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As it is a second order tensor, the stress deviator tensor also has a set of invariants, which can be obtained using the same procedure used to calculate the invariants of the stress tensor. It can be shown that the principal directions of the stress deviator tensor s i j {\displaystyle s_{ij}} are the same as the principal directions of the ...
To achieve this, it is necessary to perform a tensor transformation under a rotation of the coordinate system. From the definition of tensor, the Cauchy stress tensor obeys the tensor transformation law. A graphical representation of this transformation law for the Cauchy stress tensor is the Mohr circle for stress.
As an example, the stress state of a steel beam in compression differs from the stress state of a steel axle under torsion, even if both specimens are of the same material. In view of the stress tensor, which fully describes the stress state, this difference manifests in six degrees of freedom , because the stress tensor has six independent ...
Describing the stress, strain and deformation either in the reference or current configuration would make it easier to define constitutive models (for example, the Cauchy Stress tensor is variant to a pure rotation, while the deformation strain tensor is invariant; thus creating problems in defining a constitutive model that relates a varying ...
Figure 7.1 Plane stress state in a continuum. In continuum mechanics, a material is said to be under plane stress if the stress vector is zero across a particular plane. When that situation occurs over an entire element of a structure, as is often the case for thin plates, the stress analysis is considerably simplified, as the stress state can be represented by a tensor of dimension 2 ...
In continuum mechanics, the most commonly used measure of stress is the Cauchy stress tensor, often called simply the stress tensor or "true stress". However, several alternative measures of stress can be defined: [1] [2] [3] The Kirchhoff stress (). The nominal stress ().
This model has the general form and the isotropic form respectively =: = +. where : is tensor contraction, is the second Piola–Kirchhoff stress, : is a fourth order stiffness tensor and is the Lagrangian Green strain given by = [() + + ()] and are the Lamé constants, and is the second order unit tensor.
The complete state of stress in a body at a particular deformed configuration, i.e., at a particular time during the motion of the body, implies knowing the six independent components of the stress tensor (,,,,,), or the three principal stresses (,,), at each material point in the body at that time. However, numerical analysis and analytical ...