Fully automatic modelling of cohesive discrete crack

国际著名岩⼟类SCIEI期刊中英⽂简介国际著名岩⼟⼒学、⼯程地质学报（期刊）索引国际著名岩⼟⼒学、⼯程地质学报（期刊）索引(转载)国际著名岩⼟⼒学、⼯程地质学报（期刊）索引1.《Engineering Geology》——An International Journal, Elsevier------------《⼯程地质》——国际学报2.《The Quarterly Journal of Engineering Geology》，U.K.---------------------《⼯程地质季刊学报》3.《News Journal, International Society for Rock Mechanics》-----------《国际岩⽯⼒学学会信息学报》4.《International Journal of Rock Mechanics and Mining Sciences》---------《国际岩⽯⼒学与矿业科学学报》（包括岩⼟⼒学⽂摘）5.《Rock Mechanics and Rock Engineering》----------------------------《岩⽯⼒学与岩⽯⼯程》6.《Felsbau》[G.]---------------------------------《岩⽯⼒学》，奥地利地质⼒学学会(AGG)主办7. Geomechnik and Tunnelbau (G.)——Geomechanics and Tunnelling---------------《地质⼒学与隧道⼯程》——奥地利地质⼒学学会（ACC）主办8.《GEOTECHNIGUE》-------------------------------------《岩⼟⼒学》，英国⼟⽊⼯程师学会ICE主办9.《Journal of Geotechnical & Geoenvironmental Engineering》(formerly Journal of Geotechnical Engineering)-----------岩⼟⼯程与环境岩⼟⼯程学报》，改版前称《岩⼟⼯程学报》，美国⼟⽊⼯程师学会ASCE主办。

simulation modelling practiceSimulation modelling is a crucial tool in the field of science and engineering. It allows us to investigate complex systems and predict their behaviour in response to various inputs and conditions. This article will guide you through the process of simulation modelling, from its basicprinciples to practical applications.1. Introduction to Simulation ModellingSimulation modelling is the process of representing real-world systems using mathematical models. These models allow us to investigate systems that are too complex or expensiveto be fully studied using traditional methods. Simulation models are created using mathematical equations, functions, and algorithms that represent the interactions and relationships between the system's components.2. Building a Basic Simulation ModelTo begin, you will need to identify the key elements that make up your system and define their interactions. Next, you will need to create mathematical equations that represent these interactions. These equations should be as simple as possible while still capturing the essential aspects of the system's behaviour.Once you have your equations, you can use simulation software to create a model. Popular simulation softwareincludes MATLAB, Simulink, and Arena. These software packages allow you to input your equations and see how the system will respond to different inputs and conditions.3. Choosing a Simulation Software PackageWhen choosing a simulation software package, consider your specific needs and resources. Each package has its own strengths and limitations, so it's important to select one that best fits your project. Some packages are more suitable for simulating large-scale systems, while others may bebetter for quickly prototyping small-scale systems.4. Practical Applications of Simulation ModellingSimulation modelling is used in a wide range of fields, including engineering, finance, healthcare, and more. Here are some practical applications:* Engineering: Simulation modelling is commonly used in the automotive, aerospace, and manufacturing industries to design and test systems such as engines, vehicles, and manufacturing processes.* Finance: Simulation modelling is used by financial institutions to assess the impact of market conditions on investment portfolios and interest rates.* Healthcare: Simulation modelling is used to plan and manage healthcare resources, predict disease trends, and evaluate the effectiveness of treatment methods.* Education: Simulation modelling is an excellent toolfor teaching students about complex systems and how they interact with each other. It helps students develop critical thinking skills and problem-solving techniques.5. Case Studies and ExamplesTo illustrate the practical use of simulation modelling, we will take a look at two case studies: an aircraft engine simulation and a healthcare resource management simulation.Aircraft Engine Simulation: In this scenario, a simulation model is used to assess the performance ofdifferent engine designs under various flight conditions. The model helps engineers identify design flaws and improve efficiency.Healthcare Resource Management Simulation: This simulation model helps healthcare providers plan their resources based on anticipated patient demand. The model can also be used to evaluate different treatment methods and identify optimal resource allocation strategies.6. ConclusionSimulation modelling is a powerful tool that allows us to investigate complex systems and make informed decisions about how to best manage them. By following these steps, you can create your own simulation models and apply them to real-world problems. Remember, it's always important to keep anopen mind and be willing to adapt your approach based on the specific needs of your project.。

