Comparative study on biaxial low-cycle fatigue behaviour of three structural steels

In this study the uniaxial/biaxial low‐cycle fatigue behaviour of three structural steels (Ck45 normalized steel, 42CrMo4 quenched and tempered steel and AISI 303 stainless steel) are studied, evaluated and compared. Two parameters are considered for estimating non‐proportional fatigue lives: the coefficient of additional hardening and the factor of non‐proportionality. A series of tests of uniaxial/biaxial low‐cycle fatigue composed of tension/compression with cyclic torsion were carried out on a biaxial servo‐hydraulic testing machine. Several loading paths were carried out, including proportional and non‐proportional ones, in order to verify the additional hardening caused by different loading paths. The experiments showed that the three materials studied have very different additional hardening behaviour. Generally, the transient process from the initial loading cycle to stabilized loading cycle occurs in a few cycles. The stabilized cyclic stress/strain parameters are controlling parameters for fatigue damage. A factor of non‐proportionality of the loading paths is evaluated based on the Minimum Circumscribed Ellipse approach. It is shown that the microstructure has a great influence on the additional hardening and the hardening effect is dependent on the loading path and also the intensity of the loading.     

Evaluation of the basic formulations for the cumulative probability of brittle fracture with two different spatial distributions of microcracks

The random distribution of microcracks in terms of their size, shape, orientation and spatial location has direct impact on the cumulative probability of brittle fracture induced failure, with the effect of spatial distribution being rarely explored. Recently, two weakest link theory‐based formulations for the cumulative probability of brittle fracture induced failure have been proposed for the spatial distribution of microcracks obeying the Poisson postulates and the uniform distribution, respectively. This work compares these two new formulations with the currently commonly adopted one built on the Poisson postulates under both the uniform and the non‐uniform uniaxial loading conditions. It is concluded that under general loading conditions involving non‐uniform stress states, the existing formulation is equivalent to or closely approximate to neither of the two new formulations thus should be discarded, because of its inaccurate derivation. The new formulations are featured with unique symmetry or self‐similarity in their expressions. Their capability in revealing the size effect or the scaling law of failure is highlighted and validated by a set of published uniaxial and biaxial flexural strength data of brittle material.     

Energy dissipation capacity of fibre reinforced concrete under biaxial tension–compression load. Part I: Test equipment and work of fracture

The objective of this research was to analyse the differences in the dissipated energy under uniaxial tension and biaxial tension–compression load of fibre reinforced concretes using the Wedge Splitting Test. Under biaxial load the specimens were subjected to compressive stress ratios from 10% to 50% of the concrete compressive strength perpendicular to the direction of the tensile load.Under biaxial tension–compression load the energy dissipation capacity of the specimens decreases compared to the uniaxial tension load case on average 20–30%. It is believed that the decrease is a result of the damage mechanism of the concrete matrix and deterioration of the fibre–matrix and/or aggregate–cement paste interfaces in case the section is additionally loaded with compression stresses. This indicates that dimensioning of concrete elements under biaxial stress states using material parameters obtained from tests conducted on specimens under uniaxial tensile load is unsafe and could potentially lead to a non-conservative design.In the second part of this paper the extent of the fracture process zone under uniaxial tension and biaxial tension–compression load will be examined with the Acoustic Emission technique and the reasons for decrease of the energy dissipation capacity under biaxial load will be further discussed.     

