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Tetsuya Morimoto, Hisaya Katoh, Yuichi Ishida, Eiichi Hara, [Masahiro Kusano](https://orcid.org/0000-0002-5061-0195), [Kimiyoshi Naito](https://orcid.org/0000-0002-3334-4876), [Makoto Watanabe](https://orcid.org/0000-0002-5064-9583), Kiyoka Takagi, Keiji Arai, Koichi Hasegawa

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[Modified double cantilever beam test method for mode-I energy release rate of elastic adhesive layers](https://mdr.nims.go.jp/datasets/9d79621d-7c54-48bb-b043-ad9d223f462f)

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Modified double cantilever beam test method for mode-I energy release rate of elastic adhesive layersEngineering Fracture Mechanics 307 (2024) 110320A0(MrTMKabcAKDAEWEB1p2aAsSci(amwcrhRContents lists available at ScienceDirectEngineering Fracture Mechanicsjournal homepage: www.elsevier.com/locate/engfracmechodified double cantilever beam test method for mode-I energyelease rate of elastic adhesive layersetsuya Morimoto a,∗, Hisaya Katoh a, Yuichi Ishida a, Eiichi Hara a,asahiro Kusano b, Kimiyoshi Naito b, Makoto Watanabe b, Kiyoka Takagi c,eiji Arai c, Koichi Hasegawa cJapan Aerospace Exploration Agency (JAXA), 6-13-1 Osawa, Mitaka -City, 181-0015, Tokyo, JapanNational Institute for Materials Science (NIMS), 1-2-1 Sengen, Tsukuba-City, 305-0047, Ibaraki, JapanMitsubishi Heavy Industries, Ltd. (MHI), 1 Toyoba, Toyoyama-Cho, Nishikasugai-gun, 480-0293, Aichi, JapanR T I C L E I N F Oeywords:CB testdhesive layerlasticityinkler’s foundationnergy release rateernulli–Euler beamA B S T R A C TThis study proposes a Mode I double cantilever beam (DCB) test method for debonding thickelastic adhesive layers. The approach integrates Bernoulli–Euler beams connected via springelements akin to Winkler’s foundation. Energy release rate determination considers crack lengthcorrelation with elastic deformation in the crack-front region, accounting for both beam rotationand lateral movement. Evaluation of Ti6Al4V samples bonded with FM 309 adhesive revealedhigher initial energy release rates with the conventional beam theory and conservative valueswith the modified beam theory. The proposed method offered improved consistency, and lessconservative energy release rate values compared to the modified theory.. IntroductionThe reliability of adhesive bonding in passenger aircraft structures have been debated, necessitating a classical design sup-lemented with fastening in bonding joints [1,2]. In particular, the case of Air Transat Flight 961 ‘‘Loss of Rudder in flight’’ in005 was found to be caused by ‘‘Weak Bond’’, which is an unexpected loss of bonding performance, resulting in increased doubtsbout the reliability of adhesive bonding [3]. Based on the status of adhesive bonding in aircraft structures, the Federal Aviationdministration (FAA), USA, had established an international working group on metallic and composite structures for airplanesuitable for air transport operations, and had finalized its recommendations in the ‘‘Transport Airplane Metallic and Compositetructures Working Group – Final Recommendation Report on Structural Bonding [4]’’. This recommendation report proposes ahange in the adhesion-related sections of FAA Advisory Circular 20-107B ‘‘COMPOSITE AIRCRAFT STRUCTURE’’ as general visualnspection (GVI) is considered inadequate to detect bond failure; the most effective methods, including detailed visual inspectionDVI) and non-destructive testing using equipment, should be applied to inspect the flange side edge of the bonded area [5]. Inddition, it has been reported that local peel forces or Mode I loadings exist in the stress distribution of bonded joints, whereasost bonded joints and attachments are designed primarily to transfer shear loads. Therefore, a test method is required to assesshether adequate visual inspection or non-destructive testing is applied to cracks along the bonded line under Mode I loadingonditions. These basic data are essential for designing adhesive joints in aircraft applications.Double cantilever beam (DCB) test is a potential candidate as it is a simple yet powerful method to investigate mode I energyelease associated with crack propagation through an interface between two plate coupons. Moreover, it is already standardized by∗ Corresponding author.E-mail address: morimoto.tetsuya@jaxa.jp (T. Morimoto).vailable online 20 July 2024013-7944/© 2024 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY licensehttp://creativecommons.org/licenses/by/4.0/).ttps://doi.org/10.1016/j.engfracmech.2024.110320eceived 