Creep Constitutive Equation Research Articles

Crystallographic orientation dependence in creep deformation of micron-thick silicon films for 3-D microstructures

In this study, the steady-state creep deformation properties of 5 μm-thick single crystal silicon (Si) films at elevated temperatures were investigated using punch creep-forming tests for the design of three-dimensional (3-D) microstructured MEMS. The relationship between the creep strain rate and stress of the Si films with {100}, {110}, and {111} planes at temperatures of 1223–1323 K was derived using finite element analysis following punch creep-forming tests, and a power-law creep constitutive equation for Si films was determined. The steady-state creep properties of the Si film samples varied clearly with crystallographic orientations. Among the three crystallographic orientations, the creep strain rate of the micron-thick Si films was the fastest for the {011} plane, followed by the {111} and {001} planes, in the applied stress range of 110 MPa to 220 MPa. This is consistent with the increasing order of magnitude of the thermal activation energy. The crystallographic orientation dependence of the steady-state creep properties of Si films is discussed based on the mobility of dislocation gliding per unit lattice and scanning electron microscope observations. The sample-size dependence of the creep strain rate of Si with the {001} plane was clearly observed between the micron- and millimeter-thick samples, which was attributed to the difference in the frequency factor of the power-law creep. However, the thermal-activation energy remained constant. This study successfully revealed the steady-state creep properties of Si films and provided useful engineering data for the design of 3-D Si-MEMS fabrication by the plastic processing of Si films.

Nonlinear Nishihara model of soft rock based on damage mechanics and its parameter identification

The traditional Nishihara model is difficult to describe the nonlinear primary and accelerated creep characteristics of soft rock. Based on this, an improved nonlinear Nishihara model is established by modifying the unsteady dashpot and a nonlinear damaged viscoplastic body in series. The one-dimensional and three-dimensional creep constitutive equations of soft rock are derived. Levenberg-Marquardt algorithm is used to identify model parameters of creep test data of Cretaceous soft rock and red sandstone. The results show that the correlation coefficient between the fitted results of the improved nonlinear Nishihara model and the creep test data is above 0.97. The sensitive parameters λ,η′1 and a of the model are all within a reasonable range. It shows that the model can accurately reflect the characteristics of the unsteady primary creep stage and the nonlinear damage accelerated creep stage of soft rock, which is better than the traditional Nishihara model. The research results can provide theoretical guidance for similar soft rock creep model research.

Creep deformation and damage behavior of an alumina-forming austenitic alloy strengthened by intermetallic compounds

Based on Fe–30Ni–15Cr, a new alumina-forming austenitic (AFA) alloy, free of carbon, was designed for potential use in ultra-supercritical generator set. The creep behavior of alloy has been investigated by analyzing the creep constitutive equation, creep damage tolerance factor (λ), microstructure images, and oxide layer structure under different stresses at 750 °C. The creep microstructure and oxide morphology were obtained by electron microscopy. The minimum creep rate (ε˙m) is positively correlated with the stress, and when the stress increases from 120 MPa to 250 MPa, ε˙m increases from 6.26E-3 1/h to 6.34E-4 1/h, almost ten times of the latter. The creep constitutive equation at 750 °C was determined as: ε˙m=9.29×10−11σ3.3. B2–NiAl and L12-Ni3Al were precipitated during creep. The relationship between the misfit of nanoscale L12-Ni3Al phase and γ matrix are perfectly coherent. The L12-Ni3Al phase grows more slowly and remains at the nanoscale size after creep at 750 °C for 869h, which contribute to increase creep strength and reduce the creep rate of the material. However, the coarsened σ-FeCr phase reduced the creep productivity of the alloy. The microstructural degradation caused by precipitate coarsening was found to be responsible for creep failure, as indicated by the creep damage tolerance factor (λ) and creep fracture morphology. The elemental distribution of the oxide layer indicates the absence of a dense alumina protective scale at 750 °C. The extensive presence of oxidation cracks in the matrix suggests that the effect of oxidation cannot be ignored in the analysis of creep life. The fact that σ-FeCr is detrimental to creep damage indicates that the alloy composition needs further optimization.

