Probabilistic approach and size effect in brittle materials under multiaxial stress states

Summary The problem of evaluating the strength of brittle materials demands a statitistical approach owing to the considerable dispersion shown by these materials, particularly if subject to stress states in which the tensile component is predominant. For this purpose, the Weibull theory can be conveniently applied, which is based on the concept of the weakest link in a chain allowing the determination of both the failure probability of a model under certain loading conditions and the corresponding mean strength.In the present work the Weibull theory is applied, by means of of the Stanley's approach, to the case of multiaxial stress states in order to deal with some problems not thoroughly investigated so far, among which the effect of the shear stresses on the flexural strength and the indirect evaluation of the tensile strength by means of the Brazilian test.In conclusion, the theory is applied to assess the difference between the tensile strength values obtained by means of different tests. The results achieved are then compared with experimental data.

---

Sommario Il problema della resistenza dei materiali fragili richiede un approccio statistico a causa della forte dispersione propria di questi materiali, in particolare se soggetti a uno stato tensionale prevalentemente di trazione. A questo scopo può essere applicata la teoria di Weibull, basata sul concetto dell'anello più debole di una catena, che consente di determinare la probabilità di crisi di un modello per una determinata condizione di carico e la corrispondente resistenza media.In questo lavoro la teoria di Weibull viene applicata al caso degli stati di tensione pluriassiali con l'approccio di Stanley, allo scopo di indagare su alcuni problemi non ancora affrontati per questa via, tra cui l'influenza delle tensioni tangenziali sulla resistenza a flessione e la determinazione indiretta della resistenza a trazione mediante prova Brasiliana.A conclusione, la teoria viene impiegata per valutare la differenza tra i valori di resistenza a trazione ottenibili con diversi tipi di prova. I risultati vengono posti a confronto con dati sperimentali.

---

Signification of fundamental symbols used P _f probability of failure - applied stress - _u location parameter - ₀ scale parameter - m shape parameter (Weibull modulus) - (z) Gamma function - V total volume of the model - unit volume - _f mean tensile strength in flexure - _b mean indirect tensile strength (Brazilian test) - _t mean direct tensile strength - _tv mean direct tensile strength of unit volume - _c mean cylindrical compressive strength. Some results of this work were presented at the following meetings:- 10th National Meeting of A.I.A.S., Associazione Italiana per l'Analisi delle Sollecitazioni, Cosenza (Italy), September 22–25, 1982;- 4th I.C.A.S.P., International Conference on Applications of Statistics and Probability in Soil and Structural Engineering, Firenze (Italy), June 13–17, 1983.This work was supported by the C.N.R.     

Lüders bands propagation of 1045 steel under multiaxial stress state

Inhomogeneous plastic deformation of 1045 steel under monotonic loading was experimentally studied. Thin-walled tubular specimens were used in the experiments and custom-made small strain gages were bonded on the specimen surface to characterize the local deformation. Experiments were conducted under tension, torsion, and combined tension–torsion. During the propagation of Lüders bands, the local deformation experienced two-stage deformation: an abrupt plastic deformation stage followed by a slower deformation process. In some area of the gage section of the specimen, a small amount of initial plastic deformation occurred before the Lüders front reached. During the propagation of Lüders bands, multiple Lüders fronts can be formed. Under tension, torsion, and combined tension–torsion with a constant axial load, the Lüders front was approximately parallel to the material plane of maximum shear stress. When the combined axial-torsion followed a proportional fashion, the stress–extensometer strain responses were dependent on the axial/torsional loading ratio, and the Lüders fronts were oriented differently and propagated along the specimen axis at a different velocity. The local strain was inhomogeneous even at the work-hardening stage. The relationships between the equivalent stress and the equivalent plastic strain were found to be practically identical for all the loading cases studied.     

