On the simulation of multidimensional cyclic loading with intergranular strain

A sample of soil is subjected to multidimensional cyclic loading when two or three principal components of the stress or strain tensor are simultaneously controlled to perform a repetitive path. These paths are very useful to evaluate the performance of models simulating cyclic loading. In this article, an extension of an existing constitutive model is proposed to capture the behavior of the soil under this type of loading. The reference model is based on the intergranular strain anisotropy concept and therefore incorporates an elastic locus in terms of a strain amplitude. In order to evaluate the model performance, a modified triaxial apparatus able to perform multidimensional cyclic loading has been used to conduct some experiments with a fine sand. Simulations of the extended model with multidimensional loading paths are carefully analyzed. Considering that many cycles are simulated (\(N>30\)), some additional simulations have been performed to quantify and analyze the artificial accumulation generated by the (hypo-)elastic component of the model. At the end, a simple boundary value problem with a cyclic loading as boundary condition is simulated to analyze the model response.     

基于CLoE亚塑性模型的密砂循环荷载下力学行为模拟

亚塑性模型为模拟颗粒材料的非线性力学行为提供了一条新途径,特别是CLoE亚塑性模型在模拟应变局部化时具有一定的优势。然而该模型在模拟小幅应力-应变循环时表现出一定的锯齿效应。为了克服该效应,基于颗粒间应变张量的概念发展了修正的CLoE亚塑性模型以正确模拟循环荷载下密砂的力学行为。此外,为保证单调荷载作用下修正模型与原模型预测结果的一致性,改进了颗粒间应变率及颗粒间最大应变的定义。数值算例表明:（1）修正模型保留了克服锯齿效应的优点。（2）修正模型能够反映不同振幅条件下的卸载刚度。（3）在大振幅循环条件下,滞回圈的面积随着循环次数增加而增大。（4）修正模型能够保证单调加载条件下所得结果与原模型的一致性。（5）修正模型可以反映材料的疲劳破坏机制。     

A two-surface thermomechanical plasticity model considering thermal cyclic behavior

In thermal-related engineering such as thermal energy structures and nuclear waste disposal, it is essential to well understand volume change and excess pore water pressure buildup of soils under thermal cycles. However, most existing thermo-mechanical models can merely simulate one heating–cooling cycle and fail in capturing accumulation phenomenon due to multiple thermal cycles. In this study, a two-surface elasto-plastic model considering thermal cyclic behavior is proposed. This model is based on the bounding surface plasticity and progressive plasticity by introducing two yield surfaces and two loading yield limits. A dependency law is proposed by linking two loading yield limits with a thermal accumulation parameter n_c, allowing the thermal cyclic behavior to be taken into account. Parameter n_c controls the evolution rate of the inner loading yield limit approaching the loading yield limit following a thermal loading path. By extending the thermo-hydro-mechanical equations into the elastic–plastic state, the excess pore water pressure buildup of soil due to thermal cycles is also accounted. Then, thermal cycle tests on four fine-grained soils (natural Boom clay, Geneva clay, Bonny silt, and reconstituted Pontida clay) under different OCRs and stresses are simulated and compared. The results show that the proposed model can well describe both strain accumulation phenomenon and excess pore water pressure buildup of fine-grained soils under the effect of thermal cycles.

     

The effects of unloading on drained cyclic behaviour of Sydney sand

New excavation or tunnelling affects the stress state of soils in ground. The change of stress state due to excavation may affect the cyclic behaviour of soils. Cyclic loading, such as traffic and earthquake loading, induced ground deformation may be greater than expected if such effect is not considered. A series of cyclic triaxial tests were performed on Sydney sand with different relative densities. The effect of unloading sequence on deformation of the sand under cyclic loading was simulated by reducing lateral stress in steps between loading cycles. The dependence of strain accumulation on the magnitude of confining pressure reduction and on unloading stress paths was studied. The results indicate that the sand has a memory of stress history and the stress history of such unloading enlarges the strain accumulation during the subsequent cycles, and the greater the reduction of lateral stress, the greater the accumulated strain. Under cyclic loading, the accumulated axial strain could increase nonlinearly or linearly with the ratio of unloading magnitude to initial mean effective stress, depending on the stress state before cyclic loading. The unloading stress paths have limited effects on the final accumulated strain if the initial and final stress states are the same. The variation of strain accumulation direction attributes to the change of average stress ratio resulting from lateral stress reduction, but hardly depends on relative density and unloading stress paths. The strain accumulation direction after unloading roughly agrees with the modified Cam Clay flow rule.

     

An ISA-plasticity-based model for viscous and non-viscous clays

The ISA-plasticity is a mathematical platform which allows to propose constitutive models for soils under a wide range of strain amplitudes. This formulation is based on a state variable, called the intergranular strain, which is related to the strain recent history. The location of the intergranular strain can be related to the strain amplitude, information which is used to improve the model for the simulation of cyclic loading. The present work proposes an ISA-plasticity-based model for the simulation of saturated clays and features the incorporation of a viscous strain rate to enable the simulation of the strain rate dependency. The work explains some aspects of the ISA-plasticity and adapts its formulation for clays. At the beginning, the formulation of the model is explained. Subsequently, some comments about its numerical implementation and parameters determination are given. Finally, some simulations are performed to evaluate the model performance with two different clays, namely a Kaolin clay and the Lower Rhine clay. The simulations include monotonic and cyclic tests under oedometric and triaxial conditions. Some of these experiments include the variation of the strain rate to evaluate the viscous component of the proposed model.     

Fractional order model for granular soils under drained cyclic loading

The theory of fractional calculus has been successfully applied to model the triaxial behaviour of soils under static loading conditions. However, limited work has been carried out in using the fractional calculus to describe the cyclic behaviour of granular soils. In this paper, a fractional order constitutive model for granular soils under drained cyclic loading is proposed by incorporating the concept of fractional rate for strain accumulation. The fractional rate for strain accumulation is obtained from the analysis of the experimental data by utilizing the fractional calculus. Comparison between the test results and model predictions is presented. The key feature of the proposed model is that it can reasonably characterise the cyclic deformation of granular soils under both low and high loading cycles. Copyright © 2016 John Wiley & Sons, Ltd.     

低幅值高循环荷载作用下土体的应变累积模型

高速铁路路基上的轨道以及附近区域的结构物承受低幅值、高循环振动荷载的反复作用。在此低幅值、高循环荷载作用下土体会产生不可恢复的应变累积,导致轨道及附近区域结构物发生附加沉降。当前,描述土体的循环变形特征的理论分为两类：一类是基于经典塑性理论的应力-应变滞回模型（例如边界面模型）,另一类是基于循环三轴试验经验规律的应变累积模型（例如Bochum累积模型）。为了能够预测土体在低幅值、高循环荷载作用下的应变累积行为,在前人对土体在低幅值、高循环荷载作用下大量试验研究的基础上,在经典弹塑性理论的框架下,提出一个土体在低幅值、高循环荷载作用下的应变累积模型。该模型通过用对数律来描述塑性体应变的累积规律,并以此作为应变累积的大小度量,然后通过修正Cam-clay模型的流动准则来描述应变累积的发展方向。最后,通过多组试验结果的模拟,表明所提出的应变累积模型能够较好地预测土体在低幅值、高循环荷载作用下的应变累积行为,具有广泛的应用前景。     

A J2‐plasticity model based on bounding surface concept

This article presents the development of a J₂ small strain plasticity model based on bounding surface concept, along with numerical examples to demonstrate model behaviors and identification of model parameter using laboratory test data. The model is motivated by the need for simulating permanent deformation accumulation of asphalt concrete mixtures, which leads to rutting in flexible pavements under repeated traffic loading. The proposed model accounts for the observation that rutting is mostly caused by shearing and takes advantage of the fact that bounding surface concept allows for the progressive accumulation of plastic deformation under constant amplitude loading condition. Analytical solutions are given for typical laboratory testing conditions. The model can be calibrated using repeated simple shear test data that are typically available for asphalt concrete mixtures. It is shown that the model is easy to use and provides a promising alternative for modeling permanent deformation accumulation in materials subjected to repetitive (cyclic) loading. Copyright © 2012 John Wiley & Sons, Ltd.     

Experimental evaluation of sandstone under cyclic coupled compression-shear loading: fatigue mechanical response and failure behavior

In this study, the inclined sandstone specimen is introduced into cyclic loading tests to fulfill the coupled compression-shear loading state. 21 cyclic coupled compression-shear loading tests are conducted on inclined specimens under different loading conditions, including maximum stress levels ranging in 0.80, 0.85, 0.90 and 0.95, and amplitude levels varying in 0.40, 0.50, 0.60 and 0.70. Our testing results systematically revealed the influence of cyclic loading parameters on fatigue mechanical response of rocks under coupled compression-shear loading, regarding the deformation characteristics, energy evolution, damage variable and failure mechanism. Under higher maximum stress or cyclic amplitude, inclined rocks are characterized by larger elastic modulus and higher dissipated energy, resulting in less irreversible strain, faster damage accumulation and shorter fatigue life. Furthermore, a fatigue life prediction method is proposed based on the energy dissipation, and its reliability is verified by comparing with experimental results. In addition, the progressive cracking behavior of rocks is analyzed during the fatigue tests by virtue of digital image correlation technique. Under cyclic coupled compression-shear loading, rock specimens are featured by a prominent shear-dominated failure along its short diagonal direction combined with local tensile damage along the loading orientation.

     

An experimental database for the development,calibration and verification of constitutive models for sand with focus to cyclic loading: part II—tests with strain cycles and combined loading

For numerical studies of geotechnical structures under earthquake loading, aiming to examine a possible failure due to liquefaction, using a sophisticated constitutive model for the soil is indispensable. Such model must adequately describe the material response to a cyclic loading under constant volume (undrained) conditions, amongst others the relaxation of effective stress (pore pressure accumulation) or the effective stress loops repeatedly passed through after a sufficiently large number of cycles (cyclic mobility, stress attractors). The soil behaviour under undrained cyclic loading is manifold, depending on the initial conditions (e.g. density, fabric, effective mean pressure, stress ratio) and the load characteristics (e.g. amplitude of the cycles, application of stress or strain cycles). In order to develop, calibrate and verify a constitutive model with focus to undrained cyclic loading, the data from high-quality laboratory tests comprising a variety of initial conditions and load characteristics are necessary. It is the purpose of these two companion papers to provide such database collected for a fine sand. Part II concentrates on the undrained triaxial tests with strain cycles, where a large range of strain amplitudes has been studied. Furthermore, oedometric and isotropic compression tests as well as drained triaxial tests with un- and reloading cycles are discussed. A combined monotonic and cyclic loading has been also studied in undrained triaxial tests. All test data presented herein will be available from the homepage of the first author. As an example of the examination of an existing constitutive model, the experimental data are compared to element test simulations using hypoplasticity with intergranular strain.     

Anisotropic visco-hypoplasticity

Apart from time-driven creep or relaxation, most viscoplastic models (without plastic and viscous strain separation) generate no or a very limited accumulation of strain or stress due to cyclic loading. Such pseudo-relaxation (or pseudo-creep) is either absent or dwindles too fast with increasing OCR. For example, the accumulation of the pore water pressure and eventual liquefaction due to cyclic loading cannot be adequately reproduced. The proposed combination of a viscous model and a hypoplastic model can circumvent this problem. The novel visco-hypoplasticity model presented in the paper is based on an anisotropic preconsolidation surface. It can distinguish between the undrained strength upon triaxial vertical loading and horizontal loading. The strain-induced anisotropy is described using a second-order structure tensor. The implicit time integration with the consistent Jacobian matrix is presented. For the tensorial manipulation including numerous Fréchet derivatives, a special package has been developed within the algebra program MATHEMATICA (registered trade mark of Wolfram Research Inc.). The results can be conveniently coded using a special FORTRAN 90 module for tensorial operations. Simulations of element tests from biaxial apparatus and FE calculations are also shown.     

基于分数阶微积分的粗粒料静动力边界面本构模型

分数阶微分理论在土体静力黏弹性本构模型中得到了广泛应用,然而,其在动力弹塑性模型中的应用尚不多见。为此,基于分数阶微积分理论分析了粗粒料在循环荷载下的变形特性,提出了粗粒料在循环荷载下的分数阶应变率;并以此为基础,进一步建立了粗粒料受静动力荷载作用下的边界面塑性力学本构模型。所提出模型包含10个参数,均可以运用常规三轴试验获得。为了验证所提出模型,选取了几种已有不同文献中的不同粗粒料试验数据进行了模拟,发现,所提出的模型可以较好地模拟粗粒料在静动力加载下的应力-应变行为,对于循环荷载下的长期变形也能较好地预测。     

An experimental database for the development,calibration and verification of constitutive models for sand with focus to cyclic loading: part I—tests with monotonic loading and stress cycles

For numerical studies of geotechnical structures under earthquake loading, aiming to examine a possible failure due to liquefaction, using a sophisticated constitutive model for the soil is indispensable. Such a model must adequately describe the material response to a cyclic loading under constant volume (undrained) conditions, amongst others the relaxation of effective stress (pore pressure accumulation) or the effective stress loops repeatedly passed through after a sufficiently large number of cycles (cyclic mobility, stress attractors). The soil behaviour under undrained cyclic loading is manifold, depending on the initial conditions (e.g. density, fabric, effective mean pressure, stress ratio) and the load characteristics (e.g. amplitude of the cycles, application of stress or strain cycles). In order to develop, calibrate and verify a constitutive model with focus to undrained cyclic loading, the data from high-quality laboratory tests comprising a variety of initial conditions and load characteristics are necessary. The purpose of these two companion papers was to provide such database collected for a fine sand. The database consists of numerous undrained cyclic triaxial tests with stress or strain cycles applied to samples consolidated isotropically or anisotropically. Monotonic triaxial tests with drained or undrained conditions have also been performed. Furthermore, drained triaxial, oedometric or isotropic compression tests with several un- and reloading cycles are presented. Part I concentrates on the triaxial tests with monotonic loading or stress cycles. All test data presented herein will be available from the homepage of the first author. As an example of the examination of an existing constitutive model, the experimental data are compared to element test simulations using hypoplasticity with intergranular strain.     

Investigating the Effect of Cyclic Loading on the Indirect Tensile Strength of Rocks

This paper presents the results of laboratory experiments during the investigation of the stress–strain characteristics of Brisbane tuff disc specimens under diametral compressive cyclic loading. Two different cyclic loading methods were used: namely, sinusoidal cyclic loading and cyclic loading with increasing mean level. The first method applied the S–N curve approach to the indirect tensile strength (ITS) of rock specimens for the first time in the literature, and the second method investigated the effect of increasing cyclic loading on the ITS of rock specimens. The ITS of Brisbane tuff disc specimens was measured using the Brazilian tensile strength test. The reduction in ITS was found to be 33% with sinusoidal loading tests, whereas increasing cyclic loading caused a maximum reduction of 37%. It is believed that the fracturing under cyclic loading starts at contact points between strong grains and weak matrices, and that contact points at grain boundaries are the regions of stress concentration (i.e., indenters). Transgranular cracks emanate from these regions and intergranular cracks sometimes pass through the contact points. Once cracking begins, there is a steady progression of damage and a general ‘loosening’ of the rock, which is a precursor to the formation of intergranular cracks.     

循环加-卸载岩石本构模型研究

循环加-卸载岩石本构模型是预测压气储能洞室长期稳定性的关键,但目前还没有适用的本构模型,因此,提出了一种能够描述岩石循环加载和卸载的本构模型。鉴于岩石在循环作用下损伤不断累积,将基于Weibull分布的岩石损伤软化模型进行拓展,并用内变量疲劳本构模型描述每个循环的初始模量和卸载模量的变化,进而得到循环加-卸载作用下的岩石本构模型,然后将该模型与现有的试验结果进行对比。该模型物理意义明确,涉及的参数较少,且便于拟合。提出的循环加-卸载下岩石本构模型对试验数据拟合效果较好,能较准确地反映循环荷载上、下限值对应的轴向应变,也能反映出循环内部变形模量衰减的趋势。该模型的成功建立为循环加-卸载下岩石本构模型的研究提供了新思路。     

A two-surface plasticity model for cyclic behavior of saturated clay

This paper presents a two-surface plasticity model for describing some important features of saturated clay under cyclic loading conditions, such as closed hysteresis loops, cyclic shakedown and degradation, and different stress–strain relations for two-way loading. The model, namely ACC-2-C, is based on the elastoplastic model ACC-2 (an adapted Modified Cam Clay model with two yield surfaces) developed by Hong et al. (Acta Geotech 11(4):871–885, 2015). The small-strain nonlinearity concept is adopted to achieve the nonlinear characteristics of clay during unloading–loading stage. The new hardening law related to accumulated deviatoric plastic strain is proposed for the inner surface to describe the cyclic shakedown and degradation. Following the advantages of the ACC-2 model, the constitutive equations are simply formulated based on the consistency condition for the inner yield surface. The model is conveniently implemented in a finite element code using a stress integration scheme similar to the Modified Cam Clay model. The simulation results are highly consistent with experimental data from drained and undrained isotropic cyclic triaxial tests in normally consolidated saturated clay under both one-way and two-way loadings.

