Modeling failure in cross-anisotropic frictional materials

Three-dimensional test results for frictional materials such as soils, concrete, and rock show that the shapes of their failure surfaces are influenced by the intermediate principal stress, shear banding, and cross-anisotropy (transverse isotropy). A general 3D failure criterion for cross-anisotropic soils for both non-rotating and rotating stresses is presented and compared with experimental results for sand, clay, and sea ice. The formulation relates the loading direction to the principal directions of the cross-anisotropic microstructure of the material. The criterion is based on a function of stress, previously used as the 3D failure criterion for isotropic frictional materials, which is set equal to a scalar that varies over a sphere. The formulation is specialized for true triaxial tests and torsion shear tests and determination of material parameters is demonstrated. The failure criterion is compared with experimental results from the literature to show that it is able to capture the conditions obtained in three-dimensional experiments without and with stress rotations. The limitations of the criterion are demonstrated by its failure to capture the behavior of some cross-anisotropic materials with particular micro-structures, and the reasons for the shortcomings are discussed.     

Foam rheology: Relation between extensional and shear deformations in high gas fraction foams

A two-dimensional hexagonal foam cell model is used to derive analytic expressions for the bulk stress tensor and foam microstructure for any small homogeneous deformation. We show that calculations done for deformations where the principal axes of stress and strain coincide, such as in extension, are sufficient to provide all information about shear deformation. The stresses and foam structure for any given strain and initial cell orientation in shear bears a unique relation to a different strain and orientation in extension. Such a mapping is obtained using the assumption that the principal axes of strain and stress corotate with each other. This in turn implies that high gas fraction foams follow the Lodge-Meissner relation, i.e. the ratio of the normal-stress difference to the shear stress equals the shear strain. The spatially periodic structure of foam along with the fact that the cell centers move affinely with the bulk, makes the above assumption a justifiable one.     

The stress concentration near a rigid line inclusion in a prestressed, elastic material. Part I.: Full-field solution and asymptotics

A lamellar (zero-thickness) rigid inclusion, so-called ‘stiffener’, is considered embedded in a uniformly prestressed (or prestrained), incompressible and orthotropic elastic sheet, subject to a homogeneous far-field deformation increment. This problem is solved under the assumption of plane strain deformation, with prestress principal directions and orthotropy axes aligned with the stiffener. A full-field solution is obtained solving the Riemann-Hilbert problem for symmetric incremental loading at infinity (while for shear deformation the stiffener leaves the ambient field unperturbed). In addition to the full-field solution, the asymptotic Mode I near-tip representation involving the corresponding incremental stress intensity factor are derived and these results are complemented with the Mode II asymptotic solution. For null prestress, the full-field stress state is shown to match correctly with photoelastic experiments performed by us (on two-part epoxy resin samples containing an aluminum lamina). Our experiments also confirm the fracture patterns for a brittle material containing a stiffener, which do not obey a hoop-stress criterion and result completely different from those found for cracks. Issues related to shear band formation and evaluation of energy release rate for a stiffener growth (or reduction) are deferred to Part II of this article.     

Grain-scale processes governing shear band initiation and evolution in sands

A variety of experimental techniques have been used to advance understanding of strain localization phenomena in sands. However, all of these methods have fallen short in characterizing the evolution of the grain-scale processes that necessarily control shear band formation and growth in sands. This paper presents results of application of the non-destructive displacement measurement technique of digital image correlation (DIC) to measure two- and three-dimensional surface displacements on plane strain and axisymmetric sand specimens over short time steps. The abundance of local displacement data, high level of accuracy, and nearly continuous (spatially and temporally) record of displacement evolution afforded by the DIC technique has finally enabled a means to quantify local displacements to particulate-scale intensity. The data have been used to evaluate the local displacement mechanisms leading to the triggering of the formation of persistent shear bands, the timing of shear band formation with regard to the achievement of peak stress, and the character of displacements within fully formed shear bands. Insights are offered regarding the relation between strain localization and global stress-strain behavior, and the ensuing interpretations of shear banding as a hardening or softening phenomenon. Comparison of behavior between plane strain and triaxial tests offer additional perspective on the influences of three-dimensional stresses and boundary conditions on shear banding. The results further shed light on the micro-deformation mechanisms (i.e. buckling columns) responsible for the observed local strain non-uniformities that characterize “steady-state” shear band evolution.     