Geometric ModelingGeometric modeling is a fundamental aspect of computer-aided design (CAD) that involves the creation of digital models of physical objects or structures using mathematical algorithms. These models are used in a variety of industries, including architecture, engineering, and product design, to visualize, simulate, and analyze the behavior of complex systems. In this response, I will discuss the importance of geometric modeling, its various applications, and the challenges associated with its implementation. One of the primary benefits of geometric modeling is that it allows engineers and designers to create accurate representations of physical objects in a virtual environment. This enables them to test and refine their designs before they are built, reducing the risk of costly errors and improving the overall quality of the final product. For example, in the automotive industry, geometric modeling is used to design and simulate the behavior of car components such as engines, transmissions, and suspension systems, allowing engineers to optimize their performance and efficiency. Another important application of geometric modeling is in the field of architecture, where it is used to create digital models of buildings and other structures. These models can be used to visualize the appearance of the building, test itsstructural integrity, and analyze its energy efficiency. In addition, geometric modeling can be used to create 3D models of entire cities, allowing urban planners to simulate the impact of new developments on the surrounding environment and infrastructure. Despite the many benefits of geometric modeling, there are also several challenges associated with its implementation. One of the most significant challenges is the complexity of the mathematical algorithms used to create the models. These algorithms must be able to accurately represent the physical properties of the object being modeled, including its shape, size, and material properties. This requires a deep understanding of mathematics and physics, as well as advanced programming skills. Another challenge associated with geometric modeling is the need for high-performance computing resources. Creating and manipulating complex 3D models requires a significant amount of computational power, and many modeling applications require specialized hardware such as graphics processing units (GPUs) to achieve the necessary performance. This can bea significant barrier to entry for smaller companies or individuals who do not have access to these resources. In addition to these technical challenges, there are also ethical considerations associated with the use of geometric modeling. For example, the use of 3D modeling in the fashion industry has been criticized for promoting unrealistic body standards and perpetuating harmful stereotypes. Similarly, the use of geometric modeling in the military and defense industries raises questions about the ethics of developing advanced weapons systems and the potential consequences of their use. In conclusion, geometric modeling is a powerful tool that has revolutionized the way we design and build complex systems. Its applications are wide-ranging, from automotive engineering to architecture to urban planning. However, its implementation is not without its challenges, including the complexity of the mathematical algorithms, the need for high-performance computing resources, and ethical considerations. As we continue to develop new applications for geometric modeling, it is important to consider these challenges and work towards solutions that ensure the technology is used in a responsible and ethical manner.。

基于粘结单元的空心钢管桩沉桩过程数值模拟◎ 张琦 桂劲松 大连海洋大学通讯作者：桂劲松摘 要：随着技术的发展，钢管桩在大型海洋工程结构中的应用越来越广泛。

利用ABAQUS 有限元软件模拟实际施工过程中空心管桩与土壤之间的相互作用，并利用粘结单元来模拟土的开裂和挤压行为，通过将模拟结果与实际施工过程进行比较，验证了模拟模型的可行性。

