A statistical damage-based constitutive model for shearing of rock joints in brittle drop mode

Some rock joints exhibit significant brittleness,characterized by a sharp decrease in shear stress upon reaching the peak strength.However,existing models often fail to accurately represent this behavior and are encumbered by numerous parameters lacking clear mechanical significance.This study presents a new statistical damage constitutive model rooted in both damage mechanics and statistics,containing only three model parameters.The proposed model encompasses all stages of joint shearing,including the compaction stage,linear stage,plastic yielding stage,drop stage,strain softening stage,and residual strength stage.To derive the analytical expression of the constitutive model,three boundary conditions are introduced.Experimental data from both natural and artificial rock joints is utilized to validate the model,resulting in average absolute relative errors ranging from 3%to 8%.Moreover,a comparative analysis with established models illustrates that the proposed model captures stress drop and post-peak strain softening more effectively,with model parameters possessing clearer mechanical interpretations.Furthermore,parameter analysis is conducted to investigate the impacts of model parameters on the curves and unveil the relationship between these parameters and the mechanical properties of rock joints.Importantly,the proposed model is straightforward in form,and all model parameters can be obtained from direct shear tests,thus facilitating the utilization in numerical simulations.

Nonlinear Mixed-Effects Models for Repairable Systems Reliability

Mixed-effects models,also called random-effects models,are a regression type of analysis which enables the analyst to not only describe the trend over time within each subject,but also to describe the variation among different subjects.Nonlinear mixed-effects models provide a powerful and flexible tool for handling the unbalanced count data.In this paper,nonlinear mixed-effects models are used to analyze the failure data from a repairable system with multiple copies.By using this type of models,statistical inferences about the population and all copies can be made when accounting for copy-to-copy variance.Results of fitting nonlinear mixed-effects models to nine failure-data sets show that the nonlinear mixed-effects models provide a useful tool for analyzing the failure data from multi-copy repairable systems.

Joint inversion of gravity and magnetic data for a two-layer model

Based on the synchronous joint gravity and magnetic inversion of single interface by Pilkington and the need of revealing Cenozoic and crystalline basement thickness in the new round of oil-gas exploration, we propose a joint gravity and magnetic inversion methodfor two-layer models by concentrating on the relationship between the change of thicknessI and position of the middle layer and anomaly and discuss the effects of the key parameters. Model tests and application to field data show the validity of this method.

Joint multivariate statistical model and its applications to synthetic earthquake predic-tion

针对目前地震综合预报中的一些问题，利用近30年来迅速发展的多元统计分析中主成分分析、判别分析组成多元统计组合模型，在众多的地震预报指标(预报因子)中采用信息最大化方法，选择对中期预测信息累积贡献率大于90％地震预报指标，分别进行相关分析、预测、检验，最终应用马氏距离判别作外推综合预报；并以华北地区(30°～42°N，108°125°E)为例进行模型的应用检验，初步研究已取得了较好的效果．

Nonlinear shear behavior of rock joints using a linearized implementation of the Barton-Bandis model

Experiments on rock joint behaviors have shown that joint surface roughness is mobilized under shearing,inducing dilation and resulting in nonlinear joint shear strength and shear stress vs.shear displacement behaviors.The Barton-Bandis(B-B) joint model provides the most realistic prediction for the nonlinear shear behavior of rock joints.The B-B model accounts for asperity roughness and strength through the joint roughness coefficient(JRC) and joint wall compressive strength(JCS) parameters.Nevertheless,many computer codes for rock engineering analysis still use the constant shear strength parameters from the linear Mohr-Coulomb(M-C) model,which is only appropriate for smooth and non-dilatant joints.This limitation prevents fractured rock models from capturing the nonlinearity of joint shear behavior.To bridge the B-B and the M C models,this paper aims to provide a linearized implementation of the B-B model using a tangential technique to obtain the equivalent M-C parameters that can satisfy the nonlinear shear behavior of rock joints.These equivalent parameters,namely the equivalent peak cohesion,friction angle,and dilation angle,are then converted into their mobilized forms to account for the mobilization and degradation of JRC under shearing.The conversion is done by expressing JRC in the equivalent peak parameters as functions of joint shear displacement using proposed hyperbolic and logarithmic functions at the pre-and post-peak regions of shear displacement,respectively.Likewise,the pre-and post-peak joint shear stiffnesses are derived so that a complete shear stress-shear displacement relationship can be established.Verifications of the linearized implementation of the B-B model show that the shear stress-shear displacement curves,the dilation behavior,and the shear strength envelopes of rock joints are consistent with available experimental and numerical results.

