Rate-dependent Constitutive Model Research Articles

Towards accurate prediction of the dynamic behavior and failure mechanisms of three-dimensional braided composites: A multiscale analysis scheme

In this paper, a dynamic multiscale model is proposed for three-dimensional four-directional (3D4D) braided composites, considering the effects of strain rate and shear enhanced failure mechanism. The strain rate-dependent constitutive models are established for both resin and yarns. The accuracy of the yarn model is validated by comparing the simulated stress-strain curves with the experimental results obtained from the Split Hopkinson Pressure Bar (SHPB) tests with various strain rates. The shear enhancement coefficient of the yarn is determined from the failure envelopes with a discussion of crack propagation angles. The results show that, based on the calculated microscale properties, the mesoscale model accurately predicts both the longitudinal and transverse compressive strain-stress behavior of 3D4D braided composites at different strain rates. A comparative analysis reveals that the shear-enhanced failure model achieves a better prediction compared with other counterparts, such as Hashin-Rotem and Tsai-Wu Models. The proposed dynamic multiscale scheme for 3D braided composites is an efficient and accurate tool to characterize their multiscale features and strain rate-dependent behavior.

On thermal and strain-rate dependences of polymethacrylimide (PMI) foam materials

Polymethacrylimide (PMI) foams exhibit great potentials in engineering applications thanks to its outstanding lightweight and superior mechanical properties. However, the temperature and loading rate can greatly affect its mechanical behaviors, which are the key factors in analysis and design of the PMI foam structures for practical applications. Unfortunately, there is lack of fundamental data and constitutive models available concerning the elastic-plastic properties of the PMI foam in various stress states at different ambient temperatures and different strain rates. Therefore, this study aims to fill this knowledge gap by generating basic experimental data and providing a generic modeling solution for PMI foam materials and structures. To better reflect the actual service conditions in practice, different stress states are considered under different temperatures and strain rates. The cyclic loads are applied to characterize the actual mechanical responses of the PMI foam where the initial and subsequent yield surfaces were calibrated and analyzed. Following the experimental results, the elasticity, plasticity, viscosity and damage of the PMI foam are investigated thoroughly, especially for the initial and subsequent yielding behaviors. Further, the thermal and strain rate dependent constitutive models are established and calibrated based on the in-house experimental data for finite element (FE) modeling. A novel damage model is proposed to replicate the phenomenon of the stiffness degradation in compression. It is found that accurate predictions on the mechanical responses of PMI foam can be achieved under different temperatures and strain rates. This study is anticipated to provide fundamental experimental data and effective constitutive models of PMI foam for real engineering applications, such as lightweight design of sandwich structures under some harsh conditions.

Mott waves and fragment size in viscoplastic metals

Previous ductile fragmentation analysis involves unloading wave (Mott wave) propagations in rate-independent rigid-plastic materials. This paper studies the dynamic fragmentations of a rigid-viscoplastic material in which there exists a rigid unloading zone and a viscoplastic unloading zone. A linear diffusion equation incorporating a moving boundary condition for the viscoplastic unloading zone is presented. The fracture process of an isolated crack in the infinite medium, and that of an array of equally spaced cracks, are investigated numerically. The crack array problem reveals an optimal crack spacing upon which the material fails and unloads most rapidly. Based on the 'principle of the rapidest unloading (PRU)', this crack spacing is regarded as the measure of fragment size. Systematic numerical calculations are conducted to obtain the fragment size data as functions of strain rate and viscosity coefficient. A modified Grady-Kipp fragment size formula is proposed to accommodate the viscous effect. In the case of non-viscous material, the formula returns to the classical Grady-Kipp fragment size model. Finite element simulations using rate-dependent constitutive model show similar trend of viscous effect on the fragment sizes, and the modified fragment size formula yields better agreement with the simulated data.

