2.18 Analysis of Bend Deformation
For most engineering applications of composite components, the deformation experienced in service conditions will involve some degree of bending. As the effect of bending on ply crack formation in composite laminates is considered in Chapters 11 and 19, it is useful to describe here the essential fundamental aspects of an analysis of bend deformation for a uniform orthotropic plate.
2.18.1 Geometry and Basic Equations
A beam of rectangular cross section of length 2L, width 2W and depth h is considered within a Cartesian coordinate system such that the x1-axis is in the axial direction and the x2-axis is in the in-plane transverse direction whereas the x3-axis is in the through-thickness direction, as shown in Figure 2.2. The origin is selected to lie at the mid-point of the upper surface of the beam.
Figure 2.2 Schematic diagram of part of a rectangular orthotropic plate of length 2L and depth h, and coordinate system. The x2-axis and u2 displacement are directed out of the plane of the page, and the width is denoted by 2W.
The modelling assumes that the beam is in equilibrium such that the equilibrium equations (2.120)–(2.122) are satisfied where the stress tensor is symmetric as in (2.123). The displacement components are denoted by ui. The infinitesimal strains are then given by the relations (2.143). The stress-strain relations are assumed to be of the orthotropic form (2.196), namely,
The beam is assumed to be in a state of orthogonal bending combined with uniform through-thickness loading such that
and
where the strain parameters ε¯A, ε¯T, ε^A and ε^T, the through-thickness stress σt and the temperature difference ΔT are assumed for the moment to be known. From (2.143), (2.209) and (2.211), the in-plane strains and shear stress are given by
The expression for ε22 in (2.209) is first solved for the stress component σ22 so that
It then follows from (2.209) that
where