Discrete modeling of rock joints with a smooth-joint contact model

Structural defects such as joints or faults are inherent to almost any rock mass.In many situations those defects have a major impact on slope stability as they can control the possible failure mechanisms.Having a good estimate of their strength then becomes crucial.The roughness of a structure is a major contributor to its strength through two different aspects,i.e.the morphology of the surface(or the shape)and the strength of the asperities(related to the strength of the rock).In the current state of practice,roughness is assessed through idealized descriptions(Patton strength criterion)or through empirical parameters(Barton JRC).In both cases,the multi-dimensionality of the roughness is ignored.In this study,we propose to take advantage of the latest developments in numerical techniques.With3D photogrammetry and/or laser mapping,practitioners have access to the real morphology of an exposed structure.The derived triangulated surface was introduced into the DEM(discrete element method)code PFC3D to create a synthetic rock joint.The interaction between particles on either side of the discontinuity was described by a smooth-joint model(SJM),hence suppressing the artificial roughness introduced by the particle discretization.Shear tests were then performed on the synthetic rock joint.A good correspondence between strengths predicted by the model and strengths derived from well-established techniques was obtained for thefirst time.Amongst the benefits of the methodology is the possibility offered by the model to be used in a quantitative way for shear strength estimates,to reproduce the progressive degradation of the asperities upon shearing and to analyze structures of different scales without introducing any empirical relation.

An Error Equivalent Model of Revolute Joints with Clearances for Antenna Pointing Mechanisms

Joint clearances in antenna pointing mechanisms lead to uncertainty in function deviation. Current studies mainly focus on radial clearance of revolute joints, while axial clearance has rarely been taken into consideration. In fact, own?ing to errors from machining and assembly, thermal deformation and so forth, practically, axial clearance is inevitable in the joint. In this study, an error equivalent model(EEM) of revolute joints is proposed with considering both radial and axial clearances. Compared to the planar model of revolute joints only considering radial clearance, the journal motion inside the bearing is more abundant and matches the reality better in the EEM. The model is also extended for analyzing the error distribution of a spatial dual?axis("X–Y" type) antenna pointing mechanism of Spot?beam antennas which especially demand a high pointing accuracy. Three case studies are performed which illustrates the internal relation between radial clearance and axial clearance. It is found that when the axial clearance is big enough, the physical journal can freely realize both translational motion and rotational motion. While if the axial clearance is limited, the motion of the physical journal will be restricted. Analysis results indicate that the consideration of both radial and axial clearances in the revolute joint describes the journal motion inside the bearing more precise. To further validate the proposed model, a model of the EEM is designed and fabricated. Some suggestions on the design of revolute joints are also provided.

Identification of the Mechanical Joint Parameters with Model Uncertainty

Joint parameter identification is a key problem in the modeling of complex structures. The behavior of joint may be random due to the random properties of preload and joint geometries, contact surface and its finish, etc. A method is presented to simulate the joint parameters as probabilistic variables. In this method the response surface based model updating method and probabilistic approaches are employed to identify the parameters. The study implies that joint parameters of some structures have normal or nearly normal distributions, and a linear FE model with probabilistic variables could illustrate dynamic characteristics of joints.

