T-stresses solution and out-of-plane constraint for central cracked plate (CCP) with I-II mixed mode crack under uniaxial compression
Introduction
In the fields of aerospace, navigation ship, submarine, railway transportation and submarine pipeline, structures are usually subjected to compressive load or external pressure. It is of great scientific significance and engineering application value to carry out in-depth and systematic research on the integrity assessment and failure mechanism of pressure bearing structures containing cracks under compression or external pressure.
The crack tip constraint has a great influence on the fracture behavior of materials. In order to improve the accuracy of defect assessment and eliminate non conservatism or reduce excessive conservatism, it is necessary to develop evaluation theory and method including crack tip constraint effect, and incorporate the effect of constraint effect in fracture mechanism and fracture toughness of materials [1], [2], [3], [4]. It is essential to make a quantitative analysis of three-dimensional crack tip constraint effect, which can lay the foundation for providing a valid description of the fracture behavior of materials under different constraint states [5], [6]. The crack tip constraint can be divided into in-plane and out-of-plane constraint, which are related to specimen geometry, crack size, specimen thickness, loading mode, loading level, material parameters, etc. The in-plane constraint is mainly determined by the specimen geometry normal to the in-plane of the crack front, while the out-of-plane constraint is primarily dominated by the specimen thickness and the boundary conditions [7], [8].
The ability of single parameter K or J -integral to characterize the crack tip field is limited, and the influence of specimen geometry and load level is not considered [9], [10]. The multiparameter theories of stress-strain state at crack tip have been investigated to solve this problem. In addition to the fracture parameters K and J -integral, the commonly used parameters for in-plane constraint are T11 [4], [9], [11], [12], Q [13], [14] and A2 [15], [16], [17], and the parameters for out-of-plane constraint are T33 [7], [18], [19] and Tz [20], [21], [22] in the existing multiparameter theories.
T-stress has been taken attention in the literature of constraint parameters, and has the effects on the apparent fracture toughness and kinking angle [12]. Besides, T-stress plays a significant role in the size and shape of the plastic zone ahead of the crack tip, crack propagation path as well as propagation model [23], [24], [25], [26], [27], [28], [29]. Apart from the in-plane T-stress T11, the out-of-plane constraint parameter T33 (or Tzz) also has a prominent effect on the fracture behavior of the material [7], and the specimen thickness has a dominating influence on T33 [18], [19]. There is an interaction between in-plane constraint and out-of-plane constraint, which should be considered simultaneously [30]. T-stress solutions including T11 and T33 have been researched in through cracks and surface cracks under tension [7], [18], [31], [32]. Rice [33] and González-Albuixech et al. [7] showed that T11 and T33 must be taken into account at the same time to obtain the correct description of the stress field near the crack tip for the three-dimensional specimens. Liu et al. [18] analyzed the complete solutions of K, T11 and T33 of three-dimensional SENT specimens. In recent years, quantitative study of stress field at three-dimensional crack tip has received increasing attention [31], [34]. Thereto, it is necessary to pay more attention to the influence of various geometrical configurations, including a wide range of different relative crack lengths and thickness to width ratio, on the quantitative research of constraint effect near the crack tip [35].
The research on the constraint effect of crack tip under compression mainly focuses on T-stress component and fracture criterion modification, crack initiation angle, crack propagation path, etc. The materials include glass [36], ceramics [37], concrete [38], PMMA [39], rock [40], [41], etc. It is generally believed that T-stress includes not only the component Tx parallel to the crack surface, but also the component Ty perpendicular to the crack surface subjected to the compressive load [42]. Li et al. [42], [43] observed that the crack initiation angle of the closed crack with friction was determined by the stress intensity factor and T-stress components (Tx and Ty) under compression. Tang et al. [40], [44] investigated the influence of T-stress components (Tx and Ty) on the compression fracture of rock, and indicated that T-stress was very crucial to determine the crack propagation direction, especially for the closed crack with high friction coefficient. Liu et al. [39], [45] proposed a revised maximum tangential stress (MTS) criterion considering three T-stress components Tx, Ty and Txy, which more objectively reflected the crack initiation mechanism of rock under compression [45], and a new calculation model of crack propagation path under compression was presented [39]. Commonly used mixed mode fracture criteria, including maximum tangential stress (MTS), minimum strain energy density (MSED) and maximum energy release rate (MERR) and so on, ignore the influence of nonsingular stress term, resulting in great differences between theoretical prediction and experimental results. The crack initiation strength and initiation angle predicted by the generalized maximum tangential stress (GMTS) criterion considering T-stress are closer to the experimental results than those predicted by the traditional maximum tangential stress (MTS) [36], [37], [38]. In addition, the modified MTS-FEM criterion [41], modified MSED criterion [46], extended version of the maximum tangential strain (EMTSN) criterion [47], [48] and other related studies indicated that the mixed mode fracture criteria considering T-stress could well predict the crack initiation critical stress and crack initiation angle.
