In crystal plasticity, there are several phenomena, like size effect, dislocation patterning, dislocation core structure that cannot be modeled with traditional local plasticity theories. Pioneered by Elias Aifantis during the past 30 years, several different gradient type terms were proposed to account for the different nonlocal effects.
In the talk, we present a survey on our current understanding of the statistical physics origin of gradient terms in continuum theory of dislocations and in nonlocal elasticity theory. It is discussed what sort of terms can or cannot be justified.

In this study, we present the highlights and significant characteristics of Complexity theory, as well as we provide the new mathematical framework needed to describe the rich phenomenology arise in complex systems' behavior far from equilibrium.
Physical processes, regarding complex systems, are characterized by nonlinear dynamics and are present everywhere from the microscopic to the macroscopic level. In particular, the nonlinearity of continuous media is a basic characteristic of complexity, giving rise to the far from equilibrium rich phenomenology which is related to all the topics and manifestations of complexity theory. A non-exhaustive list of characteristics of such behavior includes: strange dynamics, anomalous diffusion, multifractal topology, intermittent turbulence, percolation, fractons, power-scaling laws, low dimensional or spatiotemporal chaos, self-organized critical behavior, self-organized nonlinear instabilities as nonequilibrium stationary states, nonequilibrium phase transition processes, nonGaussian statistics etc. In order to describe the aforementioned behavior of complex systems, current applied mathematics and statistics are inefficient and extensions have to be made in order to effectively "capture" the dynamics and its evolution.
Such extensions are the fractional extension of dynamics of continuous media, the fractional extension of Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) hierarchy of statistics and fractional Fokker Planck equation, the fractional extension of Landau Ginsburg equation, the Tsallis q-extension of Central Limit Theorem (CLT), the fractional and Tsallis q-extension of renormalization group theory (RGT) etc.
Finally, one of the most important cornerstones of complexity theory is Tsallis nonextensive statistical mechanics, an extended version of Boltzmann-Gibbs statistical mechanics and notion of entropy, which is developed and capable to describe significant features of complex systems such as multifractality, scaling and long range correlations. Basic characteristics of Tsallis statistics and its relation to the fractional extension of mathematical theory will also be given.

Nanoporous metallic actuators constitute a new class of low-voltage actuators that feature a unique combination of relatively large strain amplitudes, low operating voltages and high specific stiffness and strength. These so-called "metallic muscles" consist of ligaments and pores in the nanometer regime giving rise to a very high internal surface area. The key obstacles to the integration of nanoporous metals into current fundamental concepts and technological applications (MEMS, NEMS) are (i) the presence of the aqueous electrolyte itself that is needed to inject electronic charge in the space-charge region at the metal/electrolyte interface. (ii) the rate of actuation due to the relatively low ionic conductivity of the electrolyte, and (iii) the magnitude of the actuating displacements. Here we discuss a novel approach to generate work from metallic muscles that overcome these hurdles. From an experimental viewpoint, a new ultrafast, all-solid organometallic actuator has been designed, synthesized and tested. The tunable, semiconducting properties of conjugated polymers are exploited to inject charge into the metal. In addition, a new microstructural design based on a layered structure with enhanced actuation strokes has been developed. In the presentation, the defects and size effects of metallic muscles in particular will also be discussed.

In this study, we are going to present a method for solving the differential equation which is governed to an anisotropic finite cracked wedge. The wedge's radial boundaries are traction–traction and the circular boundary of the wedge is considered as a fixed displacement condition. Because the circular boundary of the wedge has never been fixed in previous studies, thus an appropriate mapping and semi first and second kind of finite Mellin transforms are presented in the solution procedure of the problem. According to the aforementioned mathematical operations, new complex functions based on fixed displacement boundary conditions on the circular segment of the wedge is defined. By inserting this novelty to the problem, the corresponding equations have been developed. Then, the governing singular integral equation is extracted using the complex functions regarding to radial boundary conditions. Finally, the corresponding integral equation is solved numerically and the results are plotted.

The dislocation-based fracture or distributed dislocation technique is rather well known within classical elasticity. Here, we will focus on its application to present nonsingular models of cracks in generalized continua. Nonlocal and gradient elasticity will be employed to present nonsingular fields of crack of mode III. Considering the nonstandard boundary conditions in gradient elasticity, the appropriate boundary conditions will be studied. An incompatible framework will be used for the dislocation-based approach.
References:
Mousavi SM, Paavola J, Baroudi D (2014) Distributed nonsingular dislocation technique for cracks in strain gradient elasticity, Journal of the Mechanical Behavior of Materials, 23:47-58.
Mousavi SM, Lazar M (2014) Distributed dislocation technique for cracks based on non-singular dislocations in nonlocal elasticity of Helmholtz type. Engineering Fracture Mechanics, 136:79–95.
Mousavi SM, Aifantis EC (2015) A Note on Dislocation-based Mode III Gradient Elastic Fracture Mechanics. Journal of the Mechanical Behavior of Materials, to appear.
Mousavi SM, Korsunsky AM (2015) Non-singular fracture theory within nonlocal anisotropic elasticity. Material and Design, to appear.

The main challenge of this paper is the development of a titanium implant having both favorable bulk and surface functionalities not only to certify the long-standing stability of the implant but also to moderate interactions between cells and substrate. Recently, ultrafine/nanostructured materials fabricated by severe plastic deformation (SPD) exhibits promising bulk and surface properties. Therefore, microstructural refinement and its effect on mechanical and surface functionalities of β-type titanium alloys through severe plastic deformation (SPD) processing are investigated in this study. A novel β-type titanium alloy, Ti–29Nb–13Ta–4.6Zr (TNTZ), composing of non-toxic and non-allergic Nb, Ta, and Zr alloying elements, has been subjected to high-pressure torsion (HPT). Solutionized TNTZ (TNTZST) at 1063K for 1 hour was subjected to cold rolling (TNTZCR) at reduction ratio of over 80 % and aging treatment (TNTZAT) at 723 K for 3 hours. While TNTZST and TNTZCR show coarse-grained single β structure, TNTZAT exhibits a coarse-grained β matrix and precipitated needle like α phase. TNTZCR and TNTZAT subjected to HPT processing (TNTZCHPT and TNTZAHPT) show a homogeneous microstructure consisting of nanostructured elongated β grains. The β grains exhibit nanostructured subgrains having non-uniform morphologies distorted by severe torsional deformation. Furthermore, the β grains and subgrains are surrounded by blurred and wavy boundaries in a non-equilibrium state. In addition, the needle-like α precipitates are totally refined to a nanostructure with a diameter of approximately 12 nm. TNTZCHPT and TNTZAHPT show enhanced mechanical biocompatibility, which is a greater tensile strength and a higher hardness than those of TNTZST, TNTZCR, TNTZAT, and Ti-6Al-4V ELI while maintaining relatively low Young's modulus. Furthermore, TNTZCHPT and TNTZAHPT exhibit an enhanced combination of great corrosion performance and improved cellular response in comparison to TNTZST, TNTZCR, TNTZAT, and Ti-6Al-4V ELI.

We reconsider the Toupin-Mindlin indeterminate couple stress model being one of the prominent gradient elasticity models. While abundantly used for the description of nano-sized materials, some theoretical issues concerning boundary conditions and constitutive assumptions need discussion. We provide an improved set of independent boundary conditions making it possible to uniquely identify material properties. In addition, we highlight the connections to the more general gradient elasticity models.

In this work, we employ the dual boundary element method to model crack propagation in plane micropolar elasticity. We use both - the singular and hyper-singular - boundary integral equations on the coincident parts of the boundary to model a crack. Next, we evaluate the J-integrals from the solution fields in the vicinity of a crack tip and use them to derive the stress and couple stress intensity factors and formulate the criteria of crack propagation.
The BEM-formulation allows to incorporate voids and inclusions of different micropolar materials with perfect and imperfect interfaces and analyze their effect on crack paths.
We study several benchmark problems and compare the results with classical elasticity.

In the framework of continuum mechanics, the classification of the defects, i.e. the internal sources of elastic fields, is presented. The classification is based on the dimension of the area where the eigenstrain of the defect is defined. The elastic models for the point defects, linear defects, surface defects and inclusions are demonstrated. The defects in 3D- and 2D-elastic media are compared. The elastic fields for the axisymmetric defects such as circular dilatational line and circular dilatational disk are given in the form of Lipschitz-Hankel integrals. The techniques for solutions of the boundary value problems in the theory of defects are discussed. The elastic behavior of the circular prismatic dislocation loops in elastic bodies with spherical free surfaces and interfaces is considered.

The use of a deformation version of gradient plasticity in order to obtain the analytical solution of the elastoplastic axisymmetric borehole, subjected to a far-field biaxial tension, has been provided. The constitutive model results from the basic assumption of a gradient depended flow stress, i.e. the dependence of the effective stress on the Laplacian of the effective strain. Deformation theory is formulated within an approximate strength of materials approach, utilizes an averaging process over the entire deformation history and relates the total plastic strain to the final total stress. The elastic part of the problem was considered through the classical elasticity theory. The analytical solution and the results, i.e. the distributions of stresses and strains, are illustrated and compared in the context of the size effects which contribute to the study of the borehole stability.

