On the Ohno–Wang kinematic hardening rules for multiaxial ratcheting modeling of medium carbon steel

This paper evaluates the performance of four Ohno–Wang type constitutive models in predicting ratcheting responses of medium carbon steel S45C for a set of axial/torsional loading paths. Suggestions are also made for further modification. The four models are the Ohno–Wang model, the McDowell model, the Jiang–Sehitoglu model and the AbdelKarim–Ohno model. It is shown that the Ohno–Wang model and the McDowell model overestimate the multiaxial ratcheting. Whereas, the Jiang–Sehitoglu model yields good predictions for most loading conditions used in this study with an appropriate modification of the dynamic recovery term. The AbdelKarim–Ohno model gives acceptable predictions for all considered multiaxial conditions when used with an evolution function for μ_i, but gives poor predictions of uniaxial ratcheting if the parameter μ_i is determined from a multiaxial ratcheting response. A new modified Ohno–Wang hardening rule is proposed for better adaptability under diverse situations by multiplying a factor to the dynamic recovery term, which is dependent on noncoaxiality of the plastic strain rate and back stress. This new model predicts ratcheting strain reasonably well for the test cases.     

Physical invariants for nonlinear orthotropic solids

Seven invariants, with immediate physical interpretation, are proposed for the strain energy function of nonlinear orthotropic elastic solids. Three of the seven invariants are the principal stretch ratios and the other four are squares of the dot product between the two preferred directions and two principal directions of the right stretch tensor. A strain energy function, expressed in terms of these invariants, has a symmetrical property almost similar to that of an isotropic elastic solid written in terms of principal stretches. Ground state and stress–strain relations are given. Using principal axes techniques, the formulation is applied, with mathematical simplicity, to several types of deformations. In simple shear, a necessary and sufficient condition is given for Poynting relation and two novel deformation-dependent universal relations are formulated. Using series expansions and the symmetrical property, the proposed general strain energy function is refined to a particular general form. A type of strain energy function, where the ground state constants are written explicitly, is proposed. Some advantages of this type of function are indicated. An experimental advantage is demonstrated by showing a simple triaxial test can vary a single invariant while keeping the remaining invariants fixed.     

A new free energy for plastic damage analysis

This work presents a new free energy for the ductile plastic damage analysis. The derivation here is based on the ductile plastic damage model proposed by Lemaitre and the examination of the plastic strain of a damaged material. The result given here can be used for the damage evolution law derived from the potential of dissipation.     

Nonlinear Weakly Curved Rod by Γ-Convergence

We present a nonlinear model of weakly curved rod, namely the type of curved rod where the curvature is of the order of the diameter of the cross-section. We use an approach analogous to the one for rods and curved rods and start from the strain energy functional of three dimensional nonlinear elasticity. We do not impose any constitutional behavior of the material and work in a general framework. To derive the model, by means of ??-convergence, we need to set the order of strain energy (i.e., its relation to the thickness of the body h). We analyze the situation when the strain energy (divided by the order of volume) is of the order h ⁴. This is the same approach as the one used in F?ppl-von Kármán model for plates and the analogous model for rods. The obtained model is analogous to Marguerre-von Kármán for shallow shells and its linearization is the linear shallow arch model which can be found in the literature.     

An improved anisotropy-resolving subgrid-scale model for flows in laminar–turbulent transition region

Some types of mixed subgrid-scale (SGS) models combining an isotropic eddy-viscosity model and a scale-similarity model can be used to effectively improve the accuracy of large eddy simulation (LES) in predicting wall turbulence. Abe (2013) has recently proposed a stabilized mixed model that maintains its computational stability through a unique procedure that prevents the energy transfer between the grid-scale (GS) and SGS components induced by the scale-similarity term. At the same time, since this model can successfully predict the anisotropy of the SGS stress, the predictive performance, particularly at coarse grid resolutions, is remarkably improved in comparison with other mixed models. However, since the stabilized anisotropy-resolving SGS model includes a transport equation of the SGS turbulence energy, k_SGS, containing a production term proportional to the square root of k_SGS, its applicability to flows with both laminar and turbulent regions is not so high. This is because such a production term causes k_SGS to self-reproduce. Consequently, the laminar–turbulent transition region predicted by this model depends on the inflow or initial condition of k_SGS. To resolve these issues, in the present study, the mixed-timescale (MTS) SGS model proposed by Inagaki et al. (2005) is introduced into the stabilized mixed model as the isotropic eddy-viscosity part and the production term in the k_SGS transport equation. In the MTS model, the SGS turbulence energy, k_es, estimated by filtering the instantaneous flow field is used. Since the k_es approaches zero by itself in the laminar flow region, the self-reproduction property brought about by using the conventional k_SGS transport equation model is eliminated in this modified model. Therefore, this modification is expected to enhance the applicability of the model to flows with both laminar and turbulent regions. The model performance is tested in plane channel flows with different Reynolds numbers and in a backward-facing step flow. The results demonstrate that the proposed model successfully predicts a parabolic velocity profile under laminar flow conditions and reduces the dependence on the grid resolution to the same degree as the unmodified model by Abe (2013) for turbulent flow conditions. Moreover, it is shown that the present model is effective at transitional Reynolds numbers. Furthermore, the present model successfully provides accurate results for the backward-facing step flow with various grid resolutions. Thus, the proposed model is considered to be a refined anisotropy-resolving SGS model applicable to laminar, transitional, and turbulent flows.     

