Integrated Biomaterials Science

96 Claudio Migliaresi and Alessandro Pegoretti
Equations (3.6.11) and (3.6.12) can be combined to obtain the laminate
constitutive equation, written as follows:
Matrix [A] is called the extensional stiffness matrix, because it relates the
resultant forces to the midplane strains, while matrix [D] is called the
bending stiffness matrix because it relates the resultant moments to the
laminate curvature. The so-called coupling stiffness matrix, [B], accounts for
coupling between bending and extension, which means that normal and
shear forces acting at the laminate midplane are causing laminate curvature,
or that bending and twisting moments are accompanied by midplane strain.
It is noteworthy that if the laminate is balanced (i.e., it possesses the
same number of laminae having fibers oriented at and – the coefficients
and of the matrix are zero; moreover, if the laminate contains
laminae with the same orientation at the same distance above and below the
laminate geometric midplane (i.e., it is symmetric), all the coefficient of the
matrix are equal to zero.
Equations (3.6.11) and (3.6.12) imply that forces applied in the x- or
y-direction to a symmetric and balanced laminate do not provoke shear
deformation (because the laminate is balanced) or bending (because the
laminate is symmetric).
When midplane strain and curvature are obtained through equation
(3.6.13), the strain at any point in the laminate can be easily obtained by
taking into account that:
For symmetric and balanced laminates, a set of equivalent elastic moduli
can be easily evaluated from the coefficients (Barbero,
1998):