In the majority of important practical situations, the general three-dimensional stress-strain array (Figure 2.3e) can, to an acceptable degree of accuracy, be simplified to a 4-component symmetric tensor by assuming either no stress or no strain components on one set of planes (say the z faces).

**Two Dimensional Stress
or Strain**

In the majority of important practical
situations, the general three-dimensional stress-strain array (Figure 2.3e)
can, to an acceptable degree of accuracy, be simplified to a 4-component
symmetric tensor by assuming either no stress or no strain components on one
set of planes (say the *z* faces). The first is called 'plane stress'
corresponding to free surfaces normal to the *z* axis so that ?* _{z}*
=?

'Plane strain' arises
physically in situations such as dams, pipelines, embankments, long shafts,
etc. where there are no displacements in the *z* direction nor distortion
in the *z* plane. Therefore *E** _{z}*
=?

Thus, in either case, the in-plane
tensor reduces to:

?* _{z}* and

The 2D transformation
(rotation of coordinate axes) from the *z*, *y*, *z* to a new *x*,*
y*,* z *axes can be deduced from the 3D equations [Equations (2.13)].
The* *direction cosines in terms of the positive counterclockwise angle of
axis rota-tion are shown in Figure 2.6.

Therefore,

The transformation of the 2D strain
tensor is the same by analogy, with *E** _{x}*,

The 2D characteristic equation becomes

The coefficients of the characteristic equation
must again be the same in any orientation so the 2D invariants are:

In principal space the 'octahedral'
orientation along the 'space diagonal' becomes the 'quadrahedral' orientation
along the 'square diagonal.' In this orientation the quadrahedral stress
components are:

Thus
in the quadrahedral orientation, we again see all the invariants as
indi-vidual, uncoupled stress components in themselves. There can be no doubt
that this is what nature actually feels at a point in a 2D stress field. Again
the analogous equations for the 2D strain tensor in the plane [Equations
(2.35)-(2.42)] are obtained by substituting *E** _{x}*
for ?

Perhaps it should be noted that the
elastic rotation around the *z* axis in two dimensions * _{xy}*
?2 or ?

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Civil : Principles of Solid Mechanics : Strain and Stress : Two Dimensional Stress or Strain |

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