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The experimental results show that initial damage has a clear effect on the creep behavior of rock. However, among the current creep models for rock, few consider the effect of the initial damage state. In the present study, a new nonlinear creep damage model for rock is proposed based on multi-loading creep tests of sandstone with different initial damage levels. The new model is composed of four components, a Hooke body, a Kelvin body, an improved viscous element, and a new nonlinear visco-plastic damage component. The creep damage model can not only describe the three typical creep stages (primary creep, secondary creep and tertiary creep) but also show the effect of initial damage on the creep failure stress. The parameters of the nonlinear creep damage model are obtained using the nonlinear least squares method. A unified set of creep parameters is proposed to predict the creep behavior of sandstone in different initial damage states. The agreement between the experimental data and numerical prediction demonstrates the applicability of the proposed model.

EBSD observations were conducted on the damaged materials obtained by interrupted creep tests and interrupted creep-fatigue tests for 304 austenitic stainless steel for boiler tube use in fossil power plants, and the shapes of crystal grains extracted from KAM maps and GOS maps were approximated by ellipses. Furthermore, a damage evaluation system has been developed with a neural network, which uses the information obtained by elliptic approximation as parameters. As a result, it was quantitatively found that as creep and creep-fatigue damage progress, crystal grains become elongated toward the load axis direction. Ensemble learning showed the best classification accuracy using the 20 learners obtained by changing the rank of the relative frequency of KAM. The damage evaluation system in this study was able to estimate the damage rates with a classification accuracy of 98.33% for creep test materials and 97.50% for creep-fatigue test materials using information from one of crystal grains in the EBSD image. Therefore, the system with the neural network developed in this study is effective for evaluating creep and creep-fatigue damage for 304 austenitic stainless steel.

The deleterious effect of creep damage upon fatigue endurance in elevated temperature low cycle fatigue is well established and has become known as the 'creep - fatigue interaction'. This has been identified as a narrow region where a transgranular crack becomes intergranular as creep damage develops. In other cases transgranular fatigue cracks and intergranular creep cavitation damage to some extent can take place separately ('competitive' mode) so that interaction is less pronounced. These phenomena have been incorporated into 'damage interaction diagrams' which feature in many high temperature Code cases (ASME, R5, RCC-MR etc.). Fractional creep and fatigue damage are considered separately, and failure is conceded when their sum is unity ('additive' mode). For some materials the sum is set to a value <1. It appears that purely empirical arguments have been used to arrive at total damage values <1. It is shown that such deviations from linearity can be simply predicted, assuming that creep damage accumulates uniformly during a given test. For maximum interaction, the resulting assessment curve is shown to pass through the points (total fatigue damage = 0.4; total creep damage = 0.4), or very nearly. The influence of fatigue upon creep damage is less well known. Examples are provided, and when included in the calculation, produce a more damaging curve passing through the point (0.33, 0.33). By use of 'interaction coefficients' less damaging curves for the 'competitive' model can be produced. The calculation may be simplified by use of an effective fatigue damage cycle number for use as a reference. Based on metallographic examples from power plant operation and laboratory testing, the 'competitive', 'additive' and 'true interaction' modes of creep and fatigue damage are identified. These can be related to practical applications of base load operation, fast start-up and shut-down procedures, slow start-up and shut-down procedures etc. Components might operate under base load conditions (creep) followed by temperature cycling (fatigue). These sequential events may not be interactive (as has been demonstrated in the laboratory) and thus use of the current assessment diagram would be pessimistic. Comment is made on a proposed damage diagram for the advanced ferritic steels, which is highly conservative and does not reflect their good performance in service and laboratory tests.

