A steel component has a crack of length 4 mm and is operating such that K_I = 0.8 K_c. The crack grows to 16 mm while the applied stress and geometry factor Y remain unchanged. What is the approximate new value of K_I?
CK_I = 1.6 K_c — K_I scales as √a, so it doubles when crack length quadruples; fracture occurs
DK_I = 1.13 K_c — K_I scales as the fourth root of crack length
K_I = Yσ√(πa), so K_I scales as √a. When crack length goes from 4 mm to 16 mm (a factor of 4), K_I increases by √4 = 2. The new K_I = 2 × 0.8 K_c = 1.6 K_c, which exceeds K_c — the component fractures. This square-root dependence is the key: cracks grow more dangerous faster than linearly. The common wrong answer (tripling or quadrupling) treats K as linearly proportional to crack length, which is incorrect.
Question 2 Multiple Choice
How does the stress intensity factor K differ fundamentally from the stress concentration factor K_t?
AK applies only to Mode I loading; K_t applies to all three fracture modes
BK has units of MPa√m and predicts fracture by comparison to K_c; K_t is dimensionless and describes local stress amplification from elasticity theory without predicting fracture
CK_t is the material property; K is the applied quantity — they are two halves of the same fracture criterion
DK measures crack tip displacement; K_t measures the stress ratio between tip and nominal stress
These two quantities are frequently confused because both involve 'K' and both relate to stress near a notch or crack. K_t (stress concentration factor) is a dimensionless ratio from linear elasticity — it tells you how much higher the local stress is than the nominal stress, but makes no fracture prediction. K (stress intensity factor) has dimensions of MPa√m and is compared to K_c to predict whether fracture will occur. A material without a crack can have a high K_t at a notch and still not fracture; a cracked material fails when K_I = K_c.
Question 3 True / False
According to K = Yσ√(πa), doubling the applied stress increases K by the same multiplicative factor as doubling the crack length.
TTrue
FFalse
Answer: False
Doubling the applied stress σ doubles K (linear relationship). Doubling the crack length a increases K by √2 ≈ 1.41 (square-root relationship). These are different factors. A doubling of stress is more dangerous per unit change than a doubling of crack length. This distinction matters for damage tolerance design: you can sometimes reduce stress levels more easily than eliminating cracks, and the calculations must respect the correct scaling.
Question 4 True / False
A component can fracture even when the average applied stress is well below the material's yield strength, if a crack is present and K_I reaches K_c.
TTrue
FFalse
Answer: True
This is the central insight of fracture mechanics. The stress field near a crack tip is singular — it rises steeply regardless of the average stress — and the material fails when this local intensity (K_I) reaches the material's fracture toughness (K_c). An aircraft component with a small crack can fail at loads that would be safe for an uncracked specimen. This is why damage-tolerance design evaluates components based on their crack state, not just their nominal stress levels.
Question 5 Short Answer
Explain why detecting a crack at 1 mm is disproportionately more valuable than detecting a crack at 16 mm, even if both are below the critical size at current stress levels.
Think about your answer, then reveal below.
Model answer: Because K_I scales as √a, the stress intensity at a 16 mm crack is √(16/1) = 4 times that at a 1 mm crack under the same stress and geometry. A crack that is 'safe' at 1 mm may reach K_c after modest growth. More importantly, the rate of danger accumulation is front-loaded: going from 1 mm to 4 mm quadruples crack length but only doubles K; going from 4 mm to 16 mm does the same multiplication. Each size doubling adds less marginal danger, but small cracks that are missed grow into the dangerous range through normal operation. Early detection gives the most intervention time before K_I approaches K_c.
The square-root scaling means that the marginal increase in K per unit of crack growth decreases as the crack gets longer. But the absolute level of K is still much lower at small sizes. Detecting at 1 mm and arresting growth or reducing stress gives a much larger safety margin than detecting at 16 mm when the component may already be near fracture. This is the quantitative basis for inspection interval design in damage-tolerant structures.