29 March 2024; Received in revised form 13 July 2024; Accepted 15 July 2024https://www.elsevier.com/locate/engfracmechhttps://www.elsevier.com/locate/engfracmechmailto:morimoto.tetsuya@jaxa.jphttps://doi.org/10.1016/j.engfracmech.2024.110320http://crossmark.crossref.org/dialog/?doi=10.1016/j.engfracmech.2024.110320&domain=pdfhttps://doi.org/10.1016/j.engfracmech.2024.110320http://creativecommons.org/licenses/by/4.0/Engineering Fracture Mechanics 307 (2024) 110320T. Morimoto et al.Nomenclature𝐴𝑖 Integral constants or analytical coefficients𝐵𝑖 Integral constants or analytical coefficientsBT Beam theoryCF Correlation factor𝐷𝑉 𝐼 Detailed Visual Inspection𝐸 Young’s modulus𝐸𝐵 Young’s modulus of the beam𝐸𝐹 Young’s modulus of the foundation𝐸𝑟𝑓 Error function𝑓 (𝑥) Deflection function of the beam𝑓𝐶 (𝑥) Lateral position of the cracked section𝐺 Energy release rates𝐺𝐷𝐶𝐵 Energy release rate in the DCB test form𝐺𝑉 𝐼 General Visual Inspection𝐼 Moment of inertia of the area𝐿 Hinge loading-arm length𝑙 Crack length𝑙𝑖 𝑖 th data of crack length𝑀 Moment𝑀𝐶 Moment at the crack tipMBT Modified beam theory𝑃 Lateral load𝑃𝐶 Lateral load at the crack tip𝑅𝐵𝑇 Selection parameter for beam theory𝑅𝑀𝐵𝑇 Selection parameter for modified beam theoryS.D. Standard Deviation𝑡 Thickness𝑡𝐷𝐶𝐵 Adhesive thickness of DCB test sample𝑡𝐵 Beam thickness𝑡𝐹 Thickness of the Winkler’s foundation𝑈 Strain energy of the beam𝑤 Beam width𝑥 Distance along beam from tip𝛼 Converted coefficients1∕𝛼 Proposed crack-length correlation factor𝛽 Integration constants𝜆 Compliance𝜆𝑖 𝑖th data of compliance𝜒 Crack length correlation factor𝛿 Crack opening displacements𝛥 Crack length correlation factor of the modified beam theorythe International Organization for Standardization (ISO) 15024, American Society for Testing and Materials (ASTM) D5528, D3433,and Japanese Industrial Standards (JIS) K7086. Additionally, it is widely used to obtain the energy release rate values of mode Idelamination for thin layers such as laminated composite materials [6–9]. The current DCB test standards set the scope for a crackto extend through a thin interface; thus, elastic deformation can be ignored. Therefore, there is no lateral deformation of the beamcross-section at the crack front, and the beam region where the crack does not extend shows no deformation. However, as reportedby Akhavan-Safar et al. for the mode I fracture toughness of adhesive layers [10], in adhesive layers where the thickness is notnegligible, as is often the case in engineering applications, it is questionable whether the energy release rate obtained from the DCBtest corresponds to the fracture toughness of the adhesive. Therefore, the scope of DCB test standards should be widened to includecases for the elastic layer of non-negligible thickness causing rotation and lateral deformation of the beam cross-section at the crackfront and deformation of the region where the crack has not yet extended.The theoretical background of the present DCB test was a Bernoulli–Euler beam with fixed and free ends, assuming loadingperpendicular to the longitudinal direction of the beam. Bisshopp et al. derived an exact solution for the bending displacement ofa single beam using an elliptical function and showed that the difference between the bending of a Bernoulli–Euler beam and the2exact solution was within a negligible range when the bending displacement was less than 20% of the beam length.[11]. Based onEngineering Fracture Mechanics 307 (2024) 110320T. Morimoto et al.baWetaaeeTmevatsitisttsmebcdadrolwrfApiltltcltttfthis solution, Devitt et al. determined the fracture toughness in the form of an implicit function and verified it for E-glass fiber-reinforced epoxy laminates using a fracture toughness test method; for this, they applied a devised jig to reduce the errors causedby the moment generated by the hinge loading arm [12]. Ashizawa et al. devised a hinged double-cantilever beam test method tominimize the moment [13].Contrarily, under increased loading, owing to the development of toughened epoxy resins, some moments become significantecause of the increased hinge loading arm of the high-load hinge. Moreover, the exact solution as an implicit function requiresnumerical table for the normalized parameters. These requirements can render engineering applications challenging. Therefore,ang et al. derived an approximation formula that considers the effect of the hinge loading arm as an explicit function using seriesxpansion [14] and derived a rigorous solution that considers the effect of the hinge loading arm by adding a correction term tohe