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(a) Photograph of Odontodactylus Scyllarus and its raptorial appendage showing location of dactyl club (taken from Patek et al. (2004)). (b) 3D model of the dactyl club reconstructed from CT-scanning image showing transverse and coronal sections. (c) Schematic image of transverse section of the dactyl club in which I is the impact region, II is the periodic region, and III is the striated region. (d) SEM micrograph of a fractured surface of periodic region (taken from Grunenfelder et al. (2014a)). (e) 3D schematic illustration of the Bouligand structure in the periodic region of the dactyl club.
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The Bouligand structure, which is found in many biological materials, is a hierarchical architecture that features uniaxial fiber layers assembled periodically into a helicoidal pattern. Many studies have highlighted the high damage-resistant performance of natural and biomimetic Bouligand structures. One particular species that utilizes the Boulig...
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... this study, we focus on the smashing mantis shrimp, Odontodactylus Scyllarus (Fig. 1a), which is a highly aggressive marine crustacean that uses its light-weight appendages as hammers to smash its heavily shelled preys with accelerations of a 0.22-caliber bullet and producing forces of 0.4-1.5 kilonewtons ( Weaver et al., 2012;Patek et al., 2004). Most impressive is the ability of the mantis shrimp's dactyl appendages to ...
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... appendages as hammers to smash its heavily shelled preys with accelerations of a 0.22-caliber bullet and producing forces of 0.4-1.5 kilonewtons ( Weaver et al., 2012;Patek et al., 2004). Most impressive is the ability of the mantis shrimp's dactyl appendages to endure such tremendous forces. The last segment of the appendage is the dactyl club (Fig. 1b), and has been the focus of several recent studies because of its remarkable damage tolerant properties despite the significant forces generated by each one of the thousand impacts imparted by a dactyl club between molting events (Weaver et al., 2012;Grunenfelder et al., 2014aGrunenfelder et al., , 2014bGuarín-Zapata et al., 2015;Amini ...
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... organic material and brittle minerals. The early work by Weaver et al. (2012) described the dactyl club as a multi-layered structure consisting of three main regions subsequently identified as impact (I), periodic (II), and striated regions (III). The transverse cross-section of the dactyl club highlighting these main three regions is shown in Fig. 1c. The architecture of the periodic region ( Fig. 1d) was found to follow a helicoidal arrangement of stacked layers of unidirectional chitin fibrils embedded within an amorphous mineral matrix (Weaver et al., 2012). The impact region of the club also exhibits a similar arrangement of fibers, but within a crystalline mineral matrix ( ...
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... work by Weaver et al. (2012) described the dactyl club as a multi-layered structure consisting of three main regions subsequently identified as impact (I), periodic (II), and striated regions (III). The transverse cross-section of the dactyl club highlighting these main three regions is shown in Fig. 1c. The architecture of the periodic region ( Fig. 1d) was found to follow a helicoidal arrangement of stacked layers of unidirectional chitin fibrils embedded within an amorphous mineral matrix (Weaver et al., 2012). The impact region of the club also exhibits a similar arrangement of fibers, but within a crystalline mineral matrix ( Yaraghi et al., 2016). A schematic illustration of an ...
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... arrangement of stacked layers of unidirectional chitin fibrils embedded within an amorphous mineral matrix (Weaver et al., 2012). The impact region of the club also exhibits a similar arrangement of fibers, but within a crystalline mineral matrix ( Yaraghi et al., 2016). A schematic illustration of an idealized Bouligand structure is shown in Fig. 1e. Weaver et al. (2012) carried out a detailed stress analysis of the impact event of these dactyl clubs and revealed the presence of relatively high stresses in the periodic and impact regions of the dactyl club. They also showed that the striated region consist of highly aligned chitin fiber bundles that wrap around the dactyl club. ...
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... of fracture mechanics. Assuming the fibers are strong enough, the cracks are forced to twist as they grow. These twisting cracks are assumed to occur in an idealized Bouligand structure where the fiber diameter, fiber spacing and layer thickness are constant throughout the structure, while the layered orientation follows a helicoidal architecture (Fig. 1d). As a first step approximation, we simplify the problem considering that the intrinsic anisotropy of the unidirectional fibers only affects the local strength of the material, forcing the crack to grow parallel to the fiber orientation. However, from an elasticity point of view, the Bouligand structure decreases the anisotropy to the ...