Stress relief simulation for post-weld heat treatment process of pressure equipment: Creep constitutive equation considering temperature and stress variations

Due to the limited testing methods and high costs, researches on the evolution law of residual stress in the post-weld heat treatment (PWHT) process of pressure equipment are mainly by numerical simulation, in which the selection of a reasonable and reliable creep constitutive model is crucial to ensure the accuracy of numerical predictions. It is worth noting that the temperature changes and residual stress variation during the heat treatment process result in the following creep characteristics: the creep occurs under different temperatures during the heat treatment processes of heating, holding, and cooling; the creep occurs under different residual stress level where the residual stress varies during the heat treatment stages. Thus, to accurately simulate the stress release behavior during the PWHT process, the proper creep constitutive model must involve the influences of the temperature and the stress state. Meanwhile, current heat treatment simulation usually adopts simplified qualitative approaches, and the creep constitutive models often ignore the physical mechanisms and characteristics of actual creep processes, such as the Norton, the Norton-Bailey, and the Omega models. As a result, the accuracy of heat treatment simulation is limited. In this paper, a new creep constitutive relation under varying temperature and stress is proposed. The hyperbolic-sinusoidal function is used to represent the complete three stages of creep, which compensate the inaccuracy prediction at the first and the third creep stages. Moreover, the Arrhenius function is introduced to characterize the creep response under varying temperature conditions, which enables the prediction of residual stress evolution throughout the entire heating-holding-cooling process of heat treatment. Then, a typical pressure vessel cylindrical circumferential weld is simulated. The stress release during the heat treatment is calculated, and the efficacy of the proposed model is validated through surface residual stress testing. It is found that the creep behavior is a crucial factor in numerical simulation of heat treatment, and the creep constitutive relation can significantly affect the accuracy of heat treatment simulation. The simulation utilizing the hyperbolic-sinusoidal temperature-dependent creep model is much more accurate compared to the classical simulation. The simulation results are much more consistent to the experimental results. Specifically, the traditional model often underestimates the stress state, the residual stress drops abruptly during the heating stage. The residual stress evolution mechanism is simulated using the new model: The majority of residual stress are released during the heating process which primarily relies on the creep strain transformation (contributes up to 85%). The stress state does not change during the holding stage and slight higher at the cooling stage.

Influence of constitutive equation and friction on measurement of creep strain rate based on small specimen testing technique

AbstractTwo stress regime‐dependent creep constitutive equations, that is, two‐regime Norton (2RN) model and Sin‐hyperbolic (Sinh) model, are adopted to numerically investigate the influence of constitutive equation and friction on the measurement of steady‐state creep strain rate based on five types small specimens. Results show that the influence of constitutive equation and friction depends strongly on the type of small specimen. A measurement error of more than 19 times may occur if the 2RN model is used, while the use of the Sinh model is able to generate a consistently lower measurement error. Of all small specimens, the cantilever beam testing (CaBT) is revealed to exhibit the best comprehensive performance in terms of measurement error and response to friction. To validate numerical findings, creep tests on the CaBT specimen of P91 steel are conducted at 600°C. Finally, recommendations on the choice of small specimen for obtaining creep strain rate are made.

Analysis of contact creep behaviour of nanofilled composites

An effective modelling method for analysing the contact creep behaviour of composite materials with different nanofillers was proposed. Considering both the contact and creep material nonlinearities, an augmented Lagrangian algorithm was used to treat the unilateral contact constraints, and a time-hardening creep constitutive equation was adopted to describe the deformation characteristics of the nanocomposites. The constitutive parameters were extracted through a series of linear fittings based on the multiaxial creep theory and creep test curves, and the accuracy of the constitutive equation was validated. The applications of the proposed modelling method were demonstrated using a rope-wheel contact system (RWCS), in which the wheels were made of polystyrene (PS) with carbon black (CB), multi-walled carbon nanotubes (MWCNT), or chemically reduced graphene oxide (CRGO). The results indicated that the addition of nanofillers significantly reduced the creep deformation of the contact zone. The CRGO exhibited a better anti-creep performance than that of CB and MWCNT. Compared with pure PS, the maximum deformation on the contact path of the nanocomposite wheel with 5.0 wt% CRGO sheets at 1500 s and 104 s was reduced by approximately 22 % and 52 %, respectively. The coefficient of friction μ had a strong influence on the contact creep deformation when μ < 0.3. Additionally, the anti-creep capability of the added nanofillers became more evident as the loading time increased. The contact creep deformation of the CRGO composite was less than one-tenth that of the matrix material after 30 d.

An elastic-viscoplastic creep model for describing creep behavior of layered rock

To describe the full-stage creep behavior of layered rock accurately, a new elastic-viscoplastic creep model is proposed based on fractional order theory in this manuscript, which consists of a Hooke elastomer, a fractional Abel dashpot, a Kelvin body, and a new non-linear visco-plastic component. The non-linear creep model can not only describe the changes in three creep stages (primary creep, steady-state creep and accelerating creep) but also reflect the influence of different bedding angles of rock. The constitutive equations of the non-linear creep model are deduced by the empirical model method and plastic theory method, respectively. The parameters of the non-linear creep model are identified using the Levenberg-Marquardt algorithm from Origin. It shows that the creep model in this paper are highly consistent with the experimental data under different load levels, creep stages and bedding angles, and the accuracy and rationality of the model are verified. Moreover, the creep constitutive equations for layered rock derived by the two methods have the same fitting effect on the same set of experimental data.

