Statistical distribution and size effect of residual strength of quasibrittle materials after a period of constant load

In preceding studies, the type of cumulative probability distribution functions (cdf) of strength and of static lifetime of quasibrittle structures, including their tails, was mathematically derived from atomistic scale arguments based on nano-scale cracks propagating by many small, activation energy-controlled, random breaks of atomic bonds in the nanostructure. It was shown that a quasibrittle structure (of positive geometry) must be modeled by a finite (rather than infinite) weakest-link model, and that the cdf of structural strength as well as lifetime varies from nearly Gaussian to Weibullian as a function of structure size and shape. Excellent agreement with the observed distributions of structural strength and static lifetime was demonstrated. Based on the same theoretical framework, the present paper formulates the statistics of the residual structural strength, which is the strength after the structure has been subjected to sustained loading. A strength degradation equation is derived based on Evans' law for static crack growth during sustained loading. It is shown that the rate of strength degradation is not constant but continuously increasing. The cdf of residual strength of one RVE is shown to be closely approximated by a graft of Weibull and Gaussian (normal) distributions. In the left tail, the cdf is a three-parameter Weibull distribution consisting of the (n+1)th power of the residual strength, where n is the exponent of the Evans law and the threshold is a function of the applied load and load duration. The finiteness of the threshold, which is typically very small, is a new feature of quasibrittle residual strength statistics, contrasting with the previously established absence of a threshold for strength and lifetime. Its cause is that there is a non-zero probability that some specimens fail during the static preloading, and thus are excluded from the statistics of the overload. The predictions of the theory are validated by available test data on glass–epoxy composites and on borosilicate and soda-lime silicate glasses. The size effect on the cdf of residual strength is also determined. The size effect on the mean residual strength is found to be as strong as the size effect on the mean initial strength.     

Unified nano-mechanics based probabilistic theory of quasibrittle and brittle structures: I. Strength, static crack growth, lifetime and scaling

Engineering structures must be designed for an extremely low failure probability such as 10⁻⁶, which is beyond the means of direct verification by histogram testing. This is not a problem for brittle or ductile materials because the type of probability distribution of structural strength is fixed and known, making it possible to predict the tail probabilities from the mean and variance. It is a problem, though, for quasibrittle materials for which the type of strength distribution transitions from Gaussian to Weibullian as the structure size increases. These are heterogeneous materials with brittle constituents, characterized by material inhomogeneities that are not negligible compared to the structure size. Examples include concrete, fiber composites, coarse-grained or toughened ceramics, rocks, sea ice, rigid foams and bone, as well as many materials used in nano- and microscale devices.This study presents a unified theory of strength and lifetime for such materials, based on activation energy controlled random jumps of the nano-crack front, and on the nano-macro multiscale transition of tail probabilities. Part I of this study deals with the case of monotonic and sustained (or creep) loading, and Part II with fatigue (or cyclic) loading. On the scale of the representative volume element of material, the probability distribution of strength has a Gaussian core onto which a remote Weibull tail is grafted at failure probability of the order of 10⁻³. With increasing structure size, the Weibull tail penetrates into the Gaussian core. The probability distribution of static (creep) lifetime is related to the strength distribution by the power law for the static crack growth rate, for which a physical justification is given. The present theory yields a simple relation between the exponent of this law and the Weibull moduli for strength and lifetime. The benefit is that the lifetime distribution can be predicted from short-time tests of the mean size effect on strength and tests of the power law for the crack growth rate. The theory is shown to match closely numerous test data on strength and static lifetime of ceramics and concrete, and explains why their histograms deviate systematically from the straight line in Weibull scale.Although the present unified theory is built on several previous advances, new contributions are here made to address: (i) a crack in a disordered nano-structure (such as that of hydrated Portland cement), (ii) tail probability of a fiber bundle (or parallel coupling) model with softening elements, (iii) convergence of this model to the Gaussian distribution, (iv) the stress-life curve under constant load, and (v) a detailed random walk analysis of crack front jumps in an atomic lattice. The nonlocal behavior is captured in the present theory through the finiteness of the number of links in the weakest-link model, which explains why the mean size effect coincides with that of the previously formulated nonlocal Weibull theory. Brittle structures correspond to the large-size limit of the present theory. An important practical conclusion is that the safety factors for strength and tolerable minimum lifetime for large quasibrittle structures (e.g., concrete structures and composite airframes or ship hulls, as well as various micro-devices) should be calculated as a function of structure size and geometry.     

