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A uniform reference configuration fully (a) and partially (b) embedded in the physical space of the observer
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In the present paper, modern differential-geometrical methods for modeling the incompatible finite deformations in solids are developed. The incompatibility of deformations may be caused by a variety of physical phenomena, e.g., distributed dislocations and disclinations, point defects, nonuniform thermal fields, shrinkage, growth, etc. Incompatibl...
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Context 1
... of a non-Euclidean structure of the reference shape. Let the body of B and the physical space of P be presented by two-dimensional manifolds, where, by the condition, P is Euclidean (i.e., the tor- sion and the curvature of connection on P are identical to zero). Since P is Euclidean, it can be shown by two orthogonal axes (x 1 , x 2 ) as in Fig. 3. Assuming that the physical space of P is linked to the observer A, who "sees" everything happening in P, but is unable to see what is beyond it: the observer lives in a flat world like the characters of Abbot and Burger (1976) in Flatland. We assume that the physical space of P represents a typical layer of the three-dimensional space ...
Context 2
... corresponds to time: motions in P are characterized by changes of the relative positions of points in P when the layer P is transferred along the added chronometric axis T. In the space S there exists a more powerful observer B, who sees all images of the body configu- rations of B in total, i.e., observes the global tube of the body of B (see Fig. 3). Each section of the global tube is an image of the configuration, corresponding to a certain time instant. If the body has a natural configuration, the observer A can see this configuration in one of the sections. However, in a general case, such sec- tion may not be present: the observer A makes a conclusion that deformations of the ...
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... Therefore, from the physical point of view a materially uniform simple body B is a body that can be decomposed into parts with identical response. In our considerations, we admit that these parts may be infinitesimally small and the number of them may tend to infinity [27]. In order to formalize the unique response of such particles within the whole body (without disassembling the body into parts), we use the notion of local configuration. ...
In the present paper, the finite deformations of a laminated inhomogeneous spherical shell are studied. A laminated shell can be considered as a limit case for multilayered shells when the thickness of each layer tends to zero while their quantity tends to infinity. Such a limit might be useful in modeling of multilayered structures with large amount of layers, for example, produced by layer-by-layer additive manufacturing. It is easy to explain the nature of inhomogeneity for multilayered structures (discrete inhomogeneity). It is just the result of the fact that in the stress-free state the shapes of layers do not fit to each other. Thus, they cannot be assembled without gaps or overlaps. Proper assembly becomes possible only after individual deformations of layers that cause self-equilibrated stresses in their assembly. The explanation of inhomogeneity for laminated structures (continuous inhomogeneity) is slightly more complicated. It can be given upon the idea of a continuous family of reference shapes that are free from stresses only locally. In the present paper, this approach is discussed in detail. To define measures for stresses and strains on laminated structures, one has to determine corresponding fields in some specific way. Such definitions that are obtained by formalism, adopted in the theory of smooth manifolds with non-Euclidean connection, are also given. To compare a discrete inhomogeneity with its continuous counterpart, the stress–strain states for the sequence of multilayered structures have been examined. The common factor of these structures is that they have equal final volume. Meanwhile, the number of layers increases with their order in the sequence. The measure for inhomogeneity related with non-Euclidean connection is found from a nonlinear evolutionary problem. To support improved understanding of the interplay between multilayered and laminated structures, the stresses and strains exerted in them are studied in comparison. Considerations of the reverse situation, in which some multilayered structure with discrete inhomogeneity is defined upon a given laminated structure, are also carried out. The convergence with decreasing maximal thickness for layers to the original laminated structure is illustrated numerically. In this, one can see similarity with the partitioning procedure in the theory of integrals.
... Although all elements from G are subsets of the final body B * , their topological and smooth manifold structures are not compatible with each other and with those of B * . That is, bodies B α , B β , α = β, may have slightly different topologies and smooth manifold structures [31]. In particular, it means that the topological invariants of solid with variable material composition change due to attaching of extra material. ...
