Large strain similarity solution for a spherical or circular opening excavated in elastic‐perfectly plastic media

This paper deals with the unloading problem of a spherical or circular opening excavated in elastic‐perfectly plastic media with a nonassociated Mohr–Coulomb yield criterion. A large strain similarity solution, using incremental velocity approach, is presented by replacing partial differential equations from stress equilibrium, constitutive law, consistency condition, and displacement equation with first‐order ordinary differential equations. The classical Runge–Kutta method is used to solve the first‐order ordinary differential equations. Comparisons among small and large strain solutions are made using some data sets of soil and rock. The results show that the displacements by large strain similarity solution are smaller than those by exact small strain solution and somewhat larger than those by large strain solution using total strain approach. Copyright © 2014 John Wiley & Sons, Ltd.     

Inelastic constitutive properties and shear localization in Tennessee marble

The inelastic response of Tennessee marble is modelled by an elastic plastic constitutive relation that includes pressure dependence of yield, strain‐softening and inelastic volume strain (dilatancy). Data from 12 axisymmetric compression tests at confining pressures from 0 to 100 MPa are used to determine the dependence of the yield function and plastic potential, which are different, on the first and second stress invariants and the accumulated inelastic shear strain. Because the data requires that the strain at peak stress depends on the mean stress, the locus of peak stresses is neither a yield surface nor a failure envelope, as is often assumed. Based on the constitutive model and Rudnicki and Rice criterion, localization is not predicted to occur in axisymmetric compression although faulting is observed in the tests. The discrepancy is likely due to the overly stiff response of a smooth yield surface model to abrupt changes in the pattern of straining. The constitutive model determined from the axisymmetric compression data describes well the variation of the in‐plane stress observed in a plane strain experiment. The out‐of‐plane stress is not modelled well, apparently because the inelastic normal strain in this direction is overpredicted. In plane strain, localization is predicted to occur close to peak stress, in good agreement with the experiment. Observation of localization on the rising portion of the stress–strain curve in plane strain does not, however, indicate prepeak localization. Because of the rapid increase of mean stress in plane strain, the stress–strain curve can be rising while the shear stress versus shear strain curve at constant mean stress is falling (negative hardening modulus). Copyright © 2001 John Wiley & Sons, Ltd.     

不同水化状态下的压实膨胀土应力-应变-强度特征

为探讨水化状态对饱和压实膨胀土应力-应变-强度特征的影响,以压实度为95%的荆门弱膨胀土为研究对象,开展了2种典型水化状态下的固结与三轴试验,其中第1种水化状态采用常规饱和方法,第2种水化状态为试样自由膨胀至稳定状态。结果表明：（1）受变形约束条件与渗径的影响,不同水化状态下体膨胀率有较大差别;（2）第2种水化状态下的饱和压实膨胀土具有更大的硬化指数λ与膨胀指数κ、较小的弹性剪切模量,其有效内摩擦角为第1种水化状态下的77.2%,体现出膨胀土饱和强度的变动性;（3）2种水化状态下的固结曲线均呈现出明显的屈服现象,其屈服应力分别为123.2 kPa与94.5 kPa;（4）第1种水化状态下,低围压下试样应变软化与剪胀,高围压下应变强化与剪缩;第2种水化状态下试验围压范围内均发生剪缩和轻微的应变软化;（5）2种水化状态下试样在固结与剪切过程中均表现出超固结性,这种超固结性并非完全由先期固结压力所致,而是试样受荷过程中膨胀受到约束造成的;（6）不同水化饱和状态下初始孔隙比不同,膨胀势也不同,膨胀势与外部约束条件、排水条件、应力状态相互作用,造成其应力-应变-强度特征的差异性。     

The second order characteristic condition for plastic flow

In this paper the second order characteristic (discontinuous bifurcation) condition is derived for the granular flow (fully plastic) equations. This second order bifurcation equation is shown to be formally identical to the first order localization requirement during steady elastoplastic deformation provided the elastic compliance tensor is substituted for the product of the plastic multiplier with the flow Hessian. For isotropic yield and flow functions the invariant form of the characteristic condition is given in detail, as well as an alternative expression in adapted co‐ordinates. The characteristic condition can be regarded as defining a hardening function which is maximized to identify the critical angles. When the method is applied to 3D Coulomb flow, Mohr's 3D fracture plane conditions are obtained uniquely. Copyright © 2003 John Wiley & Sons, Ltd.     