Multiaxial yield surface of transversely isotropic foams: Part I—Modeling

A new yield criterion is proposed for transversely isotropic solid foams. Its derivation is based on the hypothesis that the yielding in foams is driven by the total strain energy density, rather than a completely phenomenological approach. This allows defining the yield surface with minimal number of parameters and does not require complex experiments. The general framework used leads to the introduction of new scalar measures of stress and strain (characteristic stress and strain) for transversely isotropic foams. Furthermore, the central hypothesis that the total strain energy density drives yielding in foams ascribes to the characteristic stress an analogous role of von Mises stress in metal plasticity. Unlike the overwhelming majority of yield models in literature the proposed model recognizes the tension–compression difference in yield behavior of foams through a linear mean stress term. Predictions of the proposed yield model are in excellent agreement with the results of uniaxial, biaxial and triaxial FE analyses implemented on both isotropic and transversely isotropic Kelvin foam models.     

A two stage method for structural damage detection using a modal strain energy based index and particle swarm optimization

A two-stage method is proposed here to properly identify the site and extent of multiple damage cases in structural systems. In the first stage, a modal strain energy based index (MSEBI) is presented to precisely locate the eventual damage of a structure. The modal strain energy is calculated using the modal analysis information extracted from a finite element modeling. In the second stage, the extent of actual damage is determined via a particle swarm optimization (PSO) using the first stage results. Two illustrative test examples are considered to assess the performance of the proposed method. Numerical results indicate that the combination of MSEBI and PSO can provide a reliable tool to accurately identify the multiple structural damage.     

A Variational Model for Dislocations in the Line Tension Limit

We study the interaction of a singularly-perturbed multiwell energy (with an anisotropic nonlocal regularizing term of H ^1/2 type) and a pinning condition. This functional arises in a phase field model for dislocations which was recently proposed by Koslowski, Cuitiño and Ortiz, but it is also of broader mathematical interest. In the context of the dislocation model we identify the Γ-limit of the energy in all scaling regimes for the number N _? of obstacles. The most interesting regime is N _? ≈|ln ?|/?, where ? is a nondimensional length scale related to the size of the crystal lattice. In this case the limiting model is of line tension type. One important feature of our model is that the set of energy wells is periodic, and hence not compact. Thus a key ingredient in the proof is a compactness estimate (up to a single translation) for finite energy sequences, which generalizes earlier results from Alberti, Bouchitté and Seppecher for the two-well problem with a H ^1/2 regularization.     

Two-equation and multi-fluid turbulence models for Rayleigh–Taylor mixing

This paper presents a new, improved version of the K–L model, as well as a detailed investigation of K–L and multi-fluid models with reference to high-resolution implicit large eddy simulations of compressible Rayleigh–Taylor mixing. The accuracy of the models is examined for different interface pressures and specific heat ratios for Rayleigh–Taylor flows at initial density ratios 3:1 and 20:1. It is shown that the original version of the K–L model requires modifications in order to provide comparable results to the multi-fluid model. The modifications concern the addition of an enthalpy diffusion term to the energy equation; the formulation of the turbulent kinetic energy (source) term in the K equation; and the calculation of the local Atwood number. The proposed modifications significantly improve the results of the K–L model, which are found in good agreement with the multi-fluid model and implicit large eddy simulations with respect to the self-similar mixing width; peak turbulent kinetic energy growth rate, as well as volume fraction and turbulent kinetic energy profiles. However, a key advantage of the two-fluid model is that it can represent the degree of molecular mixing in a direct way, by transferring mass between the two phases. The limitations of the single-fluid K–L model as well as the merits of more advanced Reynolds-averaged Navier–Stokes models are also discussed throughout the paper.     

