Some general results in the theory of crystallographic slip

Crystallographic slip of a Bravais lattice is analyzed utilizing the main results of a recently constructed theory of structured solids, where explicit account is taken of the influence of dislocation density identified in terms of Curl of plastic deformationG _p . In the present paper, the scope of the subject is enlarged to also include defects (other than dislocations) such as substitutional impurities and vacancies and it is shown that these point defects may also be characterized in terms of the plastic deformation fieldG _p . Several general results pertaining to the kinematics and kinetics of Crystallographic slip are proved within the scope of an appropriate constraint theory suitable for Crystallographic slip; the latter is motivated by the well-known basic mechanism of Crystallographic slip that constrains the admissible modes of plastic deformation. The constraint responses (or forces) that are necessary to maintain the active slip systems, as well as the conditions for the transitions between the slip systems, are determined. In spite of the nature of the assumption pertaining to the mechanism of Crystallographic slip on distinct slip systems, it is shown that the yield surface does not necessarily exhibit sharp corners. Instead, the shape of the yield surface is in the form of hyperplanes joined by round corners. In fact, the presence of sharp corners is mainly a result of the use of a special set of constitutive assumptions. The predictive capability of the theoretical results is further illustrated by using a two-dimensional crystal subjected to simple shear. The effect of the initial dislocation density on the response of the sheared-crystal is studied by carrying out detailed calculations for two substantially different initial dislocation densities. The calculations show that while the response of the crystal is sensitive to the initial dislocation density in the early stages of deformation, its influence diminishes with progressively larger deformations. Furthermore, the crystal exhibits a well-defined shear band which evolves naturally due to the presence of initial dislocation distribution and is easily visible at large deformations.     

Plane constrained shear of single crystal strip with one and two active slip systems

The plane constrained shear problems of a single crystal strip with one and two active slip systems are considered within the continuum dislocation theory. Analytical solutions are found for single slip and symmetric double slip systems which exhibit the energetic and dissipative thresholds for dislocation nucleation, the Bauschinger translational work hardening and the size effect. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dislocation Structure of Grain Boundaries in Single Crystals Deforming in Simple Shear

A model for the formation of grain boundaries in single crystals having a single active slip system for the case of plane strain simple shear is proposed. It is shown that non-convexity of the condensed energy gives rise to the formation of a laminate structure, where sharp interfaces between laminate layers are interpreted as grain boundaries. Based on these results the dislocation structure of the boundaries is determined introducing a transition zone between laminate layers, where smooth functions of displacement and plastic slip connect adjacent layers. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Plane constrained uniaxial extension of a single crystal strip

Within continuum dislocation theory the plane constrained uniaxial extension of a single crystal strip deforming in single or double slip is analyzed. For the single and symmetric double slip, the closed-form analytical solutions are found which exhibits the energetic and dissipative thresholds for dislocation nucleation, the Bauschinger translational work hardening, and the size effect. Numerical solutions for the non-symmetric double slip are obtained by finite element procedures. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Toughening effect of ferroelectric ceramics induced by domain switching and dislocations

The multi-scale analysis of fracture toughness of ferroelectric ceramics under complicate mechanical–electrical coupling effect is carried out in this paper. The generalized stress intensity factor (SIF) arising from spontaneous strains and polarization transformation in switching domain zones is accurately obtained by using an extended Eshelby theory. Taking BaTiO₃ ferroelectric ceramic for example, it is discovered that the crack propagation can be induced by domain switching arising from negative electrical field when the crack surface is parallel to the isotropic plane, and the obtained critical electric displacement intensity factor (EDIF) approximates closely to that obtained by the Green’s function method. Additionally, as pinning dislocations and slip dislocations can strongly influence properties of ferroelectric devices and induce the property degradation, it is necessary to investigate the dislocation toughening effects on fatigue and fracture mechanisms. The results show that the dislocation shielding and anti-shielding effects on mode II SIF, mode I SIF and EDIF are obviously different when a dislocation locates at a position near the crack tip. Through the calculation of the critical applied EDIF for crack propagation by using mechanical energy release rate (MERR) theory, it is discovered that the slip angles obviously influence fracture toughness, and the mode II SIF arising from dislocation has little influence on fracture toughness, however, the mode I SIF and EDIF arising from dislocation have great influences on fracture toughness.     

Rate-independent plasticity and statistically stored dislocations

We present here a continuum model for the evolution of the total dislocation density in the framework of rate-independent plasticity. Three basic physical features are taken into account: (i) the role of dislocation densities on hardening; (ii) the relations between the slip velocity and the mobility of gliding dislocations; (iii) the energetics of self and mutual interactions between dislocations. We restrict attention to plastic processes corresponding to single slip. Numerical simulations showing the formation of bands are also presented.     

