Calculation of shallow polymer shell taking the creep into account

Introduction Shallow shells are widely used in the technics and particularly in the construction, so their consideration is of great interest in itself. The geometry of the middle surface of shallow shells can be identified with the geometry of the plane on which they are projected. Curvilinear coordinates, along the lines laid by the principal curvatures can be considered to coincide with the Cartesian coordinates in the plane. Problems of calculation of shallow shells in the elastic stage are fairly well understood, but many materials are characterized by not only the elastic, but also viscoelastic properties [1-4]. When creep is taken into account geometrical and static equations of shallow shells theory remain unchanged. Changes occur only in the physical equations.


Introduction
Shallow shells are widely used in the technics and particularly in the construction, so their consideration is of great interest in itself. The geometry of the middle surface of shallow shells can be identified with the geometry of the plane on which they are projected. Curvilinear coordinates, along the lines laid by the principal curvatures can be considered to coincide with the Cartesian coordinates in the plane. Problems of calculation of shallow shells in the elastic stage are fairly well understood, but many materials are characterized by not only the elastic, but also viscoelastic properties [1][2][3][4]. When creep is taken into account geometrical and static equations of shallow shells theory remain unchanged. Changes occur only in the physical equations.

Derivation of resolving equations
Let the median surface of the shell, shown at Fig.1 be described by the equation: Full shell deformations are represented as the sum of the middle surface deformations and deformations caused by the change of curvature [5]:  ; ,

Fig. 1. Shallow rectangular in plan shell
The changes of the curvatures of the middle surface are determined by the same formulas as in the plates: ; .
x y x y w w w x y x y Differentiating the first equation in (3) on x, and the second -on y and then taking their sum we will get: 2 0 2 0 2 0 2 2 .
Equilibrium equation in the case when only the vertical load is acting can be written as [5]: where , We exclude from equations (7) lateral forces: Physical equations are written in the form: We express from (9) stresses by the deformations: Bending moments and torque are defined as follows: , , .
x y N N S x y y x w ) w ) w ) w w w w We express from (12) the middle surface deformations: The second governing equation we obtain by substituting the bending moments (11) in the equilibrium equation (8):   2  2  2  2  2  4  2  2  2  2 2 .
Thus, the problem of calculation of a shallow shell has reduced to a system of two differential equations of fourth order.

Solution of the problem
The system of equations (14) and (15) can be solved numerically by the method of finite differences. To determine the creep strain the linear approximation with respect to time can be applied. This approach is used in the papers [7][8][9][10][11][12][13][14]. For polymeric materials we have the nonlinear equation of Maxwell-Gurevich, which for plane stress has the form: The relationship between the total strains and stresses at t o f takes the form: 1 .
x x y x y y y x over the area load q = 0.5 kPa. Fig. 2 is a plot of distribution of stresses σ x depending on x and y at -/ 2 z h . Net surface corresponds to the time t = 0, shaded -t = 10 h.

Summary
Obtained equations allow calculation of shallow shells of any form on any load at an arbitrary law of the connection between stresses and creep strain. For polymers, obeying the Maxwell-Gurevich law it is possible to calculate deflections and stresses in the end of the creep process, having only the solution of the elastic problem.