... On purpose to represent formulae (29) and (31) in unified form, denote K = { * } ∪ J and define the map Π : B * → K as Π(X) = * , X ∈ B * ∪ ∂ B * (B * \ B * ), π(X), X ∈ B * Then from (29) and (31) it follows that for X ∈ B * , k = Π(X) and p ∈ D k one has ...
... On purpose to represent formulae (29) and (31) in unified form, denote K = { * } ∪ J and define the map Π : B * → K as Π(X) = * , X ∈ B * ∪ ∂ B * (B * \ B * ), π(X), X ∈ B * Then from (29) and (31) it follows that for X ∈ B * , k = Π(X) and p ∈ D k one has ...
In the paper the relationship between pure geometrical concepts of the theory of affine connections, physical concepts related with non-linear theory of distributed defects and concepts of non-linear continuum mechanics for bodies with variable material composition is discussed. Distinguishing feature of the bodies with variable material composition is that their global reference shapes can not be embedded into Euclidean space and have to be represented as smooth manifolds with specific (material) connection and metric. The method for their synthesis based on the modeling of additive process are proposed. It involves specific boundary problem referred to as evolutionary problem. The statement of such problem as well as illustrative exact solutions for it are obtained. Because non-Euclidean connection is rarely used in continuum mechanics, it is illustrated from the perspective of differential geometry as well as from the point of view, adopted in the theory of finite incompatible deformations. In order to compare formal structures defined within the models of solids with variable material composition with their counterpart in non-linear theory of distributed defects, a brief sketch for latter is given. The examples for cylindrical and spherical non-linear problems are presented. The correspondences between geometrical structures that defines material connection, fields of related defect densities and evolutionary problems for bodies with variable material composition are shown.
... The first systematic theoretical study of the effects of incompatibility in surface growth was conducted by the Russian school [30][31][32][33][34][35] with the largely parallel development and subsequent extension of the theory in the West [36][37][38][39][40][41][42][43][44][45]. In particular, these studies have raised an awareness of the presence of a "historical element" in the incompatible surface growth problems, which implies that accumulated inelastic strains keep a detailed memory of the deposition process. ...
Surface growth is a crucial component of many natural and artificial processes from cell proliferation to additive manufacturing. In elastic systems surface growth is usually accompanied by the development of geometrical incompatibility leading to residual stresses and triggering various instabilities. In a recent paper (PRL, 119, 048001, 2017) we developed a linearized elasticity theory of incompatible surface growth which quantitatively linked deposition protocols with post-growth states of stress. Here we extend this analysis to account for both physical and geometrical nonlin-earities of an elastic solid. The new development reveals the shortcomings of the linearized theory, in particular, its inability to describe kinematically confined surface growth and to account for growth-induced elastic instabilities.
... The first systematic theoretical study of the effects of incompatibility in surface growth was conducted by the Russian school [30][31][32][33][34][35] with the largely parallel development and subsequent extension of the theory in the West [36][37][38][39][40][41][42][43][44][45]. In particular, these studies have raised an awareness of the presence of a "historical element" in the incompatible surface growth problems, which implies that accumulated inelastic strains keep a detailed memory of the deposition process. ...
Surface growth is a crucial component of many natural and artificial processes from cell proliferation to additive manufacturing. In elastic systems surface growth is usually accompanied by the development of geometrical incompatibility leading to residual stresses and triggering various instabilities. In a recent paper (PRL, 119, 048001, 2017) we developed a linearized elasticity theory of incompatible surface growth which quantitatively linked deposition protocols with post-growth states of stress. Here we extend this analysis to account for both physical and geometrical nonlinearities of an elastic solid. The new development reveals the shortcomings of the linearized theory, in particular, its inability to describe kinematically confined surface growth and to account for growth-induced elastic instabilities.
... If a 3 -dimensional manifold M represents reference shape in classical sense, i.e. stress free shape that can be embedded into physical Euclidean space, then the field of frames represents elementary fibers that are the object for measuring by any kind of deformation measures. In more general non-Euclidean approach such considerations remain valid only locally [25,26]. These questions will be discussed in section 2. ...
