Equivalent Energy Release Rate And Crack Stability In The End Notched Flexure With Inserted Roller Mixed Mode I/II Test

- 1 - Equivalent energy release rate and crack stability in the End Notched Flexure with inserted roller mixed mode I/II test A. Boyano*a, J. De Graciaa, A. Arreseb, F. Mujikab a,b Materials + Technologies Group, Department of Mechanical Engineering a Faculty of Engineering of Vitoria-Gasteiz, University of the Basque Country (UPV/EHU) Nieves Cano, 12, 01006 Vitoria-Gasteiz, Spain *e-mail: [email protected] // Tel: +34 945 013933 b Faculty of Engineering of Gipuzkoa - University of the Basque Country (UPV/EHU) Plaza de Europa, 1, 20018 San Sebastian, Spain Abstract The crack propagation performance of the End Notched Flexure with inserted roller test is analyzed. An equivalent energy release rate is proposed taking into account the interaction of the modes I and II, based on the linear failure criterion. Experimental results obtained with specimens of F593/T300 carbon/epoxy unidirectional composite show a good agreement with the proposed approach. The stability condition is theoretically developed based on the derivative of the equivalent energy release rate, under fixed load condition and fixed displacement condition. Experimental tests have been carried out to assess the proposed equivalent energy release rate and to evaluate the crack stability condition. An analysis of the influence of compliance measurement on the crack length and on the equivalent energy release rate is included. Based on the results obtained, test conditions with initial mode ratios between 65% and 75% are recommended. Keywords: Composites; Delamination; Mixed-mode; Energy Release Rate; Crack Stability. This is the accept manuscript of the following article that appeared in final form in Theoretical and Applied Fracture Mechanics 87: 99-109 (2017), which has been published in final form at https://doi.org/10.1016/j.tafmec.2016.11.001. © 2016 Elsevier Ltd. under CC BY-NC-ND licence (https://creativecommons.org/ licenses/by-nc-nd/4.0/) - 2 - 1 INTRODUCTION One of the main objectives in fracture mechanics is to measure the fracture toughness of materials. It is supposed that a stable crack growth is required for reliable energy release rate curves determination. Different type of test configurations have been used to investigate the mixed-mode I/II fracture of different materials, such as composite laminates [1-3], wood [4-6], or adhesively bonded joints [7-11]. However, the stability condition of the mixed-mode test is not defined in the standard. The only mention done is that in the high mode II regime, the delamination growth is often unstable, precluding propagation toughness values from being determined. It is also stated that the use of longer initial delaminations increases the tendency for stable delamination growth [12]. With respect to pure modes I and II, the Double Cantilever Beam test (DCB) is considered always stable [13], and the End Notched Flexure test (ENF) is stable if a/L>0.7 [14,15]. Phillips and Wells [16], studied the stability of transverse cracks in composites. They assumed as stability limit the derivative of stress with respect to the crack length obtaining a critical value. If the initial crack is less than the critical one, the crack propagates unstably at a stress value determined by the initial fracture energy and it grows spontaneously under decreasing load. On the contrary, cracks which are initially larger than the critical size are stable and only grow if the load increases. Allix and Corigliano [17] dealt with the problem of simulating the mixed-mode I/II crack propagation. They compared the analytical results from the Linear Elastic Fracture Mechanics (LEFM) hypotheses and numerical results concerning crack stability of the Asymmetric End Loaded Split (AELS) specimen. The analytical results stated that under load control it was always unstable and under displacement control the stability condition was a/L>0.42. By means of the nonlinear numerical model proposed, the limit for stable crack propagation under displacement control was a/L =0.435 which is very close to the analytical result. Szekrenyes [18], verified the traditional compliance based criterion, based on the derivative of the energy release rate (ERR) by experimental observations. He used transparent material to visually measure crack length and crack propagation. The transition from stability to instability was defined as the point just before a crack jump. He found that the stability of the system depends on the derivative of the critical displacement defined as the displacement of the load application point at crack initiation. The relationship between the critical displacement and the crack length was determined by experiments in many specimens of different initial crack lengths, for different test methods. The