Materials Mechanics for CAE
Plasticity, Fatigue, Fracture & Material Models
Materials are where simulation meets reality. Even the most elegant FEM formulation is useless without accurate material parameters. This article covers the mechanical behavior of engineering materials — from the initial elastic region to fracture — and explains how to represent that behavior faithfully in CAE models.
1. Reading the Stress-Strain Curve
I always see stress-strain curves in materials data sheets, but I'm not sure I'm reading them correctly for CAE input. The curve seems to show "engineering stress" but Abaqus asks for "true stress." Why the difference, and how do I convert?
Important question — using engineering stress-strain data in a large-deformation model is a common and significant error. Engineering stress uses the original cross-sectional area, but as the specimen necks, the real area decreases. True stress uses the current area and is always higher than engineering stress at the same loading. For strains above a few percent, the difference is significant. Abaqus and most modern FEM codes want "true stress vs. logarithmic plastic strain" for their plasticity input tables. Fortunately, the conversion formulas are simple.
1.1 Engineering vs. True Stress-Strain
These relations assume volume conservation (constant cross-sectional area × length) — valid for plastic deformation but not elastic.
1.2 Key Features of a Metal Stress-Strain Curve
Proportional limit: Maximum stress where Hooke's law holds exactly. Elastic limit: Maximum stress with fully recoverable deformation (very close to proportional limit for metals). Yield stress σ_y: Typically defined by 0.2% offset method for materials without clear yield point (most non-ferrous alloys, work-hardened steels). Ultimate tensile strength (UTS) σ_u: Maximum engineering stress — corresponds to onset of necking. Fracture strain ε_f: Total elongation at fracture; measure of ductility.
1.3 CAE Plasticity Input Conversion
From a tensile test, the plastic strain (for FEM input) is total strain minus elastic strain:
The first point on the FEM plasticity table must be at $\varepsilon_p = 0$ (no plastic strain at yield).
1.4 Typical Stress-Strain Data for Common Materials
| Material | E (GPa) | σ_y (MPa) | σ_u (MPa) | ε_f (%) | n (hardening) |
|---|---|---|---|---|---|
| Low carbon steel S235 | 210 | 235 | 360–510 | 26 | 0.25 |
| High strength steel S960 | 210 | 960 | 1000+ | 12 | 0.05 |
| Aluminum 6061-T6 | 68.9 | 276 | 310 | 12 | 0.06 |
| Aluminum 7075-T651 | 71.7 | 503 | 572 | 11 | 0.05 |
| Ti-6Al-4V (annealed) | 113.8 | 880 | 950 | 14 | 0.04 |
| Inconel 718 | 199.9 | 1034 | 1241 | 12 | 0.06 |
| SS316L (annealed) | 193 | 210 | 515 | 40 | 0.35 |
2. Elastic-Plastic Material Models
In Abaqus I see options like "isotropic hardening," "kinematic hardening," and something called "Chaboche." When does it matter which one I choose?
It depends whether you're loading the material monotonically or cyclically. Isotropic hardening is fine for monotonic loading — the yield surface just grows uniformly. But for cyclic loading (fatigue, low-cycle fatigue), isotropic hardening predicts that the stress-strain hysteresis loop keeps shrinking with each cycle (ratcheting and shakedown behavior all wrong). Kinematic hardening lets the yield surface translate without growing — better for cyclic response but not perfect. Chaboche combines both: the yield surface can both grow and translate, and it captures the transient hardening and saturation that real metals show under cyclic loading. For fatigue analysis of pressure vessels or turbine discs, Chaboche is worth the effort.
2.1 Rate-Independent Plasticity Framework
Three ingredients of any plasticity model:
- Yield function: $f(\boldsymbol{\sigma}, \kappa) \leq 0$ — defines the elastic domain. E.g., von Mises: $f = \sigma_\text{vM} - \sigma_y(\bar{\varepsilon}_p)$
- Flow rule: Direction of plastic strain increment: $d\boldsymbol{\varepsilon}^p = d\bar{\varepsilon}^p \frac{\partial f}{\partial\boldsymbol{\sigma}}$
- Hardening rule: How yield surface evolves with plastic strain
2.2 Isotropic Hardening
Yield surface expands uniformly; yield stress is a function of accumulated plastic strain $\bar{\varepsilon}_p$:
Power law (Ramberg-Osgood): $\sigma_y = \sigma_{y0} + K(\bar{\varepsilon}_p)^n$ — simple, widely used for monotonic loading.
