The Peierls stress in non-local elasticity

The effect of non-locality on the Peierls stress of a dislocation, predicted within the framework of the Peierls-Nabarro model, is investigated. Both the integral formulation of non-local elasticity and the gradient elasticity model are considered. A modification of the non-local kernel of the integral formulation is proposed and its effect on the dislocation core shape and size, and on the Peierls stress are discussed. The new kernel is longer ranged and physically meaningful, improving therefore upon the existing Gaussian-like non-locality kernels. As in the original Peierls-Nabarro model, lattice trapping cannot be captured in the purely continuum non-local formulation and therefore, a semi-discrete framework is used. The constitutive law of the elastic continuum and that of the glide plane are considered both local and non-local in separate models. The major effect is obtained upon rendering non-local the constitutive law of the continuum, while non-locality in the rebound force law of the glide plane has a marginal effect. The Peierls stress is seen to increase with increasing the intrinsic length scale of the non-local formulation, while the core size decreases accordingly. The solution becomes unstable at intrinsic length scales larger than a critical value. Modifications of the rebound force law entail significant changes in the core configuration and critical stress. The discussion provides insight into the issue of internal length scale selection in non-local elasticity models.     

Convergence of local atomistic stress based on periodic lattice

Local stress in an atomic system, which provides an average stress measurement within a spatial volume containing a collection of atoms, is essential for determining the mechanical properties of a nanoscale structure as well as developing a proper multiscale modeling technique. Theoretically, the smaller averaging volume where a local stress can converge, the closer this atomistic stress definition can approach the ideal continuum stress. As a result, the more accurate stress concentration can be evaluated for the inhomogeneous case. With reference to the previous studies focusing on the spherical averaging volume, dependent on the type of crystals, the convergent radius of the virial stress or Hardy stress usually spans the size of several lattice constants. In this paper, we find that, once the averaging volume is periodic, the convergence of the virial stress and Hardy stress can be accomplished within one single lattice, which is much smaller than what is required by other non-periodic volumes such as a sphere. In the final section, a cracked sodium chloride crystal is considered to demonstrate that the crack opening stress described by the periodic lattices captures the stress concentration near the crack tip.     

Relations between stress growth and stress relaxation functions

The stress relaxation function after steady shear flow and the stress growth function at inception of steady flow are derived for several constitutive equations of the integral and differential types. Relationships between these functions are deduced and discussed.     

A physical decomposition of the stress tensor for complex flows

Traditionally, the components of the stress with respect to a relevant coordinate system are used for the purpose of stress visualisation and interpretation. A case for using a flow dependent measure to interpret and visualise stress is made for two dimensional flow, together with a suggestion for extending the idea to three dimensions. The method is illustrated for Newtonian and Oldroyd B fluids in both the eccentrically rotating cylinder and flow past a cylinder benchmark problems. In the context of a generalised Newtonian fluid, the relation between the flow-dependent stress measure to other field variables under certain flow conditions, is examined and is indicative of its importance in complex flow.

Disorder,pre-stress and non-affinity in polymer 8-chain models

To assess the role of single-chain elasticity, non-affine strain fields and pre-stressed reference states we present and discuss the results of numerical and analytical analyses of a modified 8-chain Arruda–Boyce model for cross-linked polymer networks. This class of models has proved highly successful in modeling the finite-strain response of flexible rubbers. We extend it to include the effects of spatial disorder and the associated non-affinity, and use it to assess the validity of replacing the constituent chain's nonlinear elastic response with equivalent linear, Hookean springs. Surprisingly, we find that even in the regime of linear response, the full polymer model gives very different results from its linearized counterpart, even though none of the chains are stretched beyond their linear regime. We demonstrate that this effect is due to the fact that the polymer models are under considerable pre-stress in their ground state. We show that pre-stress strongly suppresses non-affinity in these unit cell models, resulting in a marked stiffening of the bulk response. Polymer networks with some degree of flexibility are thus intrinsically prestressed, and one effect of such prestresses is to reduce non-affine deformations. Combined, these findings may help explain why fully affine mechanical models, in many cases, predict the bulk mechanical response of disordered polymer networks so well.     

On the interpretation of the logarithmic strain tensor in an arbitrary system of representation

Logarithmic strains are increasingly used in constitutive modelling because of their advantageous properties. In this paper we study the physical interpretation of the components of the logarithmic strain tensor in any arbitrary system of representation, which is crucial in formulating meaningful constitutive models. We use the path-independence property of total logarithmic strains to propose different fictitious paths which can be interpreted as a sum of infinitesimal engineering strain tensors. We show that the angular (engineering) distortion measure is arguably not a good measure of shear and instead we propose area distortions which are an exact interpretation of the shear terms both for engineering and for logarithmic strains. This new interpretation clearly explains the maximum obtained in some constitutive models for the simple shear load case.     

On the shear and bending of a degrading polymer beam

In this paper we develop a model, within a general framework that has been developed to describe the response of dissipative systems, for the strain induced degradation of polymeric solids, due to scission. The theory can be generalized to include degradation due to ultraviolet radiation, oxygen diffusion etc., by incorporating an appropriate form for the rate of dissipation associated with these processes. We study the simple shear and pure bending of such degrading polymer beams.     

