## Constitutive laws and existence questions in incompressible nonlinear elasticity.(English)Zbl 0648.73013

The paper is a real contribution to the study of incompressible nonlinear elastic materials. The first part of the paper is devoted to the specification of the general concept of incompressible constitutive laws and its mathematical justification, the concept of incompressible elastic bodies, and some results regarding the existence and uniqueness of hydrostatic pressure fields. The considered constitutive law prescribes only the deviatoric part of the Cauchy stress, while its trace appears as a kind of Lagrange multiplier associated with a restricted principle of virtual work. The second part of the paper contains local existence results and regularity properties for the pure displacement problem, with dead or live loads, of an incompressible elastic body around a natural state.
It is proved that if the body force is sufficiently small, in a suitable chosen space, then there exists a unique deformation in $$W^{m+2,q}(\Omega)^ 3$$ and a unique pressure field in $$W^{m+1,q}(\Omega)^ 3/{\mathbb{R}}$$ strongly satisfying the equilibrium equations. Here m is an integer and $$q\in (1,\infty)$$, with $$(m+1)q>3$$ in the case of dead loads, and $$mq>3$$ for live loads. The inverse and implicit function theorems and variational methods are used to obtain the above existence and regularity results. We mention that these results are of different nature from those obtained by J. M. Ball [Arch. Ration. Mech. Anal. 63, 337-403 (1977; Zbl 0368.73040)].
The paper is of high mathematical level and is throwing light upon some subtle aspects regarding the behaviour of incompressible materials. The author is carefully pointing out the origins of his results and their connections with the already known results in the field.
Reviewer: Gh.Gr.Ciobanu

### MSC:

 74B20 Nonlinear elasticity 74B05 Classical linear elasticity 74A20 Theory of constitutive functions in solid mechanics 74G30 Uniqueness of solutions of equilibrium problems in solid mechanics 74H25 Uniqueness of solutions of dynamical problems in solid mechanics 74S30 Other numerical methods in solid mechanics (MSC2010)

Zbl 0368.73040
Full Text:

### References:

 [1] R. Abraham, J. Robbin, Transversal mappings and flows. New York/Amsterdam, Benjamin (1967). · Zbl 0171.44404 [2] R.A. Adams, Sobolev spaces. Academic Press New York (1975). · Zbl 0314.46030 [3] S. Agmon, A. Douglis, L. Nirenberg, Estimates near the boundary ... I. Comm. Pure Appl. Math. 12 (1959) pp. 623-727. · Zbl 0093.10401 [4] S. Agmon, A. Douglis, L. Nirenberg, Estimates near the boundary ... II. Comm. Pure Appl. Math. 17 (1964) pp. 35-92. · Zbl 0123.28706 [5] S.S. Antman, Material constraints in continuum mechanics. Atti. Acc. Naz. Lincei LXX (1981) pp. 256-264. · Zbl 0508.73002 [6] J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rat. Mech. Anal. 63 (1977) pp. 337-403. · Zbl 0368.73040 [7] P.G. Ciarlet, P. Destuynder, A justification of a nonlinear model in plate theory. Comput. Meth. Appl. Mech. Engrg. 17/18 (1979) pp. 227-258. · Zbl 0405.73050 [8] J.L. Ericksen, R.S. Rivlin, Large elastic deformations of homogeneous anisotropic materials. J. Rat. Mech. Anal. 3 (1954) pp. 281-301. · Zbl 0055.18103 [9] G. Geymonat, Sui problemi ai limiti per i sistemi lineari ellitici. Ann. Mat. Pure Appl. 69 (1965) pp. 207-284. · Zbl 0152.11102 [10] V. Girault, P.A. Raviart, Finite element approximation of the Navier-Stokes equations. Springer-Verlag Berlin (1979). · Zbl 0413.65081 [11] H. Le Dret, Quelques problèmes d’existence en élasticité non linéaire, Thèse 3ème cycle. Université Paris VI (1982). [12] P. Le Tallec, J.T. Oden, Existence and characterization of hydrostatic pressure in finite deformations of incompressible elastic bodies. J. Elasticity 11 (1981) pp. 341-358. · Zbl 0483.73035 [13] J.E. Marsden, T.J.R. Hughes, Mathematical foundations of elasticity. Prentice-Hall, Englewood Cliffs (1983). · Zbl 0545.73031 [14] G.H. Meisters, C. Olech, Locally one-to-one mappings and a classical theorem on Schlicht functions. Duke Math. J. 30 (1963) pp. 63-80. · Zbl 0112.37702 [15] F. Stopelli, Un theorema di esistenza e di unicità relativo alle equazioni dell’ elastostatica isoterma per deformazioni finite. Ricerche Matematiche 3 (1954) pp. 247-267. · Zbl 0058.39701 [16] R. Temam, Navier-Stokes equations. North-Holland (1978). · Zbl 0428.35065 [17] C. Truesdell, W. Noll, The non-linear field theories of mechanics. Handbuch der Physik, Vol. III/3, Springer-Verlag Berlin (1965). · Zbl 0779.73004 [18] T. Valent, Sulla differenziabilità dell’ operatore di Nemytsky. Rend. Acc. Naz. Lincei 65 (1978) pp. 15-26. · Zbl 0424.35084 [19] T. Valent, Local theorems of existence and uniqueness in finite elastostatics Sem. Mat. Univ. Padova (1980). [20] M. van Buren, On the existence and uniqueness of solutions to boundary value problems in finite elasticity. Thesis. Carnegie-Mellon University (1968). [21] C.C. Wang, C. Truesdell, Introduction to rational elasticity. Noordhoff Groningen (1973). · Zbl 0308.73001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.