A data-driven rate and temperature dependent constitutive model of the compression response of a syntactic foam

Polymeric syntactic foams are used in aerospace and marine applications requiring low density and low moisture absorption together with high specific strength and stiffness. Their mechanical response is highly sensitive to temperature and strain rate and such sensitivity must be modelled accurately. In this study, the uniaxial compressive response of a polymeric syntactic foam is measured at strain rates in the range [10−3, 2.5·103] /s and temperatures varying between −25°C and 100°C. The resulting dataset is used to train a neural network to predict the compressive response of the foam at arbitrary strain rates and temperatures. It is found that the surrogate model is highly effective in predicting the material response at temperature and rates not included in its training set. Finally, a stochastic version of the data-driven model to allow predictions of the variability in the stress versus strain response is proposed.

Unveiling the Strain Rate Sensitivity of G18NiCrMo3-6 CAST Steel in Tension/Compression Asymmetry

This research was devoted to unveiling the strain rate sensitivity (SRS) of G18NiCrMo3-6 cast steel in tension/compression asymmetry. For that purpose, detailed mechanical characterization tests were conducted providing a process window covering quasi-static and medium strain rate regimes (0.001, 0.1, 10 [s−1]) in tension and compression states. Through this experimental effort, the SRS of the material could be extracted by a function of strain and the strain rate which enabled us to create a mathematical expression to easily be implemented as a state variable for constitutive material modeling. Finally, a pressure- and rate-dependent constitutive material model on the basis of the Cocks’89 yield locus definition was created using a subroutine (UMAT) file and the material parameters were verified with respect to the experimental data. The UMAT file also takes into account the tension/compression asymmetry in yielding to handle the effect of the porous media plasticity concept. The predictions of the proposed material model are in line with the experimental outputs.

A rate-dependent constitutive model for saturated frozen soil considering local breakage mechanism

A rate-dependent constitutive model for saturated frozen soil is vital in frozen soil mechanics, especially when simultaneously describing the nonlinearity, dilatancy and strain-softening characteristics. The distribution of the non-uniform strain rate of saturated frozen soil at the meso-scale due to the local ice-cementation breakage is described by a newly binary-medium-based homogenization equation. Based on the field-equation-based approach of the meso-mechanics theory, the interaction expression of the strain rate at macro- and meso-scale is derived, which can give the strain rate concentration tensor at different crushed degrees. With the thermodynamics and empirical assumption, a breakage ratio in the rate-dependent form is determined. This overcomes the limitations of the existing binary-medium-based models that are difficult to simulate rate-dependent mechanical response. Based on these assumptions, a newly binary-medium-based rate-dependent model is proposed considering both the ice bond breakage and material composition characteristics of saturated frozen soil. The proposed constitutive model has been validated by the test results on frozen soils with different temperatures and strain rates.

Calibration and application of damage and rate-dependent constitutive model for SPCE steel

The SPCE steel demonstrates favorable deep-drawing and forming properties, leading to its widespread use in industrial production. However, when subjected to complex loads, it becomes crucial to develop a constitutive model that accurately characterizes its mechanical properties. Such a model serves as the foundation for engineering design and safety assessment. Both quasi-static tension tests and high strain rate tension tests were conducted on SPCE steel samples using static and dynamic tension testing machines at room temperature. The resulting test data was used to introduce three dynamic increase factor indicators (DIFs) that quantify the strain rate effect on SPCE steels. Additionally, the fracture morphology of typical post-test specimens was analyzed using a scanning electron microscope (SEM) to demonstrate the strain rate effect on the mechanical properties of SPCE steels from a microscopic perspective, revealing ductile fracture as the primary fracture mechanism. A double-power plastic constitutive model was developed to describe the material's tensile mechanical behavior prior to necking. Simulation analysis was performed to validate the model, and the parameters of the Ductile damage model were identified based on the simulation data. The proposed constitutive model underwent numerical verification, demonstrating good consistency between the simulation and experimental results. Furthermore, the same constitutive model was applied to process the test data of HC650 steel using the same methodology. The conclusion drawn from this analysis aligned with that of SPCE steel, confirming the suitability of the proposed constitutive model for characterizing the mechanical behavior of steels without yielding a platform under different strain rates.

