Rate-dependent Model Research Articles

Comprehensive analysis of mode-I cracking in ice: Exploring full-range rate dependency

Ice, as a crystalline material, demonstrates a unique response in its mode-I fracture toughness to variations in loading rates, characterized by an initial decrease and subsequent increase in toughness. This phenomenon has been explored in limited studies. In this work, an extensive numerical analysis of mode-I ice cracking is conducted by using the distinct lattice spring model (DLSM). Norton-Bailey Drucker-Prager DLSM (NB-DP-DLSM) is initially employed to investigate the classical explanations of creep and stress relaxation for the anomaly observed in ice’s fracture toughness at lower loading rates, but this approach does not successfully replicate the experimentally observed strain rate dependency. Then, two rate-dependent constitutive models are introduced to further examine the mode-I fracture mechanics of ice. Our numerical simulations of three-point bending tests show that the relationship between fracture toughness and strain rate at lower levels is more accurately captured by rate-dependent models. For higher strain rates, our numerical modeling of notched semi-circular bending tests indicates that a rate-independent constitutive model can replicate the loading rate dependency of ice’s mode-I fracture toughness. In conclusion, these observations suggest that a reverse-stage-like dynamic constitutive model for DLSM can potentially capture the full range of loading rate dependencies observed in ice.

A rate-dependent crack bridging model for dynamic fracture of CNT-reinforced polymers

Carbon nanotubes (CNTs) improve the fracture toughness of polymer-based matrix composites by bridging the crack growth path. This research presents a finite element (FE) rate-dependent crack bridging model of Mode-I dynamic crack growth in CNT-reinforced polymers accounting for rate of loading and rapid crack growth. Two distinct CNT bridging and matrix cracking damage mechanisms are taken into account in the fracture process zone (FPZ) accounting for the dissipation of fracture energy. A viscoelastic-viscoplastic material model is adapted for the matrix phase to capture the strain rate effects. The crack in the matrix phase is modeled using cohesive zone elements characterized by experimental data. A rate-dependent traction-separation law obtained from CNT pull-out simulation at relevant crack opening speeds is used to simulate the CNT bridging in the FPZ. Considering the given traction-separation law as a constitutive equation, the CNTs are replaced with non-linear spring elements to facilitate the FE simulation of crack bridging with numerous CNTs in the FPZ. The proposed rate-dependent FE model for crack bridging enables the study of the effect of key CNT factors such as length, orientation, waviness, volume fraction, and agglomeration on fracture energy dissipation at various crack speeds. The developed model can simultaneously consider the interactive effects of all CNT parameters which is useful for considering all processing-induced uncertainties in analyzing the effects of CNTs on the dynamic fracture toughness of nanocomposites.

Temperature and rate dependent shear characterization and modeling of spread-tow woven Flax/PP composite laminates

This paper investigates the in-plane shear behavior of spread-tow woven structured flax fiber-reinforced polypropylene composites, focusing on temperature and displacement rate effects. For this aim, the study includes bias-extension experiments to characterize the shear behavior of the material. Several experiments were conducted at different displacement rates and temperature levels. The temperature levels were chosen to encompass the material's behavior during solid and molten-state thermoforming, given the melting and decomposition temperatures acquired from differential scanning calorimetry and thermogravimetric analysis tests, respectively. A new fixture was designed and manufactured to allow pre-heating of the oven and rapid placement of the samples, which in turn, prevents their degradation for the time duration required to reach a uniform temperature distribution in the oven. During the experiments, the 3D digital image correlation technique accurately measured local deformations and strains on the specimen's surface under varying conditions. Further, a finite element model is developed, incorporating a fiber reorientation algorithm with a hypo-elastic shear model to simulate the material's temperature and rate-dependent behavior. The finite element shear angle distributions, as well as the shear angle-force and shear angle-displacement curves, are compared with the Digital Image Correlation (DIC) data and load measurements for different test conditions, showing good agreement between the numerical and experimental approaches.

