Incompatibility Induced Nonlocal Crystal Plasticity

Motivated by the fact that locally inhomogeneous elastic or plastic deformations may result in incompatibilities of the fictitious intermediate configuration a strain gradient crystal plasticity model is developed. Thereby incompatibilities can be accounted for and scale dependent material behavior, as also observed experimentally, is predictable. A nonlocal extension of existing local formulations is proposed which does not require additional boundary conditions and thus maintains the classical BVP structure. On the numerical side key developments are an extended FE-formulation for rate-(in)dependent strain gradient plasticity and a local FE-formulation which bases the gradient computation on an operator split combined with a smoothing algorithm. Comparative numerical studies for classical examples proove the superior efficiency of the second approach. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Mathematical modeling of the stress state of a transverse plastic layer in a round rod

We construct mathematical models of plastic deformations of a continuous round rod containing a transverse (less strong) inhomogeneous layer under an axial load. We thoroughly study the local strengthening of such layers by involving the base material of the rod in the plastic deformation process. We obtain explicit formulas for the critical stress states in the layer and the critical axial load on the rod.     

Microstructural behaviour of transversal isotropic material

We discuss and simulate transversal isotropic material under tension loading. The preferential direction of the material is inclined under 45 degrees to the direction of the tensile resultant. In this configuration the deformation of a rectangular test specimen differ from the behaviour of isotropic material in the way, that beside Poissons effect additional displacement appear perpendicular to the tension direction. In classical continuum theories, this transverse deformations describe a typical S–shape. By using a non–local continuum theory, the effect of microstructural orientation is incorporated into the numerical model. Then, it depends on a phenomenological parameter of inner structure whether the energetically favoured configuration is classical or contains microstructural behaviour. In the second case, the transverse deformation is not described by the typical S–shape, but with higher forms of it. A simple experimental model will show the connection between the inner structure of the material and the rotational parameters within the non–local continuum theory. It is evident, that these parameters are responsible for the non–classical behaviour and give the possibility to find energetically favoured solutions. The results of the finite–element–analyses can help to understand constitutive parameters for the non–local continuum theory and to apply it to other specimens. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling and numerical simulation of multiscale behavior in polycrystals via extended crystal plasticity

The purpose of this work is to exploit the algorithmic formulation of models for multiscale inelastic materials whose behavior is influenced by the evolution of inelastic microstructure and the corresponding material or internal lengthscales. The models for extended crystal plasticity are based on the formulation of rate potentials whose form is determined by (i) energetic processes via the free energy, (ii) kinetic processes via the dissipation potential, and (iii) the form of the evolution relations for the internal-variable-like quantities upon which the free energy and dissipation potential depend. Examples for these latter quantities are the inelastic local deformation or dislocation densities as GNDs. Different algorithmic implementations are discussed, namely the algorithmic variational approach and the dual mixed approach. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A finite element formulation based on the theory of a Cosserat point

The theory of Cosserat points is the basis of a 3D finite element formulation allowing for large deformations in structural mechanics, that recently was presented by [1]. First attempts have revealed, that this formulation is free of showing undesired locking or hourglassing-phenomena. It additionally shows excellent behaviour for any type of incompressible material, for large deformations and sensitive structures such as plates or shells. Within the theory of Cosserat points, the position vectors X and x , are described through director vectors D _i and d _i by use of trilinear shape functions Nⁱ for an 8-node brick element. The special choice of shape functions Nⁱ allows for director vectors with which the deformation can be split into a homogeneous and an inhomogeneous part. This split enables the use of stiffnesses that correspond to different deformation modes. Analytical solutions to the inhomogeneous deformation modes are incorporated in the formulation and avoid the undesired phenomena. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Grothendieck topologies and deformation Theory II

Starting from a sheaf of associative algebras over a scheme we show thatits deformation theory is described by cohomologies of a canonical object,called the cotangent complex, in the derived category of sheaves ofbi-modules over this sheaf of algebras. The passage from deformations tocohomology is based on considering a site which is naturally constructed outof our sheaf of algebras. It turns out that on the one hand, cohomology ofcertain sheaves on this site control deformations, and on the other hand,they can be rewritten in terms of the category of sheaves of bi-modules.     

