Multi-scale Constitutive Model Research Articles

Hierarchical Deformation and Anisotropic Behavior of (α+β) Ti Alloys: A Microstructure-Informed Multiscale Constitutive Model Study

In this study, the hierarchical deformation and anisotropic behavior of (α+β) Ti alloys are investigated using a novel microstructure-informed multiscale constitutive model. State-of-the-art crystal plasticity finite element (CPFE) models, due to their emphasis on a single length scale, are inadequate for capturing the complex hierarchical behavior of additively manufactured (AM) (α+β) titanium alloys, which are characterized by columnar grains and lamellar subgrain features at distinct length scales. To overcome this limitation, a decoupled multiscale framework was developed, integrating representative volume elements (RVEs) for both the columnar grain structure at the higher length scale and the subgrain microstructure at the lower length scale, with equal emphasis on each. The material behaviors at these scales were modeled using an anisotropic classical plasticity model and a mechanism-based CPFE model, respectively. The framework was experimentally validated for Directed Energy Deposition (DED) manufactured Ti-6Al-4V and used to investigate microscopic stress/strain fields, deformation localizations at grain and subgrain levels, and stress partitioning among neighboring grains. Insights from these studies led to the proposal of a new theory of anisotropy in AM (α+β) titanium alloys.

A multi-scale constitutive model for AlSi10Mg alloy fabricated via laser powder bed fusion

A multi-scale constitutive model for AlSi10Mg alloy fabricated via laser powder bed fusion

Micromechanism insights and constitutive modeling for deformation in a novel high manganese austenitic steel

We explore the mechanical behavior and microstructural evolution of a novel high manganese austenitic steel (HMAS), extensively applied in underground engineering and structural construction. Our investigation focuses on two variants of HMAS, namely HMAS-1.9 and HMAS-10, which exhibit distinct yield strengths and elongation behaviors. Electron backscatter diffraction (EBSD), transmission electron microscopy (TEM) and X-ray diffraction (XRD) were employed to elucidate the evolution of microstructures during plastic deformation. Based on these experimental results, we developed a new phenomenological multiscale constitutive model that captures the mechanical behavior of HMAS, integrating micromechanisms and microstructural evolutions. Optimized with the MATLAB genetic algorithm toolbox, this model was successfully implemented in a uniaxial tensile finite element model (FEM) within ABAQUS, proving its efficacy in predicting both macroscopic flow stress and microstructural evolutions. Our findings highlight the important roles of initial grain size, dislocation density, and twin fraction in determining the mechanical properties of HMAS. Throughout the deformation process, both HMAS variants demonstrate similar plastic deformation behaviors, characterized by comparable rates of increase in dislocation density and twinning fraction, accompanied by a consistent trend of grain refinement. These findings not only deepen our understanding of the complex relationship between microstructure and mechanical properties of the HMAS steel, but also provide an effective multiscale constitutive modeling approach for predicting its deformation behavior.

A multiscale constitutive model of magnesium-shape memory alloy composite

In this work, a multiscale constitutive model is established to describe the deformation behaviors of magnesium-shape memory alloy (Mg-SMA) composite in a wide temperature range and reveal the strengthening mechanism of SMA reinforcement on Mg. The model is established at the grain scale firstly and gradually transited to the macroscopic scale by employing a newly developed three-level scale transition rule. At the grain scale, the thermodynamic-consistent constitutive models of Mg and SMA are, respectively, constructed by addressing different inelastic deformation mechanisms. The basal, prismatic, pyramidal, slip systems and extension twinning system are considered for the Mg phase, and the martensite transformation (MT) and austenitic plasticity are addressed for SMA reinforcement. Thermodynamic driving forces of each inelastic deformation mechanism are derived from the dissipative inequality and the constructed Gibbs free energies. At the polycrystalline scale, to evaluate the interactions among the grains and pores, and obtain the whole responses of the polycrystalline Mg and SMA, a thermo-mechanically coupled self-consistent homogenization scheme is employed. At the mesoscopic scale, a modified thermo-mechanically coupled Mori-Tanaka's homogenization scheme is adopted to evaluate the interaction between the Mg phase and SMA phase, and predict the whole responses for the representative volume element (RVE) of the composite. According to the geometrical features and mechanical loadings applied on the specimen, a hypothesis of homogeneous stress and strain fields at the macroscopic scale is adopted to achieve the scale transition from the RVE of the composite to the whole specimen. The capacity of the multiscale model is verified by comparing the predictions with the existing experimental data (Aydogmus, 2015). Moreover, the influences of characteristic information for the microstructures at different spatial scales on the deformation behaviors of the composite are predicted and discussed.

