A micromechanical approach for mixed mode I/II failure assessment of cracked highly orthotropic materials such as wood
Introduction
Composite materials are utilized in a wide range of industries, including civil, marine, automotive, and aerospace due to their specific physical and mechanical properties. Existence of voids and inherent defects are unavoidable in the manufacturing process of these kinds of materials. In addition, development of cracks is a well-known phenomenon over lifetime of construction materials. These cracks are often subjected to normal and shear loads [1], [2], [3]. It is necessary to take these defects and cracks into consideration in the design of composite structures with a suitable fracture criterion. Fakoor and Shahsavar presented a comprehensive review on models and fracture criteria proposed for cracked composite materials [4].
Among construction materials, wood is widely employed, due to specific properties. Wood has a high strength to weight ratio, is known as a material with good thermal and electricity insulation properties, and can be prepared from natural sources [5]. Wood has highly orthotropic mechanical behaviour and significant defects can be found frequently in huge wooden structures [6], [7]. Wood specimens have also been utilized as natural composite materials in early research on mixed mode I/II fracture of orthotropic components.
Prior mixed mode fracture criteria were proposed according to curve fitting on experimental fracture data of wood species [8], [9], [10], [11]. Wu employed center notch specimens of balsa wood under mixed-mode loading. The cracks were created parallel to the fiber direction [8].
Recent proposed mixed mode fracture criteria are based on theoretical assumptions. Some of them have employed well-known isotropic failure criteria and extended them into orthotropic materials considering physical assumptions. Jernkvist extended the famous energy based mixed mode fracture theories such as strain energy release rate (SER) [12] and strain energy density (SED) [13] into the fracture of wood specimens [14]. The main assumption was self-similar crack propagation. Three kinds of wood species i.e. red spruce, Scotch pine, and Norway spruce were considered as case study materials [15]. Two different mixed mode fracture criteria for orthotropic materials based on maximum principal stress [14] and maximum shear stress [16] have been proposed with the result verification performed on wood specimens.
Van der Put proposed a new fracture mechanics theory for highly orthotropic materials such as wood in which the fibers were considered as reinforcement of the isotropic matrix [17]. Basic relations in theory of elasticity were employed to derive the fiber effects [17]. Taking the concept of reinforcement isotropic solid (RIS) model from the van der Put’s theory, some studies have been performed to investigate the mixed mode I/II fracture of composite materials in the context of linear elastic fracture mechanics (LEFM) [18], [19], [20], [21], [22], [23], [24].
It is well known that wood is a quasi-brittle material and fracture of wood components is associated with creation of a considerable damage zone in the crack tip region [25], [26]. Existence of so-called fracture process zone (FPZ) through consumption of fracture energy, prevents catastrophic failure of wooden structures, which is another advantage of wood as a construction material [27], [28]. Although this damage zone is effective for safe design of wood structures, many difficulties will arise from analytical point of view. FPZ contains several complicated toughening mechanisms such as micro-crack nucleation [29] and bridging [30]. Consideration of the effects of damage zone for fracture investigation of quasi-brittle materials needs micro-mechanical point of view and leads us to more accurate failure criterion [31]. Anaraki et al. proposed a mixed mode fracture criterion by making a micro–macro comparison on wood fracture [32]. A damage factor for investigation of the effects of process zone was introduced in this study, which was a function of wood strength properties [32]. Romanowicz and Seweryn considered the effects of FPZ in their mixed mode criterion with a non-local stress approach [33].
As suggested by the above literature, despite the serious effects of FPZ on the fracture of orthotropic materials, the assessment of crack tip damage zone fracture properties has not been considered in the literature. In this research, a new mixed mode I/II fracture criterion based on strain energy density theory is proposed for cracked composite materials. Considering the importance of crack tip damage zone on fracture investigation of quasi-brittle materials, fracture toughness is defined for a body weakened by micro-cracks, which is a new concept in fracture mechanics theory. This definition can be used for consideration of the dissipated energy in micro-mechanical damage. Comprehensive experimental studies have been performed for verification of the results. Center notch disk tension specimen is introduced for mixed mode fracture testes. The consistency of fracture limit curves with test data shows the superiority of the proposed criterion.
Section snippets
Main assumptions
Strain energy density criterion, which is a well-known approach for mixed mode I/II fracture investigation of isotropic materials, will extend into composite materials considering crack tip stress field of isotropic matrix. The fibers in this study were considered as reinforcement parts and theoretically fiber effects were defined as stress reduction factors. Reinforcement isotropic solid (RIS) model has been employed for the aforementioned modeling which is explained in the next section.
Reinforcement isotropic solid (RIS) model
Material description
Wood as a natural orthotropic material is utilized in this study. Longitudinal (L), Radial (R), and Tangential (T) directions are three mutually perpendicular defined axes for wood (see Fig. 4).
A crack may lie in one of the aforementioned planes with two possible crack propagation directions. Therefore six crack-propagation systems can be recognized, named as RL, TL, LR, TR, LT, and RT (see Fig. 5). The first letter represents perpendicular axis to the crack plane, and the second one represents
Finite element model for calculation of mode I fracture toughness
Finite element method was employed for extraction of fracture properties. Second-order standard thick shell elements (S8R Shell elements) have been recommended to analyze composite and sandwich shells1. J-Integral method was utilized for extraction of . Mesh convergence analysis was performed to ensure the number and size of the elements. Boundary conditions and load direction are shown in Fig. 14.
FE model for extraction of has been shown in Fig. 15. A fine mesh
Elastic and mechanical properties of FPZ
and which are dissipated energy due to creation of FPZ, can be calculated by integrating stress–strain curve from the yield point to the break point as shown in Fig. 13. FPZ energy which is defined as dissipated energy due to toughening mechanisms (such as bridging and micro-crack creation), releases during the crack propagation phenomenon. The non-linear part of curve of a cracked body includes the dissipated energy due to these mechanisms. Based on Fig. 13, is really
Conclusion
A new mixed mode I/II fracture criterion named NL-RIS-SED was proposed for fracture assessment of cracked orthotropic materials. This criterion is able to capture the effects of toughening mechanisms such as micro-cracks and bridging in the fracture process zone. Strain energy density theory and reinforcement isotropic solid concept were utilized to extract the mathematical expression of NL-RIS-SED criterion. The energy wasted due to creation of FPZ was considered by defining effective fracture
CRediT authorship contribution statement
Mahdi Fakoor: Conceptualization, Methodology, Supervision. Mina S. Khezri: Data curation, Writing - original draft, Software, Visualization, Validation, Formal analysis.
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.
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2022, International Journal of Solids and StructuresCitation Excerpt :This incompatibility is related to the linear fracture analysis and not considering wasted energy in FPZ. For orthotropic materials, although capturing the effects of FPZ in fracture assessment needs more complicated micro-scale investigations, it significantly leads to more accurate fracture criteria (Fakoor and Shokrollahi, 2018; Fakoor and Khezri, 2020; Anaraki and Fakoor, 2010; Anaraki and Fakoor, 2010; Mahmoudi et al., 2020). In isotropic materials, based on different fracture criteria, the crack initiation angle is estimated to derive fracture envelope curve (fracture behavior).
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2022, International Journal of Solids and StructuresCitation Excerpt :Some of these studies included the effects of the first non-singular term, T-stress, while others only considered the singular terms. We point out here that in highly anisotropic materials, such as wood and engineering composites, cracks tend to grow along the fiber direction (Cahill et al., 2014; Romanowicz, 2019; Manafi Farid and Fakoor, 2019; Fakoor and Khezri, 2020). In such cases, the fracture path is often an a priori assumption, used in the growth criteria.