are found to be optimal just above threshold; for anisotropic residual compression, straight wrinkles are present above a primary threshold and soon become unstable with respect to undulating stripes. These results account for many of the previously published experimental or numerical results on this geometry.

A constitutive theory for shape memory polymers. Part I: Large deformations

Abstract: A constitutive theory is developed for shape memory polymers It is to describe the thermomechanical properties of such materials under large deformations The theory is based on the idea, which is developed in the work of Liu et al [2006 Thermomechanics of shape memory polymers: uniaxial experiments and constitutive modeling Int J Plasticity 22, 279–313], that the coexisting active and frozen phases of the polymer and the transitions between them provide the underlying mechanisms for strain storage and recovery during a shape memory cycle General constitutive functions for nonlinear thermoelastic materials are used for the active and frozen phases Also used is an internal state variable which describes the volume fraction of the frozen phase The material behavior of history dependence in the frozen phase is captured by using the concept of frozen reference configuration The relation between the overall deformation and the stress is derived by integration of the constitutive equations of the coexisting phases As a special case of the nonlinear constitutive model, a neo-Hookean type constitutive function for each phase is considered The material behaviors in a shape memory cycle under uniaxial loading are examined A linear constitutive model is derived from the nonlinear theory by considering small deformations The predictions of this model are compared with experimental measurements

The simulation of dynamic crack propagation using the cohesive segments method

Abstract: The cohesive segments method is a finite element framework that allows for the simulation of the nucleation, growth and coalescence of multiple cracks in solids. In this framework, cracks are introduced as jumps in the displacement field by employing the partition of unity property of finite element shape functions. The magnitude of these jumps are governed by cohesive constitutive relations. In this paper, the cohesive segments method is extended for the simulation of fast crack propagation in brittle solids. The performance of the method is demonstrated in several examples involving crack growth in linear elastic solids under plane stress conditions: tensile loading of a block; shear loading of a block and crack growth along and near a bi-material interface.

On the dynamic electromechanical loading of dielectric elastomer membranes

Abstract: Dielectric elastomer actuators (DEAs) have received considerable attention recently due to large voltage-induced strains, which can be over 100%. Previously, a large deformation quasi-static model that describes the out-of-plane deformations of clamped diaphragms was derived. The numerical model results compare well with quasi-static experimental results for the same configuration. With relevance to dynamic applications, the time-varying response of initially planar dielectric elastomer membranes configured for out-of-plane deformations has not been reported until now. In this paper, an experimental investigation and analysis of the dynamic response of a dielectric elastomer membrane is reported. The experiments were conducted with prestretched DEAs fabricated from 0.5 mm thick polyacrylate films and carbon grease electrodes. The experiments covered the electromechanical spectrum by investigating membrane response due to (i) a time-varying voltage input and (ii) a time-varying pressure input, resulting in a combined electromechanical loading state in both cases. For the time-varying voltage experiments, the membrane had a prestretch of three and was passively inflated to various predetermined states, and then actuated. The pole strains incurred during the inflation were as high as 25.6%, corresponding to slightly less than a hemispherical state. On actuation, the membrane would inflate further, causing a maximum additional strain of 9.5%. For the time-varying pressure experiments, the prestretched membrane was inflated and deflated mechanically while a constant voltage was applied. The membrane was cycled between various predetermined inflation states, the largest of which was nearly hemispherical, which with an applied constant voltage of 3 kV corresponded to a maximum polar strain of 28%. The results from these experiments reveal that the response of the membrane is a departure from the classical dynamic response of continuum membrane structures. The dynamic response of the membrane is that of a damped system with specific deformation shapes reminiscent of the classical membrane mode shapes but without same-phase oscillation, that is to say all parts of the system do not pass through the equilibrium configuration at the same time. Of particular interest is the ability to excite these deformations through a varying electrical load at constant mechanical pressure.

A bio-mechanical model for coupling cell contractility with focal adhesion formation

Abstract: Focal adhesions (FAs) are large, multi-protein complexes that provide a mechanical link between the cytoskeletal contractile machinery and the extracellular matrix. They exhibit mechanosensitive properties; they self-assemble upon application of pulling forces and dissociate when these forces are decreased. We rationalize this mechano-sensitivity from thermodynamic considerations and develop a continuum framework in which the cytoskeletal contractile forces generated by stress fibers drive the assembly of the FA multi-protein complexes. The FA model has three essential features: (i) the low and high affinity integrins co-exist in thermodynamic equilibrium, (ii) the low affinity integrins within the plasma membrane are mobile, and (iii) the contractile forces generated by the stress fibers are in mechanical equilibrium and change the free energies of the integrins. A general two-dimensional framework is presented and the essential features of the model illustrated using one-dimensional examples. Consistent with observations, the coupled stress fiber and FA model predict that (a) the FAs concentrate around the periphery of the cell; (b) the fraction of the cell covered by FAs increases with decreasing cell size while the total FA intensity increases with increasing cell size; and (c) the FA intensity decreases substantially when cell contractility is curtailed.