Finite Scale Microstructures in Nonlocal Elasticity

In this paper we develop a simple one-dimensional model accounting for the formation and growth of globally stable finite scale microstructures. We extend Ericksen's model [9] of an elastic “bar” with nonconvex energy by including both oscillation-inhibiting and oscillation-forcing terms in the energy functional. The surface energy is modeled by a conventional strain gradient term. The main new ingredient in the model is a nonlocal term which is quadratic in strains and has a negative definite kernel. This term can be interpreted as an energy associated with the long-range elastic interaction of the system with the constraining loading device. We propose a scaling of the problem allowing one to represent the global minimizer as a collection of localized interfaces with explicitly known long-range interaction. In this limit the augmented Ericksen's problem can be analyzed completely and the equilibrium spacing of the periodic microstructure can be expressed as a function of the prescribed average displacement. We then study the inertial dynamics of the system and demonstrate how the nucleation and growth of the microstructures result in the predicted stable pattern. Our results are particularly relevant for the modeling of twined martensite inside the austenitic matrix.     

Energetic dislocation interactions and thermodynamical aspects of strain gradient crystal plasticity theories

This paper focuses on the unification of two frequently used and apparently different strain gradient crystal plasticity frameworks: (i) the physically motivated strain gradient crystal plasticity models proposed by Evers et al. [2004a. Non-local crystal plasticity model with intrinsic SSD and GND effects. Journal of the Mechanics and Physics of Solids 52, 2379-2401; 2004b. Scale dependent crystal plasticity framework with dislocation density and grain boundary effects. International Journal of Solids and Structures 41, 5209-5230] and Bayley et al. [2006. A comparison of dislocation induced back stress formulations in strain gradient crystal plasticity. International Journal of Solids and Structure 43, 7268-7286; 2007. A three dimensional dislocation field crystal plasticity approach applied to miniaturized structures. Philosophical Magazine 87, 1361-1378] (here referred to as Evers-Bayley type models), where a physical back stress plays the most important role and which are further extended here to deal with truly large deformations, and (ii) the thermodynamically consistent strain gradient crystal plasticity model of Gurtin (2002-2008) (here referred to as the Gurtin type model), where the energetic part of a higher order micro-stress is derived from a non-standard free energy function. The energetic micro-stress vectors for the Gurtin type models are extracted from the definition of the back stresses of the improved Evers-Bayley type models. The possible defect energy forms that yield the derived physically based micro-stresses are discussed. The duality of both type of formulations is shown further by a comparison of the micro-boundary conditions. As a result, this paper provides a direct physical interpretation of the different terms present in Gurtin's model.     

Atomistically determined phase-field modeling of dislocation dissociation,stacking fault formation,dislocation slip,and reactions in fcc systems

The purpose of the current work is the development of a phase field model for dislocation dissociation, slip and stacking fault formation in single crystals amenable to determination via atomistic or ab initio methods in the spirit of computational material design. The current approach is based in particular on periodic microelasticity (Wang and Jin, 2001, Bulatov and Cai, 2006, Wang and Li, 2010) to model the strongly non-local elastic interaction of dislocation lines via their (residual) strain fields. These strain fields depend in turn on phase fields which are used to parameterize the energy stored in dislocation lines and stacking faults. This energy storage is modeled here with the help of the ”interface” energy concept and model of Cahn and Hilliard (1958) (see also Allen and Cahn, 1979, Wang and Li, 2010). In particular, the “homogeneous” part of this energy is related to the “rigid” (i.e., purely translational) part of the displacement of atoms across the slip plane, while the “gradient” part accounts for energy storage in those regions near the slip plane where atomic displacements deviate from being rigid, e.g., in the dislocation core. Via the attendant global energy scaling, the interface energy model facilitates an atomistic determination of the entire phase field energy as an optimal approximation of the (exact) atomistic energy; no adjustable parameters remain. For simplicity, an interatomic potential and molecular statics are employed for this purpose here; alternatively, ab initio (i.e., DFT-based) methods can be used. To illustrate the current approach, it is applied to determine the phase field free energy for fcc aluminum and copper. The identified models are then applied to modeling of dislocation dissociation, stacking fault formation, glide and dislocation reactions in these materials. As well, the tensile loading of a dislocation loop is considered. In the process, the current thermodynamic picture is compared with the classical mechanical one as based on the Peach-Köhler force.     