Unified nano-mechanics based probabilistic theory of quasibrittle and brittle structures: II. Fatigue crack growth, lifetime and scaling

This paper extends the theoretical framework presented in the preceding Part I to the lifetime distribution of quasibrittle structures failing at the fracture of one representative volume element under constant amplitude fatigue. The probability distribution of the critical stress amplitude is derived for a given number of cycles and a given minimum-to-maximum stress ratio. The physical mechanism underlying the Paris law for fatigue crack growth is explained under certain plausible assumptions about the damage accumulation in the cyclic fracture process zone at the tip of subcritical crack. This law is then used to relate the probability distribution of critical stress amplitude to the probability distribution of fatigue lifetime. The theory naturally yields a power-law relation for the stress-life curve (S-N curve), which agrees with Basquin's law. Furthermore, the theory indicates that, for quasibrittle structures, the S-N curve must be size dependent. Finally, physical explanation is provided to the experimentally observed systematic deviations of lifetime histograms of various ceramics and bones from the Weibull distribution, and their close fits by the present theory are demonstrated.     

A strength reliability model by Markov process of unidirectional composites with fibers placed in hexagonal arrays

This paper proposes a strength reliability model based on a Markov process for unidirectional composites with fibers in a hexagonal array. The model assumes that a group of fiber breaking points, a so-called cluster, evolves with increased stress. The cluster evolution process branches because of various fiber-breakage paths. Load-sharing structure of intact fibers around clusters was estimated from geometric and mechanical local load-sharing rules. Composites fracture if a cluster achieves a critical size, so the model expresses a fracture criterion by setting an absorbing state. Next, the author constituted a state transition diagram concerning cluster evolutions of 1-fiber to 7-fiber breaks and analytically solved simultaneous differential equations obtained from the diagram. Results showed that, as critical cluster size increases, slope of the fracture probability distribution is given in a Weibull probability scale as follows: m_c=i×m_f (i, the number of broken fibers in a cluster; m_c and m_f, Weibull shape parameters for fracture probabilities of a critical cluster and fiber strength, respectively). This relation between m_c and m_f had been shown by Smith et al. [Proc. R. Soc. London, A 388 (1983) 353–391], but the present study demonstrated it analytically without any lower tail of the Weibull distribution used in that paper. In addition, the present model can be approximated by a one-state birth model.     

Global and local approaches to fracture normal to interfaces

The problem of a crack perpendicularly approaching a bimaterial interface is examined using both global and localapproaches to fracture. The global approach is based on the J-integral with a second parameter, Q, which scales the stress triaxiality ahead of the crack. The local approach is based on either brittle fracture(Beremin model ) or ductile fracture (Rice and Tracey model ). In the first case, the Weibull stress over the plasticzone is calculated. In the second case, the void growth rate is calculated at the tip of the crack over a representativevolume (generally associated with a characteristic length of the material ). After a brief summary of each approach,the results for a crack near an elastically homogeneous, plastically mismatched interface are presented. Thebehaviour of the bimaterial is expressed in relation to the behavior of the homogeneous material. It is shown thatthere is an effect on the crack behavior which depends on the direction of crack propagation, i.e. from the hardermaterial to the softer material or vice versa. This effect is examined as a function of change in yield strength ratioand hardening exponent, n. For the case of brittle fracture, the effect of changing the Weibull modulus, m, is also examined. The models based on the local approach show that both stress- and strain-controlledfracture mechanisms must be accounted for. This implies the necessity of using the two parameters J and Q in the global approach. This is due to the fact that the stress–strain fields ahead of the crack tip areaffected by the nature of the second material.     