Digital image correlation and infrared thermography data for seven unique geometries of 304L stainless steel

Material Testing 2.0 (MT2.0) is a paradigm that advocates for the use of rich, full-field data, such as from digital image correlation and infrared thermography, for material identification. By employing heterogeneous, multi-axial data in conjunction with sophisticated inverse calibration techniques such as finite element model updating and the virtual fields method, MT2.0 aims to reduce the number of specimens needed for material identification and to increase confidence in the calibration results. To support continued development, improvement, and validation of such inverse methods—specifically for rate-dependent, temperature-dependent, and anisotropic metal plasticity models—we provide here a thorough experimental data set for 304L stainless steel sheet metal. The data set includes full-field displacement, strain, and temperature data for seven unique specimen geometries tested at different strain rates and in different material orientations. Commensurate extensometer strain data from tensile dog bones is provided as well for comparison. We believe this complete data set will be a valuable contribution to the experimental and computational mechanics communities, supporting continued advances in material identification methods.

Machine Learning- and Finite Element-Based Temperature- and Rate-Dependent Plasticity Model: Application to the Tensile Behavior

Machine Learning- and Finite Element-Based Temperature- and Rate-Dependent Plasticity Model: Application to the Tensile Behavior

High strain-rate fracture behavior of a hollow projectile

In this paper, a hollow projectile made of AISI4340 steel was fabricated based on an arbitrary warhead shape, and high-speed impact tests were performed according to various target encounter conditions including velocity and angle. In order to characterize the dynamic fracture behavior, tensile tests were performed on various notch specimens under three strain rates (10–3 s-1, 100 s-1, 103 s-1), and the stress state histories of the specimens were calibrated through a hybrid experimental-numerical analysis. In particular, for the intermediate (100 s-1) and high (103 s-1) strain rate conditions, the local temperature rose due to adiabatic heating until fracture was experimentally measured, and the thermal softening behavior determined from the inverse optimization technique was considered for the dynamic constitutive model. For the comparison with the high-speed impact test results, a rate-dependent ductile fracture model was utilized for numerical simulation. Considering that the model parameters were calibrated with the thermal softening effect, the proposed fracture model implicitly takes temperature into account. The deformation and fracture modes of the projectile from experimental and numerical study showed very good agreement under all impact conditions. It was confirmed that the softening of a material by adiabatic heating should be considered along with strain rate hardening in dynamic fracture simulation.

A data-driven ductile fracture criterion for high-speed impact

Data-driven methods based on machine learning (ML) models offer new approaches for characterizing the fracture behavior of advanced elastoplastic materials. In this paper, a ML-based data-driven ductile fracture criterion is proposed to characterize the fracture behavior of elastoplastic materials under high-speed impact loading conditions. To reduce the required training dataset and enhance the predictability capability, several assumptions are used. Firstly, utilizing the decoupled assumption, two separate artificial neural network (ANN) models are employed to establish the fundamental fracture model and characterize the strain rate effect of ductile fracture behavior, respectively. In addition, the enhanced method with a logarithmic function is introduced to improve predictability capability of the proposed data-driven criterion under unknown high strain rates. To establish a complete numerical implementation framework, an enhanced rate-dependent data-driven constitutive model and a compatible numerical implementation algorithm are additionally introduced. Eventually, to assess the applicability of the proposed data-driven fracture criterion, numerical simulations of notched specimens and ballistic impact conditions of Ti-6Al-4V material are conducted, respectively. These investigation results demonstrate the effectiveness of the proposed data-driven ductile fracture criterion.

Investigation on the loading rate dependence of electromechanical properties of graphene-cement composites under compressive loading

In this paper, the dependence of the electromechanical properties of graphene-cement composites on the loading rate is studied by combining experiment and theory. The graphene-cement composites are prepared using a wet dispersion technique combined with surfactant assist and ultrasonication treatment. Both monotonic and cyclic compressive experiments with different loading rates are conducted on the graphene-cement composites with different graphene contents to measure their mechanical, electrical and piezoresistive properties. The optimum dosage of graphene in cement matrix is about 0.05 wt%, which can increase the compressive strength by 35.7 % and 34.1 %, and decrease the electrical resistivity to 1.43 and 3.48 × 105 Ω•cm, respectively, after 14 and 28 d curing periods. The uniform dispersion of graphene with less layers endows the graphene-cement composites with very low percolation threshold of electrical conductivity (0.007 wt%), stable piezoresistive response, and good reproducibility. Furthermore, the mechanism of the loading rate on the strain sensing behavior is discussed, and a rate dependent theoretical model of resistance change is proposed to quantitatively predict the electromechanical responses. This study provides guidelines for the fabrication of highly strain-sensitive graphene-cement composites and the rate-dependent evaluation of their electromechanical properties in the fields of intelligent construction.

