Size Effect In Strength Research Articles

Deterministic size effect in the strength of cracked concrete structures

This paper is concerned with identifying and quantifying the deterministic (as opposed to statistical) size effect in the strength of cracked concrete structures that is believed to be a result of stress discontinuities introduced by the cracks. For this, the strength of geometrically similar pre-cracked specimens of varying sizes made from three concrete mixes is measured in three-point bend and wedge splitting geometries. The true, size-independent specific fracture energy and the corresponding tension softening diagram of each of the three mixes are independently established in order to exclude their influence on the strength size effect. The test results show that the deterministic strength size effect weakens as the size of the crack reduces. This is confirmed by theoretical/computational studies based on the fictitious crack model in the range of sizes tested in the laboratory. The theoretical/computational model has been extended beyond this limited range to include cracked concrete structures in the size range 1 : 80. The computational results have been fitted by a simple strength size effect formula with appropriate asymptotic behaviour at both size extremes. The three unknown coefficients in this formula depend only on the size of the crack and they can be obtained by conducting tests on geometrically similar specimens of any shape but of varying sizes that can be conveniently handled in a laboratory. The three material properties of the concrete mix appearing in this formula, namely the Young modulus E, the direct tensile strength f t and the size-independent specific fracture energy G F must be independently measured.

Bias in the Weibull Strength Estimation of a SiC Fiber for the Small Gauge Length Case

It is known that the single-modal Weibull model describes well the size effect of brittle fiber tensile strength. However, some ceramic fibers have been reported that single-modal Weibull model provided biased estimation on the gauge length dependence. A hypothesis on the bias is that the density of critical defects is very small, thus, fracture probability of small gauge length samples distributes in discrete manner, which makes the Weibull parameters dependent on the gauge length. Tyranno ZMI Si-Zr-C-O fiber has been selected as an example fiber. The tensile tests have been done on several gauge lengths. The derived Weibull parameters have shown a dependence on the gauge length. Fracture surfaces were observed with SEM. Then we classified the fracture surfaces into the characteristic fracture patterns. Percentage of each fracture pattern was found dependent on the gauge length, too. This may be an important factor of the Weibull parameter dependence on the gauge length.

10108 SiC単繊維引張試験における破断面形態に及ぼすゲージ長の影響(微小部材の強度,OS10 先進材料の力学と強度)

It is known that the single-modal Weibull model describes well the size effect of brittle fiber tensile strength. However, some ceramic fibers have been reported that single-modal Weibull model provided biased estimation on the gauge length dependence. The objective of this study is thus to assess if the Weibull parameters are dependent on the gauge length for the case of small gauge length samples. Tyranno ZMI Si-Zr-C-O fiber has been selected as an example fiber. To analyze the dependence of the Weibull parameters on the gauge length, the tensile tests have been done on several gauge lengths. The derived Weibull parameters have shown a dependence on the gauge length. Fracture surfaces were observed with SEM. Then we categorized fracture surfaces by four kinds of fracture patterns. The percentage of each fracture pattern was found variable with the gauge length.

Empirical Modeling of Reinforced Concrete Shear Strength Size Effect for Members without Stirrups

The reduction in shear stress at shear failure as member depth of beams and slabs not containing stirrups increases is known as the size effect in shear. While many theoretical models for this phenomenon have been proposed over the years, this paper provides a purely empirical study of the effect. Twenty-four different size effect series of shear experiments are examined, and curve fit parameters are calculated for three different equation formats. Equations to estimate the curve fit parameters are presented that may be useful in generating design specifications that include size effect. The conclusions include that a size effect equation based on a power equation is not as good as a 1/d-based equation with a depth offset. Aggregate size and longitudinal percentage of reinforcement were found to be important in the estimation of shear strength. No strong evidence was found that size effect data for shear in reinforced concrete beams without stirrups should be modeled with effective depth to the power of -0.5 as would be expected from linear elastic fracture mechanics.