Emphasizing the Creep Damage Constitutive Model of Hydro-Mechanical Properties of Rocks: A Case Study of Granite Gneiss

The constitutive model of rock materials can describe the mechanical behavior of rocks in creep tests. Also, it is one of the important means to study the deformation and strength characteristics of rocks in complex stress environments. This paper is based on the analysis of the porosity variation characteristics of the internal structure under the coupling effect of rock hydro-mechanical properties. The concept of the hydro-mechanical properties variable is proposed, and the relationship between the coupling variable, damage and plastic deformation is established. By introducing the coupling variable, instantaneous damage variable and time-dependent damage variable into the yield surface equation, as well as the plastic potential energy equation and the stiffness matrix of the elastic–plastic creep constitutive equation, a hydro-mechanical properties creep damage coupling model was established to simulate the creep mechanical properties of rock under coupling. Based on the triaxial creep test results of granite gneiss, the model parameters are determined. By comparing the test results with numerical results, it was revealed that the model can better describe the creep mechanical properties of rocks under the coupling effect of hydromechanical properties.

Coupling analysis on controlling mechanisms for creep of Al-Fe-Ni alloy

The dislocation creep of Al−Fe−Ni alloy at both 300 and 400 °C was described by a coupling modification on analyzing the creep constitutive equation after introducing both threshold stress and load transfer coefficient. The quantitative results and microstructure characterizations showed that the complex interaction between dislocation and eutectic phase was the origin of threshold stress, and the modulus difference between matrix and eutectic phase induced the load transfer effect. Furthermore, the temperature effects on the modulus of matrix and eutectic phase were different, leading to the decline of threshold stress and increase of load transfer coefficient at higher temperatures. This was further evidenced by the finite element modelling analysis. Finally, the controlling mechanisms during dislocation creep of eutectic alloys were discussed in an increasingly quantitative manner.

Creep behaviour of Q245R steel at 550 ℃

ABSTRACT Q245R steel has good heat and corrosion resistant and is used in various applications, such as reactors, regenerators, reformed gas steam generators, and other equipment. In this study, the creep behaviour of Q245R steel at the temperature of 550 ℃ is investigated. In addition to the modified Kachanov-Robotnov (K-R) method, the traditional and modified θ projection methods were employed to establish the creep constitutive equations of Q245R steel. The results revealed that at small strains, the creep prediction curve was in good agreement with the test curve, and the prediction accuracy of the modified θ projection method was better than that of the other two models.

Creep lifetime of Al 6061-T6 pressurized rotating friction stir welded tube subjected to internal pressure and rotational velocity: Welding and creep lifetime optimization

Aluminum 6061-T6 alloys have numerous applications in the industry due to their unique properties such as higher corrosion resistance and strength-to-weight ratio. Friction stir welding is also a technique that can be used in aluminum joints because of its numerous advantages. In this paper, optimum friction stir welding parameters that lead to a higher failure load are obtained using tensile test data and analysis of variance technique. According to the analysis of variance results, it is found that the tool rotational speed has a major effect among the welding parameters on the failure load. Then, creep tests are carried out on non-welded and welded specimens to predict their creep behavior. The Norton power law equation is used as a creep constitutive equation to model the creep behavior of the specimens. The model is used to predict the creep lifetime of the welded and non-welded rotating pressurized vessels. The results show that an optimally welded vessel can only sustain up to 0.75 of pressure for the non-welded vessel with the same rotating speed. Since this comparison is not straightforward because of the number of affecting parameters. Hence, the reference stress method is used to produce a set of diagrams that provide design conditions and optimum creep lifetime for the friction stir welded Al 6061-T6 rotating pressurized vessels.

