Basics of Mechanics of Micropolar Shells

Eremeyev, Victor A.; Altenbach, Holm

doi:10.1007/978-3-319-42277-0_2

Cited by 29 publications

(28 citation statements)

References 118 publications

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“…Following [16,46], it can be shown that 2D strain energy can be derived from its 3D counterpart only as a certain approximation. So (21) and (24) are exact dependencies of 2D strain energy and stress resultant and surface couple stress tensors on their 3D counterparts. Nevertheless, the 2D form of U given by (11) is some direct resultant approximation of the 3D strain energy expressed by only 2D fields defined on Σ.…”

Section: Discussionmentioning

confidence: 99%

“…So in the literature this model is also known as the theory of micropolar shells, see, e.g. [5,21,23] and the reference therein. In the reference placement, the shell is represented by the base surface Σ with the position vector = (θ α ) and the unit normal vector η = η(θ α ), where θ α are surface curvilinear coordinates, α = 1, 2.…”

Section: Eshelby Tensor and Equilibrium Conditions Of Two-phase Shellsmentioning

confidence: 99%

“…In previous two sections, we have introduced the Eshelby tensors b and C as entirely independent quantities. Indeed, for shells we applied here the so-called direct approach treating a shell as a 2D elastic continuum [5,21,23]. Within the direct approach, one operates with 2D governing equations without any relation to the 3D nature of a shell, in general.…”

Section: Fig 3 Deformation Of a Shell-like Body 4 The Through-the-thmentioning

confidence: 99%

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On the correspondence between two- and three-dimensional Eshelby tensors

Eremeyev

Konopińska-Zmysłowska

2019

Continuum Mech. Thermodyn.

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We consider both three-dimensional (3D) and two-dimensional (2D) Eshelby tensors known also as energy-momentum tensors or chemical potential tensors, which are introduced within the nonlinear elasticity and the resultant nonlinear shell theory, respectively. We demonstrate that 2D Eshelby tensor is introduced earlier directly using 2D constitutive equations of nonlinear shells and can be derived also using the throughthe-thickness procedure applied to a 3D shell-like body.Keywords Eshelby tensor · Nonlinear elasticity · Nonlinear shell · Phase transformations · Through-thethickness integration IntroductionEshelby tensor known also as the Eshelby stress tensor or the energy-momentum tensor or the tensor of chemical potential was introduced originally in pioneering works by Eshelby [27][28][29]. Since it describes the energy release rate for a moving singularity, the Eshelby tensor found various applications in the theory of fracture [39,41,47,48], in the theory of stress-induced phase transitions [2,9,11,38,49], and in the modelling of the stress-assisted chemical reaction fronts propagation [32,34,58]. In particular, the properties of the Eshelby tensor result in several path-independent line integrals, see, e.g. [41,47,48]. For modelling of stress-induced phase transformations, the jump of discontinuity of the Eshelby tensor across the phase interface is used for the formulation of the thermodynamic compatibility condition which is necessary for determination of the interface position, see [2,9,33,38,64]. Motivating by the behaviour of martensitic films [12,13,30,44,50] and biological membranes [3,15,53], Eremeyev and Pietraszkiewicz [24] proposed the 2D model of thin-walled structures undergoing phase transitions within the nonlinear shell theory presented in [16,45,46,55]. In [24], the thermodynamic compatibility condition on the phase interface was derived using the stationarity of the total energy functional expressed through 2D constitutive equations. This description was further extended for more complex cases such as quasistatic deformations [26], viscoelastic properties [25], and the influence of a line tension [56]. The latter case constitutes an example of singular curves in shells discussed in [57] in detail. The analysis performed in [24,26] demonstrated that the developed 2D model can capture essential peculiarities of deformations of thin films made of shape memory alloys observed experimentally, see, e.g. [12,13,30,44,50]. Communicated by Holm Altenbach. V. A. Eremeyev · V. Konopińska-Zmysłowska (B)

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Section: Discussionmentioning

confidence: 99%

Section: Eshelby Tensor and Equilibrium Conditions Of Two-phase Shellsmentioning

confidence: 99%

Section: Fig 3 Deformation Of a Shell-like Body 4 The Through-the-thmentioning

confidence: 99%

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On the correspondence between two- and three-dimensional Eshelby tensors

Eremeyev

Konopińska-Zmysłowska

2019

Continuum Mech. Thermodyn.

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“…Following [24,26], let us briefly introduce the governing equations used within the six-parameter shell theory.…”

Section: Continuous Beam Lattice Shell: Micropolar Shellmentioning

confidence: 99%

“…Within the six-parameter shell theory, the kinematics of a shell is described by three translations and three rotations as in the rigid body dynamics or in the Cosserat (micropolar) continuum [30]. So, the model is also called the micropolar shell theory [24,26]. In the theory, only forces and moments including drilling ones are considered as stress characteristics which are also used in the static boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

Two- and three-dimensional elastic networks with rigid junctions: modeling within the theory of micropolar shells and solids

Eremeyev

2019

Acta Mech

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For two-and three-dimensional elastic structures made of families of flexible elastic fibers undergoing finite deformations, we propose homogenized models within the micropolar elasticity. Here we restrict ourselves to networks with rigid connections between fibers. In other words, we assume that the fibers keep their orthogonality during deformation. Starting from a fiber as the basic structured element modeled by the Cosserat curve beam model, we get 2D and 3D semi-discrete models. These models consist of systems of ordinary differential equations describing the statics of a collection of fibers with certain geometrical constraints. Using a specific homogenization technique, we introduce two-and three-dimensional equivalent continuum models which correspond to the six-parameter shell model and the micropolar continuum, respectively. We call two models equivalent if their approximations coincide with each other up to certain accuracy. The twoand three-dimensional constitutive equations of the networks are derived and discussed within the micropolar continua theory. The author acknowledges the support of the Government of the Russian Federation (Contract No. 14.Z50.31.0046).

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A nonlinear Lagrangian particle model for grains assemblies including grain relative rotations

Turco

Dell’Isola²,

Misra

2019

Num Anal Meth Geomechanics

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Summary We formulate a discrete Lagrangian model for a set of interacting grains, which is purely elastic. The considered degrees of freedom for each grain include placement of barycenter and rotation. Further, we limit the study to the case of planar systems. A representative grain radius is introduced to express the deformation energy to be associated to relative displacements and rotations of interacting grains. We distinguish inter‐grains elongation/compression energy from inter‐grains shear and rotations energies, and we consider an exact finite kinematics in which grain rotations are independent of grain displacements. The equilibrium configurations of the grain assembly are calculated by minimization of deformation energy for selected imposed displacements and rotations at the boundaries. Behaviours of grain assemblies arranged in regular patterns, without and with defects, and similar mechanical properties are simulated. The values of shear, rotation, and compression elastic moduli are varied to investigate the shapes and thicknesses of the layers where deformation energy, relative displacement, and rotations are concentrated. It is found that these concentration bands are close to the boundaries and in correspondence of grain voids. The obtained results question the possibility of introducing a first gradient continuum models for granular media and justify the development of both numerical and theoretical methods for including frictional, plasticity, and damage phenomena in the proposed model.

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Basics of Mechanics of Micropolar Shells

Cited by 29 publications

References 118 publications

On the correspondence between two- and three-dimensional Eshelby tensors

On the correspondence between two- and three-dimensional Eshelby tensors

Two- and three-dimensional elastic networks with rigid junctions: modeling within the theory of micropolar shells and solids

A nonlinear Lagrangian particle model for grains assemblies including grain relative rotations

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