Simulation of size-effect behaviour through sensitivity analyses

Sensitivity analysis techniques are applied to the simulation of size effect behaviour. The scale factor is included in the discretised equilibrium equations. A gradient-enhanced damage model is used. The sensitivity of the equilibrium path with respect to the loading factor is then obtained through the direct differentiation method. Particular attention is paid to the proper differentiation of constitutive internal variables. The predictive possibilities of the algorithm are illustrated by means of an example.     

结构可靠性灵敏度因子的一种新指标

通过对可靠性灵敏度的分析和推导,利用灵敏度标准差值构造了一种新的可靠性灵敏度因子。与传统灵敏度因子不同的是,新因子不仅能反映各随机变量对结构失效概率影响的重要性程度,因子的数值大小还能表征将单个变量作为确定性变量处理或将多个变量同时作为确定性量处理时所引起的可靠指标误差大小。据此结论,采用新灵敏度因子进行灵敏度分析时,只需要进行一次分析就能识别出各随机变量对结构失效整体影响的有益信息,可以有效地提高计算效率,降低分析难度。利用数值算例验证说明了所提可靠性灵敏度因子指标的有效性。     

Micromechanical modeling of intrinsic and specimen size effects in microforming

Size effect is a crucial phenomenon in the microforming processes of metallic alloys involving only limited amount of grains. At this scale intrinsic size effect arises due to the size of the grains and the specimen/statistical size effect occurs due to the number of grains where the properties of individual grains become decisive on the mechanical behavior of the material. This paper deals with the micromechanical modeling of the size dependent plastic response of polycrystalline metallic materials at micron scale through a strain gradient crystal plasticity framework. The model is implemented into a Finite Element software as a coupled implicit user element subroutine where the plastic slip and displacement fields are taken as global variables. Uniaxial tensile tests are conducted for microstructures having different number of grains with random orientations in plane strain setting. The influence of the grain size and number on both local and macroscopic behavior of the material is investigated. The attention is focussed on the effect of the grain boundary conditions, deformation rate and the grain size on the mechanical behavior of micron sized specimens. The model is intrinsically capable of capturing both experimentally observed phenomena thanks to the incorporated internal length scale and the crystallographic orientation definition of each grain.     

Effect of geometry on compressive failure of notched composites

The compressive failure of carbon fibre-epoxy laminates is investigated theoretically and experimentally. Panels with a single edge notch, a central notch or a central hole are considered. The failure mechanism is by microbuckling in the 0° plies and is accompanied by delamination and plastic deformation in the off-axis plies [1]. To predict the critical length of the microbuckle and the failure load, the microbuckle is modelled as a cohesive zone. The magnitude of the normal compressive traction across the microbuckle is assumed to decrease linearly with increasing overlap of material on either side of the microbuckle. The relative effect of the specimen size and a bridging length scale is investigated to illustrate the transition between small-scale and large-scale bridging. If the bridging length scale is small compared with the specimen dimensions, the specimen fails when the stress intensity at the notch tip equals a critical compressive stress intensity factorK _IC . When the bridging length scale is not small compared with either the initial defect size or the unnotched ligament length then it is necessary to include the details of the traction across the microbuckle to predict the failure load accurately.     

Composite scale modeling in the presence of censored data

A composite scale modeling approach can be used to combine several scales or variables into a single scale or variable. A typical application is to combine age and usage together to form a composite timescale model. The combined scale is expected to have better failure prediction capability than individual scales. Two typical models are the linear and multiplicative models. Their parameters are determined by minimizing the sample coefficient of variation of the composite scale. The minimum coefficient of variation is hard to apply in the presence of censored data. Another open issue is how to identify key variables when a number of variables are combined. This paper develops methods to handle these two issues. A numerical example is also included to illustrate the proposed methods.     

