Statistical Strength Distribution Research Articles

The Fatigue Response’s Fingerprint of Composite Materials Subjected to Constant and Variable Amplitude Loadings

This paper discusses the theoretical and experimental correlations between fatigue and static strength statistical distributions. We use a two-parameter residual strength model that obeys the qualitative strength-life equal rank assumption (SLERA) for guidance. The modeling approach consists of recovering the model’s parameters by best fitting the constant amplitude (CA) fatigue data at a given stress ratio, R, and the experimental Weibull parameters of the static strength distribution function. An extensive set of fatigue life and residual strength data for AS4 carbon/epoxy 3k/E7K8 Plain Weave Fabric with [45/−45/90/45/−45/45/−45/0/45/−45]S layups, obtained at different stress ratios, R, have been analyzed. The modeling approach consists of recovering the model’s parameters from pure tension or compression fatigue data at R = 0.1 and R = 5, respectively. Once the parameters are fixed, the model’s capabilities, potential, and limits are discussed by comparing its predictions with residual strength and fatigue life data obtained at stress ratios with mixed tension/compression loadings, namely R = −0,2 and R = −1. Moreover, from a preliminary analysis, the theoretical extension of the model’s capabilities to variable amplitude loadings is conceptualized. The application of Miner’s rule is also discussed and compared with a new damage rule to analyze the fatigue responses under variable amplitude loadings.

Evolution of Statistical Strength during the Contact of Amorphous Polymer Specimens below the Glass Transition Temperature: Influence of Chain Length.

A comprehensive study of the statistical distribution of the auto-adhesion lap-shear strength (σ) of amorphous polymer-polymer interfaces using various types of statistical tests and models is a useful approach aimed at a better understanding of the mechanisms of the self-healing interface. In the present work, this approach has been applied, for the first time, to a temperature (T) range below the bulk glass transition temperature (Tgbulk). The interest of this T range consists in a very limited or even frozen translational segmental motion giving little or no chance for adhesion to occur. To clarify this issue, the two identical samples of entangled amorphous polystyrene (PS) with a molecular weight (M) of 105 g/mol or 106 g/mol were kept in contact at T = Tgbulk - 33 °C for one day. The as-self-bonded PS-PS auto-adhesive joints (AJ) of PSs differing in M by an order of magnitude were fractured at ambient temperature, and their σ distributions were analyzed using the Weibull model, the quantile-quantile plots, the normality tests, and the Gaussian distribution. It has been shown that the Weibull model most correctly describes the σ statistical distributions of the two self-bonded PS-PS AJs with different M due to the joints' brittleness. The values of the Weibull modulus (a statistical parameter) m = 2.40 and 1.89 calculated for PSs with M = 105 and 106 g/mol, respectively, were rather close, indicating that the chain length has a minor effect on the σ data scatter. The Gaussian distribution has been found to be less appropriate for this purpose, though all the normality tests performed have predicted the correctness of the normal distribution for these PS-PS interfaces.

An efficient virtual modeling regard to the axial tensile and transverse compressive behaviors of the twisted yarns

The mechanical properties of yarns have a decisive effect on the performance of fiber-reinforced composite materials. Predictive simulations of the mechanical response of yarns are, thus, necessary for damage evaluation and geometric reconstruction of textiles. This paper proposed a quasi-fiber scale virtual modeling method regard to the axial tensile and transverse compressive behaviors of the twisted yarns. A stochastic properties model of the yarn was established for characterizing the statistical distribution of tensile strength. The variation of modeling parameters, including coefficient of friction, the amounts of virtual fibers per yarn and element length, versus calculation accuracy has been determined based on axial tensile and transverse compressive behavior of quartz fibers. The relationship between modeling parameters and mechanical behavior of yarn was established within the scope of this study. Axial tensile and transverse compressive behavior of yarns with different twists were predicted. The results show that balance between the modeling precision and computational efficiency can be achieved using the parameters, the COF of 0.35, virtual fiber count of 122 and Le of 0.3. This efficient modeling method is meaningful to be developed in further virtual weaving research.