Bernoulli–Euler beam theory [15]. Thus, it is practical to establish a DCB test method. In test standards, such as ASTM D5528nd ISO 15024, the correction formula proposed by Williams [15] and the calculation formula for approximating the compliancend crack length proposed by Berry using the 𝑛th power law [16] were presented, and the selection was assigned to the users. Tostablish the DCB test standard JIS K7086 for carbon fiber-reinforced plastic (CFRP) interlaminar fracture toughness, Kageyamat al. assumed that the elastic deformation of the crack propagation area is negligible for thin and highly rigid interlaminar CFRP.hey verified this assumption through comparisons with mechanical tests and finite-element analyses [17–20].However, when delamination occurs through thick elastic layers such as in adhesive joints, the present DCB test standardsay yield inaccurate results because of the opening and rotation at the crack tip induced by deformation across the uncrackedlastic layer. Therefore, an elastic beam on top of an elastic layer was investigated because of its unique bending behavior underarious loading conditions. Winkler developed a solution for Bernoulli–Euler beam bending for civil engineering applications, suchs railroad tracks on sleepers, assuming that the reaction of a point on the foundation solely depends on the settlement and thathe point is independent of the settlements of neighboring points [21,22]. For the elastic foundation model, Biot proposed an exactolution for an infinite Bernoulli–Euler beam under a sine wave and concentrated loading, which produced a bending moment, asn the case of an uncracked elastic layer in the DCB test [23]. Pavlović et al. analyzed a Bernoulli–Euler beam that was not fixedo Winkler’s elastic foundation, as is the case with cracked sections. Thus, the beam sections deflected from the foundation arendependent of the foundation-based loading [24].Kanninen investigated the crack propagation in DCB test specimens of isotropic materials by assuming a finite-length beam partlyupported by a Pasternak elastic foundation with elasticities of both deflection and rotation [25,26]. The rotational distortion athe crack end was considered in the analyses, and it was concluded that a beam with shear distortion in the Timoshenko beamheory can predict the numerical results. Regarding the shear distortion of an isotropic beam, Whitney employed the higher-orderhear deformation plate theory for accurate stress distribution analysis of a DCB test specimen [27]. By extending the Kanninenodels, Williams provided a factor for the crack length correction of an orthotropic beam rotating at the crack end on a Pasternaklastic foundation [28] and derived expressions for the energy release rate [29]. Kondo analyzed DCB specimens using a Timoshenkoeam supported by a Winkler foundation and concluded that the analyses accurately correlated with the predictions of Whitney [27]ompared to finite element analyses [30]. Shokrieh et al. comparatively investigated DCB analyses that considered transverse sheareformation in beams and root rotation at the crack tip [31]. Takeda developed a new model to provide the effective crack lengthnd compared it with a wedge test for the energy release rate of adhesive bonds, showing consistent results. Subsequently, theyemonstrated the applicability of the wedge test and a new model for an environmental test of bonding on the mode I energy releaseate [32].These solutions of the modified beam theory (MBT) yielded crack-length correlation factors focusing on the shear deformationr rotation at the crack front in beams, which are often derived through linear fitting between 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒1∕3 and the crackength, assuming that the factors are small compared to the crack length [6,7,9]. However, in aircraft design, it is crucial to confirmhether the correlation length is acceptably small compared to the measured crack length. This consideration arises because factorselated to the thickness and elasticity of the foundation are expected to increase the influence of slender beams on thick and elasticoundations, similar to the behavior observed in elastic adhesives applications for thin elastic materials such as advanced CFRPs.lthough advanced numerical methods display improved agreement with the data of the energy release rate, showing promisingotential for engineering design applications of adhesive joints [33,34], the importance of input parameters using mechanical testingncreases in these applications; thus, a modification of the present DCB