critical value of crack length was determined by differentiation after curve fitting of experimental data. He deduced that the point of instability was always where the critical displacement reached its minimum value. - 3 - In the present work the crack propagation performance is analyzed in a mixed mode I/II End Notched Flexure specimen with inserted roller (ENFR), recently proposed [19,20]. Taking into account the interaction between the two modes, an equivalent energy release (ERR) concept has been developed and the stability condition has been determined by means of the derivative of that equivalent energy release rate. An error analysis has been carried out in order to determine the influence of compliance measurement on the crack length and on the equivalent ERR. Analyzing the experimental data during the propagation, the proposed equivalent ERR has been assessed and the crack stability has been evaluated. Optimum test conditions for tests performed under displacement control have been proposed. - 4 - NOMENCLATURE a,  a crack length and crack increment, respectively ci distances from the support to the position of the roller b,2h width and thickness of the specimen, respectively Cspec, Cs, Cexp, compliance of the specimen, of the system and experimental compliance, respectively kS stiffness of the system  0, P0 initial displacement and initial load, respectively  displacement of the middle point of the specimen  spec,  exp calculated and experimental displacement of the middle point of the specimen, respectively Ef flexural modulus GLT, GLT’ in-plane shear modulus and out of plane shear modulus, respectively E11, E22, G12 longitudinal elastic modulus, transversal elastic modulus and shear elastic modulus respectively GI,GII , G energy release rates for mode I, mode II, and total, respectively GIc,GIIc , Gc critical energy release rate for mode I, mode II, and total, respectively Geq,Geqc equivalent energy release rate and critical equivalent energy release rate, respectively KCa, KaG error coefficients that correspond to the influence of the measurement of compliance on the crack length, and the influence of the determination of crack length on equivalent energy release rate, respectively. L half span of the test P applied Load Y force exerted by the roller R roller radius W, U, U* work done by external forces, strain energy, and complementary strain energy, respectively - 5 - 2 ANALYTICAL BACKGROUND 2.1 Compliance of ENFR test The End Notched Flexure with inserted Roller (ENFR) test configuration has been proposed [19] and experimentally assesed [20] recently. In order to get mixed mode, a roller is introduced between the two surfaces of the crack and the specimen is tested in ENF configuration. The mode II is provided by the external load and the mode I is obtained by the opening of the crack due to the insertion of the roller as shown in Fig. 1. The roller can be located at the outer side or at the inner side of the support. Fig. 1 ENFR Test configuration with the roller located at the inner side Being Y the force exerted by the roller and being it the force that generates the displacement, taking into account only bending effects, when the roller is located at the inner side of the support, the Y force is [19]: (1) 3 3 (2 ) 4 ( ) 8( ) fbh R P a ac a Ec Yc      Where R is the radius of the roller; c is the distance from the support where it is located; a is the crack length; P is the applied load; Ef the flexural modulus; b the width of the specimen; and 2h the total thickness of the specimen. The displacement of the load application point taking into account only bending effects is [19]: (2) 3 3 3 2 3 (2 ) (3 2 3 ) 8 4 ( ) f P R a c a L c ac bh aE c         - 6 - According to Eq.(2), due to the roller introduced between the specimen arms, there is an initial negative displacement without external load application. Furthermore, when the displacement is null, there is a positive load. The initial conditions can be calculated replacing P=0 and  =0 in Eq.(2) and lead to [19]: (3) 0 3 3 3 2 0 3 (2 ) 4 ( ) 2 (2 ) ( )(3 2 3 ) f R a c a c R bh a c a c a c ac E L P            The theoretical load-displacement curve before crack propagation occurs is depicted in Fig. 2.   P0 P  P Fig. 2 Load-displacement curve for ENFR test before crack propagation According to Fig. 2 the compliance of the ENFR test is [19]: (4) 3 3 3 2 0 3 0 0 3 2 3 8f CP a L c a bhP E c P          2.2 Energy release rate in the ENFR test According to Griffith [21] and Irwin [22], the energy balance in a small crack advance can be expressed as: (5) dW dU Gbda  Where dW is the work done by external forces; dU is the change in strain energy; G is the energy needed for the crack advance per unit area; b is the width of the crack; and da is the differential crack advance. The differential work done by the external forces Fi in their respective displacements  i, assuming the repeated index convention is . The complementary strain energy U* is defined as: i i dW Fd   (6) * i i U F U    Differentiating Eq.