The out-of-plane constraint is mainly determined by the specimen thickness and the boundary conditions [7]. The thickness of the specimen and structure has a momentous influence on the fracture toughness and out-of-plane constraint parameters [1]. Wang et al. [49] obtained that there was a correlation between the in-plane constraint parameters and the out-of-plane constraint parameters for the long crack specimens subjected to tensile load. Miao et al. [50] discovered that the thickness effect should be taken into account for the long crack with relative crack length of 0.7 under tension. Therefore, more attention should be paid to the influence of specimen thickness on the constraint parameters for long cracks. Matvienko [19] analyzed the T-stress of central cracked circular disc (CCCD) specimen under radial compression, and concluded that the out-of-plane constraint had a strong impact on the plastic zone at the crack tip, and the specimen thickness and mixed mode loading conditions had a significant influence on the Tzz (T33). However, the thickness to width ratio of specimens was only limited to 0.1 to 0.8. Horn et al. [51] employed the Notch Failure Assessment Diagram (NFAD) to predict the failure of three-point bending specimens with U-notch, and there was obvious out-of-plane constraint loss when thickness to width ratio of 0.5. When t/W ≤ 1.0, the lack of plane strain condition would give non conservative results in the NFAD assessment. Further investigations addressing the influence of specimen thickness on out-of-plane constraint and the loss of out-of-plane constraint under compression are highly warranted.
In addition, different specimen geometries affect the in-plane and out-of-plane constraint parameters [30]. When fracture mechanics methods are applied to engineering design and structural integrity assessment, it is essential to transfer the fracture toughness values from laboratory specimens to structural applications. Therefore, it is requisite to conduct a large-scale quantitative study on the crack tip constraint effect of different specimen shapes and sizes under compression, so as to lay a foundation for analyzing the matching of crack tip constraint between the structure and the specimen. To the best of our knowledge, most of the existing quantitative research of constraint parameters under compression were based on cracked Brazilian disc (CBD) specimens (i.e. central cracked circular disc (CCCD) specimens), commonly adopted in geotechnical mechanics. Ayatollahi et al. [36] and Matvienko [19] demonstrated that the in-plane T-stress (Txx) of CBD specimens with I-II mixed mode crack subjected to radial compressive load was always negative, and the stress intensity factors KI and KII all existed, which changed with the crack inclination angle. In the central crack plate (CCP) specimen, KI was equal to zero for closed crack [45]. The variation law of T-stress and SIF at the crack tip of CCP specimen is different from that of CBD specimen during the course of this work. Therefore, quantifying the crack tip constraint effect and the SIF for different typical specimens may provide a theoretical basis for the transferability of fracture behavior.
The research on the constraint effect of crack tip under tension has been relatively comprehensive, including in-plane constraint parameters and out-of-plane constraint parameters. However, the existing research under compression mainly focuses on in-plane constraint parameters, covering the components of T-stress and the corresponding correction and application of fracture criterion, ignoring the quantitative research on out-of-plane constraint parameters. In this paper, the three-dimensional stress-strain field near the crack tip of a typical specimen (CCP) with I-II mixed mode crack under uniaxial compression is analyzed theoretically and numerically. The material parameters and geometrical configurations are concerned to quantitatively investigate the effects of various factors on SIF, in-plane and out-of-plane constraints ahead of the crack tip. The loss of out-of-plane constraint is proposed, the empirical equations for KII, T11 and Tz are obtained as well.
Section snippets
Stress field at crack tip in classical fracture mechanics
The Westergard stress functions under mode I and mode II loading need to satisfy biharmonic functions and boundary conditions. The stress field near the crack tip of mode I and mode II crack is calculated by the sum of complex analytic functions and , respectively. In classical fracture mechanics, the nonsingular stress term and higher order term are ignored. The stress field at crack tip of I-II mixed mode crack under tension is exhibited in the Fig. 1. Stress components at
Finite element analysis model
The exploration of central cracked plate (CCP) specimen for 3D elastic finite element analysis has been performed by ABAQUS. Fig. 3 shows the finite element calculation model of CCP specimen, including geometries of CCP specimen (Fig. 3 (a)), crack surface diagram (red area in Fig. 3 (b)) and mesh of crack tip (Fig. 3 (c)). The finite element model in Fig. 3 is divided into two parts, Part 1 and Part 2, which are assembled in the interaction module, and the contact surface between them (the
Effect of material parameters on T-stress
When analyzing the influence of material parameters (elastic modulus E and Poisson's ratio ν) on T-stress, the normalized parameters are employed to express [50], [53], [55], it is shown in Eq. (11) to Eq. (13).
When a/W = 0.1, t/W = 0.1, μ = 0, the effects of material parameters on T-stress have been canvassed in Fig. 5, where E is 50, 100 and 206 GPa respectively, while ν is
Conclusions
The triaxial stress field near the crack tip plays an important role in fracture mechanics analysis. The in-plane and out-of-plane constraint parameters can be used to quantify the stress-strain field at the crack tip front. The specimen geometries affect the in-plane and out-of-plane constraint. The SIF and constraint parameters at the crack tip of the CCP specimen under compression have been investigated quantitatively, and the empirical equations for KII, T11 and Tz have been obtained, which
CRediT authorship contribution statement
Li-Zhu Jin: Conceptualization, Methodology, Formal analysis, Investigation, Writing – original draft, Writing - review & editing, Visualization. Qi Pei: Validation, Formal analysis. Chen-Yang Yu: Methodology, Validation. Le Chang: Supervision, Writing – review & editing. Xiao-Hua He: Resources, Supervision. Chang-Yu Zhou: Funding acquisition, Project administration, Resources, Supervision, Writing – review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgement
The authors gratefully acknowledge the financial support of the National Natural Science Foundation of China (51975271).
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