A Micromechanics-Based Multiscale Constitutive Model for Brittle to Ductile Behavior of Porous Geomaterial

A Micromechanics-Based Multiscale Constitutive Model for Brittle to Ductile Behavior of Porous Geomaterial

Implicit multiscale finite element analysis of polymer physics-based multiscale constitutive model for elastomers

This paper proposes a multiscale visco-hyperelastic constitutive model that can predict the material behavior bridging molecular properties to response at the continuum level and its integration within implicit finite element analysis. The proposed model incorporates all parameters having actual physical meaning obtained from molecular simulations based on the tube theory. Despite the advantages that the constitutive equation was determined from molecular network characteristics, applications of the model have been restricted to the simple shape of Finite Element (FE) models with a few cases of specific deformations, owing to its complexity and absence of the implicit formulation. Therefore, the application of the model to the 3D implicit FE analysis with a tangent stiffness was proposed by applying numerical differentiation with the complex step derivative approximation (CSDA) method. Combined with the Neo-Hookean hyperelasticity model, the results from the example analysis applied to the tensile specimen model were examined to manifest the visco-hyperelastic characteristics of elastomers under cyclic loading, stability of the analysis, and further discussion on the visco-hyperelastic constitutive model were proposed.

Grain size and orientation affected deformation inhomogeneity and local damage of hot-deformed Al-Zn-Mg alloy

The forming defects at elevated temperature caused by the microstructural evolution such as grain size, crystal orientations are the critical issue that induced deformation inhomogeneity and local damage which severely limits the application of Al-Zn-Mg alloys in thin-walled structure manufacturing. In this study, the hot deformation behavior and microstructural evolution were investigated via hot stretching tests, electron backscatter diffraction (EBSD) and the transmission electron microscopy (TEM) characterization. Then, an integrated multiscale constitutive model coupled dislocation density based crystal plasticity (CP) model and phase field method (PFM) was established through the reconstructed configuration of deformation gradient emphasized on a damaged part. The coupled CP-PFM model was numerically implemented into Düsseldorf Advanced Material Simulation Kit (DAMASK) software. Subsequently, a set of virtual polycrystalline RVE models assigned with different average grain sizes and two types of RVE models with realistic crystal orientations were constructed from EBSD data. The experimental and simulation results of the Al-Zn-Mg alloy under hot tensile loading showed that the grain size refinement leads to the increase of grain boundary density, which leads to the rapid increase of dislocation density, uniform damage distribution and better work hardening ability. In addition, the coordination of the activation of different slip systems between adjacent grains will lead to the uneven distribution of dislocation density and local damage.

Scale transition and residual fields in modeling of polycrystalline ferroelectrics based on the internal energy potential and a Voigt–Reuss approximation

A multiscale constitutive model, implemented within a semi-analytical computational framework, is presented. Accounting for domain switching in a tetragonal lattice and electromechanical intergranular interactions, it aims to accurately reproduce experimental data available from butterfly and dielectric hysteresis loops. For the sake of validation, these have been collected from various sources providing ranges of values for characteristic quantities. Adopting the fundamental ideas of a previously published model, a different homogenization approach is employed for the transition from grain to polycrystalline domain, providing both residual stress and electric fields. Furthermore, the thermodynamic driving force and constitutive equations on the grain level are based on a different thermodynamical potential. While the modeling approach inherently does not include any tuning parameters, the introduction of one and two, respectively, additional parameters is investigated, although computational results are basically in good agreement with experimental ranges.

