On the functional form of non-local elasticity kernels

The functional form of non-local elasticity kernels is studied within the context of the integral formalism. The study is limited to linear isotropic elasticity. The kernels are derived analytically based on the discrete structure of the material at the atomic scale. Atomistic simulations are used to validate the results. Materials in which the interatomic interactions are represented by pair, as well as embedded atom-type potentials are considered. The derived kernels have a range which extends up to the cut-off radius of the interatomic potential, are positive at the origin, and become negative approximately one atomic distance away, thus departing from the commonly assumed Gaussian functional form. The functional form of the potential and the radial distribution function of interacting neighbors about a representative atom fully define their shape. This new continuum model involves two material length scales that are both derived from atomistics for a Morse solid and for Al. Two applications are considered in closure. It is shown that in strained superlattices, the non-local model predicts maximum stresses that are much larger than those obtained within the local theory. This observation has implications for defect nucleation in these structures. Furthermore, the new non-local model improves upon the Gaussian one by predicting a more realistic wave dispersion relationship, with essentially zero group velocity at the boundary of the Brillouin zone.     

The non-uniqueness of the atomistic stress tensor and its relationship to the generalized Beltrami representation

The non-uniqueness of the atomistic stress tensor is a well-known issue when defining continuum fields for atomistic systems. In this paper, we study the non-uniqueness of the atomistic stress tensor stemming from the non-uniqueness of the potential energy representation. In particular, we show using rigidity theory that the distribution associated with the potential part of the atomistic stress tensor can be decomposed into an irrotational part that is independent of the potential energy representation, and a traction-free solenoidal part. Therefore, we have identified for the atomistic stress tensor a discrete analog of the continuum generalized Beltrami representation (a version of the vector Helmholtz decomposition for symmetric tensors). We demonstrate the validity of these analogies using a numerical test. A program for performing the decomposition of the atomistic stress tensor called MDStressLab is available online at http://mdstresslab.org.     

A novel constitutive model for semiconductors: The case of silicon

The photovoltaic industry relies heavily on solar-grade silicon multicrystals. Understanding their mechanical behavior requires the development of adequate constitutive models accounting for the effects of both high dislocation densities and complex loading situations in a wide range of temperature, strain rate, and impurity contents. The traditional model of Alexander and Haasen poses several limitations. We introduce in this work a novel constitutive model for covalent single crystals and its implementation into a rate-dependent crystal plasticity framework. It is entirely physically based on the dislocation generation, storage and annihilation processes taking place during plastic flow. The total dislocation density is segmented according to the dislocation mobility potential and their character. A dislocation multiplication law for the yield region more accurate than the one of Alexander and Haasen is proposed. The influence of additional dislocation sources created on forest trees, usually disregarded in models for semiconductors, is assessed. The dislocation velocity law combines three potentially rate-limiting mechanisms: the standard double kink mechanism, jog dragging and the influence of localized obstacles. The model is valid at finite strains, in multiple slip conditions and captures accurately the high temperature- and strain rate sensitivity of semiconductors. The experimental stress overshoot is qualitatively reproduced only when jog dragging is accounted for. Localized obstacles are shown not to have any significant effect on dislocation motion in silicon. The broader case of extrinsic semiconductors is discussed and the influence of dissolved oxygen on the upper yield stress of silicon monocrystals is successfully reproduced.     

A comparison of nonlocal continuum and discrete dislocation plasticity predictions

Discrete dislocation simulations of two boundary value problems are used as numerical experiments to explore the extent to which the nonlocal crystal plasticity theory of Gurtin (J. Mech. Phys. Solids 50 (2002) 5) can reproduce their predictions. In one problem simple shear of a constrained strip is analyzed, while the other problem concerns a two-dimensional model composite with elastic reinforcements in a crystalline matrix subject to macroscopic shear. In the constrained layer problem, boundary layers develop that give rise to size effects. In the composite problem, the discrete dislocation solutions exhibit composite hardening that depends on the reinforcement morphology, a size dependence of the overall stress-strain response for some morphologies, and a strong Bauschinger effect on unloading. In neither problem are the qualitative features of the discrete dislocation results represented by conventional continuum crystal plasticity. The nonlocal plasticity calculations here reproduce the behavior seen in the discrete dislocation simulations in remarkable detail.     

