Properties for Design of Composite Structures. Neil McCartney
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needed for the applications to be considered in this book, it is required to introduce in the next section strain tensors as state variables, rather than the specific volume Ω introduced in (2.65) and (2.67).
2.10 Strain Tensor
Unlike the situation for fluids, the deformation (i.e. motion) of solid media is defined relative to an initial homogeneous reference state, thus enabling the introduction of the concept of strain, which takes account of both hydrostatic deformation considered in Section 2.9 and shear deformation. Following the approach in [1], consider now a homogeneous solid body with an initial configuration that occupies a region B¯, and which deforms into a different region B after the application of applied stresses and temperature changes. Let p¯ denote the position vector of some point in the undeformed body B¯ referred to Cartesian coordinates x¯, that moves to the point p in the deformed body B referred to Cartesian coordinates x. Let i¯K denote the orthogonal unit base vectors for the coordinate system x¯, and ik denote the orthogonal unit base vectors for the coordinate system x. The position vectors p¯ and p may then be expressed in the following form
where summations over values K, k = 1, 2, 3, are implied for repeated suffices. The corresponding infinitesimal vectors are written as
As
it follows that
Any vector v may be written as
Define δKl and δkL by the relation
It then follows that
It is clear that
The time-dependent deformation that transforms the undeformed region B¯ into the region B(t) may be expressed as
It then follows that
where the increments are taken at some time t such that dt = 0. In component form,
In the undeformed body, on using (2.71) and (2.73), the increment of arc length ds¯ is such that
where ckl is Cauchy’s symmetric deformation tensor. Similarly, for the deformed body, the line increment ds¯deforms to an increment ds such that
where CKL is Green’s symmetric deformation tensor. In dyadic form
where ∇¯ denotes the gradient with respect to the material coordinates x¯, and where the symmetric Green deformation tensor C may be written as