Theoretical and numerical research on the dynamic launch response of carbon fiber composite cartridges

Understanding the dynamic response of composite material cartridges during the firing process is of great significance for improving their reliability and safety. A theoretical model describing the dynamic response of composite material cartridges is established based on the thick-walled cylinder theory and rate-dependent constitutive model of composite materials. The correctness of the theoretical model is validated through finite element simulations of cartridge deformation. The influence of chamber pressure and cartridge wall thickness on the cartridge’s deformation process and stress distribution is analyzed. The results indicate that the primary deformation of composite material cartridges inside the chamber is elastic deformation. Compared to metal cartridges, composite material cartridges require higher pressure for touching-chamber and are more prone to developing gaps after unloading to ensure smooth extraction. During the deformation process, the touching-chamber behavior of the cartridge can improve the stress distribution. Under the same chamber pressure, the touching-chamber behavior can reduce the circumferential stress by approximately 30%. The inner wall surface of the cartridge is a critical area that requires attention. The touching-chamber behavior can be facilitated by appropriately reducing the cartridge wall thickness while ensuring overall strength. This study can provide guidance for the optimization design of composite material cartridges.

Damage evolution law and failure mechanism of rock impacted by high-pressure water jet under in-situ stress condition.

To investigate the real physical mechanism of rock fragmentation subjected to water jet under in-situ stress condition, a numerical model based on the SPH algorithm was established using the rate-dependent constitutive model to simulate the rock-breaking process. First, the damage evolution law of rock impacted by high-pressure water jet under in-situ stress conditions was studied by analyzing the distribution characteristics of the damage field in the dynamic process of water jet impinging. The results showed that the damage field, widths of surface damage, maximum widths of damage and mean depths of damage of rock decreased with the increase of in-situ stress, indicating that the existence of initial in-situ stress had a strong inhibitory effect on rock fragmentation. The attenuation of the maximum widths of damage could be divided into two stages. The mean depths of damage of rock played a leading role in the number of damage elements. Furthermore, on this basis, the real physical mechanism of rock fragmentation subjected to water jet under in-situ stress condition was revealed by analyzing the stress states and damage history variables of the particles in the selected five typical regions. The study showed that the failure type of the upper rock elements in the crushing zone was brittle failure caused by a combination of compressive stress and shear stress with or without in-situ stress. However, the failure mechanisms of rock elements in crack zone were completely different with or without in-situ stress. In the absence of in-situ stress, the failure type of rock impacted by water jet was the coexistence of damage caused by compressive-shear stress and tensile stress, while in the presence of in-situ stress, the failure type of rock impacted by water jet was mainly the damage caused by compressive-shear stress.

Modelling mesh independent failure loads of a soft strain-softening clay using a rate dependent model

If no regularisation method is used during finite element modelling of strain-softening (brittle) materials, results become mesh dependent: manifesting as very thin shear bands whose thickness and orientation depends on the mesh discretisation. The objectives of this paper were to determine if mesh independent results were achieved, and to track the evolving width of the shear band when considering naturally occurring properties of soft clays, including a rate dependent stress-strain-strength relationship. The constitutive model included a destructuration process and strain-rate dependency, and the analyses allowed for local pore water flow. A bearing capacity problem was loaded to failure. Analyses were conducted with different mesh discretisation, loading rates and permeabilities. The results demonstrated mesh independent failure loads, and the strain-softening occurred in zones with thickness independent of mesh size. Local pore water flow delayed the initiation of strain softening and resulted in higher failure loads. The softening zones grew in thickness to over one meter, reinforcing that the finite element method is suitable for the determination of the bearing capacity of strain-softening sensitive clays. In conclusion, the rate dependent constitutive model is a promising method for regularising strain localisation in soft brittle clays under quasi-static loading conditions.

Study on Mechanical Failure Behavior of Steel-Wire Wound Reinforced Thermoplastic Pipe under Combined Internal Pressure and Soil Landslide Conditions.