test methods is required to derive data on elastic adhesiveayers of non-negligible thickness.Therefore, in this study, we considered the infinitesimal deformation of a slender beam on an elastic foundation, assuming thathe beam deformation is within a predictable level using the Bernoulli–Euler beam model, and that both the crack end rotation andateral movement are given by Winkler’s elastic foundation model. This approach separated the beam into cracked and foundation-op sections. We set the loading conditions as the deflectional load and moment at the end of the cracked section and linearlyonnected the other end to the top section of the foundation. The deflectional load was multiplied by the crack-length-added momentoading for the foundation-top section. The deflection and rotation at the connecting point were provided with a known solutiono Winkler’s elastic foundation model. We predicted the deflection at the loading end of the cracked section by superimposinghe connecting-point deflection, slope deflection using the connecting-point rotation and crack length, and the deformation ofhe cracked section using both the deflectional load and moment. The derived solution yielded an energy release rate with a3oundation-based correlation factor for crack length.Engineering Fracture Mechanics 307 (2024) 110320T. Morimoto et al.Fig. 1. Winkler’s linear elastic foundation model.2. Calculations of delamination in elastic foundation2.1. Lateral bending of single cantilever beam on an elastic foundationThe DCB test sample had a mirror-image relationship with a plane of symmetry in the center because it was fabricated byjoining two identical cantilevers (Fig. 1). Therefore, the model with a single cantilever and rigid base represents half of the entireDCB sample. Assuming that the reaction of the modeled upper half of the DCB sample in a single cantilever form is proportional tothe settlement and that the array of points forms a linear elastic foundation, the bending of the beam can be expressed as follows:𝐸𝐵𝐼𝑑4𝑓 (𝑥)𝑑𝑥4= −𝐸𝐹𝑤𝑡𝐹𝑓 (𝑥) (1)where 𝐸𝐵 is Young’s modulus of the beam, 𝐼 is the moment of inertia of the beam area, 𝑓 (𝑥) is the positive beam deflection functionin the upper direction, 𝑥 is the distance from the tip of the beam, 𝐸𝐹 is Young’s modulus of the elastic foundation, 𝑡𝐹 is the thicknessof the elastic foundation, and 𝑤 is the width of the beam. The thickness of the elastic foundation in the DCB test sample 𝑡𝐷𝐶𝐵 wastwice that of 𝑡𝐹 .𝑡𝐷𝐶𝐵 = 2𝑡𝐹 . (2)Eq. 1 can be rewritten as Eq. (4), which yields 𝛼 in Eqs. (3).𝛼 =(𝐸𝐹𝑤4𝐸𝐵𝐼𝑡𝐹)14(3)𝑑4𝑓 (𝑥)𝑑𝑥4= −4𝛼4𝑓 (𝑥) (4)When the lateral loading 𝑃 and moment 𝑀 are applied at tip 𝑥 = 0, the complementary solution is expressed as follows:By setting 𝑓 (𝑥) = 𝑒𝛽𝑥, Eq. (4) yields the following relationship:𝛽 = ±(1 ± 𝑖)𝛼, (5)where 𝛽 is an integral constant. Therefore, 𝑓 (𝑥) can be expressed using the integral constant 𝐴𝑖 as follows:𝑓 (𝑥) = 𝐴1𝑒(1+𝑖)𝛼𝑥 + 𝐴2𝑒(1−𝑖)𝛼𝑥 + 𝐴3𝑒−(1+𝑖)𝛼𝑥 + 𝐴4𝑒−(1−𝑖)𝛼𝑥 (6)Eq. (6) is expressed using integral constants 𝐵𝑖 by applying the relationships 𝑒𝑖𝑥 − 𝑒−𝑖𝑥 = 2𝑖 sin(𝑥) and 𝑒𝑖𝑥 + 𝑒−𝑖𝑥 = 2𝑖 cos(𝑥):𝑓 (𝑥) = 𝑒𝛼𝑥{𝐵1 cos(𝛼𝑥) + 𝐵2 sin(𝛼𝑥)} + 𝑒−𝛼𝑥{𝐵3 cos(𝛼𝑥) + 𝐵4 sin(𝛼𝑥)} (7)The modeled cantilever shown in Fig. 1 converges to 𝑓 (𝑥) = 0. in 𝑥 → ∞, which yields 𝐵1 = 𝐵2 = 0. Therefore, Eq. (7) yields thefollowing relationship:𝜕2𝑓 (𝑥)𝜕𝑥2= −2𝛼2𝑒−𝛼𝑥{𝐵4 cos(𝛼𝑥) − 𝐵3 sin(𝛼𝑥)} (8)𝜕3𝑓 (𝑥)= 2𝛼3𝑒−𝛼𝑥{(𝐵 + 𝐵 ) cos(𝛼𝑥) − (𝐵 − 𝐵 ) sin(𝛼𝑥)} (9)4𝜕𝑥3 3 4 3 4Engineering Fracture Mechanics 307 (2024) 110320T. Morimoto et al.Fig. 2. Beam deflection atop Winkler’s Foundation.Fig. 3. Cracked section at the tip of Winkler’s Foundation.Furthermore, loadings 𝑀 and 𝑃 at 𝑥 = 0 lead to the following relationships when deflection function 𝑓 (𝑥) of the beam is positivein the upper direction.𝐸𝐵𝐼[𝜕2𝑓 (𝑥)𝜕𝑥2]𝑥=0= 𝑀 (10)𝐸𝐵𝐼[𝜕3𝑓 (𝑥)𝜕𝑥3]𝑥=0= 𝑃 (11)Thereby, Eq. (7) yields the following relationship.𝑓 (𝑥) = 𝑒−𝛼𝑥2𝐸𝐵𝐼𝛼3[𝑃 cos(𝛼𝑥) +𝑀𝛼{cos(𝛼𝑥) − sin(𝛼𝑥)}] (12)Eq. (12) gives the beam deflection, as shown in Fig. 2, indicating compression into the foundation near the tip 𝑥 = 0.2.2. Lateral bending of single cantilever beam on an elastic foundation with cracked sectionAssume that a beam of length 𝑙 made of identical material with an identical cross-sectional shape is connected to the tip of thisbeam; however, it is not connected to the elastic foundation to represent the cracked section (see Figs. 3 and 4).5Engineering Fracture Mechanics 307 (2024) 110320T. Morimoto et al.Fig. 4. Lateral movements at the cracked section.The lateral bending of the foundation-top section 𝑓 (𝑥), 𝑙 ≤ 𝑥 is revised as follows:𝑓 (𝑥) = 𝑒−𝛼(𝑥−𝑙)2𝐸𝐵𝐼𝛼3⟨𝑃𝐶 