Concrete Damaged Plasticity Model (CDP)
Theory and Physics
What is the CDP Model?
Professor, what is the CDP model?
CDP (Concrete Damaged Plasticity) is a constitutive model for concrete in Abaqus. It is a combination of Plasticity (based on DP criterion) + Damage (tensile cracking + compressive crushing).
Special Characteristics of Concrete
CDP Composition
Settings in FEM
```
*CONCRETE DAMAGED PLASTICITY
dilation_angle, eccentricity, fb0/fc0, K, viscosity
*CONCRETE COMPRESSION HARDENING
stress, inelastic_strain
*CONCRETE TENSION STIFFENING
stress, cracking_strain
*CONCRETE COMPRESSION DAMAGE
damage, inelastic_strain
*CONCRETE TENSION DAMAGE
damage, cracking_strain
```
Summary
Key Points:
- CDP = Drucker-Prager plasticity + Tensile/Compressive damage — Dedicated to concrete
- Tensile softening (cracking) + Compressive softening (crushing) — Special behavior of concrete
- Stiffness recovery — Recovery under compression when tensile cracks close
- Abaqus CDP is the de facto standard in research
The Two Fathers of the CDP Model
The Concrete Damaged Plasticity (CDP) model originates from the 1989 paper "A plastic-damage model for concrete" by J. Lubliner and J. Oliver (Universitat Politècnica de Catalunya). Later in 1998, Lee & Fenves from the Abaqus team significantly improved the numerical stability of strain softening, leading to the formulation now most widely used worldwide. This Lee-Fenves version was commercialized as Concrete Damaged Plasticity in Abaqus/Standard.
Physical Meaning of Each Term
- Inertia term (mass term): $\rho \ddot{u}$, meaning "mass × acceleration". Have you ever experienced being thrown forward when slamming on the brakes? That "feeling of being carried forward" 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 so acceleration can be ignored". It cannot be omitted in impact loads or vibration problems.
- Stiffness term (elastic restoring force): $Ku$ or $\nabla \cdot \sigma$. When you pull 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 force $f_b$ (gravity, etc.) 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 it becomes "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 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 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, and 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 equilibrium 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 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
CDP Parameters
| Parameter | Typical Value | Meaning |
|---|---|---|
| Dilation angle ($\psi$) | 30–40° | Dilatancy angle |
| Eccentricity | 0.1 | Eccentricity of the hyperbola |
| $f_{b0}/f_{c0}$ | 1.16 | Biaxial/uniaxial compressive strength ratio |
| $K$ | 2/3 | Yield surface shape parameter |
| Viscosity | 0.0001–0.001 | Viscosity regularization |
Why is viscosity regularization (Viscosity) necessary?
Concrete tensile softening has strong mesh dependency. Viscosity regularization "smoothes out" localization to improve convergence. $\mu = 10^{-4} \sim 10^{-3}$ is typical. Too large makes the response inaccurate.
Summary
Tensile Strength is Only 1/10 of Compression
Ordinary concrete compressive strength is generally 24–60 N/mm², but its tensile strength is only about 1/10 of that, at 2–5 N/mm². The CDP model expresses this extreme asymmetry with independent damage variables for tension and compression (d_t, d_c). In FEM analysis, the input of the tensile stress-strain relationship most sensitively affects the final results, so the accuracy of the tensile softening curve setting determines the analysis quality.
Linear Elements (1st-order elements)
Linear interpolation between nodes. Low computational cost but low 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 important.
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 tangent stiffness matrix every iteration. Achieves quadratic convergence within convergence radius, but computational cost is high.
Modified Newton-Raphson Method
Updates tangent stiffness matrix using 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 $\epsilon = 10^{-3}$–$10^{-6}$). Displacement increment norm: $||\Delta u|| / ||u|| < \epsilon$. Energy norm: $\Delta u \cdot R < \epsilon$
Load Increment Method
Applies 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 method) than to search sequentially from the first page (direct method).
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 "flexible curves"—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
CDP in Practice
Used for seismic analysis of RC buildings, concrete dams, nuclear containment vessels, detailed analysis of PCa members.
Practical Checklist
The Great East Japan Earthquake and Seismic Analysis
After the 2011 Great East Japan Earthquake, FEM analysis using the CDP model was utilized to evaluate the seismic performance of many existing RC buildings. In commissioned research by the Ministry of Land, Infrastructure, Transport and Tourism (2012–2014), it was confirmed that the maximum load prediction by static incremental analysis (pushover analysis) using the CDP model fell within ±15% of loading test values, and it was officially recognized as a complementary method for seismic diagnosis of existing buildings.
Analogy of the Analysis Flow
The analysis flow is actually very similar to cooking. First, buy the ingredients (prepare 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 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 is far from reality. Confirm that results stabilize with 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".
Thinking 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 really fully fixed?" "Is this load really uniformly distributed?"—Correctly modeling real-world constraint conditions is actually the most important step in the entire analysis.
Software Comparison
CDP Tools
Selection Guide
Implementation Differences Between Midas and Abaqus
The CDP model is also implemented in Midas FEA NX, LS-DYNA (MAT_CDPM), and OpenSees (Concrete07) besides Abaqus. However, the yield function forms differ slightly; Abaqus uses a hyperbolic Drucker-Prager...
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