Experimental Investigation of Mode-I Fracture Properties of Parallel Strand Bamboo Composite
Aiping Zhou,*,a,c Zirui Huang,b,c Yurong Shen,a,c Dongsheng Huang,a,c and Jianuo Xu d
Parallel stand bamboo (PSB) is a high-quality wood-like bamboo composite. Failure due to cracking is a major concern in the design of PSB components for building structures. The mode-I fracture properties of PSB composite were studied. The wedge splitting method was employed as the test approach. Numerical analyses were conducted to determine the appropriate test specimen dimensions so that valid fracture toughness could be obtained. An R-curve was evaluated in accordance with the equivalent linear elastic fracture mechanics (LEFM) theory. It was found that the initial crack depth ratio should be less than 0.4 for the fracture toughness test. The fracture toughness of PSB is higher than that of commonly used woods, and their fracture behavior is similar, exhibiting quasi-brittle behavior. The R-curve of the PSB exhibits rising behavior until the critical crack length is reached. However, the post-peak R-curve exhibits a descending behavior, contrary to that of quasi-brittle materials, which present a plateau in post-peak crack extension.
Keywords: Fracture; Fracture toughness; R-curve; Energy release rate; Bamboo composite
Contact information: a: School of Civil Engineering, Nanjing Forestry University 159#, Longpan Road, Nanjing, 210037, China; b: School of Civil Engineering, Southeast University, 02, Sipailou, Nanjing, 210018, P.R. China; c: National-Provincial Joint Engineering Research Center of Electromechanical Product Packaging with Biomaterials, Nanjing Forestry University 159#, Longpan Road, Nanjing, 210037, China; d: School of Architecture, Nanjing TECH University, 30#, Puzhu Road, Nanjing, 211800, China; *Corresponding author: firstname.lastname@example.org
Parallel strand bamboo (PSB) is a wood-like bamboo-based composite fabricated from bamboo strands. To manufacture PSB, bamboo culms are first cut into strips that are approximately 2 m in length, 15 mm in width, and 2 mm in thickness and then oven dried at 80 °C until the moisture content is less than 11%. The end product is made by flattening the strips into thin strands, impregnating them with phenolic resin, and gluing them in a parallel fashion under high pressure. Because the bamboo strands are laid in a parallel fashion in the longitudinal direction and are uniformly arranged in the transverse direction, the gradient distribution of fibers in raw bamboo is eliminated; hence PSB can be reasonably treated as a transversely isotropic composite in the macroscopic sense. PSB has excellent mechanical properties, and its products are well suited for use as construction material (Huang et al. 2013, 2015a,b).
Due to the dimensional diversity of strands or bamboo fibers, some gaps between the interfaces of the strands or fibers cannot be absolutely closed in the manufacturing process. Consequently, microvoids and fine cracks are randomly scattered in the PSB composite. Once a PSB component is loaded, these microvoids will coalesce and expand to form cracks. Because the strength of the fibers is much higher than that of the matrix, these cracks usually propagate along the parallel-to-grain direction and cause mode-I, mode-II, or mixed mode fractures, depending on the loading responses of the components. Thus, the failure due to crack propagating is a major concern in the design of PSB components. Structural and fracture properties should be introduced to the failure criteria of PSB components in future designs. To this end, the present work is aimed at studying mode-I fracture properties of PSB composites. The wedge splitting method was employed to investigate the fracture toughness and failure mechanisms of PSB composites.
