We will call the original length of the side of the square X 1. Therefore we can write:. The Eigen vectors lie in the three directions that begin and end the deformation in a mutually orthogonal arrangement.
If the Eigen vectors are initially of length 1 then in the end they are length:. Strain lends itself well to geometric representation. Think of a unit sphere which has been deformed. By inspection, one could find the three orthogonal directions that have remained orthogonal in the deformation. The length of radial lines parallel to those orthogonal directions X 1 , X 2 , X 3 will have changed length such that:.
If we substitute the new dimensions into the equation for the sphere in this particular reference frame:. This is the equation of an ellipsoid, which is called the strain ellipsoid. Often geologists will use statistical studies of the shapes of pebbles, certain types of sand particles, and other natural objects which were probably originally round, to determine the strain which a particular body of rock has undergone. Unlike stress and strain, elasticity is an intrinsic property of a material.
The elastic properties of Earth materials affects everything from the variation of density with depth in the planet to the speed at which seismic waves pass through the interior. Ultimately the elastic properties of a material are governed by the arrangement and strength of the bonds between the atoms that make up the material. Elasticity is the property of "reversible deformation". If the deformation in a body under stress does not exceed a certain limit, called the elastic limit, the body will return to its initial shape when the stress is removed.
Where s is the elastic compliance and c is the elastic stiffness. In order to relate two second rank tensors, a fourth rank tensor is necessary. This can be seen if you take a square and pull on it from only one direction;.
For the three-dimensional case there are 81 terms in a fourth rank tensor. Therefore, there can be no more than 36 independent values in S ijkl. For convenience, a matrix notation is used. The subscript is broken into two parts:. To avoid the appearance of factors in the equations, the following factors are introduced into the matrix notation:.
Because of "compatibility relations," which say that material will deform continuously, the number is reduced to If the material being deformed is symmetric the number of coefficients is even further reduced. For an isotropic material, one that behaves the same in any orientation, there are only two quantities necessary. From these equations it becomes obvious that for isotropic materials the directions of the principal stresses are the same as those for the principal strains.
If a polycrystalline rock is large compared to the size of its constituent grains and does not have a preferred crystallographic orientation it will in general behave as an isotropic solid.
After Nye, The resulting stress is non-symmetric. In such a case there is no point in defining a symmetric strain and you might as well work with the whole non-symmetric deformation gradient. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Strain and stress tensor Ask Question.
Asked 8 years, 5 months ago. Active 6 years, 2 months ago. Viewed times. Improve this question. Hak hak Hak hak. It turns out that geometrically this is nothing but one half the Lie derivative of spatial metric tensor with respect to the displacement field. Metric is symmetric and so is its Lie derivative with respect to any vector field. In summary, there are different measures of deformation strain. Symmetric or "non-symmetric" tensors can be measures of strain. If one chooses a symmetric measure of strain, then its linearization will be symmetric as well symmetry is preserved under linearization.
I would like to give some background to my blog. It has some similarity with an earlier discussion in iMechanica on the concept of bending moment not properly addressed in most Theory of Elasticity courses, leading to confusion for students learning Theory of Elasticity with a Strength of Materials background. In Strength of Materials or Elements of Solid Mechanics course students are introduced to a single shear strain in 2D as the reduction in angle between two directions, which were at 90 degree in the undeformed configuration.
In Theory of Elasticity the students with Strength of Materials background find that the deformation, which was used for defining shear strain earlier, is actually a combination of shear strain and rotation. My objective in writing the above post was to get some arguments for explaing this concept better.
I also have wondered extensively about this question, and eventually concluded that the symmetry of the infinitesimal strain tensor is only a supposition. You may wish to find a memoire by Cauchy from whereby he appears to state this himself.
As it turns out, symmetry is not required for invariance in terms of Euclid objectivity or satisfaction of the principle of material frame-indifference. Both can be obtained with a skewed strain tensor. This is evident from the existence of Cosserat theory. However, I have posited that symmetry of the infinitesimal strain tensor is in fact valid and rational provided that the material undergoing the strain is isotropic and free of any micropolar moment. The corollary to this is that should the material be orthotropic or anisotropic, there is a rational and general description that suggest the infinitesimal strain tensor may be asymmetric.
This reasoning is provided in a paper authored by myself and colleagues doi. Is there a source that discusses this in greater detail?
I'm trying to understand linear elastic deformation from a Lie Theory perspective groups, algebras, derivative. Thanks, John. There is a typo where they discuss linearization of the Lagrangian strain.
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