A steel-wire wound reinforced thermoplastic pipe (SWW-RTP) has been widely utilized in many industrial areas, and a soil landslide is an inevitable hazardous extreme condition for the SWW-RTP as it is usually buried underground. It is imperative to study the mechanical failure behavior and the failure criterion of the SWW-RTP under the combination of internal pressure and soil landslide conditions, and this paper is the first study to investigate the topic. In this paper, groups of stress-strain curves of high-density polyethylene (HDPE) and steel wires were obtained by uniaxial tensile tests at different strain rates, with the help of a Digital Image Correlation device (DIC). A rate-dependent constitutive model was employed to represent the mechanical behavior of the HDPE and to help deduce the stress-strain curve of the HDPE under the required strain rate, estimated from the static simplification of the dynamic soil landslide. Afterwards, a finite element model of the SWW-RTP, embedded in a cubic of soil, was established with the software ABAQUS. The SWW-RTP model was composed of HDPE solid elements, embedded with steel-wire truss elements, and the soil was characterized with the extended Drucker-Prager model. A quartic polynomial displacement distribution was applied to the soil model to represent the soil landslide. Then, the mechanical response of the SWW-RTP was analyzed. It was found that the failure criterion of the HDPE yield was more suitable for the pipe subjected to internal pressure and soil landslide conditions, instead of the steel-wire strength failure criterion always used in traditional research on the SWW-RTP. Further, the influence of landslide width, internal pressure and steel-wire number were discussed. The larger the width of the landslide area, the gentler the deformation of the pipeline; this resulted in an increase in the maximum landslide and a decrease in the maximum curvature with the width of the landslide area. The relatively high internal pressure was beneficial to the safety of the SWW-RTP under landslide, because the internal pressure could increase the stiffness of the pipeline. The number of steel wires had a limited influence on the maximum landslide required for the SWW-RTP's failure. This work can be useful for the design and safe assessment of the SWW-RTP under internal pressure and soil landslide conditions.

Strain Rate Effect on the Thermomechanical Behavior of NiTi Shape Memory Alloys: A Literature Review

A review of experiments and models for the strain rate effect of NiTi Shape Memory Alloys (SMAs) is presented in this paper. Experimental observations on the rate-dependent properties, such as stress responses, temperature evolutions, and phase nucleation and propagation, under uniaxial loads are classified and summarized based on the strain rate values. The strain rates are divided into five ranges and in each range the deformation mechanism is unique. For comparison, results under other loading modes are also reviewed; however, these are shorter in length due to a limited number of experiments. A brief discussion on the influences of the microstructure on the strain-rate responses is followed. Modeling the rate-dependent behaviors of NiTi SMAs focuses on incorporating the physical origins in the constitutive relationship. Thermal source models are the key rate-dependent constitutive models under quasi-static loading to account for the self-heating mechanism. Thermal kinetic models, evolving from thermal source models, address the kinetic relationship in dynamic deformation.

Rate-dependent constitutive model of high-strength reinforcing steel HTRB600E in tension

High-strength reinforcing steel HTRB600E has significant potential application in reinforced concrete (RC) structures because of its high strength and ductility. Under the risk that structures with HTRB600E are subjected to severe dynamic loadings, the rate-dependent behavior and constitutive model of HTRB600E need to be investigated to provide a basis for the structural analysis. A series of tension tests were carried out to acquire the engineering stress–strain curves of HTRB600E in the strain rate range from 0.00025 to 590 s−1. Moreover, a hybrid experimental–numerical converse method was used to obtain the true stress–strain curves beyond the necking of HTRB600E at different strain rates. The test results indicate that the strain rate significantly affects the mechanical properties of HTRB600E. The effect of strain rate on yield and ultimate tensile strength, uniform elongation, and yield to tensile ratio of HTRB600E was discussed. Three commonly-used models were calibrated or modified to predict the dynamic increase factors (DIFs) for yield and ultimate tensile strength of HTRB600E. A rate-dependent constitutive model for the engineering stress–strain curves modified from the Mander model was presented for the structure's analytical model based on the section analysis technique. A new rate-dependent constitutive model for the true stress-plastic strain relationship with necking behavior was proposed for the structure's nonlinear finite element analysis model. The formulations of DIFs and constitutive models agree with the test results well.

