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Curl bounds grad on SO(3)

Ingo Münch, Patrizio Neff (2008)

ESAIM: Control, Optimisation and Calculus of Variations

Let F p GL ( 3 ) be the plastic deformation from the multiplicative decomposition in elasto-plasticity. We show that the geometric dislocation density tensor of Gurtin in the form Curl [ F p ] · ( F p ) T applied to rotations controls the gradient in the sense that pointwise R C 1 ( 3 , SO ( 3 ) ) : Curl [ R ] · R T 𝕄 3 × 3 2 1 2 D R 27 2 . This result complements rigidity results [Friesecke, James and Müller, Comme Pure Appl. Math. 55 (2002) 1461–1506; John, Comme Pure Appl. Math. 14 (1961) 391–413; Reshetnyak, Siberian Math. J. 8 (1967) 631–653)] as well as an associated linearized theorem...

Curl bounds Grad on SO(3)

Patrizio Neff, Ingo Münch (2010)

ESAIM: Control, Optimisation and Calculus of Variations

Let F p GL ( 3 ) be the plastic deformation from the multiplicative decomposition in elasto-plasticity. We show that the geometric dislocation density tensor of Gurtin in the form Curl [ F p ] · ( F p ) T applied to rotations controls the gradient in the sense that pointwise R C 1 ( 3 , SO ( 3 ) ) : Curl [ R ] · R T 𝕄 3 × 3 2 1 2 D R 27 2 . This result complements rigidity results [Friesecke, James and Müller, Comme Pure Appl. Math.55 (2002) 1461–1506; John, Comme Pure Appl. Math.14 (1961) 391–413; Reshetnyak, Siberian Math. J.8 (1967) 631–653)] as well as an associated linearized theorem saying...

Existence theorem for nonlinear micropolar elasticity

Josip Tambača, Igor Velčić (2010)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper we give an existence theorem for the equilibrium problem for nonlinear micropolar elastic body. We consider the problem in its minimization formulation and apply the direct methods of the calculus of variations. As the main step towards the existence theorem, under some conditions, we prove the equivalence of the sequential weak lower semicontinuity of the total energy and the quasiconvexity, in some variables, of the stored energy function.

On the stability of multipolar elastic materials.

N. S. Wilkes (1979)

Stochastica

In 1964, Green and Rivlin [1, 2] proposed two non-standard theories of continua. Both papers concerned non-simple materials: the first considered deformation gradients of higher order than the first as dependent variables; and the second, which generalised the first, treated materials whose kinematic state was not completely detemined by the deformation function, but was also dependent upon some multipolar deformation functions. In both theories the existence of higher order stresses is fundamental.In...

Semicontinuity theorem in the micropolar elasticity

Josip Tambača, Igor Velčić (2010)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper we investigate the equivalence of the sequential weak lower semicontinuity of the total energy functional and the quasiconvexity of the stored energy function of the nonlinear micropolar elasticity. Based on techniques of Acerbi and Fusco [Arch. Rational Mech. Anal.86 (1984) 125–145] we extend the result from Tambača and Velčić [ESAIM: COCV (2008) DOI: 10.1051/cocv:2008065] for energies that satisfy the growth of order p≥ 1. This result is the main step towards the general existence...

Stress equations of motion of Ignaczak type for the second axisymmetric problem of micropolar elastodynamics

Janusz Dyszlewicz (1997)

Applicationes Mathematicae

A second axially-symmetric initial-boundary value problem of linear homogeneous isotropic micropolar elastodynamics in which the displacement and rotation take the forms u ̲ = ( 0 , u θ , 0 ) , φ ̲ = ( φ r , 0 , φ z ) ((r,θ,z) are cylindrical coordinates; cf. [17]) is formulated in a pure stress language similar to that of [12]. In particular, it is shown how u ̲ and φ ̲ can be recovered from a solution of the associated pure stress initial-boundary value problem, and how a singular solution corresponding to harmonic vibrations of a concentrated...

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