2 Cube Stretch (20 points) Consider a cube of side length L, with uniform mass density R. Application of forces F[t] in uniaxial tension stretches the cube out in the direction of F[t], and contracts the transverse directions, as shown in Problem-Figure 2.1. X L F[t] xx Problem-Figure 2.1: Stretching of a cube. The deformation mapping representative of this is x2[X.1] =2[1]X2, x3[X,1] =2[1]X3, (2.1) with a[] R are functions of time. The applied force is also a function of time, and is assumed to result in uniform traction on the surfaces to which it is applied; the remaining surfaces of the cube are traction-free. Assume that there are no further body forces. Problem 2.1 (3 points) Compute the material and spatial velocity fields. Problem 2.2 (3 points) From mass conservation, derive an expression relating the reference and deformed densities, p and R, through the functions [t]. Problem 2.3 (6 points) Integrating over the reference body, find the total kinetic energy of the cube during deformation, K[t], as a function of time. Problem 2.4 (8 points) Integrating over the deformed body, find the total kinetic energy of the cube during deformation, K[r], as a function of time. How does this result compare to what you obtained in Problem 2.3? 2 Mass Conservation Definition 2.1 Define a Lagrangian mass density as the mass density per unit undeformed volume, R: 2 R, along with an Eulerian mass density as the mass density per unit deformed volume, p: [52] R. The mass contained in some sub-body &C 2, is given by From this, we note that we can write where the circle operator denotes composition. m[8,1] = [Rdv = pdv = [p{9[X,t],t]Jav . R=p] Q. (2.1) (2.2) Definition 5.3 The kinetic energy of a simple continuum (i.e. no rotational inertia) ios defined as [52] == || where V and v are the mean material/spatial velocity fields, respectively. The kinetic energy is an extensive variable. Thermal fluctuations do not enter K, but make up another external variable known as heat. (5.3)