If, in some orientation (x,y,z), shear stresses vanish throughout the body and the normal stresses are constant everywhere,

**Uniform Stress **

If, in some orientation (*x*,*y*,*z*),
shear stresses vanish throughout the body and the normal stresses are constant
everywhere,

and all the field
equations are satisfied. Strains are also constant giving the displacements by *finite*
linear transformation as discussed in Chapter 1. If one stress is zero, the
general case reduces to plane stress or, if say Ïƒ* _{y}*
=v(Ïƒ

Another common situation is the triaxial state where
two of the stresses are equal, say:

Unlike plane stress or
plane strain, the triaxial field really is two-dimensional, in that, because
the stresses are isotropic in the *x*-*y* plane, there is only one
Mohrâ€™s Circle throughout the body.

We create this situation in the laboratory in a
triaxial cell by first pressurizing the sample in a chamber to some confining
pressure, â€'*p*, and then increasing or decreasing the vertical stress with
weights or with a hydraulic piston (Figure 5.1). This device is particularly
useful for testing granular materials, which have little or no tensile
strength. Both invariants can be controlled independently. A pure deviatoric
test can be devised to determine the shear modulus, *G*, and

yield point. For
example, the confining pressure can be decreased while the vertical stress is
increased such that the radius of Mohrâ€™s Circle grows while the mean stress
remains constant. Conversely, we can determine the bulk modulus, *K*, by
keeping* *Ïƒ_{z =-}*
** p *throughout
the test. A uniaxial state of uniform stress is* *achieved corresponding
to an unconfined compression test if *p =* 0.

Depending on the boundary conditions and material
properties, all of these uniform stress fields can occur in the half-space
loaded by a constant surface load, *q* (Figure 5.2). Because of the
semi-infinite geometric condition, there can be no shear strains and *E** _{x}*
=

If v = 0, then the solution
reduces to a simple uniaxial field of unconfined compression or if v=0.5 .5 (incompressible), it becomes the isotropic field of hydrostatic compression
(Ïƒ_{x}
= Ïƒ_{y}
= Ïƒ_{z}
=-
*q*).

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Civil : Principles of Solid Mechanics : Linear Free Fields : Uniform Stress |

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