details of each material model are given for each model it is necessary to provide the incremental stress strain relationship in global coordinates these are required to complete the incremental formulation of the equilibrium equations described in the following pages linear orthotropic theory will be reviewed before presenting details of the nonlinear orthotropic model for a two dimensional axisymmetric problem a linear orthotropic material is defined in terms of 7 independent engineering constants such that equ1 where u03b5 strain vector the compliance matrix s may be written in terms of 7 engineering constants the compliance matrix s is expressed in fig1 for the matrix shown in fig1 a b c are principal material directions the material constants are the 3 young 39 s moduli in the principal material directions e aa e bb e cc the 3 poissons ratios u03c5 ab u03c5 ac u03c5 bc and the shear modulus in the a b plane g ab the poissons ratios are defined by in the above two relations i and j take on values of a b or c in the present formulation the 34 c 34 axis is taken to be in the circumferential or hoop direction the hoop is a principal material direction the above relation may be inverted to obtain a stress strain relation e g where c is equal to the inverse of the s matrix e g in linear theory the 7 material properties or terms as described above are assumed to be true constants however in nonlinear theory incremental material constants which represent material properties may be functions of strain and or temperature and may have different values.