Cyclic behavior of a type 304 stainless steel in biaxial stress states at elevated temperatures

The results of companion, incremental/decremental, and stepup fatigue experiments on austenitic stainless steel tube (type 304) are presented. The experiments include proportional and nonproportional loading conditions at ambient as well as elevated temperatures. Empirical relations are developed between von Mises effective stress and strain, and these relations are shown to describe the cyclic behavior during proportional companion as well as incremental/decremental tests. In case of nonproportional incremental/decremental experiments, the material behavior is not accurately modeled either by using the von Mises effective stress and strain, or by relating Tresca's maximum shear stress and strain.     

An evaluation for several kinematic hardening rules on prediction of multiaxial stress-controlled ratchetting

This study evaluates the performance of several non-linear kinematic hardening rules in predicting the various biaxial ratchetting experiments of stainless steel (SS) 304L under various stress-controlled histories performed by Hassan et al. (2008). The non-linear kinematic hardening rules proposed by 9, 32, 33 and 160, 19, 12 and 13 and the different rules of Abdel-Karim (2009) are examined and carefully scrutinized. The considered kinematic hardening rules range from the simple classical ones to more detailed rules, which incorporate additional terms and/or parameters to simulate different factors that affect ratchetting. It is shown that none of the examined kinematic hardening rules is general enough to simulate all of the ratchetting responses for the experiments under consideration.     

Effect of elastic modulus variation during plastic deformation on uniaxial and multiaxial ratchetting simulations

In order to identify different variables that affect ratchetting simulations, variation of elastic modulus during loading and unloading is considered and discussed based on the experimental observations which pointed out by Morestin and Boivin, 1996, Ishikawa, 1997, Cleveland and Ghosh, 2002, Zhou et al. (2005) and recently by Khan et al., 2009a, Khan et al., 2009b, Khan et al., 2009c. Then the effect of such variation on simulations is scrutinized from the theoretical point of view by considering simulations of ratchetting experiments conducted on stainless steel 304L by Hassan et al. (2008) using the well-known Armstrong–Frederick model. It is shown that, using two different values for the elastic modulus during loading and unloading could have a significant effect on simulations of uniaxial ratchetting. On the other hand, such significant effect hardly occurs in the case of simulations of biaxial ratchetting experiments under consideration. The importance of such findings is that the excessive ratchetting over-prediction resulting from any specific kinematic hardening rule is expected to decrease significantly by taking into consideration this effect. In this case, modeling of kinematic hardening rules could necessitate more attention and reconsideration.     

Time-dependent ratchetting experiments of SS304 stainless steel

The time-dependent strain cyclic characteristics and ratchetting behaviours of SS304 stainless steel were investigated by uniaxial/multiaxial cyclic loading tests at room and elevated temperatures (350 and 700 °C). The effects of loading rate, peak/valley strain or stress holds, ambient temperature and non-proportional loading path on the cyclic softening/hardening and ratchetting behaviours of the material were discussed. It is shown that: the cyclic deformation of the material presents remarkable time-dependence at room temperature and 700 °C; the cyclic hardening feature and ratchetting strain depend significantly on straining or stressing rate, hold-time, ambient temperature and the non-proportionality of loading path; the time-dependent ratchetting is resulted from the slight opening of hysteresis loop and visco-plasticity together, and the viscosity is a dominating factor at 700 °C; at 350 °C, abnormal rate-dependence and quick shakedown of ratchetting are observed due to the dynamic strain aging of the material at this temperature. Some significant conclusions are obtained, which are useful to construct a constitutive model to describe the time-dependent ratchetting behaviour of the material. It is also stated that the unified visco-plastic constitutive model discussed here cannot provide reasonable simulation to the time-dependent ratchetting at 700 °C, especially to that with certain peak/valley stress hold, since the effect of the high viscosity on time-dependent ratchetting cannot be properly described by using a unified visco-plastic flow rule.     

