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Rotor Fracture Analyzer

Assess fracture risk and fatigue crack growth in fiber-reinforced polymer rotors under centrifugal loading. Combines LEFM stress intensity analysis with Paris-law crack growth life prediction.

Tool Purpose & README

About This Tool

The Rotor Fracture Analyzer evaluates two tightly coupled failure modes for rotating composite components:

  • Static fracture assessment — stress intensity factor KI vs fracture toughness KIC, critical crack size
  • Fatigue crack growth — Paris law integration (da/dN = C(ΔK)m) to determine remaining life

When to Use

  • Early-stage damage tolerance assessment of composite flywheel rotors, centrifuge components, or turbine disks
  • Estimating inspection intervals for rotating components with known defects
  • Comparing material choices (CFRP, GFRP, Aramid, filled polymers) for crack resistance

Limitations

  • Assumes linear-elastic fracture mechanics (LEFM) — small-scale yielding only
  • Uses simplified geometry factors; does not account for complex 3D stress fields
  • Composite anisotropy is not modeled; properties represent effective isotropic behavior
  • Paris law constants are approximate; use test data for critical applications

Inputs

Expert Mode Show all parameters
Sample Test Cases
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Required Safety Factor ? 1.5
1.0 (Minimum) 4.0 (Conservative)

Crack Orientation

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Material Properties

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Material Knockdown Factors

UTS Knockdown ? 0%
0%80%
KIC Knockdown ? 0%
0%80%

Paris Law Parameters

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Loading & Design

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Fracture Assessment

Enter inputs and press Calculate to see results.

Stress Intensity Factor
1. General Form
$$ K_I = Y \sigma \sqrt{\pi a} $$
2. Substituted Values
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Fracture Safety Factor
1. Definition
$$ SF = \frac{K_{IC}}{K_I} $$
2. Substituted Values
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Critical Crack Size
1. Definition
$$ K_I(a_{cr}) = K_{IC} $$
2. Result
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Centrifugal Hoop Stress
1. Annular Disk (Shigley)
$$ \sigma_\theta = \frac{3+\nu}{8} \rho \omega^2 \left(r_i^2 + r_o^2 + \frac{r_i^2 r_o^2}{r^2}\right) - \frac{1+3\nu}{8} \rho \omega^2 r^2 $$
2. Solid Disk (Shigley)
$$ \sigma_\theta = \frac{3+\nu}{8} \rho \omega^2 r_o^2 - \frac{1+3\nu}{8} \rho \omega^2 r^2 $$
3. Thin Ring
$$ \sigma_\theta = \rho \omega^2 r_m^2 $$
4. Substituted Values
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Fatigue Crack Growth

Paris Law Integration
1. Paris Law
$$ \frac{da}{dN} = C (\Delta K)^m $$
2. Stress Range
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3. Initial Stress Intensity Range
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4. Initial Growth Rate
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5. Numerical Integration
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Because Y(a) changes with crack size, the integral is evaluated numerically using adaptive forward-Euler stepping from a0 to acr.

Life Fraction Used
1. Definition
$$ \text{Life fraction} = \frac{N_{design}}{N_f} $$

The life fraction compares your required service life (Ndesign) to the predicted crack-growth life (Nf). It answers: what proportion of the component's total crack-growth life does the design life consume?

2. Interpretation
  • ≤ 50% — Ample margin. The crack would need more than twice the design life to reach critical size. Standard inspection schedules are sufficient.
  • 50 – 100% — Marginal. The design life approaches or equals the predicted crack-growth life. Increase inspection frequency and consider reducing operating speed or accepting a smaller initial flaw size.
  • > 100% — Insufficient life. The crack is predicted to reach critical size before the design life is reached. The component cannot meet the service target without design changes.
3. Substituted Values
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Enter inputs and press Calculate to see crack growth results.

Rotor Cross-Section

KI vs Crack Size

What this shows: How the stress intensity factor increases with crack size. The horizontal dashed line is KIC (fracture toughness). Where KI crosses KIC, the crack becomes critical.
Contents
  1. Overview
  2. How to Think About These Problems
  3. Centrifugal Stress in Rotating Disks
  4. Fracture Mechanics Equations
  5. Fatigue Crack Growth
  6. Material Properties
  7. Limitations & Assumptions
  8. References

1. Overview

This tool applies Linear Elastic Fracture Mechanics (LEFM) to assess whether a pre-existing crack in a rotating composite component will propagate to failure. It addresses a common question in rotating machinery design: given a known or assumed defect, is the component safe to operate, and for how long?