Size Effect in Compressive Strength of Grouted Sand - Part I: Experiments

Size Effect in Compressive Strength of Grouted Sand - Part I: Experiments

Size Effect in Compressive Strength of Grouted Sand - Part II: Modeling of Experimental Data

Size Effect in Compressive Strength of Grouted Sand - Part II: Modeling of Experimental Data

Size effect of concrete members applied with flexural compressive stresses

In this study, two types of special experiments are carried out to understand flexural compressive strength size effect of concrete members. The first type is an ordinary cylindrical specimen (CS) with a fully penetrated and vertically standing plate type notch at the mid-height of the specimen, which is loaded in compression at the top surface (e.g., in the parallel direction to the notch length). The second type is a general double cantilever beam (DCB), which is compression loaded in axial direction (e.g., in the parallel direction of the notch). For CS, an adequate notch length is taken from the experimental results obtained from the compressive strength experiment of various initial notch lengths. The trial tests to select the effective initial notch length show that CS with an initial notch length approximately greater than four times the maximum aggregate size fails without an additional increased load and in stable manner under Mode I failure mechanism. Therefore, the initial notch length to the maximum aggregate size ratio of 4.0 is used for all size specimens. For DCB, the eccentricity of loading points with respect to the axial axis of each cantilever and the initial notch length are varied. In both specimens, the compressive loads apply flexural compressive stresses on the crack tip region of the specimens. These two types of specimens fail by Mode I crack opening mechanism. By testing 3 geometrically proportional size specimens for CS and DCB, the experimental datum for flexural compression size effect of concrete are obtained. Using the obtained flexural compressive strength size effect datum, regression analyses are performed using Levenberg-Marquardt's least square method (LSM) to suggest new parameters for the modified size effect law (MSEL). The analysis results show that size effect is apparent for flexural compressive strength of specimens with an initial notch. For CS, the effect of initial notch length on flexural compressive strength size effect is apparent. For DCB, flexural compressive size effect is dependent on the eccentricity of loading points with respect to the axial axis of the cantilever beam. In other words, if DCB specimen is applied with greater tensile stress at the crack tip, the size effect of concrete becomes more distinct. The results show that the flexural compressive strength size effect of initial notch length variation of DCB exists but directly dependent on the loading location. This is due to the fact that the sizes of fracture process zone (FPZ) of all DCB specimens are similar regardless of the differences in the specimen slenderness ratio, but the flexural compressive and tensile stress combinations resulting in stress concentration at the crack tip region has direct effect on size effect of concrete members.

掘削体積比エネルギーを用いた岩石強度の寸法効果の推定

It is well known that rock strength decreases with increasing of specimen size. Accurate knowledge for the size or scale effects in rock strength is very important to apply the small-scale experimental results to the large-scale underground space design. However, the investigation of the size effects requires tremendous efforts and expenses in most cases. In this study, a new method to estimate the size effects of rock based on the specific energy in excavation was proposed. The method needs minimum efforts without extra equipments. Through the careful examination of eleven excavation methods, it was found that the specific energy decreases with the excavation scale following the equation:(Specific energy) ∝ (Excavation scale)-2α with α=0.2.Cutting spacing, penetration or geometrical mean of the two was found to be appropriate as excavation scale. Specific energy is roughly proportional to square of strength. Then, the following equation stands:(Rock strength) ∝ (Excavation scale)-α.This result is very conformable to the size effects deduced by the results of the compression tests. The size of cuttings in excavation was examined and the close relation was found between the excavation scale and the size of cuttings. Relation between the specific energy and the work index in comminution was also discussed.