关键词：粘结单元；锤击沉桩；贯入度；有限元模拟如今在建筑工程领域里，钢管桩的应用颇为广泛。

钢管桩基础具有强度高、自重轻、施工快、制作运输方便等优势。

随着科技的发展，钢结构防腐技术也愈发成熟，这使得钢管桩也越来越多的在大型海洋工程结构中得以应用。

数值模拟可以通过建立模型反映实际工程中复杂的应力应变关系、空心管桩与土体的相互作用以及模拟实际情况下的施工过程，是研究和解决工程中受力和变形问题的有力工具。

1.粘结单元介绍在ABAQUS软件中，粘结单元适用于建模粘合剂、粘结界面、垫片和岩石断裂等[1]。

单元的本构响应取决于具体的应用，并基于对变形和应力状态的某些假设，这些假设适用于每个应用区域。

在以往的数值模拟中采用Mohr-Coulomb弹塑性模型模拟土的屈服准则，本文通过粘结单元模拟钢管桩在进入土后排土效应，运用A BAQUS软件建立大直径空心管桩连续锤击贯入模型，将管桩贯入模拟与实际工程进行对比来验证模型的可行性。

粘结单元的设置方式有两种，分别是在Par t模块中创建实体单元和在网格划分过后自动设置。

第一种方法需要在Par t模块中先创建实体单元，然后通过材料的赋予实体单元粘结模型的属性来设置粘结单元。

通常可以通过材料的力学参数，如剪切模量、泊松比、弹性模量等来设置粘结单元的属性。

在实际应用中，这种方法需要将粘结单元与实体单元进行Tie绑定，以保证粘结单元和实体单元的连通性和协同工作。

在这种情况下，应该注意调整材料参数和Tie约束条件，以保证模拟结果的准确性和可靠性。

另一种方法是在网格划分过后，通过A BAQUS自带的工具，选择需要设置粘结单元的面进行自动设置。

关于Cohesive模型应用的一些小结学习粘聚力单元时从各种讨论中获益匪浅，现总结自己做过的一些练习模型，希望对大家有所帮助。

里面有很多是论坛中帖子里面的知识，在此对原作者一并谢过。

错误疏漏之处请大家多指正。

这里所有的粘聚力模型都是指Traction-separation-based modeling( The modeling of bonded interfaces in composite materials often involves situations where the intermediate glue material is very thin and for all practical purposes may be considered to be of zero thickness，帮助文献目录为32.5.1-2 )。

模型中参数仅作测试用，没有实际意义。

1.引言及一些讨论粘聚力模型( Cohesive Model )将复杂的破坏过程用两个面之间的‘相对分离位移-力’关系表达。

这种粘聚力关系很大程度上是宏观唯象的，有多种表达形式，如图1-1所示。

图1-1 常见的粘聚力关系Abaqus软件中自带的粘聚力模型为线性三角形(下降阶段可以为非线性)。

其它如指数、梯形等模型主要通过用户单元子程序(UEL/VUEL)实现。

粘聚力模型的形状对某些计算结果( 例如单纯的拉开分层)影响很大。

1.1 粘聚力单元及粘聚力接触粘聚力模型可以通过使用粘聚力单元( Cohesiev Elements )或者粘聚力接触( Cohesive Surfaces )来实现。

在模型和参数都一致的时候，两类方法得到的结果略有差别。

1.2粘聚力单元Abaqus中的粘聚力单元包括3D单元COH3D8，COH3D6；2D单元COH2D4；轴对称单元COHAX4；以及相应的孔压单元。

单元的厚度(分离)方向对于粘聚力单元，一个非常重要的方面是确定单元的厚度(分离)方向( Thickness direction、Stack direction )。