The Statistical Damage Constitutive Model of Longmaxi Shale under High Temperature and High Pressure

Abstract In the exploitation of shale gas in deep and ultradeep formations, the mechanical properties of shale change under the action of high temperature and pressure. High-temperature stimulation can effectively release the damage of water phase trapping, which was caused during the drilling and completion of hydraulic fracturing of shale gas reservoirs. In this paper, the experiments have twelve groups of shale samples (three samples per group) under four target temperatures, 25, 200, 400, and 600°C as well as the confining pressure set as 0 MPa, 15 MPa, and 30 MPa. The servo testing machine is used to perform triaxial compression tests on the shale specimens that have undergone high temperature. The porosity, permeability, and velocity are also obtained under different temperatures. A statistical constitutive model of shale after temperature thermal damage under triaxial compression is established. Based on the characteristics of the random statistical distribution of rock strength and strain strength theory, apply relevant knowledge of damage mechanics as well as consider the failure of the microprotocol and the nonlinear relationship between elastic modulus and temperature. According to the test results, the relationship between the mechanical parameters of the shale and the temperature is discussed. The parameters of the statistical constitutive model considering temperature thermal damage are given also; a comparison with the results of uniaxial compression experiments shows the rationality and reliability. This work not only enriches the theory of shale failure pattern but also contributes to the deep shale development at high temperature.

A 2D image-based multiphysics model for lifetime evaluation and failure scenario analysis of self-healing ceramic-matrix mini-composites under a tensile load

We propose a multi-physics numerical model for a self-healing ceramic matrix mini-composite under tensile load. Crack averaged PDEs are proposed for the transport of oxygen and of all the chemical species involved in the healing process and studied in the dimensionless form to perform the most appropriate discretization choices concerning time integration, and boundary conditions. Concerning the fibres’ degradation, a slow crack growth model explicitly dependent on the environmental parameters is calibrated using a particular exact solution and integrated numerically in the general case. The tow failure results from the statistical distribution of the fibres’ initial strength, the slow crack growth kinetics, and the load transfer following fibres breakage. The lifetime prediction capabilities of the model, as well as the effect of temperature, spatial variation of the statistical distribution of fibres strength, and applied load, are investigated highlighting the influence of the diffusion/reaction processes (healing) on the fibre breakage scenarios.

Effect of Chain Architecture and Conformation on the Features of the Statistical Strength Distributions of Oriented Polymer Materials

Effect of Chain Architecture and Conformation on the Features of the Statistical Strength Distributions of Oriented Polymer Materials

Influence of non-uniformity in inter-fibre distance on strength distribution of unidirectional fibre-reinforced polymer composites

Unidirectional fibre-reinforced polymer (UDFRP) composites exhibit statistical variations in the material properties. It is essential to acknowledge these variations and use a statistics-based strength property for designing composite structures. In this work, the effect of non-uniformity in inter-fibre distance on the statistics of strength distribution of UDFRP composite is studied through computational micromechanics. Micromechanics-based realisations of the UDFRP composite are developed with a quantified measure of non-uniformity (MoN) in fibre distribution through an algorithm. Several realisations of these models are created for each non-uniform fibre distribution. External stress is applied to each realisation in the transverse direction till failure initiates in the matrix. The modified Drucker–Prager failure criteria are implemented for matrix failure. This externally applied stress is termed as the failure initiation strength. Statistical distribution of failure initiation strength is obtained for each fibre distribution. Variations of mean value, standard deviation and B-basis of failure initiation strength with the measure of non-uniformity in fibre distribution are studied for the fibre volume fractions of 0.5 and 0.6. In addition to the mean value, non-uniformity in fibre distribution also significantly influences the standard deviation and B-basis of strength distribution.

A general method for determining probabilistic S-N curve based on static strength data of material

The conventional group test method for obtaining probabilistic S- N curve is a time-consuming and expensive test program. This study aims to present an alternative method for determining the probabilistic S- N curve of material based on its static strength data. Firstly, the probabilistic mapping relationship between the static strength and fatigue life of material is elaborated. Subsequently, the Weibull distribution is utilized to model the static strength and fatigue life data of material, and the correlation between the distribution parameters of them is investigated. Finally, an alternative method for determining the probabilistic S- N curve of material is proposed, and the experimental data of carbon steels and composite laminates are applied to verify its validity. The results show that the proposed method is capable of determining the probabilistic S- N curve of material based on its static strength data, which can significantly save test time and reduce test cost. In engineering applications, when the parameters of static strength distribution and the median S- N curve are known, the probabilistic S- N curve with any given survival probability can be determined through a unified analytical expression.