骨料粒径对混凝土动态拉伸强度及尺寸效应影响分析

骨料粒径是影响混凝土力学性能及破坏机理的重要因素。从细观角度出发,将混凝土看作由骨料颗粒、砂浆基质及界面过渡区组成的三相复合材料,考虑细观组分的应变率效应,建立了混凝土动态拉伸破坏行为研究的细观力学分析模型,模拟研究了不同骨料粒径下混凝土动态拉伸破坏行为,并揭示了动态拉伸强度的尺寸效应规律。研究表明:低应变率下骨料不发生破坏,骨料粒径对混凝土动态拉伸破坏模式及拉伸强度影响显著,且拉伸强度的尺寸效应随骨料粒径的减小而削弱;高应变率下裂缝将贯穿骨料,骨料粒径的大小对混凝土动态拉伸强度及尺寸效应影响可忽略。最后,结合应变率效应的影响机制,建立了混凝土拉伸强度的"静动态统一"尺寸效应理论公式,该公式可以较好描述各骨料粒径下混凝土动态拉伸强度与试件尺寸的定量关系。     

The nominal tensile strength of disordered materials: A statistical fracture mechanics approach

The aim of the present paper is to study the variation of tensile strength of disordered materials in relation to the structural size. Griffith[1] gave an explanation to this phenomenon assuming the size of the most dangerous microdefect as proportional to the specimen size. Therefore, decreasing the specimen size, the size of this microdefect decreases and the tensile strength of the specimen increases.

A defect size distribution of self-similarity well be defined, for which the maximum defect size turns out to be proportional to the linear size of the body. In this way, a very general analytical expression for the tensile strength decrease with size can be obtained. Statistical Monte-Carlo simulations are performed to confirm this trend and to obtain the relation between the degree of disorder and the slope of the size effect law. The parameters of the defect size distribution—density and average size of the imperfections—can be obtained by experimental tensile tests.

In order to consider the whole scale range, a truncated statistical model will be presented. This new model assumes a truncated defect size distribution and predicts a scale effect in complete accordance with that of the Multifractal Scaling Law proposed by Carpinteri [2] and Carpinteri et al. [3–5]. The global failure probability is obtained by the composition of the n independent local failure conditions. The paper ends with the presentation of the results obtained by this truncated model and the comparison with some experimental data.     

Ductile failure of X100 pipeline steel – Experiments and fractography

Ductile failure of X100 pipeline steel is studied. In order to study the effect of anisotropy, triaxiality factor and Lode angle on the failure, a series of specimens with different geometries are tested. Measurements of deformation in three dimensions are done to check the variation of elongation and lateral deformations during loading. Extensive fractography is done using pictures taken by SEM to study the fracture surfaces completely. The results show that triaxiality causes the failure to occur sooner and Lode angle has no effect on the failure in triaxialities higher than that of simple tension. Moreover, delamination has occurred in some of the specimens. This phenomenon is studied and the causes of delamination are explained in macro scale.     

Positive size and scale effects of all-cellulose composite laminates

Negative size effects are commonly reported for advanced composite materials where the strength of the material decreases with increasing volume of the test specimen. In this work, the effect of increasing specimen volume on the mechanical properties of all-cellulose composites is examined by varying the laminate thickness. A positive size effect is observed in all-cellulose composite laminates as demonstrated by a 32.8% increase in tensile strength as the laminate thickness is increased by 7 times. The damage evolution in all-cellulose composite laminates was examined as a function of the tensile strain. Enhanced damage tolerance concomitant with increasing specimen volume is associated with damage accumulation due to transverse cracking and strain delocalisation. A transition from low-strain failure to tough and high-strain failure is observed as the laminate thickness is increased. Simultaneously, scale effects lead to an increase in the void content and cellulose crystallinity at the core, with increasing laminate thickness.     

