The second-order, partial differential field equations in terms of stress [Equations (4.4)] can obviously be satisfied by constant or linear functions of position if body forces are also constant or linear functions.

**Linear Free Fields**

The second-order,
partial differential field equations in terms of stress [Equations (4.4)] can
obviously be satisfied by constant or linear functions of position if body
forces are also constant or linear functions. If the equilibrium equations are
satis-fied, then such a constant or linear stress field is the correct solution
to the prob-lem specified by the appropriate boundary conditions.

While such elementary
solutions may appear trivial in a mathematical sense, they are extremely
important in practice since they correspond to simple
“strength-of-materials-type“ analysis on which the great majority of design
deci-sions are based. Moreover, they are self-sustaining or “free fields“ that
do not damp out by St. Venant’s Principle into some more uniform distribution.*
Each stress component or combination of them, such as the invariants, can be
visual-ized as a flat membrane, either level or tilted, which is stretched
between a closed boundary. They are, therefore, harmonic functions satisfying
the Laplace equa-tion. To achieve this condition, the boundary traction
obviously must be constant or linear on shapes dictated by the membrane of
constant slope.

**Isotropic Stress
**

Consider first the most basic stress field in three
dimensions:

Everywhere Mohr’s Circle is a point and all
directions are principal (i.e., the mem-brane is flat at the elevation, *p*,
above any cross-section). The boundary surface must also be loaded with uniform
tension (vacuum) or compression (hydrostatic pressure). The normal strains are
also isotropic and equal *p*/3*K*. Since there are no deviatoric
stress components or elastic rotations, the displacements are a lin-ear
function of the distance from the origin, the location of which is arbitrary.

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

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