A multi-continuum theory for composite elastic materials

Variational formulation of pre-stressed solid–fluid mixture theory, with an application to wave phenomena

“…Indeed the presence of this additional parameter modifies the strong ellipticity conditions which guarantee existence and uniqueness of the linear elasticity problem (see e.g. Fichera, 1972;Bedford and Stern, 1972;Iesan, 1994;Benallal and Comi, 2003). It is therefore a crucial issue the characterisation of the restriction to be imposed to such pre-stress in order to have assured well-posedness and stability.…”

On the theory of mixtures of elastic solids

“…We note that in the theory introduced by Bedford and Stern, with the exception of the defined volumetric interaction terms which are taken to depend on the constitutive variables associated with all the constituents, the other dependent constitutive variables for each constituent are taken to depend on the constitutive variables associated with that constituent only. The theory developed in [13,14,1 has been extended by Pop and Bowen [16] who established a thermodynamic theory of mixtures with long range spatial interactions. The model of interpenetrating solid continua was applied by Tiersten and Jahanmir [17] to derive a theory of composites where the relative displacement of the individual constituents is infinitesimal.…”

“…The model of interpenetrating solid continua was applied by Tiersten and Jahanmir [17] to derive a theory of composites where the relative displacement of the individual constituents is infinitesimal. Extensive reviews of the subject are given by Bowen [18], Atkin and Craine [19,20,1 and Bedford and Drumheller [15,1. In the first part of this paper, by using an idea of Bedford and Stern [13], we derive a nonlinear theory for binary mixtures of granular materials in Lagrangian description. In this theory all dependent constitutive variables depend on the constitutive variables associated with all the constituents.…”

Existence theorems in the theory of mixtures

“…In the theories for a mixture of elastic solids presented in [5]- [7] the constitutive independent variables are displacement gradients and the relative velocities, and the spatial description is used. The first theory for a mixture of elastic solids based on the Lagrangian description has been presented by Bedford and Stern [8], [9]. In this theory the independent constitutive variables are the displacement gradients and the displacement fields.…”

“…In this theory the independent constitutive variables are the displacement gradients and the displacement fields. The theory developed in [8], [9] has been extended by Pop and Bowen [ 10] who established a thermodynamic theory of mixtures with long range spatial interactions. The model of interpenetrating solid continua was applied by qSersten and Jahanmir [11 ] to derive a theory of composites where the relative displacement of the individual constituents is infinitesimal.…”