Parameter identification method for determination of elastic modulus of rock based on adjoint equation and blasting wave measurements

The purpose of this paper is to present a parameter identification method to determine the force of a blast and the elastic modulus of the ground using the measurements of a dynamic elastic wave, the adjoint equation method of optimal control theory, and the finite element method. Before the excavation of rocky ground, it is important to estimate the ground properties. In this paper, the elastic modulus is determined as the performance function is minimized using a technique based on the first‐order adjoint method. The performance function is a square sum of the discrepancies between the computed and the observed values of the velocities. After the determination of the magnitude of the blasting force, we can determine the elastic modulus of the rock. As the basic equation to calculate the velocities of dynamic elastic body, elastic equilibrium equations with linear viscosity are employed. The adjoint equation method has been utilized in order to calculate the gradient of the performance function with respect to the parameters. The gradient of the performance function is calculated using the first‐order adjoint equation. The weighted gradient method is applied for minimization. In order to solve the state equations in space and time, the finite element method and the Newmark $\frac{1}{4}$ method are used. In this paper, we tested the practical application of our proposed method for determination of the elastic modulus of rock at the Ikawa tunnel located in the Tokushima prefecture, Japan. Copyright © 2009 John Wiley & Sons, Ltd.     

The extrapolation of elastic moduli to high pressure and temperature

The elastic constants of a crystal under stress, defined as the second derivative of the crystal free energy with respect to strain, require a correction related to the static pressure at non-zero pressures. The corrections required for the elastic constants calculated by the free energy minimisation code PARAPOCS are described and tested by comparison with the elastic constants calculated numerically by applying small stresses in the appropriate orientations to simulated crystals of fluorite, forsterite, α-quartz and albite. The corrected elastic constants are then used to investigate the extrapolation of the bulk and shear moduli (and hence also the seismic wave velocities V _p and V _s) of β-spinel and forsterite to upper mantle pressures. A Murnaghan equation, thirdorder Eulerian finite strain equation, second order polynomial equation and a logistic equation were all fitted to the simulated bulk and shear moduli between 0 and 3 GPa pressure. The parameters derived for these equations are used to extrapolate the bulk and shear moduli to 14 GPa and the results are compared to the simulated high pressure moduli. Over this pressure range, the second order polynomial provides the best extrapolation of the bulk modulus, but the use of the logistic equation results in the best extrapolation of the shear modulus.     

Cavity expansion analyses of crushable granular materials with state‐dependent dilatancy

Crushability is one of the important behaviors of granular materials particularly under high stress states, and affects both the deformability and strength of the materials that are in essence associated with state‐dependent dilatancy. In this presentation, first, a new critical state model is proposed to take into account the three different modes of compressive deformation of crushable granular materials, i.e. particle rearrangement, particle crushing and pseudo‐elastic deformation. Second, the governing equations for cavity expansion in crushable granulates are introduced, in which the state‐dependent dilatancy as well as the bounding surface plasticity model are used. Then, the procedure to obtain semi‐analytical solutions to cavity expansion in the material is described in detail, in which a commercial differential equation solver is employed. Finally, cavity expansion analyses are carried out on Toyoura sand, a well‐documented granular material, to demonstrate the effects of crushability and state‐dependent dilatancy. The study shows that particle crushing does occur at both high stress and critical states and affects the stress fields and the deformation behavior of the material surrounding the cavity in association with state‐dependent dilatancy. This leads to conclusion that particle crushing and state‐dependent dilatancy have to be taken into account when cavity expansion theory is used to interpret cone penetration tests and pressuremeter tests. Copyright © 2011 John Wiley & Sons, Ltd.     

A two‐surface thermomechanical model for saturated clays

This paper presents an advanced thermomechanical model – TEAM in the framework of two‐surface plasticity for saturated clays, with emphasis put on some important thermomechanical features of natural clays evidenced experimentally such as the limited thermomechanical elastic zone, the smooth transition from elastic to plastic behavior. Two plastic mechanisms are introduced in the model: one is to reproduce the thermoplasticity involving thermal expansion and contraction observed at high over‐consolidation ratios and the second one describes the temperature effect on the yield behavior. The model adopts additional yield surfaces, namely inner yield surfaces that are associated with the two proposed plastic mechanisms to account for the plastic behavior inside the existing conventional thermomechanical yield surface namely yield surfaces. The general expressions of the yield surfaces and plastic potentials in p′–q–T space are introduced. A progressive plastic hardening mechanism associated with the inner yield surface is defined, enabling the plastic modulus to vary smoothly during thermomechanical loadings inside the yield surfaces. Several tests on natural Boom clay along different thermomechanical loading paths have been simulated by TEAM, and results show its relevance in describing the thermomechanical behavior of saturated clays. Copyright © 2015 John Wiley & Sons, Ltd.     