An alternative treatment of phenomenological higher-order strain-gradient plasticity theory

Phenomenological higher-order strain-gradient plasticity is here presented through a formulation inspired by previous work for strain-gradient crystal plasticity. A physical interpretation of the phenomenological yield condition that involves an effect of second gradient of the equivalent plastic strain is discussed, applying a dislocation theory-based consideration. Then, a differential equation for the equivalent plastic strain-gradient is introduced as an additional governing equation. Its weak form makes it possible to deduce and impose extra boundary conditions for the equivalent plastic strain. A connection between the present treatment and strain-gradient theories based on an extended virtual work principle is discussed. Furthermore, a numerical implementation and analysis of constrained simple shear of a thin strip are presented.     

Crack-tip problem in non-local elasticity

Solutions are presented for the one- and two-dimensional Griffith crack problems in non-local elasticity. The displacements and stresses are determined in an elastic plate, weakened by a sharpedged line crack. The plate is loaded by a uniform tension perpendicular to the line of the crack at infinity. The non-local elastic solutions yield a finite hoop stress at the crack tip, thus allowing for a fracture criterion based on the maximum stress hypothesis.     

On the asymptotic crack-tip stress fields in nonlocal orthotropic elasticity

The crack-tip stress fields in orthotropic bodies are derived within the framework of Eringen’s nonlocal elasticity via the Green’s function method. The modified Bessel function of second kind and order zero is considered as the nonlocal kernel. We demonstrate that if the localisation residuals are neglected, as originally proposed by Eringen, the asymptotic stress tensor and its normal derivative are continuous across the crack. We prove that the stresses attained at the crack tip are finite in nonlocal orthotropic continua for all the three fracture modes (I, II and III). The relative magnitudes of the stress components depend on the material orthotropy. Moreover, non-zero self-balanced tractions exist on the crack edges for both isotropic and orthotropic continua. The special case of a mode I Griffith crack in a nonlocal and orthotropic material is studied, with the inclusion of the T-stress term.     

A plasticity model describing yield-point phenomena of steels and its application to FE simulation of temper rolling

To describe the yield-point phenomena of steels, an extended version of the first author’s model (Yoshida, F., 2000. A constitutive model of cyclic plasticity. International Journal of Plasticity 16, 359–380) is proposed on the premise that the material behavior of sharp yield point and the subsequent abrupt yield drop result from a rapid dislocation multiplication and the stress-dependence of dislocation velocity. A specific feature of this model is that it describes well a high upper yield point, the rate-dependent Lüders strain at the yield plateau and the subsequent workhardening, as well as cyclic plasticity characteristics, such as the Bauschinger effect and rate-dependent ratcheting. Using this model, an FE simulation of temper rolling process is conducted in order to clarify its role for the elimination of the yield point of steel sheets. Particularly, the effect of upper yield point on the deformation characteristics in the process is discussed.     

A multiscale approach for modeling scale-dependent yield stress in polycrystalline metals

Modeling of scale-dependent characteristics of mechanical properties of metal polycrystals is studied using both discrete dislocation dynamics and continuum crystal plasticity. The initial movements of dislocation arc emitted from a Frank-Read type dislocation source and bounded by surrounding grain boundaries are examined by dislocation dynamics analyses system and we find the minimum resolved shear stress for the FR source to emit at least one closed loop. When the grain size is large enough compared to the size of FR source, the minimum resolved shear stress levels off to a certain value, but when the grain size is close to the size of the FR source, the minimum resolved shear stress shows a sharp increase. These results are modeled into the expression of the critical resolved shear stress of slip systems and continuum mechanics based crystal plasticity analyses of six-grained polycrystal models are made. Results of the crystal plasticity analyses show a distinct increase of macro- and microscopic yield stress for specimens with smaller mean grain diameter. Scale-dependent characteristics of the yield stress and its relation to some control parameters are discussed.     