基于三维弹性杆模型的DNA拉伸响应的研究

通过Young-Laplace方程将界面张力引入Kirchhoff方程,并结合Gibbs与Langmuir吸附方程建立了受溶液浓度影响的三维DNA弹性杆模型。基于此模型,引入DNA端部的边界条件,运用打靶法来模拟计算溶液中的DNA链段受端部拉力作用下的几何构型。进一步分析了在界面能与弹性应变能的耦合作用下,DNA链段平衡构型的形状与尺寸的变化规律。     

Heuristic search for a predictive strain-energy function in nonlinear elasticity

In this work, a new, quasi-structural model – bootstrapped eight-chain model – is proposed as a modification to the strain energy of eight-chain model [Arruda, E.M., Boyce, M.C., 1993. A three-dimensional constitutive model for the large stretch behaviour of rubber elastic materials. J. Mech. Phys. Solids 41, 389—412] that invokes the Langevin chain statistics. This development has been led to by our heuristic search into how the strain energy of eight-chain model may be adapted in order to account better for the mechanical behaviour of elastomeric materials in both linear and nonlinear elastic regimes [Treloar, L.R.G., 1944. Stress–strain data for vulcanised rubber under various types of deformation. Trans. Faraday Soc. 40, 59–70]. The eight-chain model appears to produce very similar results in predicting biaxial stress to those of a first stretch-invariant model that gives a good fit in uniaxial extension and, thus, it is shown that the former can not be significantly enhanced within the limitation of the latter. Evaluation of predictive capability for an additive invariant-separated form of strain energy shows that an explicit inclusion of a second stretch-invariant function would not work and that any thus added term ought to be dependent on both the first and second stretch-invariants of deformation tensor, and hints that an improvement is possibly needed at low strain. The composite and filament models [Miroshnychenko, D., Green, W.A., Turner, D.M., 2005. Composite and filament models for the mechanical behaviour of elastomeric materials. J. Mech. Phys. Solids 53 (4), 748–770] have their strain-energy functions in that suggested form and cope very well with predicting the experimental data of Treloar (1944). We use the form of strain energy for the filament model, that proved to be successful, to bootstrap the strain energy of eight-chain model in order to improve the performance of the latter at low strain. Thus, we derive a new model – bootstrapped eight-chain model – that requires only two material parameters – a rubber modulus and a limiting chain extensibility. The proposed model is quasi-structural due to bootstrapping and it retains the best traits and corrects the faults of the eight-chain model, conforming more closely to the classical experimental data of Treloar (1944).     

A new second‐moment closure approach for turbulent swirling confined flows

An improved anisotropic model for the dissipation rate—ε—of the turbulent kinetic energy (k), to be used together with a non‐linear pressure‐strain correlations model, is proposed. Experimental data from the open literature for two confined turbulent swirling flows are used to assess the performance of the proposed model in comparison to the standard ε transport equation and to a linear approach to model the pressure‐strain term that appears in the exact equations for the Reynolds‐stress tensor. For the less strongly swirling flow the predictions show much more sensitivity to the εtransport equation than to the pressure‐strain model. In opposition, for the more strongly swirling flow, the results show that the predictions are much sensitive to the pressure‐strain model. Nevertheless, the improved εtransport equation together with the non‐linear pressure strain model yield predictions in good agreement with experiments in both studied cases. Copyright © 2003 John Wiley & Sons, Ltd.     

An implicit consistent algorithm for the integration of thermoviscoplastic constitutive equations in adiabatic conditions and finite deformations

The so-called viscoplastic consistency model, proposed by Wang, Sluys and de Borst, is extended here to the integration of a thermoviscoplastic constitutive equation for J₂ plasticity and adiabatic conditions. The consistency condition in this case includes not only strain rate but also the effect of temperature on the yield function. Using the backward Euler integration scheme to integrate the constitutive equations, an implicit algorithm is proposed, leading to generalized expressions of the classical return mapping algorithm for J₂ plasticity, both for the iterative calculation of the plastic multiplier increment and for the consistent tangent operator when strain rate and temperature are considered also as state variables of the hardening equation. The model was implemented in a commercial finite element code and its performance is demonstrated with the numerical simulation of four Taylor impact tests.     

A variational model for fracture mechanics: Numerical experiments

In the variational model for brittle fracture proposed in Francfort and Marigo [1998. Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46, 1319-1342], the minimum problem is formulated as a free discontinuity problem for the energy functional of a linear elastic body. A family of approximating regularized problems is then defined, each of which can be solved numerically by a finite element procedure. Here we re-formulate the minimum problem within the context of finite elasticity. The main change is the introduction of the dependence of the strain energy density on the determinant of the deformation gradient. This change requires new, more general existence and Γ-convergence results. The results of some two-dimensional numerical simulations are presented, and compared with corresponding simulations made in Bourdin et al. [2000. Numerical experiments in revisited brittle fracture. J. Mech. Phys. Solids 48, 797-826] for the linear elastic model.     

