Gurson Model (Ductile Fracture)
Theory and Physics
What is the Gurson Model?
Professor, is the Gurson model a model for ductile fracture?
Gurson Model (1977) describes the ductile fracture (void nucleation, growth, coalescence) of metals. The volume fraction $f$ of internal micro-voids affects the material's yield condition.
The void volume fraction $f$ is included in the yield surface!
As $f$ increases, the yield surface contracts → material softens → eventually fractures at $f = f_F$. It represents material degradation at the continuum level due to void growth.
GTN Model (Modified Gurson)
Tvergaard and Needleman (1984) modified Gurson's model to create the practical GTN (Gurson-Tvergaard-Needleman) Model. They added correction parameters $q_1, q_2, q_3$.
Summary
The Doctoral Thesis Origin of the Gurson Model
Arthur L. Gurson first published his model for ductile fracture via void growth in his 1975 Brown University doctoral thesis titled "Plastic Flow and Fracture Behavior of Ductile Materials Incorporating Void Nucleation, Growth, and Coalescence." The 1977 paper published in the Journal of Engineering Materials and Technology analytically derived the yield function from the upper bound theorem for porous metals containing spherical voids. This rigorous mechanical derivation is a major feature of the Gurson model.
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, assuming "forces are applied slowly enough that acceleration is negligible." It 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. So, 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 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 modeling "compression"—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 vibrational 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 vibrational 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 specified otherwise): 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 behavior like plasticity or 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 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
GTN in FEM
```
*POROUS METAL PLASTICITY
q1, q2, q3
*POROUS FAILURE CRITERIA
f_N, epsilon_N, s_N, f_0, f_c, f_F
```
LS-DYNA: *MAT_120 (GTN).
Summary
GTN Extensions and Tvergaard Constants
The Gurson model evolved into the GTN model (Gurson-Tvergaard-Needleman) when Tvergaard (1981) introduced empirical correction coefficients q₁, q₂, q₃. Typical values are q₁=1.5, q₂=1.0, q₃=q₁²=2.25, widely applied to Aluminum and Steel. Furthermore, Needleman & Tvergaard (1984) added the "critical void fraction f*" and a fracture acceleration function, resulting in the current standard GTN model form capable of representing rapid ductile fracture (void coalescence).
Linear Elements (1st Order Elements)
Linear interpolation between nodes. Low computational cost but lower stress accuracy. Beware of shear locking (mitigated by reduced integration or B-bar method).
Quadratic Elements (with Mid-side 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 for the situation.
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 each iteration. Achieves quadratic convergence within the convergence radius, but computational cost is high.
Modified Newton-Raphson Method
Updates the tangent stiffness matrix using the initial value or every few iterations. Cost per iteration is low, but convergence speed is linear.
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
Applies the total load in small increments rather than all at once. 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 estimate where to open it 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 judgment should be based on total cost-effectiveness.
Practical Guide
GTN in Practice
Used for fracture prediction in sheet metal forming, ductile tearing of nuclear pressure vessels, and ductile fracture of pipelines.
Practical Checklist
Ductile Rupture Analysis of Oil Pipelines
The Gurson model is utilized in burst test analysis for oil and gas pipelines conforming to API 5L standards. DNV-ST-F101 (subsea pipeline standard) recognizes the complementary use of virtual experiments using the GTN model for the two-stage evaluation of plastic collapse and ductile rupture, positioning it as a tool to reduce the number of actual hydrostatic burst tests (costing tens of millions of yen each). Its effectiveness has been confirmed particularly in rupture evaluation of high-strength steel grades X80 and X100.
Analogy of the 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 excellent 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 will 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 computer gave the answer, so it must be correct."
How to Think About Boundary Conditions
Setting boundary conditions is the same as "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 real-world constraint conditions is often the most critical step in the entire analysis.
Software Comparison
Tools
Comparison of Abaqus GTN and LS-DYNA MAT224
In Abaqus, the GTN model is defined with the "POROUS METAL PLASTICITY" keyword, directly inputting q₁, q₂, f₀, f_N, etc. In LS-DYNA, equivalent functionality is implemented as MAT_GURSON (Material 120), and since 2020, MAT_MODIFIED_GURSON (Material 220) has been added, providing full GTN extension capabilities. HyperWorks/OptiStruct also began supporting GTN materials in 2022, offering the unique feature of coupling topology optimization with ductile fracture evaluation.
The Three Most Important Questions for Selection
- "What are you solving?": Does it support the physical models and element types required for the Gurson model (ductile fracture)? For example, for fluids, the presence of LES support; for structures, the capability to handle contact and large deformation makes a difference.
- "Who will use it?": For beginner teams, tools with rich GUIs are suitable; for experienced users, flexible script-driven tools are better. Similar to the difference between automatic (GUI) and manual (script) transmission cars.
- "How far will it be extended?": Choosing with an eye on future expansion of analysis scale (HPC support), deployment to other departments, and integration with other tools leads to long-term cost reduction.
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