J積分（弾塑性破壊力学）
Theory and Physics
What is the J-Integral?
Professor, what is the J-integral?
The J-integral (Rice, 1968) is a parameter representing the energy release rate at the crack tip. It is a concept that extends the stress intensity factor (SIF) $K$ from linear elastic fracture mechanics to elastic-plastic conditions.
$\Gamma$ is any path surrounding the crack tip. $W$ is the strain energy density, $\mathbf{T}$ is the traction vector.
Path independent! So "the value is the same no matter where the integration path is taken".
Path independence holds strictly for elastic bodies. For elastic-plastic materials, it is nearly path independent if there is no unloading (monotonic loading). In FEM, J is calculated for multiple contours (paths) surrounding the crack tip to confirm convergence.
Relationship between $J$ and $K$
For linear elasticity:
$E' = E$ (Plane stress), $E' = E/(1-\nu^2)$ (Plane strain). $J$ is proportional to the square of $K$.
Fracture Condition
$J_{Ic}$ is the critical J-integral value (material property). The test method is specified in ASTM E1820.
Summary
Key Points:
- $J$ = Energy release rate at the crack tip — Applicable to elastic-plastic conditions
- Path independent — Same value for any path surrounding the crack tip
- $J = K^2/E'$ — Relationship with linear elasticity
- Fracture when $J \geq J_{Ic}$ — $J_{Ic}$ measured per ASTM E1820
- FEM *CONTOUR INTEGRAL — Automatically calculates J at the crack tip
Rice Changed the World with a 9-Page Paper
The J-integral was proposed by James Rice (Harvard University) in a 9-page paper published in the JAppl Mech journal in 1968. Its most significant feature is "path independence," meaning it yields the same value for any integration path surrounding the crack tip, allowing the definition of an energy release rate even under elastic-plastic conditions. This discovery enabled the extension of fracture mechanics to elastic-plastic regimes, and Rice received the Timoshenko Award in 1983.
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 pulled" 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, assuming "acceleration can be ignored because the force is 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's Hooke's law $F=kx$, the essence of the stiffness term. Now a question—if you pull an iron rod and a rubber band with the same force, which stretches more? 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 force $f_b$ (e.g., gravity) and surface force $f_s$ (pressure, contact force, etc.). 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 typical pitfall here: getting the load direction wrong. Intending "tension" but ending up with "compression"—it sounds like a joke, but it actually happens when coordinate systems are rotated 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 vibration energy is converted to heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle—they intentionally absorb vibration energy to improve ride comfort. What if damping were zero? Buildings would keep swaying forever after an earthquake. Since that doesn't happen in reality, setting appropriate damping is crucial.
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: Large deformation/large rotation problems require geometric nonlinearity. Nonlinear material behavior like plasticity or creep requires constitutive law extensions.
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Displacement $u$ | m (meter) | When inputting in mm, unify load/elastic modulus to MPa/N system. |
| Stress $\sigma$ | Pa (Pascal) = N/m² | MPa = 10⁶ Pa. Be careful of unit system inconsistency 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
J-Integral in FEM
```
*CONTOUR INTEGRAL, CONTOURS=5, TYPE=J
crack_tip_node, direction_vector
```
Calculates J for 5 contours. The value should converge as the contours move away from the crack tip.
What if the values don't converge across the 5 contours?
The mesh is too coarse or the plastic zone is large. Refine the mesh or increase the number of contours. Convergence is OK if the values from the outer 3-4 contours are nearly identical.
Crack Tip Meshing
Place a concentrated mesh (Spider web mesh) at the crack tip. Elements are arranged radially from a central point (the crack tip).
- Recommended: 2nd-order elements (C3D20R) — Accurately captures the singularity at the crack tip.
- Quarter-Point elements — Move the midside node to the 1/4 point at the crack tip. Models the $1/\sqrt{r}$ singular field.
Summary
J-Integral Calculation in FEM: Virtual Crack Extension Method
The virtual crack extension method (Domain integral method) is superior in both accuracy and efficiency for J-integral calculation in FEM. It integrates only over a region 3-5 elements away from the crack tip and does not require FEM singular elements (collapsed quarter-point elements). ANSYS's FRACTURE TB command internally uses this method and automatically outputs the average value from paths 1-10, making convergence checks easy.
