Convexity conditions and existence theorems in nonlinear elasticity

1976 ◽  
Vol 63 (4) ◽  
pp. 337-403 ◽  
Author(s):  
John M. Ball
Keyword(s):  
Nonlinear Elasticity ◽  
Existence Theorems ◽  
Convexity Conditions

Cartesian currents, weak diffeomorphisms and existence theorems in nonlinear elasticity

1989 ◽  
Vol 106 (2) ◽  
pp. 97-159 ◽  
Author(s):  
Mariano Giaquinta ◽  
Giuseppe Modica ◽  
Jiří souček
Keyword(s):  
Nonlinear Elasticity ◽  
Existence Theorems ◽  
Cartesian Currents

scholarly journals Existence theorems in intrinsic nonlinear elasticity

2010 ◽  
Vol 94 (3) ◽  
pp. 229-243 ◽  
Author(s):  
Philippe G. Ciarlet ◽  
Cristinel Mardare
Keyword(s):  
Nonlinear Elasticity ◽  
Existence Theorems

A variational approach to nonlinear electro-magneto-elasticity: Convexity conditions and existence theorems

2017 ◽  
Vol 23 (6) ◽  
pp. 907-928 ◽  
Author(s):  
Miroslav Šilhavý
Keyword(s):  
Smart Materials ◽  
Existence Theorems ◽  
Deformation Gradient ◽  
Minimum Energy ◽  
Electric Displacement ◽  
The Body ◽  
Simultaneous Occurrence ◽  
Dimension 2 ◽  
The Cross ◽  
Convexity Conditions

Electro- or magneto-sensitive elastomers are smart materials whose mechanical properties change instantly by the application of an electric or magnetic field. This paper analyses the convexity conditions (quasiconvexity, polyconvexity, ellipticity) of the free energy of such materials. These conditions are treated within the framework of the general A-quasiconvexity theory for the constraints [Formula: see text] where F is the deformation gradient, d is the electric displacement and b is the magnetic induction. If the energy depends separately only on F, or on d, or on b, the A-quasiconvexity reduces, respectively, to Morrey’s quasiconvexity, polyconvexity and ellipticity conditions or to convexity in d or in b. In the present case, the simultaneous occurrence of F, d and b leads to the cross-phenomena: mechanic–electric, mechanic–magnetic and electro–magnetic. The main results of the paper are as follows. In dimension 3 there are 32 linearly independent scalar A-affine functions (and 15 in dimension 2) corresponding to the constraints (*). Therefore, an energy function ψ ( F, d, b) is A-polyconvex if and only if it is of the form [Formula: see text] where Φ is a convex function (of 31 scalar variables). Apart from the expected terms F, cof F, det F, d and b, we have the cross-effect terms Fd, Fb (and in dimension 2 also d × b). An existence theorem is proved for a state of minimum energy for a system consisting of an A-polyconvex electro-magneto-elastic solid plus the vacuum electromagnetic field outside the body.

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