Set Of Constitutive Equations Research Articles

Short-term creep approach to redefining the role of 17-4PH stainless steel for high-temperature applications

The creep response of the 17-4PH martensitic age-hardening steel in H1150 state was investigated at 427 and 482 °C. Hardness measurements of the heads of the creep samples demonstrated that the material underwent additional age hardening during the high-temperature exposure. Microstructural investigations confirmed that the additional precipitation of carbides and the G-phase occurred at the lowest temperature. A set of constitutive equations previously developed to describe the creep response of particle-strengthened alloys was successfully used to obtain a comprehensive description of the experimental data. The value of the particle strengthening term was obtained from the hardness measurements and corresponded to the Orowan stress. The model accurately described the observed minimum creep rate dependence on the applied stress and explained the occurrence of lower values of the minimum strain rate observed during variable-load experiments.

A Predictive Toxicokinetic Model for Nickel Leaching from Vascular Stents.

In vitro testing methods offer valuable insights into the corrosion vulnerability of metal implants and enable prompt comparison between devices. However, they fall short in predicting the extent of leaching and the biodistribution of implant byproducts under in vivo conditions. Physiologically based toxicokinetic (PBTK) models are capable of quantitatively establishing such correlations and therefore provide a powerful tool in advancing nonclinical methods to test medical implants and assess patient exposure to implant debris. In this study, we present a multicompartment PBTK model and a simulation engine for toxicological risk assessment of vascular stents. The mathematical model consists of a detailed set of constitutive equations that describe the transfer of nickel ions from the device to peri-implant tissue and circulation and the nickel mass exchange between blood and the various tissues/organs and excreta. Model parameterization was performed using (1) in-house-produced data from immersion testing to compute the device-specific diffusion parameters and (2) full-scale animal in situ implantation studies to extract the mammalian-specific biokinetic functions that characterize the time-dependent biodistribution of the released ions. The PBTK model was put to the test using a simulation engine to estimate the concentration-time profiles, along with confidence intervals through probabilistic Monte Carlo, of nickel ions leaching from the implanted devices and determine if permissible exposure limits are exceeded. The model-derived output demonstrated prognostic conformity with reported experimental data, indicating that it may provide the basis for the broader use of modeling and simulation tools to guide the optimal design of implantable devices in compliance with exposure limits and other regulatory requirements.

Mechanical response of 6061-T6 aluminium alloy subjected to dynamic testing at low temperature: Experiment and modelling

The aim of this research was to investigate an effect of low temperature on the mechanical properties and microstructure of 6061-T6 aluminium alloy (AA6061-T6) subjected to dynamic loading. The specimens were subjected to dynamic compression at a low temperature of −80 °C in a range of strain rates from 1.25 × 103 1/s to 3.4 × 103 1/s to compare their mechanical responses. The deformation mechanisms were analysed through EBSD observations during which dynamic recovery, was found as the dominant one. Furthermore, microstructural analysis indicated that deformation under high strain rate conditions and temperature of -80 °C enables to keep the constant initial grain size of the material after the loading applied. The material behaviour was modelled using mechanism-based viscoplastic constitutive equations. Furthermore, an accuracy of the developed model was validated by comparing it to experimental data. The set of constitutive equations proposed has been successful in modelling the stress-strain behaviour of the material for the range of strain rates and temperatures encountered in aluminium-forming processes under low-temperature conditions.

On the continuum mechanics of growing plant-like structures

The contribution at hand focuses on the introduction of a novel approach to model biological growth. The proposed formulations are chosen to represent plant like structures. Therefore, thermomechanically open systems are considered. The balance laws are presented for such systems. Furthermore, the proposed formulations are coupled with an adaptive meshing framework. Therefore, a so-called structural generator is presented and utilized in this work. Since no growth formulations within the framework of continuum mechanics exist so far for plant like systems, a novel set of constitutive equations is shown. The newly described principles are the phototropism and graviotropism. In the numerical examples, it is shown that the proposed formulation yields physically meaningful results. The combination of different growth principles results in plausible interactions of the aforementioned principles. Furthermore, results of numerical simulations are shown, which represent the growth process of plant like biological structures.