... Y Y such that each of them transforms an infinitesimal neighborhood of a point X into a uniform shape [25]. ...
... Below we will represent our view on how to solve these problems. structures, that may differ one from other [25]. ...
In the present paper a differential-geometric approach is developed to modeling of residual stresses in layered (LbL) structures obtained as a result of successive curing of thin layers of material. The objects of modeling are the structures obtained by sequential adsorption of a large number of thin layers. During this assembling, the internal (residual) stresses appear in the multilayered structure while local deformations turn out to be incompatible. This leads to accumulation of residual stresses and distortion of the final shape of the LbL structure. To reduce these factors, geometric compensation is used, the calculation of which is extremely laborious with a large number of layers. Geometric methods allow us to implement fast algorithms for obtaining the compensation. They are based on the "smoothing" of the multilayer LbL structure and its representation by introducing a smooth body-manifold with non-Euclidean connection that characterizes the incompatibility of deformations. Connection is determined from the solution of the evolutionary problem, which formalizes the course of the technological process. The classical fields of the mechanics of a continuous medium are put in correspondence with their non-Euclidean counterparts. As an example, model problems for cylindrical LbL elastic structures are considered. A model problem of structural optimization is solved to determine the optimal strategy of the technological process.
... понятием телом, которое в ряде задач удобно представить как часть неевклидова пространства (обзор работ по этой тематике приведен в [6]). Известно, что неевклидовость пространства, моделирующего тело, и несовместность деформаций, возникающая в теле либо в силу дефектной структуры, либо по причине последовательного создания в ходе технологического процесса, тесно связаны между собой [7][8][9][10][11][12][13]. Тела подобного рода будем называть структурно неоднородными (structurally inhomogeneous). ...
... Авторы настоящей работы используют результаты [6] и имеют своей целью построение мер деформаций для структурно неоднородных тел, материал которых прост (т.е. локально отклик зависит только от первого градиента деформаций [21, с. 60]); предполагается, что деформации являются вложениями тела в физическое пространство 2 , которое в общем случае неевклидово (о прикладных аспектах задач для материальных поверхностей в неевклидовом физическом пространстве см. ...
... локально отклик зависит только от первого градиента деформаций [21, с. 60]); предполагается, что деформации являются вложениями тела в физическое пространство 2 , которое в общем случае неевклидово (о прикладных аспектах задач для материальных поверхностей в неевклидовом физическом пространстве см. [6]). Общие построения иллюстрируются задачами дискретного и непрерывного роста конечного цилиндра, структурная неоднородность в котором обусловлена послойной усадкой материала в процессе его создания. ...
Настоящая статья посвящена формализации мер деформаций в неевклидовых пространствах для простого тела. Привлечение положений неевклидовой геометрии позволяет: i) определить глобальную единообразную отсчетную форму для тел со структурной неоднородностью, вызванной послойным созданием тела в ходе аддитивного процесса; ii) определить глобальную актуальную форму для тел в неевклидовом физическом пространстве, в частности, двумерных объектов на материальных поверхностях. В работе сформулированы соотношения для мер деформаций, порождаемых вложениями риманова многообразия, формализующего простое тело, в риманово многообразие, формализующее пространство. Предложен способ описания деформируемого тела переменного материального состава как семейства римановых многообразий, над которым определены операции разбиения и соединения, характеризующие структурные особенности неоднородностей, задаваемых сценарием аддитивного технологического процесса. Рассмотрены случаи дискретной и непрерывной структурной неоднородности. Предложена процедура синтезирования материальной метрики по семейству локальных конфигураций. Определен оператор вложения. С его помощью устанавливается взаимосвязь классического градиента деформации и касательного отображения, определенного над гладким многообразием, представляющим форму тела. На примере структурно неоднородного цилиндра из несжимаемого материала показаны особенности предлагаемого подхода к описанию несовместных деформаций.