Propagation of a plane wave to a materially uniform but inhomogeneous body

We produce the equations of small deformations superimposed upon large for materially uniform but inhomogeneous bodies and specialize to an isotropic material and to a homogeneous finite elastic deformation. By assuming the small deformation to be a plane wave, a set of equations for the amplitude of the wave is produced which is accompanied by an additional set of conditions. By requiring a non-trivial solution for the amplitude, we obtain the secular equation and from it a set of necessary and sufficient conditions for having a real wave speed. The second set of conditions that have to be satisfied is due to the materials inhomogeneity. Essentially, the present analysis enhances the approach of Hayes and Rivlin for materially uniform but inhomogeneous bodies. The outcome is that for such bodies the restrictions on the constitutive law for having real wave speeds for an isotropic material subjected to a pure homogeneous deformation involves the field of the inhomogeneity as well.     

Calculation of deformations in nanocomposites using the block multipole method with the analytical-numerical account of the scale effects

Local scale effects for linear continuous media are investigated as applied to the composites reinforced by nanoparticles. A mathematical model of the interphase layer is proposed that describes the specific nature of deformations in the neighborhood of the interface between different phases in an inhomogeneous material. The characteristic length of the interphase layer is determined formally in terms of the parameters of the mathematical model. The local stress state in the neighborhood of the phase boundaries in the interphase layer is examined. This stress can cause a significant change of the integral macromechanical characteristics of the material as a whole if the interphase boundaries are long. Such a situation is observed in composite materials reinforced by microparticles and nanoparticles even when the volume concentration of the inclusions is small. A numerical simulation of the stress state is performed on the basis of the block analytical-numerical multipole method with regard for the local effects related to the special nature of the deformation of the interphase layer in the vicinity of the interface.     

Multi-phase modeling of shape memory polymers

In the present work, we propose a model for shape memory polymers based on the idea of the multiplicative decomposition of the deformation gradient. Evolution equations for the several deformation components are presented that provide for the storage of the entropy-elastic strain and its recovery during the transition between the frozen phase and the active phase. First characteristic shape memory cycles for small and large deformations will be shown as a last point. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Domain effect on the behavior of solutions of a distributed kinetic system

A distributed kinetic system that is in homogeneous equilibrium in a flat circular reactor is considered. Its behavior under deformations of the circular domain is studied. It is shown that a domain deformation may lead to the formation of stable spatially inhomogeneous oscillatory solutions in the neighborhood of the homogeneous equilibrium. The possibility of developing chaotic oscillations is discussed. This mechanism of creating spatially inhomogeneous nonlinear oscillations in the distributed kinetic system is called the domain effect.     

Continuity of a Deformation in H<Superscript>1</Superscript> as a Function of Its Cauchy-Green Tensor in L<Superscript>1</Superscript>

Let $\Omega$ be a bounded Lipschitz domain in $\BBbR^n$. The Cauchy-Green, or metric, tensor field associated with a deformation of the set $\Omega$, i.e., a smooth-enough orientation-preserving mapping $\bTh\colon\Omega\to\BBbR^n$, is the $n\times n$ symmetric matrix field defined by $\bnabla\bTheta^T(x)\bnabla\bTheta(x)$ at each point $x\in\Omega$. We show that, under appropriate assumptions, the deformations depend continuously on their Cauchy-Green tensors, the topologies being those of the spaces $\bH^1(\Omega)$ for the deformations and $\bL^1(\Omega)$ for the Cauchy-Green tensors. When $n=3$ and $\Omega$ is viewed as a reference configuration of an elastic body, this result has potential applications to nonlinear three-dimensional elasticity, since the stored energy function of a hyperelastic material depends on the deformation gradient field $\bnabla\bTheta$ through the Cauchy-Green tensor.     

Coating cones and spherical caps with membranes formed of elastic cords

Two 3-D deformations are investigated of planar membranes formed of two families of continuously distributed highly elastic cords, under the assumption of no resistance to shearing. The deformations are associated with coating cones and spherical caps. For both deformations the first integration of one of the governing differential equations is obtained analytically for any material of the cords. The deformation of coating cones can be expressed explicitly in terms of the reference in the undeformed configuration for several materials, but it is not so for coating spherical caps. In the case of coating spherical caps the stretch ratios and tensions can be expressed in terms of the reference in the deformed configuration and the shape and/or size of the undeformed membrane, which is required to coat the given cap, can be found by the integration (generally numerical procedure needed) of the other governing equation.     

