Viscoplastic Strain Research Articles

Viscoelastic-viscoplastic polymer composites: Development and evaluation of two very dissimilar mean-field homogenization models

This paper deals with the micromechanical modeling of polymer composites with viscoelastic-viscoplastic (VE-VP) constituents. Two mean-field homogenization (MFH) models based on completely dissimilar theoretical approaches are extended from elasto-viscoplasticity (EVP) to VE-VP and assessed. The first approach is the incremental-secant method. It relies on a fictitious unloading of the composite at the beginning of each time step. Then, a thermoelastic-like Linear Comparison Composite (LCC) is constructed from the computed residual state directly in the time domain. The method provides naturally isotropic per-phase incremental-secant operators for isotropic VE-VP constituents. It takes into account both the first and the second statistical moment estimates of the equivalent stress micro-field. The second approach is the integral affine method. It starts by linearizing the rates of viscoplastic strain and internal variables. The linearized constitutive equations are then recast in a hereditary integral format to which the Laplace-Carson (L-C) transform is applied. A thermoelastic-like LCC is built in the L-C domain, where MFH is carried out. Finally, the composite’s response in the time domain is recovered by numerical inversions of L-C transforms. The method is able to overcome the issue of heterogeneous viscous stresses encountered by time domain MFH models. The two proposed MFH formulations are able to handle non-monotonic, non-proportional and multi-axial loading histories. Their accuracy was assessed against full-field finite element (FE) results for different microstructures and loadings. The computational cost of both methods is negligible compared to FE analyses. Overall, the incremental-secant approach is much simpler mathematically and numerically than the integral affine formulation, its accuracy ranges from acceptable to excellent, and important improvements can be expected in the future by controlling the virtual unloading time increment.

Nonlinear Elasto-Visco-Plastic Creep Behavior and New Creep Damage Model of Dolomitic Limestone Subjected to Cyclic Incremental Loading and Unloading

For many rock engineering projects, the stress of surrounding rocks is constantly increasing and decreasing during excavating progress and the long-term operation stage. Herein, the triaxial creep behavior of dolomitic limestone subjected to cyclic incremental loading and unloading was probed using an advanced rock mechanics testing system (i.e., MTS815.04). Then, the instantaneous elastic strain, instantaneous plastic strain, visco-elastic strain, and visco-plastic strain components were separated from the total strain curve, and evolutions of these different types of strain with deviatoric stress increment were analyzed. Furthermore, a damage variable considering the proportion of irrecoverable plastic strain to the total strain was introduced, and a new nonlinear multi-element creep model was established by connecting the newly proposed damage viscous body in series with the Hookean substance, St. Venant body, and Kelvin element. The parameters of this new model were analyzed. The findings are listed as follows: (1) When the deviatoric stress is not more than 75% of the compressive strength, only instantaneous deformation, transient creep, and steady-state creep deformation occur, rock deformation is mainly characterized by the instantaneous strain, whereas the irrecoverable instantaneous plastic strain accounts for 38.02–60.27% of the total instantaneous strain; (2) Greater deviatoric stress corresponds to more obvious creep deformation. The visco-elastic strain increases linearly with the increase of deviatoric stress, especially the irrecoverable visco-plastic strain increases exponentially with deviatoric stress increment, and finally leads to accelerated creep and delayed failure of the sample; (3) Based on the experimental data, the proposed nonlinear creep model is verified to describe the full creep stage perfectly, particularly the tertiary creep stage. These results could deepen our understanding of the elasto-visco-plastic deformation behavior of dolomitic limestone and have theoretical and practical significance for the safe excavation and long-term stability of underground rock engineering.

Phase transformation and viscoplasticity coupling in polycrystalline nickel-titanium-hafnium high-temperature shape memory alloys