cos{𝛼(𝑥 − 𝑙)} + (𝑀𝐶 + 𝑃𝐶 𝑙)𝛼[cos{𝛼(𝑥 − 𝑙)} − sin{𝛼(𝑥 − 𝑙)}]⟩ (13)The lateral position of the cracked section 𝑓𝐶 (𝑥), 0 ≤ 𝑥 < 𝑙, in the form of the sum of the lateral movement at the crack end isexpressed as follows 𝑓 (𝑙), slope height −[𝜕𝑓 (𝑥)𝜕𝑥]𝑥=𝑙(𝑙−𝑥), and bendings at the cracked section caused by the loading 𝑃𝐶 and moment𝑀𝐶 :𝑓𝐶 (𝑥) = 𝑓 (𝑙) −[𝜕𝑓 (𝑥)𝜕𝑥]𝑥=𝑙(𝑙 − 𝑥) +𝑃𝐶6𝐸𝐵𝐼(𝑥3 − 3𝑙2𝑥 + 2𝑙3) +𝑀𝐶2𝐸𝐵𝐼(𝑥2 − 2𝑙𝑥 + 𝑙2) (14)Where,𝑓 (𝑙) = 12𝐸𝐵𝐼𝛼3{𝑃𝐶 + (𝑀𝐶 + 𝑃𝐶 𝑙)𝛼}, (15)and𝜕𝑓 (𝑥)𝜕𝑥= − 𝑒−𝛼(𝑥−𝑙)2𝐸𝐵𝐼𝛼2⟨𝑃𝐶 cos{𝛼(𝑥 − 𝑙)} + (𝑀𝐶 + 𝑃𝐶 𝑙)𝛼[cos{𝛼(𝑥 − 𝑙)} − sin{𝛼(𝑥 − 𝑙)}]⟩− 𝑒−𝛼(𝑥−𝑙)2𝐸𝐵𝐼𝛼2⟨𝑃𝐶 sin{𝛼(𝑥 − 𝑙)} + (𝑀𝐶 + 𝑃𝐶 𝑙)𝛼[sin{𝛼(𝑥 − 𝑙)} + cos{𝛼(𝑥 − 𝑙)}]⟩= − 𝑒−𝛼(𝑥−𝑙)2𝐸𝐵𝐼𝛼2⟨𝑃𝐶 [cos{𝛼(𝑥 − 𝑙)} + sin{𝛼(𝑥 − 𝑙)}] + 2(𝑀𝐶 + 𝑃𝐶 𝑙)𝛼 cos{𝛼(𝑥 − 𝑙)}⟩, (16)thus,[𝜕𝑓 (𝑥)𝜕𝑥]𝑥=𝑙= − 12𝐸𝐵𝐼𝛼2{𝑃𝐶 + 2𝛼(𝑀𝐶 + 𝑃𝐶 𝑙)}. (17)Therefore, the lateral position 𝑓𝐶 (𝑥) at the loading point 𝑥 = 0 can be expressed as follows:𝑓𝐶 (0) =12𝐸𝐵𝐼𝛼3{𝑃𝐶 + 𝛼(𝑀𝐶 + 𝑃𝐶 𝑙)} +12𝐸𝐵𝐼𝛼2{𝑃𝐶 + 2𝛼(𝑀𝐶 + 𝑃𝐶 𝑙)}𝑙 +𝑃𝐶 𝑙33𝐸𝐵𝐼+𝑀𝐶 𝑙22𝐸𝐵𝐼= 16𝐸𝐵𝐼[𝑃𝐶{2( 1𝛼+ 𝑙)3+ 1𝛼3}+ 3𝑀𝐶( 1𝛼+ 𝑙)2](18)2.3. Calculation of energy release rateThe strain energy 𝑈 of the beam in Eq. (18) can be expressed as𝑈 = 𝑈 (𝑃𝐶 ,𝑀𝐶 , 𝑙) =12{𝑃𝐶𝑓𝐶 (0) −𝑀𝐶[𝜕𝑓𝐶 (𝑥)𝜕𝑥]𝑥=0}(19)The differential of 𝑈 (𝑃𝐶 ,𝑀𝐶 , 𝑙) can be expressed as:𝑑𝑈 = 𝜕𝑈 𝑑𝑃𝐶 + 𝜕𝑈 𝑑𝑀𝐶 + 𝜕𝑈 𝑑𝑙 (20)6𝜕𝑃𝐶 𝜕𝑀𝐶 𝜕𝑙Engineering Fracture Mechanics 307 (2024) 110320T. Morimoto et al.Fig. 5. Moment by the hinge loading arm.Therefore, for the case in which the crack length 𝑙 slightly extends to 𝑙+𝑑𝑙, differential 𝑑𝑈𝑙→𝑙+𝑑𝑙 at 𝑓𝐶 (0) is expressed as follows:𝑑𝑈𝑙→𝑙+𝑑𝑙 =12⎧⎪⎨⎪⎩𝑃𝐶𝜕𝑓𝐶 (0)𝜕𝑙𝑑𝑙 −𝑀𝐶𝜕[𝜕𝑓𝐶 (𝑥)𝜕𝑥]𝑥=0𝜕𝑙𝑑𝑙⎫⎪⎬⎪⎭(21)Eq. (18) yields the following relationship:𝜕𝑓𝐶 (0)𝜕𝑙= 1𝐸𝐵𝐼𝛼2{𝑃𝐶 + 𝛼(𝑀𝐶 + 2𝑃𝐶 𝑙)}+𝑃𝐶 𝑙2𝐸𝐵𝐼+𝑀𝐶 𝑙𝐸𝐵𝐼(22)Furthermore,𝜕𝑓𝐶 (𝑥)𝜕𝑥=[𝜕𝑓 (𝑥)𝜕𝑥]𝑥=𝑙+𝑃𝐶2𝐸𝐵𝐼(𝑥2 − 𝑙2) +𝑀𝐶𝐸𝐵𝐼(𝑥 − 𝑙), (23)thus,[𝜕𝑓𝐶 (𝑥)𝜕𝑥]𝑥=0= − 12𝐸𝐵𝐼𝛼2{𝑃𝐶 + 2𝛼(𝑀𝐶 + 𝑃𝐶 𝑙)} −𝑃𝐶2𝐸𝐵𝐼𝑙2 −𝑀𝐶𝐸𝐵𝐼𝑙, (24)providing𝜕[𝜕𝑓𝐶 (𝑥)𝜕𝑥]𝑥=0𝜕𝑙= −𝑃𝐶𝐸𝐵𝐼𝛼−𝑃𝐶𝐸𝐵𝐼𝑙 −𝑀𝐶𝐸𝐵𝐼. (25)Therefore, the energy release rate 𝐺 = 𝑑𝑈𝑙→𝑙+𝑑𝑙∕𝑤𝑑𝑙 is given by𝐺 =𝑑𝑈𝑙→𝑙+𝑑𝑙𝑤𝑑𝑙= 𝑃𝐶𝜕𝑓𝐶 (0)2𝑤𝜕𝑙−𝑀𝐶𝜕[𝜕𝑓𝐶 (𝑥)𝜕𝑥]𝑥=02𝑤𝜕𝑙=𝑃𝐶2𝑤[1𝐸𝐵𝐼𝛼2{𝑃𝐶 + 𝛼(𝑀𝐶 + 2𝑃𝐶 𝑙)} +𝑃𝐶 𝑙2𝐸𝐵𝐼+𝑀𝐶 𝑙𝐸𝐵𝐼]+𝑀𝐶2𝑤(𝑃𝐶𝐸𝐵𝐼𝛼+𝑃𝐶𝐸𝐵𝐼𝑙 +𝑀𝐶𝐸𝐵𝐼)= 12𝑤𝐸𝐵𝐼{𝑃𝐶( 1𝛼+ 𝑙)+𝑀𝐶}2(26)2.4. Moment by hinge loading armThe moment load is induced by the lateral loading 𝑃𝐶 when the hinge loading arm 𝐿 is non-negligible because of theloading-block configuration, as shown in Fig. 5.7Engineering Fracture Mechanics 307 (2024) 110320T. Morimoto et al.Fig. 6. ISO 15024 loading hinge setup for DCB Test.The induced moment 𝑀𝐶 is expressed as follows when slope[𝜕𝑓𝐶 (𝑥)𝜕𝑥]𝑥=0is close to 0:𝑀𝐶 = −[𝜕𝑓𝐶 (𝑥)𝜕𝑥]𝑥=0𝐿𝑃𝐶 (27)The following relationship is obtained by applying Eqs. (24) and (27) for[𝜕𝑓𝐶 (𝑥)𝜕𝑥]𝑥=0−𝑀𝐶𝐿𝑃𝐶= 12𝐸𝐵𝐼{𝑃𝐶 (1𝛼+ 𝑙)2 + 2𝑀𝐶 (1𝛼+ 𝑙)}(28)Therefore, moment 𝑀𝐶 is expressed as follows:𝑀𝐶 = −𝑃𝐶2( 1𝛼 + 𝑙)2𝐿2{𝐸𝐵𝐼 + ( 1𝛼 + 𝑙)𝐿𝑃𝐶}(29)2.5. Energy release rate in DCB testEqs. (26) and (29) yields the following relationship for the energy release rate:𝐺 =𝑃𝐶22𝑤𝐸𝐵𝐼⎡⎢⎢⎣( 1𝛼+ 𝑙)−𝑃𝐶2( 1𝛼 + 𝑙)2𝐿2{𝐸𝐵𝐼 + ( 1𝛼 + 𝑙)𝐿𝑃𝐶}⎤⎥⎥⎦2=𝑃𝐶22𝑤𝐸𝐵𝐼⎡⎢⎢⎢⎣(1𝛼 + 𝑙){2 +(1𝛼 + 𝑙)𝐿𝑃𝐶𝐸𝐵𝐼}2{1 +(1𝛼 + 𝑙)𝐿𝑃𝐶𝐸𝐵𝐼}⎤⎥⎥⎥⎦2 (30)For the DCB test case, the energy release rate 𝐺𝐷𝐶𝐵 was twice that of the single-cantilever model 𝐺 in Eq. (30), yielding𝐺𝐷𝐶𝐵 = 2 × 𝐺 =𝑃𝐶2𝑤𝐸𝐵𝐼⎡⎢⎢⎢⎣(1𝛼 + 𝑙){2 +(1𝛼 + 𝑙)𝐿𝑃𝐶𝐸𝐵𝐼}2{1 +(1𝛼 + 𝑙)𝐿𝑃𝐶𝐸𝐵𝐼}⎤⎥⎥⎥⎦2. (31)For extreme cases of the hinge loading arm 𝐿, where 𝐿 = 0, Eq. (31) can be simplified as follows:𝐺𝐷𝐶𝐵,𝐿→0 =𝑃𝐶2𝑤𝐸𝐵𝐼( 1𝛼+ 𝑙)2(32)Eq. (32) is attained when the axis center of the loading hinge is set to the middle of the plate thickness, as standardized in ISO15024 (see Fig. 6).With Eq. (3), 1∕𝛼 in Eq. (32) appears negligible when the adhesive is rigid compared to the beam as 𝐸𝐹 ≫ 𝐸𝐵 , or the thicknessis small as 𝑡 → 0. However, for the elastic adhesives with 𝐸 ≈ 𝐸 , the thickness appears to be an important factor that affects8𝐹 𝐹 𝐵Engineering Fracture Mechanics 307 (2024) 110320T. Morimoto et al.the accuracy of the energy release rate in Eq. (32). For example, a DCB test with rectangular cross-sectional coupons of 𝑡𝐵 = 2 mmand 𝐼 = 2𝑤∕3 leads to 1∕𝛼 ≈ (8𝑡𝐹 ∕3)1∕4; therefore, 𝑡𝐹 ≥ 3∕8 mm affects 𝐺𝐷𝐶𝐵 at the level of 10% for the crack nucleation stage of𝑙 ≈ 10 mm.2.6. Fitting procedure for DCB test dataThe crack-opening displacement in the DCB test was twice that of the lateral position 𝑓𝐶 (𝑥) at loading point 𝑥 = 0 in Eq. (18).2𝑓𝐶 (0) =13𝐸𝐵𝐼[𝑃𝐶{2( 1𝛼+ 𝑙)3+ 1𝛼3}+ 3𝑀𝐶( 1𝛼+ 𝑙)2](33)Moment 𝑀𝐶 is approximated using Eq. (29) in a well-managed DCB test; hence, compliance 𝜆(𝑙)𝐷𝐶𝐵 = 2𝑓𝐶 (0)∕𝑃𝐶 is expressedas follows:𝜆(𝑙)𝐷𝐶𝐵 = 13𝐸𝐵𝐼⎡⎢⎢⎢⎣{2( 1𝛼+ 𝑙)3+ 1𝛼3}−3𝑃𝐶(1𝛼 + 𝑙)4𝐿2{𝐸𝐵𝐼 +(1𝛼 + 𝑙)𝐿𝑃𝐶}⎤⎥⎥⎥⎦(34)When the hinge loading arm 𝐿 decreases with 𝐿 → 0, Eq. (34) is simplified as follows:𝜆(𝑙)𝐷𝐶𝐵,𝐿→0 =13𝐸𝐵𝐼[{2( 1𝛼+ 𝑙)3+ 1𝛼3}+ 𝐸𝑟𝑓](35)where 𝐸𝑟𝑓 is the error function. The coefficients 1∕(3𝐸𝐵𝐼) and 1∕𝛼 in Eq. (35) were obtained by fitting the DCB test data(𝜆𝑖, 𝑙𝑖), 𝑖 = 1, 2, 3...