Constitutive Modeling Framework Research Articles

Top-down constitutive modelling to capture nanoscale shear localization

Deformation localization as exemplified by earthquakes, landslides, shear banding in solids, and failure of engineering components is of utmost importance. In practice, differentiating the mechanical behavior in such generative narrow bands from the rest part, with difference by orders of magnitude in characteristic size, flow strength, temperature, and shearing rate, is both experimentally and computationally formidable. Here we propose a machine-learning-based constitutive modeling framework to overcome this barrier borne from conventional top-down continuum modelling approach. The model enables us to realize ultra-fine resolutions for deformation in those narrow bands with high efficiency. Taking metallic glasses (MGs) as an example, our model captures well shear localization in BMGs across a broad range of temperatures (0 K to its melting point of ∼1000 K) and strain rates (10−4 to 108/s). We verify through this model the width of shear bands (SBs) in MGs is on the order of 5–8 nanometers, which is resulted from a cascade of (intervening) events, from localized shearing to plastic heating, subsequent temperature rise to thermal softening, and accelerated flow rate to strain-rate hardening. Temperature rise in SBs is a resultant of heat flow and plastic dissipation, but strongly depend on thermal conductivity: Low thermal conductivity facilitates strain localization and great temperature rise. It helps understanding the current controversy upon experimentally measured temperature rise ranging from several K to ∼1000 K. Lastly, strain rates within SBs are approximately one to two orders of magnitude higher than externally applied strain rates, and in general shearing in adiabatic SBs is faster than that in isothermal condition.

A nonlinear continuum framework for constitutive modeling of active polymer gels

Chemo-mechanical transduction is one of the key mechanisms that has formed the basis for designing bio-inspired self-driven synthetic systems from soft materials. Polymer hydrogels that use Belousov–Zhabotinsky (BZ) reaction are a unique class of dynamical reaction–diffusion (RD) systems that can continuously transduce internal chemical energy, from the reaction, to produce sustained mechanical work. In particular, BZ gels represent a complex nonlinear chemo-mechanical system, wherein, the autocatalytic oscillatory BZ reaction drives the rhythmic mechanical deformations through polymer-solvent inter–diffusion. The objective of our work is to develop a standardized finite element (FE) framework for chemically driven active hydrogels that captures nonlinear elastic deformations with limited chain extensibility. The distinguishing feature of our approach is that, unlike other approaches, it combines reaction kinetics, solvent transport, elastodynamics of the polymeric network, and polymer-solvent friction under a unified FE framework. Moreover, we adapt our approach to a specific case of BZ gels and capture their swelling-deswelling characteristics. We first implement our FE framework in MATLAB that subsequently, forms the basis for constructing a three-dimensional user element subroutine (3D-UEL) in ABAQUS. Ultimately, through our simulations, we are able to capture all the essential features of BZ gels that includes chemically driven mechanical deformations. In addition, we also demonstrate that our 3D-UEL efficiently captures the chemo-mechanical response of “stent-shaped” BZ gels–a non-standard 3D geometry. In essence, our FE approach not only allows us to simulate BZ gels but also provides a template for other active, dynamical, RD-based systems, driven by chemo-mechanical transduction, irrespective of internal or external mechanisms.

Exploring the shear performance and predictive shear capacity of corroded RC columns utilizing the modified compression-field theory: An investigative study

Corrosion damage substantially weakens the shear strength of reinforced concrete (RC) structures, severely compromising their seismic resistance. Therefore, developing an accurate constitutive model is imperative to investigate the impact of corrosion damage on the shear resistance of RC structures. This study presents a corroded RC shear constitutive model based on the modified compression-field theory (MCFT). To capture the flexural, shear, and slip responses of corroded RC columns accurately, a nonlinear analysis model based on OpenSees is proposed. Comparing the results from four sets of quasi-static tests reveals that the proposed constitutive model and analysis framework can well simulate the seismic performance of corroded RC columns. Based on the comparative analysis, the influence of corrosion-induced damage, reinforcement ratio, and axial load on the shear performance of RC columns is discussed. The investigation findings reveal that high corrosion levels and low reinforcement ratios remarkably weaken the shear strength of RC columns. Additionally, high axial load levels accelerate the rate of post peak shear stress degradation. Moreover, this study proposes a method for estimating the shear strength of corroded RC columns based on the MCFT. The presented approach provides a solid foundation for evaluating the shear and seismic performance of existing RC structures.