Real composite components — flywheel rims, compressor impellers, centrifuge rotors, turbine blades — inevitably contain manufacturing defects: voids, delaminations, fiber misalignment, or matrix micro-cracks. Rather than assuming these defects do not exist, damage-tolerant design accepts their presence and asks whether they will grow to a dangerous size within the service life.

Static fracture check: Compares the applied stress intensity factor KI to the material's fracture toughness KIC. If KI ≥ KIC, immediate brittle fracture is predicted.
Fatigue crack growth: Integrates the Paris law da/dN = C(ΔK)m from the initial crack size to the critical crack size, yielding the number of load cycles to failure and a recommended inspection interval.

2. How to Think About These Problems

A fracture mechanics assessment follows a logical chain. Each link feeds the next, and understanding the chain helps you interpret the results and know which inputs matter most.

Step 1: Determine the Stress Field

Before asking whether a crack is dangerous, you need to know the stress at the crack location. For a rotating disk, centrifugal body forces produce a stress field that varies with radius. The hoop (tangential) stress is usually the dominant driver because it acts to open radial cracks — the most common orientation in rotors. This tool computes hoop and radial stress from the Shigley closed-form rotating-disk equations (see Section 3).

Step 2: Compute the Stress Intensity Factor

The stress intensity factor KI quantifies how severely a crack amplifies the surrounding stress field. It depends on three things: the far-field stress σ, the crack size a, and a dimensionless geometry correction factor Y that accounts for the crack shape and the component's finite width. A larger crack or higher stress both increase KI.

Step 3: Compare to Fracture Toughness

The material's fracture toughness KIC is the critical value of KI at which unstable (fast) fracture occurs. The ratio KIC / KI is the fracture safety factor. Values above the required target (commonly 2.0–3.0 for rotating equipment) indicate the crack is stable at its current size. A safety factor below 1.0 means the crack has already reached or exceeded the critical condition.

Step 4: Predict Crack Growth Life

Even when KI < KIC, cyclic loading causes the crack to grow slowly. The Paris law describes this sub-critical growth rate. By integrating from the initial crack size to the critical size, you obtain the number of cycles to failure Nf. Dividing Nf by an appropriate safety factor (typically 3) yields a recommended inspection interval — the point at which you should re-inspect the component via NDT to verify the crack has not exceeded expectations.

What Drives the Answer?

Sensitivity insight: KI scales with √a, so doubling the crack size increases KI by about 40%. Hoop stress scales with ω² (and therefore RPM²), so a modest speed increase can dramatically raise KI. When in doubt, the two most impactful inputs are rotational speed and initial crack size.
Choosing the initial crack size: Use the largest defect that could escape your inspection process. For ultrasonic NDT on composites, a typical detectable flaw size is 0.5–2 mm depending on the method and component geometry. If no inspection is planned, assume worst-case manufacturing defects from coupon testing or industry standards.

3. Centrifugal Stress in Rotating Disks

A spinning disk experiences centrifugal body forces that produce both hoop (tangential) and radial stress components. These stresses depend on the disk geometry, material density, Poisson's ratio, and angular velocity. The closed-form solutions below assume a uniform-density, constant-thickness disk (Shigley, Ch. 3).

Annular Disk — Hoop Stress $$ \sigma_\theta = \frac{3+\nu}{8}\,\rho\,\omega^2 \!\left(r_i^2 + r_o^2 + \frac{r_i^2\,r_o^2}{r^2}\right) - \frac{1+3\nu}{8}\,\rho\,\omega^2\,r^2 $$
  • ri, ro — inner and outer radii
  • r — radius at which stress is evaluated
  • ν — Poisson's ratio
  • ρ — material density
  • ω = 2π · RPM / 60 — angular velocity (rad/s)
Annular Disk — Radial Stress $$ \sigma_r = \frac{3+\nu}{8}\,\rho\,\omega^2 \!\left(r_i^2 + r_o^2 - \frac{r_i^2\,r_o^2}{r^2} - r^2\right) $$
Solid Disk (ri = 0) $$ \sigma_\theta = \frac{3+\nu}{8}\,\rho\,\omega^2\,r_o^2 - \frac{1+3\nu}{8}\,\rho\,\omega^2\,r^2 $$ $$ \sigma_r = \frac{3+\nu}{8}\,\rho\,\omega^2 \!\left(r_o^2 - r^2\right) $$
Thin Ring $$ \sigma_\theta = \rho\,\omega^2\,r_m^2 \qquad\text{where}\quad r_m = \tfrac{1}{2}(r_i + r_o) $$

The thin-ring approximation ignores radial variation and is valid when the wall thickness is small relative to the mean radius (t/rm < 0.1).

