From applied stress, crack size, and yield strength, compute the elastic CTOD based on the stress intensity factor K_I and the Dugdale strip-yield CTOD. Switch between plane stress and plane strain constraint and visualize CTOD versus applied stress in real time.
Inputs
Crack Size a (mm)10
Yield Strength σy (MPa)500
Critical CTOD δc (μm)40
While paused, move the sliders to update the result instantly.
Results
δ = K²/(m·σy·E′) (δ ∝ load²)
δ < δc: safe (no fracture)
Crack-Tip Opening (CTOD) Animation
Live Results
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Applied Stress σ (MPa)
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Stress Intensity K (MPa√m)
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CTOD δ (μm)
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Status (vs δc)
Theory & Key Formulas
$$\delta = \frac{K_I^{2}}{m\,\sigma_y\,E'}$$
Elastic CTOD (m): $\delta$ is the crack tip opening displacement. $m$ is the constraint factor — plane stress $m\approx1$, plane strain $m\approx2$.
$$K_I = Y\,\sigma\,\sqrt{\pi a}$$
Mode I stress intensity factor (Pa√m): $\sigma$ is the applied remote stress (Pa), $a$ is crack size (m), $Y$ is the geometry factor.
Dugdale strip-yield model (m). $E' = E$ (plane stress) or $E/(1-\nu^2)$ (plane strain). Dugdale CTOD diverges as $\sigma\to\sigma_y$.
What is CTOD (Crack Tip Opening Displacement)?
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Professor, I've heard the term CTOD, but what is it actually measuring? Is it different from the stress intensity factor K_IC?
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CTOD stands for Crack Tip Opening Displacement — literally how far the crack tip opens up under load. The symbol is δ, in mm (shown as μm here). K_IC characterizes the strength of the stress field at the tip, whereas CTOD measures the actual plastic deformation of the tip. That's its strength: it still works for materials that yield a lot before they break. Set the constraint state above to plane stress and drag the applied stress σ — you'll see CTOD grow steadily.
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Then why not just always use CTOD instead of K_IC? Why do we keep both?
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Good question. K_IC assumes small-scale yielding — the plastic zone at the tip must be tiny compared with the specimen. For strong, brittle materials that's fine. But ductile structural steel in a welded joint yields heavily before fracture, so K_IC simply isn't valid. CTOD measures that plastic deformation directly, so it holds up to moderate and large-scale yielding. Watch the σ/σy readout: as it approaches 1, yielding dominates and you've effectively crossed from the K_IC regime into the CTOD regime.
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Why does the result change between plane stress and plane strain? The formula looks the same.
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Because m and E′ in δ = K_I²/(m·σy·E′) change with the state. Plane stress (thin) gives m≈1, E′=E; plane strain (thick) gives m≈2, E′=E/(1−ν²). So for the same K_I, the plane-strain CTOD is roughly half the plane-stress value. Physically, a thick section constrains the crack tip and resists opening. Toggle the constraint state in the tool — elastic CTOD should drop from about 26.9 μm to about 12.3 μm. That gap is the constraint effect.
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How is a real CTOD test done, and how does it relate to this tool's numbers?
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In the lab you follow BS 7448 or ASTM E1820 with an SENB (three-point bend) or CT specimen. You measure crack mouth opening (CMOD) with a clip gauge and extrapolate to the crack tip via a hinge model to get δ. This tool shows how δ theoretically behaves via the elastic and Dugdale models. Compared against a measured critical value δ_c, it gives you an intuitive feel for how much margin a design has. Notice how the Dugdale curve shoots up as σ→σy — that's the warning flag of large-scale yielding.