Mindlin's second strain gradient theory, due to its competency in capturing the surface effects, is of particular interest. For a homogeneous, centrosymmetric and isotropic material, formulation in this framework introduces sixteen additional constants. Fifteen parameters bring up four characteristic lengths associated with bulk and another one associated with surface. Bulk characteristic lengths and Lame constants appear in equilibrium equations, while surface characteristic length enters the formulation through boundary conditions. The remaining parameter is modulus of cohesion characterizing the surface effect. To date, there are no successful experimentations measuring these internal length scales. Due to the lack of experimental and computational data, quantitative analyses based on these enriched theories have been impossible. The present work gives an accurate remedy for the calculation of these parameters by utilizing an analytical formulation in lattice dynamics combined with the first principles. From the equivalency between the atomistic crystal lattice dynamics of bulk and its counterpart in second strain gradient elasticity, bulk characteristic lengths are expressed in terms of atomic force constants. Additionally, considering a free standing film that brings the surface effect into account, analytical expressions for surface energy and change in film thickness are obtained in terms of modulus of cohesion, Lame constants, surface and bulk characteristic lengths and film thickness. The numerical values of atomic force constants, surface energy and change in film thickness are calculated using ab initio simulations and subsequently bulk and surface characteristic lengths and modulus of cohesion are evaluated from the developed analytical expressions.

Disclinations play an important role in rotation-type motions of materials constitutive parts. Such internal motions contribute both to the irreversible change of the body shape (plasticity) and to the storage of internal energy in the body (elasticity). Disclinations, together with dislocations, represent a class of linear defects in solids with plastic distortion however localized at an internal surface. In this talk, we introduce the basis for mechanics of disclinations in solids and present an overview of recent applications of disclination concept to the mechanics of materials. The following themes of the disclination concept are discussed: (a) definitions and designations; (b) geometry of disclinations in structureless and structured continua; (c) elastic properties of screened low-energy screened disclination configurations; (d) applications of disclination approach to various problems of materials mechanics. The mathematical definitions of Volterra and Somigliana dislocations including Frank (rotation) vector of a disclination, types of disclinations: wedge or twist, are discussed. Perfect and partial disclinations are introduced for the continuum with internal structure. Screened disclination configurations are defined as those where the far-filed divergence of disclination elastic strains and mechanical stresses vanishes. The screened configurations include loops, dipoles, and defects at the vicinity of a free surface and in small elastic bodies. The mathematical methods and results of calculation of disclination elastic fields and energies are presented in details. A number of qualitative and quantitative models for rotational plastic deformation of conventional and low-dimensional solids is considered. As an example, disclination models of grain boundaries and their junctions in polycrystals and graphene are presented. The bands with misorientated crystal lattice in metals and other materials are described as a result of partial wedge disclination dipole motion. Finally, the role of disclinations in relaxation of mechanical stresses in lattice mismatched thin layers placed on the bulk substrate is examined.

The crystal structure of hexagonal close-packed (HCP) crystals dictates complex deformation mechanisms, including dislocation-slip and twinning. As a result, the mechanical behavior of HCP metals displays strong anisotropy and strong orientation dependence and identifying the fundamental aspects of plastic deformation in this class of metals is necessary to help improve their performance through alloying and microstructure design. In this talk, we present a new implementation of twin boundaries (TBs) into the framework of DDD to simulate the collective evolution of dislocation and their interactions with TBs in magnesium (Mg) crystals. We report on the effect of crystal size and orientation on the deformation of Mg microcrystals. In addition, the new model is also used to investigate the mechanisms by which TBs and their size influence the yielding stress and hardening behavior of Mg single crystals. Finally, we report on the influence of dislocation interactions with TB and the glide of TB dislocations on the evolution and propagation of TBs.

Face-centered cubic (FCC) materials containing twins such as twin-induced plasticity (TWIP) steels and nano-crystalline copper and nickel have exhibited an attractive combination of properties such as strength and ductility. Recently, there has been thus a significant interest in the deformation behavior of FCC metals involving twins. Traditionally, the coherent twin boundary (CTB) is regarded as a strong barrier to dislocation penetration unless dislocations run through the boundary or transfer with easy cross-slip. Although it is well established that slip is strongly affected by twin boundaries, the detailed aspects of dislocation-twin boundary interactions are not yet fully understood.
We here present the detailed reactions between dislocations and CTB and the resultant formation of dislocation networks in several FCC metals using atomistic simulations. It is found that dislocation networks are mainly composed of sessile Frank dislocations and partially of sessile stair-rod and Hirth dislocations and glissile twinning dislocations. The density and type of dislocations in the networks were found to be dependent on the materials' factors such as generalized stacking fault energy and also external factors like loading axis. The present work could provide insight to understand the source of the huge work-hardening rate and high stability of twin boundaries exhibited in TWIP steels.

It is a long-standing challenge to reproduce numerically the appearance and evolution of the rich variety of dislocation patterns. In conventional two-dimensional models, although specific spatial correlations between dislocations always develop, typical dislocation structures seen in experiments do not seem to emerge. In the talk we demonstrate how two specific extensions of the 2D model lead to pronounced patterning.
Firstly, the case of a non-linear elastic medium is considered, that is, quadratic terms are also taken into account in the stress-strain relation of the continuum the dislocations are embedded in. This extension naturally leads to an energy difference between the vacancy and interstitial type dislocation dipoles. The systematic inclusion of this phenomenon in the continuum theory of dislocations predicts the instability of the homogeneous dislocation structure, which is demonstrated by discrete dislocation dynamics simulations. Indeed, a specific pattern develops with a well-defined characteristic length-scale, resembling a structure seen experimentally in low-amplitude fatigue experiments.
Secondly, the effect of dislocation climb is investigated. Introduction of a small climb mobility already leads to cell formation, that is, dislocation free regions are bounded by dislocation walls consisting of pure GNDs. The formation and subsequent growth of these cells are analogous to subgrain dynamics observed experimentally during recovery.

In this paper the gauge and phase transition theories are used to construct the elasto-plastic model of quasi-brittle material with dissipation. Displacements, plastic distortion tensor are selected as the independent variables. The initial Lagrangian is constructed by the requirement of invariance of the Lagrangian with respect to translation transformation. In order to take into account the phase transition effect of plastic deformation fourth and sixth power terms of distortion tensor are added to the initial Lagrangian. The differential motion equations of the material are obtained by Euler-Lagrange equation with dissipation.The generalized Hook¡¯s laws are obtained on the basis of kinematic variational principle.Several special cese are discussed.

Plastic deformation of metallic glasses is known to depend on several intrinsic and extrinsic parameters, such as mechanical strain, temperature, deformation rate, structure, sample size and composition. In recent studies, the size-dependent deformation behavior of small-scale Pd77Si23 metallic glass samples was investigated by micro compression testing of FIB-made pillars in the size range of 200--2000 nm in diameter. A size- and rate-dependent brittle-to-ductile transition during room temperature deformation was demonstrated by showing that pillars deform in an apparently homogeneous manner when they are either smaller than a critical value in diameter, or when they are tested at very low applied rates. A careful analysis of shear band patterns in SEM micrographs of pillars above the critical size reveals that the shear band spacing scales with sample size and the brittle-to-ductile transition can be explained by a geometric overlap of neighbored shear bands due to a reduction of their spacing below the typical shear band thickness of ~15 nm. However, since the strain rate sensitivity exponent remains unchanged at a value of about 0.05 regardless of sample size, deformation rate, and apparent deformation mode, it is concluded that the underlying atomistic mechanism remains the same through the brittle-to-ductile transition. The work is now extended in order to explore the physical reasons behind the scaling of the shear band spacing with size. Gradient plasticity models have successfully explained a similar scaling behavior of shear bands in nano-crystalline materials in the past. Here, we present the first results regarding the applicability of gradient plasticity models to explain the compression behavior of small scale specimens of metallic glasses.

The paper discusses experimentally detected phenomenon of internal cavity formation in icosahedral small copper particles as a result of their annealing. Icosahedral small particle has six five-fold symmetry axes and disclination-type defects inside. Particle consists of 20 tetrahedral region with local fcc atomic arrangement joined via twin boundaries.<br />Our experiments showed that the annealed icosahedral particles have large internal cavities (up to 80 volume %) surrounded by a shell made of metal oxide. The found cavities were examined in scanning electron microscope equipped with focused-ion beam gun. To investigate the structural and phase transformations occurring in the process of icosahedral particle annealing, differential scanning calorimetry (DSC) was applied.<br />It has been shown that the formation of cavities in microparticles was promoted by (i) the presence of oxygen in the surrounding atmosphere and (ii) high internal stresses in the particle interior. The role of disclination-type defects as the sources of internal stresses in the mechanism of internal cavity formation is elucidated.<br />This work has been supported by the grant No 14.B25.31.0011 of the Ministry of Education and Science of Russian Federation (resolution # 220) at Togliatti State University.