The article discusses the mathematical modeling for the evolution of the stress-strain state and fields of defects in crystals during their contact interaction with a system of rigid punches. From a macroscopic point of view, the redistribution of defects is characterized by inelastic (viscoplastic) deformation, and therefore the processes under study can be classified as elastic-viscoplastic. Elastic and inelastic deformations are assumed to be finite. To take into account inelastic deformations, it is proposed to use a differential-geometric approach, in which the evolution of the fields of distributed defects is completely characterized by measures of incompatible deformations and quantified by material connection invariants. This connection is generated by a non-Euclidean metric, which, in turn, is given by a field of symmetric linear mappings that define (inconsistent) deformations of the crystal. Since the development of local deformations depends both on thecontact interaction at the boundary and on the distribution of defects in the bulk of the crystal, the simulation problem turns out to be coupled. It is assumed that the local change in the defect density is determined by the first-order Alexander Haasen Sumino evolutionary law, which takes into account the deviatoric part of the stress field. An iterative algorithm has been developed to find coupled fields of local deformations and defects density. The numerical analysis for the model problem was provided for a silicon crystal in the form of a parallelepiped, one face of which is rigidly fixed, and a system of rigid stamps acts on the opposite face. The three-constant Mooney Rivlin potential was used to model the local elastic response.
The needle is modeled as an elastic hollow cylindrical flexible rod, partially immersed into a viscoelastic material that simulates brain tissue. The controlled force and moment are applied at the end of the rod. The insertion of the needle is modeled as sliding with friction along a channel whose walls compress the needle. The compression force varies along the axis of the embedded part of the needle and changes in time. The magnitude of the compression forces is determined from the solution of the initial-boundary-value problem. The compression stiffness of the rod is assumed to be infinite, i.e. its deformation is reduced only to bending. Along the axis, the rod moves like an absolutely rigid body. The interaction of a viscoelastic material and a needle is modeled in the linear Winkler approximation as a dynamic system “beam-viscoelastic base” with a time-variable interaction zone length.
The present paper aims to develop geometrical approach for finite incompatible deformations arising in growing solids. The phenomena of incompatibility is modeled by specific affine connection on material manifold, referred to as material connection. It provides complete description of local incompatible deformations for simple materials. Meanwhile, the differential-geometric representation of such connection is not unique. It means that one can choose different ways for analytical definition of connection for single given physical problem. This shows that, in general, affine connection formalism provides greater potential than is required to the theory of simple materials (first gradient theory). For better understanding of this inconsistence it is advisable to study different ways for material connection formalization in details. It is the subject of present paper. Affine connection endows manifolds with geometric properties, in particular, with parallel transport on them. For simple materials the parallel transport is elegant mathematical formalization of the concept of a materially uniform (in particular, a stress-free) non-Euclidean reference shape. In fact, one can obtain a connection of physical space by determining the parallel transport as a transformation of the tangent vector, which corresponds to the structure of the physical space containing shapes of the body. One can alternatively construct affine connection of material manifold by defining parallel transport as the transformation of the tangent vector, in which its inverse image with respect to locally uniform embeddings does not change. Utilizing of the conception of material connections and the corresponding methods of non-Euclidean geometry may significantly simplify formulation of the initial-boundary value problems of the theory of incompatible deformations. Connection on the physical manifold is compatible with metric and Levi-Civita relations holds for it. Connection on the material manifold is considered in three alternative variants. The first leads to Weitzenböck space (the space of absolute parallelism or teleparallelism, i.e., space with zero curvature and nonmetricity, but with non-zero torsion) and gives a clear interpretation of the material connection in terms of the local linear transformations which transform an elementary volume of simple material into uniform state. The second one allows to choose the Riemannian space structure (with zero torsion and nonmetricity, but nonzero curvature) in material manifold and it is the most convenient way for deriving of field equations. The third variant is based on Weyl manifold with specified volume form and non-vanishing nonmetricity.