The analysis of crack problems with non-local elasticity

IntroductionTheclassicalconhnuummechanicshasbeenusedtosolvemanyproblemsinmacrofracturemechanics,butencountersdifficulheswhentheeffectofITilcrocharacteristicdimensionshouldbetakenintoaccount.Thestressfieldverynearthecracktipisstillnotclear.Somephenomenaofshortcrackscannotbeexplained["']andsomemechanismoffracturehasnotbeensolvedyet.Thenon-localelashcitytheoryseemsattractivetotheseproblems.Thetheoryofnon-localelasticity,establishedanddevelopedbyEringenetal[3),connectstheclassicalcontinuummechan…     

Viscoplastic flow rule for dislocation-mediated plasticity from systematic coarse-graining

The plastic response of metals is determined by the collective, coarse-grained dynamics of dislocations, rather than by the dynamics of individual dislocations. The evolution equations at both levels are quite different, for example considering their dependence on the applied mechanical load. On the one hand, the relation between the configurational force and dislocation velocity for individual dislocations is linear. On the other hand, in phenomenological crystal plasticity models, the relation between load and plastic slip is highly non-linear and often taken of power-law form. In this work, it is shown that this difference is justified and a consequence of emergent effects. Previously, an expression for the macroscopic dislocation flux was derived by systematic coarse graining (Kooiman et al., 2015). This expression has been evaluated numerically in this paper. The resulting relation between dislocation flux and applied mechanical load is found to be of power-law form with an exponent 3.7, while the underlying Discrete Dislocation Dynamics has a linear flux–load relation.     

THE MIXED BOUNDARY-VALUE PROBLEM FOR NON-LOCAL ASYMMETRIC ELASTICITY

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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.     

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

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.     

On divergence-free stress fields and zero-energy stress functions in elasticity

Applying two identities for divergence-free non-symmetric and symmetric second-order tensors, novel type of first- and second-order stress functions are proposed for three-dimensional elasticity problems. It is shown that self-equilibrated but non-symmetric 3D stress fields can be generated by one first-order stress function vector, whereas a self-equilibrated and symmetric 3D stress field can be generated by one Airy-type second-order stress function. Assuming linearly elastic materials, the zero-energy modes of the stress functions introduced are derived and investigated. It is pointed out that the structure of the zero-energy modes of the proposed first-order stress function vector is the same as that of the rigid-body displacements in the linear theory of elasticity.     

Computing bounds to mixed-mode stress intensity factors in elasticity

A bounding procedure combined with an effective error bound method for linear functionals of the displacements and a simple two points displacement extrapolation method is presented to compute the lower and upper bounds to the stress intensity factors in elastic fracture problems. First, the displacements of two nodes (or node pairs) on the crack edges are used to construct the linear extrapolation to obtain the stress intensity factors at the crack tip, so that stress intensity factors are explicitly expressed as linear functionals of the displacements. Then, a posteriori bounding method is utilized to compute the bounds to the stress intensity factors. Finally, the bounding procedure is verified by a mixed-mode homogenous elastic fracture problem and a bimaterial interface crack problem.     

Strain hardening in 2D discrete dislocation dynamics simulations: A new ‘2.5D’ algorithm

The two-dimensional discrete dislocation dynamics (2D DD) method, consisting of parallel straight edge dislocations gliding on independent slip systems in a plane strain model of a crystal, is often used to study complicated boundary value problems in crystal plasticity. However, the absence of truly three dimensional mechanisms such as junction formation means that forest hardening cannot be modeled, unless additional so-called ‘2.5D’ constitutive rules are prescribed for short-range dislocation interactions. Here, results from three dimensional dislocation dynamics (3D DD) simulations in an FCC material are used to define new constitutive rules for short-range interactions and junction formation between dislocations on intersecting slip systems in 2D. The mutual strengthening effect of junctions on preexisting obstacles, such as precipitates or grain boundaries, is also accounted for in the model. The new ‘2.5D’ DD model, with no arbitrary adjustable parameters beyond those obtained from lower scale simulation methods, is shown to predict athermal hardening rates, differences in flow behavior for single and multiple slip, and latent hardening ratios. All these phenomena are well-established in the plasticity of crystals and quantitative results predicted by the model are in good agreement with experimental observations.