A general cyclic plasticity model taking into account yield surface distortion for multiaxial ratchetting

Most of the theoretical studies devoted to multiaxial ratchetting are focused on the location of the macroscopic yield domain in order to predict the correct direction of plastic strain rate, since normality of the plastic strain rate to the yield surface is usually assumed. To the authors' knowledge, the shape of subsequent yield surfaces was always kept constant in these models for simplicity reasons but unlike experimental observations. In a previous paper [J. Eng. Mater. Technol. 124(4) (2002) 402], the authors have explained the need to take into account yield surface distortion in macroscopic modeling and have therefore proposed such a constitutive model but only for biaxial loadings. In the present paper, the generalization of this distortional model is proposed for any loading paths. The model is then identified and validated on a large data base obtained firstly with an efficient polycrystalline model that can predict multiaxial ratchetting as well as yield surface distortion, and secondly with the experimental results of complex tests realized on a type 316L stainless steel.     

Creep rupture under multi-axial states of stress

R are presented of metallographic examinations of specimens which have been creep tested at high temperatures and under different conditions of steady applied stress. The interaction between the growth of micro-fissures or voids and the mode of final rupture is discussed. The applied-stress versus rupture-life characteristics of a number of commercial alloys are presented together with simple expressions which are shown to describe approximately the stress-sensitive nature of their rupture behaviour.     

STRENGTH CRITERION FOR PLAIN CONCRETE UNDER MULTIAXIAL STRESS BASED ON DAMAGE POISSON＇S RATIO

A new unified strength criterion in the principal stress space has been proposed for use with normal strength concrete （NC） and high strength concrete （HSC） in compressioncompression-tension, compression-tension-tension, triaxial tension, and biaxial stress states. The study covers concrete with strengths ranging from 20 to 130 MPa. The conception of damage Poisson＇s ratio is defined and the expression for damage Poisson＇s ratio is determined basically. The failure mechanism of concrete is illustrated, which points out that damage Poisson＇s ratio is the key to determining the failure of concrete. Furthermore, for the concrete under biaxial stress conditions, the unified strength criterion is simplified and a simplified strength criterion in the form of curves is also proposed. The strength criterion is physically meaningful and easy to calculate, which can be applied to analytic solution and numerical solution of concrete structures.     

An experimental study of inhomogeneous cyclic plastic deformation of 1045 steel under multiaxial cyclic loading

An experimental study was conducted on the inhomogeneous cyclic plastic deformation of 1045 steel under multiaxial cyclic loading. Thin-walled tubular specimens were used and small strain gages were bonded on the specimen surface to characterize the local deformation. The controlled loading paths included cyclic tension–compression, cyclic torsion, proportional axial-torsion, 90°-out-of-phase axial-torsion, and fully reversed torsion with a constant axial stress. The maximum stress in each experiment was lower than the lower yield stress of the material. It was found that the cyclic plastic deformation within the gage section of the specimen under multiaxial stress state followed the three-stage process that was observed from uniaxial loading, namely, incubation, propagation, and saturation. The plastic deformation was significantly inhomogeneous during the propagation stage, and the inhomogeneity continued through the saturation stage. The duration of each stage and the saturated strains were dependent on the cyclic stress amplitude and the loading path. Multiaxial stress state reduced the incubation stage. With identical equivalent stress magnitude, the nonproportional loading path resulted in the shortest incubation and propagation stages, and the saturated equivalent plastic strain magnitude was the smallest. Although the deformation over the gage section was inhomogeneous, the plastic deformation in a given local area was found to be practically isotropic.     

基于多轴应力损伤的薄板花瓣型破口形成机理研究

开展了多种应力状态下的船用钢力学特性实验,基于多轴应力状态损伤的失效准则研究了局部冲击荷载作用下圆形板的花瓣型破口形成过程,划分了花瓣型破口形成的3个阶段,分析了裂纹区域、非裂纹区域应力状态变化过程及损伤情况。得到:(1)考虑多轴应力损伤的舰船用钢失效准则能有效预测受力状态复杂的花瓣状破口;(2)花瓣型破口的形成主要分为蝶形凹陷、中心区域裂纹扩展、花瓣形成与翻转等3个阶段;(3)花瓣型破口的裂纹区和非裂纹区均受力复杂,破口预测须考虑应力状态对损伤特性的影响;(4)花瓣形成过程中,第1阶段和第3阶段均匀变形,第2阶段损伤局部化明显,花瓣卷曲会造成花瓣根部的二次损伤。     

Fatigue limit under multiaxial loadings: on the definition of the equivalent shear stress

The goal of the present paper is to propose a criterion to predict the fatigue strength of hard metals under conditions of multiaxial, non-proportional loadings. It is very simple to compute, but still provides very good results for a wide range of in-phase and out-of-phase cycling loads.     

