Viscoplastic Model Research Articles

Evaluation of rockburst risk in deep tunnels considering structural planes based on energy dissipation rate criterion and numerical simulation

Risk evaluation of rockbursts is crucial for ensuring the safety and stability of deep-tunnel construction. Structural planes play an important role in the potential for rockbursts to occur, and more related studies are needed. In this paper, a new rockburst criterion based on energy dissipation rate (EDR) index was developed and the role of structural planes in rockburst potential was studied. Firstly, a method of generating structural planes and a viscoplastic constitutive model with internal variables were introduced for numerical simulation using FLAC3D. Then, tunnelling under different hydrostatic pressures was simulated to obtain the values of EDR index and stress-strength/strength-stress ratio indices, and EDR criterion was estimated based on stress-strength/strength-stress ratio criteria. The results show that EDR criterion is: 0∼1 Pa/s (no risk), 1∼3 Pa/s (low risk), 3∼6 Pa/s (medium risk), >6 Pa/s (high risk). Subsequently, the case study of rockbursts in the drainage tunnel of Jinping II hydropower station was conducted considering a nearby fault to validate EDR criterion. The simulation results using EDR are basically consistent with actual rockbursts in terms of the potential location, intensity and time. Finally, the numerical simulations of tunnel excavation considering a joint set with different dip angles were implemented, and the influences of joints on rockbursts were investigated using EDR. Analysis results show that rockbursts are prone to occur on tunnel walls parallel and perpendicular to joints; potential rockburst locations are controlled by local joints and are less affected by geostress distribution. Joints may increase the intensity, delay the occurrence time, and control the pit boundaries of rockbursts. Obtained EDR contours can be used to infer damaged zones of strainbursts and strain-structure slip rockbursts and understand inducing mechanism of rockbursts in Jinping II drainage tunnel. The findings of this study provide insight into the role of structural planes on rockbursts. The presented methodology can be applied to rockburst evaluation and prevention during the construction of deep tunnels.

Determination of the parameters of rock viscoelastic creep model and analysis of parameter degradation

Different stress creep tests are conducted on the sandstone in this study to better describe the creep properties of rocks under different stress states. A model that describes the rock creep process is established. The various stages of creep can be described by combining the creep properties of the creep elements of the model. A new method for determining creep parameters is proposed by using the special point on the creep curve and the definition of creep deformation. The relationship between the creep parameters, stress, and time is analyzed. An improved creep model that considers the effects of stress state and time on the creep parameters is developed. This model is verified using experimental data and calculation results. Results shows that the improved creep model better describes the creep properties of rocks and provides a new method for determining future model parameters. The shear modulus of elastic model controls the instantaneous deformation. The shear modulus of viscoelastic model governs the limit of viscoelastic deformation. The shear viscoelastic coefficient of viscoelastic model increases with the increase in stress. The coefficient of viscoplastic model controls the viscoplastic creep rate. The coefficient of a nonlinear Newtonian dashpot mainly controls the accelerated creep deformation of rock. The calculation results of the proposed model agree well with the experimental data under the action of different stress levels. This model accurately reflects the creep characteristics of the primary and steady-state creep stages, and overcomes the shortcomings of the traditional Nishihara model in describing accelerated creep.

An Improved Multistage Creep Recovery Test Procedure for Viscoelastic-Plastic Damage Model Calibration

The viscoelastic-plastic damage model has been widely used to predict the long-term performance of asphalt mixtures. However, it is challenging for obtaining model parameters to separate the total strain and determine the yield function. This study proposed an improved multistage creep-recovery test to simplify the acquisition of constitutive model parameters. The method was designed to identify the viscoplastic parameters by the equilibrium condition when the specimen was fully hardened and calibrate the damage parameters through the changes of modulus in the process of plastic development. The results indicated the proposed test is an effective method to calibrate the parameters of the viscoplastic damage model, as well as the nonlinearity between the stiffness and strength. In addition, it was found that the sigmoid function may be the more satisfactory choice to fit the cohesion excluding the damage compared with the negative exponential function.