Key behavior: hoop stress in an annular disk is highest at the inner bore and decreases toward the outer rim. Radial stress peaks somewhere in the interior and is zero at both the inner and outer free surfaces. For crack assessment, the hoop stress at the crack's radial location is what drives KI for radial cracks.

4. Fracture Mechanics Equations

Stress Intensity Factor $$ K_I = Y \,\sigma \sqrt{\pi a} $$
  • Y — geometry correction factor (depends on crack type and a/W ratio)
  • σ — far-field centrifugal hoop stress at the crack location
  • a — crack half-length (through/embedded) or depth (edge/surface)
  • W — ligament width (component thickness for radial cracks, radial width for circumferential)

KI characterizes the singular stress field near the crack tip. It is the single parameter that governs whether a crack will extend — regardless of the specific loading and geometry that produced it.

Fracture Safety Factor $$ SF = \frac{K_{IC}}{K_I} $$
  • KIC — mode-I plane-strain fracture toughness (a material property)
  • SF ≥ required value ⇒ acceptable; SF < 1 ⇒ fracture predicted

The required safety factor depends on consequence of failure and confidence in inputs. Typical values: 2.0 for well-characterized materials with regular inspection, up to 4.0 for critical applications with limited testing data.

Critical Crack Size $$ K_I(a_{cr}) = K_{IC} \quad\Rightarrow\quad a_{cr} = \frac{1}{\pi}\!\left(\frac{K_{IC}}{Y\,\sigma}\right)^{\!2} $$

The critical crack size is solved iteratively because the geometry factor Y itself depends on crack size. The simplified form above applies when Y is approximately constant.

Geometry Factors Y(a/W)

The geometry factor corrects KI for the finite size of the component and the shape of the crack. As a/W approaches 1 (crack nearly through the full ligament), Y increases rapidly, reflecting the stress amplification from the diminishing remaining material.

Through Crack (Feddersen/Tada) $$ Y = \sqrt{\sec\left(\frac{\pi a}{2W}\right)} $$

Valid for a central crack of total length 2a in a plate of width 2W under uniform tension. Accuracy within 0.3% for a/W < 0.35.

Edge Crack (Tada polynomial) $$ Y = 1.12 - 0.231\frac{a}{W} + 10.55\left(\frac{a}{W}\right)^2 - 21.72\left(\frac{a}{W}\right)^3 + 30.39\left(\frac{a}{W}\right)^4 $$

Single edge crack of depth a in a plate of width W. The leading coefficient 1.12 reflects the free-surface correction. Valid for a/W < 0.7.

Surface Crack (Newman-Raju, a/c = 1 fixed) $$ Y = \frac{F}{\sqrt{Q}}, \quad F = (M_1 + M_2 \alpha^2 + M_3 \alpha^4)\,f_w $$

Semicircular surface flaw (a/c = 1) using the full Newman-Raju model. Y ≈ 0.66 at small a/W, increasing as the crack approaches the back face.

Embedded Elliptical Crack (Irwin/Green-Sneddon) $$ Y = \frac{1}{\sqrt{Q}}, \quad Q = 1 + 1.464\left(\frac{a}{c}\right)^{1.65} $$

Embedded elliptical crack. At a/c = 1 (penny-shaped), Y ≈ 2/π ≈ 0.637. Shape factor Q accounts for crack aspect ratio; elongated cracks (a/c < 1) are more severe.

Elliptical Surface Crack (Newman-Raju, NASA TM-85793) $$ Y = \frac{F}{\sqrt{Q}} $$ $$ Q = 1 + 1.464\left(\frac{a}{c}\right)^{1.65} \quad\text{for } a/c \le 1 $$ $$ F = (M_1 + M_2\,\alpha^2 + M_3\,\alpha^4)\,f_w $$
  • a — crack depth; c — half surface length; α = a/W
  • M1 = 1.13 − 0.09(a/c)
  • M2 = −0.54 + 0.89 / (0.2 + a/c)
  • M3 = 0.5 − 1/(0.65 + a/c) + 14(1 − a/c)24
  • fw = √(sec(√α · π/2)) — finite-width correction

Applicability: Most common real-world flaw shape. Manufacturing defects, porosity, delamination origins. At a/c = 1, matches the simplified surface crack model.

Examples: Composite layup voids, fiber breakage zones, impact damage sites.

Corner Crack (Quarter-Elliptical) $$ Y_{corner} = Y_{surface} \times (1.1 + 0.1\,\alpha) $$
  • Ysurface from the Newman-Raju elliptical surface formula above
  • α = a/W — crack depth ratio
  • Corner correction factor accounts for the additional free surface at the corner

Applicability: Cracks initiating at geometric transitions — holes, fillets, shoulders. Y is 10–20% higher than the equivalent surface crack due to the additional free-surface effect.