Damage–viscoplastic consistency model with a parabolic cap for rocks with brittle and ductile behavior under low‐velocity impact loading

This paper presents a damage–viscoplastic cap model for rocks with brittle and ductile behavior under low‐velocity impact loading, which occurs, e.g. in percussive drilling. The model is based on a combination of the recent viscoplastic consistency model by Wang and the isotropic damage concept. This approach does not suffer from ill posedness—caused by strain softening—of the underlying boundary/initial‐value problem since viscoplasticity provides a regularization under dynamic loading by introducing an internal length scale. The model uses the Drucker–Prager (DP) yield function with the modified Rankine criterion as a tension cut‐off and a parabolic cap surface as a compression cut‐off. The parabolic cap is smoothly fitted to the DP cone. The strain softening law in compression is calibrated with the degradation index concept of Fang and Harrison. Thereby, the model is able to capture the brittle‐to‐ductile transition and hardening behavior of geomaterials under highly confined compression, which is the prevailing stress state under a bit‐button in percussive drilling. Rock strength heterogeneity is characterized statistically at the structural level using the Weibull distribution. An explicit time integrator is chosen for solving the FE‐discretized equations of motion. The contact constraints due to the impact of an indenter are imposed with the forward increment Lagrange multiplier method that is compatible with explicit time integrators. The model is tested at the material point level with various uniaxial and triaxial tests. At the structural level confined compression, uniaxial tension tests and a rock sample under low‐velocity impact are simulated. Copyright © 2009 John Wiley & Sons, Ltd.     

基于内变量和张量函数表示定理的本构方程

针对各向同性材料,基于张量函数表示定理,建立了本构关系的张量不变性表示,其中,3个不可约基张量取决于应力的0～2次幂,且相互正交,3个系数由塑性应变增量和应力的不变量表示。基于塑性应变增量的不变量定义内变量,本构关系归结为确定内变量的演化。使用张量函数表示定理,给出了内变量演化方程的一般表达式,它取决于应力不变量的增量,因而与主轴旋转无关。讨论了如何根据试验资料和引入适当的假定,确定具体的演化方程。通过与塑性势理论和多重屈服面理论进行比较,表明所建模型是这些理论的最一般表示,且简捷直观、使用方便。     

Nonstationary flow surface theory for modeling the viscoplastic behaviors of soils

This paper presents a three-dimensional elastic viscoplastic model that can describe the time-dependent behaviors of soft clays. The constitutive model is formulated based on the nonstationary flow surface theory and incorporates new developments, including (i) an improved definition of the nonstationary flow surface that is capable of capturing the stress–strain behaviors under different loading paths, (ii) a unique stress–strain—viscoplastic-strain-rate equation that is able to explicitly describe the nonstationary flow surface, and (iii) a final stable state concept that identifies the final equilibrium state at the end of creep and stress relaxation, which is also used to simplify the loading criteria. The consistency condition is validated for the proposed model, and the viscoplastic multipliers are calculated by solving the consistency equations. The model performance is investigated and validated via simulation of both oedometer and triaxial tests. The numerical results demonstrate that the proposed model is able to reproduce the main viscoplastic behaviors of soils, including creep, undrained creep rupture, stress relaxation, rate effect and accumulated effect.     

A three‐phase soil model for simulating stress wave propagation due to blast loading

A three‐phase soil model is proposed to simulate stress wave propagation in soil mass to blast loading. The soil is modelled as a three‐phase mass that includes the solid particles, water and air. It is considered as a structure that the solid particles form a skeleton and their voids are filled with water and air. The equation of state (EOS) of the soil is derived. The elastic–plastic theory is adopted to model the constitutive relation of the soil skeleton. The damage of the soil skeleton is also modelled. The Drucker–Prager strength model including the strain rate effect is used to describe the strength of the soil skeleton. The model is implemented into a hydrocode Autodyn. The recorded results obtained by explosion tests in soil are used to validate the proposed model. Copyright © 2004 John Wiley & Sons, Ltd.     