Modeling plasticity-mediated void growth at the single crystal scale: A physics-informed machine learning approach

Modeling the evolution of voids during plastic flow as well as their effects on plastic dissipation is critical for both component manufacturing and lifetime estimation purposes. To this end, we propose a rate-dependent constitutive model to homogenize the effects of semi-randomly distributed voids on single crystal plasticity whilst capturing void interaction and plastic anisotropy. The present work focuses on the case of face centered cubic crystals to introduce an anisotropic gauge function applicable within the crystal plasticity formalism. The approach combines analytical methods to describe the micromechanics of the system in combination with symbolic regression to capture analytically intractable mechanisms from data. The hybrid framework uses a physics-informed genetic programming-based symbolic regression algorithm to solve a multiform optimization problem simultaneously producing a new gauge function and a new strain rate equation. This is also a multi-objective optimization problem with many competing objectives. A new search and selection step is introduced to the genetic algorithm that promotes convergence toward a global solution that better satisfies all the objectives. Overall, the symbolic equations produced leverage data-driven methods to achieve greater accuracy than comparable alternatives on an analytically intractable problem while maintaining model transparency.

A modified bond-based peridynamic approach for rigid projectile perforation on concrete slabs

Peridynamic (PD) has a unique advantage in describing the crack growth and fragmentation of brittle materials. Concerning the dynamic behaviors and failure patterns of concrete slabs under projectile perforations, a modified bond-based PD approach maintaining both the easy implementation and computational stability characteristics was firstly developed from the following three aspects, (i) a rate-dependent PD constitutive model was proposed for describing the dynamic behaviors of concrete; (ii) a progressive damage criterion considering the tension-compression anisotropy, softening behavior, and strain rate effect of concrete was incorporated to more accurately reproduce the damage and failure of concrete; (iii) an improved micro-modulus function related to bond length was introduced to reveal the internal length effect of bond force. Then, numerical simulations of projectile perforation on concrete slabs by utilizing the developed modified bond-based PD approach, as well as the corresponding sensitivity analyses of discretization parameters including horizon size and particle spacing were performed. Based on the recommended horizon size and particle spacing, the predicted residual velocity of projectile and failure patterns of concrete slabs exhibited an excellent agreement with the test data. Furthermore, by comparisons of the traditional bond-based PD and classical finite element methods, the superiority of developed approach in describing the perforation damage of concrete targets against projectile impact was demonstrated. Finally, the modified bond-based PD approach was employed to blind simulate the projectile normal and oblique perforating multi-layered spaced concrete target plates. It was found that the modified PD model reasonably predicted the terminal ballistic trajectory, deflection angle, and residual velocity of projectile, as well as the failure patterns of target plates. The present work provides a new way to predict the terminal ballistic effect of projectile and dynamic behaviors of concrete slabs.

Dynamic tensile behavior and rate-dependent constitutive model of 304 + Q235 bimetallic steel

This paper aims to elucidate the rate-dependent behavior of 304 + Q235 bimetallic steel under intermediate strain rates. Tensile tests were conducted on 18 sets of specimens to obtain the engineering stress-strain curves at various strain rates. The experimental results demonstrate that the strain rate significantly influences both the yield strength and ultimate strength of the 304 + Q235 bimetallic steel, albeit with different trends. The yield strength exhibits an upward trend with increasing strain rates, whereas the ultimate strength demonstrates a downward trend. Additionally, the strain-hardening effect diminishes proportionally with the increase in strain rate. The parameters of the Cowper-Symonds model associated with the dynamic yield strength increase factor (DIFy) were determined based on the experimental findings. A bilinear rate-dependent constitutive model was established to accurately predict the dynamic ultimate strength increase factor (DIFu) of 304 + Q235 bimetallic steel. The established bilinear model was utilized to enhance the Johnson-Cook model for accurate prediction of the true stress-strain behavior of 304 + Q235 bimetallic steel under intermediate strain rates. The calculated results obtained from the modified Johnson-Cook model exhibit good agreement with the experimental results.

