Creep and Fatigue Test Procedures; Three Definitions of Toughness: 21-Mat-A5 Phase Transformations and Thermal Treatment, May 2017, Q2

Question 2: Creep and Fatigue Test Procedures; Three Definitions of Toughness (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

2.1 — (a) Test procedures and how the data are presented

The material chosen here is a 2.25Cr–1Mo low-alloy steel, used for elevated-temperature pressure parts, which is subject to both modes in service.

Creep testing (ASTM E139). A round tensile specimen with threaded or shouldered ends is placed in a lever-arm frame and dead-weight loaded, so that the load stays constant while the specimen extends. It sits inside a three-zone split furnace controlled to within about $\pm 2$ °C, with thermocouples in contact with the gauge length, and is soaked at temperature for long enough to stabilise before loading. Extension is measured on the gauge length itself by a high-temperature extensometer with rods passing out of the furnace to a linear transducer, and strain is logged continuously for the duration of the test — hours for a screening test, tens of thousands of hours for design data. Two variants matter: a true creep test records strain against time and is usually stopped at a defined strain, while a stress-rupture test is run to failure and reports only the rupture life and the elongation. Because the load is constant, the true stress rises as the section necks; a servo-controlled constant-stress test removes that artefact where the mechanism is under study.

Fatigue testing (ASTM E466). A polished specimen — surface finish matters enormously, since fatigue starts at the surface — is cycled at constant amplitude either in rotating bending (the classical R. R. Moore machine, giving fully reversed loading at $R=-1$) or, more usefully for design, in axial load control on a servo-hydraulic frame where the mean stress and the ratio $R = \sigma_{\min}/\sigma_{\max}$ can be set independently. The specimen is cycled at 10–100 Hz until it separates, and the number of cycles is recorded. Since a single test yields one data point, a whole population of specimens is tested at a series of stress amplitudes, with run-outs (typically $10^{7}$ cycles) recorded as arrows. Crack-growth data, by contrast, come from a different test — a pre-cracked compact-tension specimen cycled under load control with the crack length followed by compliance or potential drop (ASTM E647) — and yield the Paris-law constants used in Question 3.

Left: a creep curve — strain against time at fixed stress and temperature, showing the instantaneous strain $\varepsilon_0$, the decelerating primary stage I, the steady-state secondary stage II whose slope is the minimum creep rate, and the accelerating tertiary stage III ending in rupture. Right: a fatigue curve — stress amplitude against the logarithm of the number of cycles to failure, showing the sharp knee and horizontal endurance limit typical of a ferrous alloy and the continuously falling curve typical of aluminium.

How the two representations differ. The contrast is instructive and is the point of the comparison. One creep test produces an entire curve — strain against time on linear axes for one combination of stress and temperature — and a family of such curves is needed to cover the design space; the number pulled out for design use is the minimum (secondary) creep rate $\dot{\varepsilon}_{\min}$ or the time to 1 per cent strain or to rupture, and those are then cross-plotted against stress on log–log axes or collapsed onto a Larson–Miller parameter $P = T(C + \log t_r)$. One fatigue test, in contrast, produces a single point, and the curve emerges only from a population; the axes are stress amplitude against $\log N$, the scatter is wide (a factor of five in life at a given stress is unremarkable), so the curve is drawn as a mean with a survival probability attached. Creep data are deterministic and time-based; fatigue data are statistical and cycle-based.

Concept & Theory

Topic. Standardised creep and fatigue test procedures, and the different nature of the data each produces.

Key relations. Minimum creep rate $\dot{\varepsilon}_{\min}=A\sigma^{n}\exp(-Q_c/RT)$; Larson–Miller parameter $P=T(C+\log t_r)$; stress ratio $R=\sigma_{\min}/\sigma_{\max}$.

Why this works. A creep test asks how much a single specimen deforms as time passes under a fixed load; a fatigue test asks how many repetitions of a fixed load a population survives. Because the underlying processes — diffusion-driven deformation versus cycle-by-cycle crack initiation and growth — are different, so are the natural axes and the statistical character of the resulting data.

Common pitfall. Drawing a creep curve on logarithmic time axes, which hides the three distinct stages; or presenting a single fatigue specimen's result as though it were the whole S–N curve, when a population is required.

Source. ASTM E139, E466; Dieter, Mechanical Metallurgy, 3rd ed., Ch. 12–13.

2.2 — (b) Three meanings of “toughness”

The word “toughness” is used for three different quantities, with three different units, and the examiner is asking for all three to be separated cleanly. What they share is that each is an energy, not a stress; what distinguishes them is the volume or area over which that energy is reckoned, and how much of the material takes part.

