physics - Stress vector - Stress tensor - Mathematics Stack Exchange
of strain. Finally, the stress-strain relations for small deformation of linearly .. stress tensor and its relation with stress vector are developed. Definition of a. rank one tensor directional depended property, e.g. wave velocity rank two tensors relationship between two vector fields, e.g. stress, strain, conductivity. For example, the stress tensor for a cylinder with cross-sectional area in uniaxial .. Finally the relationships between the stress vector and the strain vector is.
Rank of a Tensor Tensors are referred to by their "rank" which is a description of the tensor's dimension.
A zero rank tensor is a scalar, a first rank tensor is a vector; a one-dimensional array of numbers. A second rank tensor looks like a typical square matrix.
Stress, strain, thermal conductivity, magnetic susceptibility and electrical permittivity are all second rank tensors. A third rank tensor would look like a three-dimensional matrix; a cube of numbers. Piezoelectricity is described by a third rank tensor.
A fourth rank tensor is a four-dimensional array of numbers. The elasticity of single crystals is described by a fourth rank tensor. Tensor transformation As mentioned above, it is often desirable to know the value of a tensor property in a new coordinate system, so the tensor needs to be "transformed" from the original coordinate system to the new one. As an example we will consider the transformation of a first rank tensor; which is a vector.
This can be done by noting the angle between each axis of the new coordinate system and each axis of the new coordinate system; altogether there will be 9 angles, three of which are illustrated in Figure 2: If we utilize Einstein's summation convention, we can leave out the summation symbol and get: There is a similar process for transforming a second rank tensor, but calculating a formula for the transformation by the same means that we transformed the vector above would be quite laborious.
There is a more convenient shortcut. Just as the dielectric constants "maps" the electric field Ej into the electric displacement Di, we can imagine a second rank tensor Tkl that takes Ql and produces Pk in a given coordinate system: It is abbreviated as: The Stress Tensor Stress is defined as force per unit area.
If we take a cube of material and subject it to an arbitrary load we can measure the stress on it in various directions figure 4. These measurements will form a second rank tensor; the stress tensor.
Tensors, Stress, Strain, Elasticity
Therefore, it is important to be aware of which sign convention is being used. A cube with its edges parallel to the principle stress directions experiences no sheer stresses across its faces. An important property of the stress tensor is that it is symmetric: If the cube is infinitesimally small, the forces across each face will be uniform.
If the cube is to remain stationary the normal forces on opposite faces must be equal in magnitude and opposite in direction and the shear tractions which would tend to rotate it must balance each other.
Therefore, it is only necessary to find 6 of the components of the tensor. Important concepts are often used are deviatoric stress and hydrostatic pressure. The Strain Tensor Strain is defined as the relative change in the position of points within a body that has undergone deformation.
The Stress Tensor
The classic example in two dimensions is of the square which has been deformed to a parallelepiped. Let us examine the movement of a point on the corner of the square m which moves to m': In order for this analysis to work we must only consider infinitesimally small strains.
We will call the original length of the side of the square X1. We can represent this quantity by e11 or more generally: Therefore we can write: The Eigen vectors lie in the three directions that begin and end the deformation in a mutually orthogonal arrangement.
As shown in the diagram the stress on each face can be shown in terms of three seperate stress vectors, and the stresses experienced can be expressed in nine stress vectors. These nine stress vectors are usually expressed in a stress matrix such as in seen below, and is known as the Stress Tensor.
Thus there are only six independent components to the stress tensor, and this means that the stress tensor is a symmetric tensor. The cube can be orientated in such a way that the major stress acting on it, is normal to one of the planes, and also that no shear stresses are caused, only normal stresses.Stress tensor in a Newtonian fluid
This causes the stress tensor to be reduced to In this case the stress vectors s11, s22, and s33, are collectively known as the principal stresses. The Principal axes become more important in a later section; the Stress Ellipsoid.
Mean Stress The mean stress is simply the average of the three principal stresses. Deviatoric Stress Now that we can calculate the mean stress, we can break the stress tensor down into two components.
The first part or isotropic component is the mean stress, and is responsible for the type of deformation mechanism, as well as dilation. The second component is the Deviatoric stress and is what actually causes distortion of the body. When considering the deviatoric stress, the maximum is always positive, representing compression, and the minimum is alway negative, representing tensional. The Stress tensor, still applies, but the principal stresses now come into play.
In the large majority of cases, one principle stress is larger then the other two, and the remaining two also differ in magnitude. The maximum principal stress is usually called s1, the intermediate principal stress is usually called s2, and the minimum principal stress is usually called s3.
When represented visually, you get the stress ellipsoid as shown below. The kinds of stress If all three of the principal axes are of equal magnitude, then the ellipsoid simplifies to a sphere, and each of the infinite number of stress vectors are equal.
This particular type of stress is termed Hydrostatic stress. This type of stress is commonly experienced by deeply buried rocks.
Uniaxial Stress is where only one of the principal axis is non-zero.