Effect of Viscoplasticity on Localization in Saturated Clays and Plastic Silts

A numerical study is presented of the effects of viscoplasticity on localization processes in sensitive, saturated clays and plastic silts. Numerical simulations of laboratory specimens of sensitive, viscoplastic soil subjected to monotonic, undrained, direct simple shear loading were performed using a viscoplastic constitutive model. Parametric analyses evaluated the effects of soil sensitivity, postpeak strain-softening rate, and strain rate dependency, as well as specimen size, mesh discretization, and loading rate. The numerical results showed that the global strain at which a localization forms primarily depends on the strain rate dependency of the soil’s shearing resistance relative to its rate of postpeak strain softening. A regression model is subsequently presented that relates the global strain at the onset of localization to the soil’s strain rate dependency and postpeak strain-softening rate. The results indicate that the inclusion of reasonable levels of viscoplasticity significantly increases the strain that can develop before a localization develops in clays and plastic silts with modest strain-softening rates. The consistency of the numerical results with available laboratory observations is discussed. Implications for practice and future research needs are discussed.

Developing Performance-Based Mix Design Framework Using Asphalt Mixture Performance Tester and Mechanistic Models

This paper proposes a performance-based mix design (PBMD) framework to support performance-related specifications (PRS) needed to establish relationships between acceptable quality characteristics (AQCs) and predicted performance, as well as to develop fatigue-preferred, rutting-preferred, and performance-balanced mix designs. The framework includes defining performance tests and threshold values, developing asphalt mix designs, identifying available performance levels, conducting sensitivity analysis, establishing the relationships between AQCs and predicted performance, and determining performance targets and AQC values for the three PBMDs using predicted performance criteria. Additionally, the framework recommends selecting the PBMD category for each asphalt layer to minimize pavement distresses. In this study, the proposed PBMD protocol was applied to FHWA accelerated loading facility (ALF) materials using asphalt mixture performance tester (AMPT) equipment coupled with mechanistic models. The study developed nine mix designs with varying design VMAs and air voids using the Bailey method. The cracking and rutting performance of the mix designs were determined by direct tension cyclic (DTC) fatigue testing, triaxial stress sweep (TSS) testing, and viscoelastic continuum damage (S-VECD) and viscoplastic shift models for temperature and stress effects. The study found that adjusting the design VMA was the primary way to achieve required performance targets. For fatigue-preferred mix design, the recommended targets were a cracking area of 0 to 1.9%, a rut depth of 10 mm, and a design VMA of 14.6 to 17.6%. For rutting-preferred mix design, the recommended targets were a cracking area of 18%, a rut depth of 0 to 3.8 mm, and a design VMA of 10.1 to 13.1%. For performance-balanced mix design, the recommended targets were a cracking area of 8.1 to 10.7%, a rut depth of 4.6 to 6.4 mm, and a design VMA of 12.6 to 14.3%. Finally, pavement simulation results verified that the proposed PBMD pavement design with fatigue-preferred mix in the bottom layer, performance-balanced mix in the intermediate layer, and rutting-preferred mix in the surface mix could minimize bottom-up cracking propagation without exceeding the proposed rutting performance criterion for long-life.

An elastic–viscoplastic model with non-affine deformation and rotation of a distribution of embedded fibres

A large deformation continuum model for an elastically isotropic visco-plastic matrix reinforced by a distribution of fibres has been formulated. The fibre distribution is a symmetric positive-definite tensor with unit trace that is determined by an evolution equation. This evolution equation includes a competition between tendencies for affine evolution of the distribution and non-affine evolution causing the distribution to align with the elastic distortional deformation tensor or to become isotropic. Two elastic deformation measures are introduced associated with the fibre distribution. Specifically, non-affine measures of elastic distribution stretch and non-coaxiality of the distribution cause anisotropic stress response. The resulting model describes a number of effects relevant to growth and remodelling of biological tissues as well as to additive manufacturing of artificial materials. For example, the equations model the smooth transition from isotropic to anisotropic response seen in many biological tissues.