Experiments have shown that metallic materials display strong size effects at the micron and sub-micron scales, with smaller being stronger. Such size effects have been attributed to geometrically necessary dislocations (GNDs) associated with inhomogeneous plastic flow and, since the earlier seminal works of Aifantis, several continuum strain gradient plasticity (SGP) theories have been developed in order to incorporate some length-scale parameters in the constitutive equations.
These size effects are especially significant in fracture problems as the plastic zone adjacent to the crack tip is physically small and contains strong spatial gradients of deformation. Finite element (FE) simulations have shown that GNDs near the crack tip promote strain-hardening, leading to a much higher stress level in the vicinity of the crack than that predicted by classical plasticity. Moreover, recent numerical studies revealed that the domain ahead of the crack where strain gradients significantly influence crack tip fields could be one order of magnitude higher when finite strains are taken into account.
In this work, the influence of the plastic strain gradient on the fracture process of metallic materials is evaluated within the finite deformation theory by means of both mechanism-based and phenomenological SGP theories. The relation between material parameters and the physical length over which gradient effects prominently enhance crack-tip stresses is identified and quantified. Physical implications of the results are discussed and the relevance of accounting for size effects in the modelization of damage is thoroughly examined. Several numerical tools are evaluated with the aim of integrating SGP modeling in a robust numerical framework.

Narrow fine grain layers often are generated in the vicinity of frictional interfaces in metal forming processes. One of the main contributory mechanisms responsible for the generation of such layers is intensive plastic deformation. It is a challenging task to develop an adequate model for describing such material behavior. The approach proposed in the present paper is based on the strain rate intensity factor (SRIF) previously introduced for rigid perfectly plastic materials. This factor is the coefficient of the leading singular term in a series expansion of the equivalent strain rate (quadratic invariant of the strain rate tensor) in the vicinity of maximum friction surfaces. It follows from this expansion that the equivalent strain rate approaches infinity near maximum friction surfaces and that SRIF controls its magnitude within a narrow sub-surface layer. Such theoretical behavior of the equivalent strain rate is in qualitative agreement with the aforementioned experimental facts. To develop a model for the quantitative prediction of the generation of fine grain layers, it is necessary to propose and carry out a special purpose experimental program. The present paper is based on experiment on plane strain extrusion. Standard finite element packages are not capable to calculate SRIF. In the present paper, a numerical approach based on the method of characteristics is developed to find SRIF in plane strain extrusion. A correlation between the magnitude of SRIF and the experimental thickness of the fine grain layer is discussed.

The gradient theory of plasticity proposed by Aifantis in the 1980's was motivated by the Van der Waals thermodynamic theory of liquid-vapor transitions and its mechanical counterpart was advanced by Aifantis and Serrin. By allowing the flow stress to depend on the gradients (up to the second degree and order) of accumulated plastic strain, gradient plasticity was able to describe deformation patterning and the occurrence of shear bands in plastic solids. There were also other models of strain gradient plasticity proposed to consider size effect problems at the micron scale. The simple gradient plasticity model proposed by Aifantis was able to dispense with the mesh-size dependence of finite element calculations in the material softening regime, predict the thickness and spacing of shear bands, as well as account for size effects.
This work presents some modifications performed by the author and Aifantis through the use of wavelet analysis, neural networks and stochasticity. More precisely, the gradient plasticity model has been revised through the use of wavelet analysis in order to produce scale-dependent constitutive equations that have been used for modeling size effects in on tensile strength and fracture energy of brittle, heterogeneous and disordered materials, in good comparison with Carpinteri¢s Multi Fractal Scaling Law (MFSL), as well as the size effect on the yield stress and the ultimate strength of solid bars subjected to torsion and bending. In addition, the inverse Hall-Petch phenomenon has been modeled with the use of the aforementioned scale-dependent constitutive equations, as well as a gradient plasticity model enhanced with an interface energy term, the use of which is dictated by the fact that at the nanoscale interfaces play a dominant role in the mechanical behavior of nanocrystalline materials. Stochasticity-enhanced gradient plasticity models in 1D and 2D have also been implemented in cellular automata simulations for modeling size effects in metallic foams as well as mico/nanopillar compression experiments.

The concept of grain boundary (GB) design is developed for enhancement of properties of ultrafine-grained (UFG) nanostructured metals and alloys by tailoring different GBs (low angle and high angle ones, special and random, equilibrium and non-equilibrium, and so on), using severe plastic deformation (SPD). By variations of regimes and routes of SPD processing, we show for several light alloys (Al, Mg and Ti) and steels the ability to produce UFG materials with different grain boundaries, and their effect on mechanical and functional properties of the processed materials, in particular, on their strength and ductility, fatigue or superplasticity, thermostability and electric conductivity. We demonstrate several examples of this approach for attaining superior properties in various nanostructured metals and alloys. The origin of these phenomena is discussed on the basis of the results of microstructural studies and observations of deformation mechanisms. Special emphasis is laid on the innovation potential and first applications of SPD-produced nanometals.

Critical State Theory (CST) in soil mechanics considers conditions on stress and void ratios for Critical State (CS) failure without reference to orientation of particles, contact normal or void vectors at grain level, collectively called granular fabric and measured by an appropriately defined fabric tensor F. Yet it is an experimentally verified fact that at CS there is a strong fabric that induces a significant anisotropic pre-failure response of the medium.
A recently developed Anisotropic Critical State Theory (ACST) identifies the norm and direction of the fabric tensor all the way to CS and introduces an additional CS condition related to the relative orientation of the fabric tensor and loading direction by means of a Fabric Anisotropy Variable (FAV) that affects the location of the dilatancy state line in the void ratio-pressure space. ACST can address better the granular response where the classical CST fails to do so.
The Discrete Element Method is used to achieve the identification of the fabric tensor and its evolution, since actual experimental measurements are tedious. Yet the leap from discrete to continuum description for incorporation into the continuum ACST is not trivial. Appropriate rate equations of evolution of F must satisfy objectivity, and the continuum version of F must be thermodynamically consistent in regards to per volume dissipation that requires special normalization of the DEM measurements. Practical questions on the initial value of the fabric tensor in real soil deposits remain a challenge that is discussed.
Within the new ACST framework, soil plasticity constitutive models can be constructed that obey the classical CS conditions and in addition reflect realistically the strong anisotropic response before failure. It is expected to change the way CST is presented and taught.
Keywords: Critical State; Fabric; Discrete Element Method; Anisotropy; Soil Mechanics; Granular Mechanics; Soil Plasticity

The development of magnetic and electronic material has been penetrated in every field of modern technology. Owing to the trend of device miniaturization, multifunctional materials with the properties of electricity and magnetism have drawn increasing interest. Magnetoelectric (ME) structures composed of piezoelectric (PE) and piezomagnetic (PM) materials have the magneto-electro-elastic coupling property, with the ability to convert energy from one form (among magnetic, electric and mechanical energies) to the other. Because of the coupling effects, ME materials have significant application prospects in sensor technology, memory devices and smart structures. With development and industrial application of nanotechnology, more and more microminiaturized structures and systems are needed. However, the material properties may be different from that in macro-scale and also depend on material microstructures when the material characteristic size is as small as micron or nano. Such as the size-dependent phenomena of piezoelectric material were found in experiments, magnetic properties of piezomagnetic material are closely related to particle or grain size. Because no matter how to enhance ME effect, strain plays a key role in ME coupling. Motivated by this point and being eager to investigate the size-dependence phenomena, we introduce the strain gradient into the constitutive equations of multiferroic materials. Then ME fields have relation to the material internal length scales. The size-dependent phenomena in multiferroic composite may be analyzed and given an explanation.
In this paper, we firstly build the energy functional for elastic multiferroic body considering strain gradient effects. With the help of the variational principle, all the basic equations are established. Then we analyze the problem of multiferroic composite with inhomogeneities and obtain the effective ME coefficients. Finally, some numerical results of different multiferroic composites and a summary for size effects are presented. In conclusion, the internal length scales (or strain gradients) have significant influence on ME response in multiferroic composites. For the composites with different kind inhomogeneities, the effect of internal length scales is very different. By choosing appropriate inhomogeneity and controlling microstructure, high ME effects may be obtained. These results provide us potential possibilities for enhancing ME effects in the design and manufacture of multiferroic composites.

Gradient plasticity with interface energy can capture numerous phenomena at the micro and nanoscale such as the inverse Hall-Petch, and the stochasticity observed in the stress-strain response of micropillars. The physical nature of the interface energy however remains vague and limited studies have been performed to understand its relationship to other material properties. The present talk will discuss how simulations can be used to relate the interface parameter to the internal length, which is another key parameter in gradient theories.

