Strain-Rate Constitutive Models

Updated 20 November 2025

Strain-rate constitutive models are frameworks that integrate strain, strain rate, temperature, and internal variables to predict material stress responses.
They employ methodologies ranging from classical phenomenological laws to physics-informed, data-driven approaches for accurate predictions across various loading regimes.
Applications span metals, polymers, composites, and soft materials, with validation techniques including molecular dynamics, dislocation dynamics, and global least-squares fitting.

Strain-rate effects in materials necessitate advanced constitutive modeling strategies that explicitly incorporate the dependence of stress evolution on deformation rate, temperature, and underlying microstructural phenomena. Constitutive models for strain-rate effects are essential in capturing dynamic behavior in polymers, metals, composites, and soft materials under a wide spectrum of loading conditions, from quasi-static to extreme-rate regimes.

1. Fundamental Concepts and Classification

Constitutive models with strain-rate effects describe the stress response $\sigma$ of a material as a functional of strain $\varepsilon$ , strain rate $\dot\varepsilon$ , temperature $T$ , and typically internal state variables or history. The physical origins of rate dependence are diverse:

Viscoelastic/viscoplastic flow: Time-dependent molecular rearrangement or dislocation motion.
Thermally activated processes: Temperature-dependent activation of slip or bond rearrangement.
Dynamic evolution of internal structure: Defect densities, crosslink populations, phase changes.
Micro-inertial phenomena: Inertia associated with defects at high rates or small scales.

Models are often categorized as:

Phenomenological: Johnson–Cook, Ludwik–Hollomon, standard viscoelastic/viscoplastic differential forms.
Physically motivated/multiscale: Kocks–Mecking, dislocation-based, network theory for polymers.
Machine-learning and data-driven: Physics-informed surrogates trained on simulation/experimental data for complex rate-path dependence.

2. Classical and Phenomenological Rate-Dependent Models

Several widely employed constitutive models for strain-rate-sensitive metals and polymers follow structured, parameterized laws, with temperature and rate incorporated via explicit scalings.

Johnson–Cook Model: Used for metals including aluminum alloys and low-melt-point metals, the Johnson–Cook form decouples rate and temperature effects:

$\sigma(\varepsilon, \dot\varepsilon, T) = \left[ A + B\varepsilon^{n} \right] \left[ 1 + C\ln(\dot\varepsilon/\dot\varepsilon_0) \right] \left[ 1 - (T/T_m)^m \right]$

Modifications for unique behaviors—such as strain softening in Field’s metal—decompose the response into a transient (softening) component $S(\varepsilon,\dot\varepsilon,T)$ and a plateau stress $\sigma_s(\dot\varepsilon,T)$ , each with independent rate and temperature dependencies (Nguyen et al., 2020).

Ludwik–Hollomon and Zener–Hollomon Laws: The extended Ludwik–Hollomon is optimal for small-strain, low-to-moderate temperature, and strain-rate regimes, capturing the strain-hardening exponent and rate sensitivity via temperature-dependent parameters. The Zener–Hollomon law,

$Z = \dot\varepsilon\,\exp(Q/RT),\quad \sigma = f(Z)$

is used for high-temperature, rate-controlled creep where work-hardening is negligible (Roy et al., 2012).

Kocks–Mecking Model: For large-strain hardening in metals, the Kocks–Mecking formulation relates work hardening to the evolving microstructure, incorporating an Arrhenius-type dependence on rate and temperature, as well as a saturation stress and normalized activation energy (Roy et al., 2012).

Model	Optimal Regime	Typical Parameters
Johnson–Cook	Low-T, moderate rate	$A,B,n,C,m$
Ludwik–Hollomon	Stage III, low strain, low-T	$K(N(T)), M(T)$
Kocks–Mecking	Stage IV, large strain	$\Theta_0,\sigma_s0,g_0$
Zener–Hollomon	High-T, rate-controlled flow	$Q,\Phi,\eta$

Validation and Regime-Specificity: Each model should only be applied within its data-calibrated regime. Over-extrapolation, especially in the transition between work-hardening and rate-controlled flow, typically results in >20% prediction errors (Roy et al., 2012).

3. Physically-Based and Multiscale Models

Recent developments emphasize direct connection to microstructural phenomena—dislocation dynamics in metals, chain/network dynamics in polymers, or explicit hierarchical upscaling from atomistic to continuum.

Dislocation-Density-Based Plasticity: In single-crystal Cu, the coarse-grained DDD-inspired approach relates slip-system resolved shear stress to both dislocation densities and plastic shear rates (Akhondzadeh et al., 2020):

$\tau_i = \mu b \sqrt{a'_{ij}\rho_j} + s_0 \ln\left(\frac{\dot\gamma_i}{\rho_i b v_0}\right) - \tau_0$

$\dot\gamma_i = \rho_i b v_0 \exp\left(\frac{\tau_i - (\text{forest hardening}) + \tau_0}{s_0}\right)$

The model captures both orientation-dependent and strain-rate-dependent hardening, and requires statistical extraction of parameters from DDD ensembles.

Multisurface Yielding in Polymers: For amorphous polyethylene, hierarchical models utilize MD simulations to extract multisurface Drucker–Prager-type criteria, with high-rate MD yields bridged to experimental rates via exponential scaling laws informed by Bayesian updating (Vu-Bac et al., 2019). The plastic flow rule is non-associated, with hardening directly tabulated from MD.