General Form of the *-Commutator on the Grassmann Algebra

The general form of the *-commutator on the Grassmann algebra treated as a deformation of the conventional Poisson bracket is investigated. It is shown that in addition to the Moyal *-commutator, there exist other deformations of the Poisson bracket on the Grassman algebra (one additional deformation for even and odd n, where n is the number of the Grassmann algebra generators) that are not reducible to the Moyal *-commutator by a similarity transformation.     

A molecular based model for polymer viscoelasticity: Intra- and inter-molecular variability

We develop dynamic equations for rubber viscoelasticity based on a stick-slip continuum molecular-based model. The model developed is a continuum tube reptation model in which a chemically cross-linked (CC) system of molecules act as constraint box per unit volume for a physically constrained (PC) system of molecules. The CC-system carries along the PC-system during instantaneous step deformations. The subsequent relaxation of the PC-system is determined by the configuration of the CC-system, its own configuration and confirmation, and external force fields. Conversely, the deformation of the PC-system acts as an internal variable affecting the deformations of the constraining CC-system. We model the relationship between these processes to derive a model of viscoelasticity in rubber deformation. In developing a relaxation process for the PC-system, we start from the fact that the PC-system is composed of long molecular chains. The dynamics of these molecular chains are developed by modelling them as chains of beads connected by springs, which represent inter-molecular potentials. Various segments of the molecular chains relax at different rates. In addition, variability in relaxation times across molecular chains is permitted.     

On the simulation of textile reinforced composites and structures

This paper presents experimental and numerical methods to perform simulations of the mechanical behavior of textile reinforced composites and structures. The first aspect considered refers to the meso-to-macro transition in the framework of the finite element (FE) method. Regarding an effective modelling strategy the Binary Model is used to represent the discretized complex architecture of the composite. To simulate the local response and to compute the macroscopic stress and stiffness undergoing small strain a user routine is developed. The results are transfered to the macroscopic model during the solution process. The second aspect concerns the configuration of the fiber orientation and textile shear deformation in complex structural components. To take these deformations which affect the macroscopic material properties into account they are regarded in a macroscopic FE model. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Combined stepwise stressing of creeping material,obeying the theory of hardening

Irreversible creeping deformations of a physically nonlinear material under combined stepwise stressing were investigated. A new variant of the theory of local deformations was used and the physical relationships were based on the theory of hardening. The deformation components were calculated on a BÉSM-3M digital computer and were found to be in good agreement with experimental data. Curves are given representing the intensity of deformations as a function of time.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 3, pp. 393–398, May–June, 1970.     

Holographic interferometry study of deformation of cement and concrete under mechanical and thermal loads

Compressive strain of concrete is accompanied by rotation of the rigid aggregate and by local shifts of the cement matrix, which by analogy with local deformation of metals is the cause of a decrease of the real strength of the material. It is shown that deformation of concrete with haydite and granite aggregates in the presence of a heat supply (within limits of positive operating temperatures) is distinguished by damping of deformations in the first case and by local deformation of the aggregate in the second.Presented at the Ninth International Conference on Mechanics of Composite Materials, Riga. October, 1995.Kharkov State Technical Academy of Railroad Transport, Ukraine. Kharkov State Technical Construction and Architecture University, Ukraine. Kharkov Fire Safety Institute Ukraine. Translated from Mekhanika Kompozitnykh Materialov, Vol. 32, No. 2, pp. 202–208, March–April, 1996.     

Simulation of brittle crack growth in functionally graded materials

We consider a thermodynamic consistent framework for crack propagation by applying a dissipation inequality to a time dependent migrating control volume. The direction of crack growth is obtained in terms of material forces as a result of the principle of maximum dissipation. In the numerical implementation a staggered algorithm – deformation update for fixed geometry followed by geometry update for fixed deformation – is employed within each time increment. The corresponding mesh is generated by combining Delaunay triangulation with local mesh refinement. A numerical example with inhomogeneous material properties illustrates the capability of the resulting algorithm. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)