Problems of mathematical modelling of soft shells. (Q2784290)

The development of parachute engineering -- the necessity of descent of space vehicles and battle machinery by parachutes -- promoted development of the theory of soft shells. Under some conditions, fabric, elastic films and walls of biological objects behave as soft shells. The soft thin elastic shells are characterized by the inability to resist compressing effort and pure bend. Thus in the analysis of their stress-deformed states, the bending moments, torques as well as cutting efforts can be neglected.NEWLINENEWLINENEWLINEThis monograph is devoted to mathematical modelling of isotropic homogeneous soft shells. Chapter 1 introduces the well-known equations of soft shell balance within the framework of simple shell theory. These equations are written in the form of equations of three-dimensional elastic shell balance. This allows to introduce definitions and constructions similar to the theory of elastic shells. Then the author derrives a new form of equations of soft shell balance amenable to the use of monotone operator theory and implicit function theorems. Chapter 2 considers infinitely long cylindrical shells with tangential efforts along the generatrix and with a degree of lengthening. The author formulates a boundary value problem for determination of static balance of rigidly fixed shells under the action of mass forces and normally applied tracking load (elementary model of shell interaction). The chapter is finished by an existence theorem for non-stationary problem. In chapter 3, the author considers two-dimensional static problems for soft shells. By implicit function theorems, the existence of solution for soft shell balance equation is proved in a neighborhood of prestressed biaxial state. The author obtains physical conditions under which the balance equation operator is monotone in an appropriate space, and studies the functional minimization problem for total energy of homogeneous isotropic soft shells. Chapter 4 investigates mathematical models for stress-free flows around shells, when a flow of low-viscous incompressible liquid is considered in the range of Reynolds numbers \(10^4-10^6\). The plane stationary flow around an infinitely long cylindrical shell perpendicularly to its generatrix is examined, and new variant of mathematical models of non-stationary stress-free flows around shells are proposed. The stress-free boundary layer is simulated by a vortex sheet, and Il'ichev-Postolovskij hypothesis is accepted: the stress-free point moves with the velocity of fluid particles driven at the stress-free point by the body contour and descending vortex sheet. The Il'ichev-Postolovskij formulas for the stress-free model of the flow around a circular cylinder are generalized to the case of arbitrary contour. The author gives a representation of velocity field due to the considered vortex sheets by using integral operators of simple structure, and gives estimates for stress-free flows about rigid circular cylinder and rigid elliptic cylinder at various angles of attack, and also gives the corresponding estimates for a soft circular cylindrical shell. In all cases the stress-free point ``leaves'' the steady regime of behaviour.NEWLINENEWLINENEWLINEThe results given in this book could be of importance in the design and safety verification of many engineering structures, e.g. air support structures, pneumatic constructions, water wet dams, wind proof devices, thin elastomer constructions (soft reservoirs), floating bridges, parachutes, dirigibles, flexible fences of air cushion vessels, elements of propulsive device of vibration type, behaviour of heart shells, stomach shells, bowel shells, and in simulation of cell membranes.