Читать книгу Properties for Design of Composite Structures - Neil McCartney - Страница 33

When considering laminated composite materials, where each ply is reinforced with aligned straight fibres that are inclined at various angles to a global set of coordinates, there is a need to define a set of local coordinates aligned with the fibres in each ply. There is also a need to determine the properties of each ply referred to the global coordinates. For a right-handed set of global coordinates x₁, x₂ and x₃, i ₁, i ₂ and i ₃ are unit vectors for the directions of the x₁-, x₂- and x₃-axes, respectively. For laminate models, the fibres are usually assumed to be in the x₁-direction and coordinate transformations involve rotations about the x₃-axis. When modelling unidirectional plies as transverse isotropic materials the rotations would need to be taken about the x₁-axis if the fibres are in the x₁-direction. Coordinate transformations involving rotations about the x₃-axis are now considered.

A right-handed second set of local coordinates x′1,x′2andx′3 is obtained by rotating the reference set of coordinates about the x₃-axis by an angle ϕ as shown in Figure 2.1. The rotation is clockwise when viewing along the positive direction of the x₃-axis. The unit vectors in the directions of the x′1,x′2andx′3 axes are denoted by i′1,i′2andi′3, respectively. Rotating about the x₃-axis enables account to be taken of the effects of off-axis plies in laminates, as considered in Chapters 6 and 7.

Figure 2.1 Transformation of right-handed Cartesian coordinates.

Any point in space can be represented by the vector x (a first-order tensor) the value of which is wholly independent of the coordinate system that is used to describe its components so that

(2.171)

It then follows on resolving vectors that

(2.172)

Transformation of a set of Cartesian coordinates (x1,x2,x3) to (x′1,x′2,x′3) by a rotation of the x₁- and x₂-axes about the x₃-axis through an angle ϕ (as shown in Figure 2.1) leads to

(2.173)

These relationships can be established from the geometry shown in Figure 2.1 on making use of the various constructions shown as dotted lines.

The displacement vector u is a physical quantity that is wholly independent of the coordinate system that is used to describe its components. This vector may be written as (where summation over values 1, 2 and 3 is implied by repeated lowercase suffices)

(2.174)

where uk and u′k are the displacement components referred to the two coordinate systems being considered. It follows on using (2.172) that

(2.175)

The stress and strain at any point in a material is a dyadic (an array of ordered vector pairs) or second-order tensor whose value is wholly independent of the coordinate system that is used to describe its components. The second-order stress tensor σ may, therefore, be written as (where summation over values 1, 2 and 3 is implied by repeated lower case suffices)

(2.176)

where σkl and σ′kl are the stress components referred to the two coordinate systems being considered. On defining m = cos ϕ and n = sin ϕ, it follows from (2.172) that

(2.177)

Thus, from (2.176) and (2.177), because σ′12=σ′21, σ′13=σ′31 and σ′23=σ′32

(2.178)

The inverse relationships are obtained by replacing ϕ by −ϕ (i.e. n is replaced by –n) so that

(2.179)

The relationships (2.178) and (2.179) are the standard transformations, arising from tensor theory, for the rotation of stress components about one axis of a right-handed rectangular set of Cartesian coordinates. Identical transformations apply when considering the strain tensor so that

(2.180)

with inverse relations

(2.181)