# Three-Dimensional Analysis of Strain

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Mechanical Engineering
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JoVE Central Mechanical Engineering
Three-Dimensional Analysis of Strain

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Three-dimensional strain analysis uses principal stress axes to evaluate deformation in elastic, homogeneous materials.

A small cubic element, when expanded around these axes, transforms into a rectangular parallelepiped, illustrating the deformation.

The strain analysis involves rotating an element around a principal axis, like the n-axis, and evaluating strain components on faces perpendicular to this axis. This method is based on plane strain transformation.

Using Mohr's circle, strain transformation is examined as the cube rotates around principal axes. This approach identifies the maximum shearing strain at a certain point, equivalent to the diameter of the largest of the three circles.

In plane strain, the n-axis transforms into a principal axis with zero strain at the origin of Mohr's circle diagram. Principal strains on opposite sides of this origin represent the maximum and minimum normal strains.

In thin plates of structures experiencing plane stress, the n-axis turns into a principal stress axis.

The principal strain along the n-axis is associated with in-plane strains. Rotation about the m-axis defines maximum shearing strain.

## Three-Dimensional Analysis of Strain

Three-dimensional strain analysis is crucial for understanding how materials deform under stress, particularly in elastic, homogeneous materials. This method employs principal stress axes to simplify complex stress states into more understandable forms. Subjected to stress, a small cubic element within a material either expands or contracts along these axes, transforming into a rectangular parallelepiped. This transformation effectively illustrates the material's deformation. The principal stress axes are orthogonal, representing directions where the stress does not induce shear within the material.

Mohr's circle is an essential tool in strain analysis. It provides a graphical representation of the stress states at a point and evaluates strain components when the element rotates around a principal axis, such as the n-axis. This analysis focuses on plane strain transformations, where strains at the origin of the n-axis are zero, simplifying the determination of maximum and minimum strains depicted on opposite sides of Mohr's circle.

In scenarios like thin plates under plane stress, the n-axis becomes a principal stress axis. Here, the principal strain along the n-axis directly correlates with the in-plane strains of the material. Rotation about another principal axis, such as the m-axis, helps pinpoint the locations and magnitudes of maximum shearing strain, which are crucial for predicting material behavior under load and ensuring the safety and reliability of structural designs.