Energy theorems and the J-integral in dipolar gradient elasticity

Within the framework of Mindlin’s dipolar gradient elasticity, general energy theorems are proved in this work. These are the theorem of minimum potential energy, the theorem of minimum complementary potential energy, a variational principle analogous to that of the Hellinger–Reissner principle in classical theory, two theorems analogous to those of Castigliano and Engesser in classical theory, a uniqueness theorem of the Kirchhoff–Neumann type, and a reciprocal theorem. These results can be of importance to computational methods for analyzing practical problems. In addition, the J-integral of fracture mechanics is derived within the same framework. The new form of the J-integral is identified with the energy release rate at the tip of a growing crack and its path-independence is proved.The theory of dipolar gradient elasticity derives from considerations of microstructure in elastic continua [Mindlin, R.D., 1964. Microstructure in linear elasticity. Arch. Rational Mech. Anal. 16, 51–78] and is appropriate to model materials with periodic structure. According to this theory, the strain-energy density assumes the form of a positive-definite function of the strain (as in classical elasticity) and the second gradient of the displacement (additional term). Specific cases of the general theory considered here are the well-known theory of couple-stress elasticity and the recently popularized theory of strain-gradient elasticity. The latter case is also treated in the present study.     

Impact comminution of solids due to local kinetic energy of high shear strain rate: I. Continuum theory and turbulence analogy

The modeling of high velocity impact into brittle or quasibrittle solids is hampered by the unavailability of a constitutive model capturing the effects of material comminution into very fine particles. The present objective is to develop such a model, usable in finite element programs. The comminution at very high strain rates can dissipate a large portion of the kinetic energy of an impacting missile. The spatial derivative of the energy dissipated by comminution gives a force resisting the penetration, which is superposed on the nodal forces obtained from the static constitutive model in a finite element program. The present theory is inspired partly by Grady's model for expansive comminution due to explosion inside a hollow sphere, and partly by analogy with turbulence. In high velocity turbulent flow, the energy dissipation rate gets enhanced by the formation of micro-vortices (eddies) which dissipate energy by viscous shear stress. Similarly, here it is assumed that the energy dissipation at fast deformation of a confined solid gets enhanced by the release of kinetic energy of the motion associated with a high-rate shear strain of forming particles. For simplicity, the shape of these particles in the plane of maximum shear rate is considered to be regular hexagons. The particle sizes are assumed to be distributed according to the Schuhmann power law. The condition that the rate of release of the local kinetic energy must be equal to the interface fracture energy yields a relation between the particle size, the shear strain rate, the fracture energy and the mass density. As one experimental justification, the present theory agrees with Grady's empirical observation that, in impact events, the average particle size is proportional to the (−2/3) power of the shear strain rate. The main characteristic of the comminution process is a dimensionless number B_a (Eq. (37)) representing the ratio of the local kinetic energy of shear strain rate to the maximum possible strain energy that can be stored in the same volume of material. It is shown that the kinetic energy release is proportional to the (2/3)-power of the shear strain rate, and that the dynamic comminution creates an apparent material viscosity inversely proportional to the (1/3)-power of that rate. After comminution, the interface fracture energy takes the role of interface friction, and it is pointed out that if the friction depends on the slip rate the aforementioned exponents would change. The effect of dynamic comminution can simply be taken into account by introducing the apparent viscosity into the material constitutive model, which is what is implemented in the paper that follows.     

On higher order gradient continuum theories in 1-D nonlinear elasticity. Derivation from and comparison to the corresponding discrete models

Higher order gradient continuum theories have often been proposed as models for solids that exhibit localization of deformation (in the form of shear bands) at sufficiently high levels of strain. These models incorporate a length scale for the localized deformation zone and are either postulated or justified from micromechanical considerations. Of interest here is the consistent derivation of such models from a given microstructure and the subsequent comparison of the solution to a boundary value problem using both the exact microscopic model and the corresponding approximate higher order gradient macroscopic model.In the interest of simplicity the microscopic model is a discrete periodic nonlinear elastic structure. The corresponding macroscopic model derived from it is a continuum model involving higher order gradients in the displacements. Attention is focused on the simplest such model, namely the one whose energy density involves only the second order gradient of the displacement. The discrete to continuum comparisons are done for a boundary value problem involving two different types of macroscopic material behavior. In addition the issues of stability and imperfection sensitivity of the solutions are also investigated.