On the natural shape of a plastically deformed thin sheet

Feraudy, Axelle de; Queguineur, Matthieu; Steigmann, David J.

doi:10.1016/j.ijnonlinmec.2014.10.018

Cited by 6 publications

(3 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Nevertheless, the multiplicative decomposition can be used for isotropic materials where both elastic and plastic rotations do not play any role in the final boundary-value-problem [15]. The need for a multiplicative decomposition can also be circumnavigated if we assume an additive decomposition of the total strain into elastic and plastic counterparts.…”

Section: Internal Stress and Natural Shapesmentioning

confidence: 99%

“…The concepts of material uniformity, material symmetry, and inhomogeneity in elastic Cosserat surfaces, following the pioneering works of Noll [52] and Wang [72], are also firmly established [23][24][25][26]73,74], although these works have neither attempted to describe the inhomogeneity distribution in terms of the curvature and non-metricity (the notion of torsion does appear in some of these works), nor have they discussed the relevant issue of strain incompatibility. A theory of materially uniform, inhomogeneous (dislocated) thin elastic films, derived from a 3-dimensional uniform, inhomogeneous (dislocated) elastic body, has been recently proposed by Steigmann [68], and applied to determining the natural shapes of plastically deformed thin sheets [15]. Finally, we mention, only in passing, the extensive work on mechanics of topologically defective ('geometrically frustrated') liquid crystalline surfaces [7,8,10,58], which, in contrast to the local theories mentioned above, have taken a distinguished local-global (geometrical-topological) standpoint in describing the nature of defects.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On Structured Surfaces with Defects: Geometry, Strain Incompatibility, Stress Field, and Natural Shapes

Roychowdhury

Gupta

2017

J Elast

View full text Add to dashboard Cite

Given a distribution of defects on a structured surface, such as those represented by 2-dimensional crystalline materials, liquid crystalline surfaces, and thin sandwiched shells, what is the resulting stress field and the deformed shape? Motivated by this concern, we first classify, and quantify, the translational, rotational, and metrical defects allowable over a broad class of structured surfaces. With an appropriate notion of strain, the defect densities are then shown to appear as sources of strain incompatibility. The strain incompatibility relations, with appropriate kinematical assumptions on the decomposition of strain into elastic and plastic parts, and the stress equilibrium relations, with a suitable choice of material response, provide the necessary equations for determining both the internal stress field and the deformed shape. We demonstrate this by applying our theory to Kirchhoff-Love shells with a kinematics which allows for small in-surface strains but moderately large rotations.

show abstract

Section: Internal Stress and Natural Shapesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On Structured Surfaces with Defects: Geometry, Strain Incompatibility, Stress Field, and Natural Shapes

Roychowdhury

Gupta

2017

J Elast

View full text Add to dashboard Cite

show abstract

“…Note that even in the case when W 0 does not explicitly depend on X and is an isotropic function with respect to the second argument (which corresponds to a homogeneous and isotropic material), W depends on X via K and remains an isotropic function only for particular values of K. Thus, a locally homogeneous and isotropic material turns out to be inhomogeneous and anisotropic as part of a self-stressed body. This fact is characterized by the concepts of material uniformity and inhomogeneity [47][48][49][50].…”

mentioning

confidence: 99%

Incompatible Deformations in Hyperelastic Plates

Lychev,

Digilov,

Bespalov

et al. 2024

Mathematics

View full text Add to dashboard Cite

The design of thin-walled structures is commonly based on the solutions of linear boundary-value problems, formulated within well-developed theories for elastic plates and shells. However, in modern appliances, especially in MEMS design, it is necessary to take into account non-linear mechanical effects that become decisive for flexible elements. Among the substantial non-linear effects that significantly change the deformation properties of thin plates are the effects of residual stresses caused by the incompatibility of deformations, which inevitably arise during the manufacture of ultrathin elements. The development of new methods of mathematical modeling of residual stresses and incompatible finite deformations in plates is the subject of this paper. To this end, the local unloading hypothesis is used. This makes it possible to define smooth fields of local deformations (inverse implant field) for the mathematical formalization of incompatibility. The main outcomes are field equations, natural boundary conditions and conservation laws, derived from the least action principle and variational symmetries taking account of the implant field. The derivations are carried out in the framework of elasticity theory for simple materials and, in addition, within Cosserat’s theory of a two-dimensional continuum. As illustrative examples, the distributions of incompatible deformations in a circular plate are considered.

show abstract

Asymptotic Derivation of Nonlinear Plate Models from Three-Dimensional Elasticity Theory

Shirani

Steigmann

2019

Advanced Structured Materials

View full text Add to dashboard Cite

On the natural shape of a plastically deformed thin sheet

Cited by 6 publications

References 4 publications

On Structured Surfaces with Defects: Geometry, Strain Incompatibility, Stress Field, and Natural Shapes

On Structured Surfaces with Defects: Geometry, Strain Incompatibility, Stress Field, and Natural Shapes

Incompatible Deformations in Hyperelastic Plates

Asymptotic Derivation of Nonlinear Plate Models from Three-Dimensional Elasticity Theory

Contact Info

Product

Resources

About