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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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