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.     

Experiments and theory in strain gradient elasticity

Conventional strain-based mechanics theory does not account for contributions from strain gradients. Failure to include strain gradient contributions can lead to underestimates of stresses and size-dependent behaviors in small-scale structures. In this paper, a new set of higher-order metrics is developed to characterize strain gradient behaviors. This set enables the application of the higher-order equilibrium conditions to strain gradient elasticity theory and reduces the number of independent elastic length scale parameters from five to three. On the basis of this new strain gradient theory, a strain gradient elastic bending theory for plane-strain beams is developed. Solutions for cantilever bending with a moment and line force applied at the free end are constructed based on the new higher-order bending theory. In classical bending theory, the normalized bending rigidity is independent of the length and thickness of the beam. In the solutions developed from the higher-order bending theory, the normalized higher-order bending rigidity has a new dependence on the thickness of the beam and on a higher-order bending parameter, b_h. To determine the significance of the size dependence, we fabricated micron-sized beams and conducted bending tests using a nanoindenter. We found that the normalized beam rigidity exhibited an inverse squared dependence on the beam's thickness as predicted by the strain gradient elastic bending theory, and that the higher-order bending parameter, b_h, is on the micron-scale. Potential errors from the experiments, model and fabrication were estimated and determined to be small relative to the observed increase in beam's bending rigidity. The present results indicate that the elastic strain gradient effect is significant in elastic deformation of small-scale structures.     

An elementary molecular-statistical basis for the Mooney and Rivlin-Saunders theories of rubber elasticity

By relaxing the assumption that the end-to-end vectors of molecules transform as macroscopic material line elements, we arrive at a generalization of the molecular-statistical theory of rubber elasticity. This generalization includes as special cases continuum-mechanical theories proposed by Mooney and by Rivlin and Saunders as improvements upon the classical neo-Hookean theory.     

On the r-value of textured sheet metals

The r-value of a sheet metal is a measure of plastic anisotropy frequently used for prediction of performance in deep-drawing. It has also figured prominently in the literature for validation of theories where the predicted angular dependence of r is compared with the measured dependence. As plastic anisotropy in sheet metals is caused mainly by the preferred orientations of grains within the polycrystalline metal, it is natural to ask how r would depend on the orientation distribution function (ODF) w which defines the crystallographic texture of the polycrystal. In this paper a general formula relating r to w is derived for textured sheet metals whose plastic flow behavior is governed by a plastic potential f(σ, w), the anisotropic part of which depends linearly on the texture coefficients; here σ denotes the deviator of the Cauchy stress. Specific forms of this formula for orthorhombic sheets of cubic and of hexagonal metals are explicitly given.     

Constitutive framework optimized for myocardium and other high-strain, laminar materials with one fiber family

Central to this analysis is the identification of six rotation invariant scalars α_1-6 that succinctly define the strain in materials that have one family of parallel fibers arranged in laminae. These scalars were chosen so as to minimize covariance amongst the response terms in the hyperelastic limit, and they are termed strain attributes because it is necessary to distinguish them from strain invariants. The Cauchy stress t is expressed as the sum of six response terms, almost all of which are mutually orthogonal for finite strain (i.e. 14 of the 15 inner products vanish). For small deformations, the response terms are entirely orthogonal (i.e. all 15 inner products vanish). A response term is the product of a response function with its associated kinematic tensor. Each response function is a scalar partial derivative of the strain energy W with respect to a strain attribute. Applications for this theory presently include myocardium (heart muscle) which is often modeled as having muscle fibers arranged in sheets. Utility for experimental identification of strain energy functions is demonstrated by showing that common tests on incompressible materials can directly determine terms in W. Since the described set of strain attributes reduces the covariance amongst response terms, this approach may enhance the speed and precision of inverse finite element methods.     

Modeling aspects of low plastic strain amplitude multiaxial cyclic plasticity in conventional and ultrafine grain nickel

Aspects of the cyclically saturated responses of initially annealed, conventional grain size (average grain diameter of approximately 50 μm) and electrodeposited, ultrafine grain (grains from 20 to 500 nm) nickel to reversed proportional and 90° out-of-phase axial-torsional, strain-controlled cycling at a nominally constant equivalent inelastic strain amplitude of approximately 100 μ strain are reported. An anisotropic, axial-torsional subspace version of Abdel-Karim and Ohno’s kinematic hardening model is presented. Within the framework of conventional small strain, rate-independent plasticity, this approach is used to model the responses. An anhysteretic, phenomenolically based, magnetomechanical model is coupled to the rate-independent plasticity model to include the cyclic magnetostriction response. The kinematic hardening parameter determination scheme, using the proportional path responses, is described. The model correlations achieved are presented and the ability of the resulting models to capture the 90° out-of-phase responses is examined. The model parameter sets, as determined from the proportional responses, require small changes to result in more accurate correlation of the 90° out-of-phase responses and the implications of this are discussed. The relative values of the model parameters between the two materials reflect the initial microstructures. Persistent mean stresses associated with mean total strains imposed are successfully modeled for the proportional strain path responses but not for the 90° out-of-phase responses.     