Simple shear is not so simple

For homogeneous, isotropic, non-linearly elastic materials, the form of the homogeneous deformation consistent with the application of a Cauchy shear stress is derived here for both compressible and incompressible materials. It is shown that this deformation is not simple shear, in contrast to the situation in linear elasticity. Instead, it consists of a triaxial stretch superposed on a classical simple shear deformation, for which the amount of shear cannot be greater than 1. In other words, the faces of a cubic block cannot be slanted by an angle greater than 45° by the application of a pure shear stress alone. The results are illustrated for those materials for which the strain-energy function does not depend on the principal second invariant of strain. For the case of a block deformed into a parallelepiped, the tractions on the inclined faces necessary to maintain the derived deformation are calculated.     

Triaxial tension–compression tests for multiaxial cyclic plasticity

The aim of this paper is to present triaxial tension–compression tests and a new triaxial specimen devoted to the study of cyclic plasticity under non-proportional loadings. Because the stress state in the central part of the specimen is not homogeneous, the analysis of the tests need a 3D finite element computation. The constitutive equations used in the simulations have been deduced from complex tension–torsion tests. The comparison between the structural analysis and the experimental results allows to determine the accuracy of the set of constitutive equations and if needed to optimize this set by modifying some hardening rules.     

Stability and finite strain of homogenized structures soft in shear: Sandwich or fiber composites,and layered bodies

The stability theories energetically associated with different finite strain measures are equivalent if the tangential moduli are transformed as a function of the stress. However, for homogenized soft-in-shear composites, they can differ greatly if the material is in small-strain and constant elastic moduli measured in small-strain tests are used. Only one theory can then be correct. The preceding variational energy analysis showed that, for sandwich columns and elastomeric bearings, respectively, the correct theories are Engesser’s and Haringx’s, associated with Green’s and Almansi’s Lagrangian strain tensors, respectively. This analysis is reviewed, along with supporting experimental and numerical results, and is then extended to arbitrary multiaxially loaded homogenized soft-in-shear orthotropic composites. It is found that, to allow the use of constant shear modulus when the material is in small strain, the correct stability theory is associated with a general Doyle–Ericksen finite strain tensor of exponent m depending on the principal stress ratio. Further it is shown that the standard updated Lagrangian algorithm for finite element analysis, which is associated with Green’s Lagrangian finite strain, can give grossly incorrect results for homogenized soft-in-shear structures and needs to be generalized for arbitrary finite strain measure to allow using constant shear modulus for critical loads at small strain.     

Yield surfaces of various periodic metal honeycombs at intermediate relative density

Initial yield surfaces are derived for several periodic metal honeycomb cell structures with sufficiently high relative density that failure occurs by plastic yielding. Both in-plane stress states (normal stresses perpendicular to cell axes, with in-plane shear) and triaxial stress states with one principal stress direction along the cell axes are considered. Beam/column and plate/shell yield criteria are adopted to address general in-plane loading and 3D triaxial loading, respectively, accounting for combined cell wall stretching and bending as appropriate. Cell wall behavior is assumed to be elastic-perfectly plastic. The initial yield surfaces for different periodic cell structures are systematically compared. Some issues related to the initial yield surfaces of various honeycombs are discussed, including dependencies on relative density and in-plane and out-of-plane applied stresses, as well as the influence of joints between cell walls.     

基于t准则的各向异性强度准则及变换应力法

岩土材料通常呈现出成层水平分布特点, 即可将其视为横观各向同性材料, 横观各向同性对于岩土材料的变形以及强度值都会产生显著的影响. 基于已提出的t强度准则, t强度准则是基于各向同性单元体中存在有效滑移面来构建的, 并根据该空间有效滑移面上主剪应力与主法向应力的比值达到一定阈值为破坏条件. 在空间中存在有效滑移面与物理沉积面, 基于上述两个面在空间的位置关系, 用两面夹角作为表征横观各向同性对剪切强度影响程度的参量, 并假定当该夹角值越大, 则各向异性对强度贡献程度越大, 对应更大的应力比强度值, 反之, 则对应更小的应力比强度值. 基于上述思路并类比将其推广为正交三维各向异性准则, 基于三维各向异性材料的三维沉积面, 提出了三维特征沉积面的概念, 并基于空间滑移面与三维特征沉积面之间的夹角作为度量各向异性程度的变量, 提出了基于两面角作为参量考虑原生各向异性的应力比强度公式, 并利用该应力比强度公式来修正已提出的t强度准则, 最终建立了考虑各向异性影响的t准则公式. 在上述准则基础上, 考虑将各向异性应力空间转换为各向同性应力空间的思路, 在各向异性t准则基础上, 推导得到了基于各向异性强度t准则的变换应力公式, 利用变换应力公式可以将传统的以p, q为变量的各向同性本构模型转变为可考虑各向异性的三维本构模型. 通过对岩土材料的强度以及真三轴条件下的应力应变关系试验数据预测, 验证了所提的各向异性t准则及其变换应力公式的有效性及适用性.      