The objective of this study was to investigate the interactions between phase transformation and viscoplasticity during uniaxial constant force thermal cycling (UCFTC) of a Ti-rich Ni-Ti-20Hf (at.%) high-temperature shape memory alloy (HTSMA). These tests were conducted (up till failure) at 1, 10 and 50 ∘C/min, to vary the duration of exposure to high temperatures, viz. the amount of viscoplasticity, and to examine the rate-dependency of actuation. The macroscopic results from the tests were used to investigate the evolution of transformation temperatures, hysteresis, transformation and irrecoverable strains for the cycles in which the effect of potential damage mechanisms could be assumed to be negligible. The phenomena that affected the behavior were: viscoplasticity at 1 ∘C/min, transformation-induced plasticity (TRIP) at 10 and 50 ∘C/min, and accumulation of retained martensite at all the three rates. More interestingly, the response at 1∘C/min indicated a unique interplay between the effect of viscoplasticity over phase transformation and static recovery. The retained martensite was identified through a series of DSC and XRD analyses, and its contribution to TRIP strain was quantified through a UCFTC test. Furthermore, a test involving alternating isothermal creep and UCFTC at 10 ∘C/min was conducted to investigate an effect of viscoplasticity produced by creep on the behavior, while reducing the viscoplasticity during thermal cycling. The alternating test revealed an effect of phase transformation over the viscoplastic strain rate. The experimental investigations demonstrated a rate-dependent phase transformation behavior, and a two-way coupling between phase transformation and viscoplasticity, bringing out the importance of understanding viscoplastic deformations in phase-transforming materials.

Creep Rupture and Permeability Evolution in High Temperature Heat-Treated Sandstone Containing Pre-Existing Twin Flaws

Utilizing underground coal gasification cavities for carbon capture and sequestration provides a potentially economic and sustainable solution to a vexing environmental and energy problem. The thermal influence on creep properties and long-term permeability evolution around the underground gasification chamber is a key issue in UCG-CCS operation in containing fugitive emissions. We complete multi-step loading and unloading creep tests with permeability measurement at confining stresses of 30 MPa on pre-cracked sandstone specimens thermally heat-treated to 250, 500, 750 and 1000 °C. Observations indicate a critical threshold temperature of 500 °C required to initiate thermally-induced cracks with subsequent strength reduction occurring at 750 °C. Comparison of histories of creep, visco-elastic and visco-plastic strains highlight the existence of a strain jump at a certain deviatoric stress level—where the intervening rock bridge between the twin starter-cracks is eliminated. As the deviatoric stress level increases, the visco-plastic strains make up an important composition of total creep strain, especially for specimens pre-treated at higher temperatures, and the development of the visco-plastic strain leads to the time-dependent failure of the rock. The thermal pre-treatment produces thermal cracks with their closure resulting in increased instantaneous elastic strains and instantaneous plastic strains. With increasing stress ratio, the steady-state creep rates increase slowly before the failure stress ratio but rise suddenly over the final stress ratio to failure. However, the pre-treatment temperature has no clear and apparent influence on steady creep strain rates. Rock specimens subject to higher pre-treatment temperatures exhibit higher permeabilities. The pre-existing cracks close under compression with a coplanar shear crack propagating from the starter-cracks and ultimately linking these formerly separate cracks. In addition, it is clear that the specimens pre-treated at higher temperatures accommodate greater damage.

Effect of molybdenum on structural evolution and thermomechanical behavior of a heat-resistant nickel aluminide-based alloy

Powders of heat-resistant NiAl-8Cr-6Co-xMo-1 Nb-0.9Hf (x = 0, 2, 4 and 6 at% Mo) alloys were successfully obtained by high-energy ball milling (HEBM) and combustion synthesis. Powders with the grain size of 10–55 µm were characterized by high flowability (up to 30 s) and bulk density (up to 3.1 g/cm3) due to the absence of submicron-sized particles. Compacted samples having a finer-grained (<5 µm) structure were produced from the combustion-synthesized powders by hot isostatic pressing (HIP). Molybdenum doping gave rise to a duplex structure consisting of NiAl and (Cr, Mo) grains. High-resolution electron microscopy (HRTEM) and selected area electron diffraction studies of the fine structure of the alloys detected that there are nanoparticles of refractory compounds (Co2Nb, Cr2Nb, (Hfx, Nby)C, and Ni2AlHf) along the grain boundaries and in grain bulk, which improve resistance to viscoplastic strain at temperatures above 800 °C. Molybdenum was shown to affect the mechanical properties of the base alloy in compression tests at 500–1100 °C. Two predominant mechanisms of alloy deformation were identified: glide of matrix dislocations and mechanical twinning.