Biothermomechanics of skin tissues

Abstract: Biothermomechanics of skin is highly interdisciplinary involving bioheat transfer, burn damage, biomechanics and neurophysiology. During heating, thermally induced mechanical stress arises due to the thermal denaturation of collagen, resulting in macroscale shrinkage. Thus, the strain, stress, temperature and thermal pain/damage are highly correlated; in other words, the problem is fully coupled. The aim of this study is to develop a computational approach to examine the heat transfer process and the heat-induced mechanical response, so that the differences among the clinically applied heating modalities can be quantified. Exact solutions for temperature, thermal damage and thermal stress for a single-layer skin model were first derived for different boundary conditions. For multilayer models, numerical simulations using the finite difference method (FDM) and finite element method (FEM) were used to analyze the temperature, burn damage and thermal stress distributions in the skin tissue. The results showed that the thermomechanical behavior of skin tissue is very complex: blood perfusion has little effect on thermal damage but large influence on skin temperature distribution, which, in turn, influences significantly the resulting thermal stress field; the stratum corneum layer, although very thin, has a large effect on the thermomechanical behavior of skin, suggesting that it should be properly accounted for in the modeling of skin thermal stresses; the stress caused by non-uniform temperature distribution in the skin may also contribute to the thermal pain sensation.

A constitutive theory for shape memory polymers. Part II: A linearized model for small deformations

Abstract: A constitutive theory is developed for shape memory polymers. It is to describe the thermomechanical properties of such materials under large deformations. The theory is based on the idea, which is developed in the work of Liu et al. [2006. Thermomechanics of shape memory polymers: uniaxial experiments and constitutive modelling. Int. J. Plasticity 22, 279–313], that the coexisting active and frozen phases of the polymer and the transitions between them provide the underlying mechanisms for strain storage and recovery during a shape memory cycle. General constitutive functions for nonlinear thermoelastic materials are used for the active and frozen phases. Also used is an internal state variable which describes the volume fraction of the frozen phase. The material behavior of history dependence in the frozen phase is captured by using the concept of frozen reference configuration. The relation between the overall deformation and the stress is derived by integration of the constitutive equations of the coexisting phases. As a special case of the nonlinear constitutive model, a neo-Hookean type constitutive function for each phase is considered. The material behaviors in a shape memory cycle under uniaxial loading are examined. A linear constitutive model is derived from the nonlinear theory by considering small deformations. The predictions of this model are compared with experimental measurements.

Influence of single crystal orientation on homogeneous dislocation nucleation under uniaxial loading

Abstract: Atomistic simulations are used to investigate how the stress required for homogeneous nucleation of partial dislocations in single crystal copper under uniaxial loading changes as a function of crystallographic orientation. Molecular dynamics is employed based on an embedded-atom method potential for Cu at 10 and 300 K. Results indicate that non-Schmid parameters are important for describing the calculated dislocation nucleation behavior for single crystal orientations under tension and compression. A continuum relationship is presented that incorporates Schmid and non-Schmid terms to correlate the nucleation stress over all tensile axis orientations within the stereographic triangle. Simulations investigating the temperature dependence of homogeneous dislocation nucleation yield activation volumes of ≈ 0.5 – 2 b 3 and activation energies of ≈ 0.30 eV . For uniaxial compression, full dislocation loop nucleation is observed, in contrast to uniaxial tension. One of the main differences between uniaxial tension and compression is how the applied stress is resolved normal to the slip plane on which dislocations nucleate—in tension, this normal stress is tensile, and in compression, it is compressive. Last, the tension–compression asymmetry is examined as a function of loading axis orientation. Orientations with a high resolved stress normal to the slip plane on which dislocations nucleate have a larger tension–compression asymmetry with respect to dislocation nucleation than those orientations with a low resolved normal stress. The significance of this research is that the resolved stress normal to the slip plane on which dislocations nucleate plays an important role in partial (and full) dislocation loop nucleation in FCC Cu single crystals.

Internal lattice relaxation of single-layer graphene under in-plane deformation

Abstract: For noncentrosymmetric crystals, internal lattice relaxation must be considered for theoretical predictions of elastic properties. This paper develops a molecular dynamics approach for determination of the internal relaxation displacement in a single-layer graphene sheet under macroscopically homogeneous in-plane deformation. Based on an analytical interatomic potential, a generally nonlinear relationship between the internal relaxation displacement and the applied macroscopic strain is obtained explicitly from molecular dynamics simulations with a rhombic unit cell under finite deformation. A linear relationship is derived for relatively small strains, which can be conveniently incorporated into a continuum description of the elastic behavior of graphene. It is found that the internal relaxation has a strong effect on theoretical elastic moduli of graphene. In addition, the relationship between elastic properties for graphene and carbon nanotubes is discussed.