Discussion on “Lattice modelling of size effect in concrete strength” by Ince R, Arslan A, Karihaloo BL [Engineering Fracture Mechanics 2003;70:2307–20

Discussion on “Lattice modelling of size effect in concrete strength” by Ince R, Arslan A, Karihaloo BL [Engineering Fracture Mechanics 2003;70:2307–20

Lattice modelling of size effect in concrete strength by Ince R, Arslan A, Karihaloo BL [Engineering Fracture Mechanics 2003;70:2307–20

Lattice modelling of size effect in concrete strength by Ince R, Arslan A, Karihaloo BL [Engineering Fracture Mechanics 2003;70:2307–20

Evaluation of Long-Term Strength on the Basis of Approximate Solution of the Diffusion Equation

We propose a method aimed at modeling the size effect of long-term strength for rods subjected to uniaxial tension. The method is based on the analysis of the diffusion of media in the materials of the rods. We obtain approximate solutions of the diffusion equation in the axisymmetric statement for two types of boundary conditions based on the introduction of the diffusion front and estimate the error of these solutions. The analysis of the data of two available series of tests shows that the proposed method correctly describes the observed size effect. The approximate solution of the diffusion equation is responsible for an error in the evaluation of times to fracture equal, on the average, to 5%.

Lattice modelling of size effect in concrete strength

This paper uses a recently improved lattice network model to study the size effect in the strength of plain concrete structures. The several improvements made to the lattice network model are: (i) tension softening of the matrix phase is included in the material modelling; (ii) the structural response is modelled by incrementing the deformation rather than the load. This eliminates the need for introducing arbitrary scaling parameters in the beam element failure criteria and; (iii) a square rather than a triangular lattice beam network is found to be adequate for modelling concrete, thus greatly reducing the computational time. The improved square lattice network has been used to simulate the complete load–deformation response of notched three-point bend beams of different sizes with a view to checking the validity of several size effect models available in the literature. Lattice simulation was found to identify microcracking, crack branching, crack tortuosity and bridging, thus allowing the fracture process to be followed until complete failure. The improved lattice model predicted smooth structural response curves in excellent agreement with test results. The simulated nominal strengths also correlated very well with the test results, apart from that for the smallest beams (depth 38.1 mm). However, even in the relatively broad range of sizes (1:8) of the test beams, there was no clear evidence that one size effect model is superior to the other. In fact, rather surprisingly the test data would appear to be equally well described by all the available size effect models. The lattice simulations however indicated a trend which is better predicted by the multifractal scaling model.

A simple method for calculating interaction of numerous microcracks and its applications

The effects of microcrack interaction on the failure behavior of materials present one problem of considerable interest in micromechanics, which has been extensively argued but has not been resolved as yet. In the present paper, a simple and effective method is presented based on the concept of the effective field to analyze the interaction of microcracks of a large number or of a high density. To determine the stress intensity factors of a microcrack embedded in a solid containing numerous or even countless microcracks, the solid is divided into two regions. The interaction of microcracks in a circular or elliptical region around the considered microcrack is calculated directly by using Kachanov’s micromechanics method, while the influence of all other microcracks is reflected by modifying the stress applied in the far field. Both the cases of tensile and compressive loading are considered. This simplified scheme may yield an estimate for stress intensity factors of satisfactory accuracy, and therefore provide a potential tool for elucidating some phenomena of material failure associated with microcracking. As two of its various promising applications, the above scheme is employed to investigate the size effects of material strength due to stochastic distribution of interacting microcracks and to calculate the effective elastic moduli of elastic solids containing distributed microcracks. Some conventional micromechanics methods for estimating the effective moduli of microcracked materials are evaluated by comparing with the numerical results. Only two-dimensional problems have been considered here, though the three-dimensional extension of the present method is of greater interest.

Strain gradient and wavelet interpretation of size effects in yield and strength

The problem of yield stress and ultimate strength dependence on the specimen size is considered. This dependence, in otherwise geometrically similar specimens, is herein interpreted on the basis of gradient plasticity and wavelet analysis which introduce an internal length scale in the stress vs. strain relation. In particular, gradient plasticity within a simple strength of materials approach and wavelets within a simple scale-dependent constitutive equations approach, are used to derive expressions for the yield stress and the ultimate strength of solid bars subjected to torsion and bending. It is shown that these expressions depend on the size of the specimen. When the dimensions of the specimen become comparable to the internal length scale, size effects become important, while for larger specimens such dependence becomes negligible. Comparisons of the theoretical predictions to available experimental data are given.