𝑛 and applying the least-squares method when the error function 𝐸𝑟𝑓 was negligible.𝛴{𝜆𝑖 − 𝜆(𝑙𝑖)𝐷𝐶𝐵,𝐿→0}2→ 𝑚𝑖𝑛. (36)Eq. (35) is simplified as follows when 1∕𝛼𝑙𝑖 → 0:𝜆(𝑙𝑖)𝐷𝐶𝐵,𝐿→0 =13𝐸𝐵𝐼[{2( 1𝛼+ 𝑙𝑖)3+ 1𝛼3}+ 𝐸𝑟𝑓]≈𝑙𝑖33𝐸𝐵𝐼{2(1 + 1𝛼𝑙𝑖)3+(1𝛼𝑙𝑖)3}≈2𝑙𝑖33𝐸𝐵𝐼(1 + 3𝛼𝑙𝑖)(37)Therefore, Eq. (36) can be expressed in a simplified form as follows:𝛴{𝜆𝑖 −2𝑙𝑖33𝐸𝐵𝐼(1 + 3𝛼𝑙𝑖)}2→ 𝑚𝑖𝑛. (38)Eq. (38) is expressed as follows with coefficients 3∕𝛼 = 𝐴5 and 2∕(3𝐸𝐵𝐼) = 𝐵5:𝛴(𝜆𝑖 − 𝐵5𝑙𝑖3 − 𝐴5𝐵5𝑙𝑖2)2 → 𝑚𝑖𝑛. (39)For the partial differentiation of Eq. (39) for the coefficients 𝐴5 and 𝐵5 should be zero to be minimized. Therefore, thesimultaneous equations for 𝐴5 and 𝐵5 are as follows:{𝛴(𝜆𝑖 − 𝐵5𝑙𝑖3 − 𝐴5𝐵5𝑙𝑖2)(𝐵5𝑙𝑖2) = 0𝛴(𝜆𝑖 − 𝐵5𝑙𝑖3 − 𝐴5𝐵5𝑙𝑖2)(𝑙𝑖3 + 𝐴5𝑙𝑖2) = 0(40)As 𝐵5 ≠ 0, Eq. (40) is simplified as follows:{𝛴(𝜆𝑖 − 𝐵5𝑙𝑖3 − 𝐴5𝐵5𝑙𝑖2)𝑙𝑖2 = 0𝛴(𝜆𝑖 − 𝐵5𝑙𝑖3 − 𝐴5𝐵5𝑙𝑖2)𝑙𝑖3 = 0(41)Therefore, the simultaneous equations in Eq. (41) yield the following solutions for 𝐴5 and 𝐵5.⎧⎪⎪⎨⎪⎪⎩𝐴5 =3𝛼=𝛴𝑙𝑖6𝛴𝜆𝑖𝑙𝑖2 − 𝛴𝑙𝑖5𝛴𝜆𝑖𝑙𝑖3𝛴𝑙𝑖4𝛴𝜆𝑖𝑙𝑖3 − 𝛴𝑙𝑖5𝜆𝑖𝑙𝑖2𝐵5 =23𝐸𝐵𝐼=𝛴𝑙𝑖4𝛴𝜆𝑖𝑙𝑖3 − 𝛴𝑙𝑖5𝛴𝜆𝑖𝑙𝑖2𝛴𝑙𝑖4𝛴𝑙𝑖6 − 𝛴𝑙𝑖5𝛴𝑙𝑖5.(42)Therefore,⎧⎪⎪⎨⎪⎪1𝛼=𝛴𝑙𝑖6𝛴𝜆𝑖𝑙𝑖2 − 𝛴𝑙𝑖5𝛴𝜆𝑖𝑙𝑖33(𝛴𝑙𝑖4𝛴𝜆𝑖𝑙𝑖3 − 𝛴𝑙𝑖5𝜆𝑖𝑙𝑖2)1𝐸 𝐼=3(𝛴𝑙𝑖4𝛴𝜆𝑖𝑙𝑖3 − 𝛴𝑙𝑖5𝛴𝜆𝑖𝑙𝑖2)( 4 6 5 5).(43)9⎩𝐵 2 𝛴𝑙𝑖 𝛴𝑙𝑖 − 𝛴𝑙𝑖 𝛴𝑙𝑖Engineering Fracture Mechanics 307 (2024) 110320T. Morimoto et al.Fig. 7. DCB test setup:Overview.The energy release rate 𝐺𝐷𝐶𝐵,𝐿→0 was derived by substituting 𝐸𝐵𝐼 and 1∕𝛼 into Eq. (32).3. Experimental and discussionsThe proposed method considers both the beam rotation and lateral movement at the crack front, and their comparisons are madewith the MBT of the crack front rotation and simple beam theory (BT) of the fixed crack front.The theoretical backgrounds of DCB test standards, such as ISO 15024, ASTM D5528, and JIS K7086, were not significantlydifferent based on the MBT. JIS K7086 mainly introduces BT, and the crack-length correlation factor in MBT is determined bya linear plot between the cube root of the normalized compliance and normalized crack length. MBT is recommended in ASTMD5528 after round-robin testing with a compliance calibration method and a modified compliance calibration method, yielding acorrelation factor 𝛥 derived from a plot of 𝜆1∕3 versus crack length 𝑙. The energy release rate 𝐺𝐷𝐶𝐵 and Young’s modulus of beam𝐸𝐵 were determined in the MBT of ASTM D5528 as follows:𝐺𝐷𝐶𝐵 =3𝑃𝐶𝛿2𝑤(𝑙 + |𝛥|)(44)𝐸𝐵 =64𝑃𝐶 (𝑙 + |𝛥|)3𝛿𝑤(2𝑡𝐵)3(45)In crack inspection, JIS K7086 accepts temporal unloading to stop crack propagation for intensive checks, rendering GVI in manycases, acceptably accurate in measuring crack length 𝑙. Therefore, JIS K7086 was adopted for the DCB testing on a Ti-6Al-4V plate(Standard Test Piece, Japan), 150 mm along the crack elongation direction, 25 mm in width, and 2 mm in average thickness, bondedwith an FM 309 film adhesive (Solvay, USA), as shown in Fig. 7.The Ti-6Al-4V plates were surface-treated for bonding as follows: degreased with FineSoleve E acetone-substituting detergent(Sankyo Chemical Co., Japan), sandblasted with Fuji Rundum WA SMAWF080 Al2O3 powder–150–212 μm in diameter (FujiManufacturing, Japan), and boiled in purified water for 30 min to stabilize the surface conditions. The autoclave was set at 183.0◦C for 140 min to cure the specimen with a 40 mm film crack of 12.5 μm thick Kapton film (DuPont, USA). A 12 mm square loadingblock was attached for the DCB testing of samples No. 1 to 5 using a Shimadzu Autograph AG-X tensile test machine (ShimadzuSLBL 1kN type Shimadzu Autograph AG-X) , as shown in Fig. 8, with the cross-head speed of 1.0 mm/min.The specimen sides were coated in white for GVI to increase the visual contrast of the dark-appearing cracks caused by the whitecoating, as depicted in Fig. 9.Fig. 10 shows the crosshead displacement–load curves. The crosshead displacements were correlated with the zero positionsusing linear fitting, as shown in Fig. 10 to determine the compliance for the BT and MBT methods.Moment 𝑀𝐶 caused by the hinge loading arm 𝐿 = 7 (mm), which is half that of the loading block of 12 mm and Ti-6Al-4Vplate thickness of 2 mm, as shown in Fig. 8, was two to three digits smaller and thus regarded negligible against the lateral load𝑃𝐶 with Eq. (29), as presented in Table 1.The crack length correlation factor 𝛥 in MBT was defined by applying a linear relationship between crack length 𝑙 and 𝜆1∕3 asdepicted in Fig. 11.The calculated energy release rates, excluding the data points of the unstable fracture cases, were compared between