Cyclic stress-strain data analysis under biaxial tensile stress state

A new device was developed to assess fatigue life under biaxial tensile loading at elevated temperatures. It makes use of an annular disk specimen and can be easily mounted onto a standard push-pull machine so that the axial force is converted into radial forces extending across the disk specimen. Therefore, a positive ratio of the tangential to the radial stress can be imposed at the reduced section of the disk specimen; this ratio depends on the specimen configuration and may be fixed to a value ranging from 0.5 to 0.9 by varying the inner diameter of the disk. The proposed device has performed successfully and was used to study the cyclic behavior of Type-304 stainless steel subjected to various biaxial tensile stress states at room temperature and at 200°C. The data obtained from this experimental procedure have been analyzed to evaluate the effectiveness of some correlations already available for treating the biaxial cyclic stress-strain response in terms of the uniaxial behavior. This analysis shows that a successful correlation should account for all the stress components. The authors discuss the concept used in the modeling of the material cyclic behavior and the formulation of a biaxial fatigue damage parameter necessary for an effective analytical life prediction methodology.     

Plastic flow under multiaxial cyclic loading

An experimental study of the inelastic behavior of annealed aluminum alloy 6061-T6 tubular specimens subjected to combined axial and torsional stress cycles is presented. Particular attention is paid to the question of how plastic strain is developed and how the yield surface moves along the 90-deg out-of-phase stress cycle. Experimental results agree qualitatively with the predictions of the two-surface plasticity theory. Paper was presented at the 1985 SEM Spring Conference on Experimental Mechanics held in Las Vegas, NV on June 9–14, 1985.     

Experimental study of the phase transformation plasticity of 16MND5 low carbon steel under multiaxial loading

This paper is concerned with the experimental behaviour of a 16MND5 steel (french vessel steel) under complex loading. A particular attention is paid to plasticity induced by phase transformation. We present an experimental set-up to apply thermo-mechanical loads under tension-torsion. This apparatus enables us to reach temperature of 1200 °C at a maximum heating rate of 60 °C s⁻¹ and a high cooling rate of −30 °C s⁻¹. A series of tests is performed in order to show the rule of loading on transformation plasticity.     

多轴载荷下缺口试件疲劳寿命预测研究

基于临界平面法,在分析光滑薄壁圆管试件疲劳寿命预测模型的基础上,借助有限元应力应变分析,进一步将模型推广应用到了缺口试件的多轴疲劳寿命预测中,并利用坐标变换原理,明确了临界平面及有效循环变量的确定方法.在存在平均应力的情况下,分析了平均应力对疲劳寿命的影响,并对所建模型进行了平均应力修正.     

Anisotropic work hardening of steel sheets under plane stress

This work is a follow-up of the previous report by Kim and Yin [Kim, K.H., Yin, J.J., 1997. Evolution of anisotropy under plane stress. J. Mech. Phys. Solids 45, 841–851] regarding the anisotropic work hardening of cold rolled steel sheets. Tensile prestrain has been applied at angles to the rolling direction and then tensile uniaxial yield stress and R-value distributions are measured. As reported earlier, the orientations of local maxima and minima in the yield stress are altered when the prestrain axis is not in the rolling direction. This led Kim and Yin [Kim and Yin (1997)] to suggest that the orientations of orthotropy axes are altered by the tensile prestrain at angles to the rolling direction. However, R-value distribution is found to be hardly affected by the prestrain. The unchanging R-value distribution shows that the material remembers the rolling direction even after the prestrain. An attempt is made to approximate the observed yield and flow behavior based upon isotropic-kinematic hardening with the quadratic yield function (Hill, 1948). The degree of approximation raises the issues of yield point definition, flexibility of yield function, non-associated flow rules, distortional hardening and others.