Reversible magneto-elastic behavior : A multiscale approach

Abstract: Magnetic and mechanical behaviors are strongly coupled. But few models are able to describe magneto-mechanical coupling effects. We propose a multiscale approach for the modeling of the reversible magneto-elastic behavior of ferromagnetic materials. This approach stands between macroscopic phenomenological modeling and micromagnetic simulations. We detail first the definition of the magneto-elastic behavior of a single crystal, deduced from energetic considerations made at the scale of magnetic domains and hypotheses concerning the domains microstructure. This model is then applied to the description of the behavior of polycrystalline media, through a multiscale approach. The heterogeneity of stress and magnetic field is taken into account through a self-consistent localization–homogenization scheme, including crystallographic texture data. Results are discussed and compared to experimental data from the literature.

Multiscale cohesive failure modeling of heterogeneous adhesives

Abstract: A novel multiscale cohesive approach that enables prediction of the macroscopic properties of heterogeneous thin layers is presented. The proposed multiscale model relies on the Hill's energy equivalence lemma, implemented in the computational homogenization scheme, to couple the micro- and macro-scales and allows to relate the homogenized cohesive law used to model the failure of the adhesive layer at the macro-scale to the complex damage evolution taking place at the micro-scale. A simple isotropic damage model is used to describe the failure processes at the micro-scale. We establish the upper and lower bounds on the multiscale model and solve several examples to demonstrate the ability of the method to extract physically based macroscopic properties.

Force flux and the peridynamic stress tensor

Abstract: The peridynamic model is a framework for continuum mechanics based on the idea that pairs of particles exert forces on each other across a finite distance. The equation of motion in the peridynamic model is an integro-differential equation. In this paper, a notion of a peridynamic stress tensor derived from nonlocal interactions is defined. At any point in the body, this stress tensor is obtained from the forces within peridynamic bonds that geometrically go through the point. The peridynamic equation of motion can be expressed in terms of this stress tensor, and the result is formally identical to the Cauchy equation of motion in the classical model, even though the classical model is a local theory. We also establish that this stress tensor field is unique in a certain function space compatible with finite element approximations.

Inelastic deformation and energy dissipation in ceramics: A mechanism-based constitutive model

Abstract: A mechanism-based constitutive model is presented for the inelastic deformation and fracture of ceramics. The model comprises four essential features: (i) micro-crack extension rates based on stress-intensity calculations and a crack growth law, (ii) the effect of the crack density on the stiffness, inclusive of crack closure, (iii) plasticity at high confining pressures, and (iv) initial flaws that scale with the grain size. Predictions of stress/strain responses for a range of stress states demonstrate that the model captures the transition from deformation by micro-cracking at low triaxiality to plastic slip at high triaxialities. Moreover, natural outcomes of the model include dilation (or bulking) upon micro-cracking, as well as the increase in the shear strength of the damaged ceramic with increasing triaxiality. Cavity expansion calculations are used to extract some key physics relevant to penetration. Three domains have been identified: (i) quasi-static, where the ceramic fails due to the outward propagation of a compression damage front, (ii) intermediate velocity, where an outward propagating compression damage front is accompanied by an inward propagating tensile (or spallation) front caused by the reflection of the elastic wave from the outer surface and (iii) high velocity, wherein plastic deformation initiates at the inner surface of the shell followed by spalling within a tensile damage front when the elastic wave reflects from the outer surface. Consistent with experimental observations, the cavity pressure is sensitive to the grain size under quasi-static conditions but relatively insensitive under dynamic loadings.

J-integral and crack driving force in elastic–plastic materials

Abstract: This paper discusses the crack driving force in elastic–plastic materials, with particular emphasis on incremental plasticity. Using the configurational forces approach we identify a “plasticity influence term” that describes crack tip shielding or anti-shielding due to plastic deformation in the body. Standard constitutive models for finite strain as well as small strain incremental plasticity are used to obtain explicit expressions for the plasticity influence term in a two-dimensional setting. The total dissipation in the body is related to the near-tip and far-field J-integrals and the plasticity influence term. In the special case of deformation plasticity the plasticity influence term vanishes identically whereas for rigid plasticity and elastic-ideal plasticity the crack driving force vanishes. For steady state crack growth in incremental elastic–plastic materials, the plasticity influence term is equal to the negative of the plastic work per unit crack extension and the total dissipation in the body due to crack propagation and plastic deformation is determined by the far-field J-integral. For non-steady state crack growth, the plasticity influence term can be evaluated by post-processing after a conventional finite element stress analysis. Theory and computations are applied to a stationary crack in a C(T)-specimen to examine the effects of contained, uncontained and general yielding. A novel method is proposed for evaluating J-integrals under incremental plasticity conditions through the configurational body force. The incremental plasticity near-tip and far-field J-integrals are compared to conventional deformational plasticity and experimental J-integrals.

