Localization of deformation and stress–strain relation for mesoscopic heterogeneous brittle rock materials under unloading

Stress redistribution induced by excavation of underground engineering and slope engineering results in the unloading zone in parts of surrounding rock masses. The mechanical behaviors of crack-weakened rock masses under unloading are different from those of crack-weakened rock masses under loading. A micromechanics-based model has been proposed for brittle rock material undergoing irreversible changes of their microscopic structures due to microcrack growth when axial stress is held constant while lateral confinement is reduced. The basic idea of the present model is to classify the constitution relation of rock material into four stages including some of the stages of linear elasticity, pre-peak nonlinear hardening, rapid stress drop, and strain softening, and to investigate their corresponding micromechanical damage mechanisms individually. Special attention is paid to the transition from structure rearrangements on microscale to the macroscopic inelastic strain, to the transition from distribution damage to localization of damage and the transition from homogeneous deformation to localization of deformation. The closed-form explicit expression for the complete stress–strain relation of rock materials containing cracks under unloading is obtained. The results show that the complete stress–strain relation and the strength of rock materials under unloading depend on the crack spacing, the fracture toughness of rock materials, orientation of the cracks, the crack half-length and the crack density parameter.     

Analysis of the localization of damage and the complete stress-strain relation for mesoscopic heterogeneous brittle rock subjected to compressive loads

A micromechanics-based model is established. The model takes the interaction among sliding cracks into account, and it is able to quantify the effect of various parameters on the localization condition of damage and deformation for brittle rock subjected to compressive loads. The closed-form explicit expression for the complete stress-strain relation of rock containing microcracks subjected to compressive loads was obtained. It is showed that the complete stress-strain relation includes linear elasticity, nonlinear hardening, rapid stress drop and strain softening. The behavior of rapid stress drop and strain softening is due to localization of deformation and damage. Theoretical predictions have shown to be consistent with the exoerimental results.     

Study on the coalescence mechanism of splitting failure of crack-weakened rock subjected to compressive loads

It is of important significance to study the coalescence mechanism of splitting failure of crack-weakened rock masses under compressive loads. In this paper, a simplified mechanism of crack propagation, in which the crack grows along the direction of maximum principal compressive stress, is proposed. Thus, only mode I is taken into account in the formulation and solution. On the basis of the near crack line analysis method, the elastic–plastic stress field near the crack line is analyzed, and the law that the length of the plastic zone along the crack line is varied with an external loads have been established by the matching condition of the elastic- plastic fields on the boundary, the coalescence stress and the strength properties of rock masses have been determined. The solution is a function of the geometry of the crack array. The results show that the crack coalescence depends on the crack interface friction coefficient, the sliding crack spacing, orientation of the cracks, and the crack half-length. The conclusions are of important significance for rock mass engineering.     

Zonal disintegration of deep crack-weakened rock masses: A non-Euclidean model

Rock masses are characterized by the existence of distributed joints and fractures. One of behaviors of the deep rock masses is high in situ stresses. The internal space of rock-like materials subjected to high in situ stresses after deformation is treated as a non-Euclidean one. The incompatible deformation of the deep rock masses is induced by high in situ stresses within the framework of non-Euclidean geometric space. A non-Euclidean model in which effects of cracks on zonal disintegration phenomenon of the deep crack-weakened rock masses is taken into account is established. Based on the non-Euclidean model, the elastic stress-field distribution of the deep surrounding rock masses induced by compatible deformation of non-fractured zones and incompatible deformation of fractured zones is determined. The stress intensity factors at the tips of cracks is given out. The strain energy density factor is applied to investigate the occurrence of disintegration zones. It is observed from the numerical results that the magnitude and site of fractured zones depend on the value of in situ stress, mesomechanical parameters and non-Euclidean parameters.     

Rock burst of deep circular tunnels surrounded by weakened rock mass with cracks

Rock masses containing pre-existing cracks are considered as non-homogeneous geomaterials. During excavation of tunnels, pre-existing cracks may nucleate, grow and propagate through rock matrix, then secondary cracks may appear. The stress concentration at the tips of secondary cracks is comparatively large, which may lead to the unstable growth and coalescence of secondary cracks, and consequently the occurrence of fractured zones. For brittle rocks, the dissipative energy of slip and growth of pre-existing cracks and secondary crack growth is small, but the elastic strain energy storing in rock masses may be larger than the dissipative energy of slip and growth of pre-existing macrocracks and secondary crack growth. The sudden release of the residual elastic strain energy may lead to rock burst in crack-weakened rock masses. Based on this understanding, the criteria of rock burst in crack-weakened rock masses are established. The influences of the in situ stresses, micromechanical parameters and physico-mechanical parameters on the distribution of rock burst zones and area of rock burst zones are investigated in detail.     