Towards a Generalized Bounding Surface Model for Cohesive SoilsVictor N.Kaliakin 1and Andr´e s Nieto Leal 21Department of Civil &Environmental Engineering,University of Delaware,Newark,DE 19716;PH (302)831-2409;FAX (302)831-3640;email:kaliakin@ 2Department of Civil &Environmental Engineering,University of Delaware,Newark,DE 19716;PH (302)831-2409;FAX (302)831-3640;email:anieto@ ABSTRACTSince its inception over thirty years ago,the bounding surface model for cohe-sive soils has been progressively improved and expanded to a three-invariant,elasto-plastic formulation that accounts for the initial and stress induced anisotropy.More recently,a simplified version of the model served as the basis for a simplified form of the anisotropic model.In order to more accurately simulate the softening of natural clays,the associative flow rule used exclusively in all earlier versions of the model was recently extended to include a non-associative flow rule.In an effort to unify the afore-mentioned versions of the bounding surface model for cohesive soils,a generalized model framework has been developed.The present paper describes this framework.INTRODUCTIONBeginning with its inception in the 1980’s (Dafalias and Herrmann,1980,1982),the bounding surface model for cohesive soils has been progressively improved and expanded to a three-invariant,elastoplastic formulation (Dafalias,1986;Dafalias and Herrmann,1986).Notable milestones in the development of this model were the ability to account for initial and stress induced anisotropy (Anandarajah and Dafalias,1986)and then the ability to simulate the time-dependent response of isotropic cohesive soils within a combined and coupled elastoplastic-viscoplastic framework (Dafalias,1982;Kaliakin and Dafalias,1990a,b).Based on extensive experience,this version of the isotropic model was subsequently simplified (Kaliakin and Dafalias,1989).More recently,the simplified version of the model served as the basis for asimplified form of the anisotropic model (Ling et al.,2002).In order to more accurately simulate the softening of natural clays,the associative flow rule used exclusively in all earlier versions of the model was recently extended to include a non-associative flow rule (Jiang et al.,2012).In an effort to unify the aforementioned versions of the bounding surface modelfor cohesive soils,a generalized model framework has been developed.The present paper describes this framework.Following a general overview of the bounding surfaceD o w n l o a d e d f r o m a s c e l i b r a r y .o r g b y A n h u i U n i v e r s i t y o f T e c h n o l o g y o n 03/26/15. C o p y r i g h t A S CE .F o r p e r s o n a l u s e o n l y ; a l l r i g h t s r e s e r v e d .concept,the paper discusses the direct and joint invariants used in the model formula-tion,specific forms of the bounding surface,explicit expressions for isotropic and kine-matic hardening laws,and the shape hardening functions that constitute a novel aspect of bounding surface based models.A limited number of applications of the model to cohesive soils will also be presented.THE BOUNDING SURFACE IN STRESS SPACEThe bounding surface concept,originally introduced by Dafalias and Popov (1974)and independently by Krieg (1975),was motivated by the observation that any stress-strain curve for monotonic loading,or for monotonic loading followed by re-verse loading,eventually converges to certain well-defined “bounds”in the stress-strain space.These bounds cannot be crossed but may change position in the process of load-ing.For the development of constitutive models,the simplest way to describe the bounding state is by means of the concept of a bounding surface in stress space (Dafalias,1981).The bounding surface is the multiaxial generalization of the concept of “bounds”in the uniaxial stress-strain space.In this particular representation,appro-priate for soils,the bounding surface always encloses the origin and is origin-convex;i.e.,any radius emanating from the origin intersects the surface at only one point.The essence of the bounding surface concept is the hypothesis that plastic deformations can occur for stress states either within or on the bounding surface.Thus,unlike classi-cal yield surface plasticity,the plastic states are not restricted only to those lying on a surface.This fact has proven to be a great advantage of