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Continuum mechanics MAE 640
Summer II 2009
Dr. Konstantinos Sierros
263 ESB new add
kostas.sierros@mail.wvu.edu
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Principal strains
The tensors E and e can be expressed in any coordinate system much like any dyadic.
In a rectangular Cartesian system, we have;
The components ofE and e transform according to Eq. (2.5.17):
where i jdenotes the direction cosines between the barred and unbarred coordinatesystems [see Eq. (2.2.49)].
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The principal invariants of the GreenLagrange strain tensorE are;
Principal strains
dilatation
The eigenvalues of a strain tensor are called theprincipal strains, and the
corresponding eigenvectors are called theprincipal directions of strain.
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Infinitesimal strain tensor and rotation tensor
Infinitesimal Strain Tensor
When all displacements gradients are small (or infinitesimal), that is, | u
|
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Infinitesimal strain tensor and rotation tensor
Expanded form
The strain components 11, 22, and 33 are the infinitesimal normal strains and 12,
13, and 23 are the infinitesimal shear strains.The shear strains 12 = 212, 13 = 213, and 23 = 223 are called the engineering
shear strains.
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Physical Interpretation of Infinitesimal Strain Tensor Components
Infinitesimal strain tensor and rotation tensor
To gain insight into the physical meaning of the infinitesimal strain components,
Dividing by ( dS2 )
Let dX/dS= N, the unit vector in the direction ofdX. For small deformations, wehave ds + dS= ds + dS 2dS, and therefore we have
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Infinitesimal strain tensor and rotation tensor
Physical Interpretation of Infinitesimal Strain Tensor Components
The ratio of change in length per unitoriginal length for
a line element in the direction of N.
For example, consider N along theX1-direction
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Infinitesimal strain tensor and rotation tensor
Physical Interpretation of Infinitesimal Strain Tensor Components
Then we have from Figure below;
Thus, the normal strain 11 is the ratio of change in length of a line element that was
parallel to thex1-axis in the undeformed body to its original length.
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Infinitesimal Rotation Tensor
Infinitesimal strain tensor and rotation tensor
infinitesimal rotation tensor
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Rate of Deformation and Vorticity Tensors
Definitions
In fluid mechanics, velocity vectorv(x, t) is the variable of interest as opposed to the
displacement vectoru in solid mechanics.
We can write the velocity gradient tensorL v as the sum of symmetric and
antisymmetric (or skew-symmetric) tensors
rate of deformation tensor vorticity tensor
orspin tensor
L v
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Recall that the deformation gradient tensorF transforms a material vectordX at X into
the corresponding spatial vectordx, and it characterizes all of the deformation, stretch
as well as rotation, at X.
Polar decomposition theorem
Therefore, it forms an essential part of the definition of any strain measure.
Another role ofF in connection with the strain measures is discussed here with the help
of the polar decomposition theorem of Cauchy.
The polar decomposition theorem decomposes the general deformation into pure stretch
and rotation.
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Polar decomposition theorem
right Cauchy stretch tensor(stretch is the
ratio of the finallength to the original length)
V the symmetric left Cauchy stretch tensor,
orthogonal rotation tensor,
describes a pure stretch deformation in which there are three mutually
perpendicular directions along which the material element dX stretches
(i.e., elongates or compresses) but does not rotate.
Also the role ofR in R U dX is to rotate the stretched element.
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Compatibility equations
The task of computing strains (infinitesimal or finite) from a given displacement field is a
straightforward exercise.
However, sometimes we face the problem of finding the displacements from a given
strain field.
This is not so straightforward
because there are six independent partial differential equations (i.e., strain-
displacement relations) for only three unknown displacements, which would in general
overdetermine the solution.
We will have to use some conditions, known as St. Venants compatibility equations,
that ensure the computation of unique displacement field from a given strain field.
The conditions are valid for infinitesimal strains. For finite strains, the same steps may
be followed, but the process is so difficult.
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Compatibility equations
Strain compatibility
condition among the
three strains for a two-dimensional case
using index
notation
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