Triaxial compressive behavior of rock with mesoscopic heterogenous behavior: Strain energy density factor approach

The strain energy density factor approach is used in conjunction with a micromechanics model to investigate the condition and direction of shear failure for brittle rock subjected to triaxial compression. Moderate confinement in addition to localized deformation and damage are considered. Quantified are the effects of the various geometric and load parameters that involve the interaction of microcrack, friction and the confining pressure such that the path of the wing crack is taken into account. The influence of all microcracks with different orientations are introduced into the constitutive relation. The closed-form solution for the complete stress–strain relation of rock containing microcracks is obtained. It is shown that the complete stress–strain relationship includes linear, nonlinear hardening, rapid stress drop and strain softening effects. The theoretical results show that deviation of the direction of wing cracks from the line of the pre-existing crack decreases with increasing confinement pressure and friction coefficient. Theoretical predictions and experimental results show good agreement.     

Zonal fracturing mechanism in deep crack-weakened rock masses

The mechanical behaviors of deep crack-weakened rock masses are different from those of shallow crack-weakened rock masses. The surrounding rock in shallow crack-weakened rock mass engineering is classified into loose zone, plastic zone and elastic zone, while the surrounding rock in deep crack-weakened rock mass engineering is classified into fractured zone and non-fractured zone, which occur alternatively. It is assumed that the deep rock masses contain one joint set, in which the probability density function describing the distribution of sizes is assumed to follow the Rayleigh distribution, and the probability density function describing the distribution of spacing is assumed to follow the Weibull distribution. On the basis of strength criterion of deep rock mass, the near-field stress redistribution around circular opening induced by excavation is determined. The strong interaction among cracks is investigated by using the dislocation model. The nucleation, growth, interaction and coalescence of cracks were analyzed based on the strain energy density factor theory. When cracks coalesce, failure of deep crack-weakened rock masses occurs, fractured zone is formed. Then, size and quantity of fractured zone and non-fractured zone are given out. The size and quantity of fractured zone increase with decreasing strength of rock mass. The size and quantity of fractured zone increase with increasing in situ stress. Zonal fracturing phenomenon occurs once value of in situ stress is larger than the unaxial compressive strength of rock masses. The size and quantity of fractured zone decrease with increasing λ when p₂ > p₁. The size and quantity of fractured zone increase with increasing λ when p₂ < p₁.     

Nonlinear granular micromechanics model for multi-axial rate-dependent behavior

Constitutive equations for class of materials that possess granular microstructure can be effectively derived using granular micromechanics approach. The stress–strain behavior of such materials depends upon the underlying grain scale mechanisms that are modeled by using appropriate rate-dependent inter-granular force–displacement relationships. These force–displacement functions are nonlinear and implicit evolutions equations. The numerical solution of such equation under applied overall stress or strain loading can entail significant computational expense. To address the computations issue, an efficient explicit time-integration scheme has been derived. The developed model is then utilized to predict primary, secondary and tertiary creep as well as rate-dependent response under tensile and compressive loads for hot mix asphalt. Further, the capability of the derived model to describe multi-axial behavior is demonstrated through generations of biaxial time-to-creep failure envelopes and rate-dependent failure envelopes under monotonic biaxial and triaxial loading. The advantage of the approach presented here is that we can predict the multi-axial effects without resorting to complex phenomenological modeling.     

考虑裂纹闭合效应的岩石损伤本构关系

岩石中的预存裂纹只有在一定的法向压应力即裂纹闭合应力的作用下才可能闭合，其闭合过程与其方位和外加应力场有关，并且，即使对于裂纹已经完全闭合的岩石，如果裂纹闭合应力不同，则岩石的应力应交关系也不相同。本文建立了考虑裂纹闭合效应的岩石细观损伤力学模型，分析了裂纹闭合应力对岩石损伤演化过程和应力应变关系的影响。数值结果表明裂纹闭合应力显著地改变岩石的应力应变关系，表现为随裂纹闭合应力的增加，岩石的轴向应变变化较小，侧向应变和体积应变则大为增加。     

Prediction of 42CrMo steel flow stress at high temperature and strain rate

The compressive deformation behavior of 42CrMo steel was investigated at temperatures ranging from 850 to 1150 °C and strain rates from 0.01 to 50 s⁻¹ on Gleeble-1500 thermo-simulation machine. Based on the classical stress–dislocation relation and the kinematics of the dynamic recrystallization, the flow stress constitutive equations of the work hardening-dynamical recovery period and dynamical recrystallization period were established for 42CrMo steel, respectively. The stress–strain curves of 42CrMo steel predicted by the established models are in good agreement with experimental results when the strain rate is relatively low. So, the proposed deformation constitutive equations can be used to establish the hot formation processing parameters for 42CrMo steel.     