Study on the generalized prandtl-reuss constitutive equation and the corotational rates of stress tensor

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A new two-point deformation tensor and its relation to the classical kinematical framework and the stress concept

Starting from the issue of what is the correct form for a Legendre transformation of the strain energy in terms of Eulerian and two-point tensor variables we introduce a new two-point deformation tensor, namely H=(F−F^−T)/2, as a possible deformation measure involving points in two distinct configurations. The Lie derivative of H is work conjugate to the first Piola–Kirchhoff stress tensor P. The deformation measure H leads to straightforward manipulations within a two-point setting such as the derivation of the virtual work equation and its linearization required for finite element implementation. The manipulations are analogous to those used for the Lagrangian and Eulerian frameworks. It is also shown that the Legendre transformation in terms of two-point tensors and spatial tensors require Lie derivatives. As an illustrative example we propose a simple Saint Venant–Kirchhoff type of a strain-energy function in terms of H. The constitutive model leads to physically meaningful results also for the large compressive strain domain, which is not the case for the classical Saint Venant–Kirchhoff material.     

Time dependent scission and cross-linking in an elastomeric cylinder undergoing circular shear and heat conduction

When an elastomeric material is at a sufficiently high temperature, there can be time-dependent scission of macromolecular network cross-links. The affected molecules can recoil and cross-link to form a new network in a new reference configuration. The material then consists of several molecular networks. This microstructural change affects the mechanical response and leads to permanent set. A constitutive equation is presented, based on the experimental work of Tobolsky (Properties and Structures of Polymers, Wiley, New York, 1960, pp. 223-265), which can account for the influence of this temperature-dependent microstructural change on the mechanical response. It is used to study an elastomeric cylinder undergoing circular shear and transient heat conduction.     

Viscoelasticity of brain corpus callosum in biaxial tension

Computational models of the brain rely on accurate constitutive relationships to model the viscoelastic behavior of brain tissue. Current viscoelastic models have been derived from experiments conducted in a single direction at a time and therefore lack information on the effects of multiaxial loading. It is also unclear if the time-dependent behavior of brain tissue is dependent on either strain magnitude or the direction of loading when subjected to tensile stresses. Therefore, biaxial stress relaxation and cyclic experiments were conducted on corpus callosum tissue isolated from fresh ovine brains. Results demonstrated the relaxation behavior to be independent of strain magnitude, and a quasi-linear viscoelastic (QLV) model was able to accurately fit the experimental data. Also, an isotropic reduced relaxation tensor was sufficient to model the stress-relaxation in both the axonal and transverse directions. The QLV model was fitted to the averaged stress relaxation tests at five strain magnitudes while using the measured strain history from the experiments. The resulting model was able to accurately predict the stresses from cyclic tests at two strain magnitudes. In addition to deriving a constitutive model from the averaged experimental data, each specimen was fitted separately and the resulting distributions of the model parameters were reported and used in a probabilistic analysis to determine the probability distribution of model predictions and the sensitivity of the model to the variance of the parameters. These results can be used to improve the viscoelastic constitutive models used in computational studies of the brain.     

Force flux and the peridynamic stress tensor

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.     

Constitutive modeling of Aluminum Alloy 7050 cyclic mean stress relaxation and ratcheting

The challenge of describing in a generalized mathematical pattern the inelastic behavior of metals has led to the development of several constitutive models, especially in the field of cyclic plasticity, where phenomena with particular importance to low-cycle fatigue appear. Significant research efforts have been undertaken in studying and simulating the cyclic elastoplastic response of steels, while light metals, like aluminum and titanium, have attracted less attention. This paper provides a preliminary examination on the capacity of the Multi-component Armstrong and Frederick Multiplicative (MAFM) model to simulate effectively the cyclic mean stress relaxation and ratcheting of Aluminum Alloy 7050. The derived results indicate that the model is capable to describe successfully the complex cyclic plasticity phenomena exhibited by this alloy.     

Complex unloading behavior: Nature of the deformation and its consistent constitutive representation

Complex (nonlinear) unloading behavior following plastic straining has been reported as a significant challenge to accurate springback prediction. More fundamentally, the nature of the unloading deformation has not been resolved, being variously attributed to nonlinear/reduced modulus elasticity or to inelastic/“microplastic” effects. Unloading-and-reloading experiments following tensile deformation showed that a special component of strain, deemed here “Quasi-Plastic-Elastic” (“QPE”) strain, has four characteristics. (1) It is recoverable, like elastic deformation. (2) It dissipates work, like plastic deformation. (3) It is rate-independent, in the strain rate range 10⁻⁴-10⁻²/s, contrary to some models of anelasticity to which the unloading modulus effect has been attributed. (4) To first order, the evolution of plastic properties occurs during QPE deformation. These characteristics are as expected for a mechanism of dislocation pile-up and relaxation. A consistent, general, continuum constitutive model was derived incorporating elastic, plastic, and QPE deformation. Using some aspects of two-yield-function approaches with unique modifications to incorporate QPE, the model was implemented in a finite element program with parameters determined for dual-phase steel and applied to draw-bend springback. Significant differences were found compared with standard simulations or ones incorporating modulus reduction. The proposed constitutive approach can be used with a variety of elastic and plastic models to treat the nonlinear unloading and reloading of metals consistently for general three-dimensional problems.     