Determination of magnetic anisotropy and a ca 1.2 Ga palaeomagnetic pole from the Bremer Bay area,Albany Mobile Belt,Western Australia

Granulite facies tonalitic gneiss, mafic granulite and late metadolerite dykes from Bremer Bay in the Mesoproterozoic Albany Mobile Belt yield palaeomagnetic remanence that were acquired between ca 1.2 Ga and 1.1 Ga. A well‐constrained pole (66.6°N, 303.7°E) fits the ca 1.2 Ga part of the Precambrian Australian apparent polar wander path. This pole is in agreement with the high‐latitude position of Australia at ca 1.2–1.1 Ga shown on some Rodinia reconstructions. More data are required before any significance can be attributed to a second, poorly defined pole (41.8°S, 243.7°E) that falls at some distance from the ca 0.8 Ga part of the Australian apparent polar wander path. Magnetic anisotropy measurements from all samples except late granite dykes indicate northeast‐southwest elongation (i.e. parallel to the local trend of the orogenic belt) and northwest‐southeast contraction. This is in agreement with the orientation of principal strain axes deduced from structures formed during late stages of ductile deformation. The mean magnetic fabric lineation (long axis of the strain ellipsoid) is subparallel to a mineral elongation lineation and the axes of late upright to inclined folds. Short axes of the strain ellipsoid determined from magnetic fabric measurements are in a similar orientation to poles to the axial surfaces of these folds and to the associated cleavage. This mean shortening axis bisects late conjugate ductile shear zones that overprint the folds. This study has shown that structurally complex high‐grade gneisses and intrusive rocks with variable timing relationships may yield meaningful palaeomagnetic results for late stages of metamorphism. Magnetic anisotropy analysis is also seen to be a valuable tool in providing principal strain directions for late ductile deformation.     

A two-surface plasticity model for stiff clay

This paper presents a constitutive model for describing some important features of the behavior of natural stiff clay evidenced experimentally such as the limited elastic zone, the presence of strain hardening and softening, and the smooth transition from elastic behavior to a plastic one. The model, namely ACC-2, is an adapted Modified Cam Clay model with two yield surfaces: similarly to bounding surface plasticity theory, an additional yield surface—namely Inner yield surface—was adopted to account for the plastic behavior inside the conventional yield surface. A progressive plastic hardening mechanism was introduced with a combined volumetric-deviatoric hardening law associated with the Inner yield surface, enabling the plastic modulus to vary smoothly during loading paths. The main feature of the proposed model is that its constitutive equations can be simply formulated based on the consistency condition for the Inner yield surface, so that it can be efficiently implemented in a finite element code using a stress integration scheme similar to that of the Modified Cam Clay model. Furthermore, it is proved to be an appropriate model for natural stiff clay: the simulations of a set of tests along different mechanical loading paths on natural Boom Clay show good agreement with the experimental results.     

Displacement ring load Green's functions for saturated porous transversely isotropic tri‐material full‐space

Based on the Biot's poroelastic theory and using scalar potential functions both the ring load and point load displacement Green's functions for a transversely isotropic saturated porous full‐space composed of an upper half‐space, a finite thickness middle layer and a lower half‐space is analytically presented for the first time. It is assumed that each region consists of a different transversely isotropic material. The equations of poroelastodymanics in terms of the solid displacements and the pore fluid pressure are uncoupled with the help of two scalar potential functions, so that the governing equations for the potential functions are either a second order wave equation or a repeated wave‐heat transfer equation of sixth order. With the aid of Fourier expansion with respect to circumferential direction and Hankel integral transforms with respect to the radial direction in cylindrical coordinate system, the response is determined in the form of line integrals in the real space, followed by theorem of inverse Hankel integral transforms. The solutions degenerate to a single phase elastic material, and the results are compared with previous studies, where an excellent agreement may be observed with the results provided in the literature. Some examples of displacement Green's functions are finally given to illustrate the solution. Copyright © 2016 John Wiley & Sons, Ltd.     