双轴受压状态下的高延性纤维增强水泥基复合材料本构模型

基于Darwin和Pecknold考虑混凝土双轴力学行为的方法,建立一个同时考虑双轴受压状态下非线性力学行为和抗压强度变化的高延性纤维增强水泥基复合材料(ECC)二维正交各向异性本构模型。在因双轴加载而产生的正交各向异性的2个方向上引入等效单轴应变,建立非线性应力-等效单轴应变关系以考虑ECC的双轴非线性行为,并采用一条双轴强度包络线确定2个方向上的抗压强度。推导模型的显式数值算法,编写包含该算法的用户自定义材料子程序UMAT,并嵌于有限元计算程序ABAQUS v6.14中。通过对两组不同配合比的ECC试件在不同应力比下的双轴受压加载试验进行数值分析验证本模型的有效性。数值计算得到的主压应力方向上的应力-应变曲线及预测的抗压强度与试验结果吻合较好,表明该文提出的本构模型能够有效地预测ECC在双轴受压状态下的非线性力学行为和破坏强度。     

On the identification of an effective cross section for a cruciform specimen

Biaxial tensile testing of sheet metals is becoming increasingly popular for sheet metal forming. Determining equivalent stresses in biaxial tensile specimens is more complicated than in conventional uniaxial tensile specimens. In the present study, we compare four different approaches to calculate effective stresses during biaxial tensile loading of a cruciform specimen: (a) partial unloading method, where stresses are determined based on force–strain curves; (b) identification with uniaxial tensile testing; (c) an analysis of equivalent biaxial tests; and (d) numerical simulations. Considering experimental results for an AA1050 aluminium alloy and for a low‐carbon steel DC06, we show that, for the cruciform sample studied here, two methods do not yield physically reasonable results: The uniaxial approach does not properly take into account the effect of transverse loading, and the equivalent biaxial approach exhibits uncertainties in strain measurement data. The most comprehensible approach is the numerical method, because it also yields detailed information about the local stress and strain states. The numerical results are in excellent agreement with the partial unloading method in terms of the initial flow stress and of effective stress–strain curves for strains up to 0.02, with both methods predicting a similar effective cross section of 18.0 mm² for the considered specimen.     

Evolution of mechanical properties for a dual-phase steel subjected to different loading paths

The evolution of the mechanical properties of a dual-phase (DP590) steel sheet after being prestrained by uniaxial tension, plane strain and equal biaxial stretching was investigated. Specimens were first loaded using the three prestraining modes. Then, from the prestrained specimens, a few sub-sized samples were machined along the rolling direction and the transverse direction for further uniaxial tension testing. Six loading paths were provided. Equal biaxial stretching was performed using a cruciform specimen. The evolution of work hardening performance, elastic modulus, yield stress and tensile stress under the six loading paths were discussed in detail. The results indicate that loading paths can affect the latent work hardening performances, strain hardenability, yield stress and tensile stress evolution as well as the elastic modulus decrease during plastic deformation. The uniaxial tension–uniaxial tension path results in a cross-softening phenomenon, the largest yield stress enhancement and a mild maximum tensile stress increase. The equal biaxial stretching-uniaxial tension path leads to a cross-hardening phenomenon, the least yield stress enhancement and the largest tensile strength increase maximum tensile strength. The elastic modulus of DP590 steel not only changes with the accumulated plastic strain but also varies with the loading paths. The largest decrease of the elastic modulus equal biaxial stretching–uniaxial tension can reach 12.7% beyond 8% equivalent strain, which is 5.2% greater than that in the monotonic uniaxial tension path.     

The Effect of Size on the Splitting Strength of Cubic Concrete Members

In the study of concrete fractures, split‐tension specimens, such as cylinders, cubes and diagonal cubes, are frequently preferred to beams. However, experimental investigations on concrete reveal that for the same specimen geometry, the nominal strength of specimen decreases with increasing specimen size. This phenomenon is named as the size effect in the fracture mechanics of concrete. Although nominal strength is also highly affected by the width of the distributed load in the split‐tension cylinder and cube specimens, this effect can be negligible within the practical range of the load‐distributed width in the diagonal cubes. However, the number of theoretical and experimental studies with diagonal split‐tension specimens is limited. Besides, a size effect formula for estimating the split‐tensile strength of the diagonal cube specimens has not been proposed. In this study, nine series of cube and diagonal cube specimens, with three different sizes but similar geometries, were tested under different load‐distributed widths. The ultimate loads obtained from the test results are analysed by the modified size effect law. Subsequently, prediction formulas are proposed, and they are compared with historical test data from the split‐cylinder specimens.     