Correction to: Effects of random laminate misalignment on macroscopic and microscopic elastic/viscoplastic behaviors of ultrafine plate–fin structures

In this study, the effects of random laminate misalignment on the macroscopic and microscopic elastic/viscoplastic behaviors of ultrafine plate–fin structures are statistically investigated adopting Monte Carlo simulation. To this end, a multiscale method that analyzes the elastic–viscoplastic behavior of ultrafine plate–fin structures with random laminate misalignment both macroscopically and microscopically is proposed based on the homogenization theory for nonlinear time-dependent materials combined with the substructure method. Using the method, Monte Carlo simulations for elastic and elastic–viscoplastic behaviors of ultrafine plate–fin structures subjected to uniaxial tension or simple shear are performed for many cases of random laminate misalignment. The obtained results reveal that the random laminate misalignment affects the microscopic behavior of ultrafine plate–fin structures more than the macroscopic behavior; microscopic stress and viscoplastic strain exhibit considerable variability depending on the laminate misalignment, and their maximum values can become about 30–50% higher than those in the case without laminate misalignment. This suggests the importance of considering random laminate misalignment in the design and strength/life prediction of ultrafine plate–fin structures.

Effects of rate-dependent behavior of sandstone on performance of gas storage in aquifer

Strain rate is one of the main factors affecting the properties of rock masses. The effect of strain rate on performance of gas storage in aquifer is investigated combining laboratory tests and constitutive and numerical models. Triaxial tests on sandstone specimens at different strain rates were carried out and the effect of strain rate on mechanical properties of sandstones was investigated. The results showed that the peak strength, axial strain, elastic modulus, Poisson’s ratio, cohesion and internal friction angle of sandstone specimens increased with strain rate. In addition, the elastic–viscoplastic properties of sandstones have been analyzed. It indicated that the deflection angle in the viscoplastic flow direction was increased with strain rate and viscoplastic volumetric strain grew faster than the viscoplastic deviator strain. A rate-dependent constitutive model was adopted and a method for solving the parameters of the constitutive model was determined based on test data. For the constitutive model, the elastic part is a linear elastic model, the plastic part is a linear Drucker–Prager model, and the viscous part is a power law constitutive function. The model and methodology for parameter determination were validated against the test data of sandstones. The results showed that the model selected in this study can well describe effects of strain rate on the mechanical properties of sandstones. The model and method have been applied in gas injection and production capacity assessment of a gas storage in aquifer and the strain rate of the formation was determined by gas injection pressure and time indirectly in simulation. The simulation results showed that the gas injection and production rate of gas storage in aquifer will increase significantly with the strain rate of formation in the same gas injection pressure or time. It also indicated that the gas production capacity will be greatly improved with the strain rate of formation.

Development of the skew boundary condition for soil-structure interaction in three-dimensional finite element analysis

This paper extends the use of the skew boundary condition in two- and three-dimensional (3D) finite element analysis for both structural and geotechnical applications when the need of the freedom directions at the interface nodes is different from the global Cartesian directions. For instance, jack-up structures have three conical-shaped footings, whose freedom directions at the interface nodes are not suitable for Cartesian directions. In addition, there is no axisymmetric movement of the soil as the load is applied to the jack-up's hull. Thus, a more reliable solution of jack-up-soil interaction should be analyzed in a full 3D version. This paper develops a simple 3D interface model for soil-structure interaction that uses skew boundary conditions and incorporates the visco-plastic strain method into the finite element program. The work was validated by computing bearing capacity factors of conical footings with different scenarios, such as plane strain, axisymmetric and 3D analysis. The results of the present analysis are in good agreement with the existing solutions. The fields of the displacement vector and failure patterns at the collapse state are demonstrated and discussed for each apex angle, and the influence of the conical angle on the bearing capacity factor (Nc) is also examined.

Viscoplastic lattice strain during repeated relaxation of age-hardened Al alloy

We examined the repetitive stress relaxation behavior of extruded AA7075 subjected to different heat-treatments (solid-solutionized (SS), under-aged (UA), peak-aged (PA)). Compared to SS and UA, the PA showed the highest stress drop (Δσs), plastic strain change (Δεp) and re-yielding (Δσr), in conjunction with the largest reduction in mobile dislocation density right after stress relaxation. As stress relaxation cycles repeated, the stress drop decreased due to lower activities of mobile dislocations. During stress relaxation, the harder the grain orientation was, the more was the reduction in lattice strain. However, this tendency was not consistent with the order in the magnitude of the shear stresses imposed on the respective grains based on Schmid's law. The current results confirm the grain-orientation dependency on the relative activities of dislocation slip and diffusion-controlled creep during the stress relaxation at room temperature.

