Progressive Damage Analysis
Theory and Physics
What is Progressive Damage?
Professor, what is "Progressive Damage Analysis (PDA)"?
Unlike metals, composite materials may not cause the entire structure to collapse even if local failure occurs. Even if matrix cracks appear, the fibers continue to bear the load. PDA simulates this progressive damage propagation and load redistribution.
So failure criteria (Tsai-Wu, Hashin) alone are insufficient?
Tsai-Wu and Hashin predict "First Ply Failure (FPF)". However, the final failure load (Last Ply Failure, LPF) of a composite structure can be 2-3 times the initial failure load. PDA tracks the entire process from "initial failure to final collapse".
The 3 Elements of PDA
PDA consists of three elements:
1. Damage Initiation Criteria
"When does damage start?". Hashin criteria, Puck criteria, LaRC criteria, etc. Determine the onset of each failure mode.
2. Damage Evolution Law
"How does damage progress?". How to represent stiffness reduction after damage initiation.
- Instantaneous Reduction — Stiffness drops to zero immediately upon damage initiation (sudden degradation model). Simple but high mesh dependency.
- Progressive Reduction — Stiffness reduces gradually based on fracture energy. Low mesh dependency.
3. Element Deletion
Removes elements from the analysis when the damage variable reaches 1.0 (complete damage). Represents areas where the material is completely fractured.
Is progressive reduction more physically accurate?
Yes. Progressive reduction using fracture energy $G_c$ produces mesh-independent (regularized) results. Abaqus's Hashin Damage Evolution uses this method.
Damage Variable
The degree of damage is represented by the damage variable $d$ (0~1):
- $d = 0$: Intact (no damage)
- $0 < d < 1$: Partial damage
- $d = 1$: Complete failure
Are there damage variables for each of Hashin's 4 modes?
Yes. Four independent damage variables are defined: fiber tension $d_{ft}$, fiber compression $d_{fc}$, matrix tension $d_{mt}$, matrix compression $d_{mc}$.
Stiffness matrix after damage:
Damage in each mode progresses independently, reducing the corresponding components of the stiffness matrix.
This is CDM (Continuum Damage Mechanics)-based PDA. Because it treats damage within the continuum mechanics framework, it can be naturally incorporated into the standard FEM framework.
Summary
Let me organize the theory of PDA.
Key points:
- Simulates the entire process from initial failure to final collapse — From FPF to LPF.
- 3 Elements — Damage initiation criteria + Damage evolution law + Element deletion.
- Damage variable $d$ (0~1) — Tracks the damage degree of each failure mode.
- CDM (Continuum Damage Mechanics) based — Can be naturally integrated into FEM.
- Regularization using fracture energy — Eliminates mesh dependency.
So PDA is an analysis method to "squeeze out the strength of composites to the very end".
PDA, which can evaluate the post-failure strength of composites, is an essential technology for pushing the limits of lightweight design.
Integration of Fracture Mechanics and Progressive Damage
Progressive damage in composite materials is the process where local material damage sequentially expands with increasing load. In the 1990s, Chaboche, Ladevèze, and others applied Continuum Damage Mechanics (CDM) to CFRP and formulated evolution equations for the damage variable d (0=undamaged, 1=completely damaged). CDM-based progressive analysis enables the prediction of final failure loads in FEM and is used to quantify design safety margins.
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 can be ignored". 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—an iron rod and a rubber band, which stretches more under 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"—different concepts.
- External Force Term (Load Term): Body forces $f_b$ (gravity, etc.) and surface forces $f_s$ (pressure, contact forces, etc.). Think of it this way—the weight of a truck on a bridge is a "force acting on the entire volume" (body force), the force of the tires pushing on the road 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 it's actually "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, right? Because the vibration energy is converted to heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle—deliberately absorbing vibration energy to improve ride comfort. What if damping were zero? Buildings would keep shaking 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, 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, considers only equilibrium 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 inconsistency when comparing with yield stress. |
| Strain $\varepsilon$ | Dimensionless (m/m) | Note the distinction between engineering strain and logarithmic strain (for large deformations). |
| Elastic modulus $E$ | Pa | Steel: ~210 GPa, Aluminum: ~70 GPa. Note temperature dependency. |
| 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
PDA Implementation
Please teach me the specific implementation methods for PDA.
Two main implementation approaches:
1. Abaqus Built-in Hashin Damage
Abaqus's DAMAGE INITIATION (HASHIN) + DAMAGE EVOLUTION. With the settings explained on the previous page, the entire process—damage initiation → progressive reduction → element deletion—runs automatically.
2. User Subroutines (UMAT/VUMAT)
To use more advanced damage models (Puck, LaRC05, custom CDM), you create your own user subroutine. For Abaqus:
- UMAT — For implicit (Standard) solver. Requires calculation of tangent stiffness matrix.
- VUMAT — For explicit (Explicit) solver. Only stress update required. Simpler implementation.
Is VUMAT easier to implement?
VUMAT does not require derivation of the tangent stiffness matrix; it only updates stress from the given strain increment. VUMAT + Explicit is recommended for first-time PDA implementation. Implicit UMAT is difficult due to tangent stiffness derivation and also affects convergence.
Convergence Issues
PDA has convergence difficulties, right?
Stiffness reduction due to damage causes local softening, which can lead to snap-back in the load-displacement path. Countermeasures:
| Method | Characteristics |
|---|---|
| Viscous Regularization | "Smoothes" softening with tiny viscosity. $\eta \approx 10^{-5}$ |
| Explicit Solver | No convergence issues (no iterations). High computational cost. |
| Arc-length Method (Riks) | Can track snap-back. Complex setup. |
| Stabilization Method | *STATIC, STABILIZE. Verify with energy ratio. |
Is the explicit solver the most stable?
In the sense of having no convergence worries, it is the most stable. However, solving quasi-static problems with the explicit solver requires mass scaling, and inertial effects must be considered. In practice, explicit solver + mass scaling is becoming mainstream for PDA.
PDA in LS-DYNA
In LS-DYNA, MAT54 (Chang-Chang criteria) and MAT58 (Continuum Damage Mechanics) are the standards for PDA:
| Model | Characteristics |
|---|---|
| MAT54 | Sudden degradation model. Simple but high mesh dependency. |
| MAT58 | CDM-based. Progressive reduction. More accurate than MAT54. |
MAT54/58 are widely used in automotive crash analysis, right?
Crash analysis of CFRP crash boxes and bumpers uses LS-DYNA MAT54/58 as the de facto standard. Predicts energy absorption during impact.
Summary
Let me organize the numerical methods for PDA.
Key points:
- Abaqus built-in Hashin Damage is the easiest — Works with settings only.
- VUMAT (Explicit) is recommended for custom PDA — No tangent stiffness required.
- Explicit solver + mass scaling is mainstream for PDA — No convergence
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