A damage model for granite subjected to quasi‐static contact loading

AbstractAn anisotropic damage model is employed in order to simulate the fracture pattern of Bohus granite under quasi‐static (Q‐S) spherical indentation loading. The chosen damage description is added to the previously employed Drucker–Prager (DP) plasticity model with variable dilation angle. The resulting constitutive model is implemented to simulate the behavior of Bohus granite under indentation up to the load capacity of the material. The initial fragmentation, corresponding to the first small load‐drop in the force–penetration curve, is likely due to the high radial tensile stress state at or close to the contact boundary. Both predicted fracture pattern and force–penetration results from the numerical simulation are compared to experimental data and a good agreement is found. The variability in tensile strength of the material is included in the chosen damage model by taking advantage of Weibull statistics. In this work, it is suggested that indentation test results by themselves may be used to calibrate the statistical distribution of tensile strength at small size scales such as in indentation applications.

Preventing Catastrophic Failure of Microfibrillar Adhesives in Compliant Systems Based on Statistical Analysis of Adhesive Strength.

Adhesives based on fibrillar surface microstructures have shown great potential for handling applications requiring strong, reversible, and switchable adhesion. Recently, the importance of the statistical distribution of adhesive strength of individual fibrils in controlling the overall performance was revealed. Strength variations physically correspond to different interfacial defect sizes, which, among other factors, are related to surface roughness. For analysis of the strength distribution, Weibull's statistical theory of fracture was introduced. In this study, the importance of the statistical properties in controlling the stability of attachment is explored. Considering the compliance of the loading system, we develop a stability criterion based on the Weibull statistical parameters. It is shown that when the distribution in fibril adhesive strength is narrow, the global strength is higher but unstable detachment is more likely. Experimental variation of the loading system compliance for a specimen of differing statistical properties shows a transition to unstable detachment at low system stiffness, in good agreement with the theoretical stability map. This map serves to inform the design of gripper compliance, when coupled with statistical analysis of strength on the target surface of interest. Such a treatment could prevent catastrophic failure by spontaneous detachment of an object from an adhesive gripper.

Testing Weibull as a viable statistical strength distribution for Nacre

Nacre, a composite layer present in sea-shells, exhibits a remarkable combination of toughness, strength, and stiffness through its brick–mortar micro-structure, acting as a template for novel materials. Strength, one of nacre’s important properties is highly variable due to distribution of underlying material’s properties as well as various defects present in its micro-structure. Presently, researchers assume the Weakest-link hypothesis and consequently use the Weibull distribution to model this variability. However, this assumption is theoretically unproven for biological materials such as Nacre and extrapolation of the same to predict rare but catastrophic behavior would be incorrect. In this article, the suitability of Weibull distribution to account for the variability of strength in nacre is tested, using multi-scale models, developed using FEM. Through Monte Carlo based numerical experiments and Renormalization Group (RG) based arguments, it is shown that the Weakest-link hypothesis, which is commonly used to justify the use of the Weibull distribution, does not seem to hold for nacre. Micromechanics and non-local homogenization based distributions such as the one suggested by Luo and Bazant (2019) might be more appropriate for accurate extrapolation to the low probability tail.

Water saturation effect on the dynamic mechanical behaviour and scaling law effect on the dynamic strength of coral aggregate concrete

The variation in impact resistance of coral aggregate concrete (CAC) under different levels of water saturation was investigated. Results show that the free water improves not only the strain rate effect of the dynamic compressive strength of CAC but also the increasing amplitude of the dynamic elastic modulus with the strain rate. In general, the strain rate effect on the dynamic elastic modulus is less than that on the dynamic compressive strength. The higher the water saturation, the greater are the critical strain and toughness of CAC. The promoting effect of free water on crack propagation at a low strain rate and the increasing effect of free water on the damage degree at a high strain rate improve the deformation and energy absorption properties of CAC. The proposed scaling law effect model can be effectively applied to characterise the statistical distribution of the dynamic strength of CAC.