the BT(JIS K7086) method, MBT (ASTM D5528) for Eq. (44) and the proposed method for Eqs. (32) and (43), as presented in Fig. 12 andTable 2.10Engineering Fracture Mechanics 307 (2024) 110320T. Morimoto et al.Fig. 8. DCB test setup: Crack region zoom up.Fig. 9. GVI for the crack propagation in DCB Test.Table 1Moment by Hinge loading arm 𝐿 = 7 mm.Lateral Load𝑃𝐶 (×102 N)Moment(×10−1 Nm)1.99 1.151.66 1.241.41 1.281.11 1.081.02 1.180.85 1.040.82 1.190.70 1.04The energy release rate calculations presented in Table 2 demonstrate that the proposed DCB test method provides a significantlysmaller standard deviation than the current BT method in JIS K7086. Furthermore, as shown in Fig. 12, the BT method shows asloped value on the overestimation side for the crack nucleation and propagation stages and underestimation side for the final stageof a long crack. Conversely, the influence of the crack length was small in the proposed method; thus, the proposed method isexpected to provide more reliable values than the BT method.The MBT method in ASTM D5528 demonstrated the calculated energy release rates with the smallest standard deviation and themost conservative side. Therefore, the MBT is recommended for engineering applications requiring high reproducibility and a largestructural safety margin. However, the crack length correlation factors |𝛥| appear large in the values while MTB in ASTM D5528assumes the case 𝑙 + |𝛥| is slightly longer than 𝑙; therefore, the soundness of the MBT may be elusive in this experiment.11Engineering Fracture Mechanics 307 (2024) 110320T. Morimoto et al.Fig. 10. Displacement-load curves.The proposed method and the MBT method demonstrated a good agreement for the 𝐸𝐵s of ±10% in the distributions and theaverage values within 5%; therefore, the CFs of 1∕𝛼 is expected to reflect the robustness of the crack front treatment.3.1. Method selection in DCB testThe three methods, BT, MBT, and the proposed method, treat the crack front as follows:⋅ BT method: Rigid clamp. The lateral movement of the loading point is given by the Bernoulli–Euler beam theory.⋅ MBT method: Rotation is considered. The lateral movement of the loading point is given by the Bernoulli–Euler beam theory, andthe slope at the crack front multiplied by crack length.⋅ Proposed method: Both rotation and lateral movement are considered. The lateral movement of the loading point is given by theBernoulli–Euler beam theory, the slope at the crack front multiplied by the crack length, and lateral movement at the crack front.12Engineering Fracture Mechanics 307 (2024) 110320T. Morimoto et al.Fig. 11. Crack length correlation factor 𝛥 for Modified Beam Theory.Therefore, the proposed method is expected to be less effective than the MBT method when the lateral movement is negligible,and the simple BT method becomes acceptable when the crack-front rotation triggers a negligible effect. Therefore, it is reasonableto select a method that uses two parameters; 𝑅𝑀𝐵𝑇 and 𝑅𝐵𝑇 , as follows:⋅ Selection: Proposed method, when 𝑅𝑀𝐵𝑇 ↛ 0.⋅ Selection: MBT method, when 𝑅𝑀𝐵𝑇 → 0.⋅ Selection: BT method, when 𝑅𝐵𝑇 → 0.𝑅𝑀𝐵𝑇 ≡{𝑓 (𝑙)∕(−𝑙[𝜕𝑓 (𝑥)𝜕𝑥]𝑥=𝑙)}= 1 + 𝛼𝑙(46)13𝛼𝑙(1 + 2𝛼𝑙)Engineering Fracture Mechanics 307 (2024) 110320T. Morimoto et al.Fig. 12. R-Curves.𝑅𝐵𝑇 ≡−𝑙[𝜕𝑓 (𝑥)𝜕𝑥]𝑥=𝑙𝑃𝐶 𝑙3∕(3𝐸𝐵𝐼)=3(1 + 2𝛼𝑙)2(𝛼𝑙)2(47)where the lateral movement at the crack front is expressed by Eq. (15) as 𝑓 (𝑙), the rotation at the crack front is given in Eq. (17)as[𝜕𝑓 (𝑥)𝜕𝑥]𝑥=𝑙, and 𝑃𝐶 𝑙3∕(3𝐸𝐵𝐼) is the lateral bending in Bernoulli–Euler beam theory.Fig. 13 depicts the case when 𝑅𝑀𝐵𝑇 and 𝑅𝐵𝑇 are 0.1, where 𝑓 (𝑙) provides 10% level error over −𝑙[𝜕𝑓 (𝑥)𝜕𝑥]𝑥=𝑙and −𝑙[𝜕𝑓 (𝑥)𝜕𝑥]𝑥=𝑙over 𝑃 𝑙3∕(3𝐸 𝐼), yielding 𝛼𝑙 ≈ 5.42 and 𝛼𝑙 ≈ 30.5. In this experiment, 1∕𝛼 ranged from 5.11 to 10.1, as summarized in Table 2;14𝐶 𝐵Engineering Fracture Mechanics 307 (2024) 110320T. Morimoto et al.Table 2Experimental results.Sample No. Energy Release Rate (kJ/m2) CF(mm) 𝐸𝐵 (GPa) 𝑡𝐷𝐶𝐵Average S.D.