Creep-fatigue damage mechanisms and life prediction based on crystal plasticity combined with grain boundary cavity model in a nickel-based superalloy at 650°C

In this study, a dual-scale numerical procedure is developed to reveal the creep-fatigue damage mechanisms and estimate the crack initiation life for notched structures made of Inconel 718 superalloy at 650 °C. The macro-scale simulation solves the creep-fatigue deformation behavior with viscoplastic constitutive models, and the local deformation histories are supplied to the micro-scale simulation as boundary conditions. In the micro-scale simulation, the local damage evolutions are solved based on crystal plasticity combined with grain boundary cavity model. The creep damage is calculated by a special formulation in the form of cavity nucleation, growth and coalescence. The fatigue damage is represented by accumulated energy dissipation originated from crystal plasticity finite element simulation. Experimentally, the creep-fatigue tests of notched structures are carried out for Inconel 718 superalloy at 650 °C to validate the feasibility and robustness of the proposed numerical procedure. Moreover, the crack initiation behavior, including transgranular cracks under fatigue loading and intergranular cracks under creep-fatigue loading, is explained through different types of microstructure observations together with a dual-scale numerical procedure. In detail, the crack initiation sites transferred from the grain interior at notched surface to the grain boundaries at notched subsurface with an increase in hold times can be well predicted by the proposed numerical procedure. In addition, the simulated life based on the developed life prediction approach agrees well with the experimental data within an error band of ±2. Parametric studies show that the creep damage is more sensitive to grain boundary diffusion than to the external conditions of strain level and hold time.

Experimental Characterization and Numerical Modelling of the Impact Behavior of PVC Foams

BackgroundPolyvinyl chloride (PVC) foams are widely used in crashworthiness and energy absorption applications due to their low density and the capability of crushing up to large deformations with limited loads. This property is due to their particular constitutive behavior: the stress-strain curve is characterized, after an initial yield or peak stress, by a relevant plateau region followed by a steep increase due to foam densification. Furthermore, the mechanical response of PVC foam is strongly strain rate dependent.ObjectiveThis work aims to characterize the mechanical behavior of PVC foams and to develop a complete constitutive model for impact and energy absorption applications.MethodsCompressive tests are carried out at different speeds on PVC foam samples having different relative densities. Quasi-static and intermediate strain rate tests are performed by a pneumatic machine, while high strain rate tests are conducted by means of a Split Hopkinson Pressure Bar. The uniaxial stress-strain curves are used to calibrate the visco-elastic and visco-plastic constitutive model. In particular, the material behavior is divided into two parallel branches: the former describes the elasto-plastic behavior, while the latter accounts for the visco-elastic one; the plastic branch also includes a multiplicative term accounting for the strain rate sensitivity of the base material.ResultsThe tests highlight a strong compressibility of the foam with negligible lateral expansion. The energy absorption efficiency, as well as the densification strain, is evaluated. The material model is also implemented in Finite Element (FE) simulations of puncture impact tests, validating the results of the calibration procedure.ConclusionsThe calibration of the visco-elasto-plastic material model offers a physically consistent identification of the constitutive response of the PVC foams, showing an effective characterization of the impact behavior of the material.

A comparative analysis of continuum plasticity, viscoplasticity and phase-field models for earthquake sequence modeling

This paper discusses continuum models for simulating earthquake sequences on faults governed by rate-and-state dependent friction. Through detailed numerical analysis of a conventional strike-slip fault, new observations regarding the use of various continuum earthquake models are presented. We update a recently proposed plasticity-based model using a consistently linearized formulation, show its agreement with discrete fault models for fault thicknesses of hundreds of meters, and demonstrate mesh objectivity for slip-related variables. To obtain a fully regularized fault width description with an internal length scale, we study the performance and mesh convergence of a plasticity-based model complemented by a Kelvin viscosity term and the phase-field approach to cohesive fracture. The Kelvin viscoplasticity-based model can introduce an internal length scale and a mesh-objective response. However, on grid sizes down to meters, this only holds for very high Kelvin viscosities that inhibit seismic slip rates, which renders this approach impractical for simulating earthquake sequences. On the other hand, our phase-field implementation for earthquake sequences provides a numerically robust framework that agrees with a discrete reference solution, is mesh objective, and reaches seismic slip rates. The model, unsurprisingly, requires highly refined grids around the fault zones to reproduce results close to a discrete model. Following this line, the effect of an internal length scale parameter on the phase-field predictions and mesh convergence are discussed.

