Strain EngineeringandElastic Efunda
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Mechanics of Materials: StrainDirectory | Career | News | Standards | Industrial | SpecSearch
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Global 1D Strain
Consider a rod with initial length L which is
stretched to a length L'. The strain measure ,
a dimensionless ratio, is defined as the ratio of
elongation with respect to the original length,
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Infinitesimal 1D Strain
The above strain measure is defined in a global sense. The strain at each point may
vary dramatically if the bar's elastic modulus or cross-sectional area changes. To track
down the strain at each point, further refinement in the definition is needed.
Consider an arbitrary point in the bar P, which
has a position vector x, and its infinitesimal
neighbor dx. Point P shifts to P', which has a
position vector x', after the stretch. In the
meantime, the small "step" dx is stretched to
dx'.
The strain at point p can be defined the same
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as in the global strain measure,
Since the displacement , the strain can hence be rewritten as,
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General Definition of 3D Strain
As in the one dimensional strain derivation, suppose that point P in a body shifts to
point P after deformation.
The infinitesimal strain-displacement relationships can be summarized as,
where u is the displacement vector, x is coordinate, and the two indices iand j can
range over the three coordinates {1, 2, 3} in three dimensional space.
Expanding the above equation for each coordinate direction gives,
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where u, v, and ware the displacements in the x, y, and zdirections respectively (i.e.
they are the components ofu).
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3D Strain Matrix
There are a total of 6 strain measures. These 6 measures can be organized into a
matrix (similar in form to the 3D stress matrix), shown here,
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Engineering Shear Strain
Focus on the strain xy
for a moment. The expression inside the parentheses can be
rewritten as,
where . Called the engineering shear strain, xy
is a total
measure of shear strain in the x-yplane. In constrast, the shear strain xy is the
average of the shear strain on the x face along the ydirection, and on the y face along
the xdirection.
Engineering shear strain is commonly used in engineering reference books. However,
please beware of the difference between shear strain and engineering shear strain, so
as to avoid errors in mathematical manipulations.
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Compatibility Conditions
In the strain-displacement relationships, there are six strain measures but only three
independent displacements. That is, there are 6 unknowns for only 3 independent
variables. As a result there exist 3 constraint, or compatibility, equations.
These compatibility conditions for infinitesimal strain refered to rectangular Cartesian
coordinates are,
In two dimensional problems (e.g. plane strain), all z terms are set to zero. The
compatibility equations reduce to,
Note that some references use engineering shear strain ( ) when
referencing compatibility equations.
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