Dissipative interface waves and the transient response of a three-dimensional sliding interface with Coulomb friction

We investigate the linearized response of two elastic half-spaces sliding past one another with constant Coulomb friction to small three-dimensional perturbations. Starting with the assumption that friction always opposes slip velocity, we derive a set of linearized boundary conditions relating perturbations of shear traction to slip velocity. Friction introduces an effective viscosity transverse to the direction of the original sliding, but offers no additional resistance to slip aligned with the original sliding direction. The amplitude of transverse slip depends on a nondimensional parameter η=c_sτ₀/μv₀, where τ₀ is the initial shear stress, 2v₀ is the initial slip velocity, μ is the shear modulus, and c_s is the shear wave speed. As η→0, the transverse shear traction becomes negligible, and we find an azimuthally symmetric Rayleigh wave trapped along the interface. As η→∞, the inplane and antiplane wavesystems frictionally couple into an interface wave with a velocity that is directionally dependent, increasing from the Rayleigh speed in the direction of initial sliding up to the shear wave speed in the transverse direction. Except in these frictional limits and the specialization to two-dimensional inplane geometry, the interface waves are dissipative. In addition to forward and backward propagating interface waves, we find that for η>1, a third solution to the dispersion relation appears, corresponding to a damped standing wave mode. For large-amplitude perturbations, the interface becomes isotropically dissipative. The behavior resembles the frictionless response in the extremely strong perturbation limit, except that the waves are damped. We extend the linearized analysis by presenting analytical solutions for the transient response of the medium to both line and point sources on the interface. The resulting self-similar slip pulses consist of the interface waves and head waves, and help explain the transmission of forces across fracture surfaces. Furthermore, we suggest that the η→∞ limit describes the sliding interface behind the crack edge for shear fracture problems in which the absolute level of sliding friction is much larger than any interfacial stress changes.     

High-pressure strength of aluminum under quasi-isentropic loading

Under shock loading, metals typically increase in strength with shock pressure initially but at higher stresses will eventually soften due to thermal effects. Under isentropic loading, thermal effects are minimized, so strength should rise to much higher levels. To date, though, study of strength under isentropic loading has been minimal. Here, we report new experimental results for magnetic ramp loading and impact by layered impactors in which the strength of 6061-T6 aluminum is measured under quasi-isentropic loading to stresses as high as 55 GPa. Strength is inferred from measured velocity histories using Lagrangian analysis of the loading and unloading responses; strength is related to the difference of these two responses. A simplified method to infer strength directly from a single velocity history is also presented. Measured strengths are consistent with shock loading and instability growth results to about 30 GPa but are somewhat higher than shock data for higher stresses. The current results also agree reasonably well with the Steinberg–Guinan strength model. Significant relaxation is observed as the peak stress is reached due to rate dependence and perhaps other mechanisms; accounting for this rate dependence is necessary for a valid comparison with other results.     

Scaling function, anisotropy and the size of RVE in elastic random polycrystals

This article is focused on the identification of the size of the representative volume element (RVE) in linear elastic randomly structured polycrystals made up of cubic single crystals. The RVE is approached by setting up stochastic Dirichlet and Neumann boundary value problems consistent with the Hill(-Mandel) macrohomogeneity condition. Within this framework we introduce a scaling function that relates the single crystal anisotropy to the scale of observation. We derive certain exact characteristics of the scaling function and postulate others based on detailed calculations on copper, lithium, tantalum, magnesium oxide and antimony-yttrium. In deriving the above, we make use of the fact that cubic crystals and polycrystals have a uniquely determined scale-independent bulk modulus. It turns out that the scaling function is exact in the single crystal anisotropy. A methodology to develop a material selection diagram that clearly separates the microscale from the macroscale is proposed. The proposed scaling function not only bridges the length scales but also unifies the treatment of a wide spectrum of cubic crystals. Although the scope of this article is restricted to aggregates made up of cubic-shaped and cubic-symmetry single crystals, the concept of the scaling function can be generalized to other crystal shapes and classes as well as to scaling of other elastic/inelastic properties.     

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.     

Modeling transients in the mechanical response of copper due to strain path changes

When copper is deformed to large strains its texture and microstructure change drastically, leading to plastic anisotropy and extended transients when it is reloaded along a different strain path. For predicting these transients, we develop a constitutive model for polycrystalline metals that incorporates texture and grain microstructure. The directional anisotropy in the single crystals is considered to be induced by variable latent hardening associated with cross-slip, cut-through of planar dislocation walls, and dislocation-based reversal mechanisms. These effects are introduced in a crystallographic hardening model which is, in turn, implemented into a polycrystal model. This approach successfully explains the flow response of OFHC Cu pre-loaded in tension (compression) and reloaded in tension (compression), and the response of OFHC Cu severely strained in shear by equal channel angular extrusion and subsequently compressed in each of the three orthogonal directions. This new theoretical framework applies to arbitrary strain path changes, and is fully anisotropic.     