An Egg-Shaped Elastic Viscoplastic Model for Clay: Experimental Investigation and Constitutive Modelling

This paper presents the experimental investigation carried out for evaluation of the rheological behaviours of Hangzhou soft clay, and proposes a three-dimensional elastic viscoplastic model to describe the rheological characteristics under triaxial condition. Firstly, the time-dependent behaviours of Hangzhou soft clay, such as creep and strain rate effects, were observed and discussed based on the analysis of rheological tests. Subsequently, a constitutive model was formulated under the framework of Perzyna's overstress theory and incorporates other developments, including: 1) A smooth and continuous egg-shaped yield surface, whose shape can change from an ellipse to a teardrop shape, thus providing greater flexibility for better modelling; 2) An equivalent time concept, which enables the model to describe the time-dependent behaviours under different loading paths. By varying the shape parameter, the newly established model can be degenerated into modified Cam-clay (MCC) model or Yin-Graham elastic viscoplastic model. Finally, the model was solved numerically using the fourth-order convergent iterative method combined with the Runge-Kutta method. Comparison of simulation results and experimental data shows that the newly proposed model has an ability to predict creep characteristics and strain rate effects of soft clay.

Warping model of high-power IGBT modules subjected to reflow soldering process

High-power insulated gate bipolar transistor (IGBT) modules are the key devices for energy change and transmission. Reflow soldering process is a crucial process in IGBT module packaging manufacture. Residual stress and warpage of IGBT modules are induced after soldering, which can lead to degradation of device performance and working life. Combining frame-based baseplate and the key material properties of AlSiC at different temperatures, latent heat and viscoplastic intrinsic models of the solder, this study established a finite element (FE) model of the reflow soldering process of IGBT modules to predict the warpage and residual stresses. Reflow soldering experiments of IGBT modules were conducted to examine the shape of the lower surface of the baseplate before and after the soldering process, and are used to verify the modeling results. The model shows a good agreement with the experimental with an error of 5.46%. The FE model will contribute to the optimization of reflow soldering process and package structure of IGBT module.

One-dimensional elastic viscoplastic finite strain consolidation model for soft clay with uncertainty

One-dimensional elastic viscoplastic finite strain consolidation model for soft clay with uncertainty

Constitutive recasting of macromolecular-based thermoviscoplasticity as yield function-based formulation

The rate- and temperature-dependent “macromolecular” polymer plasticity relies on a thermally-activated model of the non-linear strain measure, and is thus deficient in an explicit definition of the yield surface. Such an approach, typically written in the rate-form, is only suitable for the explicit finite element solver where the consistent tangent modulus and the return mapping process are unnecessary. This paper presents a procedure to deduce the pressure-dependent yield function of the rate- and thermal-based plastic constitutive models in the continuum thermodynamics. The recasted yield model aids in deriving its consistent elasto-thermoviscoplastic tangent modulus and implementing it in the implicit finite element solver. The recasted expressions are suitable for incremental formulation using the classical radial return mapping algorithm, regardless of the complexities of the original macromolecular expression. Four viscoplastic macromolecular models are used to illustrate the recasting process: three of them are based on the classical Boyce-Parks-Argon model and one on the Bergström-Boyce model. Numerical simulations are performed on real-sized specimen geometries to establish the functionality and capabilities of the methodology. The practical advantage of the implicit-oriented forms is studied in comparison to the explicit-based implementation under quasi-static and isothermal conditions.