Tension-compression asymmetric functional degeneration of super-elastic NiTi shape memory alloy: Experimental observation and multiscale constitutive model

The functional degeneration of super-elastic NiTi shape memory alloy (SMA) rods is investigated in the cyclic tension-unloading and compression-unloading experiments. The results show that the functional degeneration exhibits a strong tension–compression asymmetry, which originates from the initial 〈111〉 and 〈110〉 mixed textures along the axial direction of the rods, the polarity of martensite transformation, and the complex interaction between martensite transformation and dislocation slipping. Moreover, the dependence of functional degeneration on the applied stress level is discussed. To understand and predict the effect of initial texture on the cyclic deformation of polycrystalline NiTi SMA, a multiscale constitutive model is constructed at finite deformation. At the mesoscopic scale, two major inelastic deformation mechanisms, i.e., the transformation between austenite and martensite phases and the transformation-induced plasticity caused by the strain mismatch at the austenite–martensite interface, are considered. The total deformation gradient is multiplicatively decomposed into the martensite transformation, transformation-induced plasticity and elastic parts. The deformation gradient parts at the mesoscopic scale are obtained by upscaling these physical quantities at a smaller spatial scale (i.e., the microscopic scale). The proposed model is formulated within the framework of irreversible thermodynamics. The driving forces for martensite transformation and transformation-induced plasticity are derived according to the established Helmholtz free energy and Clausius-Duhem inequality. The complex interaction between martensite transformation and dislocation slipping, and the forward/reverse inheritance of dislocations caused by the moving austenite–martensite interfaces are considered. To calculate the overall cyclic responses of polycrystalline NiTi SMA at the macroscopic scale, the developed constitutive model is implemented into the finite element software ABAQUS/Explicit using a user-defined material subroutine. A polycrystalline representative volume element is constructed by introducing the initial 〈111〉 and 〈110〉 mixed textures observed in the experiment. By comparing the simulated results with the experimental data, the capability of the proposed model to predict the texture-induced tension–compression asymmetric functional degeneration is verified, and the influence of texture type on the cyclic deformation is discussed.

Nonlinear multiscale model for interstitial structures of densely ordered multi-walled carbon nanotube bundles

The microscopic structural changes upon mechanical deformation of coiled multi-walled carbon nanotube (MWCNT) yarns were investigated through a multiscale constitutive model. All-atom molecular dynamics simulations reveal a strong dependence of the nonlinear mechanical behavior on the diameter of the nanotubes composing the MWCNT bundles. Accordingly, the multiscale modeling is configured to predict the stress and interstitial area change experienced by each nanotube according to the diameter and position distribution of the nanotubes constituting the microstructure. From the proposed multiscale method, representative volume elements (RVEs) of MWCNT bundles with micrometer scale lengths are developed. The RVEs are applied to characterize the deformation of each nanotube due to the longitudinal sway and the stress concentration in the cross-section due to local tightening. The results are in good agreement with those of the all-atom cross-sectional model corresponding to each longitudinal position, which proves the validity of the developed RVEs.

Bending behavior of strain hardening cementitious composites based on the combined fiber-interface constitutive model

Strain-hardening cementitious composites (SHCC) are a class of fiber-reinforced cementitious composites characterized by strain hardening behavior accompanied by multiple cracking, which is designed by tailoring matrix, fiber, and fiber/matrix interface based on the micromechanics theory. At present, there are several numerical simulation methods to study the multiple cracking behavior, such as discrete crack model, unity finite element method, 3D smeared crack model, multiscale constitutive models, and lattice model. However, it is difficult to provide a good balance between accuracy and computational efficiency due to the vast number of fiber and the complex relationship between fibers, matrix, and interface. This paper developed a combined fiber-interface constitutive model based on the single fiber pull-out theory, and established a combined unit. A three-dimensional two-phase finite element model was established, which had been proved to be effective on simulating the cracking behavior of SHCC. Furthermore, the effects of fiber volume fraction and matrix cracking strength on the four-point bending properties of SHCC were investigated. Further study showed that there was a critical matrix cracking strength for a specified fiber and matrix. This value can optimize the design of SHCC to achieve high strength and high toughness performance.