The application of conventional fracture mechanics, which deal with homogeneous, isotropic materials, and anisotropic composites, began with efforts to design load bearing structures in wood (Hearmon 1969). In particular, early aircraft structures contained wood, resulting in great interest in calculating stress concentration factors around holes (Williams 1989). By extending Lekhnitskii’s method (Lekhnitskii 1963) to the fracture analysis for anisotropic media, Sithet al. (1965) developed a model to analyze the near-tip field of an edged or centered crack in an infinite sheet. Closed solutions for the near-tip stress and displacement fields as well as the stress intensity factors (SIFs) and the energy release rate for these cases were analytically obtained. According to Sith’s theory (Sith et al. 1965), the stress intensity factors may be determined by contour integral around the crack tip. In most cases, however, the media containing a crack cannot be treated as an infinite body because the size of the crack is not small enough. The near-tip fields in these cases are strongly affected by dimensions and remote conditions and are difficult to determine due to mathematical complexity. Similar to Williams’s (1957, 1952) works on isotropic materials, Sith’s formulas for near-tip fields have only theoretical meanings. However, the relationship between the near-tip contour integral, J, and the stress intensity factor, K, proposed by Sith can be practically used to evaluate stress intensity factors for anisotropic media with various configurations.
Due to difficulties in determining the near-tip field of a crack, energy methods are widely accepted to evaluate the stress intensity factor in linear fracture mechanics. Based on the energy balance theory proposed by Griffith (1920), Iwrin (1956) defined a measure known as the energy release rate to quantify the energy balance for a unit area during crack extension. Rice (1968) investigated the energy flow around crack tips and provided a basis to evaluate the fracture toughness with an energy approach. Known as the J-integral method, this approach enjoyed great success in both anisotropic and isotropic materials. The J-integral approach provides a straightforward experimental method for evaluating J derived from its definition as a rate of change of potential energy with crack length. To estimate the energy release rate, some methods were developed to evaluate the fracture toughness without involving the near-tip field (Anderson 2005). The most common procedure among these methods may be the double cantilever beam (DCB) test. By changing the loading condition or the length ratio of the two cantilevers, the DCB specimen can be tested for mode-I, mode-II, and mixed-mode failures. For the mode-I test, the energy release rate of the DCB specimen can be inferred from beam theory. The advantages of the DCB test are obvious in that many intractable problems, such as the singularity of the crack tip, the anisotropy of materials, and the rigorous measurement of the crack length, are ignored. As far as PSB composites are concerned, the material stiffness is much lower compared to metallic materials, which results in large deformations in the DCB test and consequently leads to erroneous estimates of the energy release rate.
Compact specimen tests, recommended by the most major fracture test standards around the world, such as the American standard, ASTM E 399 (2013), and British standard, BSI 12737 (1999), are commonly accepted alternatives for the DCB mode-I fracture test. The compact specimen test methods are particularly attractive for brittle materials because the crack growth is generally stable, allowing the load-COD curve to be recorded. Test procedures and stress intensity factor with respect to crack length ratio for isotropic materials have been presented in ASTM E 399(2013). The wedge splitting method is a modification of the compact test approach. Proposed by Tschegg and Linsbauer (1986), it is a non-standard fracture test strategy, which introduces a pair of opening forces for a compact specimen by exerting a sharp wedge into a prepared crack to induce mode-I fracture. During testing, the wedge produces both opening forces perpendicular to the crack face, driving crack propagation, and vertical forces parallel to the crack direction, stabilizing crack growth. This method has some outstanding advantages (Stanzl-Tschegg et al. 1995). First, it is a very stable test approach and the specimen is compact. Moreover, geometric requirements of the sample are not very stringent. The most attractive characteristic of this method is that it is particularly suitable for testing quasi-brittle materials such as wood and bamboo composites. Considering the advantages of the wedge splitting method, it was used to investigate the mode-I fracture properties of PSB composites in the present work.