Chaboche viscoplastic material model for process simulation of additively manufactured Ti-6Al-4 V parts

For the estimation and further optimization of the residual stress and distortion state in additively manufactured structures during and after the wire arc additive manufacturing (WAAM) process, thermomechanical simulation can be applied as a numerical tool. In addition to the detailed modelling of key process parameters, the used material model and material data have a major influence on the accuracy of the numerical analysis. The material behaviour, in particular the viscoplastic behaviour of the titanium alloy Ti-6Al-4 V which is commonly used in aerospace, is investigated within this work. An extensive material characterization of the viscoplastic material behaviour of the WAAM round specimen is carried out conducting low cycle fatigue (LCF) and complex low cycle fatigue (CLCF) tests in a wide temperature range. An elasto-viscoplastic Chaboche material model is parameterised, fitted, and validated to the experimental data in the investigated temperature range. Subsequently, the material model is implemented in the thermomechanical simulation of a representative, linear ten-layer WAAM structure. To finally determine the effect of the fitted material model on the estimation accuracy of residual stress and distortion, simulation results using the standard material model and the elaborated Chaboche model from this study are compared to experimental data in the substrate. The thermomechanical simulation with the Chaboche model reveals a better agreement with the experimental distortion and residual stress state, whereby the standard material model tends to an overestimation. The estimation accuracy with respect to the maximum distortion is improved from an error of 60% with the standard model to an acceptable error of about 6% using the elaborated model. Additionally, the estimated residual stress state shows a sound agreement to the experimental residual stress in the substrate.

An efficient finite strain constitutive model for amorphous thermoplastics: Fully implicit computational implementation and optimization-based parameter calibration

In this paper, a new visco-elastic visco-plastic (three-dimensional) constitutive model is consistently formulated under isothermal conditions to describe the nonlinear behavior of thermoplastic polymers at finite strains. The constitutive equations include three well-established rheological elements to have well-defined parametric behavior and capture the experimental response. In particular, a visco-elastic generalized Maxwell element, a visco-plastic Eyring dashpot, and a plasticity-induced (nonlinear) hardening element are selected and consistently combined. A fully implicit integration algorithm is derived, and highly efficient implementation is obtained by simplifying the system of equations to a single (scalar) nonlinear residual equation. A new two-stage optimization-based calibration procedure is developed to identify the material parameters in a completely unsupervised way. The constitutive model is validated against results available in the literature for polycarbonate, accounting for temperature and strain rate dependencies. The results show the efficiency of the overall numerical strategy and demonstrate that quadratic convergence rates are attained. Despite the strongly nonlinear finite strain response, it is possible to observe an excellent agreement for all stages of deformation.

A microstructure-based three-scale homogenization model for predicting the elasto-viscoplastic behavior of duplex stainless steels

A new microstructure-informed three-scale homogenization scheme for elasto-viscoplastic heterogeneous materials is developed. It is applied to predict the mechanical behavior of cast duplex austenitic-ferritic (AF) steels, which are widely used in primary loop of pressurized water reactors (PWR). At the first scale (microscale), the elasto-viscoplastic behavior of single crystals for both phases is modeled using linear elasticity and a viscoplastic crystal plasticity model. An “affine” type formulation based on the first moments of stresses is applied to the inelastic non-linear part of the deformation. Then, at the second scale (mesoscale), an EBSD-informed two-phase austenite/ferrite laminate structure (LS) model is developed with {110}-type habit planes (HP) in ferrite and a Kurdjumov-Sachs orientation relationship (KS-OR) between both phases. At the third scale (macroscale), the model considers a single ferritic primary grain as an aggregate of spherical two-phase laminate structure domains corresponding to the 24 KS-OR variants. The elasto-viscoplastic self-consistent scheme (EVPSC) is used through the Translated Fields (TF) method to obtain the effective behavior at this scale. The TF-EVPSC scheme is also used for an ensemble of primary ferritic grains to identify materials parameters from experimental tensile curves corresponding to as received and aged specimen. From EBSD measurements, the crystallographic data are thoroughly analyzed to physically feed the three-scale model. The results are discussed in terms of stress/strain responses in ferrite and austenite, which are correlated to the different KS-OR variants and HP orientations. The effect of primary ferritic grain crystallographic orientation and aging is also studied regarding monotonic and cyclic tests to study the possible origins of backstress in this material.