Characterization of mechanical properties of in-service nickel-based alloy by continuous indentation

Accurate acquisition local material mechanical properties of key structures material plays a key role in equipment safety evaluation and life prediction. In this study, the method for the mechanical properties of nickel-based alloy 600 before and after creep based on continuous indentation test combined with numerical simulation was established, and the high stress creep constitutive equation of nickel-based alloy 600 in autoclave environment was obtained, then the influencing factors of mechanical properties are analyzed. Results show that with the increase of the creep time of alloy 600, the microstructure and macroscopic deformation become more obvious. Simultaneously, the strain strengthening caused by creep increases gradually, and the mechanical properties of nickel-base alloy 600 increase. The indentation test model verified by numerical simulation can carry out the continuous indentation test and accurately obtain the indentation load-depth curve of alloy 600 before and after creep, and the characteristic stress–strain points obtained agreement with the stress–strain curve obtained by uniaxial tensile test. The established indentation test system platform can obtain the mechanical property parameters of nickel-based alloy by using indentation method. Compared with the traditional uniaxial tensile test, the relative strength error is less than 5% and nondestructive, which provides a theoretical basis for the safety evaluation and residual life analysis of in-service structural materials.

Numerical Modelling of the Nonlinear Shear Creep Behavior of FRP-Concrete Bonded Joints.

This paper presents a numerical investigation of the shear creep behavior of the adhesive joint in concrete structures strengthened by externally bonded fibre-reinforced polymers (FRP) composites. Based on experimental data collected in a previous study, creep constitutive equations were developed for the adhesive layer and implemented into a finite element code. The proposed model extends the classical one-dimensional formulation of Burgers creep model to a fully 3D model and introduces the nonlinearity of the model parameters. This numerical approach was first used to simulate the nonlinear creep behavior of bulk epoxy samples; it was then extended to predict the nonlinear creep response of the FRP-concrete interface in double lap shear specimens. Globally, a fair agreement was obtained between numerical results and experimental evidences. As a main result, it was found that creep induces a redistribution of the interfacial shear stress along the FRP-concrete lap joint, leading both to a stress relaxation near the loaded end of the adhesive joint and to an increase in the effective transfer length.

Integrity analysis of wellbores in the bedded salt cavern for energy storage

Salt cavern used for energy (natural gas, hydrogen) storage has a significant advantage in peak shaving of gas supply due to their high injection-production efficiency and fast gas injection-delivery switching speed. As the only channel for gas injection and delivery, the wellbore is easily damaged by the alternating gas pressure and is a weak component. For salt cavern underground gas storages (UGSs) in bedded rock salt, the roof collapse can also cause tensile damage to the wellbore. Long-running UGSs may suffer leakage accidents due to wellbore failures, and wellbore safety issues need to be fully considered. This paper proposes a coupled analysis of cement sheath fatigue damage and rock salt creep, and its applicability is proved by mechanical tests. A three-dimensional numerical model was established that includes a wellbore, open well section (cavern neck), and bedded salt cavern. The creep and damage constitutive equations were used to calculate rock salt creep and cement sheath damage. The results show that the creep ability of bedded rock salt is lower than that of high-purity rock salt, confirmed by the creep test results and can be described by the creep constitutive equation in this paper. The cavity shrinkage of the bedded UGSs is generally lower than expected, resulting in better wellbore integrity. The low creep capacity of the impurity-containing rock salt will reduce the cavity shrinkage and protect the airtightness of the wellbore. Based on the integrity of the wellbore, the working gas volume of the UGSs in the bedded formation can be increased. Taking the Jintan gas storage as an example, under the operating pressure of 7–17 MPa, the damaged area of the cement sheath is limited to within 0.5 m of the lower end of the wellbore. When the lower limit pressure is reduced to 6 MPa, the damaged area is limited to within 2 m of the lower end of the wellbore, and the wellbore still maintains sufficient airtightness.

Compressive creep behavior of spherical pressure hull scale model for full-ocean-depth manned submersible

The phenomenon of room temperature creep in titanium alloys has drawn great attention for decades. With the increasing application of titanium alloys to the manned submersible pressure hull, the traditional calculation and analysis based on strength and stability cannot meet the requirement of structure design and safety evaluation for the spherical pressure hull completely. In this paper, a series of compressive creep tests were performed to reveal the change rule and distribution characteristics for creep behavior of the spherical pressure hull model. Results show that there is no measurable creep strain for pressure hull under maximum operating pressure and steady-state creep appeared in high stress areas of the pressure hull under 1.25 times maximum operating pressure, with the characteristics of internal creep strain and strain rate being greater than the external counterparts. The stress threshold value of the pressure hull model for steady-state creep is 853.7 MPa (equivalent to 0.88Rp0.2), and a creep constitutive equation is proposed based on improved Norton power law model. The results of this paper can provide support for the safety assessment and optimization design of manned submersibles for long-term service.