Analytical flow equation for irradiated low-alloy steels established by multiscale modeling

In this paper, we show how results of multiscale modeling can be condensed to establish a simple analytical flow equation for irradiated low-alloy steels. The flow equation accounts for temperature, strain rate, the initial and irradiation microstructure. Starting from a complete set of constitutive equations describing crystal plasticity of irradiated ferritique steels, simple assumptions and approximations are used to integrate the equations over the homogenized polycrystal. The final flow equation incorporates the lattice friction resistance, the Hall-Petch effect, the forest strengthening and irradiation hardening. Every component is established by multiscale modeling at lower scales. Irradiation hardening is attributed to solute clusters. Their size and density are determined in experiment, their shear resistance computed by atomistic simulations and their contribution to the flow stress assessed using dislocation dynamics simulations. The formation of solute clusters is found to increase the flow stress and the strain hardening modulus. Irradiation hardening decreases with temperature and with the yield stress of the unirradiated material. Predictions of the tensile curves and irradiation hardening are compared with many experimental results. A special section is dedicated to the interpretation of irradiation hardening measured in a surveillance program.

Constitutive equations for thermo-elasto-plastic metallic materials undergoing large temperature variations

In the present work, a framework for the development of constitutive models for metallic materials in a thermomechanical context is proposed. Such a framework provides some guidelines to deal with processes involving large temperature variations, which is typical of manufacturing operations or severe service situations. The proposed framework relies on the additive decomposition of the logarithmic strain tensor to include the contributions of elasticity, plasticity and thermal expansion to deformation. Also, the classical internal variable concept is used to describe the history effects (e.g., hardening and recovery) associated with the development of plasticity. Particular attention is given to considering the impact of temperature on thermophysical properties. For the purpose of illustration, the proposed framework is used to build a constitutive model for polycrystalline copper. The resulting set of constitutive equations allows investigating the thermomechanical behavior of this specific material for some simple deformation histories. The corresponding results are finally used to evaluate the temperature-dependence of common thermophysical properties. The importance of the different heat sources that contribute to the heat diffusion equation is also discussed.

A unified constitutive approach for creep response of AISI 316 steel produced by conventional technologies or additive manufacturing techniques

ABSTRACT The creep response of AISI 316 and AISI 316 L was analysed to provide a coherent picture of the material behaviour, valid for both conventional wrought and additively manufactured steels. Literature evidences were considered. The analysis demonstrated that the presence of a fine distribution of particles, precipitated during creep exposure in both AISI 316 and 316 L(N), should be taken into account. A recent model, expressly developed for particle-strengthened alloys, was successfully used to describe the minimum creep rate dependence on applied stress for AISI 316 and 316 L(N). The same set of constitutive equations, in combination with the composite model, was then used for describing creep data obtained by testing the AISI 316 L produced by additive manufacturing. The main difference with wrought materials was that the microstructure of the additively manufactured alloy was composed by cells (soft zones) and thick walls heavily decorated with ultrafine oxide particles (hard zones).

Thermodynamic framework for variance-based non-local constitutive models

The present work deals with the development of a framework dedicated to the construction of constitutive models with non-local internal variables. Such internal variables allow considering the impact of microstructural changes on the current state of a material point. Non-locality is introduced by considering, not only the spatial average, but also the spatial variance of an internal variable in constitutive relations. The proposed framework relies on continuum thermodynamics to construct the set of constitutive equations. Such a framework allows including some information regarding the spatial distribution of internal variables when constructing non-local models for thermomechanical applications. In contrast with gradient-type models, this strategy does not require additional equilibrium equations and boundary conditions. For the purpose of illustration, some numerical examples are presented. According to the numerical results, the proposed framework can be used to circumvent the difficulties associated with excessive spatial localization or to consider size effects.