     

A hypoplastic macroelement model for shallow foundations under monotonic and cyclic loads

In this work, a macroelement for shallow foundations on sands has been developed based on the theory of hypoplasticity. The incrementally nonlinear constitutive equations of the macromodel are defined in terms of generalized forces and displacements and are constructed based on the general approach proposed by Niemunis (Extended Hypoplastic Models for Soils. Habilitation Thesis, Bochum University, 2002). A suitable vectorial internal variable—mimicking the concept of intergranular strain introduced by Niemunis and Herle (Mech Cohes Frict Mater 2:279–299, 1997) for continuum hypoplasticity—is employed to provide the model sufficient memory of past displacement history to be able to reproduce the observed behavior under cyclic loading paths. As compared to similar macroelements formulated within the framework of the theory of elastoplasticity, the proposed approach has the advantage of a much simpler mathematical structure, which allows a straightforward implementation in existing structural analysis FE codes. The model performance has then been evaluated by comparing the model predictions with available experimental results from a series of small-scale model tests reported by Nova and Montrasio (Géotechnique 41:243–256, 1991). Overall, the model captures reasonably well the observed response under nonproportional, complex loading paths. A series of simulations reproducing the tests performed by di Prisco et al. (Shallow footing under cyclic loading: experimental behaviour and constitutive modelling. Patron, Bologna, 2003) has also demonstrated the potential of the proposed model to simulate the observed behavior of footings under cyclic loading paths, at least from a qualitative point of view.     

Simulation of cyclic strength degradation of natural clays via bounding surface model with hybrid flow rule

Strength loss of natural clays subjected to seismic loading is a critical factor contributing to earthquake‐induced ground failure and associated hazards. This work proposes a bounding surface constitutive law to simulate cyclic strength degradation of natural clays resulting from the loss of structure and attendant accumulation of excess pore pressures. The proposed model employs an enhanced plastic flow rule that can simulate accurately the development of pore pressure and explicitly incorporates soil structure effects. The validation of the model with reference to the experimental evidence available for 3 structured clays shows that with a single set of parameters the proposed model can reasonably represent the mechanical behavior of natural clays under various loading conditions (1D compression, monotonic shearing in compression and extension, cyclic loading, and postcyclic shearing). Particularly, its satisfactory performance in terms of quantification of cyclic strength degradation encourages the use of the model in simulating boundary value problems related to the stability of geotechnical facilities under earthquakes.     

ISA model: A constitutive model for soils with yield surface in the intergranular strain space

In this article, a new constitutive model for soils is proposed. It is formulated by means of plasticity, but in contrast to the precedent works, it presents a yield function describing a surface within the intergranular strain space. This latter is a state variable providing information of the recent strain history. An expression for the plastic strain rate has been proposed to guarantee the stress rate continuity. Under the application of medium or large strain amplitudes, the constitutive equation becomes independent of the intergranular strain and delivers a mathematical structure similar to some Karlsruhe hypoplastic models. Some simulations of monotonic and cyclic triaxial test are provided to evaluate and analyze the model performance. Copyright © 2015 John Wiley & Sons, Ltd.     

A two-mechanism soil–structure interface model for three-dimensional cyclic loading

A three-dimensional (3D) soil–structure interface model is proposed within the two-mechanism constitutive theory and bounding surface theory originally established for soils. The proposed model has two main characteristics: first, the model is formulated based on two different and superposed deformation mechanisms. The first mechanism is due to the stress ratio increment, and the second is due to the normal stress increment. Each mechanism induces a shear strain component and a normal strain component. The proposed model can be reduced to the conventional single-mechanism interface model. Second, the plastic modulus and stress dilatancy are defined using the bounding surface theory. The plastic flow rule under cyclic loading is modified and assumed to be dependent on both the stress state of the mapping point and the stress reversal loading direction. The proposed model was validated against the available 3D interface tests and was found to satisfactorily reflect the salient features of the interfaces under monotonic and cyclic loading paths with different normal boundaries. The problem in which the elastic normal stiffness in conventional single-mechanism interface models is often underestimated to enhance the simulation performance under varying normal stress conditions is solved by incorporating the second mechanism. And the effect of the second mechanism on the modeling behavior is discussed. The modified plastic flow direction accurately simulates the 3D cyclic shear response, and the difference between the model simulation and test result increases with the number of cycles by use of the plastic flow direction defined in conventional bounding surface theory.