o conceptually understand the difference between normal and shear strain, calculate normal strains and use them to compare the deformation of objects of different length, and calculate shear strains to show that shear strain is dependent on the axes of interest.

Displacement of an object due to applied forces can cause changes in that object$$s length along one or more axes. The amount of deformation depends on the applied loads and the dimensions of the object. Consider the case of stretching two steel wires with the same diameter, one $10-$m long and one $100-$m long. It would take significantly more force to stretch the $10-$m wire by $1$ cm than it would take to stretch the $100-$m wire by the same amount. In the first case, the strain is $0\mathrm{.}001,$ while in the latter case the strain is $0\mathrm{.}0001.$ You must know object dimensions to determine the forces from the absolute deformations. However, if you know the strain, you can determine the stress based only on the material being used without knowing its dimensions.

When there is an observed normal strain, it does not always mean there is a stress in the same direction. This is due to the Poisson effect exhibited by most materials. For these materials, when you compress them in one direction, they will expand in the direction perpendicular to the applied force. Likewise, if you stretch an object along one axis, it will contract along axes perpendicular to the applied force. This effect is described by the Poisson ratio, which is the ratio of the normal strain in the direction of the applied load to the normal strain perpendicular to the applied load.

Part A $-$ Normal strain and shear strain

Normal and shear strains describe an object$$s deformations. Normal strain is characterized by changes in length per unit length along an axis, whereas shear strain is characterized by a change in angle between two originally perpendicular axes. Examine each of the deformations below and identify the signs of the normal and shear strains for the axes shown. Note that the signs of the strains are not necessarily the same as the signs of the applied loads because of the Poisson effect.

Drag$-$and$-$drop the appropriate labels to their respective targets.

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Part B $-$ Normal strain: normalizing deformation

A clothing company is developing a new line of athletic attire designed to elastically deform during exercise. The company has developed five weaves to produce their new clothing and with some leftover strips of different sizes, decides to quantitatively measure how much the weaves stretch for a given load. Each strip is loaded into a uniaxial tensile testing machine as shown below and subject to a tensile load of $15$ N in the y$-$direction. Given the undeformed and deformed dimensions of the weaves, rank the weaves in order of the normal strain in the y$-$direction.

The figure shows a vertical strip of weave mounted between two horizontal grips. The strip broadens near the grips and has a constant width at the main part. The top grip pulls upward. The bottom grip pulls downward. The x axis is horizontally rightward. The y axis is vertically upward.

Rank the weaves from largest to smallest strain. To rank weaves as equivalent, overlap them.

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Least extensibleMost extensible

Part C $-$ Shear strain along different axes

Shear strain in an object is defined by the change in angle between two axes that lie within an object; it is$,$ therefore, possible that shear strain may be present in a deformed object depending on the axes of interest. Consider the cubes below that are subjected to forces that deform them. In the undeformed state, the sets of axes XY$,$ AB$,$ PQ$,$ and RS all are pairs of perpendicular axes. Which statement below accurately describes the relationship between the shear strains $\backslash$gamma XY

$,\backslash$gamma AB

$,\backslash$gamma PQ

$,$ and $\backslash$gamma RS

$?$