饱和超固结黏性土的三剪弹塑性本构模型研究

针对饱和超固结黏性土现有下加载面修正剑桥模型中破坏应力比为定值、土体黏聚力为零,以及不能准确反映不同应力状态下土的强度差异这些问题,基于三剪统一强度准则以及应力坐标平移法得到了扩展破坏应力比,其特点是能更好地反映应力状态变化以及土体黏聚力的影响。在此基础上提出了饱和超固结黏性土的三剪弹塑性本构模型,该模型的特点是能描述土体受力时的中间主应力效应,应力区间效应和拉压差影响,同时也能更好地考虑土体黏聚力的影响。基于该模型对ABAQUS软件进行了二次开发,并利用其模拟了饱和超固结黏性土在排水和不排水条件下的真三轴和常规三轴压缩试验特性。对常规三轴压缩条件下土体力学特性作了模拟和试验结果对比。结果表明所提模型能很好地反映不同超固结比下土体的变形、剪胀、孔隙水压力变化特性。     

A new three-dimensional Hoek-Brown strength criterion

The Hoek-Brown(HB) strength criterion has been widely applied to the estimation of strength of intact rock and rock mass,while evolving ever since.However,negligence of the effect of the intermediate principal stress still remains in the criterion’s latest version.At the same time,several three-dimensional(3D) HB strength,which can takes into account the influence of the intermediate principal stress,have already been proposed,among which the 3D HB criterion proposed by Zhang and Zhu seems to be the most reasonable one.However,the Zhang 3D HB criterion may have problems with some stress path close to triaxial extension state because of the non-convexity characteristic of its failure surface.In this paper,a new 3D HB strength criterion is presented based on a generalized form of the HB criterion,which also considers the effect of the intermediate principal stress and inherits all the merits of the original version of the HB criterion.In addition,this new criterion can remedy to some extent the shortcomings observed in the Zhang 3D HB criterion.Polyaxial tests for five different rocks from published literatures are used for evaluating this new criterion and comparing it with the Zhang 3D HB criterion.The results show that this new criterion may over-predict or underpredict the polyaxial strength of rocks but the errors are relatively small,and similar results are also found for the Zhang 3D HB criterion,which one is better depends on the type of the rock under estimation.     

Dynamic effective shear strength of saturated sand

The dynamic effective shear strength of saturated sand under cyclic loading is discussed in this paper. The discussion includes the transient time dependency behaviors based on the analysis of the results obtained in conventional cyclic triaxial tests and cyclic torsional shear triaxial tests. It has been found that the dynamic effective shear strength is composed of effective frictional resistance and viscous resistance, which are characterized by the strain rate dependent feature of strength magnitude, the coupling of consolidation stress with cyclic stress and the dependency of time needed to make the soil strength sufficiently mobilized, and can also be expressed by the extended Mohr-Coulomb's law. The two strength parameters of the dynamic effective internal frictional angle φ_d and the dynamic viscosity coefficient η are determined. The former is unvaried for different number of cyclic loading, dynamic stress form and consolidation stress ratio. And the later is unvaried for the different dynamic shear strain rate developed during the sand liquefaction, but increases with the increase of initial density of sand. The generalization of dynamic effective stress strength criterion in the 3-dimensional effective stress space is studied in detail for the purpose of its practical use. The project supported by the National Natural Science Foundation of China (10172070)     

Dilatational plastic constitutive equations and their application to ductile fracture analysis of axisymmetric specimens

In many ductile materials voids nucleate and grow under large strain and triaxial stress, which yield volumetric plastic expansion. A constitutive equation is presented, which accounts for this plastic dilatancy. The plastic moduli involved in this equation can be calibrated by using necking tests of axisymmetric bars, void model analysis and computer simulation. To verify the rationality of such a constitutive equation and adjust the values of plastic moduli, the constitutive equation with its moduli to be determined is applied to analyse the ductile fracture behaviour of axisymmetric bars.     

