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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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Strain EngineeringandElastic Efunda

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