Rate-independent plasticity and statistically stored dislocations

We present here a continuum model for the evolution of the total dislocation density in the framework of rate-independent plasticity. Three basic physical features are taken into account: (i) the role of dislocation densities on hardening; (ii) the relations between the slip velocity and the mobility of gliding dislocations; (iii) the energetics of self and mutual interactions between dislocations. We restrict attention to plastic processes corresponding to single slip. Numerical simulations showing the formation of bands are also presented.     

Are plastic yielding and work hardening sensitive to the boundary conditions?

The plane strain shear of a single crystal strip with one active slip system placed in a mixed device with one clamped and one free boundary is considered. Since dislocations pile up against only the clamped boundary, the plastic yielding and work hardening differ essentially from those of a hard device, showing clearly their sensitivity to the boundary conditions. An analytical solution to this problem within continuum dislocation theory is found explicitly which exhibits the energetic and dissipative thresholds for dislocation nucleation, the Bauschinger translational work hardening, and the size effects. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Phase-field Model of Dislocations on Parallel Slip Planes

We establish a line-tension energy for dislocation networks concentrated in finitely many parallel slip planes as the limit of a Peierls-Nabarro energy. Dislocations in different planes interact depending on their distance. At intermediate distances a two-scale dislocation microstructure is optimal. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Variational Phase Field Modeling of Laminate Deformation Microstructure in Finite Gradient Crystal Plasticity

The macroscopic mechanical behavior of many materials crucially depends on the formation and evolution of their microstructure. In this work, we consider the formation and evolution of laminate deformation microstructure in plasticity. Inspired by work on the variational modeling of phase transformation [5] and building on related work on multislip gradient crystal plasticity [9], we present a new finite strain model for the formation and evolution of laminate deformation microstructure in double slip gradient crystal plasticity. Basic ingredients of our model are a nonconvex hardening potential and two gradient terms accounting for geometrically necessary dislocations (GNDs) by use of the dislocation density tensor and regularizing the sharp interfaces between different kinematically coherent plastic slip states. The plastic evolution is described by means of a nonsmooth dissipation potential for which we propose a new regularization. We formulate a continuous gradient-extended rate-variational framework and discretize it in time to obtain an incremental-variational formulation. Discretization in space yields a finite element formulation which is used to demonstrate the capability of our model to predict the formation and evolution of laminate deformation microstructure in f.c.c. Copper with two active slip systems in the same slip plane. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A crystal plasticity framework based on the Continuum Dislocation Dynamics theory: relaxation of an idealized micropillar

The motion and interaction of dislocation lines are the physical basis of the plastic deformation of metals. Although ‘discrete dislocation dynamic’ (DDD) simulations are able to predict the kinematics of dislocation microstructure (i.e. the motion of dislocations in a given velocity field) and therefore the plastic behavior of crystals in small length scales, the computational cost makes DDD less feasible for systems larger than a few micro meters. To overcome this problem, the Continuum Dislocation Dynamics (CDD) theory was developed. CDD describes the kinematics of dislocation microstructure based on statistical averages of internal properties of dislocation systems. In this paper we present a crystal plasticity framework based on the CDD theory. It consists of two separate parts: a classical 3D elastic boundary value problem and the evolution of dislocation microstructure within slip planes according to the CDD constitutional equations. We demonstrate the evolution of dislocation density in a micropillar with a single slip plane. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Steady solutions of the Navier–Stokes equations with threshold slip boundary conditions

We establish the wellposedness of the time‐independent Navier–Stokes equations with threshold slip boundary conditions in bounded domains. The boundary condition is a generalization of Navier's slip condition and a restricted Coulomb‐type friction condition: for wall slip to occur the magnitude of the tangential traction must exceed a prescribed threshold, independent of the normal stress, and where slip occurs the tangential traction is equal to a prescribed, possibly nonlinear, function of the slip velocity. In addition, a Dirichlet condition is imposed on a component of the boundary if the domain is rotationally symmetric. We formulate the boundary‐value problem as a variational inequality and then use the Galerkin method and fixed point arguments to prove the existence of a weak solution under suitable regularity assumptions and restrictions on the size of the data. We also prove the uniqueness of the solution and its continuous dependence on the data. Copyright © 2006 John Wiley & Sons, Ltd.     

