Mechanical / fracture

Fracture mechanics assessment

A cracked component does not fail when the stress reaches yield — it fails when the stress intensity at the crack tip reaches the material toughness. That is a different question from strength, and it is the one that decides whether a pipeline, a bridge or a pressure vessel can stay in service with a known defect.

VerdictAcceptableKr max0.942J5.4 kJ/m²

Stress intensity

35.45 MPa√m

59.1% of the 60 MPa√m toughness

Assessment point

(0.56, 0.59)

Lr max 1.16

Utilisation

62.7%

inside the assessment line

Critical crack

28.6 mm

allowable 14.3 mm at a factor of two

Assessment working

Two ratios, one curve, one decision.

63%

Assessment working
Quantity	Value	Note
Applied stress	200 MPa	remote, away from the crack
Crack size	10.00 mm	half-length
Geometry factor Y	1.000	infinite plate, 2a
Stress intensity K_I	35.45 MPa√m	Y σ √(πa)
Fracture toughness	60 MPa√m	K_IC or K_mat
Toughness ratio Kr	0.591	K_I / K_mat
Reference stress	200 MPa	taken as the applied stress
Load ratio Lr	0.563	σ_ref / σ_Y
Lr cut-off	1.162	(σ_Y + σ_U) / 2σ_Y
Assessment curve Kr_max	0.942	BS 7910 Option 1
Utilisation	62.7%	below 100% is acceptable
Critical crack	28.65 mm	at this stress
Allowable crack	14.32 mm	half the critical, the usual inspection target
J-integral	5.45 kJ/m²	elastic estimate
CTOD	0.0153 mm	J / σ_Y

The two failure modes are separate: Kr measures toughness, Lr measures the approach to plastic collapse. A defect can be safe on one and unsafe on the other, which is why the assessment is a point on a diagram rather than a single number.

Failure assessment diagram

Inside the curve is acceptable. The point is the assessment; the axes are the two ways a cracked body fails.

Driving force against crack size

K_I rises with the square root of the crack. Where it crosses the toughness is the critical size.

Canonical equations

The exact relationships implemented by this engine.

K_I = Y σ √(πa) · Y depends on geometry and loading.

Kr = K_I / K_mat · Below 1 means the crack does not propagate.

Lr = σ_ref / σ_Y · Capped at (σ_Y + σ_U)/2σ_Y.

Kr_max = (1 − 0.14Lr²)(0.3 + 0.7e^{−0.65Lr⁶}) · BS 7910.

Show model notation and computationOpen the symbolic form, the variable definitions, and the step-by-step derivation when you need the detail.

A clear path from inputs to results.

A cracked component does not fail when the stress reaches yield — it fails when the stress intensity at the crack tip reaches the material toughness. That is a different question from strength, and it decides whether a pipeline, a bridge or a pressure vessel can stay in service with a known defect.

Step-by-step methodWorked exampleAssumptions explained

The formula

K_I = Y\,\sigma\sqrt{\pi a} \;;\; K_r = \frac{K_I}{K_{mat}} \;;\; L_r = \frac{\sigma}{\sigma_Y}

Reference: BS 7910:2019, Guide to methods for assessing the acceptability of flaws in metallic structures, Option 1.

Worked example

A 10 mm centre crack under 200 MPa

Inputs

200 MPa applied stress, 10 mm half-length centre crack.
Toughness 60 MPa√m, yield 355 MPa, ultimate 470 MPa.

Calculation path

1
The geometry factor for a centre crack in an infinite plate is 1.0.
2
The stress intensity is 200 × √(π × 0.01) = 35.45 MPa√m.
3
That is 59.1% of the 60 MPa√m toughness, so Kr is 0.591.
4
The load ratio is 200/355 = 0.563, and the cut-off is (355+470)/2/355 = 1.16.
5
The Option 1 curve at Lr 0.563 gives Kr max 0.942, so the assessment uses 62.7% of the allowance and the defect is acceptable.
6
The critical crack at this stress is 28.6 mm, so the allowable is 14.3 mm.

Result

K_I 35.45 · Kr 0.591 · 62.7% utilised · critical crack 28.6 mm.

Interpreting the result

The two axes are two different failure modes: Kr measures the approach to brittle fracture and Lr the approach to plastic collapse. A defect can be safe on one and unsafe on the other, which is why the answer is a point on a diagram rather than a single number — and why a toughness test matters more than the stress calculation.

Frequently asked questions

What is the difference between Kr and Lr?

Kr is the ratio of the applied stress intensity to the material toughness; Lr is the ratio of the reference stress to yield. One measures toughness exhaustion, the other the approach to plastic collapse.

Can I use this for a weld?

Only with the weld’s own toughness and a level-2 assessment that accounts for secondary stress and the strength mismatch. Option 1 on a weld is a screening check, not a design.

Why is the critical crack so sensitive to toughness?

Because K_I is linear in toughness but the critical size goes with its square. Halving the toughness quarters the critical defect size.

What is the assessment curve for?

It defines the combination of Kr and Lr at which failure occurs. Inside the curve is acceptable; the further outside, the more certain the failure.