Composite Failure Envelopes

What a first ply failure envelope shows, how it is constructed, and how to read one against your own load case.

What is a failure envelope?

A safety factor answers one question: how much more of this load can the laminate take? A failure envelope answers the wider one. Instead of scaling a single load direction, it scales every direction in a plane of the load space and records where failure occurs in each. Joining those points traces a closed curve, and that curve is the laminate's strength boundary: inside it the laminate survives, outside it a ply has failed.

The construction rests on the same load multiplier as the safety factor, usually written R in this context, the strength ratio. For each direction the whole load vector is scaled by R until the first ply reaches FI = 1, and the resulting load is one point on the envelope. Because CLT is linear in the applied load, ply stress is proportional to R, which is what makes each point an exact solution rather than something found by trial and error.

Your own load case is plotted on the same axes. The dashed line from the origin through that point to the boundary is the direction the load would grow in if you scaled it up, and R is how far you can scale it before the first ply fails. R = 1 puts you on the boundary.

Load planes

A laminate carries three in-plane force resultants, so its full strength boundary is a surface in three dimensions. A chart can show two at a time, which is why the envelope is swept in a plane and you pick which one:

• N_x-N_y is the biaxial plane, the one most design cases live in. It is also the only plane with a quadrant where both direct loads are compressive.
• N_x-N_xy and N_y-N_xy combine one direct load with shear, which is where fiber-dominated layups and matrix-dominated layups part company.

The plot below is the same laminate, the same load case and the same three criteria as the one further up the page. Only the plane picker moved, to N_x-N_xy. That one control is the whole difference between them, and it draws a boundary of a different shape against a different second axis.

Quasi-isotropic [0/±45/90]s, N_x-N_xy plane. The load case carries an N_y, and this plane holds N_y at zero, so the chart tells you the envelope was swept without it and withholds R for the marker. The marker is still drawn, projected onto the plane: it shows roughly where you sit, not a margin you can read off.

In each plane the third in-plane component is held at zero. That is not a simplification of convenience: it keeps every load path proportional, and proportionality is what lets R come out of a single pass rather than a root solve per direction. Hold the third component at some fixed nonzero value and ply stress becomes affine in R instead of proportional, which is the same reason moments and hygrothermal loads sit outside the sweep.

Reading an envelope

• Your margin is a distance, and it has a direction. Plot the applied load as a point inside the envelope. How far it sits from the boundary along the direction the load would grow is the margin that matters. A load point comfortably far from the boundary in one direction can be very close to it in another, which a single safety factor number cannot show you.
• The shape tells you which criterion you are looking at. Non-interactive criteria check each stress component separately, so their envelopes come out as polygons with corners. Interactive criteria combine all components into one polynomial, so theirs are smooth closed curves.
• Asymmetry about the origin is real, not a glitch. Composites are much weaker in transverse tension than in transverse compression, so an envelope is normally offset rather than centred. A criterion whose envelope is symmetric, Tsai-Hill for instance, is telling you it does not distinguish tension from compression.
• A dent or a corner is a change of governing mode. Where the boundary changes character, a different ply or a different failure mode has taken over as the critical one. Those transitions are worth noting, because a design sitting near one is sensitive to small changes in the load ratio.

Inside the boundary the chart reports no first ply failure for the criterion you picked. On or outside it, the first ply has failed, and R has dropped to 1 or below.

Quasi-isotropic [0/±45/90]s, overloaded. The marker sits outside the Max Stress boundary, so R is below 1 and the chart reports first ply failure. The verdict follows the criterion in the picker, which is why the ray is drawn to that curve rather than to whichever one happens to be furthest out.

Comparing criteria on one envelope

A table of safety factors tells you the criteria disagree. It does not tell you where, or by how much, or whether the disagreement matters for the load you actually apply. Drawing several envelopes on the same axes does, because the gap between two curves is the disagreement, measured in the units of your load.

How wide that gap gets depends on the laminate. A unidirectional ply loaded transversely has very little strength to spend, so the interaction terms dominate and the criteria spread out. A quasi-isotropic layup spreads the load across four fiber directions and the same criteria land close together. That is a useful thing to know before you argue about which criterion to certify against: sometimes the choice moves the answer by a factor, and sometimes it moves it by a few percent.

Two comparisons are worth making a habit of. Draw Maximum Stress alongside whichever criterion you are designing to, because the polygon is the no-interaction reference and the gap to it is the interaction your criterion is claiming. And in the biaxial compression quadrant, read Hashin or Puck rather than Tsai-Wu: Tsai-Wu's F₁₂ term is an estimate rather than a measured value, and it stretches the curve out where both direct loads are compressive.

Load and strain views

The same sweep can be plotted two ways. The load view shows the force resultants at first ply failure, in N/mm. The strain view shows the midplane strains (ε⁰) those same failure loads produce, so a boundary can be read against a strain allowable instead of a load one. No extra solve happens between them: it is one set of failure points, plotted against a different pair of axes.

Cross-ply [0/90]s, strain view. The same failure points as the load view, with the midplane strains at failure on the axes instead of the force resultants.

What an envelope leaves out

That last point has a practical consequence. If you enter a load case that carries moments, hygrothermal effects, or a component outside the plane on screen, the dashboard still draws the envelope but withholds R for your load point, and says why. The plies in that case already carry stress the boundary was swept without, so the distance from your point to the curve overstates the margin. Surface plies under a bending moment, or a cured-in ΔT, can overstate it by a lot.

Author: Rick Schrijver

References

1. Jones, R.M., Mechanics of Composite Materials, 2nd Ed., Taylor & Francis, 1999.
2. Tsai, S.W. & Wu, E.M., "A General Theory of Strength for Anisotropic Materials", J. Composite Materials, Vol. 5, 1971. doi:10.1177/002199837100500106
3. Tsai, S.W., Theory of Composites Design, Think Composites, 1992.
4. Hashin, Z., "Failure Criteria for Unidirectional Fiber Composites", J. Applied Mechanics, Vol. 47, 1980, pp. 329-334. doi:10.1115/1.3153664
5. Puck, A. & Schurmann, H., "Failure Analysis of FRP Laminates by Means of Physically Based Phenomenological Models", Composites Science and Technology, Vol. 62, 2002, pp. 1633-1662. doi:10.1016/S0266-3538(01)00208-1
6. Daniel, I.M. & Ishai, O., Engineering Mechanics of Composite Materials, 2nd Ed., Oxford University Press, 2006.

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