Mechanics of elastic micropolar shells

L. M. Zubov ; V. A. Eremeyev

Dalʹnevostočnyj matematičeskij žurnal, Tome 4 (2003) no. 2, pp. 182-225

Résumé

The general static theory of micropolar shells under finite deformations is presented. The micropolar shell or Cosserat's shell is a material surface each point of which have six degrees of freedom of the rigid body. The various statements of boundary value problems of a nonlinear statics of elastic shells are given and their variational statements are formulated. The six variational principles are considered. The nonlinear equations of compatibility of strains of elastic Cosserat's shells are obtained and deformation boundary conditions are introduced. The torsion and bending of micropolar shell are considered by using semi-inverse method. The mathematical definition of the property of surface anisotropy is given. The universal deformations of micropolar shell are introduced. These universal deformations are solutions of static problem which satisfy the equilibrium equations for any constitutive equation of orthotrophic or isotropic shell. The theory of isolated and continuously distributed dislocations in elastic micropolar shell is developed. The stress-induced phase transitions of martensitic type are considered within the framework of continuum mechanics methods. The thermodynamical equilibrium relations are investigated. The phase equilibrium conditions are established by using Lagrange's variational principle. These relations consist of static balance equations of impulse and angular moment on a phase separation line and additional thermodynamical relation. The latter is necessary to determine an a priori unknown phase line. For elastic shell of Cosserat type, the expressions of energy-impulse tensors are given. From the linear thermodynamic of irreversible processes point of view the kinetic equation of propagating phase line are formulated. This equation describes also the motion of linear defects of other nature in shells. For equilibrium deformations, energy changes are determined with regard to phase line motion. The application of theory of the micropolar shells to the the mathematical modelling of the biological or lipidic membranes is discussed. From the mechanical properties of cellular membranes point of view the constitutive equations of liquid elastic micropolar shell are proposed. The obtained governing equations are equations of two-dimensional liquid which have a property of orientation elasticity and resist to bending. The presented model is compared with the smectic liquid crystals.