Materials testing

The photograph shows a universal testing machine set up for a tensile test on a metallic test specimen, with its four main components labelled. Describe the function of each element in the key: which applies the load and how it is transmitted to the test specimen, which holds it, and which measures its elongation. Also explain why the specimen is shaped like a "dumbbell", narrower in its central region.

In a tensile test, stress $\sigma$ and strain $\varepsilon$ are obtained. State the units of each in the International System of Units and explain why $\varepsilon$ is dimensionless and often expressed as a percentage.

In a tensile test, a cylindrical steel specimen with a diameter of $10\ \text{mm}$ and an initial length of $100\ \text{mm}$ supports a load of $20\ \text{kN}$ and elongates by $0{,}12\ \text{mm}$. a) Calculate the applied stress $\sigma = \frac{F}{S}$ in $\text{MPa}$ (remember that $1\ \text{MPa} = 1\ \text{N/mm}^2$). b) Calculate the strain $\varepsilon = \frac{\Delta L}{L_0}$. c) Reason whether this elongation would be visible to the naked eye.

The figure shows the stress-strain curve obtained when subjecting a steel test specimen to a tensile test. a) State which point on the curve is marked by the question mark and what it means. b) Explain what happens to the specimen if the load is removed from point P. c) Name the two shaded regions A and B.

Some materials do not show a clear yield point on their curve. Explain what the $0{,}2\,\%$ proof stress ($R_{p0{,}2}$) is: how it is obtained by drawing a line parallel to the elastic region with a permanent strain of $0{,}002$, and why this international convention was adopted.

During a standardized tensile test on a cylindrical metallic alloy specimen ($d_0 = 10\ \text{mm}$, $L_0 = 50\ \text{mm}$), the following pairs of force and elongation in the elastic regime are recorded: for $F_1 = 15{,}7\ \text{kN}$, $\Delta L_1 = 0{,}050\ \text{mm}$ is measured, and for $F_2 = 31{,}4\ \text{kN}$, $\Delta L_2 = 0{,}100\ \text{mm}$ is measured. Upon exceeding the yield strength and reaching a load of $48{,}0\ \text{kN}$, the specimen accumulates a total engineering strain $\varepsilon = 0{,}008$, at which point the machine is fully unloaded.

1. a) Determine the elastic modulus $E$ of the material in $\text{GPa}$ and identify, with justification, whether the base material is aluminium ($E \approx 70\ \text{GPa}$), titanium ($E \approx 110\ \text{GPa}$), or steel ($E \approx 210\ \text{GPa}$).
2. b) Knowing that the unloading follows a straight line with the same initial elastic slope, predict analytically the remaining (plastic) strain $\varepsilon_p$ and the final length of the specimen after unloading.

After a specimen with an initial length of $100\ \text{mm}$ and a diameter of $12\ \text{mm}$ breaks, the two halves together measure $118\ \text{mm}$ and the diameter at the fracture area is $8\ \text{mm}$. a) Calculate the percentage elongation $A = \frac{L_f - L_0}{L_0} \cdot 100$. b) Calculate the percentage reduction of area $Z = \frac{S_0 - S_f}{S_0} \cdot 100$ using the circular cross-sections. c) Reason what both values indicate about the ductility of the material.

Two tested materials yield these curves: material 1 reaches $800\ \text{MPa}$ with an elongation of $2\,\%$ and fracture without necking; material 2 reaches $400\ \text{MPa}$ with an elongation of $25\,\%$ and marked necking. Classify them as brittle or ductile, state which absorbs more energy before breaking (area under the curve), and choose one for a crane hook, justifying your choice.

Explain what a standardised test specimen is and why it has a narrower calibrated central gauge length between two thick ends. What would happen if the test were carried out on any random bar without controlled geometry?

Describe what an indentation hardness test consists of and explain the basic difference between the Brinell, Rockwell, and Vickers methods (indenter and reading). Why do you think hardness testing is the fastest mechanical test to perform in a workshop?

Impact toughness is measured in joules. Explain what that value represents physically (energy absorbed upon fracturing the notched test specimen) and why a material can be very hard and, at the same time, have low impact toughness.

Explain what the Charpy pendulum impact test consists of: what a notched test specimen is, what the pendulum measures exactly, and what property of the material is obtained. Two steels give impact toughnesses of $5\ \text{J}$ and $60\ \text{J}$ at room temperature: which would you choose for the structure of a bridge in an area with cold winters, and why?

Two independent laboratories test specimens from the same batch of steel to approve its required minimum ductility ($A \ge 22\,\%$) and its yield strength ($R_e \ge 355\ \text{MPa}$). Laboratory A uses standardized cylindrical specimens ($L_0 = 5{,}65\sqrt{S_0}$) with generous shoulder transition radii and tests at a standardized quasi-static rate, obtaining $R_e = 360\ \text{MPa}$ and $A = 24\,\%$. Laboratory B tests non-proportional specimens with sharp, unpolished shoulder transition radii and at a strain rate ten times higher than the standard, obtaining $R_e = 398\ \text{MPa}$ and $A = 15\,\%$, ruling that the batch must be rejected for being brittle.

1. a) Justify physically and mechanically why the excessive test speed in Laboratory B overestimated the material's yield strength.
2. b) Analyze the effect that the defective fillet radii and the lack of geometric proportionality had on stress concentration and on the measurement of elongation at break $A\,\%$.
3. c) Determine with reasoning which of the two technical reports must be accepted and which principles of standardization and reproducibility have been violated.

A solid railway axle made of steel operates under rotating bending with a cyclic stress amplitude $\sigma_a = 280\ \text{MPa}$. The base steel has a tensile strength $R_m = 650\ \text{MPa}$ and a fatigue limit on a polished specimen of $\sigma_D = 325\ \text{MPa}$. However, after $10^5$ rotation cycles, the axle suffers a catastrophic transverse fracture at a shoulder where machining left surface marks and a sharp fillet radius (stress concentration factor $K_t = 1{,}7$).

1. a) Interpret the fracture mechanism and justify how it is possible that the failure occurred if the nominal working stress ($280\ \text{MPa}$) was lower than both the yield strength and the catalog fatigue limit of the material.
2. b) Technically refute the claim of an inspector who asserts: "Since steels have a fatigue limit below which they do not fail, if the axle passed the static load tests without plastic deformation, its service life should be indefinite."
3. c) Design and justify two improvements in the axle geometry and one surface treatment technique that drastically increase its fatigue resistance.

To determine the characteristic yield strength of a batch of structural steel rebar, a laboratory tests five specimens and obtains: $255$, $261$, $249$, $258$, and $302\ \text{MPa}$. The technical standard establishes that the characteristic yield strength $f_{yk}$ corresponds to the $5\,\%$ percentile, estimated according to the expression $f_{yk} = \bar{x} - 1{,}645\, s$ (where $\bar{x}$ is the sample mean and $s$ is the sample standard deviation), with the requirement for this steel being $f_{yk} \ge 240\ \text{MPa}$.

1. a) Analyze the sample: detect the value suspected of being an outlier and explain what possible anomalies in the material or in the test could give rise to such a disparate value.
2. b) Calculate the value of $f_{yk}$ considering all five specimens and recalculate it after excluding the anomalous specimen of $302\ \text{MPa}$.
3. c) Interpret the result obtained: why can the presence of a specimen with a higher yield strength paradoxically reduce the regulatory characteristic yield strength of the batch? Justify the importance of penalizing statistical dispersion in safe structural design.