Properties for Design of Composite Structures. Neil McCartney

(2.202)

      such that σ=kTΔV/V when ΔT=0.

      For isotropic materials, EA=ET=E, νA=νt=ν and μA=μt=μ so that

(2.203)

      and so that (2.198) has the following form

(2.204)

      It is clear that the elastic constants of an isotropic material are fully characterised by just two independent elastic constants, such as one of the following combinations: (E,ν), (μ,ν) and (E,μ). One of Lamé’s constants λ (the other is the shear modulus μ) and the bulk modulus k are often used as elastic constants for isotropic materials. These are related to Young’s modulus E, the shear modulus μ and Poisson’s ratio ν as follows (see (2.161)):

(2.205)

      The inverse form is

      (2.206)

      It is sometimes convenient to characterise an isotropic material using the two elastic constants μ and ν in which case, in addition to the relation (2.204),

      (2.207)

      More frequently, and as required for Chapter 3, it is useful to express Young’s modulus E and Poisson’s ratio ν in terms of the bulk modulus k and the shear modulus μ. On using (2.204) and (2.205)

      (2.208)

      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,

(2.209)

      The beam is assumed to be in a state of orthogonal bending combined with uniform through-thickness loading such that

(2.210)

      and

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