Properties for Design of Composite Structures. Neil McCartney

equals epsilon 22 comma epsilon 33 left-parenthesis x right-parenthesis identical-to StartFraction partial-differential u 3 Over partial-differential x 3 EndFraction equals epsilon 33 comma 2nd Row epsilon 12 left-parenthesis x right-parenthesis identical-to one-half left-parenthesis StartFraction partial-differential u 1 Over partial-differential x 2 EndFraction plus StartFraction partial-differential u 2 Over partial-differential x 1 EndFraction right-parenthesis equals epsilon 12 comma epsilon 13 left-parenthesis x right-parenthesis identical-to one-half left-parenthesis StartFraction partial-differential u 1 Over partial-differential x 3 EndFraction plus StartFraction partial-differential u 3 Over partial-differential x 1 EndFraction right-parenthesis equals epsilon 13 comma 3rd Row epsilon 23 left-parenthesis x right-parenthesis identical-to one-half left-parenthesis StartFraction partial-differential u 2 Over partial-differential x 3 EndFraction plus StartFraction partial-differential u 3 Over partial-differential x 2 EndFraction right-parenthesis equals epsilon 23 period EndLayout"/>(2.136)

This implies that the displacement fields differ by a rigid rotation about some axis. Such a rigid rotation will not affect the values of the strain and stress fields. The form (2.135) for the displacement field is preferred as it is then much easier to describe the meaning of the shear strains ε12, ε13 and ε23. When ε12 is the only non-zero strain component, it follows that u1 is the only non-zero displacement component and that its value is proportional to x₂ so that the shear plane is normal to the x₂-axis. Similarly, when ε13 is the only non-zero strain component, it again follows that u1 is the only non-zero displacement component and that its value is proportional to x₃ so that the shear plane is normal to the x₃-axis. When ε23 is the only non-zero strain component, it follows that u2 is the only non-zero displacement component and that its value is proportional to x₃ so that the shear plane is again normal to the x₃-axis.

      When using cylindrical polar coordinates (r,θ,z), the displacement components ur,uθ,uz are related to the Cartesian components u1,u2,u3 as follows

      (2.137)

      having inverse

      (2.138)

      and the strain–displacement relations are given by

      (2.139)

      When using spherical polar coordinates (r,θ,ϕ), the displacement components ur,uθ,uϕ are related to the Cartesian components u1,u2,u3 as follows:

      (2.140)

      having inverse

      (2.141)

      and the strain–displacement relations are given by

(2.142)

      2.14 Constitutive Equations for Anisotropic Linear Thermoelastic Solids

      As we are concerned in this book with various types of composite material, it is necessary to define a set of constitutive relations that will form the basis for the development of theoretical methods for predicting the behaviour of anisotropic materials. Consider a general homogeneous infinitesimal strain εkl (applied to a unit cube of the composite material) defined in terms of the displacement vector uk and the position vector xk by (2.107), namely,

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