An atomistic and non-classical continuum field theoretic perspective of elastic interactions between defects (force dipoles) of various symmetries and application to graphene

Force multipoles are employed to represent various types of defects and physical phenomena in solids: point defects (interstitials, vacancies), surface steps and islands, proteins on biological membranes, inclusions, extended defects, and biological cell interactions among others. In the present work, we (i) as a prototype simple test case, conduct quantum mechanical calculations for mechanics of defects in graphene sheet and in parallel, (ii) formulate an enriched continuum elasticity theory of force dipoles of various anisotropies incorporating up to second gradients of strain fields (thus accounting for nonlocal dispersive effects) instead of the usual dispersion-less classical elasticity formulation that depends on just the strain (c.f. Peyla, P., Misbah, C., 2003. Elastic interaction between defects in thin and 2-D films. Eur. Phys. J. B. 33, 233-247). The fundamental Green's function is derived for the governing equations of second gradient elasticity and the elastic self and interaction energies between force dipoles are formulated for both the two-dimensional thin film and the three-dimensional case. While our continuum results asymptotically yield the same interaction energy law as Peyla and Misbah for large defect separations (∼1/rⁿ for defects with n-fold symmetry), the near-field interactions are qualitatively far more complex and free of singularities. Certain qualitative behavior of defect mechanics predicted by atomistic calculations are well captured by our enriched continuum models in contrast to classical elasticity calculations. For example, consistent with our atomistic calculations of defects in isotropic graphene, even two dilation centers show a finite interaction (as opposed to classical elasticity that predicts zero interaction). We explicitly find the physically consistent result that the self-energy of a defect is equivalent to half the interaction energy between two identical defects when they “merge” into each other. The atomistic, classical elastic and the enriched continuum predictions are thoroughly compared for two types of defects in graphene: Stone-Wales and divacancy.     

On the stress singularities generated by anisotropic eigenstrains and the hydrostatic stress due to annular inhomogeneities

The problems of singularity formation and hydrostatic stress created by an inhomogeneity with eigenstrain in an incompressible isotropic hyperelastic material are considered. For both a spherical ball and a cylindrical bar with a radially symmetric distribution of finite possibly anisotropic eigenstrains, we show that the anisotropy of these eigenstrains at the center (the center of the sphere or the axis of the cylinder) controls the stress singularity. If they are equal at the center no stress singularity develops but if they are not equal then stress always develops a logarithmic singularity. In both cases, the energy density and strains are everywhere finite. As a related problem, we consider annular inclusions for which the eigenstrains vanish in a core around the center. We show that even for an anisotropic distribution of eigenstrains, the stress inside the core is always hydrostatic. We show how these general results are connected to recent claims on similar problems in the limit of small eigenstrains.     

THE TENSOR DENOTATION OF BELTRAMI SPHERICAL VORTICES AND THEIR SYMMETRY ANALYSIS

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Shear and normal stress relaxation in polymer blends: stress predictions from interface velocity and Laplace pressure terms

We derived for the first time the relationships among shear stress and normal stress differences for ellipsoidal interfaces under large step shear strains considering interface velocity term and Laplace pressure term in the expression of the stress tensor for mixtures of two Newtonian fluids. In the derivation, orientation angle of the interface is assumed to be given by the affine deformation assumption and is independent of time based on experimental results for blends with 0.048 ≤ K ≤ 0.54 where K is the ratio of droplet viscosity to matrix viscosity. For ellipsoidal droplets, the shear stress is only proportional to the first normal stress difference. On the other hand, for spheroidal droplets, proportionality among the shear stress, the first and the second normal stress differences was derived, and the ratio of the second normal stress difference to the first normal stress difference was given as a function of step strain. The shear stress and the first normal stress difference obtained experimentally satisfy the derived relationship, indicating applicability of the stress expression for polymer blends.     

Stress measurement of SS400 steel beam using the continuous indentation technique

An analytical model is presented for determining surface residual stress using continuous indentation. The elastic residual stress is assumed to have no influence on contact area or hardness and to be uniform over a volume that is several times larger than the indentation mark. A step-by-step analysis for the residual-stress-induced load difference at a given depth is outlined here and such concepts as stress interaction, stress-sensitive contact morphology, and reversible contact recoveries during a stress relaxation are described. Finally, the proposed method is applied to the interpretation of the continuous indentation results obtained from an SS400 steel beam in which controlled bending stresses are generated. The stress estimated, however, showed a high scatter due to plastic pile-up deformation. When the optically measured contact area is used as an alternative of the contact area calculated from the unloading curve, the re-evaluated stress agrees well with the already known applied stress.