Analysis of MSCR test results for asphalt binders with improved accuracy

The multiple stress creep recovery (MSCR) test has been regarded by many as the best available test method to evaluate the high-temperature performance of asphalt binders. However, recent studies have shown that the 9-s recovery time may not allow a full recovery of viscoelastic strain for some modified asphalt binders. In that case, some potential viscoelastic recovery is mistakenly reported as unrecovered strain, leading to an underestimation of the asphalt binder rutting resistance. In order to address the issue of incomplete recovery, this paper developed a mechanistic modeling approach to analyze the MSCR test results of six types of asphalt binders. Before using the mechanistic modeling approach, the steady-state strain response at each stress level was firstly determined for all asphalt binders. It was found that the three neat asphalt binders reached a steady-state condition from the beginning of the last 10 cycles, whereas the SBS-modified asphalt binders reached a relatively steady state in the last 5 cycles. Afterwards, a mechanistic modeling approach was employed to analyze the shear strain data in the last 5 cycles at each stress level based on the Boltzmann superposition principle and the Schapery’s nonlinear viscoelastic theory. The viscoplastic strain at the end of the recovery portion and the viscoelastic strain at the end of the creep portion were utilized to calculate the non-recoverable creep compliance and percent recovery. It was demonstrated that the approach adopted in this study was capable of analyzing the MSCR test results of asphalt binders with improved accuracy.

Effects of random laminate misalignment on macroscopic and microscopic elastic/viscoplastic behaviors of ultrafine plate–fin structures

In this study, the effects of random laminate misalignment on the macroscopic and microscopic elastic/viscoplastic behaviors of ultrafine plate–fin structures are statistically investigated adopting Monte Carlo simulation. To this end, a multiscale method that analyzes the elastic–viscoplastic behavior of ultrafine plate–fin structures with random laminate misalignment both macroscopically and microscopically is proposed based on the homogenization theory for nonlinear time-dependent materials combined with the substructure method. Using the method, Monte Carlo simulations for elastic and elastic–viscoplastic behaviors of ultrafine plate–fin structures subjected to uniaxial tension or simple shear are performed for many cases of random laminate misalignment. The obtained results reveal that the random laminate misalignment affects the microscopic behavior of ultrafine plate–fin structures more than the macroscopic behavior; microscopic stress and viscoplastic strain exhibit considerable variability depending on the laminate misalignment, and their maximum values can become about 30–50% higher than those in the case without laminate misalignment. This suggests the importance of considering random laminate misalignment in the design and strength/life prediction of ultrafine plate–fin structures.

A unified viscoplastic model and strain rate–temperature equivalence of frozen soil under impact loading

Based on the double-pulse shaping technology, a method for obtaining long rising edge loading pulse by using the structural response of a pulse shaper is proposed. The traditional split Hopkinson pressure bar (SHPB) was improved to be suitable for testing the dynamic mechanical properties of low wave impedance materials. The dynamic mechanical properties of frozen soil at different temperatures and strain rates were tested using the improved SHPB equipment. The increase in the strength of frozen soil was associated with a decrease in temperature and increase in strain rate. The strain rate–temperature equivalence of frozen soil is proposed by regression analysis of the experimental data. The relationship between the temperature and strain rate of frozen soil satisfied the Arrhenius equation and could be explained by the thermal activation mechanism. The thermal softening characteristics of frozen soil caused by the adiabatic heating under impact loading were analyzed, and described by the damage variable driven by the adiabatic heating. The damage dynamic constitutive model is proposed based on the unified viscoplastic theory. The comparison between the model and experimental results indicate the applicability of the model.

A unified viscoplastic model for characterizing the softening behavior of the Sn3.0Ag0.5Cu solder under monotonic and cyclic loading conditions

A viscoplastic model based on the Chaboche framework was developed to consider the behaviors of Sn3.0Ag0.5Cu solder under tensile, creep, relaxation and cyclic conditions at a wide range of temperatures. The experimentally-observed strong softening responses, especially at lower strain rates and higher temperatures, were modeled by incorporating evolutionary dynamic recovery parameters, which depend both on the accumulated viscoplastic strain and the strain memory surface. Temperature rate terms were also included to consider the thermomechanical response. The viscoplastic model parameters were obtained by using a two-step procedure and validated to experimental data by using Runge-Kutta and finite element simulations.