基于Copula函数的并联结构系统可靠度分析

不完备概率信息条件下变量联合分布函数的确定及其对结构系统可靠度的影响还缺少系统地研究,该文目的在于研究表征变量间相关性的Copula函数对结构系统可靠度的影响规律。首先,简要介绍了变量联合分布函数构造的Copula函数方法。其次,提出了并联系统失效概率计算方法,并推导了相应的计算公式。最后以几种典型Copula函数为例研究了Copula函数类型对结构并联系统可靠度的影响规律。结果表明:表征变量间相关性的Copula函数类型对结构系统可靠度具有明显的影响,不同Copula函数计算的系统失效概率存在明显的差别,这种差别随构件失效概率的减小而增大。当并联系统的失效区域位于Copula函数尾部时,Copula函数的尾部相关性对系统可靠度有明显的影响,计算的失效概率比没有尾部相关性的Copula函数的失效概率大。当组成并联系统的两构件功能函数间正相关时,系统失效概率随相关系数的增大而增加;当构件功能函数间负相关时,系统失效概率随相关系数的增大而减小。此外,无论构件失效概率和变量间相关系数如何变化,Copula函数计算的失效概率都位于系统失效概率的上下限内。     

Statistics of strength of ceramics: finite weakest-link model and necessity of zero threshold

It is argued that, in probabilistic estimates of quasibrittle structure strength, the strength threshold should be considered to be zero and the distribution to be transitional between Gaussian and Weibullian. The strength histograms recently measured on tough ceramics and other quasibrittle materials, which have been thought to imply a Weibull distribution with nonzero threshold, are shown to be fitted equally well or better by a new weakest-link model with a zero strength threshold and with a finite, rather than infinite, number of links in the chain, each link corresponding to one representative volume element (RVE) of a non-negligible size. The new model agrees with the measured mean size effect curves. It is justified by energy release rate dependence of the activation energy barriers for random crack length jumps through the atomic lattice, which shows that the tail of the failure probability distribution should be a power law with zero threshold. The scales from nano to macro are bridged by a hierarchical model with parallel and series couplings. This scale bridging indicates that the power-law tail with zero threshold is indestructible while its exponent gets increased on each passage to a higher scales. On the structural scale, the strength distribution except for its far left power-law tail, varies from Gaussian to Weibullian as the structure size increases. For the mean structural strength, the theory predicts a size effect which approaches the Weibull power law asymptotically for large sizes but deviates from it at small sizes. This deviation is the easiest way to calibrate the theory experimentally. The structure size is measured in terms of the number of RVEs. This number must be convoluted by an integral over the dimensionless stress field, which depends on structure geometry. The theory applies to the broad class of structure geometries for which failure occurs at macro-crack initiation from one RVE, but not to structure geometries for which stability is lost only after large macro-crack growth. Based on tolerable structural failure probability of <10^?6, the change from nonzero to zero threshold may often require a major correction in safety factors.     

Age-dependent size effect and fracture characteristics of ultra-high performance concrete

This paper presents an investigation of the age-dependent size effect and fracture characteristics of ultra-high performance concrete (UHPC). The study is based on a unique set of experimental data connecting aging tests for two curing protocols of one size and size effect tests of one age. Both aging and size effect studies are performed on notched three-point bending tests. Experimental data are augmented by state-of-the-art simulations employing a recently developed discrete early-age computational framework. The framework is constructed by coupling a hygro-thermo-chemical (HTC) model and the Lattice Discrete Particle Model (LDPM) through a set of aging functions. The HTC component allows taking into account variable curing conditions and predicts the maturity of concrete. The mechanical component, LDPM, simulates the failure behavior of concrete at the length scale of major heterogeneities. After careful calibration and validation, the mesoscale HTC-LDPM model is uniquely posed to perform predictive simulations. The ultimate flexural strengths from experiments and simulations are analyzed by the cohesive size effect curves (CSEC) method, and the classical size effect law (SEL). The fracture energies obtained by LDPM, CSEC, SEL, and cohesive crack analyses are compared, and an aging formulation for fracture properties is proposed. Based on experiments, simulations, and size-effect analyses, the age-dependence of size effect and the robustness of analytical-size effect methods are evaluated.     