Transversely isotropic layered rock is widely distributed in nature. To better describe the time-dependent entire creep characteristics for transversely isotropic rock, a simple nonlinear damage creep model is derived based on fractional order theory, which consists of a Hooke elastomer, a fractional Abel dashpot, a fractional nonlinear damage dashpot, and can effectively describe the characteristics of primary creep, steady-state creep and accelerating damage creep. Assuming that Poisson's ratio is constant, the creep equation of isotropic rock is extended to transversely isotropic rock, and the nonlinear damage creep model for transversely isotropic rock is established. Step-wise loading triaxial creep tests of phyllite specimens with three kinds of bedding angles (0°, 45° and 90°) are carried out, and it is found that there are significant differences in creep deformation and failure characteristics under different bedding angles. The parameters of the creep model at each bedding angle are identified using the Universal Global Optimization method. By comparing the Nishihara model, the modified Nishihara model and experimental data, it shows that the creep model in this paper are highly consistent with the experimental data under different bedding angles, load levels and creep stages, and the accuracy and rationality of the model are verified.HighlightsA simple nonlinear damage creep model is derived based on fractional order theory.By assuming that Poisson's ratio is constant, the creep equation of isotropic rock is extended to transversely isotropic rock, and the nonlinear damage creep model for transversely isotropic rock is established.There are significant differences in creep deformation and failure characteristics of phyllite specimens with different bedding angles.Parameters of the proposed creep model at each bedding angle are identified by using the Universal Global Optimization, and the accuracy and rationality of the model are verified.

Dolomitic limestone is the main surrounding rock material in Yangzong tunnel engineering; the instantaneous mechanical properties and creep behaviors of limestone are significant for stability evaluation during the stages of tunnel excavation and long-term maintenance. Herein, four conventional triaxial compression tests were carried out to explore its instantaneous mechanical behavior and failure characteristics; subsequently, the creep behaviors of limestone subjected to multi-stage incremental axial loading at the confinements of 9 MPa and 15 MPa were studied by employing an advanced rock mechanics testing system (i.e., MTS815.04). The results reveal the following. (1) comparing the curves of axial strain–, radial strain–, and volumetric strain–stress under different confining pressures shows that these curves present a similar trend, whereas the stress drops during the post-peak stage decelerate with the increase in confining pressure, suggesting that the rock transits from brittleness to ductility. The confining pressure also has a certain role in controlling the cracking deformation during the pre-peak stage. Besides, the proportions of compaction- and dilatancy-dominated phases in the volumetric strain–stress curves differ obviously. Moreover, the failure mode of the dolomitic limestone is a shear-dominated fracture but is also affected by the confining pressure. (2) When the loading stress reaches a creep threshold stress, the primary and steady-state creep stages occur successively, and a higher deviatoric stress corresponds to a greater creep strain. When the deviatoric stress surpasses an accelerated creep threshold stress, a tertiary creep appears and then is followed by creep failure. Furthermore, the two threshold stresses at 15 MPa confinement are greater than that at 9 MPa confinement, suggesting that the confining pressure has an obvious impact on the threshold values and a higher confining pressure corresponds to a greater threshold value. Additionally, the specimen’s creep failure mode is one of “abrupt” shear-dominated fracturing and is similar to that under a conventional triaxial compression test at high confining pressure. (3) A multi-element nonlinear creep damage model is developed by bonding a proposed visco-plastic model in series with the Hookean substance and Schiffman body, and can accurately describe the full-stage creep behaviors.

The study of the time-dependent deformation behavior of soft rock under water–rock interaction is of great significance to reveal slope creep deformation behavior. In this study, the saturation-loss cycle's triaxial creep test was performed, and the creep deformation, creep rate, and long-term strength of silty mudstone were analyzed. A two-step improved modeling method was also proposed. Firstly, by introducing fractional Abel viscous pot and aging factor, a one-dimensional creep constitutive model was established based on the Burgers model. And then the three-dimensional creep damage constitutive was derived through Laplace transform. Finally, the Levenberg–Marquardt (L–M) algorithm method was used to identify the creep curve and calibrate the model parameters. The new constitutive model could reflect the soft rock's creep characteristics during saturation-loss cycle.

Water is crucial to the durability and safety of underground engineering. To investigate the effect of long-term goaf water infiltration on creep characteristics and long-term stability of the No.3 anthracite pillar in the Qinshui coalfield in Shanxi Province, conventional uniaxial compression and uniaxial creep tests were conducted under dry and saturated conditions. In addition, based on the test results, a nonlinear damage viscoelastic–plastic creep model was developed to describe each creep stage in detail. The experimental results reveal the following. (1) When the loading stress exceeds the long-term strength, the anthracite creep in dry and saturated states undergoes decay, steady-state and accelerated creeps; (2) the total instantaneous elastic strain of the anthracite sample increased linearly with the stress level, and the growth rate of the saturated sample exceeded that of the dry sample; (3) the average of the long-term strength of anthracite under dry and saturated conditions is 7.604 MPa and 4.489 MPa respectively, and the corresponding long-term strength to instantaneous strength ratios are 0.73 and 0.68, respectively; (4) the developed nonlinear damage viscoelastic–plastic creep model can simultaneously describe the two-stage creep characteristics of anthracite under low stress and the three-stage creep characteristics under high stress; (5) the failure form of the dry coal sample after the creep test was a shear failure, whereas the failure form of the saturated coal sample was ductile owing to water softening. When anthracite was transformed from the dry to the saturated state, its instantaneous and long-term strengths decreased by ~ 37% and ~ 41%, respectively.

