#                  PRINCIPAL STRESSES - 3 D

#This file calculates the three stress invariants I1,I2,I3
#from equations (1.16 - 1.18), the three principal stresses 
#s1,s2,s3 from equations (1.12 -- 1.14) and the Von Mises
#equivalent tensile stress sE from equation (1.19).
#
#To use this file in cylindrical polar coordinates or spherical
#polar coordinates, uncomment the appropriate three lines below
#and comment the present expressions for I1,I2,I3.
#
I1:=sRR+stt+sbb;
I2:=sRR*stt+stt*sbb+sbb*sRR-stb^2-sbR^2-sRt^2;
I3:=sRR*stt*sbb-sRR*stb^2-stt*sbR^2-sbb*sRt^2+2*stb*sbR*sRt;
#
#I1:=srr+stt+szz;
#I2:=srr*stt+stt*szz+szz*srr-stz^2-srz^2-srt^2;
#I3:=srr*stt*szz-srr*stz^2-stt*srz^2-szz*srt^2+2*stz*srz*srt;
#
#I1:=sxx+syy+szz;
#I2:=sxx*syy+syy*szz+szz*sxx-syz^2-szx^2-sxy^2;
#I3:=sxx*syy*szz-sxx*syz^2-syy*szx^2-szz*sxy^2+2*syz*szx*sxy;
#
phi0:=(1/3)*arccos((2*I1^3-9*I1*I2+27*I3)/(2*(I1^2-3*I2)^(3/2)));
#
s1:=I1/3+2*sqrt(I1^2-3*I2)*cos(phi0)/3;
s2:=I1/3+2*sqrt(I1^2-3*I2)*cos(phi0+(2*Pi/3))/3;
s3:=I1/3+2*sqrt(I1^2-3*I2)*cos(phi0+(4*Pi/3))/3;
#
sE:=sqrt(I1^2-3*I2);