扩展的D-P材料模式.doc

11.3.1Extended Drucker-Prager modelsProducts: ABAQUS/StandardABAQUS/ExplicitThe extended Drucker-Prager models are defined by using the *DRUCKER PRAGER option together with the *DRUCKER PRAGER HARDENING option and, optionally, the *RATE DEPENDENT, the *DRUCKER PRAGER CREEP, and the *TRIAXIAL TEST DATA options.扩展的DruckerPrager材料模式是由*DRUCKER PRAGER选项连同*DRUCKER PRAGER HARDENING及*RATE DEPENDENT, *DRUCKER PRAGER CREEP, *TRIAXIAL TEST DATA定义的。References 参考资料 *DRUCKER PRAGER *DRUCKER PRAGER HARDENING *RATE DEPENDENT *DRUCKER PRAGER CREEP *TRIAXIAL TEST DATA “Material library: overview,” Section 9.1.1 “Inelastic behavior,” Section 11.1.1 “Rate-dependent yield,” Section 11.2.3 “Rate-dependent plasticity: creep and swelling,” Section 11.2.4Overview 概要The extended Drucker-Prager models:扩展的D-P模式： are used to model frictional materials, which are typically granular-like soils and rock, and exhibit pressure-dependent yield (the material becomes stronger as the pressure increases); 可以用来模拟摩擦材料，典型的是粒状岩土材料及表现压力相关屈服材料（由于压力的增加材料强度增高）； are used to model materials in which the compressive yield strength is greater than the tensile yield strength, such as those commonly found in composite and polymeric materials; 可以用来模拟抗压屈服强度大于抗拉屈服强度的材料，如通常所说的复合材料； allow a material to harden and/or soften isotropically; 容许材料等向硬化和/或软化 generally allow for volume change with inelastic behavior: the flow rule, defining the inelastic straining, allows simultaneous inelastic dilation (volume increase) and inelastic shearing; 通常容许材料的无弹性行为的体积变化：流动法则，定义的无弹性应变，容许同时发生无弹性膨胀（体积增加）和无弹性剪切； can include creep in ABAQUS/Standard if the material exhibits long-term inelastic deformations; 在ABAQUS/Standard中如果材料表现长期的无弹性变形则可以包含蠕变； can be defined to be sensitive to the rate of straining, as is often the case in polymeric materials; 可以定义为对应变率灵敏的材料，如在复合材料中常用到； can be used in conjunction with either the elastic material model (“Linear elastic behavior,” Section 10.2.1) or, in ABAQUS/Standard if creep is not defined, the porous elastic material model (“Elastic behavior of porous materials,” Section 10.3.1); and 可以用来弹性材料模式或者在如果在ABAQUS/Standard下未定义蠕变的多孔弹性材料模式的协作； are intended to simulate material response under essentially monotonic loading. 可以用来模拟材料在单一荷载作用下的相应。Yield criteria屈服准则The yield criteria for this class of models are based on the shape of the yield surface in the meridional plane. In ABAQUS/Standardthe yield surface can have a linear form, a hyperbolic form, or a general exponent form; in ABAQUS/Explicit only the linear form is available. These surfaces are illustrated in Figure 11.3.11. The stress invariants and other terms in each of the three related yield criteria are defined later in this section. 这类材料的屈服准则基于子午面（pi平面）上的屈服面的形状。在ABAQUS/Standard中屈服面可以是线性的、双曲线形的或者是通常的指数形的；在ABAQUS/Explici中仅采用线形的。图11.3.11解释了这些面的含义。在本节后面解释了在这三个相关的屈服准则中的应力不变和其它术语。Figure 11.3.11 Yield surfaces in the meridional plane.在子午面中的屈服面The linear model (Figure 11.3.11a) provides for a possibly noncircular yield surface in the deviatoric plane (-plane) to match different yield values in triaxial tension and compression, associated inelastic flow in the deviatoric plane, and separate dilation and friction angles. Input data parameters define the shape of the yield and flow surfaces in the meridional and deviatoric planes as well as other characteristics of inelastic behavior such that a range of simple theories is providedthe original Drucker-Prager model is available within this model. However, this model cannot provide a close match to Mohr-Coulomb behavior, as described later in this section.