Examples: Bolt holes in composite flanges, corner of rectangular cutouts, fillets at disk-to-shaft transitions.

Double Edge Crack (Tada/Isida) $$ Y = 1.122 - 0.561\frac{a}{W} - 0.205\left(\frac{a}{W}\right)^2 + 0.471\left(\frac{a}{W}\right)^3 - 0.190\left(\frac{a}{W}\right)^4 $$
  • a — crack depth (each side); W — half-width of specimen
  • Valid for a/W < 0.7

Applicability: Symmetric edge flaws in flat specimens or thin sections. Better validity range than single-edge for symmetric loading configurations.

Examples: Double-notch tensile specimens, symmetric machining damage, edge delaminations on both sides of a composite panel.

5. Fatigue Crack Growth

Even when KI is well below KIC, cyclic loading causes the crack to extend incrementally with each load cycle. The Paris-Erdogan law describes the stable (Region II) growth rate as a power-law function of the stress intensity factor range:

Paris Law $$ \frac{da}{dN} = C \left(\Delta K\right)^m $$
  • C, m — experimentally determined material constants
  • ΔK = Kmax − Kmin = (1 − R) · Kmax
  • R = σmin / σmax — stress ratio

The total number of cycles to failure is obtained by integrating the Paris law from the initial crack size a0 to the critical crack size acr:

Cycles to Failure $$ N_f = \int_{a_0}^{a_{cr}} \frac{da}{C\left(\Delta K(a)\right)^m} $$

This integral is evaluated numerically because ΔK(a) = Y(a) Δσ √(πa) changes with crack size through both the √a term and the geometry factor Y(a).

Inspection Interval

The recommended inspection interval is Nf / 3. This factor of 3 provides margin for uncertainties in the Paris law constants, crack measurement accuracy, and load estimation. In practice, the interval should be validated against the specific NDT method's probability of detection (POD) curve and any regulatory requirements.

Stress Ratio Effects

The stress ratio R = σmin / σmax describes the cyclic loading range. In this tool, σmax is the centrifugal stress at the crack location at operating speed — specifically the hoop stress for radial cracks or the radial stress for circumferential cracks, as computed from the rotating disk equations. The minimum stress σmin = R × σmax represents the stress at the lowest speed in the duty cycle.

For a rotor that starts from rest each cycle, R = 0 (zero-to-max loading) and the full stress range drives crack growth: Δσ = σmax. If the rotor maintains a minimum idle speed, R > 0 and the effective stress range is reduced to Δσ = σmax(1 − R), which lowers ΔK and slows crack growth. This tool uses the basic Paris model with ΔK = Y Δσ √(πa).

Loading Frequency Limitation

The Paris law is purely cycle-based — it counts cycles (da/dN), not time. The rate at which stress is applied (how fast the rotor spins up) does not appear in any equation and has no effect on the results. In reality, loading frequency can matter for polymer composites:

  • Viscoelastic rate sensitivity — Polymers are stiffer and more brittle at high strain rates, which shifts the effective KIC and Paris constants.
  • Environmental interaction — At lower frequencies, each cycle gives more time for moisture or chemical attack at the crack tip (corrosion fatigue), accelerating growth.
  • Hysteretic heating — At very high frequencies, internal heat generation in polymers can soften the crack-tip material and alter fracture behavior.

The preset Paris constants in this tool are representative of moderate test frequencies (1–10 Hz) in laboratory air. If the actual duty cycle involves very slow thermal cycling (hours per cycle) or high-frequency vibration (hundreds of Hz), the constants C and m may not be applicable. Accounting for this requires frequency-dependent crack growth models, which are beyond the scope of this tool.

Vibratory Fatigue (Not Modeled)

This tool models low-cycle fatigue from start-stop centrifugal loading only. In real rotating machinery, cracks can also be driven by vibratory stresses — alternating loads from resonances, blade-passing excitation, or rotor imbalance — superimposed on the steady centrifugal stress. In that scenario the cyclic stress parameters become:

  • σmax = σcentrifugal + σvibratory
  • σmin = σcentrifugal − σvibratory
  • R = σmin / σmax

Vibratory fatigue often dominates crack growth even when the alternating stress amplitude is small, because cycle counts can reach millions per hour at resonant frequencies compared to thousands of start-stop cycles over a service life. Estimating σvibratory typically requires modal finite-element analysis or strain-gauge test data, which is beyond the scope of this tool.