Physical Model & Key Equations
CTOD can be estimated elastically by linking the stress intensity factor K_I to the material yield strength. The governing relations are:
where $\delta$ is the crack tip opening displacement (m), $K_I$ is the Mode I stress intensity factor (Pa√m), $\sigma$ is the applied remote stress (Pa), $a$ is the crack size (m), $Y$ is the geometry factor, and $\sigma_y$ is the yield strength (Pa). The constraint factor $m$ and effective modulus $E'$ take $m\approx1,\ E'=E$ for plane stress and $m\approx2,\ E'=E/(1-\nu^2)$ for plane strain.
As yielding proceeds, the crack-tip plastic zone can no longer be ignored. The Dugdale strip-yield model places a fictitious strip ahead of the crack that closes at the yield stress, giving a closed-form elastic-plastic CTOD:
For small $\sigma/\sigma_y$ this reduces to the elastic (plane-stress) CTOD, but as $\sigma\to\sigma_y$ the argument of $\sec$ approaches $\pi/2$ and CTOD diverges logarithmically. This represents the limit of large-scale yielding (LSY) in a perfectly-plastic material, where general yielding drives the tip to open without bound.
Standards & Assumptions
Standards and models cited
BS 7448 / ASTM E1820: standards for CTOD (δ) fracture toughness testing. CMOD is measured on SENB/CT specimens and converted to crack-tip δ via a hinge model, defining critical values δ_c (brittle initiation), δ_u (instability) and δ_m (maximum load).
Dugdale (1960) strip-yield / BCS model: an elastic-plastic model assuming a strip-shaped plastic zone ahead of the crack acting at the yield stress, the source of the closed form $\delta=(8\sigma_y a/\pi E)\ln\sec(\pi\sigma/2\sigma_y)$.
Assumptions
A single through-thickness crack — center through-crack or single-edge crack in a near-infinite plate — corrected by the geometry factor Y.
Small-to-moderate yielding. The constraint factor m is the plane-stress / plane-strain idealization (m≈1 / 2); in reality it varies continuously with specimen thickness and constraint.
The Dugdale model is valid for $\sigma<\sigma_y$ and a perfectly-plastic strip-yield zone.
Scope and limits
This tool is for educational and reference estimates; it is not a substitute for a standardized CTOD test record.
Near $\sigma\to\sigma_y$, large-scale yielding (LSY) causes the Dugdale CTOD to blow up. In real materials, work hardening and finite-size effects keep it bounded.
Real-World Applications
Integrity assessment of welded structures: Welded joints in ships, bridges, and offshore structures combine lower toughness with high residual stress, making CTOD the workhorse of fracture assessment. Flaw-assessment procedures such as BS 7910 estimate the δ applied to the structure and compare it with the material δ_c to decide whether a flaw size is acceptable. Because ductile fracture is involved, K_IC tends to mislead and CTOD is preferred.
Pipeline and pressure-vessel design: For natural-gas pipelines and pressure vessels, the CTOD transition temperature is controlled to avoid low-temperature brittle fracture. δ_c is measured separately for base metal, weld metal, and the heat-affected zone to confirm adequate opening margin (δ) even at the minimum service temperature.
Material selection for low-temperature service: In LNG tanks and cold-climate structures, falling temperature increases crack-tip constraint toward plane strain. The fact that CTOD drops when you switch the constraint state to plane strain in this tool mirrors exactly that tendency to open less and behave more brittle in the cold.
Failure analysis and remaining-life assessment: When a structure fails unexpectedly, engineers back-calculate δ from the crack size inferred on the fracture surface and the service stress, then compare with δ_c to find the root cause. For in-service equipment where a crack has been found, the gap between the current δ and the critical δ_c estimates how many more years the part can run.
Common Misconceptions and Points to Note
First, it is a mistake to read CTOD simply as "bigger is better, smaller is dangerous." CTOD itself is a measure of how far the tip opens; the relevant pairing is the material's critical value δ_c. Assessment is always the comparison "applied δ ≤ critical δ_c." A large applied δ is fine if that material's δ_c is even larger. Never judge "safe or unsafe" from the tool's number alone — always read it together with a tested critical value.