A bounding surface viscoplastic damage model for time-dependent rock deformation

A novel bounding surface viscoplastic damage (BS-VPD) framework is proposed in this study for capturing the time-dependent rock deformation driven by subcritical cracking. Our constitutive creep model has been established via imposing effective control over stress-state evolution with respect to the characteristic surfaces while enforcing strictly the thermodynamic constraint. In particular, a new viscoplastic flow rule is first developed by upscaling the microscopic subcritical cracking towards macroscopic creep characterization based on the bounding surface concept. A continuum damage criterion has then been deduced from the maximum dissipation principle for capturing realistically the strain hardening and damage softening during sustained creep loading. By further completing the model with a novel strategy for regulating the interactive surface evolution, we have thus reproduced successfully the progressive transition from stable subcritical cracking characteristic of decelerating creep into unstable propagation during accelerating rupture. In addition, a hierarchical procedure has also been formulated in this study to facilitate straightforward calibration via conventional laboratory testing, and the applicability of our BS-VPD model is then verified against constant strain-rate and creep tests of brittle rocks representing distinct lithologies (i.e., granite, sandstone, and gneiss). The close agreement between model simulation and experimental measurement suggests that the proposed BS-VPD framework is able to capture faithfully the trimodal creep behaviour of brittle rocks across primary, secondary, and tertiary regimes.

Numerical modelling of stress‐strain response and deformation‐induced martensite in metastable austenitic stainless steels under monotonic tensile loading

AbstractA constitutive model describing the deformation‐induced martensite transformation in metastable austenitic CrNi steels is presented. In line with the previous work of Stringfellow et al. [1], the material is considered to have a composite response of the underlying austenite and the evolving martensite phases. The stresses and strains in the individual phases are described using viscoplastic models. The effective properties of the material are then computed by a homogenization method. Strain hardening in the individual phases is represented using separate flow rules. Furthermore, based on the plastic strain in the austenitic phase, a transformation kinetics model describes the evolution of the martensite volume, in the two‐phase composite.A numerical implementation of the model is done in the finite element program FEAP [2]. The parameters in model are identified by experimental data using the least‐squares optimization. Finally, some results are presented using an illustrative boundary value problem of a structured surface.

A finite strain gradient theory for viscoplasticity by means of micromorphic regularization

AbstractStretching of polycarbonate films leads to the formation of shear bands in the necking zone [1]. Standard viscoplastic material models render mesh size dependent results, which requires a mathematical regularization. To this end, we present a finite strain gradient theory for a viscoplastic, isotropic material model where we extend the model presented in [2] to a micromorphic model by introducing a new micromorphic variable as an additional degree of freedom with its first gradient [3, 4]. The variable here has the meaning of a micro plastic strain, and is coupled with the macro plastic by a micro penalty term, forcing the macro‐plastic strain to be close to the micro‐plastic strain for the targeted shear band regularization effect. We have implemented the model equations as a three dimensional initial boundary value problem in an in house FE‐tool, to simulate different geometries with different thickness and to compare it the experimental tests. The analysis is performed for a uniaxial tensile geometry as well as for a biaxial tensile geometry. The numerical examples show the ability of the model to regularize the shear bands and solve the problem of localization.

The strain rate sensitive and anisotropic behavior of rare-earth magnesium alloy ZEK100 sheet

To overcome the limitation in formability at room temperature, manufacturers have developed magnesium alloys with remarkable properties by adding rare-earth elements. The rare-earth magnesium alloys behave differently from the conventional alloys, especially with respect to their coupled anisotropic and strain rate sensitive behavior. In the current work, such behavior of the rare-earth Mg alloy ZEK100 sheet at room temperature is investigated with the aid of the elastic viscoplastic self-consistent polycrystal plasticity model. Different strain rate sensitivities (SRSs) for various deformation modes are employed by the model to simulate the strain rate sensitive behaviors under different loading directions and loading rates. Good agreement between the experiments and simulations reveals the importance and necessity of using different SRSs for each deformation mode in hexagonal close-packed metals. Furthermore, the relative activities of each deformation mode and the texture evolution during different loadings are discussed. The anisotropic and strain rate sensitive behavior is ascribed to the various operating deformation modes with different SRSs during loading along different directions.