(6), and replacing in the energy balance equation of Eq.(5) it results: (7) * i i dU dF Gbda    - 7 - In spite of the crack advance is an irreversible process, it is assumed that an elemental variation the complementary strain energy is an exact differential. Since the state variables are Fi and a, thus: (8) * * * i i U U dU dF da F a       Identifying the first terms in Eq.(7) and Eq.(8) it results the theorem of Engesser-Castigliano. Identifying the second summands it results: (9) * 1 i F cte U Gb a         In the ENFR test the work is done by two forces, Y and P. The work carried out by Y is related to the finite displacement imposed by the roller, and the work carried out by P is related to the application of the load. According to Eq.(9) the complementary strain energy must be used for determining G if forces are used as state variables. When the roller is positioned at the inner side of the crack tip, taking into account only bending effects, the energy release due to each fracture mode can be expressed as follows [19]: (10) 2 3 2 2 4 2 2 3 2 2 2 3 33 3 4( ) 4 ( ) 16 9 16 f I f II f R E h PRc P c Ga c b a c E b h P a GE b h       The values of GI and GII of Eq. (10) agree with those obtained by William´s partition method [23]. In the case that c=0 and taking into account only bending effects, they agree with those obtained by Szekrenyes [24] for this test configuration. In many cases G is determined based on the compliance, using the Irwin-Kies approach [25]. In the present case, the calculation of G based on the derivative of the compliance is: (11) 2 2 2 2 2 2 3 2 3 9 3 2 16 16 f f P dC P a P c Gb da E b h E b h    In Eq. (11) two summands corresponding mode I of Eq. (10) are not included. Then, in the present case the approach based on the compliance is not valid. It can be concluded that it is also valid in cases where the external work is carried out by a unique force. - 8 - 3 EQUIVALENT ENERGY RELEASE RATE Crack propagation only occurs if the energy available for a crack extension, G, is enough to provide all the energy that is required for crack growth [26]. The energy required for crack growth is called critical energy release rate Gc. For pure modes, the critical condition for crack propagation is: (12)   I Ic II IIc G G G G However, when mode I and mode II are involved, the energy release rate is . Then, the   I II G G G critical value of the energy release rate Gc when crack propagates depend on the mode ratio GII/G. For instance, in a mixed mode test with a great mode ratio, Gc must be close to GIIc and when the mode ratio is low Gc must be close to GIc. Thus, in spite of GIc and GIIc being material properties, Gc is not. In the previous study concerning the experimental assessment of ENFR [20], it was found that the linear criterion is suitable for representing the crack propagation in an ENFR test, where mode ratio varies during the test. Therefore, the condition for the crack propagation can be defined as: (13) 1 I II Ic IIc G G G G   Expressing Eq.(13) in similar form to Eq.(12), the crack propagation condition leads to: (14)   I IIc II Ic Ic IIc G G G G G G The left member of Eq. (12) is the energy available for a crack extension and the right member is the critical value for crack advance. Comparing Eq.(12) and Eq.(14) an equivalent energy release rate Geq and an equivalent critical value Geqc are defined as: (15)        1/2 1/2 eq I IIc II Ic eqc Ic IIc G G G G G G G G Therefore, it can be stated then that crack propagates when: (16)  eq eqc G G To the best knowledge of the authors, equivalent values of energy release rate given in Eq. (15) have not been previously defined. - 9 - 4 ANALYSIS OF CRACK STABILITY 4.1 Stability definition A general definition for stability is that a system is stable if a finite change in the input parameters does not cause an infinite change in the output values [27]. Regarding fracture tests carried out in universal testing machines, the input parameters are the load or the displacement applied by the testing machine and the output value is the crack length. Therefore, a fracture test is stable during crack growth when an infinitesimal change of the load or the displacement does not cause an infinite change in the crack length. According to Schwalbe et al [28], a crack extension under displacement control is stable when the crack stops when the applied displacement is held constant. In the present study, the stability condition is applied to the equivalent energy release rate Geq: (17)   eq eqc eq eqc dG dG and G G da da Assuming that Geqc is constant during crack growth, the stability criterion from Eq.(17) simplifies to: (18) 0 eq dG da  Replacing Eq. (15) in the condition given in Eq.