A hyper-viscoelastic uniaxial characterization of collagenous embolus analogs in acute ischemic stroke

PurposeAcute ischemic stroke is a leading cause of death and morbidity worldwide. Despite advances in medical technology, nearly 30% of strokes result in incomplete vessel recanalization. Recent studies have demonstrated that clot composition correlates with success rates of mechanical thrombectomy procedures. To understand clot behavior during thrombectomy, which exerts considerable strains on thrombi, in vitro studies must characterize the rate-dependent high-strain behavior of embolus analogs (EAs) with different formation conditions, which can be used to fit models of hyper-viscoelasticity. MethodsIn this study, the effect of collagen infiltration as a carotid-induced collagen-rich thrombosis surrogate is considered as a contributor to embolus analog high-strain stiffness, when compared to 40% hematocrit EAs. ResultsEA high-strain stiffnesses, characterized on a uniaxial load frame, increase by an order of magnitude for collagenous clot analogs. Chandler loop analogs show high-strain stiffnesses and clot compositions commensurate with previous reports of stroke patient clots, and collagenous clots show significant increase in stiffness when compared to stroke patient clots. Finally, hyper-viscoelastic curve fitting demonstrates the asymmetry between tension and compression. Nonlinear, rate-dependent models that consider clot-stiffening behavior match the high strain stiffness of clots fairly well. Furthermore, we demonstrate that the stability of the elastic energy needs to be considered to obtain optimal curve fits for high-strain, rate dependent data. ConclusionThis study provides a framework for the development of dynamically formed EAs that mimic the mechanical and structural properties of in vivo clots and provides parameters for numerical simulation of clot behavior with hyper-viscoelastic models.

Experimental study on dilatancy behavior of soft rock under dynamic loading

Correct understanding of rock dilatation plays a non-negligible role in the safety of rock engineering and the efficiency of geological resources extraction. Although the significance of dilatation under quasi-static loading or unloading conditions is well-studied, its behavior under dynamic loads remains poorly understood. This is largely because conventional dynamic testing system fails to capture the volumetric strain evolution of rock under confinement. This study develops a transparent confining chamber in SHPB system with which high-speed 3D-DIC is capable of measuring both axial and circumferential strain history of dynamically compressed rock. The progressive failure stages and volumetric change of red sandstone under confining pressure of 3.5 MPa and strain rates of 129∼292 s−1 were studied and compared with those in quasi-static case. Results show that, the normalized stress thresholds of crack initiation and dilatancy in dynamic cases are much smaller than those in the quasi-static one. The volumetric variation of soft rock is characterized by four distinct stages: elastic contraction, pre-peak dilatancy, strain-softening dilatancy and strain-recovery dilatancy. The pre-peak elastic contraction is negligible compared to the post-peak dilation, and the ultimate dilation observed in dynamic compression is significantly greater than that in quasi-static compression. Increasing strain rate delays the onset of dilatancy, decreases the dilatant rate but remarkably extends the strain-softening dilatant process which is dominant in total dilation. A rate-dependent piecewise model incorporating key strain thresholds and a dilatancy rate factor was established to characterize the dynamic dilatancy behavior. The research results provide a new insight and possible method in dilation prediction for underground engineering subjected to dynamic loading.