Voce law: $R = R_\infty(1 - e^{-b\bar{\varepsilon}_p})$ — saturates at $R_\infty$; better for materials that strain-harden to a plateau.
2.3 Kinematic Hardening (Prager/Ziegler)
Yield surface translates in stress space without changing size. The back stress tensor $\boldsymbol{\alpha}$ defines the center:
2.4 Chaboche Combined Hardening
The industry-standard model for low-cycle fatigue and ratcheting:
Typically 2–3 backstress components are used. Parameters $(C_k, \gamma_k, R_\infty, b)$ are identified from cyclic strain-controlled tests at multiple strain amplitudes.
2.5 Hyperelastic Models (Rubber/Soft Materials)
For rubber, silicone, and soft tissues, the strain energy density $W$ determines the stress state. Common models:
Neo-Hookean: $W = C_{10}(I_1 - 3)$ (one parameter)
Mooney-Rivlin: $W = C_{10}(I_1 - 3) + C_{01}(I_2 - 3)$ (two parameters)
Ogden: $W = \sum_p \frac{2\mu_p}{\alpha_p^2}(\lambda_1^{\alpha_p}+\lambda_2^{\alpha_p}+\lambda_3^{\alpha_p}-3)$ (up to 6 parameters)
3. Creep and Viscoelasticity
I'm analyzing a turbine blade that operates at 900°C. My colleague says I need to include creep. What is creep exactly, and when does it become important?
Creep is slow, time-dependent plastic deformation under constant stress — it doesn't require any additional force to keep occurring. The rule of thumb for metals: creep becomes significant when the temperature exceeds about 0.3–0.4 times the melting temperature (in Kelvin). For steels (melting ~1500°C = 1773 K), that's above about 270–440°C. For nickel superalloys in turbine blades, they're operating at 900°C / 1600°C melting ≈ 0.56 T_melt — creep is absolutely the dominant life-limiting mechanism. If you ignore creep at these temperatures, your predicted life will be wildly optimistic.
3.1 Creep Stages and Constitutive Laws
A standard creep curve at constant stress shows three stages:
- Primary (transient) creep: Decreasing strain rate as dislocations rearrange. $\dot{\varepsilon} \propto t^{-n}$
- Secondary (steady-state) creep: Constant minimum strain rate. Most design-relevant stage.
- Tertiary creep: Accelerating strain rate → fracture (grain boundary cavitation, necking).
Norton power law for secondary creep:
where A and n are material constants, Q is activation energy, R is gas constant, T is absolute temperature. Typical n values: 3–7 for metals.
3.2 Viscoelasticity (Polymers and Composites)
For polymers, the stress-strain relationship is time-dependent even at room temperature. The generalized Maxwell model (Prony series):
where $G_\infty$ is the long-time (equilibrium) shear modulus, $G_k$ are instantaneous moduli of each Maxwell element, and $\tau_k = \eta_k/G_k$ are relaxation times. The Prony series parameters are fit from dynamic mechanical analysis (DMA) tests or stress relaxation tests.
4. Fatigue Failure (S-N, ε-N, Goodman)
I've heard that parts can fail at stress levels way below yield stress if they're cycled enough times. How does fatigue analysis work in CAE?
Fatigue failure is one of the most common failure modes in engineering — responsible for maybe 80-90% of all mechanical failures. The key insight is that microscopic cracks initiate at stress concentrations even at low nominal stress, then grow incrementally with each cycle until they reach critical size and cause sudden fracture. In CAE fatigue analysis, you first compute stress history from a static or dynamic simulation, then post-process those stresses through an S-N or ε-N curve to predict life. Tools like FE-SAFE, nCode DesignLife, or Ansys Mechanical Fatigue do this automatically.
4.1 Stress-Life (S-N) Method — High Cycle Fatigue
For N > 10⁴ cycles (stress below yield at all times). The Basquin relation:
where $\sigma_f'$ is the fatigue strength coefficient, $N_f$ is cycles to failure, and $b$ is the Basquin exponent (typically -0.05 to -0.12 for metals).
Equivalently in log-log space: $\log\sigma_a = \log C - m\log N_f$, the classic S-N curve slope.
