Properties for Design of Composite Structures. Neil McCartney

identity tensor defined by (2.11) does not exhibit the same symmetry as the stiffness and compliance tensors, which are such that

      

(2.13)

      It is noted that

      

(2.14)

      indicating that Iijmn≠Ijimn and Iijmn≠Ijinm.

      A symmetric fourth-order identity tensor may be defined by

(2.15)

      so that

      

(2.16)

      The definition (2.15) is used to define the fourth-order identity tensors used in this book which are denoted by I or I.

      2.4 Displacement and Velocity Vectors

      Consider a continuous elastic medium that is being deformed from some homogeneous initial state as a result of loading. At some time t, a material point at x will have moved from its initial location x¯ in the material. The motion of the medium can be described by the following transformation g and its inverse G

(2.17)

      The vector x¯ defines ‘material coordinates’, associated with the motion of the medium that, together with the function g, can be used to describe the spatial variation during deformation of any physical quantity with respect to its original configuration. The transformation (2.17) is assumed to be single-valued and possess continuous partial derivatives with respect to their arguments. It is also assumed that the inverse function G exists locally, and this is always the case when the Jacobian J is such that

(2.18)

      The displacement of a material point x¯ is denoted by u(x¯,t) when using material coordinates, and is defined by

      

(2.19)

      The velocity v of a material point x¯ may be calculated using the relation

(2.20)

      2.5 Material Time Derivative

      The concept of a material time derivative (often named the substantive derivative) associated with the motion of the material is fundamental to the mechanics of continuous media. If ϕ is any scalar, vector or tensor quantity such that ϕ(x,t)≡ϕ¯(x¯,t), then its material time derivative is defined by

      

(2.21)

On using (2.20) the material time derivative may then be written as

(2.22)

      where ∇ denotes the gradient with respect to the coordinates x.

      Consider now a region V of the system bounded by the closed surface S enclosing a sample of the medium which is moving such that the velocity distribution at time t is denoted by v(x,t). It follows that, for any extensive property ϕ(x,t) of the system, the material time derivative (associated with the mean motion) of the integral of ϕ(x,t) over the region fixed region V may be written as

(2.23)

      From reference [1, Equation (2.4.9)], for example,

(2.24)

      On substituting (2.24) into (2.23) it follows on using (