线性模式(Figure 11.3.11a)提供了一个在偏平面（平面）里可能的非圆弧屈服面，以符合在三轴拉伸或压缩的不同的屈服面，偏平面里的无弹性流及各自的膨胀角和摩擦角。输入数据参数定义子午面和偏平面中屈服面和流动面的形状及无弹性行为的其它特性，例如提供了一个简单的理论范围原始的D-P模式在该模式中仍然是有效的。然而，该模式不能提供一个接近于莫尔库仑行为的模式，像本节后面阐述的一样。The hyperbolic and general exponent models use a von Mises (circular) section in the deviatoric stress plane. In the meridional plane a hyperbolic flow potential is used for both models, which, in general, means nonassociated flow. These models are available only in ABAQUS/Standard.双曲线和通用指数模式为在偏应力平面里的vonmis（圆形）截面。在子午面中一个双曲线流可能用于两个模型，通常意味着为非关联流。这些模式仅使用于ABAQUS/Standard中。The choice of model to be used depends largely on the analysis type, the kind of material, the experimental data available for calibration of the model parameters, and the range of pressure stress values that the material is likely to experience. It is common to have either triaxial test data at different levels of confining pressure or test data that are already calibrated in terms of a cohesion and a friction angle and, sometimes, a triaxial tensile strength value. If triaxial test data are available, the material parameters must be calibrated first. The accuracy with which the linear model can match these test data is limited by the fact that it assumes linear dependence of deviatoric stress on pressure stress. Although the hyperbolic model makes a similar assumption at high confining pressures, it provides a nonlinear relationship between deviatoric and pressure stress at low confining pressures, which may provide a better match of the triaxial experimental data. The hyperbolic model is useful for brittle materials for which both triaxial compression and triaxial tension data are available, which is a common situation for materials such as rocks. The most general of the three yield criteria is the exponent form. This criterion provides the most flexibility in matching triaxial test data. ABAQUS will determine the material parameters required for this model directly from the triaxial test data. ABAQUSA least-squares fit that minimizes the relative error in stress is used for this purpose.材料模式的选择很大程度上取决于分析类型、材料类型、可以用于模型参数判据的试验数据及材料可能经历的压应力范围。通常具有不同限制压力水平下的三轴试验资料或者已经按照一个粘聚力和摩擦角校准了的试验数据，甚至有时候具有一组三轴抗拉强度值。如果三轴试验数据可以利用，那么材料参数必须首先被校准。线性材料模式能够与试验数据相匹配的精度由偏应力与压应力的线性相关程度限制。尽管双曲线模式假定为在高限制压力，但是它提供了一个在低限制压力下偏应力和压应力非线性相关关系，这一点可以较好的满足与三轴试验数据相吻合。双曲线材料模式适用于三轴抗压强度及三轴抗拉强度均可利用的脆性材料，这是类似岩石材料的共性。三个屈服准则中最一般的是指数模式。该准则在与三轴试验数据相匹配方面最具有适应性。将从三轴试验数据中直接得到模型需要的材料参数。为实现这个目的使用最小二乘法使得应力相对误差最小。For cases where the experimental data are already calibrated in terms of a cohesion and a friction angle, the linear model can be used. If these parameters are provided for a Mohr-Coulomb model, it is necessary to convert them to Drucker-Prager parameters. The linear model is intended primarily for applications where the stresses are for the most part compressive. If tensile stresses are significant, hydrostatic tension data should be available (along with the cohesion and friction angle) and the hyperbolic model should be used.在试验数据已经根据粘聚力和摩擦角校准了的情况下，可以使用线性模式。如果这些参数提供给莫尔库仑模式，有必要将他们转换为D-P参数。线性模式主要用于应力很多程度上为压应力的情况下。如果拉应力为主要的，应该使用静水拉应力数据（连同粘聚力和摩擦角）并采用双曲线模式。Calibration of these models is discussed later in this section.这些模式准则在本节的后面进行探讨。