Experiments on elastic wave propagation in granular media are tremendous and have covered different systems from the simple one-dimensional chain of disks, to the disorder induced wave localization problems in three dimensions. Recent experiments have explored the method to find the internal structure of granular media by passing waves through the media and determining the 'network structure' through which the waves had traveled from one point to the other. This implies the determination of the arrangement geometry of the particles, where the granular network also has certain topological properties. The "internal structure is defined by the mechanical equilibrium, the geometry and topology" of the network. The determination of internal structure can be approached as a mathematical problem. Given the information on transmitted and received waves of finite amplitude and frequency range through measurements in many points on the enclosed boundary, we can establish the co-ordinates of the particles within the confining boundary, in analogy to the case for the celebrated inverse problem of "hearing the shape of drum". In the present work the discrete calculus equations proposed for mechanically stable static granular media in two dimensions are applied to the wave propagation problem, to deduce the internal structure by imposing extra constraints on the system. To satisfy these equations, infinitesimal perturbations of the granular particle network are considered to only allow the propagation of elastic disturbances. It is found that, based on the number of constraints on the system and its variation, the number of measurements required on the boundary to build a bijective mapping between the structure and wave properties could be properly defined. The internal structure, wave propagation and inverse approach relate to the statistical mechanics framework of granular media, where the temperature or entropy could be expressed as a function of wave characteristics of the network.

In the field of transmission electron microscopy fundamental and practical reasons still remain that hamper a straightforward correlation between microscopic structural information and deformation mechanisms in materials. We argue that one should focus in particular on in-situ rather than on postmortem observations of the microstructure. In this contribution, this viewpoint will be exemplified with in-situ TEM nanoindentation and in-situ straining studies on crystalline and non-crystalline metallic materials. In-situ TEM displacement-controlled indentations in crystalline alloys show that many dislocations are nucleated prior to the initial macroscopic yield point and that the macroscopic yield event is associated with the rearrangements of the dislocations. Also size effect, or the lack thereof, during deformation of nano-sized metallic glasses (MGs) has recently drawn great attention. An intriguing question is why and how nucleation and propagation of these shear bands (SBs) are affected by the size of the system. Therefore, we have carried out quantitative in-situ TEM deformations of metallic glass pillars with diameters ranging from 50 nm to 500 nm. A micromechanical model based on quantitative description of shear banding events explains the size-dependent deformation behavior and a statistical analysis of strength reveals the physical picture defined by the interactions between stress fields of flow defects. Implications of our findings for applications in nanosized systems will be illustrated.

In this talk we give definitions for Iso-Probability of the first, second and third kind,Iso-Brownian motion of the first, second and third kind, iso-white noise of the first and second kind. They are deducted the main classes iso-martingales. It is formulated and proved the iso-large number law. As an application we deduct Paley-Wiener-Zygmund -iso-integral and Ito's iso-integral and their properties. The paper describes the main classes of isotopic elements so that we can deduct the iso-Stratonovic integral.

In this talk we give definition for iso-stochastic differential equations of the first, second and third kind. We formulate and prove general existence and uniqueness theorems for these classes iso-stochastic differential equations. Dependence on parameters and initial data are investigated. The steps are given for finding of the iso-derivative with respect a parameter for the main classes iso-stochastic equations. Various cases are considered for Lyapunov and structural stability of the solutions of the main classes iso-stochastic equations.

Large scale three-dimensional discrete dislocation dynamics (DDD) simulations are utilized to study the plastic response of Nickel superalloys. In DDD, individual dislocations are tracked and physics-based precipitate penetration rules as well as glide rules in both gamma and gamma phases are implemented. Starting with a general distribution of gamma phases dispersed in a single crystal gamma phase, the dislocation microstructure is evolved in time under the applied load and the material behavior is computed accordingly. DDD simulations for both shearable and non-shearable gamma phases are discussed. The influence of volume fraction, channel width, initial dislocation density and crystal sizes on the strain hardening and density evolution are assessed. The simulations are validated with detailed comparisons with experiments.

The achievements of nanotechnology, the new nano-instruments creation and new nano-servers require thorough investigation of the problems of strength and stability for nano-constructions. It is possible to use the methods of classical continuum mechanics taking into account some modification [1] M.E.Gurtin and A.I.Murdoch Arch.Ration Mech. Anal. 57,291. (1975); [2] Ya.S.Podstrigach and Yu.Z.Povstenko Introduction to the mechanics of surface phenomena in deformable solids. Naukova Dumka Kiev 1985 (in Russian); [3] H.L.Duan, J.Wang, Z.P.Huang and B.L.Karihaloo. J. Mech. Phys. Solids 58. 1574 (2005).
This approach makes possible to solve some well-known problems for nanosized situation:
a) Theory of shells and plates with nanosized thickness [4], V.A.Eremeyev, H.Altenbach, N.F.Morozov. The Influence of surface tension on the effective stiffness of nanosize plates. Doklady Physics 2009, vol.54 N2, pp.98-100,
b) The problem of Kirsch,
c) The problem of stability of plane equilibrium for nanosized plates etc.
In result, it is possible to estimate the corrections, which gives nanosize of objects.

Our discovery of the electric current spike (ECS) that emerges during nanodeformation of semiconductors, was first highlighted in the Letters to Nature Nanotechnology (Nowak et al. Vol.4 /2009/; Chrobak, Nowak, et al. Vol.6 /2011/) and offers an enhanced understanding of the link between nanoscale mechanical deformation and electrical properties. Our conclusions point to key advances in pressure-sensing, pressure-switching and unique phase-change applications in next-generation electronics. It is a very encouraging demonstration of the way in which nanomechanics contribute to electronics and optoelectronics developments.
The onset of plasticity is traditionally understood in terms of dislocation nucleation and motion. The extensive study of nanoscale deformation has proven that initial displacement transient events occurring in metals are the direct result of dislocation nucleation in stressed nanovolumes. Our research, however, shows that this is not always true. Instead of dislocation activity, nanoscale deformation may simply be due to transition from one crystal structure to another, as predicted for GaAs by our earlier atomistic calculations. The presented results lead to a major shift in our understanding of the elastic-plastic transition as well as inherent formation of a Schottky barrier in semiconductors under localized high pressures. The results showing the dramatic impact of crystal imperfections on the functional properties of GaAs motivated our further study into the onset of incipient plasticity in Si nanoparticles. Molecular Dynamics calculations and supporting experimental results reveal that the onset of plasticity in Si nano-orbs with <130 nm diameter is governed by dislocation-driven mechanisms, in striking contrast to bulk Si where incipient plasticity is dominated by phase transformations. We established the previously unforeseen role of "nanoscale confinement" governing a transition in mechanical response from "bulk" to "nanovolume" behavior, shedding new light on the dilemma concerning origin of incipient plasticity in nanoscale volumes of semiconductors, debated extensively up to present.

Materials with structure realized statistically at diverse scales show interesting mechanical properties that can be tailored to different applications. In addition to being able to tailor their mechanical properties, their strength reliability can be adjusted and in certain cases near perfect strength reliability can be achieved. The key is that spatial scales interact with each other and this interaction offers unique and extensive possibilities for mechanical properties and strength reliability. In certain cases, a certain scale or a tight range of scales dominate the material properties. For such cases where also the heterogeneity is statistically homogeneous, a spatial correlation distance can be identified that is closely relevant to the length scales appearing in gradient theories developed by Aifantis.

Work hardening during plastic deformation of crystalline solids is associated with significant changes in dislocation microstructure. The increase in dislocation density on the specimen scale is accompanied by a spontaneous emergence of regions of low dislocation density and clusters of high density which, to a large extent, persists upon unloading ("dislocation patterns").
Despite a significant degree of morphological variation depending on slip geometry and loading mode (e.g. cell or labyrinth structures, dislocation accumulation in veins or walls), these patterns are characterized by fairly universal scaling relationships, commonly referred to as 'similitude principle', relating the characteristic length of the dislocation patterns to the applied stress and to their average dislocation density. Despite long-standing efforts in the materials science and physics of defects communities, there is no general consensus regarding the physical mechanism leading to the formation of dislocation patterns.
We present for the first time dislocation patterning results from a continuum theory capturing the coupled dynamics of statistically stored and geometrically necessary dislocations while accounting for the specific kinematics of curved dislocations. The resulting patterns are automatically consistent with the similitude principle.
A surprisingly minimalistic set of 'ingredients' is already sufficient to create patterns: starting with an idealized model in single slip configuration, which allows to comprehend the basic mechanisms of pattern formation, we then turn to more realistic 3D multislip models where latent hardening terms couple dislocation densities on different slip systems through short-range interaction stresses. Our simulations explain how complex cell structures matching experimentally observed structures form - again being consistent with similitude. In addition, we show how data from dedicated discrete dislocation dynamics simulations can be extracted and used for gauging and parametrizing continuum models (e.g. in terms of short range dislocation interactions and reactions). Together with the mechanisms for dislocation pattern formation, we have the key ingredients for the formation of persistent slip bands under cyclic loading conditions.