Size effect in quasibrittle failure: Analytical model and numerical simulations using cohesive zone model

The recent rewriting of the Ba?ant’s size effect law (Morel, 2008) which has suggested the existence of an additional asymptotic regime for intermediate structure sizes is now compared to numerical simulations of fracture of geometrically similar notched structures of different sizes extending over 2.4 decades. The quasibrittle fracture behavior is simulated through cohesive zone model (bilinear softening) using a constant set of cohesive parameters whatever the specimen size D is. The R-curves resulting from the load–displacement responses are estimated and appear as size-independent. On this basis, the different asymptotic regimes expected for the size effect on fracture properties at peak load such as the relative crack length, the resistance to crack growth and the nominal strength are shown in fair agreement with the size effect observed on the results obtained from numerical simulations.     

Effect of source and obstacle strengths on yield stress: A discrete dislocation study

The combined effect of dislocation source strength τ_s, dislocation obstacle strength τ_obs, and obstacle spacing L_obs on the yield stress of single crystal metals is investigated analytically and numerically. A continuum theory of dislocation pileups emanating from a finite-strength source and impinging on asymmetric obstacles gives a closed-form expression for the yield stress. A 2d discrete dislocation model for a single-source/obstacle problem agrees well with the analytic model over a wide range of material parameters. Discrete dislocation simulations for a full tensile bar with statistically distributed sources and obstacles show that the distribution of obstacles plays a significant role in controlling the yield stress. Over a wide range of parameters, the simulations agree well with the analytic model using an effective obstacle spacing L_obs^* chosen to capture the strength-controlling statistically weaker pileup configurations. The analytic model can thus be used to guide the choice of source and obstacle parameters to obtain a desired yield stress. The model also shows how different combinations of internal source and obstacle parameters can generate the same macroscopic yield stress, and points to several internal length scales that could relate to size-dependent plasticity phenomena.     

Interplay of size effects in concrete specimens under tension studied via computational stochastic fracture mechanics

We attempt the identification, study and modeling of possible sources of size effects in concrete structures acting both separately and together. We are particularly motivated by the interplay of several identified scaling lengths stemming from the material, boundary conditions and geometry. Methods of stochastic nonlinear fracture mechanics are used to model the well published results of direct tensile tests of dog-bone specimens with rotating boundary conditions. Firstly, the specimens are modeled using microplane material law to show that a large portion of the dependence of nominal strength on structural size can be explained deterministically. However, it is clear that more sources of size effect play a part, and we consider two of them. Namely, we model local material strength using an autocorrelated random field attempting to capture a statistical part of the complex size effect, scatter inclusive. In addition, the strength drop noticeable with small specimens which was obtained in the experiments is explained by the presence of a weak surface layer of constant thickness (caused e.g., by drying, surface damage, aggregate size limitation at the boundary, or other irregularities). All three named sources (deterministic-energetic, statistical size effects, and the weak layer effect) are believed to be the sources most contributing to the observed strength size effect; the model combining all of them is capable of reproducing the measured data. The computational approach represents a marriage of advanced computational nonlinear fracture mechanics with simulation techniques for random fields representing spatially varying material properties. Using a numerical example, we document how different sources of size effects detrimental to strength can interact and result in relatively complex quasibrittle failure processes. The presented study documents the well known fact that the experimental determination of material parameters (needed for the rational and safe design of structures) is very difficult for quasibrittle materials such as concrete.     

Estimation of strengths in large annealed glass panels

A new model of the Log-Normal form for predicting the cumulative probabilistic distribution of strength in annealed glass panels is presented in this paper. The proposed model, which is supported by experimental evidences, shares certain features that are common with predictions by the Weibull’s model. However, as the dimension of the panel is above a certain limit, the strength of glass as predicted by the new model is much less sensitive to any further increase in the panel dimension than strength predicted by the Weibull’s model. This has important implications to the engineering design and risk assessments of glass facades in the future. The proposed alternative model was derived from results obtained from Monte Carlo simulations of non-interacting Griffith flaws based on principles of fracture mechanics. As interactions between flaws have been neglected in the analyses presented in the paper, the proposed model is intended to be applicable to glazing panels which contain widely spaced flaws. Results from physical experimentation in support of the simulation model have been presented in the paper.     