the bounding surface concept.If the material state is defined in terms of the effective stress σij (external vari-ables),and proper plastic internal variables q n ,then the bounding surface in effective stress space is defined analytically byF ¯σ ij ,q n =0(1)where a bar over stress quantities indicates an “image”point on the bounding surface.The actual stress point σij lies always within or on the surface.To each effective stressstate σ ij a unique “image”stress point ¯σij is assigned by a properly defined “mapping”rule that becomes the identity mapping if σij is on the surface.The Role of the Bounding SurfaceThe bounding surface is instrumental in defining the direction (or vector)of plastic loading-unloading L ij and the plastic modulus K p that enter into a typical elastoplasticity formulation.The expression for L ij at σij is defined as the gradient of F at the “image”point;viz.,L ij =∂F ∂¯σij(2)For any stress increment dσij causing plastic loading,a corresponding imageD o w n l o a d e d f r o m a s c e l i b r a r y .o r g b y A n h u i U n i v e r s i t y o f T e c h n o l o g y o n 03/26/15. C o p y r i g h t A S CE .F o r p e r s o n a l u s e o n l y ; a l l r i g h t s r e s e r v e d .stress increment ¯dσij occurs as a result of the hardening of the bounding surface by means of the internal variables q n .The following relations are thus required to com-plete the general bounding surface formulation:A scalar loading index that is definedin terms of equation (2),the stress increments dσij ,¯dσij ,the plastic modulus K p (as-sociated with σij )and a “bounding”plastic modulus ¯K p (associated with ¯σij )in thefollowing manner:L =1K p ∂F ∂¯σij dσij =1¯K p ∂F ∂¯σij d ¯σij (3)A bounding plastic modulus ¯Kp that is obtained from the consistency condition dF =0.Finally,a state-dependent relation between K p and ¯Kp that is established as a function of the Euclidean distance δbetween the current stress state and its “image”stress;viz.,K p =¯K p +ˆH (σij ,q n )δr −δ(4)Equation (4)embodies the meaning of the bounding surface concept.If δ<r and ˆH is not approaching infinity,the concept allows for plastic deformations to occur for points either within or on the surface at a progressive pace that depends upon δ.The closer to the surface is the actual stress point (σij ),the smaller is K p (itapproaches the corresponding ¯Kp ),and the greater is the plastic strain increment for a given stress increment.The σij may eventually reach (possibly asymptotically)the bounding surface in the course of plastic loading;it remains on the surface if loading continues,and detaches from the surface and moves inwards upon unloading.Thus,forstates within the bounding surface,the function ˆHalong with its associated parameters are intimately related to the material response,and as such,constitute important “new”elements of the present formulation with regard to ones based upon classical yield surface elastoplasticity.ELEMENTS OF A GENERALIZED BOUNDING SURFACE MODELA generalized bounding surface model for cohesive soils is described in termsof the following quantities:1)a properly defined strain decomposition,2)suitably cho-sen stress invariants,3)a set of internal variables,4)an appropriate definition of the elastic response,5)an explicit definition of the bounding surface,6)a failure crite-rion,7)suitable hardening rules,8)a shape hardening function,and 9)a set of model parameters,whose values are determined by matching numerical simulations with ex-perimental results.Details pertaining to each of the aforementioned quantities are dis-cussed below.Strain DecompositionThe general coupling that exists between plastic and viscoplastic processes re-quires the following additive strain decomposition:D o w n l o a d e d f r o m a s c e l i b r a r y .o r g b y A n h u i U n i v e r s i t y o f T e c h n o l o g y o n 03/26/15. C o p y r i g h t A S CE .F o r p e r s o n a l u s e o n l y ; a l l r i g h t s r e s e r v e d .