ENG5312 – Mechanics of Solids II Strain Transformation Plane Strain The general state of strain at a point in a body is composed of three components of normal strain ( , and ), and three components of shear strain ( , and ). As for stress the given strains may be transformed to components along alternate co-ordinate directions. One common use of this technique is to transform strains measured along particular directions on a strain gauge rosette into other directions. Consider only strain in the plane (i.e. , and ): Note: Plane stress does not cause plane strain. If an element is subjected to normal stresses and , 11
Transcript
ENG5312 – Mechanics of Solids II
Strain Transformation
Plane Strain
The general state of strain at a point in a body is composed of three components of normal strain ( , and ), and three components of shear strain ( , and ).
As for stress the given strains may be transformed to components along alternate co-ordinate directions.
One common use of this technique is to transform strains measured along particular directions on a strain gauge rosette into other directions.
Consider only strain in the plane (i.e. , and ):
Note: Plane stress does not cause plane strain. If an element is subjected to normal stresses and , normal strains , and are produced. Normal strain results from the Poisson effect.
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ENG5312 – Mechanics of Solids II
General Equations of Plane-Strain Transformation
Sign Convention
o and are positive if they cause elongation along the and axes, respectively.
o is positive if the interior angle AOB becomes smaller than 90o.
The goal will be to obtain strain transformation to the plane at an angle to the plane.
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ENG5312 – Mechanics of Solids II
To determine we need to find the change in length of due to , and .
Normal strain will cause a change in length of :
Normal strain will cause a change in length of :
Shear strain will cause a change in length of :
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ENG5312 – Mechanics of Solids II
Adding the three components to give the total change in length ( ) of :
The normal strain in the direction is defined as , and since and :
(14)
Similarly:
The normal strain in the direction is , and since and :
(15)
To evaluate we need to determine the amount of rotation line segments and undergo due to , and .
For small angles: , and using and :
(16)
The angle , where and :
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ENG5312 – Mechanics of Solids II
(17)
The total change in angle will give and since and are measured in opposite directions:
(18)
Substitution of the trigonometric identities: ; ; into Eqs. (14), (15) and (18) gives:
(19)
(20)
(21)
Eqs. (19) to (21) are the general equations of plane-strain transformation. Note the similarity to the equations for plane-stress transformation, Eqs. (1) to (3). Note: , , and correspond to , , and while
and correspond to and .
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ENG5312 – Mechanics of Solids II
Principal Strains
As for plane stress, an element may be oriented such that the plane strain is represented by two normal strains only (i.e. principal strains), and no shear strains.
Due to the similarity between the plane-stress and plane-strain transformation equations, the orientation of the principal axes and the principal strains are:
(22)
(23)
Maximum In-Plane Shear Strain
Similarly, the maximum in-plane shear strain is;
(24)
(25)
(26)
Again the axes for the maximum in-plane shear strain are directed 45o to the axes for the principal strains.
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ENG5312 – Mechanics of Solids II
Mohr’s Circle for Plane Strain
Due to the similarity between the plane-stress and plane-strain transformation equations, Mohr’s circle may be used to solve graphically for plane-strain transformation.
Following the procedure used for developing the equations used in Mohr’s circle for plane stress gives the following equations for Mohr’s circle for plane strain:
(27)
(27a)
(27b)
Eq. (27) is the equation of a circle with radius that is offset from the origin in the direction by on the axes.
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ENG5312 – Mechanics of Solids II
Point A is the known state of strain and .
Point C is the center, offset by .
Radius
Rotation of an element by 180o will give and . To obtain this result on Mohr’s circle requires a rotation of 360o, therefore, a rotation of an element by requires an angle of on Mohr’s circle.
Principal normal strains are located at points B ( , i.e. maximum) and D (, i.e. minimum), i.e. points of no shear strain. These strains occur at
angles and as measured from .
Maximum in-plane shear strain ( ) occurs at E and F, with the associated normal strain .
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ENG5312 – Mechanics of Solids II
State of strain at an orientation of counter-clockwise from the known state is given by point P (at counterclockwise from ). and
can be read from Mohr’s circle or found from trigonometry.
Note: points E and F occur at 900 to points B and D, as expected since the axes of maximum in-plane shear stress are oriented 45o to the principal axes.
Absolute Maximum Shear Strain (3D)
If a body is subjected to a general 3D state of stress, an element at a point within the body may be oriented such that it is subjected to normal stresses only (i.e. the principal stresses , and ). These normal stresses would cause the principal strains (with no shear strain) since there is no shear stress on the principal planes.
Assuming , and correspond to the , and directions, respectively:
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ENG5312 – Mechanics of Solids II
This state of strain can be illustrated on Mohr’s circle:
The absolute maximum shear strain will occur on the largest Mohr’s circle (here the plane) and it has magnitude:
(28)
With an associated normal strain:
(29)
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ENG5312 – Mechanics of Solids II
Absolute Maximum Shear Strain (Plane Strain)
Consider an element exposed to plane strain ( plane). Define , and ( ) in the , and directions, respectively. Assuming and have the same sign (positive here):
The absolute maximum shear strain is located out of plane (i.e. not on the plane) and it has magnitude:
(30)
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ENG5312 – Mechanics of Solids II
If the in-plane principal strains have opposite signs:
Then:
(31)
i.e. when the principal strains have opposite signs the absolute maximum shear strain occurs in-plane (here ) and is equivalent to the maximum in-plane shear strain.
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ENG5312 – Mechanics of Solids II
Strain Gauge Rosettes
A strain gauge can be used to measure the normal strain in a specimen loaded in tension (i.e. measurement of the change in resistance in a thin wire as it is stretched).
The normal strains on the surface of a body are measured with a strain gauge rosette, and this information is used to specify the state of strain at a point.
Using the strain transformation equation (Eq. (14) for ) for each gauge:
(32)
Solving Eq. (31) simultaneously will give the state of strain ( , and ). Then the plane-strain transformation equations, Eqs. (19) to (21), can be used to determine the strains along other co-ordinate directions.
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ENG5312 – Mechanics of Solids II
Often strain gauges rosettes are arranged in 45o or 60o configurations.
For the 45o rosette, , , and , and Eq. (32) gives:
(33)
For the 60o rosette, , , and , and Eq. (32) gives:
