Viscoplastic Body Research Articles

Seepage Evolution Model of the Fractured Rock Mass under High Seepage Pressure in Dam Foundation

The seepage of the fractured rock mass in dam foundations involves complex fluid‐structure coupling behavior, due to practical hydrogeological conditions. In this work, the seepage characteristics of the fractured rock mass and their correlations with the structural permeable mediums are experimentally explored to reveal the cracking effect on the hydromechanical properties firstly. Subsequently, the tangential and the compression creep damage constitutive models are, respectively, established by introducing a nonlinear viscoplastic body to improve the Nishihara model. Afterwards, an innovative evolution equation of the permeability coefficient considering the creep damage is proposed. It can indicate the time effect of the porosity, the permeability, and damage variables of the fractured rock mass under the long‐term infiltration action of the hydraulic pressures. Ultimately, the proposed methods are applied to the seepage simulation on the dam foundation of the Longyangxia hydropower station and the significantly increased leakage is in good agreement with the measured values during the storage period. It was further confirmed that the crack expansion and penetration in the rock masses can be constantly intensified by the seepage pressures. The research results can provide a reference for engineering repair and supervision through controlling the permeability performance for long‐term operations.

Features and Constitutive Model of Gypsum’s Uniaxial Creep Damage considering Acidization

Acidic fluids cause rock erosion and further endanger the safety of rock engineering, especially the corrosion of pillars by acidic or weakly acidic groundwater. In this paper, the rock samples in the gypsum mining area were taken as the research object. Uniaxial compression creep tests were carried out under neutral water,pH=6andpH=5hydrochloric acid solutions, respectively. Meanwhile, the specimens before and after saturation were observed by an electron microscope scanner. The results show that (1) the gypsum specimens withpH=5hydrochloric acid were damaged at the first stress level, while that withpH=6andpH=7were destroyed at the second stress level. The failure modes of the three groups were basically the same, with cleavage and end damage of different degrees. The difference is that the failure time of the former is earlier than that of the latter, which indicates that the stronger acidity causes greater corrosion on the creep of the samples. (2) From the perspective of microstructure, the samples saturated in the neutral aqueous solution and dry state are compact and complete in structure, and the whole is relatively homogeneous. However, after saturating in the acid solution, the samples significantly increased dense pores with large size and loose structure. Due to the rapid increase, the surfaces of the samples are almost like “holes”. (3) A new nonlinear creep constitutive model was established by connecting Burgers model with nonlinear viscoplastic body (NVPB) model in series, which can well describe the creep characteristics of gypsum rocks under acid corrosion.

A Nonlinear Creep Damage Model Considering the Effect of Dry-Wet Cycles of Rocks on Reservoir Bank Slopes

Water has a crucial effect on the time-dependent behavior of rocks. The long-term cyclical fluctuations of reservoir water level lead to dry–wet (DW) cycles of rocks on reservoir bank slopes, making this influential factor more complex. To deeply understand the time-dependent behavior of rocks under DW cycles, argillite from the reservoir bank slope of Longtan Hydropower Station was used to perform a series of triaxial creep tests. Subsequently, based on analysis of creep test results after different DW cycles, a damage nonlinear Burgers viscoelastic-plastic (DNBVP) model considering the effect of saturation–dehydration cycles was proposed by introducing a nonlinear viscoplastic body and a damage variable describing DW cycles. Then, the three-dimensional creep equations of the new model were derived and its creep parameters were identified. Comparison between the theoretical curves and the test results shows that the theoretical curves of the DNBVP model were able to describe rock creep tests results after different DW cycles. Furthermore, by comparing classical creep models with the proposed model, it was found that the DNBVP model can accurately reflect the nonlinear characteristics of rocks at the accelerated creep stage. Finally, the sensitivity of the DNBVP model was analyzed and discussed, and three-dimensional central difference expressions necessary for secondary development of the new model were also derived in detail. The proposed new model with secondary development may provide a basis for improving the geotechnical design of reservoir bank slopes and the control of reservoir bank landslides.

Fractional-order visco-elastoplastic constitutive model for rock under cyclic loading

The creep property of rock under cyclic loading is very important in civil engineering. In order to establish a novel constitutive equation for rock under cyclic loading, a fractional-order viscoplastic body under cyclic loading was constructed based on fractional-order viscous element. A fractional-order visco-elastoplastic model (FVEPM) for rock was established by connecting constructed fractional-order viscoplastic body with Burgers model. The model was a Burgers model when the maximum value of cyclic loading was less than the critical strength of rock; otherwise, it was a FVEPM which can be used to reflect the transient, steady-state, and tertiary creep phases of rock. The cyclic loading was decomposed into a static load and a cyclic loading with a zero average stress. According to rheological mechanics theory, the rheology constitutive equation of rock under the static load can be derived. According to viscoelastic mechanics theory, the constitutive equation under cyclic loading with a zero average stress was established by introducing the variation parameters of energy storage and energy dissipation compliance caused by rock damage and fracture. Finally, a new dynamic constitutive equation of rock cyclic loading can be obtained by superimposing the constitutive equation under static load and cyclic loading with a zero average stress. Compared with existing test results of rock under cyclic loading, the proposed constitutive model can be used to describe the creep characteristics of rock under cyclic loading and reflect the presented fluctuation of strain curve of rock under cyclic loading.

Tykhonov well-posedness of a viscoplastic contact problem<sup>†</sup>

We consider an initial and boundary value problem \begin{document}$ {{\mathcal{P}}} $\end{document} which describes the frictionless contact of a viscoplastic body with an obstacle made of a rigid body covered by a layer of elastic material. The process is quasistatic and the time of interest is \begin{document}$ \mathbb{R}_+ = [0,+\infty) $\end{document} . We list the assumptions on the data and derive a variational formulation \begin{document}${{\mathcal{P}}}_V $\end{document} of the problem, in a form of a system coupling an implicit differential equation with a time-dependent variational-hemivariational inequality, which has a unique solution. We introduce the concept of Tykhonov triple \begin{document}$ {{\mathcal{T}}} = (I,\Omega, {{\mathcal{C}}}) $\end{document} where \begin{document}$ I $\end{document} is set of parameters, \begin{document}$ \Omega $\end{document} represents a family of approximating sets and \begin{document}${{\mathcal{C} }} $\end{document} is a set of sequences, then we define the well-posedness of Problem \begin{document}${{\mathcal{P}}}_V $\end{document} with respect to \begin{document}$ {{\mathcal{T}}}$\end{document} . Our main result is Theorem 3.4, which provides sufficient conditions guaranteeing the well-posedness of \begin{document}$ {{\mathcal{P} }}_V $\end{document} with respect to a specific Tykhonov triple. We use this theorem in order to provide the continuous dependence of the solution with respect to the data. Finally, we state and prove additional convergence results which show that the weak solution to problem \begin{document}$ {{\mathcal{P}}} $\end{document} can be approached by the weak solutions of different contact problems. Moreover, we provide the mechanical interpretation of these convergence results.

Overview of the rheological models for rocks

This paper introduces the models of rock rheological element model, including classical models and some new models proposed based on the basic models. The classical models are Maxwell model, Kelvin model, Burgers model, and Xi-yuan model. The improvements in the new models mainly focus on using the nonlinear Newtonian body to replace the linear Newtonian body in viscoplastic body. The establishment method and constitutive equations of the new models are also presented. These new models can represent the properties of the corresponding type of rock and soil, and the feasibility of the models has been verified by experiments and modelling. All these researches provide more models in representing the rock rheology behaviour of soil, which is of vital importance in the study of soil and rock mechanics.

Adhesive contact problem for viscoplastic materials

We examine a mathematical model that describes a quasistatic adhesive contact between a viscoplastic body and deformable foundation. The material’s behaviour is described by the rate-type constitutive law which involves functions with a non-polynomial growth. The contact is modelled by the normal compliance condition with limited penetration and adhesion, a subdifferential friction condition also depending on adhesion, and the evolution of bonding field is governed by an ordinary differential equation. We present the variational formulation of this problem which is a system of an almost history-dependent variational–hemivariational inequality for the displacement field and an ordinary differential equation for the bonding field. The results on existence and uniqueness of solution to an abstract almost history-dependent inclusion and variational–hemivariational inequality in the reflexive Orlicz–Sobolev space are proved and applied to the adhesive contact problem.

A new transversely isotropic nonlinear creep model for layered phyllite and its application

Phyllite, which is a low-grade metamorphic rock with well-developed foliation planes, is encountered frequently during tunnel construction in western China. Its creep behavior is affected significantly by the foliation planes and has a crucial influence on the long-term safety of tunnel structures. Uniaxial compressive creep testing was conducted to analyze the time-dependent features of phyllite obtained from the Zhegu mountain tunnel on the Wenma expressway, China. A new creep model that connects a Maxwell body, a Kelvin body, and a nonlinear visco-plastic body was proposed to describe both the full creep process (including the transient, steady, and accelerated creep stages) and the transversely isotropic characteristics of phyllite. The creep model was also applied to investigate the long-term safety of a cracked tunnel lining in phyllite bedrock. The results showed that the creep strength and corresponding axial strain of phyllite exhibited maximum and minimum values at θ (the angle between the loading direction and the weak planes) = 90° and 30°, respectively. Good agreement was found between the calculated and experimental creep curves, indicating that the creep model replicates the physical creep process of phyllite well. The safety of the cracked lining was affected mainly by the damage degree of cracks and the creep behavior of the surrounding rock. Uncracked sections, because of their greater stiffness, were more sensitive to creep load than cracked ones. The inclination angle of foliation planes influenced the location of unsafe sections (those with a safety factor less than one), and this effect was weakened as the number of pre-existing cracks increased.