Creep damage coupling model of concrete based on the statistical damage theory

The creep behavior is one of the key issues in the design and construction of hydraulic concrete structures. However, it is an enormous challenge to accurately predict the development of creep deformation in concrete structures. Since the creep deformation behavior of concrete under long-term load action originates from the interaction between creep and damage, a new creep damage coupling model was developed by introducing the damage variable into the creep constitutive equation. This model utilized the statistical damage theory framework and assumed that the damage variable follows the random distribution described by the Weibull function. Then Weibull distribution parameters F0 and m were derived to determine the evolution equation of the damage variable D. Considering stress under complex stress conditions may change, the stress-strain correlation of concrete under complex stress conditions was deduced using the elastic creep theory. Moreover, the results of uniaxial compressive concrete creep tests with the corresponding numerical simulations under different levels of sustained loads, and the accuracy and applicability of the creep damage coupling model were verified. Consequently, the nonlinear creep properties of hydraulic concrete were analyzed by combining it with acoustic emission techniques.

Construction of nonlinear creep damage model of frozen sandstone based on fractional-order theory

The frozen wall of a coal mine vertical shaft undergoes creep deformation under long-term load, and excessive deformation induces damage to the frozen wall. The creep behaviour of frozen rock is key in controlling the long-term stability of frozen walls. In this study, Cretaceous saturated frozen sandstone was used as the research object to conduct creep mechanics tests under different confining pressure conditions. Based on the Riemann-Liouville-type integral function, a 6-element frozen sandstone nonlinear creep constitutive equation considering temperature–damage–stress coupling was established and verified by experimental data. The results show that: compared with the rock creep test at 20 °C, the creep process of saturated frozen sandstone also presents an attenuation creep stage, a steady-state creep stage, and an accelerated creep stage. A nonlinear creep damage equation is established based on the fractional-order theory. Comparing the theoretical formula with the experimental data, it was found that the agreement was higher. The research results can provide a theoretical and experimental basis for evaluating creep-induced instability and failure of frozen walls.

Rapid estimation of creep constitutive equation using digital image correlation and FEM

Estimation of the creep constitutive equation requires uniaxial creep tests under various stress conditions, which is costly and time consuming. In this study, a rapid estimation method of a creep constitutive equation is investigated based on an inverse analysis using digital image correlation (DIC) and FEM. First, a steady-state creep strain field that develops non-uniformly near the stress raiser was measured by DIC. Next, an inverse analysis was performed to minimize the difference between the strain measured by DIC and the strain calculated by FEM, and the creep constitutive equation in the steady creep region was estimated. The strain rate calculated from the estimated creep constitutive equation were consistent with the results obtained from uniaxial creep tests at high stress side, though the strain rates were overestimated at lower stress condition, where the duration of transient creep is relatively longer.

Nonlinear Viscoelastic‐Plastic Creep Model of Rock Based on Fractional Calculus

A rock creep constitutive model is the core content of rock rheological mechanics theory and is of great significance for studying the long‐term stability of engineering. Most of the creep models constructed in previous studies have complex types and many parameters. Based on fractional calculus theory, this paper explores the creep curve characteristics of the creep elements with the fractional order change, constructs a nonlinear viscoelastic‐plastic creep model of rock based on fractional calculus, and deduces the creep constitutive equation. By using a user‐defined function fitting tool of the Origin software and the Levenberg–Marquardt optimization algorithm, the creep test data are fitted and compared. The fitting curve is in good agreement with the experimental data, which shows the rationality and applicability of the proposed nonlinear viscoelastic‐plastic creep model. Through sensitivity analysis of the fractional order β2 and viscoelastic coefficient ξ2, the influence of these creep parameters on rock creep is clarified. The research results show that the nonlinear viscoelastic‐plastic creep model of rock based on fractional calculus constructed in this paper can well describe the creep characteristics of rock, and this model has certain theoretical significance and engineering application value for long‐term engineering stability research.

Thermal Fatigue Life Prediction of BGA Solder Joint Incorporating Microstructural Coarsening Effect due to Thermal Diffusion.

A creep constitutive equation incorporating the effect of microstructure coarsening was introduced into the Finite Element Method analysis using a user subroutine to predict thermal fatigue life of solder joint. Dispersed particles in the microstructure grew due to thermal diffusion with increasing fatigue cycles, which resulted in the inelastic strain range and the inelastic strain energy density range increased. This increase in the inelastic strain range reduced the fatigue life by approximately 77% compared to the case where no microstructure coarsening effect was incorporated. Thus, it is necessary to incorporate the microstructure coarsening effect in the fatigue analysis of solders. Thermal fatigue life was found to be significantly affected by the microstructure coarsening effect. Thus, it is necessary to incorporate the microstructure coarsening effect in the fatigue analysis of solders.