(i) Toughness in elastic deformation — the modulus of resilience. If the deformation is entirely elastic, the only energy the material can absorb is the recoverable strain energy stored in the stretched bonds. Per unit volume this is the area under the elastic part of the stress–strain curve up to yield,

$$U_r \;=\; \int_0^{\varepsilon_y}\sigma\,\mathrm{d}\varepsilon \;=\; \tfrac{1}{2}\sigma_y\varepsilon_y \;=\; \frac{\sigma_y^{2}}{2E}$$

with units of joules per cubic metre (equivalently, pascals). It is the property that matters for a spring: the design goal is to store and return as much energy as possible without permanent set, so a spring material wants a high $\sigma_y$ and a low $E$, which is why hard-drawn spring steel, beryllium copper and fibre-reinforced polymers all appear in leaf and coil springs. Note that resilience says nothing about resistance to failure — it measures energy stored, all of which is given back.

(ii) Toughness in plastic deformation — tensile toughness or the work of fracture. Once the material yields, the energy going into it is largely dissipated rather than stored, and the relevant measure is the total area under the nominal stress–strain curve to fracture,

$$U_T \;=\; \int_0^{\varepsilon_f}\sigma\,\mathrm{d}\varepsilon \;\approx\; \left(\frac{\sigma_y+\sigma_{\text{UTS}}}{2}\right)\varepsilon_f$$

again in joules per cubic metre. Because $\varepsilon_f$ can be tens of per cent while $\varepsilon_y$ is a fraction of one per cent, plastic toughness is typically two or three orders of magnitude larger than resilience — it is dominated by ductility, not by strength. This is the quantity a Charpy or Izod impact test estimates (reported as an absorbed energy in joules rather than per unit volume, since the deforming volume is fixed by the specimen), and it is what “tough” means in the everyday engineering sense of a material that gives warning, absorbs a crash, and deforms rather than shattering. It also embodies the classic strength–ductility trade-off: every strengthening mechanism in Question 4 raises $\sigma_y$ and lowers $\varepsilon_f$, so the product passes through a maximum.

(iii) Toughness in fast fracture — fracture toughness. Neither of the first two definitions helps once a sharp crack is present, because then almost all of the material is elastic and only a small process zone at the crack tip does any work. The correct measure is the energy dissipated per unit area of new crack surface, the toughness or critical strain-energy release rate $G_c$, in joules per square metre. Its stress-based equivalent is the critical stress-intensity factor,

$$K \;=\; Y\sigma\sqrt{\pi a}, \qquad K = K_c \ \text{at fracture}, \qquad G_c \;=\; \frac{K_c^{2}}{E'}$$

with $E'=E$ in plane stress and $E/(1-\nu^{2})$ in plane strain; $K_c$ has the unusual units MPa m${}^{1/2}$. Under plane-strain constraint the value falls to a geometry-independent minimum, $K_{Ic}$, which is the number quoted as a material property (ASTM E399). This is the toughness that governs the fast, unstable fracture of a cracked structure, and — as Question 3(a) of this paper shows — it also fixes the crack length at which a fatigue crack stops growing stably and runs.

Concept & Theory

Topic. The three energy-based definitions of toughness and the design contexts each governs.

Key relations. Resilience $U_r=\sigma_y^{2}/2E$ (J m$^{-3}$); tensile toughness $U_T=\int_0^{\varepsilon_f}\sigma\,d\varepsilon$ (J m$^{-3}$); fracture toughness $K=Y\sigma\sqrt{\pi a}$, $G_c=K_c^{2}/E'$ (J m$^{-2}$).

Why this works. Each definition answers a different question about how much energy a material can absorb before it stops resisting further deformation or fracture, and the correct one to use is set entirely by whether the material is defect-free and behaving elastically, defect-free and behaving plastically, or already cracked.

Common pitfall. Using the word “toughness” loosely without specifying which of the three is meant, or quoting resilience (a small number, since $\varepsilon_y$ is tiny) when the design question is really about impact energy or fracture toughness.

Source. Callister & Rethwisch, Materials Science and Engineering, 10th ed., §6.6 (resilience and toughness), §8.5 (fracture toughness); Dieter, Mechanical Metallurgy, 3rd ed., Ch. 12.9–12.11.

Question 2 of 8: Creep and Fatigue Test Procedures; Three Definitions of Toughness

Question 2: Creep and Fatigue Test Procedures; Three Definitions of Toughness (20 marks)

2.1 — (a) Test procedures and how the data are presented

2.2 — (b) Three meanings of “toughness”