Strain hardening in crystals and the accompanying dislocation pattern evolution (in the form of cell-like structures) are among the most difficult self-organizing behaviors to predict and explain. Screw character dislocation cross-slip has been typically presumed to play the main role in dislocation cell structure formation. However, many open questions remain regarding this mechanism. Recent molecular dynamics simulations showed that two cross-slip mechanisms, namely, surface and intersection mediated cross-slip mechanisms, exhibit a considerably lower activation energy than the traditionally accepted Friedel-Escaig cross-slip mechanism. In this work, we present the results of implementing these newly identified cross-slip mechanisms into discrete dislocation dynamics (DDD) simulations of nickel microcrystals, ranging in size from 0.5 to 20 microns in diameter. The conditions for each mechanism are discussed, along with their statistics and frequencies. The results show that dislocation cell structures form in simulation cells having diameters greater than 5 microns, as the dislocation density increases with increasing plastic strain. Smaller simulations cells, however, do not show any considerable cell formation at small strains as compared to the lager cells. The findings agree with recent experimental observations.

A computationally intensive multi-scale (macro, meso-and micro-mechanically) physically-based model is developed and implemented to describe physical phenomena associated with friction and wear in heterogeneous solids, particularly under high velocity impact loading conditions. Emphasis is given on the development of fundamental, thermodynamically consistent theories to describe high velocity material wear failure processes in combination with both ductile and brittle materials for wear damage related problems. The wear failure criterion is based on dissipated energies due to plastic strains at elevated temperatures. Frictional coefficients can be identified for the contact surfaces based on temperature, strain rates and roughness of the surfaces. In addition, failure models for micro-structural effects, such as shear bands and localized deformations, are studied.
The computations are carried with Abaqus Explicit as a dynamic temperature-displacement analysis. The contact between sliding against each other surfaces is specified as surface to surface contact on the master-slave basis. The tangential behavior is defined as kinematic contact with finite sliding. The validation of computations utilizing the novel approach presented in this proposal is going to be conducted on the continuum level while comparing the obtained numerical results with the experimental results.

Multi-modal microscopy is a term that refers to combining different imaging and mapping modes applied to the same object in order to obtain complementary information about material structure, function and properties. Alongside the well-established modalities, such as optical microscopy (including using polarised light) and scanning electron microscopy (including EDX and EBSD), multi-modal microscopy includes the use of TEM and STEM, AFM, as well as focused beams of ions (FIB), neutrons and X-rays.
The advent of tight (sub-micron) focusing of X-rays has opened up a vast range of possibilities in terms of full field imaging (including tomography), as well as scanning transmission X-ray microscopies (STXM) that can be used in the WAXS or SAXS regimes, and also for spectroscopic analysis (XAS).
My particular interest lies in the tight integration of these techniques with materials modelling across the scales. As examples, in my lecture I shall draw on our studies of the structure and thermo-mechanical response of human dental tissues (dentine and enamel); the structure and residual stress of carbon monofilament cores used in SiC fibre composites for aerospace applications; and some studies of materials for Li-ion batteries.
In terms of modelling, the use of non-singular solutions will be discussed in the context of dislocation and fracture mechanics, and the use of eigenstrain basis for residual stress analysis will be overviewed.

We review in this presentation recently proposed approaches to reduce the computational expense associated with multi-scale modelling of fracture. In light of two particular examples, we show connections between algebraic reduction (model order reduction and quasi-continuum methods) and homogenisation-based reduction. We open the discussion towards suitable approaches for machine-learning and Bayesian statistical based multi-scale model selection. Such approaches could fuel a digital-twin concept enabling models to learn from real-time data acquired during the life of the structure, accounting for “real” environmental conditions during predictions, and, eventually, moving beyond the “factors of safety” era.[1] O. Goury et al. (2013) A partitioned model order reduction approach to rationalise com- putational expenses in nonlinear fracture mechanics Comp. Meth. App. Mech. Engng.. 256, 169-188.<br />[2] L.A.A.Beex et al.(2014)Quasicontinuum-basedmultiscaleapproachesforplate-likebeam lattices experiencing in-plane and out-of-plane deformation Comp. Meth. App. Mech. En- gng. . 279, 348-378.<br />[3] A. Akbari (2015) Error Controlled Adaptive Multiscale Method For Fracture Modelling in Polycrystalline materials Philosophical Magazine. in press

In this work, a finite strain elasto-viscoplastic model coupled to anisotropic damage is developed through a thermodynamics admissible framework. A hyperelastic-based model using the multiplicative decomposition of the deformation gradient tensor into elastic and plastic parts is used and the resulting plastic intermediate configuration is connected with a fictitious undamaged configuration. This work is motivated by the inappropriate kinematical definition of the stress measure work conjugated with the plastic velocity gradient in the effective configuration, i.e. "effective stress", obtained by classical pull-back/push-forward operations. Usually, the pull-back of the Mandel-type stress tensor, resulting from the multiplicative decomposition of the deformation gradient, in the fictitious undamaged configuration leads to effective and nominal stress measures very close to each other. When compared to previously published works, the physical interpretation of the material degradation, represented by a second order tensor, has been enhanced by using a novel definition of the pull-back/push-forward operations which bridge effective and nominal configurations. The emphasis of this work is placed on the description of the interesting properties of the proposed formulation as well as its consequences in a thermodynamics point of view. Within this framework, a specific constitutive model with plastic and damage flow rules deduced from the restrictions imposed by the entropy balance is discussed with an application on an asphalt concrete material where the anisotropic evolution of the damage is highlighted.

While the deleterious effects of hydrogen on metals and alloys are well known, the precise role of hydrogen in the underlying microscopic mechanisms is still not well understood and as of yet, the modeling attempts on hydrogen embrittlement and hydrogen induced cracking has not led to a proper method for life-time prediction.
This work aims at the development of a robust numerical strategy in order to solve the non-linear coupled problem which accounts for diffusion of hydrogen, diffusion of heat and large elastoviscoplastic deformations. Specifically, problems related to macroscale shear-band formation due to the underlying mechanism of hydrogen enhanced localized plasticity (HELP) will be pointed out and a comparison between the numerical results and the available experimental data will be demonstrated.

Materials such as Si, Al and Sn are considered as active sites during the formation of Li-alloys. During the Li insertion, the volume of the active sites expands over 100% at maximum capacity. As a result, large internal stresses are produced. The effect of stresses is widely acknowledged. However, as the size of these particles is small, the gradient effects cannot be ignored. In this paper, we present for the first time the size effects within a purely gradient elasticity framework for a composite model with an inner Sn particle and a C matrix. Each of these constituents has different internal lengths. As the constitutive equation involves higher order gradient terms, the conventional finite element method is not suitable. We employ B-spline functions with the framework of the iso-geometric analysis spatial discretization. The effect of internal characteristic length on the radial and hydrostatic stresses is studied. It is observed that the internal length has significant impact on the stresses amplitudes.

It is well known that the introduction of higher-order gradients in the constitutive equations for plasticity and elasticity provides the ability to capture size effects, i.e. the dependence of strength and other properties on the size of the specimen. This inherent ability of the gradient models is based on the interaction between intrinsic constitutive lengths associated with the gradient coefficients and a characteristic geometrical length of the specimen under consideration. Moreover, it has been shown that the gradient plasticity model proposed by Aifantis and co-workers is able to predict the thickness of shear bands which is directly related to the gradient coefficients. In this case, the problem remains well-posed upon the onset of softening and the numerical simulations do not exhibit the spurious discretization sensitivity of the classical theories. The rate-dependent (i.e. viscoplastic) counterpart of this model has been also used to provide estimates for the spacing of Portevin-Le Châtelier (PLC) bands and adiabatic shear bands.
The present work deals with a rather overlooked ability of the gradient theories, i.e. the suppression of instabilities at quite small length scales. Two benchmark problems are discussed concerning thermoviscoplastic and diffusional instabilities, respectively. In the former case, the competition of thermal and strain gradient terms on the onset of instability and its dependence on specimen size is illustrated. It is shown that heat conduction promotes the instability initiation in the hardening part of the homogeneous stress-strain graph, while the strain gradient term favors the occurrence of this initiation in the softening regime. This behavior is size dependent, i.e. small specimens can support stable homogeneous deformations even in the softening regime by suppressing the onset of the instability. In the latter case, a thermodynamically consistent model of gradient elastodiffusion is developed by coupling the standard Cahn-Hilliard-type of diffusion with a simple gradient elasticity model that includes the gradient of volumetric strain in the expression of the Helmholtz free energy density. Then, an initial-boundary-value problem is derived in terms of concentration and displacement fields and linear stability analysis is employed to determine the contribution of concentration and strain gradient terms on the instability that leads to spinodal decomposition. It is shown that the theoretical predictions are in accordance with the experimental trends in intercalation materials (as, for example, Li-ion battery material LiFePO4), i.e. the spinodal concentration range shrinks (i.e. the tendency for phase separation is reduced) as the crystal size decreases. Moreover, for crystals smaller than a critical size there is no spinodal region at all.

