Question

Consider the state of stress \(\sigma_{x x}=\sigma_{y y}=\sigma_{z z}=\sigma\) and \(\sigma_{x y}=\sigma_{y x}=\tau, \sigma_{x z}=\sigma_{z x}=\sigma_{y z}=\sigma_{z y}=0\) Determine the yield conditions on the basis of the Tresca and von Mises criteria. How does hydrostatic loading affect plastic yielding?

Short answer

Tresca: \(\tau = \frac{\sigma_Y}{2}\); von Mises: \(\tau = \frac{\sigma_Y}{\sqrt{3}}\). Hydrostatic stress doesn't affect yielding.

Step-by-step solution

Understanding the Tresca Criterion

The Tresca yield criterion states that yielding begins when the maximum shear stress in a material reaches a critical value, known as the yield shear stress of the material. For a general state of stress, Tresca criterion can be expressed as: \[ \tau_{max} = \frac{\sigma_Y}{2} \]where \(\sigma_Y\) is the uniaxial yield stress. Since the stress state in this problem is given, we will compute the maximum shear stress and equate it to \(\frac{\sigma_Y}{2}\).

Calculate Maximum Shear Stress for Tresca

In the principal stress space, the principal stresses are \(\sigma_1 = \sigma + \tau\), \(\sigma_2 = \sigma - \tau\), and \(\sigma_3 = \sigma\). Thus, the maximum shear stress is:\[ \tau_{max} = \frac{1}{2} (\sigma_1 - \sigma_2) = \frac{1}{2} ((\sigma + \tau) - (\sigma - \tau)) = \tau \]So, for yielding according to the Tresca criterion, the condition is \(\tau = \frac{\sigma_Y}{2}\).

Understanding the von Mises Criterion

The von Mises yield criterion suggests that yielding occurs when the second deviatoric stress invariant reaches a critical value. This can be expressed as:\[ \sigma_{v} = \sqrt{\frac{1}{2}((\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2)} = \sigma_Y \] where \(\sigma_Y\) is again the yield stress in simple tension.

Calculate von Mises Stress

Using \(\sigma_1 = \sigma + \tau\), \(\sigma_2 = \sigma - \tau\), and \(\sigma_3 = \sigma\), substitute into von Mises equation:\[\sigma_{v} = \sqrt{\frac{1}{2}((2\tau)^2 + (\sigma - \sigma)^2 + (\sigma - \sigma - 2\tau)^2)} = \sqrt{3\tau^2} = \sqrt{3} \tau\]Thus, the von Mises yield condition is \(\sqrt{3} \tau = \sigma_Y\).

Effect of Hydrostatic Stress on Yielding

Hydrostatic stress is the isotropic part of the stress tensor and does not influence the shear stress state, as it does not change the size of the shear stress circle in the Mohr’s circle representation. Hence, both the Tresca and von Mises criteria are unaffected by hydrostatic stress, implying hydrostatic loading does not affect yielding behavior.

Key concepts

Tresca Criterion

The Tresca Criterion is one of the earliest yield criteria used to predict the onset of plastic deformation in materials. It is based on the idea that yielding begins when the maximum shear stress within the material reaches a critical value, known as the yield shear stress. This criterion is particularly useful for analyzing ductile materials where shear deformation is key.
In mathematical terms, the Tresca Criterion can be expressed as follows:

• Yielding occurs if: \[ \tau_{max} = \frac{\sigma_Y}{2} \]

Here, \( \sigma_Y \) is the material's yield stress in uniaxial tension, while \( \tau_{max} \) denotes the maximum shear stress experienced by the material.
For the given problem,

• The principal stresses are: \( \sigma_1 = \sigma + \tau \), \( \sigma_2 = \sigma - \tau \), \( \sigma_3 = \sigma \)
• The Tresca Criterion simplifies to: \[ \tau = \frac{\sigma_Y}{2} \]

This form of analysis is critical when designing components typically subjected to torsional or shearing loads.

Von Mises Criterion

The Von Mises Criterion, also known as the Maximum Distortion Energy Criterion, is another widely used theory to predict yielding in ductile materials. Unlike the Tresca Criterion, it considers the effect of multi-axial stress states by focusing on the second deviatoric stress invariant. This makes it a more universally applicable criterion under complex loading conditions.
According to Von Mises, yielding occurs when the equivalent von Mises stress reaches the material's yield stress:

• Yielding occurs if:\[ \sigma_{v} = \sqrt{\frac{1}{2}((\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2)} = \sigma_Y \]

For the stress state in the given problem:

• Substituting the principal stresses:\[ \sigma_{v} = \sqrt{3\tau^2} = \sqrt{3} \tau \]
• Thus, yielding occurs if:\[ \sqrt{3} \tau = \sigma_Y \]

The Von Mises Criterion is often preferred over Tresca for its mathematical smoothness, which makes it more suitable for numerical methods and its ability to provide a more conservative estimate for yielding.

Plastic Yielding Conditions

Plastic yielding conditions describe when a material transitions from elastic behavior, where deformation is reversible, to plastic behavior, where deformation becomes permanent. This transition is crucial in materials science, particularly in structural and mechanical engineering.
The major criteria, Tresca and Von Mises, are used for predicting this transition:

• Tresca Criterion: Focuses on the maximum shear stress reaching a critical level.
• Von Mises Criterion: Based on distortion energy, offering a comprehensive measure of yielding under complex loadings.

Understanding these conditions allows engineers to design safer and more efficient structures and components by predicting failure accurately. Factors such as material properties, type of loading, and environmental conditions must all be considered in this analysis.

Hydrostatic Stress

Hydrostatic stress refers to the part of the stress state in a material that is isotropic or uniform in all directions. It is a state of stress that does not change the shape of the material but can change its volume. In many cases, especially in solid mechanics, hydrostatic stress does not affect yielding.

• Effect on Yielding: Both Tresca and Von Mises criteria are unaffected by hydrostatic stress since it does not contribute to shear stress. This is because shear stress and hence yielding are influenced by differences between principal stresses rather than their absolute values.

In practical terms, understanding hydrostatic stress is essential in applications such as pressure vessels and underwater structures, where resistance to deformation rather than shape change is a priority.

Shear Stress Calculations

Shear stress calculations are crucial in determining how forces acting parallel to an area can cause deformation in a material. Accurate determination of shear stress is essential for assessing the strength and reliability of materials used in various engineering applications.
To calculate shear stress efficiently:

• Identify the loaded surface area and the forces acting parallel to it.
• Use the formula: \[ \tau = \frac{F}{A} \]where \( F \) is the force applied, and \( A \) is the area it acts upon.
• In problems involving principal stresses, like the given exercise, calculate shear stress using:\[ \tau_{max} = \frac{1}{2}(\sigma_1 - \sigma_2) \]

Shear stress calculation is essential in designing materials and structures to withstand forces without experiencing excessive deformation or failure, thereby ensuring safety and longevity.