Corrosion fatigue behaviour of 304 stainless steel under proportional and non‐proportional multiaxial loading condition

Corrosion fatigue and electrochemical tests under proportional loading and non‐proportional loading were conducted on 304 stainless steel in 0.63 mol L^?1 NaCl solution at room temperature. Two biaxial loading paths were applied to study the effect of proportional loading and non‐proportional loading on corrosion fatigue behaviour. Surface and fractographic observations of multiaxial corrosion fatigue specimens were carried out by scanning electron microscopy. It was shown that proportional loading had a more significant effect on the occurrence of local corrosion compared with non‐proportional loading because the continuous rotation of the principal stress plane under non‐proportional loading inhibits the pit formation.     

钢纤维混凝土多轴损伤比强度准则EI北大核心CSCD

以损伤比强度理论为基础,建立了钢纤维混凝土真三轴损伤比强度准则,并根据钢纤维混凝土试验资料,推荐了钢纤维混凝土损伤比变量表达式中的6个经验参数。利用钢纤维混凝土在单轴、双轴和三轴受力状态下的应力-应变曲线试验结果验证了损伤比取值合理性,对比了单轴受拉、单轴受压和双轴等压等典型受力状态下钢纤维混凝土和普通混凝土损伤比变量取值的差异。通过与国内外共104组钢纤维体积率为0.5%~2.5%的钢纤维混凝土三轴强度试验资料的比较,表明六经验参数钢纤维混凝土损伤比强度准则的三维破坏包络面接近已有认识;通过与国内外强度准则比较,表明损伤比强度准则与钢纤维混凝土三轴试验数据有较高的吻合度。对于围压三轴受力状态,提出简化的钢纤维混凝土常规三轴强度准则,并与已有常规三轴强度准则进行比较分析。此外,对于材料处于二轴受力,推荐了简化的损伤比二轴强度准则中的经验参数取值。     

Load-carrying capacity of circular cylindrical shells in cyclic loading with internal pressure

Results are presented of experimental examination of the strength and endurance of smooth cylindrical circular shells in cyclic loading with internal pressure with a constant amplitude in uniaxial and biaxial tensile loading. It is shown that the level of fracture stresses in these shells in the biaxial stress state can be evaluated using the criterion of static strength on the basis of a fatigue curve constructed in uniaxial tensile loading.Translated from Problemy Prochnosti, No. 1, pp. 18–21, January, 1990.     

考虑细观组分影响的混凝土宏观力学性能理论预测模型

混凝土宏观力学性能与其内部细观结构构造密切相关。该文建立了一类能够考虑细观组分影响的混凝土宏观力学性能理论预测模型。首先,采用细观力学数值试验法对理论模型中的参数进行了标定;进而,基于该模型对混凝土断裂能和单轴抗拉强度在材料层次的尺寸效应行为进行了分析。结果表明:混凝土断裂能和单轴抗拉强度均随骨料级配（即最大骨料粒径）发生变化,且受到界面特性的影响。当界面过渡区力学性能相对薄弱时,混凝土强度较低,断裂能和单轴抗拉强度随骨料级配增大而呈现减小的趋势;当界面过渡区力学性能较强时,混凝土强度较高,断裂能和单轴抗拉强度随骨料级配增大亦呈现增大的趋势。计算结果与试验结果吻合良好,验证了该文建立的理论预测模型的准确性和合理性。     

Prediction of the size effect in concrete structures using an analytical approach to the weakest link and localization method (WL2)