Electroviscous effect on non-Newtonian fluid flow in microchannels

Understanding non-Newtonian flow in microchannels is of both fundamental and practical significance for various microfluidic devices. A numerical study of non-Newtonian flow in microchannels combined with electroviscous effect has been conducted. The electric potential in the electroviscous force term is calculated by solving a lattice Boltzmann equation. And another lattice Boltzmann equation without derivations of the velocity when calculating the shear is employed to obtain flow field. The simulation of commonly used power-law non-Newtonian flow shows that the electroviscous effect on the flow depends significantly on the fluid rheological behavior. For the shear thinning fluid of the power-law exponent n < 1, the fluid viscosity near the wall is smaller and the electroviscous effect plays a more important role. And its effect on the flow increases as the ratio of the Debye length to the channel height increases and the exponent n decreases. While the shear thickening fluid of n > 1 is less affected by the electroviscous force, it can be neglected in practical applications.     

A multi-scale algorithm for dislocation creep at elevated temperatures

Dislocation creep at elevated temperatures plays an important role for plastic deformation in crystalline metals. When using traditional discrete dislocation dynamics(DDD) to capture this process, we often need to update the forces on N dislocations involving ～N~2 interactions. In this letter, we introduce a multi-scale algorithm to speed up the calculations by dividing a sample of interest into sub-domain grids:dislocations within a characteristic area interact following the conventional way, but their interaction with dislocations in other grids are simplified by lumping all dislocations in another grid as a super one. Such a multi-scale algorithm lowers the computational load to ～N 1.5. We employed this algorithm to model dislocation creep in Al-Mg alloy. The simulation leads to a power-law creep rate in consistent with experimental observations. The stress exponent of the power-law creep is a resultant of dislocations climb for ～5 and viscous dislocations glide for ～3.     

Extensional rheology of concentrated emulsions as probed by capillary breakup elongational rheometry (CaBER)

Elongational flow behavior of w/o emulsions has been investigated using a capillary breakup elongational rheometer (CaBER) equipped with an advanced image processing system allowing for precise assessment of the full filament shape. The transient neck diameter D(t), time evolution of the neck curvature κ(t), the region of deformation l _def and the filament lifetime t _c are extracted in order to characterize non-uniform filament thinning. Effects of disperse volume fraction ?, droplet size d _sv , and continuous phase viscosity η _c on the flow properties have been investigated. At a critical volume fraction ? _c , strong shear thinning, and an apparent shear yield stress τ _y,s occur and shear flow curves are well described by a Herschel–Bulkley model. In CaBER filaments exhibit sharp necking and t _c as well as κ _max ?=?κ (t?=?t _c ) increase, whereas l _def decreases drastically with increasing ?. For ? <?? _c , D(t) data can be described by a power-law model based on a cylindrical filament approximation using the exponent n and consistency index k from shear experiments. For ??≥?? _c , D(t) data are fitted using a one-dimensional Herschel–Bulkley approach, but k and τ _y,s progressively deviate from shear results as ? increases. We attribute this to the failure of the cylindrical filament assumption. Filament lifetime is proportional to η _c at all ?. Above ? _c, κ _max as well as t _c /η _c scale linearly with τ _y,s . The Laplace pressure at the critical stretch ratio ε _c which is needed to induce capillary thinning can be identified as the elongational yield stress τ _y,e , if the experimental parameters are chosen such that the axial curvature of the filament profile can be neglected. This is a unique and robust method to determine this quantity for soft matter with τ _y ?< 1,000 Pa. For the emulsion series investigated here a ratio τ _y,e /τ _y,s = 2.8 ± 0.4 is found independent of ?. This result is captured by a generalized Herschel–Bulkley model including the third invariant of the strain-rate tensor proposed here for the first time, which implies that τ _y,e and τ _y,s are independent material parameters.     