主应力轴旋转对压实黄土动变形特性的影响

实际工程中的地基或路基的土体往往处于复杂初始应力状态,在地震或其他动荷载的作用下会出现变形和沉降,而常规的室内土工试验无法真实再现这种固结应力条件下土的动力特性。通过对原有的DTC 199型周期扭转荷载三轴仪进行简单的改造后可进行土体在主应力轴发生旋转时的压实黄土动变形的试验研究。结果表明,在其他固结条件不变的情况下,初始主应力方向角α对压实黄土的动剪切模量有一定的影响,随着α的增加,动剪切模量有减小的趋势,最大动剪应力也逐渐减小,但是α对最大动剪切模量的影响不太显著。初始主应力方向角α对压实黄土的阻尼比基本没有影响,在λ-lgγd的半对数坐标图中,阻尼比随动剪应变的增加有逐渐增大的趋势,并且表现出较好的相关性。     

ANISOTROPIC STRENGTH CRITERION BASED ON T CRITERION AND THE TRANSFORMATION STRESS METHOD 1)

岩土材料通常呈现出成层水平分布特点, 即可将其视为横观各向同性材料, 横观各向同性对于岩土材料的变形以及强度值都会产生显著的影响. 基于已提出的t强度准则, t强度准则是基于各向同性单元体中存在有效滑移面来构建的, 并根据该空间有效滑移面上主剪应力与主法向应力的比值达到一定阈值为破坏条件. 在空间中存在有效滑移面与物理沉积面, 基于上述两个面在空间的位置关系, 用两面夹角作为表征横观各向同性对剪切强度影响程度的参量, 并假定当该夹角值越大, 则各向异性对强度贡献程度越大, 对应更大的应力比强度值, 反之, 则对应更小的应力比强度值. 基于上述思路并类比将其推广为正交三维各向异性准则, 基于三维各向异性材料的三维沉积面, 提出了三维特征沉积面的概念, 并基于空间滑移面与三维特征沉积面之间的夹角作为度量各向异性程度的变量, 提出了基于两面角作为参量考虑原生各向异性的应力比强度公式, 并利用该应力比强度公式来修正已提出的t强度准则, 最终建立了考虑各向异性影响的t准则公式. 在上述准则基础上, 考虑将各向异性应力空间转换为各向同性应力空间的思路, 在各向异性t准则基础上, 推导得到了基于各向异性强度t准则的变换应力公式, 利用变换应力公式可以将传统的以p, q为变量的各向同性本构模型转变为可考虑各向异性的三维本构模型. 通过对岩土材料的强度以及真三轴条件下的应力应变关系试验数据预测, 验证了所提的各向异性t准则及其变换应力公式的有效性及适用性.     

The growth of unstable thermoplastic shear with application to steady-wave shock compression in solids*

The catastrophic growth of unstable thermoplastic shear following the transition from homogeneous deformation to heterogeneous localized deformation through distributed shear banding is studied through approximate analytic and computational methods. The calculations provide expressions for shear band widths, spacing, catastrophic growth times and the rate of stress communication between shear bands. The optimum shear band width and spacing are found to be consistent with a minimum work principle. The model predicts that the product of the energy dissipated and the localization time in the shear localization process is invariant with respect to changes in the driving strain rate. Such behavior has been noted in the steady-wave shock compression of a number of solids. The calculations are applied to heterogeneous shear localization observed in the shock compression of aluminum.     

Effect of porosity on the properties of strain localization in porous media under undrained conditions

Within the general framework of mixture theory and by introducing the fictitious “fluid phase” as a mixture of a liquid and a gas, the conditions for localization of deformation into a shear band in the incremental response of partially saturated and fully saturated elastic–plastic porous media under undrained conditions are derived. The effect of porosity is included in the derivation. The explicit analytical expressions of the direction of shear band initiation and the corresponding hardening modulus of the porous media for the plane strain case are deduced, and a parametric analysis is made of the influence of the porosity on the properties of strain localization based on Mohr–Coulomb yield criterion. It is found that the dependence of the shear banding properties of partially saturated porous media on the porosity is related to the stress states and Poisson's ratio. However, the properties of the strain localization for the fully saturated porous media are almost independent of Poisson's ratio. Finally, on the basis of Mohr–Coulomb yield criterion, some solutions of the shear banding orientation for water-saturated granular materials are obtained, which are proved to be in good agreement with the experimental results reported by other researchers.