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... As a first step, we assume a linear elastic isotropic solid with a growing twisting crack similar to that shown in Fig. 3c. Even though this is a three dimensional problem, we assume that the specimen is laterally constrained. This constraint mimics the effect of the striated region (Region III in Fig. 1c, (Weaver et al., 2012)), and also avoids the finite thickness effect, which can affect the stress fields and resulting fracture toughness (Anderson, 2005;Narasimhan and Rosakis, 1990;Barsom and Rolfe, 1987). We will later examine these assumptions with ancillary numerical simulations to validate these results. We expect the study of a twisting crack under ...
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... of the crack front of the twisting crack causes sudden change in plane derivation from the initial flat crack, which results in a kinked crack. Such discontinuity due to the kinked crack results in the jump of G G / 0 from the initial flat crack front (which is always equal to 1) to the first increment ( Cotterell and Rice, 1980). Consequently, Fig. 10 highlights the results starting at the first increment ( = X l / 0.05) and the dotted lines in Fig. 10d indicate the uncertainty of G G / 0 between the initial flat crack and the first increment of the crack propagation due to such ...
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... crack, which results in a kinked crack. Such discontinuity due to the kinked crack results in the jump of G G / 0 from the initial flat crack front (which is always equal to 1) to the first increment ( Cotterell and Rice, 1980). Consequently, Fig. 10 highlights the results starting at the first increment ( = X l / 0.05) and the dotted lines in Fig. 10d indicate the uncertainty of G G / 0 between the initial flat crack and the first increment of the crack propagation due to such ...
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... particular, the comparison shown in Fig. 10a reveals that the difference in G G / 0 between these three models increases at large values of X l / and Z t / . In part this difference is due to the fact that the numerical simulations deviate from the assumptions that Fig. 10d), our analytical model is clearly closer to the numerical results than the Faber & Evans model (with only a 0.52% ...
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... particular, the comparison shown in Fig. 10a reveals that the difference in G G / 0 between these three models increases at large values of X l / and Z t / . In part this difference is due to the fact that the numerical simulations deviate from the assumptions that Fig. 10d), our analytical model is clearly closer to the numerical results than the Faber & Evans model (with only a 0.52% ...
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... similar comparison in terms of Fig. 11a-c. The comparisons show that in general our analytical model has a better correlation with the numerical results than the Faber & Evans model. In particular, the average difference between our model and the numerical results in modes I, II, and III (among all data points shown in Fig. 11) ...
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... similar comparison in terms of Fig. 11a-c. The comparisons show that in general our analytical model has a better correlation with the numerical results than the Faber & Evans model. In particular, the average difference between our model and the numerical results in modes I, II, and III (among all data points shown in Fig. 11) ...
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... there is no increase in fracture resistance due to the strain energy release rate required to propagate or create a crack is equal to the material property G C m . The value of G G / C C m for the continuous twisting crack can be determined from our analytical model by using Eq. (12), Eq. (13), and Eq. (16). Fig. 12a shows G G / C C m as a function of ϕ and α*, which indicates that G G / C C m increases as the magnitudes of ϕ and α* become larger. Moreover, the values of G G / C C m at = ϕ 0°, 30°, 60°, and 90° are plotted against α*and are shown in Fig. 12b, which provides a better visualization that the G G / C C m increases with the magnitudes ...
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... twisting crack can be determined from our analytical model by using Eq. (12), Eq. (13), and Eq. (16). Fig. 12a shows G G / C C m as a function of ϕ and α*, which indicates that G G / C C m increases as the magnitudes of ϕ and α* become larger. Moreover, the values of G G / C C m at = ϕ 0°, 30°, 60°, and 90° are plotted against α*and are shown in Fig. 12b, which provides a better visualization that the G G / C C m increases with the magnitudes of ϕ and α*. This means that the twisting crack requires more applied force to propagate, especially at higher degrees of crack twisting, and subsequently leads to the increase in fracture resistance ...
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... to find the mathematical expression of ′ z , which is a tangent vector of the crack front. Therefore, we outline the guideline to obtain the mathematical expression of ′ z for an arbitrary crack front shape as follows: We first define the projection of the crack front on the XZ-plane, mathematically expressed as = X f Z ( ), as illustrated in Fig. 13a. The mathematical expression of the crack front on the twisting crack plane is then ...