Experimental Analysis and Constitutive Modeling of Cyclic Behavior of 304 L Stainless Steel: Introduction of Isotropic Hardening Fading Effect

Abstract In this study, the mechanical behavior of austenitic stainless steel 304 L under low cycle fatigue was investigated under different uni-axial strain-controlled loadings of 0.5%, 0.8%, 1.0%, 1.2%, and 1.5%. The analysis of the experimentally determined strain versus stress hysteresis curves was carried out to achieve stress quantities such as amplitude stress, peak effective stress, and peak back stress. It was observed that in the early stage of cyclic loading, material underwent initial hardening, followed by softening phenomena which were more considerable in the lower strain range. Before the failure, the secondary hardening was observed at the final stage. In addition to accumulated plastic strain, it was shown that the peak back stress and peak effective stress which is associated with isotropic hardening and kinematic hardening behavior, respectively, are influenced by the strain range effect. Therefore, the coefficient of recall term that appeared in the Armstrong–Frederick nonlinear kinematic hardening model was considered to be dependent on the radius of the memory surface. Furthermore, to increase the ability of the plasticity constitutive model to show a smooth transition between various hardening stages, the radius of the yield surface which is associated with the isotropic hardening rule was equipped with the fading effect. Finally, by the comparison of numerical and experimental results, the capability of the rate-dependent constitutive model over classical rate-independent plasticity in the prediction of mechanical behavior of steel 304 L under strain-controlled cyclic loading was revealed.

Strengthening and toughening behaviors and dynamic constitutive model of carbon-based hierarchical fiber modified concrete: Cross-scale synergistic effects of carbon nanotubes and carbon fiber

Considering the synergistic effects (filling, activating, bridging, anti-cracking, and interface-optimizing) of combined carbon fiber and carbon nanotubes, the strategy of using carbon nanotubes-carbon fiber cross-scale reinforcement (CNTs-CF) to enhance concrete was proposed. The strengthening and toughening behaviors of CNTs-CF modified concrete (CCMC) were experimentally investigated. The results showed that CCMC had good crushing resistance capacity, and the addition of an appropriate amount of CNTs-CF was conducive to alleviating the brittle failure characteristics of concrete materials. Compared with ordinary concrete, the static and dynamic strength properties and toughness indexes of CCMC were greatly improved. When the volume content of CNTs-CF in concrete was 0.3%, CCMC showed the most superior strengthening and toughening behaviors. The hierarchical assembly and synergistic effect of carbon fiber and carbon nanotubes realized the combination of high strength and high toughness of concrete materials. The proposed rate-dependent constitutive model with damage based on Weibull statistical theory in this study can be used to accurately predict the dynamic compressive stress-strain relationship curve of concrete under different fiber content and strain rate.

Dynamic compressive response of porcine muscle measured using a split Hopkinson bar system with a pair of PVDF force transducers

The classic metallic Split Hopkinson Pressure Bar (SHPB) cannot capture the transmitted signal accurately when measuring soft biological tissue, because of the very low wave impedance and strength of this material. So the dynamic compressive response of porcine muscle has been investigated by using a modified SHPB. The forces on both ends of the sample measured using Polyvinylidene fluor (PVDF) transducers were applied to calculate the stress in the specimen instead of the strain gauge signal on the transmitted bar. Moreover, a circular cardboard disk pulse shaper was applied for generating a suitable incident pulse to achieve stress equilibrium and constant strain rates in the specimens. Then, the dynamic mechanical properties of porcine muscle parallel and perpendicular to the fiber directions were measured, and the stress equilibrium process during loading was analyzed, as well as the inertia-induced extra stress being corrected. Furthermore, quasi-static tests were conducted at two different strain rates to investigate the strain rate dependence using a universal material testing machine. The results show that the stress-strain curves are sensitive to strain rate in the two different loading directions. The compressive stress perpendicular to the fiber direction is stiffer than that parallel to the fiber direction. In addition, a strain rate-dependent constitutive model was developed based on the mechanical response of the muscle at different strain rates and fitted to the experimental data. The results show that the overall fit is good, and the constitutive model could describe the muscle's dynamic mechanical properties.