Thermodynamically consistent nonlinear viscoplastic formulation with exact solution for the linear case and well-conditioned recovery of the inviscid one

In this work, a consistent viscoplasticity formulation is derived from thermodynamical principles and employing the concept of continuum elastic corrector rate. The proposed model is developed based on the principle of maximum viscoplastic dissipation for determining the flow direction. The model uses both the equivalent viscoplastic strain and its rate as state variables. Power balance and energy balance give, respectively, separate evolution equations for the equivalent viscoplastic strain rate and the viscoplastic strain, the former written in terms of inviscid rates. Several key points distinguish our formulation from other proposals. First, the viscoplastic strain rate (instead of a yield function) consistently distinguishes conservative from dissipative behaviours during reverse loading; and the discrete implicit integration algorithm is an immediate discretization of the continuum theory based on the mentioned principles considered as separate equations. Second, the inviscid solution is recovered in a well-conditioned manner by simply setting the viscosity to zero. Indeed, inviscid plasticity, viscoelasticity and viscoplasticity are particular cases of our formulation and integration algorithm, and are recovered just by setting the corresponding parameters to zero (viscosity or yield stress). Third, the linear viscoplasticity solution is obtained in an exact manner for proportional loading cases, independently of the time step employed. Four, general nonlinear models (Perzyna, Norton, etc) may be immediately incorporated as particular cases both in the theory and the computational implementation.

The strain characteristics and corresponding model of rock materials under uniaxial cyclic load/unload compression and their deformation and fatigue damage analysis

The fatigue characteristics of rock materials are usually studied by cyclic load–unload tests, and the deformation and damage development reflect their weakening characteristics. In this paper, according to the mechanical characteristics of rock materials during load/unload cycles, the total strain can be separated into three types, that is, elastic strain, viscoelastic strain, and viscoplastic strain. The elastic strain is linear with stress, and viscoelastic strain exhibits a special behavior after unloading, the viscoplastic strain also displays its own unique features and reflects the damage in rocks. Based on their unique characteristics, we establish elastic, viscoelastic, and viscoplastic submodels, then an elastic-visco-plastic model can be obtained by connecting three submodels in series, which can reflect the development of the law of different strains. In order to verify the reliability of the model, red sandstone samples are selected for cyclic load/unload tests. The results show that the collected strain–time data are well fitted by the model. In addition, the characteristics of strain–time curves imply the deformation and damage development of rocks during load/unload cycles.

Formulation of non-linear viscoelastic–viscoplastic constitutive equation for polyamide 6 resin

In this study, a non-linear viscoelastic–viscoplastic constitutive equation for polyamide 6 (PA6) is formulated and a new model is suggested for the viscoplastic part of the equation. The suggested model is empirical but can accurately predict the viscoplastic strain. In this study, creep and recovery tests are conducted to evaluate the viscoplastic strain. Using a non-linear dashpot, a viscoplastic strain is formulated and its parameters are identified for PA6. In addition, a stress relaxation test is conducted, and the relationship between the viscoelastic strain and stress is identified when considering the viscoplastic strain. In this study, the time–temperature superposition principle is thoroughly applied to include the effect of elevated temperature on the viscoelastic–viscoplastic behavior. All material constants in the non-linear viscoelastic–viscoplastic constitutive equation including the time–temperature superposition principle for PA6 are presented in this study.

Failure prediction of magnesium alloys based on improved CDM model

The aim of this contribution is to improve the behavior and failure prediction accuracy for magnesium alloy sheet forming simulation by using advanced fully coupled Continuum Damage Mechanics (CDM) model. Starting from known fully coupled constitutive equations at large inelastic (plastic or viscoplastic) strains, three aspects are enhanced on the CDM model developed by Saanouni (2012). The first one concerns the accurate description of tension-compression asymmetry in yielding and hardening of magnesium alloys induced by combined slip and twinning deformations. The second concerns introducing the stress state dependence in damage evolution to improve the fracture prediction under various loading paths. The third deals with temperature and strain rate (viscosity) effects for metal forming processes at elevated temperatures. The improved fully coupled constitutive equations were implemented into finite element code ABAQUS/Explicit via user subroutine VUMAT including the specific local integration scheme to compute the state variables at each quadrature point. Various kinds of tensile tests at different temperatures and strain rates are used to determine the material parameters, and the complete identification procedure is given in details. As validation, the improved model is applied to simulate the three-point bending and circular cup deep drawing tests, the predicted results show good agreements with the experimental observations.