Visco-plastic deformation analysis of rock tunnels based on fractional derivatives

The squeezing of rock tunnels in soft strata is often encountered during and after excavation, which is closely related to the viscoplasticity of rock mass. However, previous studies involved multiple creep elements in the calculation of visco-plastic deformation, which cost much time to identify creep parameters. In this paper, a novel four-element fractional visco-plastic (FVP) model, which includes two Abel dashpots, one spring and one visco-plastic body, is established using fractional derivatives. Three-dimensional creep equation of the FVP model is first derived using the von Mises equivalent plastic stress-strain relationship. Then the closed-form solution of visco-plastic deformation of the surrounding rock around circular tunnels is derived by the FVP model combining with the non-associated flow rule and the Hoek-Brown criterion. Next, taking rock tunnels that are excavated in three different quality rock salt as examples, parametric investigations of time-dependent parameters and time-independent parameters are carried out. Visco-plastic deformation curves of rock tunnels, which includes (a) the displacement of tunnel wall versus time; (b) the displacement of visco-plastic zone versus depth, are analyzed. Finally, the auxiliary tunnel of Jinping II hydropower station is taken as a typical example for studying the time-dependent deformation of rock tunnels. The analytical results are in good agreement with the experimental and monitoring data. This paper may blaze a new trail to quickly evaluate the visco-plastic deformation of rock tunnels during the early design process.

Study of the 1‐D consolidation behavior of two‐layered soft soils: Parametric studies using a rheological model with viscoplastic body

SummaryThe governing equations for the coupled processes of consolidation and creep of two‐layered soft soils are established. The Nishihara rheological model is adopted to simulate the elasto‐viscoplastic characteristics of soft soils, disregarding the effects of the soil self‐gravity. A semi‐analytical theory combined with numerical and analytical methods is introduced to solve the governing equations of the one‐dimensional rheological model. The computational procedure and the approximate solutions for two‐layered soft soils subjected to surface loading are obtained for two drainage conditions. The solutions and the computational procedure are used to study the effects of the two layers and constitutive parameters on rheological consolidation behavior of soft soils. It can be concluded that two layers affect the rate of excess pore water pressure dissipation and settlement development. The parametric studies show that when the parameters of the upper layer remain constant, increases in the permeability and elastic modulus in the lower layer accelerate the dissipation of the excess pore water pressure, and meanwhile increases in the viscosity coefficient and viscoplastic limit slows down the dissipation of the excess water pressure.

Variational–hemivariational approach to quasistatic viscoplastic contact problem with normal compliance, unilateral constraint, memory term, friction and damage

The aim of this paper is to study a mathematical model which describes the quasistatic frictional contact between a viscoplastic body and a foundation. The material’s behavior is modeled with a rate-type constitutive law with internal state variable. The contact is modeled with normal compliance, unilateral constraint and memory term, friction and damage. We present the classical formulation of the problem and we derive its variational–hemivariational formulation. Finally, we prove its unique weak solvability. The proof is based on abstract results for a class of history-dependent variational–hemivariational inequalities.

Uncoupling the widespread occurrence of anti-NMDAR1 autoantibodies from neuropsychiatric disease in a novel autoimmune model

The aim of this paper is to study a mathematical model which describes the quasistatic frictional contact between a viscoplastic body and a foundation. The material’s behavior is modeled with a rate-type constitutive law with internal state variable. The contact is modeled with normal compliance, unilateral constraint and memory term, friction and damage. We present the classical formulation of the problem and we derive its variational–hemivariational formulation. Finally, we prove its unique weak solvability. The proof is based on abstract results for a class of history-dependent variational–hemivariational inequalities.

The substantiation of the elastic-viscoplastic model of the human spine for modeling the correction process of kyphoscoliotic deformation.

Purpose:The relevance of the problem is caused by an increase in the number of spine-related diseases among children, including scoliosis. Currently, there are no methodologies for the treatment of scoliosis, which ensure an unambiguous positive result. The purpose of the article is to justify the spinal model as an elastic viscoplastic body for further mathematical modeling of the process of spine correction and search for its optimal conditions.Methodology:The leading approach to the study of this problem is the development of techniques for the surgical treatment of deformities of the vertebral column with the aid of an external fixation device for the spine, providing for a rigid connection of the elements of the apparatus with each other and with the spine. The rigid connection between the elements of the external fixation device increases the degree of static indeterminacy of the design, which leads to the occurrence of additional dangerous stresses in the details of the apparatus and in the vertebrae. The control actions in such devices do not provide an adequate result for the process of correction of the vertebral column.Results:The main result is the substantiation of the spine model as an elastic viscoplastic body. This will allow more detailed consideration of the medical and biological features of the spine and the physical and mechanical properties of human bone and soft tissues. The proposed model will allow developing an adaptive design of the device, taking into account specific features of the organism and more effectively managing the correction process.Value:The materials of the article can be useful for scientists, doctors and specialists in conducting scientific research on the problem of spine deformation correction and the development of appropriate technical means.

