Overdetermination of the speed in rectilinear motion of non-Newtonian
fluids
Jerald L. Ericksen
Quarterly of Applied Mathematics
Vol. 14, No. 3, pages 318--321
1956
Consider fluid flowing through an infinitely long tube (a cylinder).
In classical Newtonian fluid mechanics (the Navier-Stokes equations) it's possible to show that every particle of the fluid moves along a straight line parallel to the cylinder's axis. That holds true regardless of the tube's cross-sectional shape.
This paper shows that in the case of a non-Newtonian fluid the particles do not move along straight lines unless the tube's cross-section is circular, or the fluid's constitutive equation is of a rather special form.
Steady flow of non-newtonian fluids through tubes
A. E. Green and R. S. Rivlin
Quarterly of Applied Mathematics
Vol. 14, No. 3, pp. 299-308
1956
This is a follow-up to Ericksen's paper (above). It is shown that the fluid particle paths are something like helices by constructing an approximate solution. The case of an elliptical cross section is studied in detail.
On non-existence of rectilinear motion in
plastic solids and non-newtonian fluids
D. E. Stone
Quarterly of Applied Mathematics
Vol. 15, No. 3, pages 257-262
1957
This is yet another take on Ericksen's result. It is shown that straight paths down the tube are possible for constitutive equations of certain type. Viscoelastic and plastic materials are included.
Determination of the stretch and rotation in the polar decomposition
of the deformation gradient
Anne Hoger and Donald E. Carlson
Quarterly of Applied Mathematics
Vol. 42, No. 1, pp. 113-117
1984
We know that the right Cauchy-Green strain tensor C is defined in terns of the polar decomposition of deformation gradient F = RU through C = F^T F = (RU)^T (RU) = U^T R^T R U = U^2. We know that calculating C is rather trivial—it's a matter of composing F^T and F. On the other hand, calculating U is challenging since it requires finding the square root of C, which in turn requires calculating C's eigenvalues and eigenvectors.
In this paper the authors present a trick for calculating U which does not involve the eigenvectors of C.
Aside: Anne Hoger was a graduate student of Donald Carlson's. She graduated in 1984, so I suspect that this was a part of her thesis.
New exact solutions in non-linear elasticity
K. R. Rajagopal and A. S. Wineman
International Journal of Engineering Science
Vol. 23, No. 2, pp. 217-234,
1985
Consider an elastic material occupying the domain
−∞ < x < ∞,
−∞ < y < ∞,
−h < z < h.
We attach rigid plates to the top and bottom faces of the domain and then twist the plates parallel to themselves. We wish to determine the elastic material's deformation.
Finite Amplitude Motions in Some Non-Linear Elastic Media
Ph. Boulanger and M. Hayes
Finite Amplitude Motions in Some Non-Linear Elastic Media
Proceedings of the Royal Irish Academy
Vol. 89A, No. 2, pages 135-146
1989
This paper considers an elastic material that fills up the entire 3D space. The material is deformed in such a way that ellipses map onto ellipses. (The mapping is given.) The resulting stresses and strains are computed.
After analyzing the static deformation, the paper proceeds to analyze the dynamic (periodic in time) oscillations.
A comparison of hyperelastic constitutive models
applicable to brain and fat tissues
L. Angela Mihai, LiKang Chin, Paul A. Janmey, and Alain Goriely
Journal of The Royal Society Interface
Vol. 12: 20150486
This article presents experimental data toward finding the constitutive equations of biological structures such as brain and fat tissues. These are modeled as homogeneous, isotropic, incompressible, hyperelastic materials.
This paper offers what the author calls a "simple proof" of the polar decomposition theorem of n×n matrices.
This is pretty much a math (linear algebra) paper.