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**Examples of state of stress**

Specifying a state of stress means providing sufficient information to compute the components of the stress tensor with respect to some basis. As discussed before, knowing t_{(n)} for three independent pairs of {**t _{(n)} , n**} we can construct the stress tensor

The state of stress is said to be uniform if the stress tensor does not depend on the space coordinates at each time t, when the stress tensor is represented using Cartesian basis vectors.

If the stress tensor has a representation

at some point, where

Evidently, the traction is along (or opposite to) n. This stress Ïƒ characterizes either pure tension (if Ïƒ > 0) or pure compression (if Ïƒ < 0).

If we have a uniform stress state and the stress tensor when represented using a Cartesian basis is such that Ïƒ

If the stress tensor has a representation

at any point, where n and m are unit vectors such that n Â· m = 0, then it is said to be in equibiaxial stress state. On the other hand, if the stress tensor has a representation

at any point, where n and m are unit vectors such that n Â· m = 0, then it is said to be in pure shear stress state. Post-multiplying (4.32) with the unit vector n we obtain

**Ïƒn **= **Ï„ (n **âŠ—** m + m **âŠ—** n)n **

** = Ï„ [(m Â· n)n + (n Â· n)m] = Ï„m =** (4.33)

Evidently, is tangential to the surface whose outward unit normal is along (or opposite to) **n**.

More generally, if the stress tensor has a representation**Ïƒ** = Ïƒ** _{1}n** âŠ—

at any point, where n and m are unit vectors such that n Â· m = 0, then it is said to be in plane or biaxial state of stress. That is in this case one of the principal stresses is zero. A general matrix representation for the stress tensor corresponding to a plane stress state is:

Here we have assumed that e

Next, we consider 3D stress states. Analogous to the equibiaxial stress state in 2D, if the stress tensor has a representation

at some point, we say that it is in a hydrostatic state of stress and p is called as hydrostatic pressure. It is just customary to consider compressive hydrostatic pressure to be positive and hence the negative sign. Post-multiplying (4.36) by some unit vector n, we obtain

Thus, on any surface only normal traction acts, which is characteristic of (elastic) fluids at rest that is not able to sustain shear stresses. Hence, this stress is called hydrostatic. Any other state of stress is called to be triaxial stress state. Many a times the stress is uniquely additively decomposed into two parts namely an hydrostatic component and a deviatoric component, that is

(4.38)

Thus, the deviatoric stress is by definition,

(4.39)

and has the property that tr(

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