Periodic magnetohydrodynamic mixed convection flow along a cone embedded in a porous medium with variable surface temperature

The entrained flow and heat transfer in Newtonian fluid along the thermally and electrically conducting cone embedded in porous medium by encountering the variable surface temperature is carried out. The proposed process is time dependent that is the source of oscillation in all considered variables. The primary phenomenon is given a mathematical form in partial differential equations which are non-linear and coupled. By means of dimensionless formulation, the resulting set of constitutive equations is reduced to non-dimensional form. A suitable formulation that produces the smoothness in algorithm is utilized for the transformation of dimensionless form of equations to smooth ones. The pertinent material properties of interest such as velocity, temperature and magnetic profiles along with their derivatives are portrayed in graphical way under the impact of pertinent parameters. The main finding about the current communication is that for elevating the variable surface temperature parameter ζ, the main stream velocity and temperature field get declined and opposite attitude is noted in magnetic profile. Furthermore, maximum value for velocity and temperature are secured at ζ=0.1 and ζ=8.0 respectively but reverse scenario is seen in magnetic profile.

A new set of physically-based unified viscoplastic constitutive equations for superplastic deformation of Al-Zn-Mg-Cu alloy

A new set of physically-based unified viscoplastic constitutive equations using internal state variable(ISV) method during the superplastic deformation of Al-Zn-Mg-Cu alloy at temperatures ranging from 480 to 530 ℃ and strain rates ranging from 3×10−4s−1 to 3×10−3s−1 has been proposed. The damage evolution equation in this set of constitutive equations takes into account the critical cavity nucleation strain. This set of constitutive equations can predict microstructure evolution and flow stress during superplastic deformation of the investigated Al-Zn-Mg-Cu alloy. The set of constitutive equations have been coupled into crystal plasticity finite element method for determining the evolution of grain size and cavity volume fraction and the lattice rotation field map. The Material parameters are determined by genetic algorithm. The influences of strain, strain rate and temperature on grain size evolution and cavitation behavior were analyzed. The predicted stress-strain behavior, grain size and damage evolution are in good agreements with the experimental results.

Revisiting copper as a case study of creep in pure metals: Prediction of creep response in pure Cu in half-hard and friction-stir-processed states

A physical model was used to obtain a pre-assessment of the dependence of minimum creep rate on stress and temperature for ETP copper under two different initial conditions, i.e., in the R240 half-hardened state and after friction stir processing. Although the R240 samples contained a high dislocation density owing to pre-straining, the material that underwent friction stir processing (FSP) had a much finer grain size and a far lower dislocation density.The original Sandström physically based model developed for copper was modified to take into account the strengthening role of the grain boundaries, and its declining importance with increasing temperature and/or decreasing applied stress. In addition, the grain growth effect was estimated using appropriate empirical equations. The model curves obtained by introducing a few initial information, such as the grain size and the ultimate tensile strength at room temperature, in the resulting set of constitutive equations, largely overlapped. This finding was somewhat in conflict with the traditional view, which assumes that a fine grain size is an ineffective source of creep strengthening, if not detrimental.Subsequent creep experimental testing carried out between 178 and 355 °C substantially confirmed the picture provided by the pre-assessment, and the minimum creep rate values for the two materials largely overlapped on the same curves. Only at 355 °C did the FSP samples exhibit a somewhat higher creep rate, when compared with the R240 samples and the pre-assessment curve. The experimental data obtained for R240 copper at 475 °C were not properly described by the physical model, as expected on the basis of the previous literature reports.

Modeling of heat and mass transfer in a non-conventional rotary dryer

This study analyzed the simultaneous heat and mass transfer between soybean seeds and the air in a non-conventional rotary dryer developed by our research group, named as roto-aerated dryer. A two-phase model, with an appropriate set of constitutive equations, was used to describe the drying process. The system of equations was numerically solved using the normal collocation technique. The simulated results shown good agreement with the experimental data. The average difference between experimental and simulated results was of 1.5% for seed moisture content and of 6.3% for soybean temperature.