Deep learning-based modeling of the strain rate-dependent thermomechanical processing response for a novel HIPed P/M nickel-based superalloy

Understanding and modeling the thermomechanical processing (TMP) response of a novel HIPed powder metallurgy (P/M) superalloy is crucial for its industrial application. This research characterizes the TMP-responding behavior of a novel HIPed FGH4113A P/M superalloy at 1000–1150 ℃ with strain rates of 0.01–5 s−1. The results depict that the adiabatic temperature rise during high-strain-rate TMP will accelerate the stress softening. A two-stage deep learning (DL)-based constitutive model framework, coupled with the deformation-induced temperature rise and its influence on flow stress, was first proposed and compared with the traditional Arrhenius-type model. Two types of DL approaches including the backpropagation (BP) and long short-term memory (LSTM) neural networks were trained and compared separately. It is noted that the LSTM neural network, considering previous deformation history in its unique gate structure, depicts the best prediction accuracy and generalization ability under large strains than other models. Then the strain rate-dependent hot workability and flow softening mechanisms were modeled and discussed. High-strain-rate deformation (5 s−1) in certain temperatures (1110–1150 ℃) can avoid unstable defects and transform the generated deformation heat into the driving force for the activation of multiple dynamic recrystallization mechanisms. Therefore, the near fully recrystallized microstructure can be achieved similarly to that at 0.01 s−1, indicating the feasibility and superiority of the high-strain-rate TMP of P/M superalloys in a certain temperature range.

Anisotropy in the Liquefaction Resistance of Fibre Reinforced Sand.

Adding discrete fibres to sand has been seen as a feasible technique to improve sand's strength as well as liquefaction resistance. Considering the anisotropic distribution of fibre orientations, the anisotropy in the liquefaction resistance of the reinforced sand is also introduced using fibres. Here, the triaxial compression and extension test results of unreinforced and fibre-reinforced sand in different density states are provided, from which the anisotropy in the liquefaction resistance of fibre-reinforced sand is demonstrated. Fibre reinforcement improves the liquefaction resistance of sand by introducing both the densifying effect and the confining effect. The inclusion of fibres increases both the slope and the intercept of the strength envelope in comparison with the unreinforced sand under triaxial compression, while the strength envelope is not affected by fibres under triaxial extension. Stress contribution of fibres makes the ESP of the composite under undrained loading reverse its direction to develop even though the phase transformation is absent. The stress ratio initiating the ESP reversal is irrespective of the fibre content but dependent on the density state under triaxial compression. Under triaxial extension, the stress ratio initiating the ESP reversal remains the same in the samples with varied density states and fibre contents. The mechanism correlating to the strength envelope and ESP reversal of the fibre-reinforced sand was demonstrated following a rule of mixture based constitutive modelling framework. By introducing an alternatively defined pore pressure ratio that incorporates the stress contribution of fibres, the liquefaction state of the fibre reinforced sand is reasonably assessed. Liquefaction remains absent in the sand once the fibres are mixed. The anisotropy in the liquefaction resistance of fibre-reinforced sand arises, as the predominant role played by the fibres to suppress the liquefaction is different when varied loading paths are involved, which is sourced from the anisotropic distribution of fibre orientations.