(×10−1) 1∕𝛼(𝑇 𝑜𝑝), |𝛥|(Middle) (mm)1 Proposed Method 1.35 1.02 7.92 118 0.611MBT(ASTM D5528) 1.21 0.606 17.7 118 –BT(JIS K7086) 1.42 1.89 – – –2 Proposed Method 1.36 1.09 8.23 128 0.571MBT(ASTM D5528) 1.21 0.858 19.1 134 –BT(JIS K7086) 1.44 1.32 – – –3 Proposed Method 1.38 1.05 10.1 136 0.649MBT(ASTM D5528) 1.23 0.521 20.7 135 –BT (JIS K7086) 1.89 1.45 – – –4 Proposed Method 1.26 3.20 5.23 111 0.536MBT(ASTM D5528) 1.21 1.92 14.6 113 –BT(JIS K7086) 1.87 1.62 – – –5 Proposed Method 1.45 1.08 5.11 113 0.631MBT(ASTM D5528) 1.23 0.868 19.8 133 –BT(JIS K7086) 1.52 1.92 – – –(S.D. of Sample Averages)Average Proposed Method 1.36 0.610 7.32 121 0.600of MBT(ASTM D5528) 1.22 0.0970 18.4 127 –1 to 5 BT(JIS K7086) 1.63 2.09 – – –Fig. 13. DCB test method selection parameters.therefore, the proposed method is recommended when the crack length 𝑙 < 54.7(mm) and MBT is recommended over BT when𝑙 < 308(mm), which is true because of the limitation of the specimen length, that is, 150 mm.Therefore, the existing BT or MBT methods provide an acceptable option when the limitation in the crack length 𝑙 is not an issuein DCB testing; however, the proposed method provides a reasonable option when 𝑙 is limited owing to engineering requirements,such as DCB testing for the crack initiation stage and assessment of nondestructive testing if the detection limit of 𝑙 is acceptable.4. ConclusionsWe propose a modified DCB test method to consider the infinitesimal deformation of a slender beam on Winkler’s elasticfoundation, assuming that the beam deformation is predictable using the Bernoulli–Euler beam model. Moreover, the crack-enddeformation was determined using both the slope and lateral movement of Winkler’s elastic foundation, yielding an energy releaserate with a foundation-based correlation factor for the crack length.The modified method was assessed based on the DCB test results of Ti6Al4V samples, showing reasonable consistency, particularlyfor the short cracking of the nucleation and growth stages of the energy release rate.15Engineering Fracture Mechanics 307 (2024) 110320T. Morimoto et al.oCRediT authorship contribution statementTetsuya Morimoto: Writing – original draft, Visualization, Validation, Supervision, Resources, Project administration, Method-logy, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization. Hisaya Katoh: Methodology, In-vestigation. Yuichi Ishida: Methodology, Investigation. Eiichi Hara: Methodology, Investigation. Masahiro Kusano: Validation,Investigation, Funding acquisition, Conceptualization. Kimiyoshi Naito: Investigation, Funding acquisition, Conceptualization.Makoto Watanabe: Funding acquisition, Conceptualization. Kiyoka Takagi: Project administration, Funding acquisition, Conceptu-alization. Keiji Arai: Project administration, Funding acquisition, Conceptualization. Koichi Hasegawa:Methodology, Investigation,Conceptualization.Declaration of competing interestThe authors declare the following financial interests/personal relationships which may be considered as potential competinginterests: Tetsuya Morimoto reports financial support was provided by Acquisition Technology and Logistics Agency. If there areother authors, they declare that they have no known competing financial interests or personal relationships that could have appearedto influence the work reported in this paper.Data availabilityData will be made available on request.AcknowledgmentsThis study was supported by the Innovative Science and Technology Initiative for Security Grant Number JPJ004596, ATLAJapan. We would like to thank Editage (www.editage.jp) for the English language editing.Appendix A. Energy release rate of extreme-long hinge loading arm caseFor the extreme case of the hinge loading arm 𝐿 as 𝐿 → ∞, Eq. (31) can be simplified as follows:𝐺𝐷𝐶𝐵,𝐿→∞ =𝑃𝐶24𝑤𝐸𝐵𝐼( 1𝛼+ 𝑙)2(A.1)The case in Eq. (A.1) is equivalent to a DCB test with the crack-tip rotation[𝜕𝑓𝐶 (𝑥)𝜕𝑥]𝑥=0constrained to zero, which is realized forhalf of the center-cracked double beam, as shown in Fig. A.1 or by using rotation-constraining gripping, as shown in Fig. A.2.In addition, Eq. (34) is simplified as follows.𝜆(𝑙𝑖)𝐷𝐶𝐵,𝐿→∞ = 13𝐸𝐵𝐼[{2( 1𝛼+ 𝑙𝑖)3+ 1𝛼3}− 32( 1𝛼+ 𝑙𝑖)3+ 𝐸𝑟𝑓]≈𝑙𝑖33𝐸𝐵𝐼{12(1 + 1𝛼𝑙𝑖)3+(1𝛼𝑙𝑖)3}≈1𝑙𝑖36𝐸𝐵𝐼(1 + 3𝛼𝑙𝑖)(A.2)Therefore, coefficients 1∕𝛼 and 1∕(𝐸𝐵𝐼) are obtained as follows:⎧⎪⎪⎨⎪⎪⎩1𝛼=𝛴𝑙𝑖6𝛴𝜆𝑖𝑙𝑖2 − 𝛴𝑙𝑖5𝛴𝜆𝑖𝑙𝑖33(𝛴𝑙𝑖4𝛴𝜆𝑖𝑙𝑖3 − 𝛴𝑙𝑖5𝜆𝑖𝑙𝑖2)1𝐸𝐵𝐼=6(𝛴𝑙𝑖4𝛴𝜆𝑖𝑙𝑖3 − 𝛴𝑙𝑖5𝛴𝜆𝑖𝑙𝑖2)(𝛴𝑙𝑖4𝛴𝑙𝑖6 − 𝛴𝑙𝑖5𝛴𝑙𝑖5).(A.3)The energy release rates 𝐺𝐷𝐶𝐵,𝐿→∞ were derived by substituting 𝐸𝐵𝐼 and 1∕𝛼 into Eq. (A.1).Appendix B. Crack inspection in DCB testIn many cases, GVI provides acceptable accuracy in measuring the length of stable cracks in temporal unloading in JIS K7086,if needed, with the aid of increasing the visual contrast of dark cracks by white coating of the specimen sides, as depicted in Fig. 9or using a microscope as shown in Fig. B.1.When temporal unloading is not accepted, the crack gauge provides an option for DVI of the crack length in real time. Crackgauges were set on the two sides of the DCB specimen to provide the average crack length, as depicted in Fig. B.2. 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Energy Release Rate of Extreme-Long Hinge Loading Arm Case Appendix B. Crack Inspection in DCB Test References