6. Material Properties

Preset materials and their fracture properties used in this tool:

Interpreting Material Properties

  • KIC (fracture toughness) — Higher is better. Carbon fiber composites have substantially higher toughness than short-fiber-filled polymers, reflecting the energy absorbed by fiber pullout and bridging during crack propagation.
  • Paris C — Sets the baseline crack growth rate. Smaller values (e.g., 1e-10) mean slower growth per cycle. C has units that depend on m, so values cannot be compared across different m exponents.
  • Paris m — Controls the sensitivity of growth rate to ΔK. Higher m (4–6 for polymers) means crack growth accelerates more steeply as the crack lengthens, leaving less warning before failure.
  • ρ (density) — Directly affects centrifugal stress. Lower density is advantageous for rotors because it reduces the body force at any given speed.
  • σuts (ultimate tensile strength) — Used for reference only in this tool; the fracture assessment is entirely K-based, not strength-based.
Note: Paris law constants are approximate and can vary significantly with fiber orientation, temperature, frequency, and environment. Always use test data for critical applications. The values here are representative of room-temperature, laboratory-air conditions at moderate frequencies (1–10 Hz).

7. Limitations & Assumptions

  • LEFM validity: Assumes small-scale yielding — the plastic (or damage) zone at the crack tip must be small relative to the crack length, ligament width, and component thickness. Most fiber-reinforced composites satisfy this condition due to their brittle matrix behavior.
  • Idealized crack geometry: Geometry factors are for canonical crack shapes (through, edge, surface, embedded, elliptical surface, corner, double edge) in flat plates. Real defects in curved rotors may differ; consult FEA-based K solutions for critical designs.
  • Isotropic material model: Composite materials are treated as effectively isotropic. This is a reasonable first approximation for quasi-isotropic layups or short-fiber composites, but underestimates the directional dependence in strongly anisotropic layups (e.g., unidirectional or hoop-wound composites where delamination toughness differs from transverse toughness).
  • Constant-amplitude loading: The Paris law integration assumes each cycle applies the same Δσ. Variable-amplitude loading (e.g., different operating speeds) requires cycle-counting methods such as rainflow counting and Miner's rule.
  • No threshold or closure effects: The tool does not model the fatigue threshold ΔKth below which cracks do not grow, nor crack closure effects that reduce the effective ΔK. This makes predictions conservative for low-R, low-ΔK conditions.
  • Uniform disk geometry: Centrifugal stress uses Shigley closed-form equations for uniform-density, constant-thickness disks. Profiled disks, bladed rotors, or non-uniform density distributions require FEA.
  • No environmental effects: Stress corrosion cracking, moisture absorption in composites, and elevated-temperature effects are not modeled. These can significantly accelerate crack growth in service.

8. References

  • T.L. Anderson, Fracture Mechanics: Fundamentals and Applications, 4th ed., CRC Press, 2017 — Comprehensive treatment of LEFM, EPFM, and fatigue crack growth. Chapters 2 and 10 are directly relevant.
  • H. Tada, P.C. Paris & G.R. Irwin, The Stress Analysis of Cracks Handbook, 3rd ed., ASME Press, 2000 — Standard reference for stress intensity factor solutions by crack/geometry type.
  • P. Paris & F. Erdogan, "A Critical Analysis of Crack Propagation Laws", J. Basic Eng., vol. 85, pp. 528–534, 1963 — Original paper establishing the power-law fatigue crack growth relationship.
  • J.C. Newman & I.S. Raju, "Stress Intensity Factor Equations for Cracks in Three-Dimensional Finite Bodies Subjected to Tension and Bending Loads", NASA TM-85793, 1984 — Parametric SIF equations for surface and corner cracks widely used in aerospace damage tolerance.
  • R.G. Budynas & J.K. Nisbett, Shigley's Mechanical Engineering Design, 11th ed., McGraw-Hill, 2020 — Rotating disk stress equations (Ch. 3) used for centrifugal hoop and radial stress computation.
  • ASTM E399-22, "Standard Test Method for Linear-Elastic Plane-Strain Fracture Toughness of Metallic Materials" — Defines KIC testing methodology; adapted procedures exist for polymer composites.
  • ASTM E647-23, "Standard Test Method for Measurement of Fatigue Crack Growth Rates" — Standard method for determining da/dN vs. ΔK and the Paris law constants C and m.

Parameter Sensitivity Study

Run a baseline calculation first, then sweep a parameter to see how results change.

2D Contour Sensitivity

Run a baseline calculation first, then select two parameters to generate a 2D contour map.