Non-quadratic defect energy: A comparison of gradient plasticity simulations to discrete dislocation dynamics results

A small-deformation strain gradient plasticity (GP) model for single-crystals has been proposed in [1], including a grain boundary (GB) yield condition without hardening. It has been extended by a hardening term for the GBs after a comparison to discrete dislocation dynamics (DDD) results in [2]. Differences between the strain gradients of the GP results and the DDD results motivate the consideration of a non-quadratic defect energy [3] in the GP model. It is shown that the gradients in the GP model can be improved using an exponent different from two. Remaining discrepancies in the strain profiles, compared to the DDD results, are attributed to the neglect of the individual gradients of plastic slip and due to the lack of a mechanism for the misorientation-dependent elastic interactions of dislocations across GBs [4] in the GP model. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global analysis of multi-strains SIS, SIR and MSIR epidemic models

We consider SIS, SIR and MSIR models with standard mass action and varying population, with n different pathogen strains of an infectious disease. We also consider the same models with vertical transmission. We prove that under generic conditions a competitive exclusion principle holds. To each strain a basic reproduction ratio can be associated. It corresponds to the case where only this strain exists. The basic reproduction ratio of the complete system is the maximum of each individual basic reproduction ratio. Actually we also define an equivalent threshold for each strain. The winner of the competition is the strain with the maximum threshold. It turns out that this strain is the most virulent, i.e., this is the strain for which the endemic equilibrium gives the minimum population for the susceptible host population. This can be interpreted as a pessimization principle.     

Simulation of stage I-crack growth using a 3D-model

A three-dimensional model for stage I-short crack propagation on a single slip plane is presented. It considers elastic plastic material behaviour by allowing sliding on the active slip plane in a defined slip direction. A crack propagation law based on the crack tip slide displacement is implemented to simulate crack propagation. The model is solved numerically using the dislocation loop technique. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling of coupled deformation-induced twinning and dislocation slip using incremental energy minimization

The complex interplay between dislocations and deformation-induced twinning leads to a relatively poor formability of magnesium at room temperature. For understanding the complicated behavior of this metal, a novel model is presented. It is based on a variational principle. Within this principle based on energy minimization, dislocation slip is modeled by crystal plasticity theory, while the phase decomposition associated with twinning is considered by sequential laminates. The proposed model captures the transformation of the crystal lattice due to twinning in a continuous fashion by simultaneously taking dislocation slip within both, possibly co-existent, phases into account. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Experimental investigation of elastic turbulence and boundary slip of elastomers over a broad temperature interval

Ethylene-propylene copolymer, a typical stereo rubber, has been investigated by capillary viscometry. Ethylene-propylene copolymer possesses high thermooxidative stability, which has made it possible to study its viscosity properties, determine the onset of elastic turbulence and boundary slip, and measure the slip rate over a very broad temperature interval, from room temperature to 260° C. The flow of elastomers differs from that of thermoplastics in that at relatively low strain rates flow is complicated by the boundary slip effect. The mean boundary slip velocities of the copolymer at shear stresses above 10⁶ dynes/cm² are measured in tens of centimeters per second. As the temperature rises, they rapidly increase.Mekhanika Polimerov, Vol. 4, No. 2, pp. 336–342, 1968     

Modelling thermoplastic material behaviour of dual-phase steels on a microscopic length scale

On a microscopic length scale dual-phase steels exhibit a polycrystalline microstructure consisting of ferrite and martensite. In this work a material model for the temperature dependent hardening behaviour of the ferritic phase is presented. As the dislocation structure determines the resistance to dislocation glide, dislocation densities are introduced as state variables to capture the dependence of the material behaviour on the loading history. Motivated by the elementary processes of multiplication by the Frank-Read-mechanism and annihilation by cross-slip, evolution equations for the dislocation densities are introduced. Based on the interaction of dislocations on different slip systems and the Peierls-stress, the resistance to dislocation motion with its temperature dependence is formulated to describe the hardening behaviour. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Development of the local strain function and the slip function in statistical theories of plasticity

The development of the concept of a local strain function is examined. The principal forms and methods of approximation of the local strain function are described and compared. A generalization of the local strains theory in differential form is proposed. Yosimura's criticism of the slip theory is shown to be invalid. Certain aspects of the local strains and slip theories are compared.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 3, pp. 422–430, May–June, 1969.     

The destabilizing effect of boundary slip on Bénard convection

We investigate the influence of slip boundary conditions on the onset of Bénard convection in an infinite fluid layer. It is shown that the critical Rayleigh number is a decreasing function of the slip length, and therefore boundary slip is seen to have a destabilizing effect. Chebyshev‐tau and compound matrix formulations for solving the eigenvalue problem are presented. Copyright © 2005 John Wiley & Sons, Ltd.