Static Fatigue of SiC/SiC Minicomposites at High Temperatures Up to 1200 °C in Air: Multiscale Approach

The present paper investigates the static fatigue behavior of Hi-Nicalon fiber-reinforced SiC–SiC minicomposites at high temperatures in the 900–1200 °C range, and under tensile stresses above the proportional limit. The stress–rupture time relation was analyzed with respect to subcritical crack growth in filaments and fiber tow fracture. Slow crack growth from flaws located at the surface of filaments is driven by the oxidation of free carbon at the grain boundaries. Lifetime of the reinforcing tows depends on the statistical distribution of filament strength and on structural factors, which are enhanced by temperature increase. The rupture time data were plotted in terms of initial stresses on reinforcing filaments. The effect of temperature and load on the stress–rupture time relation for minicomposites was investigated using results of fractography and predictions of minicomposite lifetime using a model of subcritical growth for critical filaments. The critical filament is the one whose failure by slow crack-growth triggers unstable fracture of the minicomposite. This is identified by the strength–probability relation provided by the cumulative distribution function for filament strength at room temperature. The results were compared to the fatigue behavior of dry tows. The influence of various factors related to oxidation, including multiple failures, load sharing, and variability, was analyzed.

Temperature‐dependent tensile properties of hybrid carbon/glass thermoplastic composite rods

AbstractHybrid carbon/glass composite rods consisting of unidirectional polyacrylonitrile (PAN)‐based carbon fiber (T700SC), braided E‐glass fibers, and thermoplastic epoxy matrix were fabricated with differing carbon/glass ratios (24K1P, 24K2P, and 24K3P). The cross‐sectional morphologies of the hybrid rods were observed using a digital microscope. Tensile properties and fracture behavior of the hybrid rods were investigated over a range of temperatures: −50°C, 0°C, 23°C (RT), 50°C, and 80°C. The tensile modulus and strength increased with increasing carbon fiber volume fraction, but decreased with increasing temperature. The Weibull statistical distribution of tensile strength for the hybrid rods was examined. The Weibull modulus increased with increasing tensile strength and decreasing void volume fraction of the hybrid rods. A Curtin‐type global load‐sharing model was applied to the hybrid rods to predict mechanical properties at different temperatures.

Features of Statistical Strength Distributions in Mono- and Polyfilament Ultraoriented High-Strength Fibers of Ultrahigh-Molecular-Weight Polyethylene

The regularities of statistical distributions of breaking strength of high-strength ultraoriented monofilaments and polyfilament fibers consisting of several hundreds single ultrahigh-molecular-weight polyethylene (UHMWPE) fibers were analyzed using the Weibull and Gauss models based on a large array of measurements (50 identical samples in each series). It was shown that the distribution of strength of polyfilament UHMWPE fibers can be described both in terms of the normal Gauss distribution and using the standard Weibull distribution function. At the same time, the strength distribution in UHMWPE monofilaments drawn to an ultimate draw ratio of 120 is governed by normal distribution. The reasons for the detected difference in the types of statistical distributions of breaking strengths in high-strength UHMWPE mono- and polyfilaments are discussed.

Особенности статистических распределений прочностей моно- и полифиламентных ультраориентированных высокопрочных волокон сверхвысокомолекулярного полиэтилена

The patterns of the statistical distributions of tensile strength of high-strength ultraoriented mono-and multifilament fibers, consisting of several hundred single fibers of ultra-high-molecular-weight polyethylene (UHMWPE), have been analyzed using the Weibull and Gaussian models by considering a large number of measurements (50 identical samples in each series). It has been shown that the strength distribution of the multifilament UHMWPE fibers can be described both within the normal Gaussian distribution and the standard Weibull distribution function. At the same time, the strength distribution of the UHMWPE monofilaments oriented to a final draw ratio of 120 is not subject to the normal distribution. The reasons for the revealed differences in the types of the statistical strength distributions of the mono- and multifilaments of high-strength UHMWPE during fracture are discussed.

Partial safety factors for laminated glass

AbstractGlass plies can be bonded together by polymeric interlayers to form laminated glass. Thanks to the capacity in finding alternative stress paths after partial breakage, it is commonly accepted that the more the layers are, the higher the safety level is. However, a negative aspect is that the tensile strength of glass carries a size effect in terms of surface area, which increases with lamination. Here we evaluate these competing factors by calibrating the partial safety factors to be used in semi‐probabilistic (level‐I) design, through comparison with probabilistic (level‐III) methods in paradigmatic case‐studies under wind, snow, and dead weight. Starting from a two‐parameter Weibull distribution for glass strength, the “failure modes” approach determines the statistical distribution of strength for a multi‐laminate as a function of the number of plies, which interferes with the statistics for actions. As a function of the target probability of failure for the assigned class of consequence, we introduce a new coefficient in the verification formula of level‐I, to account for the effects of lamination with dependence upon the number of plies. We find that there is strong gain when passing from a monolith to a two‐ply laminate, but the advantage fades by increasing the number of layers. Verification formulas of this type could avoid overconservative design.

