Tsai-Wu破壊基準
Theory and Physics
What is the Tsai-Wu Criterion?
Professor, what is the Tsai-Wu failure criterion?
The Tsai-Wu criterion (1971) is one of the most widely used criteria for predicting layer-by-layer failure in composite materials. It is the equivalent of the von Mises criterion for metals.
Von Mises is for isotropic materials, right? Can't von Mises be used for composites?
It cannot. Composite materials have different strengths in tension and compression ($X_t \neq X_c$), and their strength also varies with direction ($X \neq Y$). Von Mises cannot handle these asymmetries.
Tsai-Wu Criterion Formula
The Tsai-Wu criterion for a 2D stress state (Plane Stress):
Here, $\sigma_1$ is the fiber direction stress, $\sigma_2$ is the transverse direction stress, and $\tau_{12}$ is the in-plane shear stress.
Each coefficient:
$X_t, X_c$ are the tensile/compressive strengths in the fiber direction, $Y_t, Y_c$ are in the transverse direction, and $S$ is the shear strength, right?
Correct. The coefficients are determined from five material strength values. The problem is the value of $F_{12}$ (the interaction term). This needs to be determined from biaxial stress tests, but because such experiments are difficult, $F_{12} = -0.5\sqrt{F_{11}F_{22}}$ (Tsai's recommended value) is often used.
Failure Index
How do you determine failure?
Calculate the Tsai-Wu index $FI$:
- $FI < 1$: No failure
- $FI = 1$: Failure limit
- $FI > 1$: Failure (stress exceeded)
$FI$ is conceptually similar to "the inverse square of the safety factor," right?
Strictly speaking, it's not (because it's a quadratic equation), but intuitively, yes. $FI = 0.5$ means "using about 70% of the strength," and $FI = 1.0$ means "failure."
Advantages and Disadvantages of the Tsai-Wu Criterion
| Advantages | Disadvantages |
|---|---|
| Accounts for strength differences in tension/compression | Does not distinguish failure modes (fiber/matrix) |
| Handles multiaxial stress states | Determination of $F_{12}$ is difficult |
| Single equation for failure determination | Not suitable for progressive damage |
| Easy to implement | Cannot handle delamination |
Is "not distinguishing failure modes" its biggest weakness?
Yes. The Tsai-Wu criterion only tells you "failed/not failed," but not whether it's fiber breakage or matrix cracking. For progressive damage analysis (load redistribution after failure), criteria like Hashin or Puck are needed.
Summary
Let me organize the Tsai-Wu criterion.
Key points:
- The most widely used failure criterion for composites — Accounts for strength differences in tension/compression
- Formulated with 5 material strength values + $F_{12}$
- Safe when $FI \leq 1$ — The failure index is the design judgment value
- Does not distinguish failure modes — Use Hashin to discriminate fiber/matrix failure
- Evaluated in the material coordinate system (1, 2 directions) — Cannot use global coordinate stresses
If von Mises is the standard for isotropic materials, then Tsai-Wu is the standard for composites.
Exactly. However, Tsai-Wu is a criterion for "predicting initial failure," and "whether the structure collapses" is a separate issue. To evaluate the characteristic of composites that can still bear load after initial failure, progressive damage analysis is required.
Tsai-Wu Criterion: Generalization of the Failure Surface
The Tsai-Wu failure criterion (1971) is a generalization of the Hill criterion adapted to the tension-compression asymmetry of CFRP strength, where F1σ1+F2σ2+F11σ1²+F22σ2²+F66τ12²+2F12σ1σ2≥1 is the failure condition. The cross-term F12 represents "biaxial load interaction," and its determination was the biggest technical challenge. Tsai and Wu proposed a method to identify F12 using biaxial tensile tests, but even today, methods for determining F12 continue to be debated among various institutions.
Physical Meaning of Each Term
- Inertia Term (Mass Term): $\rho \ddot{u}$, i.e., "mass × acceleration." Have you ever experienced being thrown forward when slamming on the brakes? That "feeling of being carried away" is precisely the inertial force. Heavier objects are harder to set in motion and harder to stop once moving. Buildings shake during earthquakes because the ground moves suddenly while the building's mass "gets left behind." In static analysis, this term is set to zero, which is the assumption that "acceleration can be ignored because forces are applied slowly." It absolutely cannot be omitted for impact loads or vibration problems.
- Stiffness Term (Elastic Restoring Force): $Ku$ or $\nabla \cdot \sigma$. When you stretch a spring, you feel a "force trying to return it," right? That is Hooke's law $F=kx$, and it's the essence of the stiffness term. So, a question—an iron rod and a rubber band, which stretches more when pulled with the same force? Obviously, the rubber band. This "resistance to stretching" is the Young's modulus $E$, which determines stiffness. A common misconception: "High stiffness ≠ strong." Stiffness is "resistance to deformation," strength is "resistance to failure"—they are different concepts.