Steady elongational flow from rotarance theory

The physics of macromolecular orientation has been used to explain the elasticity of polymeric liquids. Specifically, by first sculpting a rigid bead–rod likeness of the macromolecule, we can then derive its hydrodynamic resistance to orientation. The solution for the orientation distribution function has then been used, by integration in phase space, to get rheological material functions in both (i) small- and (ii) large-amplitude oscillatory shear flow, including its limiting case, and (iii) steady shear flow. However, rheological material functions in steady homogeneous extension from rigid bead–rod theory remain elusive. In this paper, we derive the orientation distribution function, and the rheological material functions, for suspensions of general rigid bead–rod structures. We focus on the time-steady viscosities in extension, and we first do so for general extensional kinematics. We then obtain the viscosities in steady extension for (iv) uniaxial extension, (v) planar extension, and (vi) biaxial extension. We close with a worked example, in which we use our new result for the steady uniaxial extensional viscosity to build a bridge between the macromolecular theory and the Oldroyd framework for rheological constitutive models. We, thus, arrive at a constitutive equation whose parameters are deducible from the moments of inertia of the macromolecule, and thus, deducible from macromolecular architecture alone. Our model is accurate up to third order for time-independent flows and is accurate to second order for time-dependent ones.

Functional non-uniformity of periodontal ligaments tunes mechanobiological stimuli across soft- and hard-tissue interfaces

The bone-periodontal ligament-tooth (BPT) complex is a unique mechanosensing soft-/hard-tissue interface, which governs the most rapid bony homeostasis in the body responding to external loadings. While the correlation between such loading and alveolar bone remodelling has been widely recognised, it has remained challenging to investigate the transmitted mechanobiological stimuli across such embedded soft-/hard-tissue interfaces of the BPT complex. Here, we propose a framework combining three distinct bioengineering techniques (i, ii, and iii below) to elucidate the innate functional non-uniformity of the PDL in tuning mechanical stimuli to the surrounding alveolar bone. The biphasic PDL mechanical properties measured via nanoindentation, namely the elastic moduli of fibres and ground substance at the sub-tissue level (i), were used as the input parameters in an image-based constitutive modelling framework for finite element simulation (ii). In tandem with U-net deep learning, the Gaussian mixture method enabled the comparison of 5195 possible pseudo-microstructures versus the innate non-uniformity of the PDL (iii). We found that the balance between hydrostatic pressure in PDL and the strain energy in the alveolar bone was maintained within a specific physiological range. The innate PDL microstructure ensures the transduction of favourable mechanobiological stimuli, thereby governing alveolar bone homeostasis. Our outcomes expand current knowledge of the PDL's mechanobiological roles and the proposed framework can be adopted to a broad range of similar soft-/hard- tissue interfaces, which may impact future tissue engineering, regenerative medicine, and evaluating therapeutic strategies. Statement of significanceA combination of cutting-edge technologies, including dynamic nanomechanical testing, high-resolution image-based modelling and machine learning facilitated computing, was used to elucidate the association between the microstructural non-uniformity and biomechanical competence of periodontal ligaments (PDLs). The innate PDL fibre network regulates mechanobiological stimuli, which govern alveolar bone remodelling, in different tissues across the bone-PDL-tooth (BPT) interfaces. These mechanobiological stimuli within the BPT are tuned within a physiological range by the non-uniform microstructure of PDLs, ensuring functional tissue homeostasis. The proposed framework in this study is also applicable for investigating the structure-function relationship in broader types of fibrous soft-/hard- tissue interfaces.

A microstructure-informed constitutive model for hierarchical materials with subgrain features

Additively manufactured metal alloys often have a hierarchical microstructure consisting of subgrain features. The subgrain features, being smaller in size than grains, significantly influence the mechanical properties and anisotropy of the material. Recognizing the critical role of subgrain features, this work proposes a novel generalized microstructure-informed constitutive modeling framework for hierarchical materials using a mechanism-based crystal plasticity formulation. Considering direct energy deposited (DED) stainless steel (SS) 316 L as the reference material we first categorize subgrain features as a network of solidification cells and a scattered distribution of secondary phases. Then we advanced the state-of-the-art mechanism-based crystal plasticity formulation by explicitly accounting for the geometric and mechanistic effects of both cellular network and scattered secondary phase subgrain features. Homogenization schemes are developed to implement the constitutive model on a finite element analysis (FEA) framework. The anisotropy is realized as a natural outcome of a preferred alignment of cells in the build direction. The model is then calibrated using experimental data. Limited validation of the model is also performed. Results show that explicitly accounting for the subgrain features in a constitutive model allows one to accurately capture the macroscopic and microscopic response of a hierarchical material.