Bioinspired microstructure design simultaneously enhances strain-rate stiffening and toughening of composites

Keratinous biological materials, such as baleen, pangolin scales, and human hair, have similar constituent materials. However, their strain-rate dependent mechanical properties vary due to differences in their microstructures. Inspired by the microstructures observed in baleen, here we present novel microstructural designs of three types of lamellar-tubular fibers, namely the lamellar-tubular fiber (LTF), mineralized lamellar-tubular fiber (MLTF) and hollow mineralized-lamellar-tubular fiber (HMLTF). We systematically investigated their strain-rate dependent mechanical behaviors with comparison to the conventional cylinder-fiber (CF) reinforced composites. Through the finite element analysis (FEA), we found that the baleen-inspired composites have superior strain-rate stiffening and toughening effects than the conventional fiber reinforced composite. Rate-dependent constitutive models decoupling elastic and inelastic regimes were constructed for these bioinspired composites. Based on the FEA results, three constitutive parameters were obtained to quantitatively characterize the rate-dependent mechanical behaviors of these composites, especially the microstructure-induced difference. Furthermore, our study found that the baleen-inspired tubular microstructure improves stiffness and strain-rate stiffening through raising the stress levels of all phases and improves toughness and strain-rate toughening through enlarging the deformation in the inelastic region. This novel bioinspired design is hopeful to pave ways for the development of advanced composites with simultaneously enhanced strain-rate stiffening and toughening.

Bayesian calibration and uncertainty quantification of a rate-dependent cohesive zone model for polymer interfaces

This study presents a rate-dependent cohesive zone model for the fracture of polymeric interfaces and performs a Bayesian calibration, an uncertainty quantification, and a sensitivity analysis for the model. The proposed cohesive zone model accounts for both reversible elastic and irreversible rate-dependent separation sliding deformation at the interface. The viscous dissipation due to the irreversible opening at the interface is modeled using elastic-viscoplastic kinematics that incorporates the effects of strain rate. Inverse calibration of parameters for such complex models through trial and error is challenging due to the large number of parameters of the model. Moreover, the calibrated parameter values are often non-unique and uncertain when the available experimental data is limited. To tackle this challenge, we employ a Bayesian calibration approach to identify parameters from experimental data, the resulting parameters significantly enhance the accuracy of the model. To quantify the uncertainty associated with the inverse parameter estimation, a modular Bayesian approach is employed to calibrate the unknown model parameters, accounting for the parameter uncertainty of the cohesive zone model. The advantages of the Bayesian calibration over a deterministic parameter fit are demonstrated. Further, to quantify the model uncertainties, such as incorrect assumptions or missing physics, a discrepancy function is introduced, which significantly improves the model’s prediction. Finally, the total uncertainty of the model is quantified in a predictive setting. A sensitivity analysis is performed to assess how changes in the input variables of the model affect the peak load, facilitating the identification of a concise set of highly influential parameters. The present approach can be used for calibration and uncertainty quantification for other complex computational mechanics models. It should also facilitate the designing of interface materials under uncertainty.

A model for capturing the rate-dependent mechanical behaviour of liquid crystal elastomers

A modelling framework is adopted and adapted here to model the rate-dependent deformation behaviour of liquid crystal elastomers (LCEs). On using an isotropic hyperelastic strain energy function WI1,I2 of binomial form, previously employed for capturing the (quasi-static) deformation of polydomain LCEs, an extension is presented for application to the rate-dependent behaviour of LCEs by incorporating the deformation rate into the model parameters. That is, the model parameters of the core hyperelastic strain energy function W are considered to evolve with, i.e., be a function of, the deformation rate. A specific measure of deformation rate ɛ̇ is utilised, and a particular functional dependency of the hyperelastic model parameters on ɛ̇, of a skewed exponential form, is devised and presented. The ensuing rate-dependent model Wr is then applied to a range of extant experimental datasets, encompassing various LCE specimens and across different deformation rates, under uniaxial loading and including both tension and compression deformations. The model is shown to capture and predict the data favourably. Given the simplic of devising and implementing the presented modelling framework, the flexibility it provides for incorporating different choices of strain energy W function as well as the functional form(s) of the dependency of the model parameters on the deformation rate, and the accuracy of the modelling results, the application of this modelling approach to capturing the rate-dependent deformation behaviour of LCEs is hereby presented.

