# Formulas for Yield Stress

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To manage problems related to yield stress, engineers and scientists rely on a variety of formulas dealing with the mechanical behavior of materials. Ultimate stress, whether it is tension, compression, shearing or bending, is the highest amount of stress a material can withstand. Yield stress is the stress value at which plastic deformation occurs. An accurate value for yield stress can be difficult to pinpoint.

#### TL;DR (Too Long; Didn't Read)

A range of formulas apply to yield stress, including Young's Modulus, stress equation, the 0.2 percent offset rule and the von Mises criteria.

## Young's Modulus

Young's Modulus is the slope of the elastic portion of the stress-strain curve for the material being analyzed. Engineers develop stress-strain curves by performing repeated tests on material samples and compiling the data. Calculating Young's Modulus (E) is as simple as reading a stress and strain value from a graph and dividing the stress by the strain.

## Stress Equation

Stress (sigma) is related to strain (epsilon) through the equation:

\sigma = E\times \epsilon

This relationship is only valid in regions where Hooke's Law is valid. Hooke's Law states that a restorative force is present in an elastic material that is proportional to the distance the material has been stretched. Since yield stress is the point where plastic deformation occurs, it marks the end of the elastic range. Use this equation to estimate a yield stress value.

## The 0.2 Percent Offset Rule

The most common engineering approximation for yield stress is the 0.2 percent offset rule. To apply this rule, assume that yield strain is 0.2 percent, and multiply by Young's Modulus for your material:

\sigma = 0.002\times E

To distinguish this approximation from other calculations, engineers sometimes call this the "offset yield stress."

## Von Mises Criteria

The offset method is valid for stress that occurs along a single axis, but some applications require a formula that can handle two axes. For these problems, use the von Mises criteria:

(\sigma_1-\sigma_2)^2+\sigma_1^2+\sigma_2^2=2\sigma (y)

where σ1 = x-direction max shear stress, σ2 = y-direction max shear stress and σ(y) = yield stress.

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