Modelling the creep behavior of an AlSi10Mg alloy produced by additive manufacturing

AlSiMg alloys produced by additive manufacturing possess an extremely fine and complex microstructure that in many ways defies the most widely used phenomenological models, which, in fact, have turned out to be poorly suited for predicting their mechanical properties. The underlying rationale for the peculiar properties of these alloys has been qualitatively established, however the need for a constitutive model with better predictive capability is still strong. To this aim, the ultra-fine microstructure was described by using a model-material (MM) consisting of soft and hard zones deforming under a similar strain rate. A physically-based set of constitutive equations which took into account also the coarsening/ripening phenomena of the Si-particles was used to predict the creep behavior of the MM. In parallel, the creep response of an AlSi10Mg alloy produced by additive manufaturing and tested in the as-deposited condition was investigated at temperatures ranging from 150 to 225 °C. The minimum creep rate curves obtained for the MM by the constitutive model were then compared with the experimental data obtained by testing the real alloy under constant load in different initial states. The excellent correlation between model curves and experimental results was discussed, taking into account the evolution of the microstructure during creep.

Plastic and fracture behavior of a dual phase steel sheet under quasi-static and dynamic loadings

Crack prediction is an important step of car design: its accuracy is crucial to avoid additional development costs and delays. The prediction of steel or aluminium sheets tearing is particularly challenging, and its simulation is not always reliable yet. This could be explained by the use of too simple fracture criteria based on a critical plastic strain that are uncoupled with the constitutive behavior (i.e.the material response is not affected by damage). To overcome this problem, coupled damage models are proposed in the literature. To select the appropriate constitutive equations to model the plastic and fracture behavior of a given material, various characterization tests are needed. A comprehensive experimental campaign that covers a wide range of stress states and loading rates was conducted. All tests can be performed on a tensile machine. From the results, an original set of constitutive equations was chosen from the literature to propose an extended Gurson-based damage model. A practical parameter identification procedure is proposed to calibrate the model on a few relevant specimen geometries. The model is then validated on another set of specimen geometries: comparisons between test results and simulations show a good agreement in terms of load–displacement curves for both quasi-static and dynamic loading conditions.

Analytical study of the direct initiation of gaseous detonations for small heat release

An analysis of the direct initiation of gaseous detonations in a spherical geometry is presented. The full set of constitutive equations is analysed by an asymptotic analysis in the double limit of Mach number close to unity (small heat release) and large thermal sensitivity. The quasi-steady curvature-induced quenching phenomenon is first revisited in this limit. Considering a realistic decrease rate of the rarefaction wave, the unsteady problem is reduced to a single nonlinear hyperbolic equation. The time-dependent velocity of the lead shock is an eigenfunction of the problem when two boundary conditions are imposed to the flow at the lead shock and at the burnt gas side. Following (Linan et al. , C. R. Mec. , vol. 340, 2012, pp. 829–844), the boundary condition in the quasi-transonic flow of burnt gas is expressed in terms of the curvature. Focusing our attention on successful initiation, the time-dependent velocity of the lead shock of a detonation approaching the Chapman–Jouguet regime is the solution of a nonlinear integral equation investigated for stable and marginally unstable detonations. By comparison with the quasi-steady trajectories in the phase space ‘propagation velocity versus radius’, the solution exhibits the unsteady effect produced upon the detonation decay by the long time delay of the upstream-running mode for transferring the rarefaction-wave-induced deceleration across the inner detonation structure from the burnt gas to the lead shock. In addition, a new and intriguing phenomenon concerning pulsating detonations is described. Even if the results are not quantitatively accurate, they are qualitatively relevant for real detonations.

Macro-mesoscale microstructural evolution modeling under hot forging of a Ti-17 alloy with a lamellar (α+β) starting microstructure

Microstructural conversion mechanisms under hot forging process (at temperatures ranging from 750 °C to 1050 °C and strain rates ranging from 10–3 s–1 to 1 s–1) of a Ti-5Al-2Sn-2Zr-4Mo-4Cr (Ti-17) alloy with a lamellar starting microstructure were experimentally identified in this work. After that, constitutive formulae for predicting the microstructural evolution were established followed by calculation using finite-element (FEM) analysis. In the α phase, a lamellae kinking is the dominant mode in the higher strain rate region and dynamic globularization frequently occurs at higher temperatures. On the other hand, continuous dynamic recrystallization is the dominant mode below the transition temperature, Tβ (880~890 °C) in the β phase. And, at conditions of lower strain rates and higher temperatures, dynamic recovery tends to be more active. For microstructural prediction, a set of constitutive equations modeling the microstructural evolution and forging properties are established by optimizing the experimental data followed by implementation in the DEFORM-3D software package. Herein, microstructural evolution on dynamic globularization process, dynamic recrystallization behavior are predicted according to both approaches of physical model and artificial neural network model followed by FEM simulation. In these calculated results, there is a satisfactory agreement between the experimental and simulated results, indicating that the established series of constitutive models can be used to reliably predict the properties of a Ti-17 alloy after forging.