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... illustrate this procedure, we look at a specific curved crack front growing on the twisting surface that follows a Bouligand structure as shown in Fig. 13. The projection of such a curved crack front on the XZ -plane may follow a parabolic pattern which can be written ...
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... the tangent vector (Eq. (21) Fig. 13b. We would like to emphasize that the local coordinate system is always unique at each location on the twisting crack regardless of the crack front shape. The rest of the derivation for the local stress intensity factors and stress energy release rate follow the same procedure described in Sections 2.1 and ...
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... α*). The degree of mode-mixity is found to be the mechanism behind the local fracture resistance behavior in the twisting crack as a result of Eq. (12) and Eq. (13). We also quantify the fracture resistance behavior in terms of the local toughening factor (G G / C C m ), which is unique for each location on the twisting crack surface as shown in Fig. ...
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... Faber and Evans (1983) (a rotation of a plane around a global X -axis by an angle ϕ with respect to the orientation of the initial flat crack front with the assumption of straight crack front as shown in Fig. 4). By drawing a surface connecting the initial flat crack and the twisting crack front, we can obtain a twisting surface ABCD as shown in Fig. C1. Fig. C1. A twisting surface ABCD as a result of a twisting crack described by (Faber and Evans ...
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... Evans (1983) (a rotation of a plane around a global X -axis by an angle ϕ with respect to the orientation of the initial flat crack front with the assumption of straight crack front as shown in Fig. 4). By drawing a surface connecting the initial flat crack and the twisting crack front, we can obtain a twisting surface ABCD as shown in Fig. C1. Fig. C1. A twisting surface ABCD as a result of a twisting crack described by (Faber and Evans ...
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... constraint, (3) = t R / 0 100 with free lateral surfaces, and (4) 2D plane strain finite element solution. In all cases, we employed the same type of finite element mesh as those in section 3.2 (e.g., hexahedral and quadrilateral elements with the smallest element size of − R 10 4 0 near the crack front). The geometry of these cases is shown in Fig. F1. The asymptotic K I field is attained by applying the displacement field described by Eq. (14). ...
Citations
... In the context of altering the stacking sequence, which aim to replicate forms, functions, and principles from the natural world, have led to the creation of lightweight, stronger composite materials capable of withstanding damage. Numerous intricate biological structures in nature, like nacre [15] and the dactyl club of the mantis shrimp [16], prove to be sources of inspiration for bio-inspired structures that offer increased strength and also toughness. Researchers discovered that mantis shrimp clubs possess an inner structure arrangement resembling a spiral, called a helicoidal structure (see Figure 1) [17][18][19][20]. ...
... (a) Mantis shrimp; (b) Dactyl club; (c) Schematic illustrating the helicoidal structure, providing clarity on the origins of the aligned fiber layers' helicoidal arrangement; (d) SEM image depicting a fracture surface, unveiling a fibrous arrangement organized in a helicoidal pattern. (Adapted from[16,33]) ...
Biostructures found in nature exhibit remarkable strength, toughness, and damage resistance, achieved over millions of years. Observing nature closely might help develop laminates that resemble natural structures more closely, potentially improving strength and mimicking natural principles. Bio-inspired Carbon Fiber-Reinforced Polymers (CFRP) investigated thus far exhibit consistent pitch angles between layers, whereas natural structures display gradual variations in pitch angle rather than consistency. Therefore, this study explores helicoidal CFRP laminates, focusing on the Non-Linear Rotation Angle (NLRA) or gradual variation to enhance composite material performance. In addition, it compares the strength and failure mechanisms of the gradual configuration with conventional helicoidal and unidirectional (UD) laminates, serving as references while conducting transverse tensile tests (out-of-plane tensile). The findings highlight the potential of conventional and gradual helicoidal structures in reinforcing CFRP laminates, increasing the failure load compared to unidirectional CFRP laminate by about 5% and 17%, respectively. In addition, utilizing bio-inspired configurations has shown promising improvements in toughness compared to traditional unidirectional laminates, as evidenced by the increased displacement at failure. The numerical and experimental analyses revealed a shift in crack path when utilizing the bio-inspired helicoidal stacking sequence. Validated by experimental data, this alteration demonstrates longer and more intricate crack propagation, ultimately leading to increased transverse strength.
