3. Cyclic plasticity and constitutive models

Durability simulation under LCF and thermo-mechanical fatigue lives or dies by the constitutive model. Elastic FEA gives you a stress field; only a cyclic plasticity model tells you how that stress redistributes as the material yields, hardens, softens, and creeps forward cycle after cycle. This chapter builds the modeling chain from the Bauschinger effect up to the Chaboche framework and shows how to calibrate and validate it against ratcheting data.

Why Isotropic Hardening Is Not Enough

Under monotonic loading, isotropic hardening — a yield surface that expands uniformly — captures the stress–strain curve well. Reverse the load and it fails immediately. Real metals yield at a lower magnitude in reverse than they did in the forward direction: the Bauschinger effect. Isotropic hardening predicts symmetric yield, so it cannot reproduce this asymmetry, and it cannot produce a mean-stress relaxation or ratcheting response at all.

The fix is to let the yield surface translate in stress space. The center of the yield surface is tracked by a backstress tensor α, and the yield condition becomes:

f = J₂(σ − α) − R − σ_y0 = 0

where J₂ is the von Mises equivalent stress of the shifted stress, R is the isotropic drag (size change), and σ_y0 is the initial yield stress. Translation of α is kinematic hardening; growth of R is isotropic hardening. Most durability models combine both.

Prager and Armstrong–Frederick: The First Generation

Prager's linear kinematic hardening sets dα = (2/3)·C·dεᵖ, giving a constant plastic modulus C. It captures the Bauschinger effect but predicts a fixed hysteresis loop shape and, critically, produces no ratcheting under asymmetric stress control — the loop closes exactly. Armstrong–Frederick (A–F) added a recall (dynamic recovery) term:

dα = (2/3)·C·dεᵖ − γ·α·dp

where dp is the accumulated plastic strain increment. The recall term makes the backstress saturate at α_sat = C/γ, which is far more realistic for stabilized cyclic loops. But A–F over-predicts ratcheting badly — often by an order of magnitude — because a single backstress variable cannot simultaneously fit the smooth transition and the ratcheting rate.

The Chaboche Decomposition

Chaboche's insight was to superpose several A–F backstress components, each with its own C_i and γ_i, plus an isotropic variable:

α = Σ α_i ,  i = 1..n
dα_i = (2/3)·C_i·dεᵖ − γ_i·α_i·dp
R = b·(Q − R)·dp

Typical practice uses n = 2 to 4. The first component with a large γ₁ captures the sharp elastic-to-plastic knee; the last component with a small γ_n (often set to zero, i.e. linear) controls the long-range ratcheting slope. The isotropic term R handles cyclic hardening or softening toward saturation Q with rate b. This is the model implemented in Abaqus (*PLASTIC, HARDENING=COMBINED), Ansys, and most fatigue codes.

Ratcheting: The Durability Killer

Ratcheting (cyclic creep) is the progressive accumulation of plastic strain under stress-controlled cycling with a nonzero mean stress. Each cycle the hysteresis loop does not close; it shifts by an increment dε_r. Over thousands of cycles this can mean millimeters of growth — the classic failure mode of pressurized pipes, bolted joints, and thermal sleeves.

Three regimes matter for calibration:

• Shakedown — ratcheting decays to zero, the loop closes elastically. Safe.
• Constant ratcheting — a steady increment per cycle. Life is finite and predictable.
• Accelerating ratcheting — increment grows each cycle, leading to plastic collapse. Must be avoided by design.

The Chaboche model reproduces all three, but only if the parameters are calibrated against ratcheting data — not just stabilized loop data. Fitting only to a symmetric strain-controlled loop will give you a model that looks perfect and predicts ratcheting wrong by 5–20×.

Calibrating Chaboche from Test Data

A practical calibration sequence for a ductile alloy (example parameters below are illustrative, not measured):

1. Initial yield — read σ_y0 from the first deviation from linearity on the monotonic curve. Example: σ_y0 = 320 MPa.
2. Stabilized loop — from a strain-controlled test at ±0.8%, extract the loop tips and the plastic modulus. Fit C₁, γ₁ to the sharp knee and C₂, γ₂ to the mid-range. Example: C₁ = 85000 MPa, γ₁ = 1200, C₂ = 42000 MPa, γ₂ = 180.
3. Ratcheting slope — from a stress-controlled test at σ_m = 50 MPa, σ_a = 350 MPa, measure dε_r/dN after stabilization. Add a third, nearly linear component to match it. Example: C₃ = 6000 MPa, γ₃ = 5.
4. Isotropic term — if the material hardens cyclically, fit Q and b from the peak-stress evolution over the first ~100 cycles. Example: Q = 60 MPa, b = 8.

Run the calibration as a single-element simulation (one C3D8 or CPS4 element, displacement- and force-controlled) before touching the full model. This isolates constitutive behavior from mesh and boundary effects and takes seconds per iteration.

Verification and Common Failure Symptoms

From Constitutive Response to Durability

Once the model reproduces the stabilized loop, the mean-stress relaxation, and the ratcheting rate, the FEA stress–strain history is trustworthy enough to feed a fatigue post-processor. The stabilized plastic strain amplitude drives the strain-life (Coffin–Manson) damage, while the ratcheting increment feeds a ductility-exhaustion or plastic-collapse check. Getting the constitutive layer right is what separates a durability prediction that correlates with test from one that merely produces colorful contours.