Properties for Design of Composite Structures. Neil McCartney

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СКАЧАТЬ equals negative nu subscript text A end text end subscript over E subscript text A end text end subscript sigma subscript r r end subscript minus nu subscript text A end text end subscript over E subscript text A end text end subscript sigma subscript theta theta end subscript plus 1 over E subscript text A end text end subscript sigma subscript z z end subscript plus alpha subscript text A end text end subscript capital delta T comma epsilon subscript r theta end subscript equals fraction numerator sigma subscript r theta end subscript over denominator 2 mu subscript text t end text end subscript end fraction. end cell end table"/>(2.199)

      When the fibres are aligned in a direction parallel to the x₁-axis, as required in Chapters 6 and 7 concerning laminates and their plies, the transverse isotropic stress-strain relations, resulting from the orthotropic form (2.196), are given by

(2.200)

      where, again, the relation (2.198) must be satisfied.

      For plane strain conditions such that ε11≡0, it follows from (2.200) that

      (2.201)

      When ΔT=0, the term ε22+ε33 is the change in volume per unit volume ΔV/V for the plane strain conditions under discussion when an equiaxial transverse stress σ is applied such that σ2=σ3=σ. It then follows that a plane strain bulk modulus kT can be defined by

      (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)

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