An experimental investigation of shock wave propagation in periodically layered composites

In heterogeneous media, scattering due to interfaces/microstructure between dissimilar materials could play an important role in shock wave dissipation and dispersion. In this work, the influence of interface scattering on finite-amplitude shock waves was experimentally investigated by impacting flyer plates onto periodically layered polycarbonate/6061 aluminum, polycarbonate/304 stainless steel and polycarbonate/glass composites. Experimental results (obtained using velocity interferometer and stress gage) show that these periodically layered composites can support steady structured shock waves. Due to interface scattering, the effective shock viscosity increases with the increase of interface impedance mismatch, and decreases with the increase of interface density (interface area per unit volume) and loading amplitude. For the composites studied here, the strain rate within the shock front is roughly proportional to the square of the shock stress. This indicates that layered composites have much larger shock viscosity due to the interface/microstructure scattering in comparison with the increase of shock strain rate by the fourth power of the shock stress for homogeneous metals. Experimental results also show that due to the scattering effects, shock propagation in the layered composites is dramatically slowed down and the shock speed in composites can be lower than that either of its components.     

Models for stiffening in cross-linked biopolymer networks: A comparative study

In a recent publication, we studied the mechanical stiffening behavior in two-dimensional (2D) cross-linked networks of semiflexible biopolymer filaments under simple shear [Onck, P.R., Koeman, T., Van Dillen, T., Van der Giessen, E., 2005. Alternative explanation of stiffening in cross-linked semiflexible networks. Phys. Rev. Lett. 95, 178102]. These simulations make use of a geometrically nonlinear finite-element technique, taking into account the discreteness of the biopolymer network. As an alternative to the prevalent view, these computations relate the stiffening to nonaffine network reorientations. However, this discrete-network model neglects any interaction of the filaments with the surrounding fluid, which is the origin of entropic stiffening in single filaments and in biopolymer networks, according to MacKintosh et al. For this reason, this article is devoted to a thorough study of the difference between both approaches on the 2D single-filament level. In addition, we investigate the deviation from affine deformation behavior, by comparing the discrete calculations with an affine-network model.     

Experimental investigation and modeling of non-monotonic creep behavior in polymers

The deformation behavior of two unfilled engineering thermoplastics, ultra high molecular weight polyethylene (UHMWPE) and polycarbonate (PC), has been investigated in creep test conditions. It has been found that a loading history (prior to the creep test) comprising of loading to a maximum stress or strain value followed by partial unloading to arrive at the target stress value can greatly modify the strain-time behavior. Under such a test protocol, while the expected increase in strain during creep (constant tensile load) is observed, at relatively low creep stresses specimens have also demonstrated a monotonic decrease in strain. In an intermediate stress range, specimens have demonstrated time dependent behavior comprising of a transition from decreasing to increasing strain during creep in tension. This paper presents experimental results to delineate these findings and explore the effect of prior strain rate on the qualitative and quantitative changes in the output (strain-time) behavior. Furthermore, modification of the viscoplasticity theory based on overstress (VBO) model into a double element configuration is introduced. These changes confer upon the model the ability to yield non-monotonic behavior in creep, and supporting simulation results have been included. These changes, therefore, allow the model to simulate strain rate sensitivity, creep, relaxation, and recovery behavior, but more importantly address the issue of non-monotonic changes in creep and relaxation when a loading history involves some degree of unloading.     

Free-energy density functions for nematic elastomers

The recently proposed neo-classical theory for nematic elastomers generalizes standard molecular-statistical Gaussian network theory to allow for anisotropic distributions of polymer chains. The resulting free-energy density models several of the novel properties of nematic elastomers. In particular, it predicts the ability of nematic elastomers to undergo large deformations with exactly zero force and energy cost—so called soft elasticity. Although some nematic elastomers have been shown to undergo deformations with unusually small applied forces, not all do so, and none deform with zero force. Further, as a zero force corresponds to infinitely many possible deformations in the neo-classical theory, this non-uniqueness leads to serious indeterminacies in numerical schemes. Here we suggest that the neo-classical free-energy density is incomplete and propose an alternative derivation that resolves these difficulties. In our approach, we use the molecular-statistical theory to identify appropriate variables. This yields the choice for the microstructural degrees of freedom as well as two independent strain tensors (the overall macroscopic strain plus a relative strain that indicates how the deformation of the elastomeric microstructure deviates from the macroscopic deformation). We then propose expressions for the free-energy density as a function of the three quantities and show how the material parameters can be measured by two simple tests. The neo-classical free-energy density can be viewed as a special case of our expressions in which the free-energy density is independent of the overall macroscopic strain, thus supporting our view that the neo-classical theory is incomplete.     

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.