Hardening and rate dependence 硬化和率相关For granular materials these models are often used as a failure surface, in the sense that the material can exhibit unlimited flow when the stress reaches yield. This behavior is called perfect plasticity. The models are also provided with isotropic hardening.In this case plastic flow causes the yield surface to change size uniformly with respect to all stress directions. This hardening model is useful for cases involving gross plastic straining or in which the straining at each point is essentially in the same direction in strain space throughout the analysis. Although the model is referred to as an isotropic “hardening” model, strain softening, or hardening followed by softening, can be defined.对于粒状材料这些模式常常用作破坏面，在这种意义上当材料的应力达到屈服时，材料将表现为无穷流。这种行为称为理想塑性。 模式也提供了等向硬化。在这种情况下塑性流引起屈服面在各个应力方向均匀的变化。当包含总塑性应变或者在整个分析过程中各个点的应变在应变空间中的同一方向上时，这种硬化是有用的。尽管模式被称为等向硬化模式，应变软化或者伴随有软化的硬化也可以被定义。As strain rates increase, many materials show an increase in their yield strength. This effect becomes important in many polymers when the strain rates range between 0.1 and 1 per second; it can be very important for strain rates ranging between 10 and 100 per second, which are characteristic of high-energy dynamic events or manufacturing processes. The effect is generally not as important in granular materials. The evolution of the yield surface with plastic deformation is described in terms of the equivalent stress , which can be chosen as either the uniaxial compression yield stress, the uniaxial tension yield stress, or the shear (cohesion) yield stress: 随着应变率的增加，多数材料的屈服强度也随之增加。在多数复合材料中，当应变率在每秒钟从0.1到1范围变化时这种效应是重要的；当应变率变化在每秒钟从10到100变化时，这种效应将变的异常重要，这是高能动态事件或者制造夜过程的特性。在粒状材料中这种效应不是那么重要。屈服面随着塑性变形的发展是根据等效应变来描述的，这可以被看作为单轴压缩屈服应力，单轴拉伸屈服应力或者剪切（粘聚）屈服应力。where is the equivalent plastic strain rate, defined for the linear Drucker-Prager model as = if hardening is defined in uniaxial compression;= if hardening is defined in uniaxial tension;= if hardening is defined in pure shear, and defined for the hyperbolic and exponential Drucker-Prager models in ABAQUS/Standardas is the equivalent plastic strain;is temperature; andare other predefined field variables.式中是等效塑性应变率，定义为线性D-P模式，如果硬化由单轴压缩中定义则=。如果硬化由单轴拉伸则定义为=；当硬化由纯剪切定义则= ，并且在ABAQUS/Standard中定义为双曲线和指数D-P模式。为等效塑性应变；是温度；为其它预定义场变量。The functional dependence includes hardening as well as rate-dependent effects. The material data can be input either directly in a tabular format or by correlating it to static relations based on yield stress ratios.相关函数包含硬化和率相关效应。材料数据可以通过表格的形式直接输入或者通过与基于屈服应力率的静态关系相关联。When using the Drucker-Prager material model, ABAQUS allows the user to prescribe initial hardening by defining initial equivalent plastic strain values, as discussed below along with other details regarding the use of initial conditions.在使用D-P材料模式的时候，ABAQUS允许用户通过定义初始等效塑性应变来指定初始硬化，如下关于使用初始条件的详细阐述。Direct tabular data 直接表格数据Test data are entered as tables of yield stress values versus equivalent plastic strain at different equivalent plastic strain rates; one table per strain rate. Compression data are more commonly available for geological materials, whereas tension data are usually available for polymeric materials. The guidelines on how to enter these data are provided in “Rate-dependent yield,” Section 11.2.3Usage:*DRUCKER PRAGER HARDENING, RATE=试验数据作为在各个等效塑性应变率时的屈服应力及之相应的等效塑性应变表格输入；每个应变率一个表格。压缩数据对于地质材料来说更具有使用价值，而拉伸数据对于复合材料来说具有使用价值。.关于如何输入这些数据的指导方针在“Rate-dependent yield,” Section 11.2.3.中给予了阐明。用法：*DRUCKER PRAGER HARDENING, RATE=Yield stress ratios 屈服应变率Alternatively, the strain rate behavior can be assumed to be separable, so that the stress-strain dependence is similaror, in ABAQUS/Explicit, identicalat all strain rates: 应变率行为可以假定为单独的，以便应力应变关系相似或者在ABAQUS/Explicit一样对所有的应变率：where is the static stress-strain behavior and is the ratio of the yield stress at nonzero strain rate to the static yield stress (so that ). 