Functionally Graded Materials in Engineering - Structural Interfaces and Attachments in Biology - page 22

The constitutive relations for thin-walled structures are reduced from three-
dimensional to two-dimensional formulation by integrating the stresses given
by ( 2.1 ) over the thickness. This integration eliminates the dependence on the
z -coordinate and replaces the stresses with a system of equivalent stress resultants,
i.e., forces per unit width of the cross section, and stress couples, i.e., moments per
unit width of the cross section. Thus the stress resultant and stress couple vectors are
defined by

ð

fNMg¼

z sf 1

zg d z (2.8)

The mathematical formulation should employ assumptions related to the
deformed shape of an infinitesimal element detached from the structure. These
assumptions reflect kinematics of the structure, i.e., the transformation of the
displacement tensor uðxÞ in the course of deformation and strain-displacement
relationships that can reflect geometrically linear or nonlinear formulations.
In particular, in a thin structure characterized by the so-called classical or technical
theory, we rely on the Kirchhoff-Love assumptions, i.e., the transverse shear strains
as well as the axial strain in the thickness direction are assumed equal to zero. These
assumptions imply that the thickness of the structure remains constant, while
normal lines to the undeformed middle surface remain straight and perpendicular
to this surface upon deformation. The validity of these assumptions becomes
questionable for very thick structures or in the vicinity to such discontinuities as
cut-outs, rivet holes, etc. In composite materials, the effect of transverse shear
strains has to be sometimes accounted for due to low shear stiffness. However,
the Kirchhoff-Love assumptions are usually applicable for relatively thin compos-
ite plates and shells manufactured from conventional composites, if their in-plane
size to thickness ratio is larger than a factor in the range from 30 or 40.
In case of a thin-walled structure characterized by the classical theory, the vector
of strain in ( 2.1 ) is represented by a sum of two components, i.e., the strains in the
middle surface ð« 0 Þ and the changes of curvature and twist k :

« ¼ « 0 þ zk (2.9)

where « 0 ¼ «ðz ¼ 0 Þ . The so-called first-order and higher-order theories accounting

for the rotation and warping of cross sections perpendicular to the middle surface
during deformation include additional terms proportional to integer powers of the
z -coordinate.
The substitution of ( 2.1 ), ( 2.7 ), and ( 2.9 ) into ( 2.8 ) results in the constitutive
relations relating the vectors of stress resultants and stress couples to the vectors of
middle surface strains and the changes of curvature and twist:

¼

N
M

AB
BD

« 0
k

(2.10)