Load-Bearing Bond Network in Glassy Polymers: Atomistic stress decomposition isolates highly stretched “load-bearing” bonds whose upper-tail stretches dominate hardening. A continuum model encodes this mechanism via a parallel network with orientation-induced back stress and Eyring-type viscosity, capturing rate–temperature coupling over wide dynamic ranges (Zhao, 12 Nov 2024):

$\sigma_{\mathrm{hard}} = \mu_b\,\text{dev}[b_{\mathrm{lb}}^*],\quad \eta_b(T) = \eta_{b,g0}\, a_b(T) \ldots \sinh(Q_b m_{\mathrm{eq}}/T)$

Empirical validation requires that these microstructurally motivated parameters are extracted from targeted MD data.

4. Strain-Rate-Sensitive Polymeric and Soft Network Models

For gels, elastomers, and soft polymers, explicit representation of network dynamics, transient crosslinks, or microstructural kinetics is essential.

Finite-Strain Micro-Mechanism Decomposition: Polyurethane-urea and PBS models utilize Kröner–Lee–type multiplicative decompositions for each microphysical mechanism (e.g., hard/soft segments, coordinate and entanglement crosslinks) and track independent elastic and inelastic flows (Lee et al., 2021, Konale et al., 2023). Free energy for each branch is specified (e.g., Arruda–Boyce network, logarithmic elasticity, time-convolution for transient bonds) and rate-dependence enters through viscosity, crosslink kinetics, or reptation flows.

Multiple relaxation times spanning orders of magnitude (e.g., $\tau_{\mathrm{bd}} \sim 3\,$ s, $\tau_{\mathrm{el}} \sim 5\times10^{-4}\,$ s in PBS) enable the model to accurately track six decades of strain rate.
Validation is demonstrated both in homogeneous (tension/compression) and inhomogeneous (indentation, impact) regimes.

Kelvin–Voigt and QLKV Models in Soft Gels: Quadratic-Law Kelvin–Voigt models, incorporating both strain-stiffening elasticity and viscous dissipation, are critical for capturing the dynamic stress fields during cavitation/histotripsy events. The quadratic term ensures that elastic resistance at high strain is captured realistically, a regime where simple Neo-Hookean models fail (Mancia et al., 2021).

5. Thermodynamically Consistent, Gradient, and Nonlocal Models

For high-rate, high-gradient, or size-dependent phenomena, models that incorporate thermodynamic consistency, nonlocality, and micro-inertia are required.

Micro-Inertia Strain Gradient Plasticity: At extreme strain rates, the inertia of accelerating defects (e.g., dislocations) is macroscopic and contributes to flow stress through micro-force balances (Rahaman et al., 2016):

$\rho \ell^2\,\ddot y^p + ... = \nabla\cdot\xi + \tau - \pi$

Relaxation-time (Maxwell–Cattaneo) terms for all dissipative fluxes avoid unphysical infinite propagation speeds. The resulting nonlocal flow rules admit both size effects and delayed plastic flow, as validated in high-rate plate impact simulations.

6. Data-Driven and Physics-Informed Surrogates

The increasing complexity of real heterogeneous and path-dependent materials has driven the use of hybrid, physics-informed, data-driven constitutive models.

Physics-Informed Surrogate Modeling: Surrogate models encode classical constraints (objectivity, thermodynamic consistency, stress-free reference) into machine-learning architectures. Stress is decomposed into interpretable components (volumetric, isochoric elastic, rate-dependent overstress), with regression (e.g., Gaussian Process) mapping strain/strain-rate invariants onto coefficients in an integrity tensor basis (Upadhyay et al., 2023).

Surrogates trained only on single-mode data can extrapolate to unseen loading paths while maintaining nonnegative dissipation and frame indifference.
Comparative benchmarking evidences that physics-informed ML surrogates outperform both classical and black-box data-driven models in extrapolation and constraint satisfaction.

Hybrid Recurrent Neural Networks: Emerging architectures embed internal-variables–based constitutive models directly within recurrent neural networks, capturing both finite-strain kinematics and history dependence. For path- and rate-dependent composites, performance gains of three orders of magnitude with respect to high-fidelity micromodels are reported, with accurate extrapolation to unseen loading scenarios (cyclic, relaxation, varying strain rates) (Maia et al., 5 Apr 2024). Details on the embedded constitutive equations, return mapping, and viscosity laws depend on the full technical exposition of the referenced work.

7. Parameter Identification and Validation

All credible strain-rate-sensitive constitutive models require identification/calibration of rate- and temperature-dependent parameters through rigorous procedures:

Direct MD or DDD extraction: Parameters are tabulated directly from simulation.
Bayesian updating: Combines high-rate (simulation) and low-rate (experiment) data to yield posterior parameter distributions (Vu-Bac et al., 2019).
Global least-squares fitting: Used extensively in classic and extended Johnson–Cook models for metals and soft matter (Nguyen et al., 2020).
Model-Based Metrology: Ensures intrinsic character by predicting both homogeneous and spatio-temporally varying loading paths with one parameter set, as demonstrated in “probation tests” for semi-crystalline polymers (André et al., 2020).
Efficiency considerations: Models for rate-stiffening or transient-network polymers may require memory-efficient numerical strategies to avoid exponential time/space scaling with loading path duration (Konale et al., 2023).