Free vibration analysis of functionally graded panels and shells of revolution

The aim of this paper is to study the dynamic behaviour of functionally graded parabolic and circular panels and shells of revolution. The First-order Shear Deformation Theory (FSDT) is used to study these moderately thick structural elements. The treatment is developed within the theory of linear elasticity, when the materials are assumed to be isotropic and inhomogeneous through the thickness direction. The two-constituent functionally graded shell consists of ceramic and metal that are graded through the thickness, from one surface of the shell to the other. Two different power-law distributions are considered for the ceramic volume fraction. For the first power-law distribution, the bottom surface of the structure is ceramic rich, whereas the top surface is metal rich and on the contrary for the second one. The governing equations of motion are expressed as functions of five kinematic parameters, by using the constitutive and kinematic relationships. The solution is given in terms of generalized displacement components of the points lying on the middle surface of the shell. The discretization of the system equations by means of the Generalized Differential Quadrature (GDQ) method leads to a standard linear eigenvalue problem, where two independent variables are involved without using the Fourier modal expansion methodology. Numerical results concerning eight types of shell structures illustrate the influence of the power-law exponent and of the power-law distribution choice on the mechanical behaviour of parabolic and circular shell structures. Preliminary results were presented by the authors at the XVIII° National Conference of Italian Association of Theoretical and Applied Mechanics (AIMETA 2007) (Tornabene and Viola 27).     

A unified periodical boundary conditions for representative volume elements of composites and applications

An explicit unified form of boundary conditions for a periodic representative volume element (RVE) is presented which satisfies the periodicity conditions, and is suitable for any combination of multiaxial loads. Starting from a simple 2-D example, we demonstrate that the “homogeneous boundary conditions” are not only over-constrained but they may also violate the boundary traction periodicity conditions. Subsequently, the proposed method is applied to: (a) the simultaneous prediction of nine elastic constants of a unidirectional laminate by applying multiaxial loads to a cubic unit cell model; (b) the prediction of in-plane elastic moduli for [±θ]_n angle-ply laminates. To facilitate the analysis, a meso/micro rhombohedral RVE model has been developed for the [±θ]_n angle-ply laminates. The results obtained are in good agreement with the available theoretical and experimental results.     

金属材料的强度与应力-应变关系的球压入测试方法

压入法获取材料单轴应力-应变关系和抗拉强度对服役结构完整性评价有重要的基础意义.假定材料均匀连续、各向同性、应力应变关系符合Hollomon律,基于能量等效假定,即代表性体积单元(representativevolume element, RVE)的vonMises等效和有效变形域内能量中值等效假定,本文提出了关联材料载荷、深度、球压头直径和Hollomon律的四参数半解析球压入(semi-analyticalspherical indentation,SSI)模型.通过球压入载荷-深度试验关系获得材料的应力-应变关系和抗拉强度.考虑压入过程中的损伤效应,针对金属材料提出了用于球压入测试的材料弹性模量修正模型.对11种延性金属材料完成了球压入试验,采用本文提出的球压入试验方法测到的弹性模量、应力-应变关系和抗拉强度与单轴拉伸试验结果吻合良好.      

Modelling matrix damage and fibre–matrix interfacial decohesion in composite laminates via a multi-fibre multi-layer representative volume element (MRVE)

A three-dimensional multi-fibre multi-layer micromechanical finite element model was developed for the prediction of mechanical behaviour and damage response of composite laminates. Material response and micro-scale damage mechanism of cross-ply, [0/90]_ns, and angle-ply, [±45]_ns, glass-fibre/epoxy laminates were captured using multi-scale modelling via computational micromechanics. The framework of the homogenization theory for periodic media was used for the analysis of the proposed ‘multi-fibre multi-layer representative volume element’ (M²RVE). Each layer in M²RVE was represented by a unit cube with multiple randomly distributed, but longitudinally aligned, fibres of equal diameter and with a volume fraction corresponding to that of each lamina (equal in the present case). Periodic boundary conditions were applied to all the faces of the M²RVE. The non-homogeneous stress–strain fields within the M²RVE were related to the average stresses and strains by using Gauss’ theorem in conjunction with the Hill–Mandal strain energy equivalence principle. The global material response predicted by the M²RVE was found to be in good agreement with experimental results for both laminates. The model was used to study effect of matrix friction angle and cohesive strength of the fibre–matrix interface on the global material response. In addition, the M²RVE was also used to predict initiation and propagation of fibre–matrix interfacial decohesion and propagation at every point in the laminae.     

