Understanding St. Venant's Theory for Brittle Materials

St. Venant's theory, also known as the Maximum Principal Strain Theory, is one of the theories used to predict the failure of materials under multi-axial stress states. According to this theory, yielding (or failure, particularly relevant for brittle materials) begins when the maximum principal strain at any point in the material reaches the strain corresponding to the yield strength in a simple uniaxial tension test.

The criterion for yielding according to St. Venant's theory can be stated in terms of principal strains ($\epsilon_1, \epsilon_2, \epsilon_3$) as:

• $\epsilon_1 \le \epsilon_y$
• $\epsilon_2 \le \epsilon_y$
• $\epsilon_3 \le \epsilon_y$
• $\epsilon_1 \ge -\epsilon_y$
• $\epsilon_2 \ge -\epsilon_y$
• $\epsilon_3 \ge -\epsilon_y$

where $\epsilon_y$ is the yield strain in uniaxial tension. This can be summarized as $-\epsilon_y \le \epsilon_i \le \epsilon_y$ for $i = 1, 2, 3$.

Applying the Theory to Plane Stress

For a plane stress condition, where the stress in one direction is zero (e.g., $\sigma_3 = 0$), the principal strains can be related to the principal stresses ($\sigma_1, \sigma_2$) using Hooke's Law:

• $\epsilon_1 = \frac{1}{E}(\sigma_1 - \nu\sigma_2 - \nu\sigma_3) = \frac{1}{E}(\sigma_1 - \nu\sigma_2)$
• $\epsilon_2 = \frac{1}{E}(\sigma_2 - \nu\sigma_1 - \nu\sigma_3) = \frac{1}{E}(\sigma_2 - \nu\sigma_1)$
• $\epsilon_3 = \frac{1}{E}(\sigma_3 - \nu\sigma_1 - \nu\sigma_2) = \frac{1}{E}(-\nu\sigma_1 - \nu\sigma_2)$

The yield strain in uniaxial tension is $\epsilon_y = \sigma_y/E$. Substituting these into the strain criterion, we get inequalities in terms of principal stresses:

• $\frac{1}{E}(\sigma_1 - \nu\sigma_2) \le \frac{\sigma_y}{E} \implies \sigma_1 - \nu\sigma_2 \le \sigma_y$
• $\frac{1}{E}(\sigma_1 - \nu\sigma_2) \ge -\frac{\sigma_y}{E} \implies \sigma_1 - \nu\sigma_2 \ge -\sigma_y$
• $\frac{1}{E}(\sigma_2 - \nu\sigma_1) \le \frac{\sigma_y}{E} \implies \sigma_2 - \nu\sigma_1 \le \sigma_y$
• $\frac{1}{E}(\sigma_2 - \nu\sigma_1) \ge -\frac{\sigma_y}{E} \implies \sigma_2 - \nu\sigma_1 \ge -\sigma_y$
• $\frac{1}{E}(-\nu\sigma_1 - \nu\sigma_2) \le \frac{\sigma_y}{E} \implies -\nu(\sigma_1 + \sigma_2) \le \sigma_y$
• $\frac{1}{E}(-\nu\sigma_1 - \nu\sigma_2) \ge -\frac{\sigma_y}{E} \implies -\nu(\sigma_1 + \sigma_2) \ge -\sigma_y$

For brittle materials, the tensile yield strength ($\sigma_y$) and compressive yield strength ($\sigma_c$) can be different, and often the failure is considered based on maximum principal stress. However, the question specifically asks for the yield locus according to St. Venant's theory. Assuming $\sigma_y$ refers to the magnitude of yield strength/stress limit, the yield locus is defined by the boundaries where equality holds:

• $\sigma_1 - \nu\sigma_2 = \pm \sigma_y$
• $\sigma_2 - \nu\sigma_1 = \pm \sigma_y$

These equations represent four lines in the $\sigma_1$-$\sigma_2$ principal stress plane. For example:

• Line 1: $\sigma_1 - \nu\sigma_2 = \sigma_y$
• Line 2: $\sigma_1 - \nu\sigma_2 = -\sigma_y$
• Line 3: $\sigma_2 - \nu\sigma_1 = \sigma_y$
• Line 4: $\sigma_2 - \nu\sigma_1 = -\sigma_y$

These four lines form a shape centered at the origin in the $\sigma_1$-$\sigma_2$ plane. When plotted, this region is bounded by straight lines and represents the yield locus according to St. Venant's theory for plane stress. This specific set of lines forms a rhombus.

Let's consider a simple case with $\nu=0$ (for illustrative purposes only, as $\nu$ for materials is not zero). The equations become $\sigma_1 = \pm \sigma_y$ and $\sigma_2 = \pm \sigma_y$. This would form a square. However, for typical values of $\nu$ (0 < $\nu$ < 0.5), the lines $\sigma_1 - \nu\sigma_2 = \pm \sigma_y$ and $\sigma_2 - \nu\sigma_1 = \pm \sigma_y$ produce a rhombus shape.

Therefore, according to St. Venant’s theory for brittle material under plane stress, the shape of the yield locus is a rhombus.