Problem 27 - An Introduction to the Mechanics of Solids in SI Units

Question

A batch of 2024-T4 aluminum alloy yields in uniaxial tension at the stress \(\sigma_{o}=330 \mathrm{MN} / \mathrm{m}^{2}\). If this material is subjected to the following state of stress, will it yield according to (a) the Mises criterion, and (b) the maximum shearstress criterion? $$ \begin{array}{ll} \sigma_{x}=138 \mathrm{MN} / \mathrm{m}^{2} & \tau_{x y}=138 \mathrm{MN} / \mathrm{m}^{2} \\ \sigma_{y}=-69 \mathrm{MN} / \mathrm{m}^{2} & \tau_{y z}=0 \\ \sigma_{z}=0 & \tau_{z x}=0 \end{array} $$

Step-by-Step Solution

Answer

Our calculations for both criteria will provide a yes or no answer about whether this material yield according to each criterion.

Step 1: Determine the deviatoric stress invariant

First, we need to compute the second deviatoric stress invariant which for a 3D state of stress is given by \(J_2=\frac{1}{6}\) [(\(\sigma_{x}-\sigma_{y}\))\(^2\) + (\(\sigma_{y}-\sigma_{z}\))\(^2\) + (\(\sigma_{z}-\sigma_{x}\))\(^2\) + 6\(\tau_{xy}\)\(^2\) + 6\(\tau_{yz}\)\(^2\) + 6\(\tau_{zx}\)\(^2\)]. So we substitute in the given values to get the value of \(J_2\).

Step 2: Apply the Mises criterion

Following the Mises criterion, the material will yield if the square root of 2 times \(J_2\) is greater than or equal to the uniaxial yield stress, \(\sigma_{o}\). We use the value of \(J_2\) which we calculated in the previous step, apply the formulas, and compare the result with \(\sigma_{o}\).

Step 3: Determine the maximum shear stress

We should now determine the maximum shear stress, \(\tau_{max}\), from the provided state of stress. The formula is as follows: \(\tau_{max}=\frac{1}{2}\) max [ | \(\sigma_{x}-\sigma_{y}\) | , | \(\sigma_{y}-\sigma_{z}\) | , | \(\sigma_{z}-\sigma_{x}\) | ].

Step 4: Apply the maximum shear stress criterion

The maximum shear stress theory states that yielding of the material happens when maximum shear taken becomes more than half of the uniaxial tensile yield stress. If the \(\tau_{max}\) is greater than or equal to \(\sigma_{o}/2\), then the material will yield according to the maximum shear stress criterion. Use the calculated values and compare them in order to come out to conclusion.

Key Concepts

Mises Criterion

The Mises Criterion, often called the von Mises yield criterion, is a fundamental concept used in material science to predict the onset of plastic deformation. It is centered around the idea of energy.
The criterion asserts that yielding begins when the equivalent stress, a combination of the material's stresses, reaches a critical value similar to the material's uniaxial tensile yield stress. The mathematical expression used involves the second deviatoric stress invariant, often denoted by \(J_2\).When you apply this criterion, you calculate \(J_2\) using stresses from different directions:

• \((\sigma_x - \sigma_y)^2\)
• \((\sigma_y - \sigma_z)^2\)
• \((\sigma_z - \sigma_x)^2\)
• \(6\tau_{xy}^2\)
• \(6\tau_{yz}^2\)
• \(6\tau_{zx}^2\)

The material yields when the computed stress based on these invariants matches or exceeds the actual yield stress. This means it doesn’t matter how the stress is distributed; the criterion remains focused on the energy involved.
In practical scenarios, the Mises Criterion is highly reliable and widely used because of its ability to accurately predict yielding across various multiaxial stress conditions.

Maximum Shear Stress Criterion

The Maximum Shear Stress Criterion, often known as the Tresca Criterion, predicts when a material will yield under complex stress conditions. This concept revolves around shear stress, the force attempting to slide layers within a material.To apply this criterion, you assess the maximum shear stress present in the stress state. It's computed using the principal stresses, with the formula:- \( \tau_{\text{max}} = \frac{1}{2} \text{ max} [ \left| \sigma_x - \sigma_y \right|, \left| \sigma_y - \sigma_z \right|, \left| \sigma_z - \sigma_x \right| ] \)
Essentially, it looks at the absolute differences between the stresses along different axes.
The principle is straightforward: yielding happens when this shear stress exceeds half the material's tensile yield stress. This direct approach focuses on the maximum internal sliding within the material.
While simple, the Maximum Shear Stress Criterion provides valuable insights, particularly for materials where sliding motions strongly influence yielding. Its simplicity also makes it easy to apply, offering quick and often reliable predictions in various engineering situations.

Stress Invariants

Stress invariants are crucial components in understanding and predicting material behavior under load. Essentially, they are mathematical combinations of the stress components within a material that remain constant, regardless of how the material is oriented.These invariants help simplify the complex stress state a material experiences into values that can be readily used for calculations and predictions.### Commonly Used Stress Invariants1. The first invariant, \(I_1\), is the sum of the principal stresses: \[ I_1 = \sigma_1 + \sigma_2 + \sigma_3 \] 2. The second invariant, \(I_2\), involves sums of products of principal stresses: \[ I_2 = \sigma_1\sigma_2 + \sigma_2\sigma_3 + \sigma_3\sigma_1 \] 3. The third invariant, \(I_3\), is the product of the principal stresses: \[ I_3 = \sigma_1\sigma_2\sigma_3 \]These invariants play a pivotal role in criteria like Mises and Tresca as they form a basis for predicting yielding.
While it might sound complex, think of stress invariants as the backbone of simplifying stress analysis. They give us a 'steady frame' to analyze and predict how and when materials will deform or fail under various loads.