PSB composite made from 4-year-old Phyllostachys sp., a common bamboo species cultivated in southeast China, was the test material in this study. Because of the inherently oriented structure of raw bamboo culms (Zhou et al. 2012), PSB composite is consequently characterized with orthotropic features and its mechanical properties are very sensitive to fiber orientation (Huang et al. 2013). Therefore, in this study, it was treated as transversely isotropic material. A notation scheme adopted by the American standard ASTM E 399 (2013) was introduced to report the orientation of the material and fractures. Following the definition of ASTM E 399, the letters L, T, and S denote the longitudinal (parallel-to-grain), transverse, and short transverse directions of PSB composite, respectively. The orientation of a crack is identified by two letters; the first letter indicates the direction of principal tensile stress, which is always perpendicular to the fracture surface in the mode-I test, while the second denotes the direction of crack propagation. For example, T-L orientation corresponds to loading in perpendicular-to-grain direction and crack propagation in parallel-to-grain direction. Since the fracture of a wedge splitting specimen is usually treated as plane stress problem and the mechanical properties are identical in all directions on the S-T plane, cracks oriented in the T-L and S-L directions may not be identified for mode-I fracture problems; hence the present work only studies T-L oriented fractures. For the purpose of mathematical convenience, the letters L and T were replaced with 1 and 2, respectively, to express the orientations of material and fracture. Mechanical properties of PSB composites involved in this study have been investigated in previous studies and their elastic parameters are presented in Table 1 (Huang et al. 2015a).
Table 1. Elastic Parameters of PSB Composite Evolved in this Study
The wedge splitting method (BSI 12737 1999) was employed to study mode-I fracture properties of PSB composite in this work. The principle of the test methodology is schematically illustrated in Fig. 1. A test specimen with a rectangular groove and a starter notch at the bottom of the groove are placed on a narrow support fixed on the test machine.
Fig. 1(a). The sketch of wedge splitting test
Fig. 1(b). The test setup of wedge splitting test
Needle bearings fixed on load transmission pieces are installed between the wedge and needle bearings. The driving force, F, is transmitted from the wedge to the specimen via the load transmission pieces. Obviously, F can be decomposed into two components, namely the horizontal component, FH, inducing mode-I fracture and the vertical component, FV, which helps to stabilize the crack path. The angle of the wedge should be as small as possible so that the influence of the vertical component on the stress of near crack tip may be neglected. Suppose that the wedge angle is 2a and the friction between the wedge and bearings is ignored. Then the crack opening force can be calculated by . Simultaneous recordings of the driving force, F, the crack opening displacement (COD), and the load-COD curve, i.e., the curve, can be plotted, where .
Determination of KIc
From the viewpoint of theoretical linear elastic fracture mechanics (LEFM), if the plasticity of the near crack tip can be omitted, the energy release rate, G, and the values of the J-integral are identical for isotropic materials (Rice 1968). It is well known that the relationship between stress intensity factor, KI (N.mm-3/2), and the value of the J-integral, JI(N.mm1/2), of mode-I fracture for isotropic materials can be expressed as Eq. 1 (Gdoutos 2005),
where for plane stress (MPa) and for plane strain, E is Young’s modulus (MPa) and is Poisson’s ratio. Hence the stress intensity factor of isotropic materials for plane problems may be determined by the J-integral. Virtually all test samples were designed with special configurations so that the energy release rate could be easily calculated. The expressions of stress intensity factors associated with the crack depth ratio, , for standard test samples can be seen in ASTM E 399 (2013), where a is the initial crack length (mm) and W is the length of total ligament of test specimen (mm), as shown in Fig. 1.
The stress intensity factors for wedge splitting specimens of isotropic material have been numerically investigated by Guinea et al. (1996). They offered an expression for calculating KI using least squares fitting for numerical results. Based on numerous numerical studies, Guinea et al. concluded that the stress intensity factors of wedge splitting specimens are very close to those of standard compact specimens provided identical specimen geometry. Thus, the stress intensity factor of the isotropic wedge splitting samples may be calculated by ASTM E 399 (2013) (Eqs. 2 and 3),
where is the crack depth ratio.
As an anisotropic composite, the constitutive law of PSB is distinct from isotropic materials. Therefore, Eq. 2 cannot be directly used to compute the stress intensity factor of mode-I fracture in PSB specimens. Kageyama (1989) proposed a method to estimate the stress intensity factor for orthotropic materials, which can be expressed as Eq. 4,
where and are mode-I stress intensity factors of orthotropic and isotropic materials, respectively, is the orthotropic factor which depends on the crack depth ratio, , and the material orthotropy, and . However, Kageyama did not provide any method for the orthotropic factor determination.
