Rank-one convexity implies polyconvexity in isotropic planar incompressible elasticity. (English. French summary) Zbl 1426.74047
The notion of polyconvexity was introduced into the context of nonlinear elasticity theory by John Ball [Arch. Ration. Mech. Anal. 63, 337–403 (1977; Zbl 0368.73040); Nonlin. Anal. Mech., Heriot-Watt Symp., Edinburgh 1976, Vol. I, 187–241 (1977; Zbl 0377.73043)]. On the other hand, rank-one convexity plays an important role in the existence and uniqueness theory for linear elastostatics and elastodynamics.
In this work, the authors study the relation between rank-one convexity and polyconvexity of objective and isotropic functions \(W\) on \(SL(2)= \{X\in \mathbb{R}^{2\times 2},\det X= 1\}\). These convexity properties play an important role in the theory of nonlinear hyperelasticity, where \(W(\nabla\varphi)\) is interpreted as the energy density of a deformation \(\varphi\). In particular, energy functions on the domain \(SL(2)\) are used for modelling of incompressible materials, since, in this case, the deformation \(\varphi\) is subjected to an additional constraint \(\det(\nabla\varphi)= 1\).
This work is interesting and may find readers working in the area of nonlinear hyperelasticity.
In this work, the authors study the relation between rank-one convexity and polyconvexity of objective and isotropic functions \(W\) on \(SL(2)= \{X\in \mathbb{R}^{2\times 2},\det X= 1\}\). These convexity properties play an important role in the theory of nonlinear hyperelasticity, where \(W(\nabla\varphi)\) is interpreted as the energy density of a deformation \(\varphi\). In particular, energy functions on the domain \(SL(2)\) are used for modelling of incompressible materials, since, in this case, the deformation \(\varphi\) is subjected to an additional constraint \(\det(\nabla\varphi)= 1\).
This work is interesting and may find readers working in the area of nonlinear hyperelasticity.
Reviewer: Hilmi Demiray (Istanbul)
MSC:
| 74B20 | Nonlinear elasticity |
| 74G65 | Energy minimization in equilibrium problems in solid mechanics |
| 26B25 | Convexity of real functions of several variables, generalizations |
Keywords:
quasiconvexity; nonlinear elasticity; Morrey’s conjecture; volumetric-isochoric split; nonlinear hyperelasticityReferences:
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