The dislocation structure and plastic deformation of α/β brass after friction in boundary lubrication (BL) conditions were studied. Pin-on-disk tests at different loads were performed. The microstructure was studied using SEM, TEM and AFM techniques. The friction of brass was attributed to severe plastic deformation (SPD) in thin surface layers of the α-phase grains. The SPD of surface layers is accompanied by formation of thin shear bands practically parallel to the direction of friction. The thickness of wear particles was found to be close to the thickness of shear bands. Intragranular slip, as the accommodating mechanism, occurred in α-phase, whereas little deformation was observed in the β-phase. The accommodation of sliding was accompanied by growth and coalescence of voids, formation and propagation of cracks, leading finally to delamination of the wear particles. The equivalent strain vs. the depth was evaluated. Large deformation of thin surface layers (ƒO = 10-12) led to formation of the nanocrystalline structure (d = 35 nm).
Keywords: Severe plastic deformation; Nanocrystalline structure; Shear bands; Friction

Although Classical Continuum Mechanics (CCM) is very successful in handling numerous structural mechanics problems that we encounter, it is not capable of capturing physical phenomena occurring at various length scales since it does not have a length scale parameter. Moreover, due to the differentiation in spatial term of its governing equation, it is not valid if there is any discontinuity in the structure such as a crack. Hence, it is necessary to use other techniques when CCM starts to break down. Such a technique, peridynamics (PD), has recently been introduced. PD has a length scale parameter called horizon which defines the range of the domain of influence of material points located distance apart, so that the technique can be used at different scales. The finite size of interaction domain gives PD a non-local characteristic. Furthermore, PD is based on integro-differential equations which is always valid regardless of discontinuities. Hence, it is also very suitable for failure prediction of materials and structures. In this study, the superior features of peridynamics are demonstrated by investigating the granular fracture in polycrystalline materials.

The modeling approach adopted here for nanocrystalline materials is based on the phase-mixture model introduced by Kim, Estrin & Bush (2000). In this model, different deformation mechanisms are assumed to operate in the grain interior and the grain boundaries in parallel. The deformation mechanism in the grain boundaries is associated with the diffusional mass transport along the boundaries, while in the grain interior dislocation glide mechanism as well as diffusion mechanisms are considered. The model is capable of correctly predicting the transition of the flow stress from the Hall-Petch behavior in the conventional grain size range to an inverse Hall-Petch relation for nanocrystalline materials. The original model showed an increase of the strain rate sensitivity with decreasing grain size. However, this type of behavior is only observed in face-centered cubic (fcc) materials. By contrast, nanocrystalline body-centered cubic (bcc) materials show a decrease of the strain rate sensitivity with decreasing grain size. To account for experimental observations for bcc materials, we have modified the original phase-mixture model in the nanocrystalline regime. We assume that plastic deformation in bcc materials is governed by the Peierls mechanism, while the constraints put on the dislocation kink formation by the small grain size are considered to be responsible for a decrease of the strain rate sensitivity with grain refinement. The model for dislocation glide is modified accordingly. Additionally, we look into the effect of strain gradients on the mechanical response of nanocrystalline materials. The phase mixture model is augmented with gradient terms of higher order, namely second and fourth order. Differences in the mechanical behavior of fcc and bcc nanocrystalline materials are discussed in terms of the numerical results obtained with this gradient enhanced model. Details and challenges of the numerical implementation of higher order gradient terms will be provided.

Current strategies of computational crystal plasticity that focus on individual atoms or dislocations are impractical for real-scale, large-strain problems even with today's computing power. Dislocation-density based approaches are a way forward but most schemes published to-date give a heavier weight on the consideration of geometrically necessary dislocations (GNDs), while statistically stored dislocations (SSDs) are either ignored or treated in ad hoc manners. In reality, however, the motions of GNDs and SSDs are intricately linked through their mutual (e.g. Taylor) interactions, and in fact, GNDs and SSDs are indistinguishable on a microstructural level, notwithstanding the fact that the GNDs are simply the portion of dislocations associated with the overall shape change of the crystal. A correct scheme for dislocation dynamics should therefore be the one commonly used in discrete dislocation dynamics (DDD) simulations, namely, an "all-dislocation" treatment that is equally applicable for all dislocations comprising both the GNDs and SSDs, with a rigorous description of the interactions between them.
In this paper, a new scheme for computational dynamics of dislocation-density functions, based on the above "all-dislocation" principle, is discussed. The dynamic evolution laws for the dislocation densities are derived by coarse-graining the individual density vector fields of all the discrete dislocation lines in the system, without distinguishing between GNDs and SSDs. The mutual elastic interactions between dislocations are treated in full by generalizing the elastic interactions between dislocation segments for dislocation densities, and reducing the Hirth-Lothe line-integral formulation into an algebraic form comprising only elementary functions which are straightforward enough for efficient numerical implementation. Other features in the model include forest (Taylor) hardening, generation due to the connectivity nature of dislocations, and dipole annihilation. Numerical implementation is by means of the finite volume method (FVM), which is well suited for high gradients often encountered in dislocation plasticity.
As a first case study, the model is utilized to predict vibration-induced softening and dislocation pattern formation experimentally observed in crystalline metals. The simulations reveal the main mechanism for subcell formation under oscillatory loadings to be the enhanced elimination of SSDs by the oscillatory stress, leaving behind GNDs with low Schmid factors which then form the subgrain walls. The depletion of the SSDs also accounts for the softening, and this occurs because the oscillatory loading brings reversals into the motions of SSDs which then increase their chance of meeting up and annihilation. This is the first simulation effort to successfully predict the cell formation phenomenon under vibratory loadings, and this example highlights the importance of a rigorous "all-dislocation" treatment since both the SSDs and GNDs have significant roles to play.
A second case study concerns size effects in crystal plasticity. The new model is found capable of capturing a number of key experimental features including the Hall-Petch relation in polycrystalline states, and power-law relation between strength and size in micro-crystals. In the former, dislocation pile-ups at grain boundaries, and in the latter case, low dislocation storage and jerky deformation, are predicted.

Experimental studies of the non-linear internal friction is an efficient tool to derive reversible (microplastic or anelastic) dislocation or twin-related strain as a function of stress amplitude or temperature. We analyze here the anelastic behaviour (both dislocation and twin boundary related) of different crystalline solids based on the non-linear internal friction data obtained by different techniques. We give experimental evidence that the microplastic reversible strain is heterogeneous in space and intermittent in time, just like its macroscopic irreversible counterpart. It is shown that the stress - anelastic strain response for different materials can demonstrate both a scale-free power law (jamming-like dynamics) and typical depinning critical behaviour. We found that in the case of jamming-like dynamics, the stress exponent can vary substantially for the dislocation microplastic strain in different crystals, but is universal for microplastic strain related to twin boundary motion in a variety of martensitic structures.

Deformation bands are the basic microstructural elements in metal single crystals and polycrystals where the dominating deformation mechanism is a dislocation glide. The deformation bands are detected in a form of elongated misoriented domains separated by roughly parallel families of geometrically necessary boundaries. From the continuum mechanics point of view the deformation bands are spontaneous deformation instabilities, Biot (1965). The deformation bands are a consequence of anisotropy of hardening. The anisotropy causes that it is energetically less costly to flow the material through the crystal lattice buckled by the deformation bands with a decreased number of the active slip systems than to flow it through a lattice deformed homogeneously by multislip. In our approach a symmetric double slip model of a plane strain compression, a spontaneous formation of the deformation bands is presented in a variational form suitable for an energetic consideration. We adopt the rigid-plastic, rate-independent approximation, which turns out to be the optimal viewpoint pointing to the essential features of the formation of the deformation bands. A simple version of the model based on the standard hardening rule reveals the conditions for the formation of the deformation bands. It is shown that in the simplified case the predicted bands have extreme properties: the band orientation is perpendicular to the direction of the compression and their width tends to zero. Except for a modification of the hardening rule of the advanced model, which incorporates additionally the higher plastic strain gradients, we follow the classical crystal plasticity framework. The gradients represent hardening caused by the incremental work needed to build-up the boundaries and to overcome the dislocation bowing (Orowan) stress. The proposed model provides a unified interpretation of the band orientation, their width, the misorientation across the band boundaries, their dislocation composition, and the band reorientation occurring at large strains in agreement with observations.

We propose a new structural model of the mechanical twinning applicable for description of severe plastic deformation in 1D and 2D formulation. The model includes the equations of kinetics and dynamics of twins based on the energetic and structural data about the process of twinning. The growth of twins is considered as a process of collective slipping of the twinning dislocations. The model of twinning is a complement to the dislocation plasticity model and requires only one additional parameter, which is the stacking fault energy characterizing the material tendency for twinning. The twinning model together with the dislocation plasticity model is applied to model the structure of the shock wave front in thin metal foils, the distribution of dislocations and twins over the cross section of shock-loaded metal specimens at plate impact and dynamical axis-symmetric Taylor tests.
For the plate collision problem, it was found that the volume fraction of twins in vicinities of the loaded surface and a spall crack is, in order of magnitude, higher than that at the center of the target. The features of twins arising in different parts of the target are discussed.
The application of models to describe the plastic deformation in the dynamical Taylor anvil on rod tests leads to interesting results, which have a quantitative and qualitative correspondence with the experimental data about the form of the compacted samples. Calculations led us to reveal contributions of the dislocation plasticity and twinning on different stages of the deformation process and to explain the observed complex ("mushrooming") shape of the deformed copper rods at high impact velocities.
This study was supported by grants of the President of the Russian Federation (MD-286.2014.1), and the Ministry of Education and Science of the Russian Federation (competitive part of State Task of NIR CSU No. 3.1334.2014/K).