Modelling the flow of self‐compacting concrete

A Lagrangian particle‐based method, smooth particle hydrodynamics (SPH), is used in this paper to model the flow of self‐compacting concretes (SCC) with or without short steel fibres. An incompressible SPH method is presented to simulate the flow of such non‐Newtonian fluids whose behaviour is described by a Bingham‐type model, in which the kink in the shear stress vs shear strain rate diagram is first appropriately smoothed out. The viscosity of the SCC is predicted from the measured viscosity of the paste using micromechanical models in which the second phase aggregates are treated as rigid spheres and the short steel fibres as slender rigid bodies. The basic equations solved in the SPH are the incompressible mass conservation and Navier–Stokes equations. The solution procedure uses prediction–correction fractional steps with the temporal velocity field integrated forward in time without enforcing incompressibility in the prediction step. The resulting temporal velocity field is then implicitly projected on to a divergence‐free space to satisfy incompressibility through a pressure Poisson equation derived from an approximate pressure projection. The results of the numerical simulation are benchmarked against actual slump tests carried out in the laboratory. The numerical results are in excellent agreement with test results, thus demonstrating the capability of SPH and a proper rheological model to predict SCC flow and mould‐filling behaviour. Copyright © 2010 John Wiley & Sons, Ltd.     

Fuzzy parameters analysis of time‐dependent fracture of concrete dam models

In order to apply the mechanical properties (measured on material specimens or laboratory‐sized models) to large structures (such as concrete dams), a non‐linear theory able to predict the size‐scale effect has to be used. One of these theories was first proposed by Hillerborg and co‐workers (fictitious crack model) and is based on the earlier works by Barenblatt and Dugdale for metals (cohesive crack model). It is based on the existence of a fracture process zone (FPZ), where the material undergoes strain softening. The behaviour of the material outside the FPZ is linear elastic. A large number of short‐time laboratory tests were executed, by varying the load, under crack mouth opening displacement control. Since concrete exhibits a time‐dependent behaviour, an interaction between creep and micro‐crack growth occurs in the FPZ. Therefore, different testing conditions can be applied: rupture can be achieved by keeping the load constant before peak value (pre‐peak tests), or after peak value and after an unloading and reloading procedure (post‐peak tests). The crack propagation rate is shown to be small enough to neglect inertial forces and large enough to keep the time‐dependent behaviour of the process zone as dominant compared to the behaviour of the undamaged and viscoelastic zone. Due to the variability in material microstructure from one specimen to another, experimental data show large ranges of scatter. Well established methods in probability theory require sufficient experimental data in order to assume a probability density distribution. The objective of this study is to investigate the ranges of variation of the time response under constant load in simple structural elements associated with pre‐selected variation (fuzziness) in the main material parameters. For situations where the values of the material parameters are of a non‐stochastic nature, the fuzzy set approach to modelling variability has been proposed as a better and more natural approach. Copyright © 2002 John Wiley & Sons, Ltd.     

基于统一状态参数模型的砂中柱孔扩张分析

目前关于不同初始状态砂土砂中柱孔扩张的研究结果还缺乏更深层次的分析,并且由于没有考虑砂土屈服面形状因素的影响,造成许多研究成果难以在不同类型砂土中推广。采用统一状态参数模型（clay and sand model,CASM）和Rowe剪胀方程来描述砂的弹塑性变形特点,结合大变形理论并引入辅助变量,推导了基于拉格朗日描述的弹塑性区内砂土体积和有效应力的一阶偏微分方程组,在此基础上结合弹塑性区的边界条件和柱孔扩张弹性解,建立了饱和砂中的排水柱孔扩张半解析解。结果表明,CASM可以通过改变应力状态参数n和间距比r*的值使砂的屈服面形状发生改变,进而使文中解答能够用于不同类型饱和砂中的排水柱孔扩张计算,其中n、r*值越大,松砂初次屈服时的偏应力和后续砂中的扩孔压力越大,但中密、密实砂土中的情况与松砂完全相反。极限扩孔压力随砂土初始状态参数的减小而增大,相应的砂土体积也从一直剪缩变为先剪胀后剪缩,弹塑性区半径先减小后增大,硬化行为从一直硬化变为先软化后硬化。静止侧压力系数增大时,极限扩孔压力也增大,但对砂的体积变化规律影响不大。本研究可为相关岩土工程问题分析提供可靠理论支持。     

The Study of Propagation of Linear Cracks Intersection Hole

In this paper, the superimposing principle and the theory of Mellin transforms are used to reduce the controlling equations and boundary conditions of linear cracks intersecting hole into dual integral equations in an infinite elastic body. With finite Hilbert transforms, the dual integral equations can be reduced into a Fredhole integral equation of the second kind which is related to S(t2 ). The dynamic stress intensity factor can be expressed with the solution of integral equation and its numerical results in obtained. In the particular case, the numerical results are the same as those in reference[2] . the intensity factor decreases with the propagation of cracks. With experiments, the propagation velocity of these linear cracks are measured.