QUICK ASSESSMENT METHODOLOGY FOR RELIABILITY OF SOLDER JOINTS IN BALL GRID ARRAY (BGA) ASSEMBLY——PART Ⅰ: CREEP CONSTITUTIVE RELATION AND FATIGUE MODEL

In this study, a new unified creep constitutive relation and a mod- ified energy-based fatigue model have been established respectively to describe the creep flow and predict the fatigue life of Sn-Pb solders. It is found that the relation successfully elucidates the creep mechanism related to current constitutive relations. The model can be used to describe the temperature and frequency dependent low cycle fatigue behavior of the solder. The relation and the model are further employed in part Ⅱ to develop the numerical simulation approach for the long-term reliability assessment of the plastic ball grid array (BGA) assembly.

The distribution of deep source rocks in the GS Sag:joint MT-gravity modeling and constrained inversion

The coal-bearing strata of the deep Upper Paleozoic in the GS Sag have high hydrocarbon potential. Because of the absence of seismic data, we use electromagnetic （MT） and gravity data jointly to delineate the distribution of deep targets based on well logging and geological data. First, a preliminary geological model is established by using three-dimensional （3D） MT inversion results. Second, using the formation density and gravity anomalies, the preliminary geological model is modified by interactive inversion of the gravity data. Then, we conduct MT-constrained inversion based on the modified model to obtain an optimal geological model until the deviations at all stations are minimized. Finally, the geological model and a seismic profile in the middle of the sag is analysed. We determine that the deep reflections of the seismic profile correspond to the Upper Paleozoic that reaches thickness up to 800 m. The processing of field data suggests that the joint MT-gravity modeling and constrained inversion can reduce the multiple solutions for single geophysical data and thus improve the recognition of deep formations. The MT-constrained inversion is consistent with the geological features in the seismic section. This suggests that the joint MT and gravity modeling and constrained inversion can be used to delineate deep targets in similar basins.

Equivalent continuum modeling of beam-like truss structures with flexible joints

The paper investigated the equivalent continuum modeling of beam-like repetitive truss structures considering the flexibility of joints,which models the contact between the truss member and joint by spring-damper with six directional stiffnesses and dampings.Firstly,a two-node hybrid joint-beam element was derived for modeling the truss member with flexible end joints,and a condensed model for the repeating element with flexible joints was obtained.Then,the energy equivalence method was adopted to equivalently model the truss structure with flexible joints and material damping as a spatial viscoelastic anisotropic beam model.Afterwards,the equations of motion for the equivalent beam model were derived and solved analytically in the frequency domain.In the numerical studies,the correctness of the presented method was verified by comparisons of the natural frequencies and frequency responses evaluated by the equivalent beam model with the results of the finite element method model.

Joint modelling of location and scale parameters of the skew-normal distribution

Joint location and scale models of the skew-normal distribution provide useful ex- tension for joint mean and variance models of the normal distribution when the data set under consideration involves asymmetric outcomes. This paper focuses on the maximum likelihood estimation of joint location and scale models of the skew-normal distribution. The proposed procedure can simultaneously estimate parameters in the location model and the scale model. Simulation studies and a real example are used to illustrate the proposed methodologies.

A five-parameter constitutive model for hysteresis shearing and energy dissipation of rock joints

Rock joints exhibit hysteresis shearing behavior and produce energy dissipation under shear cyclic loads,which however cannot be accurately depicted by existing constitutive models. This paper establishes a constitutive model for hysteresis shearing and associated energy dissipation of rock joints. Analytical expressions of the model during cyclic shearing processes are derived. Derivation of the model indicates no energy dissipation in the elastic stage. When the shear load exceeds elastic boundary, nonlinear energy dissipation takes place. Validations with experiments show that the proposed model provides good conformities with direct shear curves and hysteresis loops, and can predict the energy dissipation characteristics of rock joints under different working conditions. Compared to the constitutive models using Weibull's distribution, the proposed one is smooth at the elastic boundary and can accurately capture the maximum shear stress. Unlike the existing incremental-type models, the proposed one provides clear and direct analytical expressions for both shear stress and energy dissipation during the whole displacement domain, which is more convenient in application.