Modeling Lifetime Performance of Monopile Foundations for Offshore Wind Applications

This paper explores the application of a numerical method for modeling pseudorandom cyclic loading, at very large cycle numbers, to the design of offshore wind turbine foundations. The work expands the development of a novel constitutive modeling framework, the hyper-plastic accelerated ratcheting model (HARM), for which the key constitutive equations and the calibration method are presented. HARM captures both the nonlinear hysteretic behavior during cycling and the accumulation of permanent deformation (ratcheting) with large cycle numbers in a rigorous, yet computationally efficient manner, enabling the computation of foundation response over a lifetime of loading. This paper demonstrates how the approach can be applied to the cyclic pile field testing from the pile soil analysis (PISA) project. Following calibration, the model is used to assess pile response to three load signals representative of operational and extreme loads throughout the lifetime of a full-scale wind turbine foundation: (1) a short storm, (2) a 35-h storm, and (3) lifetime loading. The paper discusses how computational efficiency can be achieved while maintaining a high level of calculation accuracy.

Model-driven identification framework for optimal constitutive modeling from kinematics and rheological arrangement

Soft materials such as biological tissues or magnetorheological elastomers present complex mechanical behaviors that include large deformations, numerous nonlinearities, time- or even external field (magnetic)-dependent responses. The description of their constitutive modeling is challenging and often time-consuming. Numerical algorithms to automatically calibrate model parameters have provided invaluable tools to help this purpose. However, these are mostly limited to the fitting of a set of pre-defined parameters associated with the model used. In this work, we go a step further by developing a machine learning framework capable of automatically identifying not only such model parameters but also the optimal kinematics and rheological model. To this end, we present a multiphysics model-driven framework that optimally selects the most suitable model kinematics, its rheological components and their arrangement for a given set of experimental curves. Subsequently, it calibrates all the material constants belonging to such a model, independent of its complexity. We demonstrate the versatility and capabilities of this framework with examples on hyperelastic, viscohyperelastic and magneto-viscohyperelastic materials. The present work opens new routes to not only fit model parameters but to identify the constitutive ingredients and underlying mechanisms needed to describe nonlinear responses of soft active materials.

Correction: An asynchronous parallel high-throughput model calibration framework for crystal plasticity finite element constitutive models

Correction: An asynchronous parallel high-throughput model calibration framework for crystal plasticity finite element constitutive models

Robust deep learning framework for constitutive relations modeling

Robust deep learning framework for constitutive relations modeling

An asynchronous parallel high-throughput model calibration framework for crystal plasticity finite element constitutive models

An asynchronous parallel high-throughput model calibration framework for crystal plasticity finite element constitutive models

A constitutive model for elastomers tailored by ionic bonds and entanglements

Over the past decade or two, the concept has emerged of using multiple types of weak interactions simultaneously to enhance the mechanical properties of elastomers. These weak interactions include physical entanglements, hydrogen bonds, metal-coordination bonds, dynamic covalent bonds, and ionic bonds. The combination of entanglements and ionic bonding has been minimally explored and is particularly exciting because of the broad application space for polyelectrolytes. In this work, a constitutive model framework is developed to describe the response of elastomers with both ionic bonds and entanglements. We formulate a micromechanical model that couples together chain stretching, ionic bond slipping, and entanglement evolution. The ionic bonds provide toughness by enabling plastic deformation in comparison to covalently crosslinked material and add strength compared to a linear polymer. Evolution of the entanglement density is taken as a key mechanism that can govern stiffness, toughness, and self-recovery in elastomers. The model is used to match bulk polyelectrolytes with different fractions of ionic components under a variety of loading histories. The variations in material parameters are then used to help understand the relative importance of different governing mechanisms in the bulk polymers. We show that the theoretical framework can explain our experimental uniaxial tensile experimental results for polyelectrolytes. This model can help to design better material with high stiffness and toughness. We expect that our model can be extended to explain the mechanical behavior of other polyelectrolytes and other soft materials with a wide range of dynamic bonds.