Endurance limit (S_e): For ferrous alloys (steels, cast iron), the S-N curve flattens below a threshold — the endurance limit. For steel: $S_e \approx 0.5\sigma_u$ (up to σ_u = 1400 MPa). Above this level, infinite life is assumed. Non-ferrous alloys (aluminum, copper, titanium) do not have a true endurance limit — the S-N curve continues downward.
4.2 Mean Stress Correction (Goodman, Gerber, Soderberg)
Fatigue data is typically taken at zero mean stress (fully reversed, R = -1). A tensile mean stress reduces fatigue life; compressive mean stress increases it.
Modified Goodman line (linear, conservative):
Gerber parabola (fits ductile metal data better):
Safety factor against fatigue failure:
4.3 Strain-Life (ε-N) Method — Low Cycle Fatigue
When stresses exceed yield (LCF: N < 10⁴), strain is the better damage indicator. Morrow equation:
where $\varepsilon_f'$ is the fatigue ductility coefficient and $c$ is the fatigue ductility exponent (typically -0.5 to -0.7).
4.4 Damage Accumulation: Miner's Rule
For variable amplitude loading, linear damage accumulation:
where $n_i$ is the number of cycles at stress amplitude $\sigma_{a,i}$ and $N_{f,i}$ is the life at that amplitude. In practice, Miner's rule overestimates life in high-low loading sequences and underestimates in low-high sequences — a 20-50% scatter is common. More sophisticated damage models (damage parameter approach, energy-based) exist but require more test data.
5. Fracture Mechanics (K, G, J-integral, Paris Law)
I've seen "fracture toughness K_IC" on material data sheets. What does that number actually tell me and when should I use fracture mechanics instead of just checking von Mises stress against yield?
K_IC tells you the critical crack size — how large a crack can be before catastrophic fracture occurs at your design stress. If you're designing a pressure vessel that must never fail catastrophically (even if a crack develops from corrosion or welding defects), fracture mechanics is mandatory. Classic von Mises check tells you nothing about cracks. The fundamental question in fracture mechanics is: given this crack size a and this applied stress, will the crack grow stably (safe — you can detect it before failure) or unstably (catastrophic fracture)? K_IC draws that dividing line.
5.1 Linear Elastic Fracture Mechanics (LEFM)
Near a crack tip, the stress field is singular:
The stress intensity factor $K_I$ (Mode I opening) characterizes the severity of the stress field:
where $\sigma$ is nominal stress, $a$ is crack length, and $Y$ is a geometry correction factor (equal to 1.0 for a through crack in infinite plate). Failure when: $K_I = K_{IC}$ (plane strain fracture toughness).
5.2 Typical Fracture Toughness Values
| Material | K_IC [MPa·√m] | σ_y [MPa] | Notes |
|---|---|---|---|
| High-strength steel 4340 | 46–66 | 1,570 | Classic high-strength low-toughness tradeoff |
| Structural steel A36 | 200+ | 250 | Very tough, large plastic zone |
| Aluminum 7075-T651 | 24–29 | 503 | Aerospace alloy, relatively brittle |
| Aluminum 2024-T351 | 35–40 | 345 | Tougher, widely used in fuselage |
| Ti-6Al-4V | 55–110 | 880 | Depends strongly on processing |
| Silicon carbide (SiC) | 3–4 | — | Ceramic — extremely brittle |
| Soda-lime glass | 0.7–0.8 | — | Always handle as pre-cracked |
5.3 J-Integral (Elastic-Plastic Fracture Mechanics)
When the plastic zone is not small compared to crack length (ductile materials, high toughness), LEFM is invalid. The J-integral is a path-independent contour integral that generalizes K to elastic-plastic problems:
where $W$ is strain energy density and $\mathbf{T}$ is traction on contour Γ. In FEM, J is computed by the domain integral (equivalent area/volume form). Fracture when $J = J_{IC}$.
Relation to K in the elastic case: $J = K^2(1-\nu^2)/E$ (plane strain).
5.4 Fatigue Crack Growth: Paris Law
Under cyclic loading, crack growth rate per cycle:
Typical Paris law exponents: m ≈ 3 for steels, m ≈ 3–4 for aluminum. Residual life from initial crack size $a_0$ to critical $a_c$:
For constant amplitude loading with integer m: this integrates analytically. For variable amplitude, use cycle-by-cycle integration.
6. Composite Materials
I'm working on a carbon fiber composite structure. I know the material is anisotropic, but I'm not sure how to set up the material properties in the FEM software. What are the key differences from isotropic metals?