Effects of thickness on mechanical properties of conducting polythiophene films

Owing to the rapid development of conducting polymers [1, 2], it has become possible to control the electrical property of polymers over the range from insulation to high conductivity. Some polymeric materials have successively been synthesized with conductivity similar to such metals as copper [3]. Polymers and electrical conductivity are no longer mutually exclusive. In addition, conducting polymers possess some other superior properties such as low mass density, ease of fabrication, flexibility in design, and resistance to corrosion. Therefore, they can be used in various applications as substitutes of metals [4]. The promise of combining these properties with good electrical properties in polymers has prompted extensive interest in the last two decades. However, most of the previously synthesized conducting polymers are brittle, insoluble, intractable, and often decomposed before melting [5]. The preparation of conducting polymers with high strength and good chemical stability remains still a great challenge. To date, there is still a lack of investigation on the relationship between the macroscopic properties and microstructures of conducting polymer films. Recently, polythiophene thin films with high conductivity and chemical stability have been synthesized by an electrochemical method [6, 7]. The mechanical properties of such kinds of films are experimentally investigated in this paper by using an electronic speckle pattern interferometry method and the SEM technique. Significant effects of thickness on the strength and Young’s modulus of films are found. The conducting polythiophene films used in this study were synthesized in a one-compartment cell with the use of an EG&G potentiostat model 283 under computer control [6, 7]. Two AISI 304 stainless steel sheets with a spacing of 5 mm are used as the working and counter electrodes. The anodic potential is referred to the third electrode made of Ag/AgC1. Polythiophene films were yielded by electrolysis of 0.03 M thiophene in freshly distilled boron trifluoride diethyl etherate at a constant potential of 1.3 V. The film thickness was controlled by adjusting the total charge passed in the cell. Thus, polythiophene films with different thicknesses ranging from one to hundreds of microns were successively grown/deposited on the base of a stainless steel plate by controlling the strength of the electric current and the synthesis time. These films behave like metal thin films and can be cut easily into various shapes of specimens. An electronic speckle pattern interferometry (ESPI) method [8] was used to measure the displacement of a film specimen under strain. When subjected to an externally applied force, a polythiophene film may undergo a considerable shape change, especially in the longitudinal direction due to the large ratio of the length L to the width W (L/W ≥ 10). The shape change of a film can be detected by the ESPI method. Several tensile stress-strain curves of polythiophene films of different thickness are shown in Fig. 1. Following the linear elastic deformation, plastic deformation occurs at a threshold stress, referred to as the yield stress. During plastic deformation, an evident strain-hardening curve is observed, that is, the applied stress has to be increased continuously for further development of plastic strain. The maximum stress that a polythiophene film of thickness less than 10 μm can bear is similar to that of an aluminum thin film. It can be seen from Fig. 1 that the strength of a polythiophene film exhibits an evident dependence upon its thickness. As the film thickness increases, the tensile strength decreases. This size effect of tensile strength is especially evident when the film thickness is less than 10 μm. However, the film strengths approach to a constant value when the thickness is larger than 15 μm. No evident difference has been observed in the yield stress among films when their thickness is larger than about 15 μm. In our measurement, films of thickness 4 μm possess the highest strength. The physical micromechanisms of such a size effect will be analyzed below. The elastic modulus of a film can also be obtained by the same experimental system, from which both strain

Scale-dependent constitutive relations and the role of scale on nominal properties

