Problem 2 - An Introduction to Materials Engineering and Science for Chemical and Materials Engineers

Question

Experimentally, it has been observed for single crystals of a number of metals that the critical resolved shear stress is a function of the dislocation density, \(\rho_{D}\) : $$ \tau_{c r}=\tau_{0}+A\left(\rho_{D}\right)^{0.5} $$ where \(\tau_{0}\) and \(A\) are constants. For copper, the critical resolved shear stress is \(2.10 \mathrm{MPa}\) at a dislocation density of \(10^{5} \mathrm{~mm}^{-2}\). If it is known that the value of \(A\) for copper is \(6.35 \times 10^{-3} \mathrm{MPa} \cdot \mathrm{mm}\), compute the critical resolved shear stress at a dislocation density of \(10^{7} \mathrm{~mm}^{-2}\).

Step-by-Step Solution

Answer

The critical resolved shear stress for \( \rho_D = 10^7 \text{ mm}^{-2} \) is 20.18 MPa.

Step 1: Understand the Formula

The problem involves the formula \( \tau_{cr} = \tau_0 + A(\rho_D)^{0.5} \), where \( \tau_0 \) and \( A \) are constants, \( \rho_D \) is the dislocation density, and \( \tau_{cr} \) is the critical resolved shear stress.

Step 2: Analyze Given Values

We are given \( \tau_{cr} = 2.10 \text{ MPa} \) at \( \rho_D = 10^5 \text{ mm}^{-2} \), and \( A = 6.35 \times 10^{-3} \text{ MPa} \cdot \text{mm} \). We need to find \( \tau_0 \) before calculating the new shear stress.

Step 3: Calculate \( \tau_0 \)

Substitute the given values into the formula: \( 2.10 = \tau_0 + 6.35 \times 10^{-3} \times (10^{5})^{0.5} \). Calculate \( (10^{5})^{0.5} = 316.23 \). Then, \( 2.10 = \tau_0 + 6.35 \times 10^{-3} \times 316.23 \). Solve for \( \tau_0 \).

Step 4: Solve for \( \tau_0 \)

Calculate the term: \( 6.35 \times 10^{-3} \times 316.23 \approx 2.01 \). Therefore, \( 2.10 = \tau_0 + 2.01 \). Solve for \( \tau_0 \) to obtain \( \tau_0 = 2.10 - 2.01 \approx 0.09 \text{ MPa} \).

Step 5: Compute Critical Resolved Shear Stress for New Density

Now that we have \( \tau_0 = 0.09 \text{ MPa} \), use the same formula for \( \rho_D = 10^7 \text{ mm}^{-2} \): \( \tau_{cr} = 0.09 + 6.35 \times 10^{-3} \times (10^{7})^{0.5} \).

Step 6: Calculate New \( \tau_{cr} \)

Compute \( (10^7)^{0.5} = 3162.28 \). Now, \( \tau_{cr} = 0.09 + 6.35 \times 10^{-3} \times 3162.28 \). Multiply to get \( 6.35 \times 10^{-3} \times 3162.28 \approx 20.09 \). Add \( 0.09 \) to obtain \( \tau_{cr} = 0.09 + 20.09 = 20.18 \text{ MPa} \).

Key Concepts

Dislocation Density

Dislocation density is an important concept in materials science, particularly when discussing the mechanical properties of metals. It refers to the number of dislocations, which are defects, in a given volume of material. In simpler terms, imagine dislocations as tiny imperfections in a metal's crystal structure that disrupt the orderly arrangement of atoms.

Dislocation density is often measured in terms like dislocations per square millimeter. Higher dislocation densities usually mean more mechanical interactions within the material, which can affect its strength and ductility. Here's why dislocation density is crucial:

• Metal Strength: As dislocation density increases, metals generally become stronger because dislocations have a more difficult time moving through the material.
• Increased Hardness: Greater dislocation density usually leads to harder metals, as the crystal lattice structure becomes more resistant to deformation.
• Toughness vs. Brittleness: A high dislocation density can make a material tougher but potentially more brittle if other factors, such as strain hardening, are not managed.

Understanding dislocation density helps us predict and manipulate the behavior of metals under stress, especially as we change conditions to achieve desired mechanical properties.

Critical Resolved Shear Stress

Critical resolved shear stress (CRSS) is a key factor in determining when a metal begins to plastically deform or yield under an applied stress. To simplify, it's the minimum stress required to move dislocations and cause the material to permanently change shape.

It's calculated by considering the external force applied to the metal and its orientation concerning the crystal structure's planes and directions. In exercise scenarios, CRSS is often linked to dislocation density, with the expression:\[\tau_{cr} = \tau_{0} + A (\rho_{D})^{0.5}\]where:

• \( \tau_0 \) is the base stress required without dislocations,
• \( A \) is a material constant,
• \( \rho_D \) is the dislocation density.

This relation shows that CRSS increases with higher dislocation densities, emphasizing the material's inherent resistance to deforming. Knowing CRSS helps engineers design materials that withstand specific stresses without failing.

Mechanical Properties of Metals

The mechanical properties of metals encompass their behavior under mechanical loads. These properties include, but are not limited to:

• Strength: The ability of a metal to withstand an applied force without deforming or breaking.
• Ductility: The capacity of a metal to deform under stress, often characterized by its ability to be stretched into a wire.
• Hardness: A measure of a metal's resistance to deformation, indentation, or scratching.
• Toughness: Represents how much energy a metal can absorb before fracturing.

In materials science, these properties are crucial because they determine how a metal can be used in various applications. By altering factors such as dislocation density or heat-treating, we can tailor these properties to suit specific engineering needs. For example:

• Increasing dislocation density typically enhances strength and hardness but may reduce ductility.
• Heat treatment processes can improve the toughness and relieve internal stresses.

Understanding these properties allows engineers and scientists to create materials that are both reliable and efficient for their intended functions. This ensures not only performance but also safety in applications ranging from everyday objects to critical structural components.