On Material Conservation Laws for a Consistent Plate Theory

Bose, Dipak K.; Kienzler, Reinhold

doi:10.1007/s00419-006-0022-z

Cited by 9 publications

(4 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Another field of application of this correspondence is the theory of fracture of thin-walled structures with the use of 2D analogues of energy release rate relations such as discussed in [41,47,48] and path-independent integrals for plates and shells of specific geometry like cylindrical, spherical, etc., see [14,41,54] for the use of the latter within the plate and shell theories. For example, using 2D formulations based on 2D Eshelby tensor one can perform rapid assessments of the stress concentration for thin-walled structures.…”

Section: Discussionmentioning

confidence: 99%

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

Eremeyev

Konopińska-Zmysłowska

2019

Continuum Mech. Thermodyn.

View full text Add to dashboard Cite

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)

show abstract

Section: Discussionmentioning

confidence: 99%

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

Eremeyev

Konopińska-Zmysłowska

2019

Continuum Mech. Thermodyn.

View full text Add to dashboard Cite

show abstract

“…It is possible that using the intrinsic approach, certain three-dimensional effects may be missing in the derived shell model. Another approach which is also related to the derivation approach is the uniform-approximation technique, mostly motivated by engineering intuition [95,94,12,46,47,4,86]. It uses polynominal expansions in the thickness direction both for the displacements and for the stresses and then it truncates the series expansions.…”

Section: Different Approaches To Shell Modellingmentioning

confidence: 99%

The isotropic Cosserat shell model including terms up to $O(h^5)$. Part I: Derivation in matrix notation

Ghiba¹,

Bîrsan²,

Lewintan³

et al. 2020

Preprint

View full text Add to dashboard Cite

We present a new geometrically nonlinear Cosserat shell model incorporating effects up to order O(h 5 ) in the shell thickness h. The method that we follow is an educated 8-parameter ansatz for the three-dimensional elastic shell deformation with attendant analytical thickness integration, which leads us to obtain completely two-dimensional sets of equations in variational form. We give an explicit form of the curvature energy using the orthogonal Cartan-decomposition of the wryness tensor. Moreover, we consider the matrix representation of all tensors in the derivation of the variational formulation, because this is convenient when the problem of existence is considered, and it is also preferential for numerical simulations. The step by step construction allows us to give a transparent approximation of the three-dimensional parental problem. The resulting 6-parameter isotropic shell model combines membrane, bending and curvature effects at the same time. The Cosserat shell model naturally includes a frame of orthogonal directors, the last of which does not necessarily coincide with the normal of the surface. This rotation-field is coupled to the shell-deformation and augments the well-known Reissner-Mindlin kinematics (one independent director) with so-called in-plane drill rotations, the inclusion of which is decisive for subsequent numerical treatment and existence proofs. As a major novelty, we determine the constitutive coefficients of the Cosserat shell model in dependence on the geometry of the shell which are otherwise difficult to guess.

show abstract

“…Instead, in the case of two-dimensional (2D) structures such as shells, one faces a problem of homogeneity as a shell is an inhomogeneous 2D medium, since its geometry is point-dependent, in general. The 3D conservation laws could be transformed for plane geometry, i.e., for plates (see results for first-order shear-deformable linear plates [10,11], linear second-order plate theory [12], and von Ka´rma´n plates [13]). As a result, unlike to plate theory conservation laws for shells were established for particular geometries, such as spherical, cylindrical, or shells of revolution (see the previous works [5,14,15]).…”

Section: Introductionmentioning

confidence: 99%

Minimal surfaces and conservation laws for bidimensional structures

Eremeyev

2022

Mathematics and Mechanics of Solids

View full text Add to dashboard Cite

We discuss conservation laws for thin structures which could be modeled as a material minimal surface, i.e., a surface with zero mean curvatures. The models of an elastic membrane and micropolar (six-parameter) shell undergoing finite deformations are considered. We show that for a minimal surface, it is possible to formulate a conservation law similar to three-dimensional non-linear elasticity. It brings us a path-independent J-integral which could be used in mechanics of fracture. So, the class of minimal surfaces extends significantly a possible geometry of two-dimensional structures which possess conservation laws.

show abstract

On Material Conservation Laws for a Consistent Plate Theory

Cited by 9 publications

References 17 publications

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

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

The isotropic Cosserat shell model including terms up to $O(h^5)$. Part I: Derivation in matrix notation

Minimal surfaces and conservation laws for bidimensional structures

Contact Info

Product

Resources

About