This paper proposes a refined Weibull effective volume (WEV) approach in order to model the size effect in concrete structures. Both the Highly Stressed Volume (HSV) and Weibull Weakest Link and Localization (WL2) methods are presented. An analytical probabilistic approach to running WL2 is then developed, knowing that this method takes into account the energetic-statistical size effect. The approach employed recognizes the inelastic phase before the peak load; it provides an analytical and fast estimation of the structural tensile strength at various scales. This approach depends on a scale length, which accounts for the spatial randomness of the concrete tensile strength, and is identified on a series of concrete specimens under uniaxial tension by use of an inverse analysis. The Weibull modulus estimation is also discussed herein. Moreover, it is shown that the analytical probabilistic approach to WL2 implementation yields the size effect prediction in both average and dispersion for various experimental series, from laboratory tests to large massive structures. The experimental test series discussed in this paper consists of concrete specimens under uniaxial tension and 3-point bending loading.     

Experiments on concrete under uniaxial impact tensile loading

A problem of practical importance for designing of structural elements is discussed in this paper—the behaviour of concrete subjected to uniaxial impact tensile loading. The “Split Hopkinson Bar” technique was adopted for testing concrete in uniaxial tension at stress rates between 2 and 60 N/mm²/ms. A remarkable increase in tensile strength was observed due to high stress rate. The ratio of impact and static tensile strength varied between 1.33 and 2.34 for various concrete mixes. The influence of maximum aggregate size, water-cement ratio, cement content, cement type and quality, specimen humidity, static compressive strength and loading/casting direction upon the uniaxial impact tensile strength was studied. The high stress rate resulted in an increase of the modulus of elasticity of concrete in uniaxial tension. An explanation for the observed phenomena is suggested.     

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.     

混凝土试样在静态载荷作用下断裂过程的数值模拟研究

提出了一个模拟混凝土断裂过程的细观力学模型,并应用该模型从混凝土的细观非均匀性结构出发,对混凝土试样在单轴和双轴静态载荷作用下的断裂过程进行了数值模拟,给出双轴载荷作用下混凝土的强度包络面。数值模型结果较好地模拟了混凝土试样从裂纹萌生、扩展到宏观裂纹形成的整个断裂过程,与实验结果表现出较好的一致性。     

On the estimation of the material fatigue properties required to perform the multiaxial fatigue assessment

To accurately perform the fatigue assessment of engineering components subjected to in‐service multiaxial fatigue loading, the adopted design criterion must properly be calibrated, the used information usually being the fatigue strength under both pure uniaxial and pure torsional fatigue loading. Because of the complex fatigue response of metallic materials to multiaxial loading paths, the only reliable way to generate the necessary pieces of calibration information is by running appropriate experiments. Unfortunately, because of a lack of both time and resources, very often, structural engineers are requested to perform the multiaxial fatigue assessment by guessing the necessary fatigue properties. In this complex scenario, initially, the available empirical rules suitable for estimating fatigue strength under both pure axial and pure torsional fatigue loading are reviewed in detail. Subsequently, several experimental results taken from the literature and generated by testing metallic materials under a variety of proportional and non‐proportional multiaxial loading paths are used to investigate the way such empirical rules affect the accuracy in estimating fatigue strength, the damage extent being evaluated according to the modified Wöhler curve method. Such a systematic validation exercise allowed us to prove that under proportional loading (with both zero and non‐zero mean stresses), an adequate margin of safety can be reached even when the necessary calibration information is directly estimated from the material ultimate tensile strength. On the contrary, in the presence of non‐proportional loading, the use of the empirical rules reviewed in the present paper can result, under particular circumstances, in a non‐conservative fatigue design.     