... toughness 37,38 . The crossed lamellar architecture is also a bioinspired structure with excellent toughness, featuring variations in alignment orientations and an interlocking arrangement across adjacent layers ( Fig. 2c) [39][40][41] . ...
Fabrication of composite hydrogels can effectively enhance the mechanical and functional properties of conventional hydrogels. While ceramic reinforcement is common in many hard biological tissues, ceramic-reinforced hydrogels lack a similar natural prototype for bioinspiration. This raises a key question: How can we still attain bioinspired mechanical mechanisms in composite hydrogels without mimicking a specific composition and structure? Abstracting the hierarchical composite design principles of natural materials, this study proposes a hierarchical fabrication strategy for ceramic-reinforced organo-hydrogels, featuring (1) aligned ceramic platelets through direct-ink-write printing, (2) poly(vinyl alcohol) organo-hydrogel matrix reinforced by solution substitution, and (3) silane-treated platelet-matrix interfaces. Unit filaments are further printed into a selection of bioinspired macro-architectures, leading to high stiffness, strength, and toughness (fracture energy up to 31.1 kJ/m²), achieved through synergistic multi-scale energy dissipation. The materials also exhibit wide operation tolerance and electrical conductivity for flexible electronics in mechanically demanding conditions. Hence, this study demonstrates a model strategy that extends the fundamental design principles of natural materials to fabricate composite hydrogels with synergistic mechanical and functional enhancement.
... The literature on mantis shrimp dactyl clubs has focused on their failure mechanisms and their capability to absorb impact energy [27,28]. Weaver et al. [29] studied the microstructural characteristics of shrimp dactyl clubs, including their Young's modulus, graded microstructure, and hardness of the outer layer. ...
Recent progress in additive manufacturing, also known as 3D printing, has offered several benefits, including high geometrical freedom and the ability to create bioinspired structures with intricate details. Mantis shrimp can scrape the shells of prey molluscs with its hammer-shaped stick, while beetles have highly adapted forewings that are lightweight, tough, and strong. This paper introduces a design approach for bioinspired lattice structures by mimicking the internal microstructures of a beetle's forewing, a mantis shrimp's shell, and a mantis shrimp's dactyl club, with improved mechanical properties. Finite element analysis (FEA) and experimental characterisation of 3D printed polylactic acid (PLA) samples with bioinspired structures were performed to determine their compression and impact properties. The results showed that designing a bioinspired lattice with unit cells parallel to the load direction improved quasi-static compressive performance, among other lattice structures. The gyroid honeycomb lattice design of the insect forewings and mantis shrimp dactyl clubs outperformed the gyroid honeycomb design of the mantis shrimp shell, with improvements in ultimate mechanical strength, Young's modulus, and drop weight impact. On the other hand, hybrid designs created by merging two different designs reduced bending deformation to control collapse during drop weight impact. This work holds promise for the development of bioinspired lattices employing designs with improved properties, which can have potential implications for lightweight high-performance applications.
... The forewing of a beetle is a prime example of the hierarchical adaptation of the hexagonal honeycomb structure, displaying an energy absorption capacity that is five times greater than that of conventional hexagonal honeycomb structures [51,52]. Contrary to the limited deformation mechanism of the re-entrant structure, Bouligand-auxetic hybrid structures exhibited multi-directional compressive strength, thereby enhancing their resistance to impact [53][54][55][56]. The structure inspired by the corrugated design of a mantis shrimp's shell proved to be exceptionally efficient in crashworthiness applications by exhibiting outstanding impact resistance [57,58]. ...
Biological thin-walled cellular structures, such as hierarchical structures of bone, exhibit unique internal structures that offer lightweight characteristics and high energy absorption capabilities. The compact bone shell can protect against local damage during high-impact events, while the porous cancellous bone is essential in absorbing energy. This research proposed five novel bioinspired cellular structures inspired by the well-arranged structure of femur bones. These structures comprise four distinct cell types: hexagonal honeycomb, re-entrant, hybrid, and hierarchical hybrid cells. A comprehensive numerical model, validated with experimental data, was employed to assess the performance of these structures under uniaxial compression. Some key characteristics were revealed, including peak elastic load, plateau load, energy absorption capacity, Poisson's ratio, and the effect of hierarchical cell size. The results demonstrated that the novel bioinspired structures surpassed the energy absorption performance of traditional designs, such as hexagonal design, re-entrant, and trabecular-bone-inspired structures. This enhanced performance was due to the progressive buckling and collapse mechanisms, showing promising implications for future engineering applications, particularly where energy absorption is of paramount importance.