Permanent deformation performance and a nonlinear constitutive model with thermal-mechanical coupling of PCPMA mixture

Polymers cracking products modified asphalt (PCPMA) pavement is a new warm mixing pavement technology. The permanent deformation properties of three types of PCPMA mixture were studied using repeated load triaxial (RLT) tests at different temperatures and deviator stress levels. The half sine wave intermittent load was adopted to simulate the vehicle load on the actual pavement. The test results indicate that the permanent deformation of the PCPMA mixture is increased gradually with the increase of temperature and stress level at the same load times. A robust skeleton structure can improve a specific resistance to the deformation of the PCPMA mixture. A ‘three-stage’ deformation characteristics, which are transfer section, stabilization section, and destruction section in proper order, exhibited under high temperature and significant stress level. For the condition of lower temperature and smaller stress levels, the permanent deformation only accumulates in the first two stages. The PCPMA was only damaged at 50 °C and 0.7 MPa, which has better durability than the matrix asphalt mixture. Accumulative residue viscoelastic strain and viscoplastic strain are derived from RLT test results. And then, a new viscoplastic mechanical model is proposed to predict the permanent deformation of the PCPMA mixture at different conditions, considering the nonlinear characteristics. Furthermore, this new mechanic model, which realized the unification of three and two stages deformation law of PCPMA mixture, may be effectively used for mechanical analysis and permanent deformation calculation of PCPMA pavement.

Numerical Implementation of a Unified Viscoplastic Model for Considering Solder Joint Response under Board-Level Temperature Cycling

An implicit integration scheme was developed for simulating the viscoplastic constitutive behavior of Sn3.0Ag0.5Cu solder and programmed into a user material subroutine of the finite element software ANSYS. The numerical procedure first solves the essential state variables by using a three-level iterative procedure, and updates the remaining stress and state variables accordingly. The numerical implementation was applied to consider the responses of solder joints in an electronic assembly under temperature cycling condition. The viscoplastic strain energy density accumulation over one temperature cycle was identified as a feasible parameter for evaluating the thermomechanical reliability of the solder joints.

Triaxial Creep Behavior of Red Sandstone in Freeze-Thaw Environments

To investigate the time-dependent mechanical properties of rock masses in cold regions under the effects of freeze-thaw cycling and long-term loading, triaxial multilevel loading and unloading creep tests were performed on saturated red sandstone samples subjected to different numbers of freeze-thaw cycles. The effects of freeze-thaw cycles and confining pressure on the creep properties, long-term strength, and creep failure mode of the rock were analyzed. The effect of freeze-thaw cycles on the microstructure of the rock was analyzed using scanning electron microscopy. The results showed that as the number of freeze-thaw cycles increased, the rock particle boundaries became more distinct, and more pores formed. The effect of freeze-thaw cycles on the creep deformation of red sandstone was related to the loading stress level. At low stress levels, the rock viscoelastic strain increased gradually and almost linearly with an increasing number of freeze-thaw cycles; in contrast, at high stress levels, the rock viscoelastic strain increased nonlinearly. The viscoplastic strain increased almost linearly with increasing freeze-thaw cycles. The fourth loading stress level (70% σ c ) corresponded to the transition of the creep deformation of the red sandstone. When the confining pressure was low, a higher stress level caused the confining pressure to have a more significant effect on the creep strain. However, as the confining pressure continued to increase, the effect of the confining pressure on the creep strain eventually disappeared. The long-term strength of the red sandstone decreased approximately linearly with an increase in the number of freeze-thaw cycles. When the number of freeze-thaw cycles and the confining pressure were high, the rock samples formed a transverse shear plane and were more fragmented than those without a transverse shear plane. These results provide a reference for construction in rock mass engineering and long-term stability analysis in cold regions.