2. Material models for automotive applications

Automotive structural simulation relies on a small set of constitutive model families, each chosen to reproduce a specific physical mechanism observed in the component under its actual loading history. Selecting the wrong family, or calibrating the right family with the wrong test data, produces results that may look plausible in a contour plot but fail badly in a crash, durability, or NVH assessment. This section maps the five model families used most often in body, chassis, and powertrain work — elastoplasticity, viscoelasticity, hyperelasticity, damage, and failure criteria — to the parameters they require and the physical meaning of those parameters.

Elastoplasticity: Permanent Deformation in Metals

Elastoplastic models describe metals such as DP590, AA6061-T6, and cast aluminum beyond the yield point. The elastic branch is defined by Young's modulus E and Poisson's ratio ν. The plastic branch requires a hardening law and a yield surface. For von Mises plasticity, the yield surface is a cylinder in principal stress space, and the hardening curve σ_y(ε_p) defines how the flow stress grows with equivalent plastic strain.

Two hardening formulations dominate automotive practice. Isotropic hardening expands the yield surface uniformly and is adequate for monotonic loading such as a single crash event. Kinematic hardening translates the yield surface and captures the Bauschinger effect, which matters for springback prediction and for low-cycle fatigue where the material is loaded in tension and compression alternately. The combined isotropic–kinematic (Chaboche) model uses a backstress tensor with parameters C and γ per term; C controls the initial kinematic hardening modulus and γ controls the rate at which that modulus saturates.

Calibration inputs are typically a uniaxial tensile test to fracture, a shear test, and a stack compression or bulge test to extend the curve to large strains. The tensile test alone under-predicts the true stress–strain response beyond necking because the gauge section is no longer uniform; a correction such as the Bridgman or Hill model is applied. The calibrated output is a tabulated true stress versus true plastic strain curve, plus C and γ if kinematic hardening is active. A practical check: run a single-element simulation of the calibration test and confirm the reaction force matches the experimental curve within a few percent before applying the model to a full assembly.

Viscoelasticity: Time and Rate Dependence

Viscoelastic models apply to polymers, elastomeric bushings, adhesives, and some composites. The defining feature is that stress depends on strain rate and on load history, not on strain alone. The generalized Maxwell model represents this as a spring in parallel with several Maxwell elements, each element being a spring and dashpot in series. Each Maxwell branch contributes a relaxation time τ_i = η_i / E_i and a modulus E_i.

Calibration uses stress relaxation tests (apply a fixed strain, record decaying stress) or dynamic mechanical analysis (DMA) at multiple frequencies and temperatures. The Prony series coefficients g_i and τ_i are fitted from these curves. Because automotive elastomers operate over a wide temperature range, a Williams–Landel–Ferry (WLF) shift function with reference temperature T_ref and constants C1, C2 is usually calibrated alongside the Prony series so that a master curve at the reference temperature can be shifted to service conditions. A common failure mode is fitting a Prony series to a single temperature and then applying it at 80 °C, where the real material has relaxed almost completely.

Hyperelasticity: Large-Strain Rubber Behavior

Hyperelastic models describe rubber mounts, seals, and air springs, where strains routinely exceed 100% and the material is nearly incompressible. The strain energy density function W is expressed in terms of invariants I1, I2, I3 or principal stretches. The Neo-Hookean model needs only C10; Mooney–Rivlin adds C01; Yeoh uses C10, C20, C30; Ogden uses μ_i and α_i pairs. The incompressibility constraint is enforced through the bulk modulus D1 (or K), which is often set from a volumetric compression test rather than assumed.

Calibration requires multiple deformation modes because a single test cannot uniquely determine the coefficients. A standard set is uniaxial tension, planar tension (pure shear), and equibiaxial tension. Fitting only uniaxial data with a Mooney–Rivlin model frequently produces unstable behavior in compression, where the model may predict a negative slope. The practical rule is to fit at least three modes and to verify the fitted model against a fourth independent test, such as a simple shear or a bulge test.

Damage and Failure Criteria

Damage models degrade stiffness as micro-cracks accumulate. The GISSMO (Generalized Incremental Stress State Dependent Damage Model) and Johnson–Cook damage models are common in crash simulation. GISSMO requires a failure strain curve as a function of stress triaxiality η and Lode angle parameter θ̄, plus a damage exponent n and a critical damage value D_crit at which element deletion occurs. Calibration uses specimens that span the triaxiality range: smooth round bars (high positive η), notched bars (moderate), shear specimens (near zero), and compression specimens (negative).

Failure criteria such as the forming limit diagram (FLD) or the CrachFEM criterion are used when the goal is to predict onset rather than progressive degradation. The FLD is calibrated from Nakajima or Marciniak tests and is expressed as major versus minor strain at necking. It is appropriate for sheet metal forming but not for thick-section crash components, where a stress-based or damage-based criterion is more suitable.

Choosing the Right Model and Avoiding Common Calibration Errors

The model family should follow the dominant physics, not the analyst's familiarity. A front rail in a 64 km/h offset crash is dominated by plastic flow and ductile fracture, so elastoplasticity plus GISSMO is appropriate. An engine mount under vibration is dominated by viscoelastic dissipation, so a Prony series with WLF shift is appropriate. A door seal under compression is dominated by large-strain hyperelasticity, so a Yeoh or Ogden model is appropriate. Mixing families — for example, using hyperelasticity for a metal — produces a model that fits one test but has no predictive power elsewhere.

Three calibration errors recur in automotive projects. First, extrapolating a hardening curve beyond the calibrated strain range without a validated extrapolation law; the extrapolated branch often controls the crash result. Second, fitting a damage model to specimens that do not span the triaxiality range present in the component; a model calibrated only on smooth bars will under-predict failure in shear-dominated regions. Third, ignoring strain-rate sensitivity in the base material while calibrating a rate-dependent damage model; the two effects are coupled and must be fitted together.

Parameter Summary and Physical Interpretation

Each parameter in this table has a direct physical interpretation and a corresponding test that can identify it. The calibration task is to design a test matrix that spans the operating range of the component, fit the model to that matrix, and verify the fit against an independent condition. When the model family, the test matrix, and the verification case are aligned with the component's actual loading, the calibrated parameters become predictive rather than merely descriptive.