A Mixed Variational Formulation of a Contact Problem with Wear

We consider a mathematical model which describes the sliding frictional contact between a viscoplastic body and an obstacle, the so-called foundation. The process is quasistatic, the material’s behavior is described with a viscoplastic constitutive law with internal state variable and the contact is modelled with normal compliance and unilateral constraint. The wear of the contact surfaces is taken into account, and is modelled with a version of Archard’s law. We derive a mixed variational formulation of the problem which involve implicit history-dependent operators. Then, we prove the unique weak solvability of the contact model. The proof is based on a fixed point argument proved in Sofonea et al. (Commun. Pure Appl. Anal. 7:645–658, 2008), combined with a recent abstract existence and uniqueness result for mixed variational problems, obtained in Sofonea and Matei (J. Glob. Optim. 61:591–614, 2014).

A quasistatic viscoplastic contact problem with normal compliance, unilateral constraint, memory term and friction

The goal of this paper is to deal with a mathematical model which describes the quasistatic frictional contact between a viscoplastic body and a foundation. The contact is modeled with normal compliance, unilateral constraint and memory term. We present the classical formulation of the problem together with the list of assumptions on the data. Then we derive the variational–hemivariational formulation of the model and we prove its unique weak solvability. The proof is based on a recent abstract result of a class of history-dependent variational–hemivariational inequalities.

Experimental study on time-dependent behavior of silty mudstone from the Three Gorges Reservoir Area, China

Short-term and creep tests were performed on saturated silty mudstone obtained from the Three Gorges Reservoir area. Based on the triaxial creep test results, the variations of creep strain and creep rate of silty mudstone with time were analyzed in detail, and the differences between the axial and lateral creep parameters were quantitatively studied at different deviatoric stress levels. The longterm strength of silty mudstone was obtained. The results demonstrate that the lateral creep characteristics are more significant than the axial creep characteristics for silty mudstone in the tests. Analysis of the lateral creep characteristics of the rock is a reasonable approach to determine whether creep failure occurs in the silty mudstone. The long-term strength of silty mudstone decreases remarkably, and it is actually the lateral long-term strength of the rock. According to the triaxial creep curves of silty mudstone, a new nonlinear viscoelasto-plastic creep model was proposed by connecting in series the nonlinear viscoplastic body with the sixcomponent model. Comparing the model curves with test results, this model was determined to reflect the creep properties of silty mudstone accurately. Finally, the selection of the creep model parameters was achieved by considering the axial and lateral creep characteristics of silty mudstone.

A mixed variational formulation of a contact problem with adhesion

A mathematical model describing the contact between a viscoplastic body and a deformable foundation is analyzed under small deformation hypotheses. The process is quasistatic and in normal direction the contact is with adhesion, normal compliance, memory effects and unilateral constraint. We derive a mixed-variational formulation of the problem using Lagrange multipliers. Finally, we prove the unique weak solvability of the contact problem.

A mixed variational problem with applications in contact mechanics

We provide an existence result in the study of an abstract mixed variational problem which involves a history-dependent operator and two nondifferentiable functionals depending on the solution. The proof relies on generalized saddle point theory and a fixed point argument. Then we consider a new mathematical model which describes the quasistatic frictionless contact process between a viscoplastic body and an obstacle. The contact is modeled with a multivalued normal compliance condition and unilateral constraint. We use our abstract result to prove the weak solvability of this contact problem.

Numerical solution of a contact problem with unilateral constraint and history-dependent penetration

A numerical method is presented for a mathematical model which describes the frictionless contact between a viscoplastic body and an obstacle, the so-called foundation. The process is quasistatic, and the contact is modeled with normal compliance and unilateral constraint, in such a way that the stiffness coefficient depends on the history of the penetration. A solution algorithm is discussed and implemented. Numerical simulation results are reported, illustrating the mechanical behavior related to the contact condition.

On the behavior of the solution of a viscoplastic contact problem

We consider a mathematical model which describes the frictionless contact between a viscoplastic body and an obstacle, the so-called foundation. The process is quasistatic and the contact is modeled with normal compliance and unilateral constraint. We provide a mixed variational formulation of the model which involves a dual Lagrange multiplier, and then we prove its unique weak solvability. We also prove an estimate which allows us to deduce the continuous dependence of the weak solution with respect to both the normal compliance function and the penetration bound. Finally, we provide a numerical validation of this convergence result.