式中为静态应力应变行为，为非零应变率下的屈服应力率（）。To specify such a definition, the *RATE DEPENDENT option must be used in conjunction with the *DRUCKER PRAGER HARDENING option. Two methods are offered by the *RATE DEPENDENT option to define : specifying an overstress power law or defining the variable directly as a tabular function of .为指定这样一个定义，必须使用*RATE DEPENDENT选项及*DRUCKER PRAGER HARDENING选项。为定义*RATE DEPENDENT选项提供了两种方法：指定一个overstress power law或者指定的值直接作为的一个表函数。Overstress power lawThe overstress power law has the form where and are material parameters that can be functions of temperature and, possibly, of other predefined field variables. Usage: Use both of the following options: *DRUCKER PRAGER HARDENING*RATE DEPENDENT, TYPE=POWER LAW用法：使用下面两个选项*DRUCKER PRAGER HARDENING*RATE DEPENDENT, TYPE=POWER LAWTabular functionWhen is entered directly, it is entered as a tabular function of the equivalent plastic strain rate, ; temperature, ; and predefined field variables, .表函数Tabular function当直接输入时，它被作为一个等效塑性应变率、温度、预确定场函数值的表函数。Usage: Use both of the following options: *DRUCKER PRAGER HARDENING*RATE DEPENDENT, TYPE=YIELD RATIO用法：使用下面两个选项*DRUCKER PRAGER HARDENING*RATE DEPENDENT, TYPE=YIELD RATIORate dependence as described so far is most suitable for moderate- to high-speed events in ABAQUS/Standard. Time-dependent inelastic deformation at low deformation rates can be better represented by creep models. Such inelastic deformation, which can coexist with rate-independent plastic deformation, is described later in this section. However, the existence of creep in an ABAQUS/Standard material definition precludes the use of rate dependence as described here.到目前为止所谈到的率相关使用于ABAQUS/Standard中的中等到高速事件。与时间相关的非弹性变形在低变形率下的能够通过蠕变模式来很好的显现。这种可以与率无关的塑性变形共存的非弹性变形，将在本节后面进行介绍。但是ABAQUS/Standard材料定义中的蠕变模式不包含这里所说的率相关。Stress invariants 应力不变量The yield stress surface makes use of two invariants, defined as the equivalent pressure stress, and the Mises equivalent stress, where is the stress deviator, defined as In addition, the linear model also uses the third invariant of deviatoric stress, 应力屈服面使用两个不变量，定义为等效压应力：和米斯等效应力：式中为应力偏量，定义为：此外，线性模式也使用第三偏应力不变量：Linear Drucker-Prager model 线性DruckerPrager模式The linear model is available in ABAQUS/Standard and ABAQUS/Explicit. The model is written in terms of all three stress invariants. It provides for a possibly noncircular yield surface in the deviatoric plane to match different yield values in triaxial tension and compression, associated inelastic flow in the deviatoric plane, and separate dilation and friction angles.线性模式在ABAQUS/Standard 和ABAQUS/Explicit中均适用。该模式根据三个应力不变量来表达。它提供了一个在偏应力平面中的可能非圆弧屈服面，使得符合三轴拉伸和压缩条件下的不同屈服值，偏平面中的关联无弹性流及各自的膨胀角和摩擦角。Yield criterion 屈服准则The linear Drucker-Prager criterion (see Figure 11.3.11a) is written as where is the slope of the linear yield surface in the stress plane and is commonly referred to as the friction angle of the material;is the cohesion of the material; and is the ratio of the yield stress in triaxial tension to the yield stress in triaxial compression and, thus, controls the dependence of the yield surface on the value of the intermediate principal stress (see Figure 11.3.12).线性D-P准则（见Figure 11.3.11a）写为：式中：线性屈服面在应力平面中的坡度，通常指的是材料的摩擦角；为材料的粘聚力；三轴拉伸下的屈服应力与三轴压缩下的屈服应力的比率，因而，控制了屈服面对中间主应力的依赖关系（Figure 11.3.12）。