Sith et al. (1965) extended Lehnitskii’s (Lekhnitskii 1963) complex expressions for isotropic plane problems to the infinite anisotropy sheet and offered general solutions for crack tip stress fields in anisotropic bodies. By implementing the J-integral calculation for the near tip field of the cracks, they concluded that JI and KI for orthotropic materials have the same form as that of isotropic materials but the material constant in Eq. 1 should be replaced by that in Eq. 5,
where and are Young’s moduli along direction-1 and -2, respectively, is Poisson’s ratio in the 1-2 plane, and is the shear modulus in the 1-2 plane. Eq. 1 and 5 provide a practical method to determine SIFs for orthotropic composites.
For the wedge splitting samples, SIFs may be determined using Eq.1, provided that the curve is plotted via test, i.e., by determining the critical opening force, , through experiments, the critical energy release rate can be calculated using Eq. 6,
where, is the strain energy density (MPa) in the domain enclosed by the integral contour , (N) and (mm) are components of traction and displacements on the contour, respectively, and ( ) are the components of the outward unit vector normal to . In these notations, the subscript, , represents the items corresponding to the critical opening force, (N). Furthermore, the parameter can be calculated using Eq. 1 and 5 by replacing with .
Determination of crack depth ratio
To obtain valid fracture toughness, the initial crack depth ratio, a/W, should be chosen carefully so that the effects of remote conditions may be neglected. To this end, near-tip contour integrals for diverse crack depth ratios were implemented using a numerical approach with ABAQUS® software. In the ABAQUS® model (FEM model), the wedge-splitting specimen was treated as a plane stress problem and the material was considered to be orthotropic. The longitudinal direction of the material, i.e., direction-1, was orientated on the y-axis of the model and the transverse direction was along the x-axis, as can be seen in Fig. 2. Material constants were adopted according to Table 1. Cracks were simulated by pre-specified seams, which were supposed to have the same lengths and locations as the starter notches of the test specimens. Because the seam defines an edge of a face with overlapping nodes that can be separated during the analysis process, there is no need to remesh the crack faces as they are separated. To simulate the effect of load transmission, crack opening forces were applied to the real position where the test load subjected and the distributed coupling technique in ABAQUS was adopted. This technique allows for a concentrated load on a reference point to produce an equivalent response of a distributed pressure. An 8-node bi quadratic plane stress element (CPS8R) was chosen to model the specimens. The contour route enclosing the crack tip for the J-integral calculation was specified as shown in Fig.2. Mesh sizes were determined such that the analysis results had little sensitivity to the element density. The vertical component produced by the wedge was omitted due to the small angle of the wedge.
Samples with crack depth ratios ranging from 0.20 to 0.80 were analyzed. The dimensions of the analyzed specimens are illustrated in Fig. 3. In order to evaluate the effect of material anisotropy on SIFs, FEM analyses for both the PSB specimens and their equivalent specimens were implemented to compute and . An equivalent specimen for an anisotropic specimen is defined by having the same dimensions but having isotropic material properties.
In this study, the longitudinal Young’s modulus, , and Poisson’s ratio, , of the PSB composites were assigned as the Young’s modulus, , and Poisson’s ratio, for the equivalent specimens, respectively. The results of the SIFs of both PSB specimens and their equivalent samples are compared in Fig. 4. Discrepancies between the SIFs for isotropic and anisotropic materials can be observed in the case where . The SIFs for both the isotropic and anisotropic materials calculated by the FEM model are distinctly different from that calculated by ASTM E 399 when the crack depth ratio is less than 0.4. Once the crack depth ratio exceeds 0.4, the SIFs of both the anisotropic obtained by the
Fig. 2. FEM model for J-integral