(18), the condition for stability can be expressed as follows: (19)   1/2 10 2 eq I II I IIc II Ic IIc Ic dG dG dG G G G G G G da da da           If the result of Eq.(19) is positive, then the crack growth is unstable, because the energy released is more than that needed to create a new surface area. If it is negative external work must be done to keep the crack moving. It is worth noting that the equivalent energy release rate proposed is based on the fulfillment of the linear criterion. If that criterion is not suitable the present stability analisis is not valid. 4.2 Fixed load condition Considering that GI = GI (a,P)and GII = GII (a,P), the total derivative of the ERR due to each mode fracture with respect to the crack length, can be expressed as follows: - 16 - 0 0.5 1 1.5 2 20 30 40 50 60 a (mm) KaG ENF R0.5-c0 R0.5-c10 R1.5-c0 Fig. 8 Influence of crack length relative error on the relative error of Geq Fig. 8 shows the absolute value of the coefficient KaG for different test conditions. In the ENF test, KaG =1 for any crack length since in Eq.(28) GI=0. In the case of R0.5-c0, when the crack length a>24mm the absolute value of KaG is below 1. Moreover, there is a minimum value when a=30mm. In the case of R0.5-c10, the curve has a similar shape but the crack length should be a>39mm to get a value of KaG below 1, and the minimum value corresponds to a=45mm. In the case of R1.5-c10 when a>43mm it gives an absolute value of KaG below 1, being the minimum value when a=52mm. Therefore when R or c increase, the minimum crack length to get KaG<1 increases. According to Eq.(28), it is possible to analyze directly the influence of the relative input error of the compliance in the relative output error of Geq as it follows (29) eq aG Ca a eq a GC K K G C                 Fig. 9 Influence of compliance relative error on the relative error of Geq - 17 - Fig. 9 shows the absolute value of the product of two coefficients KaG KCa. The absolute values of the product KaG KCa. lower than 1 take place in similar crack lengths to those obtained in the analysis of KaG. In order to determine an optimum test condition, the product of KaG KCa should be taken into account. Fig. 10 Absolute value of KaG KCa versus mode ratio GII/G Fig. 10 shows the absolute value of KaG KCa versus mode ratio GII/G. When the mode ratio is greater, which is related to a longer crack length for fixed vaues of c, R, the absolute value of KaG KCa is lower. Thus it means that the relative error of Geq is reduced with respect to the relative error of the experimental compliance. Table 2 summarizes the optimum crack lengths depending on the different criteria. Table 2 Optimum crack lengths in mm. STABILITY CONDITION ERROR ANALYSIS TEST NOMENCLATURE Load Control Displacement control KCa<1 KaG<1 KaG KCa<1 ENF Not stable a > 42 a > 42 KaG=1 a > 42 R0.5-c0 a < 30 stable a > 42 a > 24 a > 27 R0.5-c10 a < 45 stable a > 42 a > 38 a > 39 R1.5-c0 a < 51 stable a > 42 a > 43 a > 43 In order to define optimum test conditions, the crack length has to satisfy the stability condition and to minimize the relative error in the determination of Geq. - 18 - 6 EXPERIMENTAL 6.1 Materials and test apparatus T6T/F593 prepregs provided by Hexcel Composites with a 55% volume-content of fibre were used to produce laminates. The plates were manufactured by hot press molding. Sixteen-layered unidirectional laminates, [0]16, were made with a Teflon film introduced centered during the piling up process in order to make the initial crack. The specimens were cut with a diamond disc saw, being the nominal thickness and width 3 mm and 15 mm, respectively. The edges of the laminate were discarded for the preparation of the specimens. Tests were performed on an MTS-Insight 100 electromechanical testing machine equipped with a 5kN load cell, operating in a displacement controlled mode. In order to avoid the influence of the resin rich area the specimens were precracked in mode II by a ENF test, increasing the cracked length around 5 mm. All the specimens were tested using a procedure based on three-point bending tests at five different spans proposed by Mujika [29], in order to obtain the flexural modulus, Ef and the out of plane shear modulus GLT’, which is equal to the in-of-plane shear modulus GLT assuming that the material is transversely isotropic. The mean values corresponding to five specimens were: Ef = 107.4 (±1.4) GPa GLT’ = 4.3 (±0.4) GPa 6.2 Experimental test conditions Several mixed mode tests have been performed in order to compare the results with the ones obtained applying the theoretical stability condition. The experimental displacement is defined as 0 when the contact between the load nose and the specimen without roller occurs, as it is shown in Fig. 11. After inserting the roller, there is an initial negative displacement  0 for the zero load