Abstract Size effects in strength and fracture energy of heterogeneous materials is considered within a context of scale-dependent constitutive relations. Using tools of wavelet analysis, and considering the failure state of a one-dimensional solid, constitutive relations which include scale as a parameter are derived from a ‘background’ gradient formulation. In the resulting theory, scale is not a fixed quantity independent of deformation, but rather directly dependent on the global deformation field. It is shown that strength or peak nominal stress (maximum point at the engineering stress–strain diagram) decreases with specimen size while toughness or total work to fracture per nominal area (area under the curve in the engineering stress–strain diagram integrated along the length of the considered one-dimensional specimen) increases. This behavior is in agreement with relevant experimental findings on heterogeneous materials where the overall mechanical response is determined by variations in local material properties. The scale-dependent constitutive relations are calibrated from experimental data on concrete specimens.

Size Effects on Tensile Strength and Fracture Energy In Concrete: Wavelet vs. Fractal Approach

The problem of size effect on tensile strength and fracture energy of brittle, heterogeneous and disordered materials is considered within two, apparently different approaches: one based on wavelets and scaledependent constitutive relations and another based on fractal arguments. Using wavelet analysis, scaledependent constitutive relations are derived from an underlying gradient formulation. By varying the scale parameter, size effects in tensile strength and fracture energy can be captured. Using fractal analysis, a fractal dimension of the damaged material surface which the stress acts upon is defined and a corresponding sizedependent tensile strength and fracture energy is obtained. These two approaches are then compared. Both of them give results with similar trend and their divergence emerges only in the limit of infinitesimally small specimens, where different deformation mechanisms and random responses may prevail.

Effect of specimen size on strength reliability of SiC/Al composite

A tensile test was carried out for SiC/Al composite specimens which have different width. A Monte Carlo simulation using finite element method was performed with changing the width of SiC/Al composite model. The size effect of composite strength and its reliability was examined and discussed. The results showed that composite strength does not appear to have large change but its reliability is improved with increasing the composite specimen width. Effect of specimen width on composite strength and its reliability is dependent on the both factors, that is, (a) the accumulating of fracture process and (b) fibre strength and its scatter of the weaker fibres in the composite. The accumulation of the fracture process is related to the fibre stress distribution in the composite.

Size effect in tensile strength caused by stress fluctuations

The randomness of microstructure heterogeneous materials leads to creation of microscopic random stress fields within the bulk of the material under loading. Although in average the microscopic stresses coincide with the macroscopic (e.g., externally applied) stress, the local differences (stress fluctuations) can be high, the magnitude increasing with the volume of the heterogeneous material. In the case of uniform macroscopic loading, Gaussian stress fluctuations lead to a size effect in which the tensile strength reduces as square root of logarithm of the sample size. In practice, however, the macroscopic tensile stress fields are usually nonuniform. In this case, failure is determined by the maximum value of the macroscopic stress with the scale effect controlled by the minimum degree of the macroscopic stress decrease from its maximum. Therefore, a second model is proposed which accounts for a linear stress variation. Comparison of both models with the experimental data on macroscopic strength and stress variations in dog-bone shaped samples (scale range of 1:32), shows that the model based on the assumption of uniform macroscopic stresses can only explain part of the experimental data with unrealistic values of the fitting parameters. The model which takes into account the linear part of the macroscopic stress distribution offers reasonably good accuracy. This serves as another indication that macroscopic stress nonuniformity plays a crucial role in the mechanism of size effect.

Analytical simulation of strength size effect in composite materials

This paper presents a theoretical analysis of the strength size effect associated with the length, width and thickness of laminated composites. The presented expressions for prediction of the reliability, use the Equal Load Sharing Rule (ELS Rule) which are determined by sequential multi-step failure of single lamina or sublaminate. The developed equations are validated using two sets of experimental data obtained from published data. The influence of the length, width, and thickness on the ultimate strength of laminated composites is demonstrated. Strength size effect is analyzed for the in-plane strength parameters. The developed size effect equations are implemented in a user-defined subroutine UMAT in the finite element code ABAQUS to demonstrate their applicability in numerical analysis.