Читать книгу Properties for Design of Composite Structures - Neil McCartney - Страница 17

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)