Quantitative Metal Magnetic Memory Reliability Modeling for Welded Joints

Metal magnetic memory（MMM） testing has been widely used to detect welded joints. However, load levels, environmental magnetic field, and measurement noises make the MMM data dispersive and bring difficulty to quantitative evaluation. In order to promote the development of quantitative MMM reliability assessment, a new MMM model is presented for welded joints. Steel Q235 welded specimens are tested along the longitudinal and horizontal lines by TSC-2M-8 instrument in the tensile fatigue experiments. The X-ray testing is carried out synchronously to verify the MMM results. It is found that MMM testing can detect the hidden crack earlier than X-ray testing. Moreover, the MMM gradient vector sum K_（vs） is sensitive to the damage degree, especially at early and hidden damage stages. Considering the dispersion of MMM data, the K_（vs） statistical law is investigated, which shows that K_（vs） obeys Gaussian distribution. So K_（vs） is the suitable MMM parameter to establish reliability model of welded joints. At last, the original quantitative MMM reliability model is first presented based on the improved stress strength interference theory. It is shown that the reliability degree R gradually decreases with the decreasing of the residual life ratio T, and the maximal error between prediction reliability degree R_1 and verification reliability degree R_2 is 9.15%. This presented method provides a novel tool of reliability testing and evaluating in practical engineering for welded joints.

Elastodynamic modeling and joint reaction prediction for 3-PRS PKM

To gain a thorough understanding of the load state of parallel kinematic machines(PKMs), a methodology of elastodynamic modeling and joint reaction prediction is proposed. For this purpose, a Sprint Z3 model is used as a case study to illustrate the process of joint reaction analysis. The substructure synthesis method is applied to deriving an analytical elastodynamic model for the 3-PRS PKM device, in which the compliances of limbs and joints are considered. Each limb assembly is modeled as a spatial beam with non-uniform cross-section supported by lumped virtual springs at the centers of revolute and spherical joints. By introducing the deformation compatibility conditions between the limbs and the platform, the governing equations of motion of the system are obtained. After degenerating the governing equations into quasi-static equations, the effects of the gravity on system deflections and joint reactions are investigated with the purpose of providing useful information for the kinematic calibration and component strength calculations as well as structural optimizations of the 3-PRS PKM module. The simulation results indicate that the elastic deformation of the moving platform in the direction of gravity caused by gravity is quite large and cannot be ignored. Meanwhile, the distributions of joint reactions are axisymmetric and position-dependent. It is worthy to note that the proposed elastodynamic modeling method combines the benefits of accuracy of finite element method and concision of analytical method so that it can be used to predict the stiffness characteristics and joint reactions of a PKM throughout its entire workspace in a quick and accurate manner. Moreover, the present model can also be easily applied to evaluating the overall rigidity performance as well as statics of other PKMs with high efficiency after minor modifications.

Analytical model of shear mechanical behaviour of bolted rock joints considering influence of normal stress on bolt guide rail effect

Rock bolts have been widely used in slopes as a reinforcement measure.Modelling the shear mechanical behaviours of bolted rock joints is very complicated due to the complex factors that affect the axial force and shear force on the bolts.Rock bolts under shear action exhibit the guide rail effect;that is,the rock mass slides along the rock bolt as if the rock bolt is a rail.The normal stress can inhibit the guide rail effect and reduce the axial force on bolts.However,this factor is not considered by the existing analysis models.Shear tests of bolted joints under different normal stresses were carried out in the laboratory.During the test,the axial force on each point monitored on the bolt was recorded by a strain gauge,and the attenuation trend of the strain was studied.An analytical model that considers the inhibition of the bolt rail effect due to an increase in the normal stress was proposed to predict the shear mechanical behaviour of rock bolted joints.The new model accommodates the bolt shear behaviours in the elastic stage and plastic stage,and the estimated values agree well with the results of the direct shear tests in the laboratory.The validation shows that the proposed model can effectively describe the deformation characteristics of the bolts in the shear tests.