Metallic and non-metallic whiskers have been the subject of research interests for more than 50 years. Numerous studies devoted to whiskers proved that they have unique mechanical and electrical properties. Whiskers are already widely used in electronics and chemical industry. Therefore, the study of structure and formation mechanisms of whiskers and methods for their creation is an important scientific task. This research is devoted to the investigation of metal whiskers obtained by annealing in the air of electrolytic coatings and monolayers of icosahedral small copper particles. A set of contemporary physical methods, such as scanning and transmission electron microscopy, X-ray diffraction, local energy-dispersive X-ray spectroscopy and low-temperature gas adsorption are used for this study. It is shown that the whiskers are needle-like nanocrystals of copper oxide CuO with length up to 15 um and less than 100 nm in diameter. The whisker demonstrates high mechanical properties. Throughout the process of electrocrystallisation and subsequent annealing, long-range mechanical stresses appear in coatings, which, in the presence of oxygen in the atmosphere, intensify the formation of whiskers. The disclination-type defects in electrodeposited particles, as well as in the substrate, influence the formation and growth of whiskers. The growth of whiskers is provided by copper cation transport from the depth of the coating to the surface. The delivery of metal cations to the whiskers is due to the pipe-diffusion through the disclination and dislocation cores or nanopore channels. This work has been performed under the support of RFBR research project No. 13-02-00221 and the grant No. 14.B25.31.0011 from the Ministry of Education and Science of Russian Federation (resolution # 220) at Togliatti State University.

The structure transformations occurring upon severe plastic deformation (SPD) of materials attract a great interest since, in many respects, they do not follow the regularities typical of conventional macroscopic deformation. First of all, the following phenomenon must be stand out at the SPD: fragmentation, the absence of work hardening, abnormally high atomic diffusibility, dynamical recrystallization, precipitation or solution of unstable phases, amorphization. Unfortunately, most people studying SPD are limited to the study of finite structures and properties of materials and do not analyze the physical processes occurring directly at huge degrees of plastic flow. There is still no clear physical criterion of the transition from conventional deformation to the SPD stage.
The purpose of this presentation is to bring the structure formation processes occurring upon SPD into a single physical picture which should be capable to consistently explain all the experimental data accumulated to date. A general approach to the description of the basic law of structural and phase transformations is proposed within the concept of the appearance of additional channels for the dissipation of mechanical energy supplied to the solid state. It is shown that the active involvement of cyclic processes of low-temperature dynamic recrystallization and of the crystal-amorphous state phase transformations in combination with additional thermal effects under the conditions of insufficient efficiency of dislocation and disclination accommodation processes can consistently explain almost all the main experimental results obtained to date for SPD.
The authors are grateful to the Russian Scientific Foundation for financial support (grant 14-12-00170).

The density of geometrically necessary dislocations (GNDs) obtained from the lattice curvature was studied in different strain paths up to extreme large strains in commercially pure copper. Its evolution shows a maximum at medium large strains followed by a constant decrease when the stationary limiting stage of grain refinement is reached. At the same time, the total dislocation density is also decreasing. It is suggested using polycrystal modeling of the evolution of the crystallographic texture that the low quantity of GNDs at extreme large strains leads to a near Taylor-type homogeneous behavior of the polycrystalline ultra-fine grained structure. A new analytic modeling scheme is also proposed which is able to reproduce the evolution of the GND density for all stages of the severe plastic deformation processes.

The celebrated Walgraef-Aifantis (WA) model for dislocations has motivated many studies ranging from the continuum to the discrete micro-scale level. Here, we perform a computational-assisted analysis of the (W-A) model of dislocation patterning in one dimensional finite domain. We perform linear stability analysis with respect to the size of the finite domain and we extract the critical size at which the solution disappears. We investigate symmetric properties of the non homogeneous solutions and construct the bifurcation diagram with respect to the domain size.
Furthermore, by considering the Lattice Boltzmann discretization of the WA, we exploit the Equation Free method (EFM) in order to reconstruct the free energy and calculate the first time passage between metastable states related to the different material states under loading (i.e. from veins to persistent sleep bands).
Finally, we study a 2D discrete dislocation model and we link the different scales of descriptions, with the use of Diffusion Maps. Within this framework, we reduce the high dimensional representation to a few macroscopic variables by passing an explicit statistical mechanics hierarchy and reduction. Possible connections between the reduced macroscopic description and the W-A model are also discussed.

Dislocation based modeling of plasticity is one of the central challenges at the crossover of materials science and continuum mechanics. Developing a continuum theory of dislocations requires the solution of two long standing problems: (i) to represent dislocation kinematics in terms of a reasonable number of variables and (ii) to derive averaged descriptions of the dislocation dynamics (i.e. material laws) in terms of these variables. The kinematic problem (i) was recently solved through the introduction of continuum dislocation dynamics (CDD), which provides kinematically consistent evolution equations of dislocation alignment tensors, presuming a given average dislocation velocity [1, 2].
In the current talk we demonstrate how a free energy formulation may be used to solve the dynamic closure problem (ii) in CDD. We do so exemplarily for the lowest order CDD variant for curved dislocations in a single slip situation [2]. In this case, a thermodynamically consistent average dislocation velocity is found to comprise five mesoscopic shear stress contributions. For a postulated free energy expression we identify among these stress contributions a back-stress term and a line-tension term, both of which have already been postulated for CDD. The back-stress term is a second order strain gradient term strongly resembling a term introduced in a phenomenological strain gradient theory in a seminal paper by E. Aifantis [3]. A new stress contribution occurs, which contains a first order strain gradient. Such a stress contribution is found to be missing in earlier CDD models including the statistical continuum theory of straight parallel edge dislocations by Groma and co-workers [4]. Two entirely new stress contributions arise from the curvature of dislocations.
[1] T. Hochrainer, S. Sandfeld, M. Zaiser and P. Gumbsch, 2014., JMPS 63, 167–178.
[2] T. Hochrainer, 2015, Philos. Mag. 95 (12), 1321–1367
[3] Aifantis, E. C., 1987. Int. J. Plast. 3, 211–247.
[4] I. Groma, F. Csikor and M. Zaiser (2015), Acta Mater. 51, 1271–1281

Strain gradient plasticity is a successful framework to explain the size effect of mechanical behavior in microscale and nanoscale. Most of the components of Micro-Electro-Mechanical System (MEMS) system are often subjected to a cyclic loading, thus, fully understanding and modeling the cyclic deformation of material in microscale and nanoscale is urgent and important. Here, we proposed a thermodynamically consistent strain gradient cyclic plasticity model. The cyclic response with isotropic and kinematic hardening is predicted. Specifically, the isotropic hardening is supposed to be related with the dissipative part of plastic work of the material, while the stored energy in the plastic deformed material is supposed to contribute to the kinematic hardening. The evolution of back-stress during cyclic response is an extension of classical Armstrong-Frederick model through the inclusion of plastic strain gradient. The new established strain gradient cyclic plasticity model could predict the size effect in twisting of wires. Then, the abnormal Bauchinger effect observed experimentally, i.e. torque-twist curve shows reverse plasticity at positive torque, can be captured using the new established strain gradient plasticity model. Furthermore, the closed expression of the cyclical torsion behavior of thin wires is obtained, and the cyclic hardening response is then systematically investigated.

This abstract was withdrawn by the authors

We present the results on theoretical and experimental studies of threading dislocations (TDs) behavior in III-nitride layers grown in polar orientation. TDs are defects formed during epitaxial growth of layered electronic and optoelectronic materials. The effect of TDs on the functional properties of most III-nitride layers is deleterious. Optoelectronic devices fabricated from high dislocation density layers demonstrate poorer performance than those made from relatively defect-free layers.
We develop a general methodology for the reduction of TD density in III-nitride layers fabricated in (0001) polar growth orientation. In such layers, the majority of TDs in as grown state are parallel to c-axis, i.e. dislocation lines are parallel to [0001] crystallographic direction. The methodology proposes to use the intentional inclination of dislocation lines from this [0001] direction by exploring various techniques: changing growth conditions, growth surface roughening and faceting, layer patterning, introduction of stressed layers, porous layer formation etc.
In our studies, the reaction-kinetics approach is used to predict the behavior of TD ensemble with inclined defect lines. For example, this approach is realized when accounting for the reactions among dislocations in the ensemble. In case of III-nitride layer grown on porous template, an inclination of dislocations happens under the influence of pores. In addition, TDs can be trapping into pores. In case of the growth of the layer with faceted surface morphology, the dislocation inclination is caused by TD interaction with surface facets. The modeling results are supported with experimental data on TD density evolution in real optoelectronic device III-nitride structures.