Continuum modeling of a porous solid with pressure-sensitive dilatant matrix

Abstract: The pressure-sensitive plastic response of a material has been Studied in terms of the intrinsic sensitivity of its yield stress to pressure and the presence and growth of cavities This work focus

Surface stress effects on the resonant properties of metal nanowires: The importance of finite deformation kinematics and the impact of the residual surface stress

Abstract: We utilize the recently developed surface Cauchy–Born model, which extends the standard Cauchy–Born theory to account for surface stresses due to undercoordinated surface atoms, to study the coupled influence of boundary conditions and surface stresses on the resonant properties of 〈 1 0 0 〉 gold nanowires with { 1 0 0 } surfaces. There are two major purposes to the present work. First, we quantify, for the first time, variations in the nanowire resonant frequencies due to surface stresses as compared to the corresponding bulk material which does not observe surface effects within a finite deformation framework depending on whether fixed/free or fixed/fixed boundary conditions are utilized. We find that while the resonant frequencies of fixed/fixed nanowires are elevated as compared to the corresponding bulk material, the resonant frequencies of fixed/free nanowires are reduced as a result of compressive strain caused by the surface stresses. Furthermore, we find that for a diverse range of nanowire geometries, the variation in resonant frequencies for both boundary conditions due to surface stresses is a geometric effect that is characterized by the nanowire aspect ratio. The present results are found to agree well with existing experimental data for both types of boundary conditions. The second major goal of this work is to quantify, for the first time, how both the residual (strain-independent) and surface elastic (strain-dependent) parts of the surface stress impact the resonant frequencies of metal nanowires within the framework of nonlinear, finite deformation kinematics. We find that if finite deformation kinematics are considered, the strain-independent surface stress substantially alters the resonant frequencies of the nanowires; however, we also find that the strain-dependent surface stress has a significant effect, one that can be comparable to or even larger than the effect of the strain-independent surface stress depending on the boundary condition, in shifting the resonant frequencies of the nanowires as compared to the bulk material.

On the formulations of higher-order strain gradient crystal plasticity models

Abstract: Recently, several higher-order extensions to the crystal plasticity theory have been proposed to incorporate effects of material length scales that were missing links in the conventional continuum mechanics. The extended theories are classified into work-conjugate and non-work-conjugate types. A common feature of the former is that existence of higher-order stresses work-conjugate to gradients of plastic strain is presumed and an extended principle of virtual work involving such an additional virtual work contribution is formulated. Meanwhile, in the latter type, the higher-order stress quantities do not appear explicitly. Instead, rates of crystallographic slip are influenced by back stresses that arise in response to spatial gradients of the geometrically necessary dislocation densities. The work-conjugate type and the non-work-conjugate type of theories have different theoretical backgrounds and very unlike mathematical representations. Nevertheless, both types of theories predict the same kind of material length scale effects. We have recently shown that there exists some equivalency between the two approaches in the special situation of two-dimensional single slip under small deformation. In this paper, the discussion is extended to a more general situation, i.e. the context of multiple and three-dimensional slip deformations.

An atomistic-based finite-deformation shell theory for single-wall carbon nanotubes

Abstract: A finite-deformation shell theory is developed for single-wall carbon nanotubes (CNTs) based on the interatomic potential. The modified Born rule for Bravais multi-lattice is used to link the continuum strain energy density to the interatomic potential. The theory incorporates the effect of bending moment and curvature for a curved surface, and accurately accounts for the nonlinear, multi-body atomistic interactions as well as the CNT chirality. It avoids the amibiguous definition of nanotube thickness, and provides the constitutive relations among stress, moment, strain and curvature in terms of the interatomic potential.

Size effects in two-dimensional Voronoi foams: A comparison between generalized continua and discrete models

Abstract: In view of size effects in cellular solids, we critically compare the analytical results of generalized continuum theories with the computational results of discrete models Representatives are studied from two classes of generalized continuum theories: the strain divergence theory from the class of higher-grade continua and the micropolar theory from the class of higher-order continua In the former, the divergence of strain is proposed as an additional deformation measure, while in the latter the microrotation gradients act as the source for extra internal energy We analytically solve a range of basic boundary value problems (simple shear, pure bending and the strain concentration around a hole) and compare the results with discrete, numerical calculations that are based on a Voronoi representation of the cellular microstructure By comparing both the local deformation fields and the overall elastic response, we critically assess the capabilities of both theories in capturing size effects in cellular solids