A skew–normal mixture of joint location, scale and skewness models

Normal mixture regression models are one of the most important statistical data analysis tools in a heterogeneous population. When the data set under consideration involves asymmetric outcomes, in the last two decades, the skew normal distribution has been shown beneficial in dealing with asymmetric data in various theoretic and applied problems. In this paper, we propose and study a novel class of models： a skew-normal mixture of joint location, scale and skewness models to analyze the heteroscedastic skew-normal data coming from a heterogeneous population. The issues of maximum likelihood estimation are addressed. In particular, an Expectation-Maximization （EM） algorithm for estimating the model parameters is developed. Properties of the estimators of the regression coefficients are evaluated through Monte Carlo experiments. Results from the analysis of a real data set from the Body Mass Index （BMI） data are presented.

An Anisotropic Geomechanical Model for Jointed Rock Masses at Taishan Nuclear Power Station, Southern China

An anisotropic geomechanical model for jointed rock mass is presented. Simultaneously with deriving the orthotropic anisotropy elastic parameters along the positive axis, the equivalent compliance matrix for the deflection axis orthotropic anisotropy was derived through a three- dimensional coordinate transformation. In addition, Singh＇s analysis of the stress concentration effects of intermittent joints was adopted, based on two groups of intermittent joints and a set of cross- cutting joints in the jointed rock mass. The stress concentration effects caused by intermittent joints and the coupling effect of cross-cutting joints along the deflection-axis are also considered. The proposed anisotropic mechanics parameters method is applied to determine the deformation parameters of jointed granite at the Taishan Nuclear Power Station. Combined with the deterministic mechanical parameters of rock blocks and joints, the deformation parameters and their variability in jointed rock masses are estimated quantitatively. The computed results show that jointed granite at the Taishan Nuclear Power Station exhibits typical anisotropic mechanical characteristics; the elastic moduli in the two horizontal directions were similar, but the elastic modulus in the vertical direction was much greater. Jointed rock elastic moduli in the two horizontal and vertical directions were respectively about 24% and 37% of the core of rock, showing weakly orthotropic anisotropy; the ratio of elastic moduli in the vertical and horizontal directions was 1.53, clearly indicating the transversely isotropic rock mass mechanical characteristics. The method can be popularized to solve other rock mechanics problems in nuclear power engineering.

Joint Variable Selection of Mean-Covariance Model for Longitudinal Data

In this paper we reparameterize covariance structures in longitudinal data analysis through the modified Cholesky decomposition of itself. Based on this modified Cholesky decomposition, the within-subject covariance matrix is decomposed into a unit lower triangular matrix involving moving average coefficients and a diagonal matrix involving innovation variances, which are modeled as linear functions of covariates. Then, we propose a penalized maximum likelihood method for variable selection in joint mean and covariance models based on this decomposition. Under certain regularity conditions, we establish the consistency and asymptotic normality of the penalized maximum likelihood estimators of parameters in the models. Simulation studies are undertaken to assess the finite sample performance of the proposed variable selection procedure.

Determination of key parameters of Al–Li alloy adhesively bonded joints using cohesive zone model

The key parameters of the adhesive layer of a reinforcing patch are of great significance and affect the ability to suppress crack propagation in an Al–Li alloy patch-reinforced structure.This paper proposes a method to determine the key parameters of the adhesive layer of adhesively bonded joints in the Al–Li alloy patch-reinforced structure.A zero-thickness cohesive zone model(CZM)was selected to simulate the adhesive layer’s fracture process,and an orthogonal simulation was designed to compare against the test results.A three-dimensional progressive damage model of an Al–Li alloy patch-reinforced structure with single-lap adhesively bonded joints was developed.The simulation’s results closely agree with the test results,demonstrating that this method of determining the key parameters is likely accurate.The results also verify the correctness of the cohesive strength and fracture energy,the two key parameters of the cohesive zone model.The model can accurately predict the strength and fracture process of adhesively bonded joints,and can be used in research to suppress crack propagation in Al–Li alloy patch-reinforced structures.