˙εij =˙εe ij +˙εi ij =˙εe ij +˙εv ij +˙εp ij(5)where εij represents the total infinitesimal strain tensor and the superscripts e ,i ,vand p denote its elastic,inelastic,viscoplastic (delayed)and plastic (instantaneous)components,respectively.Choice of invariantsInvariance requirements under superposed rigid body rotation require that thebounding surface be a function of the direct and joint isotropic invariants of ¯σij and q n .The joint invariants characterize material anisotropy,and somewhat different forms of these invariants have been used in previous anisotropic versions of the model (Anan-darajah and Dafalias,1986;Ling et al.,2002;Jiang et al.,2012).These invariants are not required for isotropic versions of the model.Choice of internal variablesIn all of the previous versions of the bounding surface model for cohesive soils,the surface is assumed to undergo isotropic hardening along the hydrostatic (I 1)axis.The hardening is controlled by a single scalar internal variable that measures the in-elastic change in volumetric strain;viz.,˙εi kk =˙εv kk +˙εp kk(6)Definition of the elastic responseElastic isotropy has been assumed in all of the previous versions of the bound-ing surface model for cohesive soils.Furthermore,this isotropy is assumed to be inde-pendent of the rate of loading and to be unaltered by inelastic deformation.Explicit definition of the bounding surfaceEquation (1)gives the general analytical form of the bounding surface in stressinvariant space,which may assume many particular forms provided it satisfies certain requirements concerning the shape of the surface (Dafalias and Herrmann,1986).Two specific forms of the bounding surface have traditionally been associated with the mod-eling of cohesive soils.These are the “composite”form of the surface and the single ellipse.Motivated by the ideas of critical state soil mechanics,a specific form of thesurface consisting of two ellipses and a hyperbola with continuous tangents at their connecting points was developed (Dafalias and Herrmann,1980).Dafalias and Her-rmann (1986)present explicit expressions for F =0,which define each portion of thisD o w n l o a d e d f r o m a s c e l i b r a r y .o r g b y A n h u i U n i v e r s i t y o f T e c h n o l o g y o n 03/26/15. C o p y r i g h t A S CE .F o r p e r s o n a l u s e o n l y ; a l l r i g h t s r e s e r v e d .“composite”surface,along with derivatives of F with respect to the stress invariants.In an effort to simplify the formulation,a bounding surface consisting of a single ellipse has also been developed (Kaliakin and Dafalias,1989).This surface has been used in the more recent versions of the model (Kaliakin and Dafalias,1990a,b;Ling et al.,2002;Jiang et al.,2012).The failure criterionA critical state failure criterion has been assumed in all of the previous ver-sions of the bounding surface model for cohesive soils.For a specific value of the Lode angle θ,the failure surface is assumed to be straight and to coincide with the critical state line.In I 1−J space,the slope of the critical state line is denoted by N .The variation of N with θhas traditionally been described by a relation having the form N (θ)=g (θ,k )N c ,where k =N e /N c ,with N e =N (−π/6)and N c =N (π/6)be-ing the values of N associated with axisymmetric triaxial extension and compression,respectively.The dimensionless function g (θ,k )must take on the values g (−π/6,k )=k and g (π/6,k )=1.A simple form of this function,attributed to Gudehus (1973),and used by Dafalias and Herrmann (1986)in conjunction with bounding surface models for clays is g (θ,k )=2k/[1+k −(1−k )sin 3θ].Hardening rulesIn all of the previous versions of the bounding surface model for cohesivesoils,the surface is assumed to undergo isotropic hardening along the hydrostatic (I 1)axis.The hardening is controlled by a single scalar internal variable that mea-sures the inelastic change in volumetric strain given by equation (6).The simulation of material anisotropy requires that the surface also undergo rotational and possibly distortional hardening.The original rotational hardening rule proposed by Anandara-jah and Dafalias (1986)has been progressively simplified (Ling et al.,2002;Jiang et al.,2012).The distortional hardening rule originally proposed by Anandarajah and Dafalias (1986)has been removed from subsequent models (Ling et al.,2002;Jiang et al.,2012).Shape hardening functionThe hardening function ˆHdefines the shape of the response curves during in-elastic hardening (or softening)for points within the bounding surface.It relates theplastic modulus K p to its “bounding”value ¯Kp in the following general manner:K p =¯Kp +ˆH (σij ,q n )ˆf (δ)(7)This expression differentiates the bounding surface formulation from standard elasto-plasticity models.It is a generalization of equation (4);in that equation,ˆf(δ)=D o w n l o a d e d f r o m a s c e l i b r a r y .o r g b y A n h u i U n i v e r s i t y o f T e c h n o l o g y o n 03/26/15. C o p y r i g h t A S C E . F o r p e r s o n a l u s e o n l y ; a l l r i g h t s r e s e r v e d .