- External Force Term (Load Term): Body forces $f_b$ (e.g., gravity) and surface forces $f_s$ (e.g., pressure, contact forces). Think of it this way—the weight of a truck on a bridge is a "force acting on the entire volume" (body force), while the force of the tires pushing on the road surface is a "force acting only on the surface" (surface force). Wind pressure, water pressure, bolt tightening force... all are external forces. A common mistake here: getting the load direction wrong. Intending "tension" but it becomes "compression"—it sounds like a joke, but it actually happens when coordinate systems rotate in 3D space.
- Damping Term: Rayleigh damping $C\dot{u} = (\alpha M + \beta K)\dot{u}$. Try plucking a guitar string. Does the sound continue forever? No, it gradually fades away. That's because the vibrational energy is converted into heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle—they intentionally absorb vibrational energy to improve ride comfort. What if damping were zero? Buildings would continue shaking forever after an earthquake. Since that doesn't happen in reality, setting appropriate damping is important.
Assumptions and Applicability Limits
- Continuum assumption: Treats material as a continuous medium, ignoring microscopic heterogeneity
- Small deformation assumption (for linear analysis): Deformation is sufficiently small compared to initial dimensions, and the stress-strain relationship is linear
- Isotropic material (unless otherwise specified): Material properties are independent of direction (anisotropic materials require separate tensor definitions)
- Quasi-static assumption (for static analysis): Ignores inertial and damping forces, considering only the balance between external and internal forces
- Non-applicable cases: For large deformation/large rotation problems, geometric nonlinearity is required. For nonlinear material behaviors like plasticity and creep, constitutive law extensions are needed.
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Displacement $u$ | m (meter) | When inputting in mm, unify loads and elastic modulus to MPa/N system |
| Stress $\sigma$ | Pa (Pascal) = N/m² | MPa = 10⁶ Pa. Be careful of unit system inconsistencies when comparing with yield stress |
| Strain $\varepsilon$ | Dimensionless (m/m) | Note the distinction between engineering strain and logarithmic strain (for large deformation) |
| Elastic modulus $E$ | Pa | Steel: ~210 GPa, Aluminum: ~70 GPa. Note temperature dependence |
| Density $\rho$ | kg/m³ | In mm system: tonne/mm³ (= 10⁻⁹ tonne/mm³ for steel) |
| Force $F$ | N (Newton) | Unify as N in mm system, N in m system |
Numerical Methods and Implementation
Tsai-Wu Implementation in FEM
How is the Tsai-Wu criterion used in FEM?
The basic approach is to calculate the failure index in post-processing. The Tsai-Wu index is calculated from linear analysis results (stresses in each layer).
Nastran
Select the failure criterion in the FT (Failure Theory) field of the PCOMP card:
```
PCOMP, 1, , , TSAI, SYM
```
TSAI = Tsai-Wu criterion. Output is recorded as FAILURE INDEX in the f06 file.
Abaqus
In Abaqus's standard functionality, the Tsai-Wu failure index is typically implemented via a user subroutine (USDFLD) or calculated in the post-processor. Abaqus's built-in failure criteria are mainly the Hashin criterion and Max Stress/Max Strain.
Ansys
The Tsai-Wu criterion is built into Ansys ACP. After laminate analysis, the Failure Index is automatically calculated and visualized in ACP Post.
Does Abaqus not have Tsai-Wu as a standard feature?
Abaqus focuses more on progressive damage (damage mechanics based on Hashin) than on failure criteria. Tsai-Wu is for "predicting initial failure," but Abaqus's philosophy is to "simulate damage progression." They serve different purposes.
Relationship with Safety Factor
How do you calculate the safety factor from $FI$?
The safety factor (Strength Ratio, $SR$) is defined somewhat inversely to $FI$, but since Tsai-Wu is a quadratic equation, it's not a simple inverse.
Failure condition when loads are multiplied by $\lambda$:
Solving this for $\lambda$ yields the safety factor (the smallest positive $\lambda$). In Nastran output, this is displayed as STRENGTH RATIO.
So, safe if $SR > 1$, failure if $SR < 1$.
Yes. $SR = 2.0$ means "will not fail even if loads are doubled." $SR = 0.8$ means "failure at 80% of the current load." It can be directly compared with the design safety factor.
Summary
Let me organize the numerical methods for Tsai-Wu.
Key points:
- Calculate $FI$ in post-processing — From the layer stresses of linear analysis results
- Nastran is the most direct — Specify via the FT field in PCOMP
- Abaqus does not support Tsai-Wu by default — Damage mechanics based on Hashin is primary
- Visualization in Ansys ACP — Contour display of Failure Index
- Strength Ratio = Safety Factor — Safe when $SR > 1$
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