高强高性能混凝土三轴拉压压力学性能试验研究

采用大型动静真三轴伺服液压试验系统,对单轴压强度为90.6 MPa的高强高性能混凝土进行三轴拉压压等比例试验研究,获得了各应力比下试件的破坏模式、多轴强度及应力-应变曲线。试验结果表明:高强高性能混凝土三轴拉压压应力状态下的破坏为典型的受拉破坏;最大主应力方向的极限强度远低于其单轴压强度,中间主应力效应不明显;拉应力的存在对最大主应力方向应力-应变曲线影响十分明显,呈现出明显的线性特征。基于试验结果提出了适用于高强高性能混凝土的强度准则,为高强高性能混凝土本构关系的建立提供了试验和理论依据。     

虚拟裂缝尖端拉应力大小的确定

针对单边切口的混凝土轴心受拉构件,基于虚拟裂缝模型提出一种计算极限承载力的解析模型,并在此基础上确立了虚拟裂缝尖端拉应力与混凝土轴心抗拉强度之间的关系。结果表明:二者的比值随初始缝高比的增大呈线性增加,但对混凝土强度等级的变化不敏感。其原因是由于所有的混凝土试件都存在初始缺陷,导致截面上存在明显的应力梯度,因而得到的混凝土轴心抗拉强度值是截面应力的平均值,而虚拟裂缝尖端拉应力为截面上的最大应力。很显然,轴心受拉构件的初始缺陷越长,截面的应力梯度越大,虚拟裂缝尖端拉应力与平均应力的比就越大。通常情况下,虚拟裂缝尖端的拉应力大小约为混凝土轴心抗拉强度值的1.22倍,约等于混凝土的抗折强度。     

A stereological analysis of aggregate grading and size effect on concrete tensile strength

One of the most important topics in solid mechanics is the study of the so-called size effects, whose importance has been widely recognised during the last decades. Size effects are particularly strong in quasi-brittle (i.e., concrete-like) materials. In this paper we focus our attention on the tensile strength decrease associated with the size of concrete structures. An original explanation of this well-known size effect was proposed by the first Author based on the assumption of a fractal-like damage localisation at the mesostructural level. This hypothesis leads to a multifractal scaling law (MFSL) for concrete tensile strength. The present contribution provides a scaling law for concrete tensile strength based on its aggregate size distribution. Since the weakest link in normal strength concrete is represented by the interface between the cementitious matrix and the aggregates, it seems reasonable to look for a relationship between the aggregate grading and the material strength. Based on the hypothesis that the strength depends on the largest flaw, we compute the strength of a concrete specimen as a function of its size. Differently from other statistical approaches, we use a truncated distribution (namely the Füller distribution) in order to describe realistically the flaw population inside the specimen. Calculating the distribution of the largest flaw size by means of statistics of extremes, and relating it to the specimen size, we obtain a scaling law for concrete tensile strength whose trend strictly agrees with the MFSL. Finally, we pay particular attention to the computation of the power law exponent characterising the strength scaling at the smallest sizes and present a comparison with available experimental data.     

A stereological analysis of aggregate grading and size effect on concrete tensile strength

One of the most important topics in solid mechanics is the study of the so-called size effects, whose importance has been widely recognised during the last decades. Size effects are particularly strong in quasi-brittle (i.e., concrete-like) materials. In this paper we focus our attention on the tensile strength decrease associated with the size of concrete structures. An original explanation of this well-known size effect was proposed by the first Author based on the assumption of a fractal-like damage localisation at the mesostructural level. This hypothesis leads to a multifractal scaling law (MFSL) for concrete tensile strength. The present contribution provides a scaling law for concrete tensile strength based on its aggregate size distribution. Since the weakest link in normal strength concrete is represented by the interface between the cementitious matrix and the aggregates, it seems reasonable to look for a relationship between the aggregate grading and the material strength. Based on the hypothesis that the strength depends on the largest flaw, we compute the strength of a concrete specimen as a function of its size. Differently from other statistical approaches, we use a truncated distribution (namely the Füller distribution) in order to describe realistically the flaw population inside the specimen. Calculating the distribution of the largest flaw size by means of statistics of extremes, and relating it to the specimen size, we obtain a scaling law for concrete tensile strength whose trend strictly agrees with the MFSL. Finally, we pay particular attention to the computation of the power law exponent characterising the strength scaling at the smallest sizes and present a comparison with available experimental data.