Tensile-shear transition in mixed mode I/III fracture

The propensity of the transition of fracture type in either brittle or ductile cracked solid under mixed-mode I and III loading conditions is investigated. A fracture criterion based on the competition of the maximum normal stress and maximum shear stress is utilized. The prediction of the fracture type is determined by comparing τ_max/σ_max at a critical distance from the crack tip to the material strength ratio τ_C/σ_C, i.e., (τ_max/σ_max)<(τ_C/σ_C) for tensile fracture and (τ_max/σ_max)>(τ_C/σ_C) for shear fracture, where σ_C (τ_C) is the fracture strength of materials in tension (shear). Mixed mode I/III fracture tests were performed using circumferentially notched cylindrical bars made of PMMA and 7050 aluminum alloy. Fracture surface morphology of the specimens reveals that: (1) for the brittle material, PMMA, only tensile type of fracture occurs, and (2) for the ductile material, 7050 aluminum alloy, either tensile or shear type of fracture occurs depending on the mode mixity. The transition (in ductile material) or non-transition (in brittle material) of the fracture type and the fracture path observed in experiments were properly predicted by the theory. Additional test data from open literature are also included to validate the proposed theory.     

Simulation of two generalised Newtonian fluids flow in micropore with dead end

This research deals with the numerical simulation of Carreau and power-law fluids flow in an open capillary of a reservoir. The capillary is connected to a dead end. The finite volume method (FVM) on a structured and co-located grid has been used. The numerical method has been validated through the comparison of numerical results against the analytical solutions of power-law fluid flow in a planar channel. The effects of fluids, the operating conditions and the aspect ratio of dead end at the low Reynolds (Re) numbers on the oil sweeping from the dead end are investigated. The simulation results show that by increasing the power-law exponent in the case of power-law fluids, the swept depth in the dead end increases. However, according to the results, the effect of Re number on the flow pattern and the oil sweeping from the dead end is insignificant at the investigated conditions. In the case of Carreau model, at the conditions investigated, the swept area increases as the power-law exponent increases, but the Reynolds number has still minor effects on the flow pattern. Also, as the aspect ratio of dead end increases, the sweep efficiency increases.     

On thermal boundary layer of a non-Newtonian fluid on a power-law stretched surface of variable temperature with suction or injection

 Heat transfer characteristics of a non-Newtonian fluid on a power-law stretched surface of variable temperature with suction or injection were investigated. Similarity solutions of the laminar boundary layer equations describing heat transfer and fluid flow in a quiescent fluid were obtained and solved numerically. Velocity and temperature profiles as well as the Nusselt number, Nu, were studied for two thermal boundary conditions; uniform surface temperature and variable surface temperature, for different parameters; Prandtl number Pr, temperature exponent b, velocity exponent m, injection parameter d and power-law index n. It was found that decreasing injection parameter d, and power-law index n and increasing Prandtl number Pr and surface temperature exponent b enhance the heat transfer coefficient. Received on 27 April 2000     

Analysis of Seepage for Power-Law Fluids in the Fractal-Like Tree Network

This work studies the flow characteristics of power-law fluids in the fractal-like tree network. A fractal model is developed for the permeability of power-law fluid flow in fractal-like tree network based on straight capillary model, generalized Darcy’s law and constitutive equation for power-law fluids. Analytical expression for permeability of power-law fluids in the network is presented and found to be a function of network microstructural parameters such as the branching diameter ratio, the branching length ratio, the total number of branching levels, the bifurcation angle, the branching number, the diameter of the zeroth branching level and the power exponent of power-law fluids. Both the phase permeabilities and the relative permeabilities are also derived and found to be a function of power exponent for the wetting phase and non-wetting phase, the saturation and other microstructural parameters and independent of the bifurcation angle.