Second, do not treat the constraint factor m as "always either 1 or 2." m≈1 (plane stress) and m≈2 (plane strain) are idealized extremes; for real specimen thicknesses and geometries m varies continuously. Some references also use the flow stress (the average of yield and tensile strength) in place of σ_y. The plane-stress / plane-strain switch in this tool is an educational guide; in practice it is combined with constraint corrections such as the T-stress or Q parameter.
Finally, do not misread the Dugdale divergence as "CTOD really becomes infinite in real materials." The sharp rise of Dugdale CTOD as $\sigma\to\sigma_y$ is a consequence of the perfectly-plastic, infinite-plate idealization. Real materials work-harden and specimens are finite, so CTOD stays bounded even at general yield. The correct reading is a warning: you have entered large-scale yielding and should move from linear-elastic fracture mechanics to an elastic-plastic assessment.
Frequently Asked Questions
CTOD (Crack Tip Opening Displacement) is an elastic-plastic fracture mechanics parameter that measures how far the crack tip opens under load. Its symbol is δ, with units of mm (shown in μm in this tool). It serves as a fracture toughness measure for ductile materials such as structural steels that yield substantially before fracture, where the linear-elastic parameter K_IC no longer applies.
K_IC is a linear-elastic parameter valid only under small-scale yielding, where the crack-tip plastic zone is much smaller than the specimen dimensions. For ductile materials like welded structural steel that yield extensively before fracture, K_IC is invalid. CTOD directly measures the crack-tip plastic deformation, so it remains applicable up to moderate and large-scale yielding, giving a more realistic fracture assessment.
Elastic CTOD is δ = K_I²/(m·σ_y·E′), where the constraint factor m and effective modulus E′ depend on the state. In plane stress m ≈ 1 and E′ = E; in plane strain m ≈ 2 and E′ = E/(1−ν²). As a result, for the same K_I the plane-strain CTOD is about half the plane-stress value, reflecting that thicker sections are more constrained and open less.
Standard CTOD testing follows BS 7448 or ASTM E1820 using SENB (three-point bend) or CT specimens. Crack mouth opening displacement (CMOD) is measured with a clip gauge and extrapolated to the crack tip via a hinge model to obtain δ. For ductile fracture, critical values such as δ_c, δ_u, and δ_m are defined from characteristic points on the load–CMOD record.
How to Use
Choose plane stress (thin, m≈1) or plane strain (thick, m≈2) under "Constraint State".
Set applied stress σ, crack size a, yield strength σy, Young's modulus E, Poisson ratio ν, and geometry factor Y with the sliders or number inputs.
K_I, elastic CTOD (δ), Dugdale CTOD, and σ/σy are computed instantly.
Use the CTOD-vs-stress chart to watch the gap between elastic and Dugdale CTOD widen as σ/σy increases.
Worked Example
For σ = 300 MPa, a = 10 mm, σy = 500 MPa, E = 210 GPa, ν = 0.3, Y = 1.0 in plane stress, the tool returns K_I = 53.17 MPa√m, elastic CTOD δ = 26.93 μm, and Dugdale CTOD = 32.22 μm. Switching to plane strain at the same inputs, the m = 2 and E′ = E/(1−ν²) terms drop the elastic CTOD to about 12.25 μm, reflecting the opening suppression of a constrained thick section. σ/σy = 0.6 corresponds to small-to-moderate yielding.
Practical Notes
The elastic CTOD here is a small-to-moderate-yielding estimate. Above σ/σy ≈ 0.8, prefer the Dugdale (elastic-plastic) value or a measured δ.
The constraint factor m is the plane-stress / plane-strain ideal value; for real thicknesses combine it with T-stress or Q-parameter constraint corrections.
Dugdale CTOD diverges as σ→σy. This is a large-scale-yielding warning; real materials stay bounded due to work hardening and finite size.
Always compare the computed δ with the material's critical value δ_c (BS 7448 / ASTM E1820 test data).