In order to better understand the shear creep behavior of weak interlayers (carbonaceous shale) under the coupling effect of the rainfall dry–wet cycle and blasting vibration, as well as quantitatively characterize the coupled damage of the rainfall dry–wet cycle and blasting vibration, a series of shear creep tests were carried out. The results show that the combined damage of the rainfall dry–wet cycle and blasting vibration greatly intensifies the creep effect of carbonaceous shale, leading to an increase in deceleration creep time, an increase in steady-state creep rate, and a decrease in long-term strength. The coupling damage of the rainfall dry–wet cycle and blasting vibration in carbonaceous shale was quantitatively characterized. Based on the fractional-order theory, a fractional-order creep-damage constitutive model (DNFVP) was established by introducing the Abel dashpot to describe the coupled damage of the rainfall wet–dry cycle and blasting vibration and the nonlinear creep acceleration characteristics. The three-dimensional creep equation of the model was derived. The effectiveness of the DNFVP model was verified through the inversion of model parameters and fitting of experimental data, providing a basis for in-depth research on the long-term stability of high slopes in mines with weak interlayers.

The objective of this work is to investigate the creep damage mechanism of granite from a nuclear power station. A series of multi-step loading and unloading cycles creep tests of granite were carried out under different confining pressures. For the creep test, the creep differential stresses (i.e., 60%, 70%, 80%, 90%, 100% and even larger than 100% of σp) were applied according to the instantaneous peak deviatoric stress σp. To study the elastic-visco-plastic creep deformations of the granite specimens, the total strain was divided into instantaneous and creep strains, including the instantaneous elastic strain, plastic strain, visco-elastic strain, and visco-plastic strain. The results show that the short-term strength, crack damage threshold, and Young’s modulus all increased with the increasing of confining pressure. Due to an existing critical stress, the creep strain stabilizes when the applied stress is less than the stress level. An exponential function is proposed, which can well describe the relationship between the visco-plastic strain and stress ratio. The long-term strength of the granite corresponds to the stress level at the stress ratio range from 0.86 to 0.95. Besides, the damage mechanism of the failed granite specimens is analyzed using an X-ray micro-CT scanning system. Finally, a time-dependent damage model is proposed to quantitatively characterize the creep damage and deformation behaviors of granite.HighlightsInvestigate long-term creep mechanical behavior of granite under different confining pressureExplore the internal damage fracture behavior by X-ray micro-CT observationPropose a time-dependent damage model to describe the creep damage evolution of granite

During the operation of the Jinping II hydropower station, the rocks surrounding the headrace tunnels have undergone failure because of the time-dependent creep effect that occurs due to the high geostress present and seepage. However, a few studies of the creep deformation of deep rocks have taken into account the coupling effect between excavation damage and pore pressure. This paper reports the results of a series of triaxial creep experiments conducted on Jinping deep marble in which the coupling between excavation damage and high pore pressure was studied. The results show that the magnitude of the creep deformation, the long-term strength, and failure mode of the excavation-damaged marble vary considerably with pore pressure. That is, the pore pressure has a significant effect on the marble. A new nonlinear viscoelastic-plastic damage (VEPD) model incorporating the effect of pore pressure was subsequently developed based on the experimental data. The values of the parameters appearing in the model were then obtained using a method of Universal Global Optimization. Good consistency is found between our theoretical predictions and test data which indicates that the VEPD model can be reliably used to simulate and predict the complete creep process in excavation-damaged deep rocks. It also gives a good account of the effect of pore pressure on the creep behavior. As a result, the model provides us with a better understanding of the long-term stability of underground engineering projects.HighlightsThe coupling between excavation damage and high pore pressure was investigated during creep deformation of deep marble.The time-dependent behavior and mechanism of excavation-damaged marble subjected to various pore pressures were analyzed.A new creep model that is applicable to excavation-damaged deep marble was proposed.