Linear Elements (1st-order elements)
Linear interpolation between nodes. Low computational cost but lower stress accuracy. Beware of shear locking (mitigated with reduced integration or B-bar method).
Quadratic Elements (with midside nodes)
Can represent curved deformation. Stress accuracy improves significantly, but degrees of freedom increase by about 2-3 times. Recommended when stress evaluation is critical.
Full integration vs Reduced integration
Full integration: Risk of over-constraint (locking). Reduced integration: Risk of hourglass modes (zero-energy modes). Choose appropriately.
Adaptive Mesh
Automatic refinement based on error indicators (e.g., ZZ estimator). Efficiently improves accuracy in stress concentration areas. Includes h-method (element subdivision) and p-method (order increase).
Newton-Raphson Method
Standard method for nonlinear analysis. Updates the tangent stiffness matrix every iteration. Achieves quadratic convergence within the convergence radius but has high computational cost.
Modified Newton-Raphson Method
Updates the tangent stiffness matrix using the initial value or every few iterations. Lower cost per iteration but linear convergence speed.
Convergence Criteria
Force residual norm: $||R|| / ||F_{ext}|| < \epsilon$ (typically $\epsilon = 10^{-3}$ to $10^{-6}$). Displacement increment norm: $||\Delta u|| / ||u|| < \epsilon$. Energy norm: $\Delta u \cdot R < \epsilon$
Load Increment Method
Instead of applying the full load at once, it is applied in small increments. The arc-length method (Riks method) can trace beyond limit points on the load-displacement curve.
Analogy: Direct Method vs Iterative Method
The direct method is like "solving simultaneous equations accurately with pen and paper"—reliable but takes too long for large-scale problems. The iterative method is like "repeatedly guessing to approach the correct answer"—starts with a rough answer but improves accuracy with each iteration. It's the same principle as looking up a word in a dictionary: it's more efficient to open it at an estimated location and adjust forward/backward (iterative) than to search sequentially from the first page (direct).
Relationship Between Mesh Order and Accuracy
1st-order elements are like "approximating a curve with a ruler"—represented by straight line segments, so accuracy is limited. 2nd-order elements are like a "flexible curve"—can represent curved changes, dramatically improving accuracy even at the same mesh density. However, computational cost per element increases, so judge based on total cost-effectiveness.
Practical Guide
J-Integral in Practice
Used in pressure vessel crack assessment (API 579 FFS-1), pipeline defect assessment, and nuclear fracture mechanics evaluation (R6 method).
ASTM E1820 Test
Test for J-R curve (J vs. crack extension $\Delta a$). Conducted using CT (Compact Tension) specimens. Obtain $J_{Ic}$ (critical value for crack initiation) and the J-R curve.
Practical Checklist
Elastic-Plastic Fracture Assessment of Pressure Vessel Nozzle Junctions
ASME Sec.XI Code uses the J-integral for crack assessment at pressure vessel nozzle fillet regions. Using the J-R curve (J vs Δa) of SA-508 Cl.3 steel for nuclear-grade piping, the condition where an initial crack transitions to unstable growth (Ji = stable growth initiation point) is calculated. For considering 60-year extended operation, analysis is required to prove that safety margins exist even if the post-irradiation embrittlement JIc is conservatively reduced by 50%.
Analogy for Analysis Flow
The analysis flow is actually very similar to cooking. First, you buy the ingredients (prepare the CAD model), do the prep work (mesh generation), apply heat (solver execution), and finally plate it (visualization in post-processing). Here's an important question—which step in cooking is most prone to failure? Actually, it's the "prep work." If the mesh quality is poor, the results will be a mess no matter how good the solver is.
Pitfalls Beginners Often Fall Into
Are you checking mesh convergence? Do you think "the calculation ran = the result is correct"? This is actually the most common trap for CAE beginners. The solver will always return "some answer" for the given mesh. But if the mesh is too coarse, that answer can be far from reality. Confirm that results stabilize across at least three levels of mesh density—neglecting this leads to the dangerous assumption that "the answer must be correct because the computer produced it."
Thinking About Boundary Conditions
Setting boundary conditions is like "writing the problem statement" for an exam. If the problem statement is wrong? No matter how accurately you calculate, the answer will be wrong. "Is this surface truly fully fixed?" "Is this load truly uniformly distributed?"—Correctly modeling the real-world constraints is often the most critical step in the entire analysis.
Software Comparison
J-Integral Tools
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