... Extensive mechanical analyses have been conducted to reveal the composition-structure-property relationship of Bouligand composites, especially their toughening mechanisms. Suksangpanya et al. developed a theoretical model to provide additional insights into the local stress intensity factors at the crack front of twisting cracks formed within the Bouligand structure [12]. A numerical method based on the crack driving force was used to depict the effect of inhomogeneous and anisotropic material behavior on fracture properties [1]. ...
... The laminas are stacked along X axis. The twisting crack can be written as [12] ...
... It can be get by rotating (X , Y , Z ) around Z axis with an angle α * , and then rotating around x axis with an angle φ. The kinked angle α * is given by [12] ...
The Bouligand structure has been observed in a variety of biological materials, such as lamellar bone and arthropod cuticles. It is a hierarchical architecture that exhibits excellent damage-resistant performance, which arouse many interests for the mechanists and structure engineers. However, there still lacks a deep understanding of the toughening mechanisms in the Bouligand structure. For the purpose of revealing the toughening effect of twisting cracks, this paper developed a multiscale fracture mechanics model with considering the non-homogeneity and anisotropic properties. Firstly, the macro and micro constitutive properties of the Bouligand structure are analyzed. Then, a multiscale fracture model is established to characterize the energy release rates and the local stress intensity factors at the crack front of twisting cracks which are formed within the Bouligand structure. Based on the model, serious of digital calculations are carried out. The digital results demonstrate that the decrease of the local energy release rate can be attributed to two mechanisms. One is that the multiscale structure causes the stress release of the crack tip nearby. The other is that the twisting crack leads to the loading mode transformation from the single-mode to the mixed-mode, which is the main reason of the fracture toughness increasing. The research results shown in this paper can provide structure engineers some suggestive guidelines for the design of high-performance composites.
... Classification of bio-inspired structures for rebar-free concrete construction[58][59][60][61][62]. ...
... Carbon-fibre-reinforced polymer composites (CFRPCs) are widely used in many applications in the aerospace, automotive, defence and energy industries. In these engineering applications, development studies aimed at improving out-of-plane structural performance of the laminated composite beams/plates/shells can be achieved by using the bio-inspired helicoidal structures (BIHSs), which were found because of examining the physical properties of living creatures (insect cuticles, osteons in mammalian bones, crustacean exoskeletons, snail shell and etc.) in nature [1][2][3][4][5][6][7][8][9][10]. ...
... Bouligand structures, which have been observed in lamellar bone [4,5] and the exoskeleton of crus-helicoidal laminates, are the main source of its excellent toughness [11] . Suksangpanya et al. conducted a systematic analysis to the toughening mechanism of the twisted crack in the Bouligand structure [12] . Yang et al. proposed a theoretical modeling approach to reveal the effect of the material anisotropy on the fracture toughness of crustacean-inspired Bouligand structures [13] . ...
... Yang et al. proposed a theoretical modeling approach to reveal the effect of the material anisotropy on the fracture toughness of crustacean-inspired Bouligand structures [13] . Some researchers also show that the remarkable toughness of these structures can be attributed to the combining effect of crack twisting and crack-bridging [12] . Subsequently, Song et al. developed a hybrid model considering crack twisting and crackbridging, which successfully investigates the competitive toughening mechanism [14] . ...
... Also, this kind of crack is more common and dangerous in engineering. As Fig. 2 shows, a semi-finite crack is defined [12] . We also assume that the crack front is always straight and along the orientation of the fibers, and the crack propagates along the interface between the adjacent fibers. ...
The Bouligand structure has been observed in a variety of biological materials, such as lamellar bone and exoskeleton of lobsters. It is a hierarchical and non-homogeneous architecture that exhibits excellent damage-resistant performance. This paper presents a multiscale fracture model considering the material inhomogeneity, the multiscale property, and the anisotropy to reveal the toughening mechanisms in the Bouligand structure. Firstly, the macro and micro constitutive properties of this composite are derived. Then, a multiscale fracture model is developed to characterize the local stress intensity factors and the energy release rates at the crack front of twisted cracks. Our results demonstrate that the decrease in the local energy release rate can be attributed to two-step mechanisms. The first mechanism is that the multiscale structure and the material inhomogeneity cause a release of stress near the initial crack tip. The second mechanism is that the twisted crack leads to the transformation from single-mode loading to mixed-mode loading, which enhances the fracture toughness. These results can not only reveal the toughening mechanism of the Bouligand structure but also provide guidelines for the design of high-performance composites.