Dislocation Density-Based Modeling of Dynamic Recrystallized Microstructure and Process in Friction Stir Spot Welding of AA6082

This work mainly focuses on a series of microstructural analysis and predictions regarding dynamic recrystallization behavior, change in grain size, and dislocation density. Additionally, this study includes the shape prediction of the stir zone formed during friction stir spot welding. Microstructure analysis of the joint reveals that the mechanism of dynamic recrystallization in the stir zone is geometric dynamic recrystallization. A set of constitutive equations based on dislocation density is established and implemented in DEFORM-3D software to predict dynamic recrystallization during friction stir spot welding of AA6082. From the experimental and model predictions, it is observed that the original microstructure in the stir zone is completely replaced by a recrystallized fine grained microstructure. There is satisfactory agreement between the experimental grain size and the simulated results. In addition, the predicted shape of the stir zone fits quite well with the experimental shape as well.

Direct and converse nonlinear magnetoelectric coupling in multiferroic composites with ferromagnetic and ferroelectric phases

In this paper, we develop a theoretical principle to calculate the direct and converse magnetoelectric (ME) coupling response of ferromagnetic/ferroelectric composites with 2-2 connectivity. We first present an experimentally based constitutive equation for Terfenol-D, and then build the mechanism of domain switch for the ferroelectric phase. In the latter, the change of Gibbs free energy, thermodynamic driving force and kinetic equations for domain growth are also established. These two sets of constitutive equations are shown to capture the experimental data of Terfenol-D and PZT, respectively, well. For the direct effect under an applied magnetic field, the induced electric field and the overall ME coupling coefficient are determined. For the converse effect under an applied electric field, the induced magnetization and the excited magnetic field are obtained. Both the induced electric filed under direct effect and the excited magnetic field under converse effect are shown to display the hysteretic characteristics, and also in good agreement with experiments. We conclude that the developed theory can both qualitatively and quantitatively reflect the essential features of nonlinear direct and converse ME coupling of the multiferroic composites.

Prediction of irradiation hardening in austenitic stainless steels: Analytical and crystal plasticity studies

The flow stress in irradiated Austenitic Stainless Steels (ASSs) is analyzed in terms of the microstructure features and the fundamental mechanisms of plastic deformation. A set of constitutive equations is proposed to assess the contributions of dislocation network, grain size effect, dislocation loops and solute clusters, accompanied with an appropriate superposition rule. Using simple equations for the evolution of densities of dislocations and radiation defects, it is possible to integrate numerically the tensile stress-strain curves over a wide range of irradiation doses. It is shown that irradiation hardening can be obtained analytically without adjustable parameters. Simulation results are challenged in comparison with experimental results and found in excellent agreement with all the reported experimental trends in ASSs irradiated at conditions close to those prevailing in pressurized water reactors.

A theory of biological composites undergoing plastic deformations

Natural biological composites such as bone, dentin, nacre and enamel exhibit anisotropic microstructures, giving rise to orientation-dependent mechanical properties. Although the mechanical properties of these materials have been studied extensively, there is limited progress on modeling the common features associated with the orientation-dependent plastic deformation of biological composites. In this study, we develop a continuum theory for elastic-viscoplastic deformations of anisotropic biological composites. The pressure-sensitive and plastically dilatant plastic flow is incorporated into the theory, and the plastic spin related to the kinematics of the underlying substructure during macroscopic plastic deformation is explicitly taken into account. A special set of constitutive equations are implemented in a finite element program. Furthermore, the material parameters have been calibrated and numerical simulations of elastic-plastic deformation in bone are performed. It is found that the theory can capture the major features of plastic deformation of biological composites. The numerical simulations are in good agreement with experiments, demonstrating that the model is capable of predicting the complex plastic deformation of bone.