Cohesive phase-field model for dynamic fractures in coal seams

A novel rate-dependent cohesive phase-field model that can simulate fluid-driven dynamic quasibrittle fracture in soft coal rocks is presented. The coal matrix and fractures are described by a unified fluid continuity equation, and the Poiseuille flow in the fractures is modeled by deformation-dependent permeability. Dynamic evolution equations for porosity, permeability and the Biot coefficient during phase-field evolution are developed. The phase-field evolution equation is derived from the total energy generalization based on the Francfort–Marigo variational principle. To simulate the rate-dependent dynamic fracture problem accurately, the effects of the phase-field evolution dissipation energy and inertia energy are considered. The above multifield coupled system is solved via a staggered approach within the finite element framework via the Newton–Raphson iterative algorithm. The independence and convergence of the proposed model were verified, and its accuracy was validated via an analytical solution for crack width and a benchmark test for dynamic crack propagation. The propagation law of single/multicluster hydraulic fractures in coal seams with interlayers was investigated via the proposed model. The simulations revealed that the number of clusters and the mechanical properties of the interlayers significantly influence the fracture morphology. Finally, hydraulic fracturing of fracture-developed coal seams was simulated, confirming the potential of the proposed model.

Influence of rate effects on delamination: From Crack Leap Shear tests to low-velocity impacts

This paper is devoted to the study of different aspects of rate effects in the case of delamination. Two rate dependent cohesive zone models are compared for which the shear rate dependency is identified using Crack Leap Shear experiments. For both models, the key factor appears to be the evolution of the critical energy release rate as function of the crack opening. Moreover, if for very low velocity (1 m/s) rate effects can be neglected, it is not the case for quite low velocity (10 m/s). In the case of a 16 plies quasi-isotropic plate for the size and the shape of the delamination are significantly influenced by the rate effects.

Extension of Flow Behaviour and Damage Models for Cast Iron Alloys with Strain Rate Effect

Cast iron alloys with low production cost and quite good mechanical properties are widely used in the automotive industry. To study the mechanical behavior of a typical ductile cast iron (GJS-450) with nodular graphite, uni-axial quasi-static and dynamic tensile tests at strain rates of 10−4, 1, 10, 100, and 250 s−1 were carried out. In order to investigate the influence of stress state on the deformation and fracture parameters, specimens with various geometries were used in the experiments. Stress strain curves and fracture strains of the GJS-450 alloy in the strain rate range of 10−4 to 250 s−1 were obtained. A strain rate-dependent plastic flow model was proposed to describe the mechanical behavior in the corresponding strain-rate range. The available damage model was extended to take the strain rate into account and calibrated based on the analysis of local fracture strains. Simulations with the proposed plastic flow model and the damage model were conducted to observe the deformation and fracture process. The results show that the strain rate has obviously nonlinear effects on the yield stress and fracture strain of GJS-450 alloys. The predictions with the proposed plastic flow and damage models at various strain rates agree well with the experimental results, which illustrates that the rate-dependent plastic flow and damage models can be used to describe the mechanical behavior of cast iron alloys at elevated strain rates. The proposed plastic flow and damage models can be used to describe the deformation and fracture analysis of materials with similar properties.

A crystal plasticity based strain rate dependent model across an ultra-wide range

Numerous studies have investigated the strain rate sensitive behaviors of materials, consistently reporting enhanced stress values and increased dislocation density with rising strain rates. Behind these phenomena lies the intrinsic nature of dislocation activity. In this context, we introduce an analysis method within a crystal-plasticity (CP) framework, incorporating molecular dynamics insights for a comprehensive range of strain rates (7.5 × 10−5/s to 5 × 107/s). This approach offers a refined understanding of strain rate sensitive behaviors, mainly influenced by dislocation movement laws and strain-rate-dependent saturation of dislocation density. We elucidate the impact of deformation loading conditions on Schmidt factors and active slip systems, which are also crucial for understanding variations in SRS. Ultimately, this study underscores the CP method's effectiveness in comprehensive SRS analysis, seamlessly integrating experimental observations with theoretical predictions for advanced material characterization.