A fracture damage model for Jointed rock masses and its application to the stability analysis of dam abutments

Based on continuum damage mechanics, for jointed rock masses, a fracture damage model is presented in this paper. First, the damage tensors are defined through the elastic-flexibility of intact rock and the equivalent elastic-damage flexibility for rock mass. Then, by the self-consistent principle of solid mechanics, the equivalent elastic-damage flexibility tensors involving the interaction between multicracks are deduced. The damage evolution law is proposed involving the mechanism of crack propagation process: frictional sliding, crack kinking, growing of branched tension cracks, interlinking of the microcracks near branched crack tips leading to the breakthrough of macro-cracks and finally the failure of rock mass. Thus the evolution of damage variables reasonably unified with the process of crack propagation is given. Finally, a plastic-brittle damage constitutive relation including brittle coupled strain rate, developed and applied to the stability analysis of complicated rock foundation of a dam in China, is described in this paper.     

ZONAL DISINTEGRATION MECHANISM OF DEEP CRACK-WEAKENED ROCK MASSES UNDER DYNAMIC UNLOADING

Size and quantity of fractured zone and non-fractured zone are controlled by cracks contained in deep rock masses. Zonal disintegration mechanism is strongly dependent on the interaction among cracks. The strong interaction among cracks is investigated using stress superposition principle and the Chebyshev polynomials expansion of the pseudo-traction. It is found from numerical results that crack nucleation, growth and coalescence lead to failure of deep crack- weakened rock masses. The stress redistribution around the surrounding rock mass induced by unloading excavation is studied. The effect of the excavation time on nucleation, growth, interaction and coalescence of cracks was analyzed. Moreover, the influence of the excavation time on the size and quantity of fractured zone and non-fractured zone was given. When the excavation time is short, zonal disintegration phenomenon may occur in deep rock masses. It is shown from numerical results that the size and quantity of fractured zone increase with decreasing excavation time, and the size and quantity of fractured zone increase with the increasing value of in-situ geostress.     

Constitutive relationship of brittle rock subjected to dynamic uniaxial tensile loads with microcrack interaction effects

A micromechanical model is proposed to describe both stable and unstable damage evolution in microcrack-weakened brittle rock material subjected to dynamic uniaxial tensile loads. The basic idea of the present model is to classify the constitution relationship of rock material subjected to dynamic uniaxial tensile loads into four stages including some of the stages of linear elasticity, pre-peak nonlinear hardening, rapid stress drop, and strain softening, and to investigate their corresponding micromechanical damage mechanisms individually. Special attention is paid to the transition from structure rearrangements on microscale to the macroscopic inelastic strain, to the transition from distribution damage to localization of damage and the transition from homogeneous deformation to localization of deformation. The influence of all microcracks with different sizes and orientations are introduced into the constitutive relation by using the statistical average method. Effects of microcrack interaction on the complete stress-strain relation as well as the localization of damage for microcrack-weakened brittle rock material are analyzed by using effective medium method. Each microcrack is assumed to be embedded in an approximate effective medium that is weakened by uniformly distributed microcracks of the statistically-averaged length depending on the actual damage state. The elastic moduli of the approximate effective medium can be determined by using the dilute distribution method. Micromechanical kinetic equations for stable and unstable growth characterizing the ‘process domains’ of active microcracks are taken into account. These ‘process domains’ together with ‘open microcrack domains’ completely determine the integration domains of ensemble averaged constitutive equations relating macro-strain and macro-stress. Theoretical predictions have shown to consistent with the experimental results.     

ANALYSIS OF THE LOCALIZATION OF DAMAGE AND THE COMPLETE STRESS-STRAIN RELATION FOR MESOSCOPIC HETEROGENEOUS ROCK UNDER UNIAXIAL TENSILE LOADING

The mechanical behavior of rock under uniaxial tensile loading is different from that of rock under compressive loads. A micromechanics-based model was proposed for mesoscopic heterogeneous brittle rock undergoing irreversible changes of their microscopic structures due to microcrack growth. The complete stress-strain relation including linear elasticity, nonlinear hardening, rapid stress drop and strain softening was obtained. The influence of all microcracks with different sizes and orientations were introduced into the constitutive relation by using the probability density function describing the distribution of orientations and the probability density function describing the distribution of sizes. The influence of Weibull distribution describing the distribution of orientations and Rayleigh function describing the distribution of sizes on the constitutive relation were researched. Theoretical predictions have shown to be consistent with the experimental results.     