Figure 11.3.12 Typical yield/flow surfaces of the linear model in the deviatoric plane.Figure 11.3.12偏平面中典型的线性模式屈服/流动面In the case of hardening defined in uniaxial compression, the linear yield criterion precludes friction angles 71.5 ( 3), which is unlikely to be a limitation for real materials.单轴压缩条件下的硬化工况，线性屈服准则不包括摩擦角71.5 ( 3)的情况，对真实材料来说这样一个限制摩擦角是未必的。When , , which implies that the yield surface is the von Mises circle in the deviatoric principal stress plane (the -plane), in which case the yield stresses in triaxial tension and compression are the same. To ensure that the yield surface remains convex requires .当时，意味着在偏主应力平面（平面）中屈服面为Mise圆，此时三轴拉伸和压缩下的屈服应力相同。为保证屈服面为外凸的需要满足。The cohesion, , of the material is related to the input data as 材料的粘聚力与输入数据通过下式相联系：Plastic flow 塑性流is the flow potential, chosen in this model as where is the dilation angle in the plane. A geometric interpretation of is shown in the diagram of Figure 11.3.13. In the case of hardening defined in uniaxial compression, this flow rule definition precludes dilation angles 71.5 ( 3). This restriction is not seen as a limitation since it is unlikely this will be the case for real materials. 为流势，本模式中表达为：式中：为平面中的膨胀角。 图（Figure 11.3.13）对做了一个几何说明。在单轴压缩条件下的硬化，该流动准则不包含膨胀角71.5 ( 3)的情况。由于真实材料的膨胀角不可能超过该值因此这个约束不能看作为一个限制。Figure 11.3.13 Linear Drucker-Prager model: yield surface and flow direction in the plane.Figure 11.3.13线性D-P模式：平面中屈服面和流动方向For granular materials the linear model is normally used with nonassociated flow in the plane, in the sense that the flow is assumed to be normal to the yield surface in the -plane but at an angle to the -axis in the plane, where usually , as illustrated in Figure 11.3.13. Associated flow results from setting . The original Drucker-Prager model is available by setting and . Nonassociated flow is also generally assumed when the model is used for polymeric materials. If , the inelastic deformation is incompressible; if , the material dilates. Hence, is referred to as the dilation angle.对于粒状材料线性模型通常用于平面中的非联合流，在这种情况下假定非塑性流垂直于平面内的屈服面但是与平面内的轴成角，这里通常为，如Figure 11.3.13所示。联合流来自设定。通过设定和可以得到原始的DP模式。当材料模式用于复合材料时通常假定为非联合流。如果，无弹性变形为不可压缩的；如果，材料膨胀。因此，被定义为膨胀角。The relationship between the flow potential and the incremental plastic strain for the linear model is discussed in detail in “Models for granular or polymer behavior,” Section 4.4.2 of the ABAQUS Theory Manual.流势和塑性应变增量的关系在“Models for granular or polymer behavior,” Section 4.4.2 of the ABAQUS Theory Manual.中将进行详细的讨论。Usage:*DRUCKER PRAGER用法：*DRUCKER PRAGERNonassociated flow 非联合流Nonassociated flow implies that the material stiffness matrix is not symmetric; therefore, the unsymmetric matrix storage and solution scheme should be used in ABAQUS/Standard (see “Procedures: overview,” Section 6.1.1). If the difference between and is not large and the region of the model in which inelastic deformation is occurring is confined, it is possible that a symmetric approximation to the material stiffness matrix will give an acceptable rate of convergence and the unsymmetric matrix scheme may not be needed.非关联流表明材料的刚度矩阵时非对称的。因此，非对成矩阵的存储和求解方案应该在ABAQUS/Standard中用到。如果和 差别不是太大并且非弹性变形区的发生受到限制，那么使用对称近似的材料刚度将得到一个可以接受的收敛速度，并且不需要非对称矩阵解决方案。Hyperbolic and general exponent models in ABAQUS/StandardABAQUS/Standard中的双曲线和通用指数模式The hyperbolic and general exponent models available in ABAQUS/Standardare written in terms of the first two stress invariants only.ABAQUS/Standard中使用的双曲线和通用指数模式仅仅根据前两个应力不变量的到的。Hyperbolic yield criterion 双曲线屈服准则The hyperbolic yield criterion is a conti