List and explain briefly the conditions that induce brittle fracture in an otherwise ductile metal.

Which of the two tests, tension or compression, would require a higher capacity of testing machine, and why?

Explain why the difference between engineering strain and true strain becomes larger as strain increases. Does this difference occur for both tensile and compressive strains? Explain.

A paper clip is made of wire $0.5 \mathrm{~mm}$ in diameter. If the original material from which the wire is made is a rod 25 $\mathrm{mm}$ in diameter, calculate the longitudinal engineering and true strains that the wire has undergone during processing.

If a material has a target hardness of $300 \mathrm{HB}$, what is the expected indentation diameter?

A metal has the following properties: UTS $=70,000 \mathrm{psi}$ and $n=0.20$. Calculate its strength coefficient, $K$.

A cable is made of two strands of different materials, $A$ and $B$, and cross-sections, as follows: For material $A, K=60,000 \mathrm{psi}, n=0.5, A_0=0.6 \mathrm{in}^2$; for material $B, K=30,000 \mathrm{psi}, n=0.5, A_o=0.3$ in $^2$. Calculate the maximum tensile force that this cable can withstand prior to necking.

You are given the $K$ and $n$ values of two different metals. Is this information sufficient to determine which metal is tougher? If not, what additional information do you need?

Consider the circumstance where a Vickers hardness test is conducted on a material. Sketch the resulting indentation shape if there is a residual stress on the surface.

Consider the circumstance where a Vickers hardness test is conducted on a material. Sketch the resulting indentation shape if there is a residual stress on the surface.

Which hardness tests and scales would you use for very thin strips of metal, such as aluminum foil? Explain.

Determine whether each first-order differential equation is separable, linear; both, or neither:

$1.$ $\frac{dy}{dx} + e^x y = x^2y^2$

$2.$ $y+\sin x=x^3 y'$

$3.$ $\ln x - x^2y = x y'$

$4.$ $\frac{dy}{dx} + \cos y = \tan x$

What role, if any, does friction play in a hardness test? Explain.

Consider an elastomer, such as a rubber band. This material can undergo a large elastic deformation before failure, but after fracture it recovers completely to its original shape. Is this material brittle or ductile? Explain.