Three main differences: (1) Anisotropy — a UD carbon/epoxy lamina has E₁ ≈ 135 GPa along fibers but only E₂ ≈ 9 GPa transverse. You need to define all orthotropic elastic constants and the fiber direction in the model. (2) Layup dependence — the actual component stiffness depends on the stacking sequence and orientations of your laminae; classical lamination theory (CLT) or shell FEM handles this. (3) Failure modes are completely different from metals — fiber fracture, matrix cracking, delamination, and fiber-matrix interfacial failure each have different criteria (Tsai-Wu, Hashin, Puck). You can't just check von Mises. Most FEM codes have built-in composite failure criteria, but you need to understand what they're calculating to interpret the results.
6.1 Orthotropic Elastic Constants (Unidirectional Lamina)
A transversely isotropic (UD) lamina requires 5 independent constants: E₁, E₂, G₁₂, ν₁₂, G₂₃.
The compliance matrix for plane stress in the material coordinate system:
Typical values for carbon/epoxy UD lamina:
| Property | Value |
|---|---|
| E₁ (fiber direction) | 135 GPa |
| E₂ (transverse) | 9 GPa |
| G₁₂ (in-plane shear) | 5 GPa |
| ν₁₂ | 0.30 |
| ρ | 1,550 kg/m³ |
| F₁t (tensile strength, fiber) | 1,500 MPa |
| F₂t (tensile strength, transverse) | 50 MPa |
| F₁₂s (shear strength) | 70 MPa |
6.2 Composite Failure Criteria
Tsai-Wu criterion (smooth failure envelope, accounts for tension-compression asymmetry):
Hashin criteria (separate criteria for fiber failure vs. matrix failure, mode-dependent):
7. Material Testing and CAE Parameter Identification
My company uses a material that isn't in any database — a special aluminum alloy for a heat exchanger. How do I get the right material constants for my FEM model?
You need to run the right combination of physical tests and fit the model parameters. For basic plasticity, a monotonic tensile test gives E, σ_y, and the hardening curve. For cyclic plasticity (Chaboche), you need strain-controlled cyclic tests at two or three strain amplitudes plus hold periods. For creep, run constant-load tests at multiple stress levels and temperatures — from the Norton equation slope, you extract n and A. The fitting is often done with Python (scipy.optimize) or proprietary tools like nCode, LS-OPT, or Abaqus material calibration. The key principle: test conditions must match simulation conditions — temperature, strain rate, loading mode. Material properties measured at 20°C and quasi-static loading may not apply at 200°C under dynamic impact.
7.1 Required Tests for Common Material Models
| Material Model | Required Tests | Parameters Extracted |
|---|---|---|
| Linear elastic | Monotonic tension | E, ν (+ Poisson test) |
| Isotropic hardening | Monotonic tension (to fracture) | σ_y, K, n (Ramberg-Osgood) |
| Kinematic hardening | Cyclic tension-compression, stabilized loop | C, γ (one backstress) |
| Chaboche | Strain-controlled cyclic at 3+ amplitudes | C₁, γ₁, C₂, γ₂, R∞, b |
| Norton creep | Constant-load creep at 3+ stress levels and 2+ temps | A, n, Q |
| Hyperelastic (Mooney-Rivlin) | Uniaxial + biaxial + planar shear tension | C₁₀, C₀₁ |
| S-N fatigue | Fatigue test at 5+ stress levels (R=-1) | σ'_f, b |
- Convert engineering stress-strain to true stress and log plastic strain before inputting plasticity tables to FEM
- Isotropic hardening for monotonic loading; kinematic/Chaboche for cyclic loading and fatigue
- Fatigue life: S-N for HCF (σ < σ_y); ε-N for LCF (σ > σ_y); Goodman correction for mean stress
- Fracture mechanics required when cracks may exist: $K_I < K_{IC}$ for no unstable fracture
- Paris law: $da/dN = C(\Delta K)^m$ — integrate to get residual life from detected crack to critical size
- Composites need all orthotropic constants + fiber direction + mode-specific failure criteria
Related Interactive Tools
- S-N Curve Generator — Basquin's law parameters → fatigue life at any stress amplitude
- Goodman Diagram — Mean stress + alternating stress → safety factor plot
- Paris Law Crack Growth — Initial crack size → residual life calculation
- Stress Concentration Factor Kt — Notch/hole geometry → Kt value