A Generalized Microstructure-Informed Constitutive Modeling Framework for Hierarchical Materials Containing Cellular Network and Scattered Subgrain Features

A Generalized Microstructure-Informed Constitutive Modeling Framework for Hierarchical Materials Containing Cellular Network and Scattered Subgrain Features

Strain energy density as a Gaussian process and its utilization in stochastic finite element analysis: Application to planar soft tissues

Data-based approaches are promising alternatives to the traditional analytical constitutive models for solid mechanics. Herein, we propose a Gaussian process (GP) based constitutive modeling framework, focusing on planar, hyperelastic and incompressible soft tissues. Specifically, the strain energy density of soft tissues is modeled as a GP, which can be regressed to experimental stress–strain data obtained from biaxial stretching experiments. Moreover, the GP model can be weakly constrained to be convex. A key advantage of a GP-based model is that, in addition to the mean value, it provides a probability density (i.e. associated uncertainty) for the strain energy density. To simulate the effect of this uncertainty, a non-intrusive stochastic finite element analysis (SFEA) framework is proposed. The proposed framework is verified against an artificial dataset based on the Gasser–Ogden–Holzapfel model and applied to a real experimental dataset of a porcine aortic valve leaflet tissue. The results show that the proposed framework can be trained with limited experimental data and fits the data better than several existing models. The SFEA framework provides a straightforward way of using the experimental data and quantifying the resulting uncertainty in simulation-based predictions.

Evolving asymmetric and anisotropic hardening of CP-Ti sheets under monotonic and reverse loading: Characterization and modeling

Commercially pure titanium (CP-Ti) sheets are promising to manufacture high-performance and lightweight components in many fields such as aerospace, marine, energy, electrics and healthcare. However, due to its low symmetric hexagonal close-packed (HCP) structure and complex thermal-mechanical processing history, CP-Ti sheets may exhibit significant asymmetry and anisotropy under monotonic and reverse loading. Therefore, the coupling effect of loading-path dependent asymmetric and anisotropic hardening behaviors of CP-Ti sheets need to be systemically characterized and accurately described under monotonic and reverse loading paths. In this study, firstly, the rheological tests were performed to systemically explore the evolving asymmetric and anisotropic hardening behavior of CP-Ti sheet under monotonic and reverse loading modes. With the help of anti-buckling support fixture and DIC system in tension and compression tests, the coupling effect of asymmetric and anisotropic hardening under monotonic and reverse loading of CP-Ti sheet was identified. Secondly, a loading-path dependent constitutive modeling framework was constructed in which the stress invariants-based model (SIM) integrated with distorted hardening (DH) under reverse loading (RL) conditions (SIM+DH+RL) were considered. Within the proposed modeling framework, a judging criterion of stress reversal for explicit solution method and an activation criterion of different reverse loading modes were introduced. Finally, the proposed SIM+DH+RL model was implemented, calibrated and further applied to simulate typical V-/U-bending of CP-Ti sheets. The results show that the proposed SIM+DH+RL model can significantly improve the prediction accuracy of V-/U-bending springback with the relative errors less than 7.8% and 2.1%, compared with 15.2% and 12% relative errors of Mises+IH, 19.5% and 7.3% relative errors of SIM+DH, 15.1% and 8.4% relative errors of Yoon2014+IH, 14.5% and 5.2% relative errors of CPB06+IH, and 17.3% and 11.5% relative errors of Hill48+IH. It is noted that for the thickness distributions after V-/U-bending springback, the above used models have similar prediction accuracies with the relative errors less than 1.8%.

