Strain-Transformation equations are based on the geometry of the deformation of deformable bodies(including some small-angle approximations).
External strain , or normal strain, is defined as a ratio of a total elongation to an original length .
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Shear strain is defined as a change in angle between two originally perpendicular line segments that intersect at a point. When , angle the sheared line makes with its original orientation, equals 90 degrees, the shear strain is infinte.
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Normal and Shear Strain |
The extensional strains and are determined by examining the change in length of short, mutually othogonal line segments and ; and the shear strains and are determined by the changes in right angles that originally exist between these lines. |
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General Equations |
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Principal Strains |
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In-Plane Principal Strains: |
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In-Plane Principal Directions: |
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Maximum Shear Strains |
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Maximum In-Plane Shear Strains: |
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In-Plane Shear Directions: |
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Mohr's circle for two-dimensional Strain |
Like the stress-transformation equations, the strain-transformation equations can be simplified bu introducing the double-angle trigonometric identities. This yields |
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Therefore |
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Equation of a circle in the plane with center at and radius R, with the angle being a parameter is |
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