... The direction of the fibers aligned with the x-axis is denoted as ϕ = 0°, meanwhile the fibers twist counterclockwise around the z-axis [16][17][18] . This configuration increases the crack surface area and contributes to the reorientation of the fibers in response to external stresses, such as tensile, flexural, and impact loads 29 . Meanwhile, the resulting modulus oscillations within the Bouligand geometry enhance crack torsion (Supplementary Fig. 1) 16,24,29 . ...
... This configuration increases the crack surface area and contributes to the reorientation of the fibers in response to external stresses, such as tensile, flexural, and impact loads 29 . Meanwhile, the resulting modulus oscillations within the Bouligand geometry enhance crack torsion (Supplementary Fig. 1) 16,24,29 . Incidentally, the Bouligand chiral array results in in-plane quasi-isotropic mechanical properties (Fig. 1c, d and Supplementary Fig. 2), which overcome the typical limitations of materials with traditional unidirectional 3D fiber structures 16,[28][29][30][31] . ...
... Meanwhile, the resulting modulus oscillations within the Bouligand geometry enhance crack torsion (Supplementary Fig. 1) 16,24,29 . Incidentally, the Bouligand chiral array results in in-plane quasi-isotropic mechanical properties (Fig. 1c, d and Supplementary Fig. 2), which overcome the typical limitations of materials with traditional unidirectional 3D fiber structures 16,[28][29][30][31] . ...
Ceramic aerogels are often used when thermal insulation materials are desired; however, they are still plagued by poor mechanical stability under thermal shock. Here, inspired by the dactyl clubs of mantis shrimp found in nature, which form by directed assembly into hierarchical, chiral and Bouligand (twisted plywood) structure exhibiting superior mechanical properties, we present a compositional and structural engineering strategy to develop strong, superelastic and fatigue resistance ceramic aerogels with chiral fibers array resembling Bouligand architecture. Benefiting from the stress dissipation, crack torsion and mechanical reinforcement of micro-/nano-scale Bouligand array, the tensile strength of these aerogels (170.38 MPa) is between one and two orders of magnitude greater than that of state-of-the-art nanofibrous aerogels. In addition, the developed aerogels feature low density and thermal conductivity, good compressive properties with rapid recovery from 80 % strain, and thermal stability up to 1200 °C, making them ideal for thermal insulation applications.
... 目前， 基于生物结构材料开发设计的仿生异质结 构材料优越的力学性能已经激发了各种领域应用的 高性能材料设计，如航空航天、汽车和土木工程等 [8,9,41,58,59] [7,16,17,34] ...
... Multi-scale mechanical models: From components to representative volume elements to fracture toughening to optimized design [7,16,17,34] [42] ; (b)应力传递过程 [42] ; (c)张剪链模型与剪滞模型 的应力分布 Figure 7 Basic mechanical models of brick-and-mortar representative volume elements. (a) Model of the brick-and-mortar structure [42]; (b) Stress transfer process [42]; (c) Schematic of stress distribution in tension-shear chain models and shear-leg models. ...
... (a) Criteria for crack deflection along or across interfaces [4];(b) Crack deflection morphology along inclined interfaces in natural seashells and 3D-printed seashell-inspired interlocking structures [27,122]; (c) Relationship between energy and interface inclination angle for crack deflection or penetration in 3D-printed seashell-inspired structures; inset shows schematic of crack propagation speed and inclination angle affecting crack deflection and penetration [122,123]. Page 17 of 29 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 1 2 2 2 c 1 2 3 c 1 [34] ; (b) 以归一化笛卡尔坐标表示的归一化扭转裂纹几何及以扭转角 和倾斜角 表示 的等效几何 [34] ; (c)不同扭转角 和倾斜角 下的局部和整体断裂能之比 [34] ; (d)同时考虑裂纹扭转和桥联的断裂力学模型 示意图 [127] ; (e)裂纹尖端无量纲局部能量释放率和裂纹表面形貌 [127] ; (f)不同无量纲纤维长度下扭转纤维结构的有效断裂能 和旋转角间的关系 [128] ...