Quality by design approach for development of suspension nasal spray products: a case study on budesonide nasal suspension

The objective of this study was to provide quality by design (QbD) approach for development of suspension type nasal spray products. Quality target product profile (QTPP) of test product budesonide nasal suspension (B-NS) was defined and critical quality attributes (CQAs) were identified. Critical formulation, process and delivery device variables were recognized. Risk assessment was performed by using failure mode and effect analysis (FMEA) methodology. Selected variables were further assessed using a Plackett Burman screening study. A response surface design consisting of the critical factors was used to study the interactions between the study variables. Formulation variable X₂: median particle size of budesonide (D₅₀) (µ) has strikingly influenced dissolution (%) (Y₁), while D₅₀ droplet size distribution (µm) (Y₂) was significantly impacted by formulation variable X₁: Avicel RC 591 (%) and process variable X₄: homogenization speed (rpm). A design space plot within which the CQAs remained unchanged was established at lab scale. A comprehensive approach for development of B-NS product based on the QbD methodology has been demonstrated. The accuracy and robustness of the model were confirmed by comparability of the predicted value generated by model with the observed value.     

Multi-objective imperfect preventive maintenance optimisation with NSGA-II

Maintenance optimisation is a multi-objective problem in nature, and it usually needs to achieve a trade-off among the conflicting objectives. In this study, a multi-objective maintenance optimisation (MOMO) model is proposed for electromechanical products, where both the soft failure and hard failure are considered, and minimal repair is performed accordingly. Imperfect preventive maintenance (IPM) is carried out during the preplanned periods, and modelled with a hybrid failure rate model and quasi-renewal coefficient. The initial IPM period and the total number of IPM periods are set as the decision variables, and a MOMO model is developed to optimise the availability and cost rate concurrently. The fast elitist non-dominated sorting genetic algorithm (NSGA-II) is applied to solve the model. A case study of wind turbine’s gearbox is provided. The results show that there are 30 optimal solutions in the MOMO’s Pareto frontier that can maximise the availability and minimise the cost rate simultaneously. Compared with the single-objective maintenance optimisation, it can provide more choices for maintenance decision, and better satisfy the resource constraints and the customer’s preference. The results of the sensitivity analysis show that the effect of age reduction factor on optimisation results is greater than that of failure rate increase factor.     

A two scale damage concept applied to fatigue

The ductile type of damage is a phenomenon now well understood. Once the fully coupled set of constitutive equations is identified, Damage Mechanics is a powerful tool to predict failure. Brittle materials do not exhibit such a damageable macroscopic behavior. Nevertheless, they still fail. On the idea that damage is localized at the microscopic scale, a scale smaller than the mesoscopic one of the Representative Volume Element (RVE), we propose a three-dimensional failure modeling for monotonic as well as for fatigue loading. We develop a two scale model of what we shall call brittle damage: at the microscopic scale, micro-cracks or micro-voids exhibit a damageable plastic-like behavior with no effect on the global (mesoscopic) elastic behavior. Microscopic failure is assumed to coincide with the RVE failure. This model turns out to represent quite well physical phenomena related to high cycle fatigue such as the mean stress effect, the nonlinear accumulation of damage, initial strain hardening or damage effect and the nonproportional loading effect for bi-axial fatigue. The model has been implemented as a post-processor computer code. A simplified identification procedure for the determination of the material properties is given. This revised version was published online in July 2006 with corrections to the Cover Date.     

The role of delamination in failure of fibre-reinforced composites

The mechanisms by which delamination contributes to the failure of fibre-reinforced composites are reviewed. Through-thickness failure owing to interlaminar stresses is considered first, and the effect of delamination in impact and compression after impact. The way in which in-plane failure can occur by delamination and matrix cracks joining up to produce a fracture surface without the need to break fibres is considered next. Examples of quasi-isotropic laminates loaded at different off-axis angles, and with different numbers and thicknesses of ply blocks show large differences in unnotched tensile strength controlled by delamination from the free edge. Similar mechanisms determine the strength of notched specimens and give rise to the hole size effect, whereby tensile strength increases with decreasing hole diameter owing to increased delamination and splitting. Open hole tension and over-height compact tension tests with constant in-plane dimensions show a transition in failure mode with increasing ply block thickness from fibre-dominated fracture to complete delamination. In all these cases, the critical factor controlling strength is the relative propensity to delaminate.     