Application of dislocation-based constitutive model into research of dynamic mechanical behavior under shock loading

To understand the plastic deformation mechanism of an FCC metal (pure aluminum) under shock loading and describe its dynamic mechanical behavior accurately, a multiscale constitutive model based on the dislocation substructure is developed, which comprehensively considers the controlling mechanisms of dislocation motion and dislocation evolution. Then, the model is extended to the loading of strong shock waves by incorporating the homogeneous nucleated dislocation within the constitutive framework. The model parameters are successfully determined by the normal plate impact experiments with different thicknesses of specimens. Additionally, shock front perturbation decay experiments are performed using a line velocity interferometer system for any reflector, where the modulated surface of the specimen is subjected to a laser-driven loading. Then, the model is applied to reproduce the perturbation decay of shock fronts in experiments. During the post-process of simulated results, the method based on the pressure gradient is used to determine the amplitude and the location of distributed shock fronts. The extended model shows promise as an effective method to figure out the role of strength (shear response) on the evolution of perturbation amplitude.

Multi-scale constitutive model of human trabecular bone

The present study aims to formulate a new multiscale constitutive model of human trabecular bone. The trabecular bone was modelled as a nonlinear viscoelastic material. The viscoelastic effects of single trabeculae were considered by means of a hereditary integral in which stress depends on time and strain, while the elastic response was described by the hyperelastic Mooney–Rivlin model. The cuboid bone sample was extracted from the femoral head during the hip replacement surgery. The material constants in the constitutive equation were identified based on the stress relaxation test performed on the cuboid sample and the microindentation tests performed on trabeculae using the curve-fitting procedure. The microindentation tests were performed using a spherical tip instead of Vickers or Berkovich tip to minimize plastic effects during trabecular deformation. In order to validate formulated constitutive model, results from a FE simulation of stress relaxation test and uniaxial compression test were compared to the results of the corresponding experiments conducted on a macroscopic bone sample. Good agreement was observed between numerical and experimental results. The viscoelastic behaviour predicted by the proposed constitutive equation corresponds well to the response of human trabecular bone under various types of load conditions. This demonstrates the high ability of our constitutive model to simulate the behaviour of trabecular bone on a micro- and macroscopic scale. Thus, we conclude that the model, which was formulated for a single trabecula, can be successfully applied to simulate mechanical behaviour of the tissue in a macroscale.

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.

A data-driven multi-scale constitutive model of concrete material based on polynomial chaos expansion and stochastic damage model

Nonlinearity and randomness are two intrinsic characteristics of the mechanical behavior of concrete material. The structural response under large excitation can barely be predicted without considering these two characteristics. Brilliant works have been done for decades in the material science and computational stochastic mechanics. However, the existed numerical methods are usually parameter dependent and the key mechanical properties of concrete material are determined by empirical recognition. Therefore, in this paper, a data-driven multi-scale constitutive model is proposed for representing the mechanical behavior of concrete material based on the polynomial chaos expansion and stochastic damage model. Several groups of compressive stress–strain data of concrete material are applied to train the proposed model. By cross validation of the prediction and the concrete stress–strain experimental data, the proposed model is firstly verified to have a robust performance to gain accurate prediction results. Afterwards, the proposed method is compared with a neural network method, the results shows that the proposed method is more robust and accurate than the neural network method.

A multiscale constitutive model for metal forming of dual phase titanium alloys by incorporating inherent deformation and failure mechanisms

Ductile metals undergo a considerable amount of plastic deformation before failure. Void nucleation, growth and coalescence is the mechanism of failure in such metals. α–β titanium alloys are ductile in nature and are widely used for their unique set of properties such as specific strength, fracture toughness, corrosion resistance and resistance to fatigue failures. Voids in these alloys have been reported to nucleate on the phase boundaries between α and β phase. Based on the findings of crystal plasticity finite element method investigations of the void growth at the interface of α and β phases, a void nucleation, growth, and coalescence model has been formulated. An existing single-phase crystal plasticity theory is extended to incorporate underlying physical mechanisms of deformation and failure in dual phase titanium alloys. Effects of various factors [stress triaxiality, Lode parameter, deformation state (equivalent stress), and phase boundary inclination] on void nucleation, growth and coalescence are used to formulate a phenomenological constitutive model while their interaction with a conventional crystal plasticity theory is established. An extensive parametric assessment of the model is carried out to quantify and understand the effects of the material parameters on the overall material response. Performance of the proposed model is then assessed and verified by comparing the results of the proposed model with the RVE study results. Application of the constitutive model for utilisation in the design and optimisation of the forming process of α–β titanium alloy components is also demonstrated using experimental data.

