Finite Strain Shape Memory Alloys Modeling - Scuola di Dottorato in ...

⎛ρ⎞0 1where ρ= ⎜φ⎜⎝ −J ⎟⎠ and 1b=B φ − represent the density and the body force perunit of volume in the current placement, respectively. In a Cartesian coordinatesystem,∂σσ = (3.74)a ∂ xab( div )b3.3.6 ObjectivityAn important concept in solid mechanics is the notion of objectivity. This conceptcan be explored by studying the effect of a rigid body motion superimposed on thedeformed configuration. From the point of view of an observer attached to androtating with the body, many quantities describing the behavior of the solid willremain unchanged. Such quantities, like for example the distance between any twoparticles and, among others, the state of stress in the body, are said to be objective.Although the intrinsic nature of these quantities remains unchanged, their spatialdescription may change. To express these concepts in a mathematical framework,consider an elemental vector dX in the initial configuration that deforms to dx andis subsequently rotated to dx . The relationship between these elemental vectors isgiven asdx = Qdx=QFdX(3.75)where Q ( t)is a proper orthogonal transformation depending only on time anddescribing the superimposed rigid body rotation. Although the vector dx is differentfrom dx , their magnitudes are obviously equal. In this sense it can be said that dx isobjective under rigid body motion. This definition is extended to any vector that48