A structure for an internal state viable description of inelastic deformation of crystals is developed. The deformation gradient is multiplicatively decomposed into an elastic deformation resulting from externally applied loads and the deformations associated with each density of defects to be included as internal state variables. The deformations associated with these defects or foreign atoms such as diffusing species, may further be decomposed into elastic and plastic parts depending upon the structure of the defect. In many cases, either the elastic or plastic part of a particular deformation gradient will be negligible, depending upon the situation. The appropriate strain-like variable associated with the defects is included in the free energy resulting in conjugate thermodynamic forces (internal stresses) that must be included in the dissipation inequality. In addition, these forces (stresses) are required to satisfy micro or meso scale linear and angular balance laws. All transport equations (e.g. heat conduction or diffusing species) are derived from a combination of the energy balance and these force balance laws. This is in contrast to classic state variable theories in which only temporal evolution equations were specified for the internal state variables. Constraint equations are required for the extra kinematic degrees of freedom that are introduced. These are based upon the physics of the associated defect density/state variable an example given by the flow rule or plastic velocity gradient based upon the Orowan equation relating plastic strain rate to mobile dislocation density and velocity. This structure leads to a natural length scale bridging as conjugate forces to the state variables are required to satisfy balance laws at their respective length scale and information is passed from scale to scale via the constraint equations.
Examples are given for the construction of such theories ranging from simple statistically stored dislocations, coupled transport theories for hydrogen and micropolar type theories for asymmetric defects.

The present research is based on the ultra-sonic emissions from unstable local phenomena like fracture, turbulence and buckling, occurring in solids and fluids at different scales. At the tectonic scale, Acoustic Emission (AE) prevails, as well as Electro-Magnetic Emission (EME) at the intermediate scales, and Neutron Emission (NE) at the nano-scale. TeraHertz pressure waves are in fact produced at the last extremely small scale, and fracture experiments on natural rocks have recently demonstrated that these high-frequency waves are able to induce nuclear fission reactions with neutron emissions. The same phenomenon appears to have occurred in several different situations, in particular in the chemical evolution of Earth and Solar System through seismicity (rocky planets) and storms (gaseous planets). It can also explain puzzles related to the history of our planet like the ocean formation or the primordial carbon pollution, as well as scientific mysteries like the so-called cold nuclear fusion or the correct radio-carbon dating of organic materials.
On the other hand, at the tectonic scale, the different forms of emission might be used as earthquake precursors. At the laboratory scale they could be accompanied by heat generation, whereas at the nano-scale they could explain some cell biology open problems. Therefore, the present proposal intends to apply recently discovered and interdisciplinary phenomena to the solution of open problems representing the following objectives: (i) the environmental protection against seismicity and carbon pollution, (ii) the heat generation from innovative and sustainable systems, and (iii) a better comprehension of some biological mechanisms at the cellular and protein levels.

Duality is one of the oldest and most beautiful concepts in human knowledge with a simple origin from the oriental philosophy tracing back 5000 years. Canonical duality theory [1,2] is a newly developed, breakthrough powerful methodology, which can be used to model complex systems with a unified solution to a wide class of discrete and continuous problems in engineering sciences. The associated triality theory reveals an interesting multi-scale duality pattern in complex systems, which can be used to identify both global and local extrema and to design powerful algorithms for solving challenging problems in computational mechanics.
In this talk, the speaker will first present some fundamental principles for modeling complex systems. Based on the definitions of objectivity and canonical duality in continuum physics, he will show why the complex systems can be modelled within a unified framework, how the canonical duality theory is naturally developed and the fundamental reasons that lead to challenging problems in different fields, including chaotic dynamics, phase transitions of solids, multi-solutions in post-buckling analysis and NP-hard problems in computational sciences. By using the phase transitions of the Ericksen's bar, he will show magic to obtain a unified analytic solution for general finite deformation problems and to identify both global and local optimality conditions from infinitely many local solutions. A movie will illustrate a truth that for many nonconvex potential variational problems, the global optimal solutions are usually nonsmooth, and cannot be captured by any traditional Newton-type direct approaches[3]. Applications will be illustrated by certain well-known challenging problems in finite deformation theory (such as phase transitions and control of chaotic systems) as well as NP-hard problems in computational mechanics (such as topology optimization and post-buckling of large deformed beam). A set of complete analytical solutions to 3-D nonlinear elasticity will be presented [4,5]. Finally, some open problems and possible methodologies will be addressed.
This talk will bring some fundamentally new insights into modern mechanics, complex systems, and computational science.
References:
[1] Gao, D.Y. (2000). Duality Principles in Nonconvex Systems: Theory, Methods and Applications. Kluwer Academic Publishers, Boston/Dordrecht/London, 2000, xviii+454pp.
[2] Gao, D.Y., Ruan, N., and Latorre, V. (2015). Canonical duality-triality: Bridge between nonconvex analysis/mechanics and global optimization, Math. Mech. Solids.
[3] Gao, D.Y. and Ogden, R.W. (2008) Multiple solutions to non-convex variational problems with implications for phase transitions and numerical computation, Quarterly J. Mech. Appl. Math. . 61 (4), 497-522
[4] Gao, D.Y. (2015). Analytic solutions to general anti-plane shear problems in finite elasticity, Continuum Mech. Thermodyn
[5] Gao, D.Y. and Hajilarov, E. On analytic solutions to 3-d finite deformation problems governed by St Venant–Kirchhoff material. Math. Mech. Solids (2015)

In this study, we present results concerning the experimental data analysis of different physical complex systems and processes such as: plastic deformation in materials, space plasmas, seismogenesis, climate, DNA and others. In particular, we study serrations from stress-strain graphs, ion fluxes and magnetic field, earthquake time series, geopotential height, DNA sequences and others.
The analysis is based on modern theoretical and mathematical tools of Complexity theory, such as nonlinear time series analysis (e.g. estimation of correlation dimension, spectrum of Lyapunov exponents, Hurst exponent), turbulence analysis (flatness coefficient, structure function etc) and Tsallis non-extensive statistics (Tsallis q-triplet). The analysis is capable of capturing significant features of the dynamics of each system in study, giving significant information identifying and characterizing the dynamical characteristics and the ability of complexity and self-organization (e.g low dimensional chaotic or self-organized critical, intermittent turbulence etc).
The results clearly show that even though the systems we study are completely different in their details, they generate common universal properties and features as far as the complexity and self-organization is concerned. Also, the results can be related to fractional dynamical nonlinear processes with non-extensive statistical character.

Gradient-elasticity and more generally gradient-enhanced continuum models have been extensively developed since the beginning of the twentieth century. These models have shown the ability to account for the effect of the underlying material heterogeneity at the macroscopic scale of the continuum. Despite of a great theoretical interest, gradient-enhanced models are usually difficult to interpret physically and even more to identify experimentally. This paper proposes an attempt to validate and identify a gradient-elasticity model for a material with a periodic micro-structure from experimental data. A set of dedicated experimental and numerical tools is developed for this purpose: first, the design of an experiment, then two-scale displacement field measurements by digital image correlation and dedicated pro-processing techniques and finally a model updating technique. This paper ends up with the full set of first and second-order elastic constants of a gradient-elasticity model which macroscopic kinematic has been validated by investigating the deformation of the unit cells at the microscopic scale.

Generalized continuum models are nowadays recognized to be a useful tool for the macroscopic description of the mechanical behavior of materials with heterogeneous microstructures showing exotic properties and/or size effects.
In particular, a recently introduced generalized continuum model, which we called "relaxed micromorphic" has been shown to be well-adapted to describe very exotic behaviors of micro-structured materials in the dynamic regime. In particular, a relaxed micromorphic model is, to our knowledge, the only generalized continuum model which is able to describe complete band gaps with respect to wave propagation.
We study dispersion relations for the considered relaxed medium and we are able to disclose precise frequency ranges for which propagation of waves is inhibited (frequency band-gaps). We explicitly show that band-gaps phenomena cannot be accounted for by classical micromorphic models of the Mindlin-Eringen type as well as by Cosserat and second gradient ones.
We finally point out that such relaxed micromorphic model also gives rise to some very intriguing mathematical questions regarding its well-posedness.

The question in the title is the central one for constructing plasticity of polycrystals based on the mechanisms of micro-deformation. The first suggestions were made by Nye, Bilby and Kroner in mid-50ies. Since then new proposals keep coming. Unfortunately, all proposed macroscopic characteristics have some intrinsic flaws. In this work, the macroscopic parameters describing dislocation networks are determined from the following proposition: macroscopic parameters must determine the value of energy of dislocation network. This "postulate" seems self-evident, but until recently there were no technical tools to apply it. An advancement in understanding of the variational principle for probabilistic measure of random structures allows us to apply this variational principle to dislocation networks and get the answer to the question posed.

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