We show that certain mixed displacement/traction problems (including live pressure tractions) of nonlinear elastostatics that are solved by a homogeneous deformation admit no other classical equilibrium solution under suitable constitutive inequalities and domain boundary restrictions. This extends a well-known theorem of Knops and Stuart on the pure displacement problem.
KnopsRStuartCA. Quasiconvexity and uniqueness of equilibrium solutions in nonlinear elasticity. Arch Ration Mech Anal1984; 86: 233–249.
2.
SivaloganathanJSpectorSJ. On the uniqueness of energy minimizers in finite elasticity. J Elast2018; 133: 73–103.
3.
BallJM. Convexity conditions and existence theorems in nonlinear elasticity. Arch Ration Mech Anal1976; 63: 337–403.
4.
SilhavyM. The mechanics and thermodynamics of continuous media. Berlin: Springer Science & Business Media, 2013.
5.
BallJMJamesRD. Fine phase mixtures as minimizers of energy. Arch Ration Mech Anal1987; 100: 13–52.
6.
RosakisP. Characterization of convex isotropic functions. J Elast1997; 49: 257–267.
7.
RosakisP. Ellipticity and deformations with discontinuous gradients in finite elastostatics. Arch Ration Mech Anal1990; 109: 1–37.
8.
MizelVJ. On the ubiquity of fracture in nonlinear elasticity. J Elast1998; 52: 257–266.
9.
HealeyTJSimpsonHC. Global continuation in nonlinear elasticity. Arch Ration Mech Anal1998; 143(1): 1–28.
10.
RosakisPSimpsonHC. On the relation between polyconvexity and rank-one convexity in nonlinear elasticity. J Elast1994; 37(2): 113–137.
11.
KnowlesJKSternbergE. On the failure of ellipticity and the emergence of discontinuous deformation gradients in plane finite elastostatics. J Elast1978; 8(4): 329–379.
12.
HorganCOMurphyJG. Some analytic solutions for plane strain deformations of compressible isotropic nonlinearly elastic materials. In: GilatRBank-SillsL (eds) Advances in mathematical modeling and experimental methods for materials and structures: The Jacob Aboudi volume. New York: Springer, 2009, pp. 237–247.