Snowpack properties associated with fracture initiation and propagation resulting in skier-triggered dry snow slab avalanches

This study investigates snowpack properties associated with skier-triggered dry slab avalanches, with a particular view on snowpack conditions favoring fracture propagation. This was done by analyzing a data set of over 500 snow profiles observed next to skier-triggered slabs (including remotely triggered slab avalanches and whumpfs) and on skier-tested slopes that did not release a slab avalanche. The relation of the snowpack variables with fracture initiation and fracture propagation, both of which are required for skier-triggering, was investigated. Specific snowpack characteristics, including hardness difference and difference in crystal size across the failure layer, associated with skier-triggered dry slab avalanches were identified and the frequency of skier-triggering was determined. In order to assess snowpack variables favouring fracture propagation, variables from failure layers associated with skier-triggered slabs that were not remotely triggered and relatively small were contrasted with snowpack variables from failure layers of remotely triggered slab avalanches, whumpfs and relatively large slab avalanches. The properties of the slab overlying the weak layer, as well as the layer above the weak layer, were found to affect fracture propagation. Stiffer slabs were associated with large avalanches as well as whumpfs and remotely triggered avalanches. Furthermore, a correlation analysis of snowpack variables with the size and width of the investigated slab avalanches further accentuated the importance of these slab properties with regards to fracture propagation.     

Statistical analysis of fracture in graphite

A statistical model is proposed to study the fracture of graphite. This model, based on a more general one proposed by She et al., uses a local fracture criterion for a microcrack, a distribution function for microcracks and the weakest link principle to predict failure probability as a function of applied loading and distribution of microcracks. The model considers the effect of the three-dimensional stress state and therefore gives a general representation of both the effects of complex loading and microcrack distribution. The inputs to the model can be determined by either studying the microstructural features of the graphite or by choosing inputs to make the model prediction fit a set of experimental results. The latter method is used to apply the model to failure results of Rose and Tucker. One set of results is used to calibrate the model which is then applied to predict the failure behavior of a second set of experimental results. Finally, the effect of the various input variables on failure probability is studied first by considering a graphical representative of failure probability as a function of input variables and second by writing the equations in terms of nondimensional variables.     

Reliability–sensitivity analysis using dimension reduction methods and saddlepoint approximations

Reliability–sensitivity, which is considered as an essential component in engineering design under uncertainty, is often of critical importance toward understanding the physical systems underlying failure and modifying the design to mitigate and manage risk. This paper presents a new computational tool for predicting reliability (failure probability) and reliability–sensitivity of mechanical or structural systems subject to random uncertainties in loads, material properties, and geometry. The dimension reduction method is applied to compute response moments and their sensitivities with respect to the distribution parameters (e.g., shape and scale parameters, mean, and standard deviation) of basic random variables. Saddlepoint approximations with truncated cumulant generating functions are employed to estimate failure probability, probability density functions, and cumulative distribution functions. The rigorous analytic derivation of the parameter sensitivities of the failure probability with respect to the distribution parameters of basic random variables is derived. Results of six numerical examples involving hypothetical mathematical functions and solid mechanics problems indicate that the proposed approach provides accurate, convergent, and computationally efficient estimates of the failure probability and reliability–sensitivity. Copyright © 2012 John Wiley & Sons, Ltd.     

Parameter Recovery Studies With a Diagnostic Bayesian Network Model

This paper describes a Bayesian network model for a candidate assessment design that had four proficiency variables and 48 tasks with 3–12 observable outcome variables per task and scale anchors to identify the location of the subscales. The domain experts’ view of the relationship among proficiencies and tasks established a complex prior distribution over 585 parameters. Markov Chain Monte Carlo (MCMC) estimation recovered the parameters of data simulated from the expert model. The sample size and the strength of the prior had only a modest effect on parameter recovery, but did affect the standard error of estimated parameters. Finally, an identifiability issue involving relabeling of proficiency states and permutations of the matrixes is addressed in the context of this study.