On capturing rate-dependent anisotropic behavior of sheet metal forming using a viscoplastic computational framework

Manufacturing of metal sheets—of critical importance in the automotive, aerospace, and railway industries; is notoriously prone to uncertainty and inaccuracy driven by the lack of control in responding to anisotropic behavior developed during a rate-dependent forming process. Various expensive efforts have been made to evaluate these complex behaviors largely based on trial-and-error approaches, implying a high-fidelity predictive numerical tool is inevitable to promote a cost-effective solution. In this work, a high fidelity anisotropic-viscoplastic constitutive model framework based on Hill 48 yield function, isotropic hardening, and Cowper-Symonds viscoplasticity is developed to investigate the rate-dependent anisotropic behavior of sheet metals. This model is holistically implemented as a user-defined material subroutine (VUMAT) in ABAQUS software. The quality of numerical results is evaluated through comparison with experimental works. In the quasi-static and dynamic tensile test, the stress-strain curves obtained from numerical simulation using the developed constitutive model are in excellent agreement with experimental results. Furthermore, numerical simulation of cup drawing test using the developed constitutive model can accurately predict the earing height compared to experimental results. In addition, some novel relationships in terms of plastic rate-dependent anisotropic behavior of sheet metal forming are successfully established through the extended use of the validated numerical model.

Innovations of the UH model associated with a clue of stress–strain chain

One of the most fundamental issues in geotechnical engineering is deformation calculation. Within the elastoplastic constitutive model framework, a new concept of stress–strain chain is proposed for the analysis of the computational process of plastic strain. The connotation of the chain is then demonstrated through seven stages, starting with six-component stress and ending with six-component plastic strain. The inadequacies of previous research on plastic strain computation induced by several factors such as anisotropy, dilatancy, hardening law, and hardening parameter are evaluated utilizing the stress–strain chain. Meanwhile, based on the stress–strain chain, innovations of the UH model such as the transformed stress method, the unified normalized compression line, and the unified hardening parameter are introduced and validated. It can be seen with the stress–strain chain that plastic volumetric strain and its assignment criterion are the keys to deformation calculation. Representative innovations of the UH model facilitate the efficiency of plastic strain calculation by accurately determining plastic volumetric strain and appropriate ratio relations. Through a foundation pit excavation case, the values and ratios of strain components predicted by the UH model are more reasonable and reliable. This demonstrates that the UH model enhances the accuracy of deformation calculations greatly.

Prediction and Validation of Stress Triaxiality Assisted by Elasto-Visco-Plastic Polycrystal Model

The ΔEVPSC numerical code based on the elasto-visco-plastic HEM (Homogeneous Effective Medium) provides a multiscale constitutive modeling framework that is suitable for describing a wide range of mechanical behaviors of polycrystalline metals. In this study, an AA6061-T6 aluminum sample was chosen to validate the predictive capability of the ΔEVPSC stand-alone (ΔEVPSC-SA) code on stress triaxiality and its evolution until fracture. The model parameters were calibrated by fitting the uniaxial flow stress-strain curve, and the initial crystallographic orientation distribution (COD) was obtained using X-ray diffraction and electron backscattered diffraction (EBSD) methods. The statistical representativeness of the COD was further examined by comparing the experimental R-values with model predictions based on a set of CODs obtained via the two mentioned diffraction methods. The results suggest that the X-ray scan does not represent the texture very well, and instead, an entire cross-sectional EBSD scan is required, even though the texture gradient along the through-thickness direction is not very significant. The model-calculated triaxiality based on the ΔEVPSC-SA code was verified by comparison with the experimental results from the uniaxial tension, the notched tension, and the plane strain tests. The results were in good agreement with the ΔEVPSC finite element (FE) simulation results and other similar experimental results reported in the literature.