δ/ r −δ .The plastic moduli K p and ¯Kp both have units of stress cubed (i.e.,F 3L −6).The “bounding”plastic modulus (¯Kp )is computed from the consistency condition.Since ˆf(δ)is dimensionless it follows that ˆH must also have units of stress cubed.Dif-ferent functional forms for ˆHhave been used in the past.For example,the bounding surface formulation for isotropic cohesive soils consisting of a single ellipse (Kaliakin and Dafalias,1989)used the following functional form of H :ˆH =(1+e in )λ−κP a 9(F,¯I 1)2+13(F,¯J )2h (θ)z 0.02+h o 1−z 0.02 f (8)In equation (8)e in denotes the initial void ratio.The critical state param-eters λand κare taken to denote the slopes of the isotropic consolidation andswell/recompression lines,respectively,in a plot of void ratio versus the natural log-arithm of I 1.The quantity P a denotes the atmospheric pressure,and z =J/J 1=JR/NI o is a dimensionless variable.The dimensionless quantity h (θa )defines the degree of hardening for points within the bounding surface,except those within the immediate vicinity of the I 1-axis where z →0.Of all the hardening quantities,h (θa )has the most fundamental and significant role.It is a function of the Lode angle θand varies in magnitude from a value of h c =h (π/6)(corresponding to a state of triaxial compression)to a value of h e =h (−π/6)(corresponding to a state triaxial extension).More precisely,this interpolation is given by h (θa )=g (θa ,k )h c =2µ/[1+µ−(1−µ)sin 3θa ]h c ,where µ=h e /h c .The quantity h o is the average of h c and h e .Finally,the function f is defined in the following manner:f =12a +sign (n p )(|n p |)1w where a and w are dimensionless model parameters.Finally,n p =3F,¯I 19(F,¯I 1)2+13(F,¯J )2is the component in the p -direction of the unit outward normal to the bounding surface in triaxial stress space.As such,it is a dimensionless quantity.Although the adoption of a single ellipse simplifies the explicit definition of the bounding surface,it requiresthe modification of previous functional forms of ˆH.More precisely,if a single ellipse is used instead of a hyperbola (which is closer to the critical state line),undesirably high levels of J will be attained at large overconsolidation ratios.To prevent this from occurring,the quantity f was added to the expression for H .SAMPLE APPLICATIONTo illustrate the capabilities of arguably the simplest form of the bounding sur-face model for cohesive soils,the isotropic model with associative flow (Kaliakin andD o w n l o a d e d f r o m a s c e l i b r a r y .o r g b y A n h u i U n i v e r s i t y o f T e c h n o l o g y o n 03/26/15. C o p y r i g h t A S CE .F o r p e r s o n a l u s e o n l y ; a l l r i g h t s r e s e r v e d .Dafalias,1989)was used to simulate the response of isotopically consolidated Lower Cromer Till (LCT)based on the work of Gens (1982).LCT is classified as a low-plasticity sandy silty-clay (liquid limit =25%and plasticity index =13%),with the main clay minerals being calcite and illite.The tests on LCT were all performed on samples consolidated from a slurry with an initial water content of 31%.Although the bounding surface model has traditionally been applied to rather soft clays with lager liquid limits and plasticity indices,the choice of LCT is motivated by its use in the verification of other clay models (Dafalias et al.,2006).Figure 1.Undrained stress paths for Lower Cromer Till.Values for the traditional critical state parameters were computed from the data of Gens (1982).In particular,λ=0.066,κ=0.0077,M c =1.18,M e =0.86,and ν=0.20.A value of 2.30for the parameter R defining the shape of the elliptical bounding surface was determined from the experimental undrained stress paths for normally consolidated samples in compression and extension.A value of 0.60for the projection center parameter C was determined by the shapes of the undrained stress paths for lightly overconsolidated samples.The rather “stiff”nature of these undrained stress required a value of 2.0for the elastic nucleus parameter s p .Finally,the following values were