脆性岩石断裂破坏机理的边界配位法分析

针对裂纹表面承受载荷时的应力条件，提出了新的应力函数，该应力函数对于各种裂纹模型、各种边界条件、各种边界形状、裂纹表面自由或承受均布载荷等均适用．并利用边界配位法，计算了在压缩载荷下，岩石内部裂纹的应力强度因子（ＳＩＦ），给出了关于岩石断裂破坏的一些新结论     

混凝土率型内时损伤本构模型

混凝土是一种典型的率敏感材料,为了更好地描述混凝土结构在动力、冲击荷载作用下的强度和变形特征,本文结合内时理论和损伤理论建立了一种考虑混凝土率效应的内时损伤本构模型。该模型的特点:将混凝土材料的受力软化效应分解为密实状态的塑性效应和由微裂缝扩展引起的刚度退化效应。前者由内时理论来描述,这使该模型摆脱了一般弹塑性模型中屈服面的概念,从而更符合混凝土的变形特性,并且简化了非线性计算过程;后者由损伤理论来描述,根据混凝土的动力试验结果建立了增量型的损伤演变方程,从而使该模型能够较好地反映混凝土的动力特性。最后,应用本文建议的模型对一钢筋混凝土简支梁进行了非线性分析,结果表明:当结构承受快速荷载作用时,应变率对结构的受力性能影响较大,在进行结构分析时必须予以考虑。     

An interacting micro-crack damage model for failure of brittle materials under compression

A model is developed for brittle failure under compressive loading with an explicit accounting of micro-crack interactions. The model incorporates a pre-existing flaw distribution in the material. The macroscopic inelastic deformation is assumed to be due to the nucleation and growth of tensile “wing” micro-cracks associated with frictional sliding on these flaws. Interactions among the cracks are modeled by means of a crack-matrix-effective-medium approach in which each crack experiences a stress field different from that acting on isolated cracks. This yields an effective stress intensity factor at the crack tips which is utilized in the formulation of the crack growth dynamics. Load-induced damage in the material is defined in terms of a scalar crack density parameter, the evolution of which is a function of the existing flaw distribution and the crack growth dynamics. This methodology is applied for the case of uniaxial compression under constant strain rate loading. The model provides a natural prediction of a peak stress (defined as the compressive strength of the material) and also of a transition strain rate, beyond which the compressive strength increases dramatically with the imposed strain rate. The influences of the crack growth dynamics, the initial flaw distribution, and the imposed strain rate on the constitutive response and the damage evolution are studied. It is shown that different characteristics of the flaw distribution are dominant at different imposed strain rates: at low rates the spread of the distribution is critical, while at high strain rates the total flaw density is critical.     

Dynamic damage constitutive relation of mesoscopic heterogenous brittle rock under rotation of principal stress axes

A micro-mechanics-based model is developed to investigate microcrack damage mechanism of four stages of brittle rock under rotation of the principal stress axes. They consist of linear elastic, non-linear hardening, rapid stress drop and strain softening. The frictional sliding crack model is applied to analyze microcracks nucleation, propagation and coalescence. The strain energy density factor approach is applied to determine the critical condition of microcrack nucleation, propagation and coalescence. The inelastic strain increments are formulated within the framework of thermodynamics with internal variables. Rotation of principal stress axes affect the dynamic damage constitutive relationship and the failure strength of brittle rock.     

Dynamic limit analysis formulation for impact simulation of structural members

This paper introduces an extended concept of limit analysis to deal with the dynamic equilibrium condition considering the inertia and strain-rate effect for dynamic behavior of structures. The conventional limit analysis method has been applied to only static collapse analysis of structures without consideration of dynamic effects in the structural behavior. A dynamic formulation for the limit analysis has been derived for incremental analysis dealing with time integration, strain and stress evaluation, strain hardening, strain-rate hardening and thermal softening. The time dependent term in the governing equation is integrated with the WBZ-α method. The dynamic material behavior is described by the Johnson–Cook model in order to consider strain-rate hardening and thermal softening as well as strain hardening. Simulations have been carried out for impact analysis of a Taylor bar and an S-rail and their numerical results are compared with elasto-plastic explicit analysis results by LS-DYNA3D. Comparison demonstrates that the dynamic finite element limit analysis can predict the crashworthiness of structural members effectively with less effort and computing time than the commercial code compared. The crashworthiness of a structure with the rate-dependent constitutive model is also compared to that with the quasi-static constitutive relation in order to investigate the dynamic effect on deformation of structures.     

Finite viscoelasticity and viscoplasticity of semicrystalline polymers

Observations are reported on low-density polyethylene in uniaxial tensile and compressive tests with various strain rates and in tensile and compressive relaxation tests with various strains. A constitutive model is developed for the time-dependent response of a semicrystalline polymer at arbitrary three-dimensional deformations with finite strains. A polymer is treated as an equivalent network of chains bridged by junctions (entanglements between chains in the amorphous phase and physical cross-links at the lamellar surfaces). Its viscoelastic behavior is associated with separation of active strands from temporary junctions and merging of dangling strands with the inhomogeneous network. The viscoplastic response is attributed to sliding of junctions between chains with respect to their reference positions. Constitutive equations are derived by using the laws of thermodynamics. The stress–strain relations involve 6 material constants that are found by matching the observations.