By Bettina Albers

This moment a part of Continuum Thermodynamics is designed to compare nearly one-to-one the chapters of half I. this is often performed in order that the reader learning thermodynamics can have a deepened realizing of the topics lined partly I. The goals of the publication are specifically: the representation of easy good points of a few basic thermodynamical versions equivalent to excellent and viscous fluids, non-Newtonian fluids, nonlinear solids, interactions with electromagnetic fields, and diffusive porous fabrics. yet another objective is the representation of the above topics by way of examples and straightforward strategies of preliminary and boundary difficulties in addition to uncomplicated routines to enhance talents within the building of interdisciplinary macroscopic models.

Readership: fabrics scientists and physicists.

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**Example text**

54) and its cosines of angles with the axes of coordinates change in the following way: √ 1 1 λ 1 1 1 . 3). It is a deformation in which points of planes parallel to the xy-plane move parallel to the y-axis. 18) have the values A = 1, B = 1, C = tan ϕ, D = 0 and E = 1. 3. 3: Geometry of a simple shear. The corresponding deformation gradient and the corresponding quantities are 1 0 0 1 0 0 −1 F = 0 1 tan ϕ , det F =1, F = 0 1 − tan ϕ . 57) 0 0 1 0 0 1 The polar decomposition for this problem can be described by 1 0 1 0 √ 0 √ 0 1 − R2 , 1 − R2 , U= 0 √ R R= 0 √R 2 2 2 0 − 1−R R 0 1−R −R R 1 0 0 √ 1 − R2 , V = 0 R +√R2 (1 − R2 ) 2 0 1−R R with R := 2 4 + tan2 ϕ ≤ 1.

Then the potential has the form φ = A · grad ln r = A·n . 29) Again the boundary condition for the surface of the cylinder yields A = −R2 v0 ⇒ φ=− R2 v0 ·n. 28)2 . The above examples demonstrate the typical dependence of the potential on the distance in the infinite medium: for the three-dimensional case it is 1/r2 and for the twodimensional case it is 1/r. Such a contribution cannot appear in three-dimensional flows due to the incompressibility. The potential −a/r would yield a constant mass flow through the surface of an arbitrary sphere of radius, say, R: 4πρa, where a is a constant.

Geometry of deformations of solids • right Cauchy-Green: 1 0 0 1 tan ϕ , C= 0 0 tan ϕ 1 + tan2 ϕ • Green-St. 64) 0 0 0 1 0 0 tan ϕ . 65) 1 these results agree with the classical results of materials sciences. 62) (2) 1 1 λC tan2 ϕ ± (3) = 1 + 2 2 λC 1 0 1 and R ≈ 0 0 − 12 ϕ 4 tan2 ϕ + tan4 ϕ = 0 1 ϕ . 68) = 1, and the corresponding eigenvectors are r(1) = e1 , r(3) 1 cos α 1 √ r(2) = − √ √ e2 + √ 1 − sin αe3 , 2 1 − sin α 2 1 cos α 1 √ =√ √ e2 + √ 1 + sin αe3 . 69) that the real eigenvector of R coincides with the direction e1 (x-axis) and that the angle α, defining the complex eigenvalues exp(±iα) of R, describes the angle of rotation of the eigenvectors of U around this axis.