Page 61 - Mechanics Analysis Composite Materials

46 Mechanics and analysis of composite materials

E, is the modulus of elasticity in the x-direction (x,y,z); v.~”the Poisson’s ratio
that determines the strain in the x-direction induced by normal stress acting in the
orthogonal y-direction (x,y,z); Gxvthe shear modulus in the xy-plane (x,y,z); v.~.,=
the extension-shear coupling coefficient indicating normal strain in the x-direction
induced by shear stress acting in the yz-plane (x,y,z); q.vy.zthe shear-extension
coupling coefficient characterizing shear strain in the xy-plane caused by normal
stress acting in the z-direction (x,y, z); and %x.,.,J= the shear-shear coupling coefficient
that determines the shear strain taking place in the xy-plane under shear stress acting
in the yz-plane (x,y,2).
Having constitutive equations, Eq. (2.46), we can now write the finite expression
for elastic potential, U. Substituting stresses into Eq. (2.41) and integrating it with
respect to strains we get after some transformation with the aid of Eq. (2.46) the
following equation
.
u = (%E, + 0JE.V + O,&, + ~.Y?.lJ,,. + T.YzY.Y, + yJz) (2.51)
Potential energy of the body can be found as

w = JJJmv (2.52)
C’
Compliance matrix, Eq. (2.49), containing 21 independent elastic constants
corresponds to the general case of material anisotropy that practically never
appears in real materials. The most common particular case corresponds to an
orthotropic (orthogonally anisotropic) material which has three orthogonal
orthotropy (coordinate) axes such that normal stresses acting along these axes do
not induce shear strains, while shear stresses acting in coordinate planes do not
cause normal strains in the direction of these axes. As a result, the stiffness and
compliance matrices become uncoupled with respect to normal stresses and strains
on one side and shear stresses and strains on the other side. For the case of an
orthotropic material, with axes x, y, and z coinciding with the orthotropy axes,
Eq. (2.49) acquires the form

0 0 0

0 0 0

0 0 0
[CI = (2.53)