Multiscale Constitutive Modeling of the Mechanical Properties of Polypropylene Fibers from Molecular Simulation Data

We present a multiscale approach to create a constitutive model that predicts the mechanical properties of polypropylene fibers based on chemical and physical characteristics. The development of this method relies on validation with experimental stress–strain curves from nine different isotactic polypropylene (iPP) fibers with their varying molecular weight characteristics, Hermans orientation factors, and crystallinity. Complementary molecular models were built by using molecular dynamics (MD) simulations with united atom models. Tensile deformation simulations adapting a quasi-static procedure resulted in stress–strain curves that aligned well with the experimentally measured ones. A neural network model was trained on the MD simulation data to create correlations that predict parameters for a chosen constitutive model that describes the mechanical properties of the polypropylene fibers. This computational approach is amenable to be applied to polymer fiber systems and aims to aid in the design of polymeric materials to achieve targeted mechanical properties.

The Multiscale Characterization and Constitutive Modeling of Healthy and Type 2 Diabetes Mellitus (T2DM) Sprague Dawley (SD) Rat Skin

The Multiscale Characterization and Constitutive Modeling of Healthy and Type 2 Diabetes Mellitus (T2DM) Sprague Dawley (SD) Rat Skin

Parametrically-upscaled continuum damage mechanics (PUCDM) model for multiscale damage evolution in bending experiments of glass-epoxy composites

This paper demonstrates the effectiveness of parametrically-upscaled continuum damage mechanics (PUCDM) models for multiscale damage analysis of S-2 glass fiber/SC-15 epoxy matrix composite beams subjected to single-edge notched bending (SENB) tests. The SENB experiments with [012/9012] double-ply and [08/908/08] triple-ply composite beams use synchrotron X-ray phase-contrast imaging for in-situ damage characterization. PUCDM models represent a class of thermodynamically-consistent multiscale constitutive models that bridge length-scales through explicit incorporation of representative aggregated microstructural parameters (RAMPs) in constitutive coefficients. Functional forms of RAMPs are generated through machine learning tools, operating on a micromechanical analysis database. The efficient PUCDM-based FE simulations reveal multiscale damage phenomena in the SENB experiments.

A multiscale constitutive model coupled with martensitic transformation kinetics for micro-scaled plastic deformation of metastable metal foils

The mechanical behavior of metastable austenitic foils at the size scale from micron to submillimeter is strongly affected by the coupling between size effect and strain-induced martensitic transformation (SIMT), which remains to be a pressing issue to be explored. In this research, the focus is on developing a multiscale constitutive model to reveal the mechanical behavior of metastable foils and more accurately predict the size effect on SIMT. In tandem with this, the martensitic transformation and hardening behavior of SUS304 foils with different thicknesses and grain sizes were explored. The results figured out that the SIMT is promoted by the increase in grain size and foil thickness. Furthermore, the onset and end of stages II of work-hardening behavior are advanced and the work-hardening rate in stage II increases faster with increasing grain size and foil thickness. The SIMT kinetic model was coupled with the intermediate mixture law and the iso-work hypothesis to identify the stress-strain relationship of individual austenite and martensite at the surface and interior layers, which was used to construct the multiscale constitutive model. The multiscale model was developed based on the framework of the surface layer model and the intermediate mixture law to represent the coupling between the size effect and the SIMT. Through finite element simulation by using the proposed multiscale constitutive model, the dispersion hardening mechanism in micro-scaled deformation of metastable austenitic foils caused by the non-homogeneous plastic deformation at the interface between austenite and martensite was revealed. The multiscale model was validated via the corroboration of finite element simulation with experiments and therefore can provide a robust analysis of the micro-scaled deformation behavior of metastable austenitic foils.