The statistical analysis of tensile and compression properties of the as-cast AZ91-X%B4C composites

The statistical distribution of the ultimate tensile strength (UTS) and compression strength (UCS) values of the cast B4C containing Mg matrix composite and its matrix alloy (AZ91) was assessed using different amounts of B4C (1, 3, 5, and 7 wt%) by the use of two-parameter Weibull analysis. Microstructural observations of the composites revealed a reduction in the amount of the Mg17Al12 intermetallic phase, a rather clean interface between B4C particles and the matrix and a good distribution of the B4C reinforcement in the matrix. It was found that the addition of B4C particles to the matrix alloy resulted in the reduction in UTS and elongation values, whereas the compressive strength enhanced slightly. Weibull modulus of the castings was found to be ~ 10 for AZ91 alloy, falling to ~ 6 in the AZ91–7%B4C, but in the compression test, it varied from ~ 52 to ~ 87. The fracture study of the tensile specimens revealed that oxide films and porosity are the main factors for the decreased strength and scattered results, which are increased by using more B4C contents. If the composite can be manufactured with less amount of oxides, it is seen to be more reliable for compressive applications.

An empirical statistical constitutive relationship for rock joint shearing considering scale effect

The scale effect of rock joint shearing is of great significance in rock engineering. Most existing shear constitutive models could describe the pre- and post-peak deformation of rock joints, but only in one particular scale, that is, those existing models will fail to depict the rock joint shearing in different length scales. Therefore, this study aims to establish a constitutive relationship for rock joints with considering the scale effect. Based on the assumption of a random statistical distribution of rock material strength and statistical mesoscopic damage theory, damage variables are defined as the ratio of the number of damaged elements to the total number in the shear process. Together with the nonlinear relationship between the microelement failure and the joint scale, an empirical statistical constitutive relationship for joint is established. And then, the determination method of constitutive relationship parameters and the variation laws with the scale are discussed. Results show that the predicted results of the proposed empirical relationship not only agree well with the experimental results but also fully describe nonlinear deformation, pre-peak softening, post-peak softening, residual stage, and other mechanical properties of the shear deformation of joint with different dimensions, thereby demonstrating the rationality of the constitutive relationship. The physical meaning of the constitutive relationship parameters is clear, and the expressions of the constitutive relationship parameters can be deduced from the experimental results. In addition, the influence of scale effect on the shear deformation of rock joints can be quantified using parameters, which help accurately describe the action form of scale effect.

Stochastic micromechanical modeling of transverse punch shear damage behavior of unidirectional composites

Punch shear in unidirectional composites is induced by transverse shear loading that progressively perforates the laminate within a narrow shear annulus. At lower micromechanical length scales, punch shear loading creates unique micromechanical damage mechanisms dominated by transverse fiber shear failure, fiber–matrix interphase debonding and large inelastic deformation and cracking of the matrix. A new punch shear experimental method has been developed to test unidirectional S glass/DER353 epoxy composite ribbons at sub-millimeter length scale. The experimental data consist of a statistical measurement of the continuum response (load-deformation and punch shear strength) and the characterization of micromechanical damage modes. A simplified 2D micromechanical finite element model incorporating Weibull fiber strength distribution has been developed and correlated with the experimental data. The 2D micromechanical finite element model can simulate the punch shear failure of the ribbon incorporating mixed mode fiber fracture, and fiber–matrix debonding mechanisms using zero thickness cohesive elements. Results from stochastic simulations of punch shear experiments show that an equivalent 2D micromechanical finite element model can predict the micromechanical damage mechanisms and the statistical distribution of punch shear strength of the continuum with favorable correlation with the experiments. This paper presents a combined experimental and computational approach in simulating the stochastic non-linear progressive punch shear behavior of unidirectional composites for the first time in the literature.