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MEEM 5150

Spring 2008

Topic 3: Strain

One-dimensional strain measures L

A

l

a

Take a 1D bar which can only be

streched axially. The bar had an

initial length L and area A that is

stretched to a final length of l and

area a. The engineering strain εE

can be given as

E

l l L

L Lε

∆ −= =

The true strain εT can be given as

E

l l L

l lε

∆ −= =

For both of these measures, if strain is small

the small strain quantity is recovered. ( )l L≈

/l lε = ∆

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For the small strain case, they all converge to small strain definition.

As an example, for the Green strain

2 2

2

2 2

2

2 2 2

2

( )( )

2

( )

2

1 2

2

G

L L Ll L

L

l l l

l

l l l l l

l

l

l

ε+ ∆ −

≈ ≈

+ ∆ −≈

+ ∆ + ∆ −=

∆≈

0

Deformation concept

X2

X1

X3

E1

E2

E3

X

e2

e1

e3

x1

x3

x2

x

b

u

t=0t=t

P0

P

O

o

Vector b serves to locate the origin o wrt O. From the figure

u = b + x – X

Very often in continuum mechanics it is possible to consider

the coordinate systems OX1X2X3 and ox1x2x3 superimposed so

that b = 0. So

u = x - X

material coordinates

spatial coordinates

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Strain concept

Lagrangian Tensor� Position function of a specific point:

� Displacement gradient tensor with respect to material coordinates

� Deformation gradient tensor

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Lagrangian Tensor� The change in the square of the length of the vector dX

� Lagrangian strain tensor:

Lagrangian Tensor� Infinitesimal lagrangian strain sensor (Green stain tensor)

� Right Cauchy-Green strain tensor

� Left Cauchy-Green strain tensor

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Physical stain components� Deformation of an orthogonal triad

� Extensional strain components

� Shear strain components

Physical stain components� Extensional strain components

� Shear strain components

� When

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Volume dilation� Volume dilation is defined as

� For the case of infinitesimal stains,

Principal Strains� The values of principal strains are the three roots of the determinant equation:

Where

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Principal Strains� Three principal strain directions are obtained from these equations:

� If (x,y,z) are principal strain axes, the three invariants are reduced as

Normal Strain Transformation

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Shear Strain Transformation

Strain-displacement relations for

small displacement theory

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Strain-displacement relations for

orthogonal curvilinear coordinates� Cylindrical coordinates (3D)

� Spherical Coordinates (3D)

� Polar Coordinates (2D)

Strain Compatibility� 3D

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Strain measurements� Strain gage

� Circuit bridges

� Gage mounting and types

Example 2.8, 2.9, 2.10, 2.11