determined for the shape hardening parameters:h c =h e =1.0,a =6.0,and w =5.0.It is timely to note that the latter value has been used almost exclusively inD o w n l o a d e d f r o m a s c e l i b r a r y .o r g b y A n h u i U n i v e r s i t y o f T e c h n o l o g y o n 03/26/15. C o p y r i g h t A S CE .F o r p e r s o n a l u s e o n l y ; a l l r i g h t s r e s e r v e d .Figure 2.Deviator stress-axial strain response for Lower Cromer Till.previous simulations and can thus be taken as a model constant (as opposed to a model parameter).As evident from Figures 1and 2,even the simplest form of the bounding surface model for cohesive soils gives quite reasonable simulations for the undrained stress paths and stress-strain curves.CONCLUSIONA generalized definition of the bounding surface model for cohesive soils provides a convenient and elegant framework for unifying various versions of the model for both isotropically and anisotropically consolidated cohesive soils.This generalized defini-tion is described in terms of the following quantities:1)a properly defined strain de-composition,2)suitably chosen stress invariants,3)a set of internal variables,4)an appropriate definition of the elastic response,5)an explicit definition of the bounding surface,6)a failure criterion,7)suitable hardening rules,8)a shape hardening function,and 9)a set of model parameters,whose values are determined by matching numeri-cal simulations with experimental results.In the present paper a form of the model employing an associative flow rule,and formulated for isotropically consolidated soils has been shown to give quite good simulations for a low-plasticity sandy silty-clay.It is also timely to note that this simple form of the model has recently beenD o w n l o a d e d f r o m a s c e l i b r a r y .o r g b y A n h u i U n i v e r s i t y o f T e c h n o l o g y o n 03/26/15. C o p y r i g h t A S CE .F o r p e r s o n a l u s e o n l y ; a l l r i g h t s r e s e r v e d .used to successfully simulate a rather wide range of cohesive soils,subjected to both monotonic and cyclic loading histories.In particular,the response of Keuper Marl low plasticity silty clay (Yasuhara,et al.2003),Cloverdale clay (Zergoun,1991),a Kaoli-nite Clay (Li and Meissner,2002),and Bogota clay (Camacho and Reyes,2005)has also been successfully simulated.ACKNOWLEDGEMENTThe second authors graduate studies are supported by the Fulbright Colombia-COLCIENCIAS Scholarship.This support is gratefully acknowledged.REFERENCESAnandarajah,A.and Dafalias,Y .F.(1986).“Bounding Surface Plasticity III:Applica-tion to Anisotropic Cohesive Soils,”Journal of Engineering Mechanics,ASCE ,112(12),1292-1318.Camacho,J.and Reyes,O.(2005).“Aplication of modified cam-clay model to recon-stituted clays of the sabana of Bogota,”Ciencia e Ingeniera Neogranadina .Bogota,Colombia,20(1).Dafalias,Y .F.and Popov,E.P.(1974).“A Model of Nonlinearly Hardening Materialsfor Complex Loadings,”Acta Mechanica ,21(3),173-192.Dafalias,Y .F.(1981).“The Concept and Application of the Bounding Surface in Plas-ticity Theory,”IUTAM Symposium on Physical Non-Linearities in Structural Analysis ,Hult,J.and Lemaitre,J.eds.,Senlis,France,Springer-Verlag,Berlin-Heidelberg-New York,56-63.Dafalias,Y .F.(1982).“Bounding Surface Elastoplasticity -Viscoplasticity for Par-ticulate Cohesive Media,”Deformation and Failure of Granular Materi-als ,IUTAM Symposium on Deformation and Failure of Granular Materials,Delft,the Netherlands edited by P.A.Vermeer and H.J.Luger,Rotterdam:A.A.Balkema,97-107.Dafalias,Y .F.(1986).“Bounding Surface Plasticity.I:Mathematical Foundation andthe Concept of Hypoplasticity,”Journal of Engineering Mechanics,ASCE ,112(9),966-987.Dafalias,Y .F.and Herrmann,L.R.(1980).“A Bounding Surface Soil PlasticityModel,”Proceedings of the International Symposium on Soils Under Cyclic and Transient Loading ,edited by G.N.Pande and O.C.Zienkiewicz,Rotter-dam:A.A.Balkema,335-345.Dafalias,Y .F.and Herrmann,L.R.(1982).“A Generalized Bounding Surface Consti-tutive Model for Clays,”Application of Plasticity and Generalized Stress-Strain in Geotechnical Engineering ,edited by R.N.Yong and E.T.Selig,New York,ASCE,78-95.D o w n l o a d e d f r o m a s c e l i b r a r y .o r g b y A n h u i U n i v e r s i t y o f T e c h n o l o g y o n 03/26/15. 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