A rate-dependent constitutive model of piezoelectric thermoelasticity and structural thermo-electromechanical responses analysis to multilayered laminated piezoelectric smart composites

Thermo-electromechanical coupling analysis at micro/nano-seconds appears to be particularly important, where strain and electric relaxation effects will increase significantly in such case. So far, although temperature-rate-dependent theory of piezoelectric thermoelasticity has been historically proposed, it may be no longer hold anymore as relaxation effects both in electrical and strain fields have not been fully considered yet. This work mainly contributes to constitutive modeling of a novel rate-dependent piezoelectric thermoelasticity by considering time-derivatives terms of the elastic strain, relative temperature, and electric intensity associated with related relaxation time parameters. Constitutive and field equations are strictly derived via extended thermodynamics. Newly developed theoretical model is applied to analyze structural thermo-electromechanical responses of multilayered laminated piezoelectric smart composites accounting for contact imperfection and material parameters via a semi-analytical integrated transformation technique. Dimensionless results reveal that the properly selecting thermal/stain/electric relaxation time parameter or considering non-idealized boundary conditions at interface of multilayered laminated piezoelectric smart composites will maximally enhance the electric energy harvesting, realize the displacement control, and improve the harmful stress isolation.

High-temperature and dynamic mechanical characterization of closed-cell aluminum foams

Metallic foams exhibit great application prospects in some extreme working conditions such as high temperatures and/or high-velocity impact environments, but predominantly, their mechanical behavior has remained to be fully understood to date. In this study the uniaxial and biaxial tests are conducted on closed-cell aluminum foams at elevated temperatures. It is indicated that both drop stress and initial failure strength exhibit approximately linear relationship with the applied temperature, while the densification strain keeps almost at a constant value. The higher foam density or lower applied temperature, the larger the initial failure surface characterized in the von Mises - mean stress plane. However, the normalized failure surfaces nearly do not depend on foam density or temperature, and can be well fitted in terms of elliptical or parabolic function. Dynamic compressive tests under constant strain-rates reveal that the compressive strength is correlated to strain-rate positively, but the densification strain negatively. A novel rate-dependent constitutive model is proposed to describe the compressive constitutive behavior of Al foams. Finally, the crushable foam model with the tabular input of yield ratio approach can produce satisfactory numerical predictions compared to dynamic experimental results. This study provides new insights into the intrinsic mechanical properties of metallic foams under some extreme conditions.

Understanding of scratch behavior of an automotive coating system: Experiments and finite element analysis

As one of the most important components of a vehicle, the automotive coating system is typically a multi-layer polymeric structure with complex scratch damage patterns. This work is the first one to systematically investigate the scratch performance of an automotive coating system via experimental and numerical approaches. To achieve this end, uniaxial tensile tests are firstly carried out to acquire the basic mechanical properties of paint films at different strain rates. Then, a series of scratch tests are conducted in accordance with ASTM D7027/ISO 19252 standards, and the damage features are identified. In addition, the effects of scratch speed, loading rate, and stylus type are experimentally investigated. Afterwards, a finite element approach for scratch simulations is developed, which includes a rate-dependent constitutive model calibrated using the uniaxial tensile tests, a tied interface algorithm for efficient connection between coating and substrate, and an interior contact method to avoid excessive element distortions. Eventually, with the aid of the presented numerical approach, the damage patterns observed in scratch tests are numerically explained, and the scratch damage process is reasonably surmised. The presented approaches and findings can contribute to the development of anti-scratch automotive coating structures.

Numerical Simulation Study on the Strengthening Mechanism of Rock Materials under Impact Loads

Based on an analysis of real stress in rock specimens, the mechanisms of inertia and cracks on the dynamic properties of concrete material are discussed in this paper. A dynamic enhancement factor is introduced on the basis of static mesodamage constitutive theory, and a rate-dependent three-dimensional rock dynamic constitutive model is established. The orthogonal design and range method are used to analyze the sensitivity of the dynamic damage constitutive parameters, and a differential evolution algorithm is used to invert the relevant parameters of the constitutive model. Based on the drop hammer impact model established by the rock dynamic damage constitutive model, the AE distribution, crack propagation mode, and load-displacement curve of the rock sample are obtained. The numerical simulation results show that rock specimens under dynamic loading have different failure forms, mechanical response mechanisms, and energy evolution than those under static loading. Under dynamic loading, the rock specimen forms more cracks and releases more elastic energy. Furthermore, it shows greater strength characteristics. Inhomogeneities and inertial effects of rock materials are the fundamental reasons for the increase in the dynamic strength of rock materials. A reduction in inertial effects is the main reason for strength reduction in rock samples with hole defects.