condition, as seen in Fig. 2. Actually, the contact in the testing machine has been defined when the load was 0.5 N. - 19 - Fig. 11 Initial Conditions in Load-Displacement curve The determination of the crack length at every point where P and  are measured, is based on the variation of the compliance during the crack advance, based on the Beam Theory including Bending Rotation effects (BTBR) method developed for the ENF test by Arrese et al [30]. For the experimental analysis bending rotation effects have not been included because the support roller radii are 2.5mm and thus the influence is negligible [30]. In a previous work [20], it was verified that in the ENFR test the crack length can be determined without optical methods. In spite of only bending effects have been presented for simplicity in the analytical background, the calculations concerning the experimental part have been carried out including also shear effects. Since the contribution of each fracture mode varies during the crack propagation, the mode ratio at the initiation point is the parameter chosen to define the type of mixed mode test. To determine that initial mode ratio, the value of energy release rate for each mode contribution has been determined substituting the value of the crack length in Eq.(10), when a=0.25mm, considering it as nonlinearity point (NL). The NL point is one of the definitions of crack initiation presented in the standard [12]. The nomenclature that has been used in order to identify each test condition is ai-Rj-ck. ai is the nominal initial crack length; Rj is the inserted roller radius; and ck is the value of the c distance that defines the position of the roller. For instance, a40-R1-c8 is the test with initial nominal crack length of 40 mm, an inserted roller of 1mm radius and positioned at 8mm at the inner side of the support. The nominal geometric dimensions are span 2L=120mm, width b=15mm and thickness 2h=3mm. - 20 - All the test conditions are summarized in Table 3. For simplicity, a number will be used in the legends of the graphics. Table 3 Summary of experimental test conditions ID NOMENCLATURE INITIAL GII/G (%) Test 1 a40-R0.5-c0 96 Test 2 a31-R0.5-c0 90 Test 3 a45-R1.5-c0 74 Test 4 a43-R0.9-c8 66 Test 5 a38-R0.7-c8 59 6.3 Results and discussion 6.3.1 Energy release rate curves The determination of the crack length at any point of the test where P and  are measured, allows the calculation of the energy release rate at any point during the crack propagation. The experimental values of GI and GII of the tests presented in Table 3, have been obtained substituting the value of the crack length, in Eq.(10). In Fig. 12 the experimental energy release rate curves due to each mode can be seen. In the legend of the figures besides the identifier of the test the initial mode ratio is included in brackets. 0 2 4 6 8 10 12 200 400 600 800 1000 Test 1 (96%) Test 2 (90%) Test 3 (74%) Test 4 (66%) Test 5 (59%) Crack advance (mm) GI or GII (J/m2) Upper Curves GII Lower Curves GI - 21 - Fig. 12 Experimental Energy Release Curves As it can be seen in Fig. 12, the upper curves correspond to GII, and the lower ones to GI. The curve of GI and the curve of GII that correspond to the same test condition are drawn with the same color. When crack propagates, the curves of GI decrease slowly. At the same time, the curves for GII increase. This means that the contribution of each mode is changing during crack propagation. In Fig. 13 the total energy release rate G= GI + GII is depicted. 0 2 4 6 8 10 12 0 200 400 600 800 1000 Test 1 (96%) Test 2 (90%) Test 3 (74%) Test 4 (66%) Test 5 (59%) Crack advance (mm) G=GI+GII (J/m2) Fig. 13 Experimental total Energy Release Rate When the mode II is predominant the total energy release rate tends to a constant value. When the mode I is more important the total G increases with crack advance. Those trends agree qualitatively with the results obtained in R-curves of the pure modes concerning ENF and DCB tests, respectively: a plateau in ENF tests [30] and the increase of G in DCB tests for the same material [31]. 6.3.2 Linear criterion, normalized mode ratios and equivalent energy release rate curves In the previous study concerning the experimental assessment of ENFR [20], it was found that the linear criterion of Eq.(23) is suitable for representing the crack propagation in an ENFR test, where mode ratio varies during the test. The value of GIIc is four times GIc in the case of the material used in this study as it can be seen in Table 1. Therefore, in order to analyze the direct contribution of each mode to failure, instead of GI and GII, normalized mode ratios have been defined [20]: - 22 - (30) 0 0   I II I II Ic IIc G G G G G G These normalized mode ratios of Eq.(30) are the summands of the linear criterion. The initial values of the total and the normalized mode ratios given in Eq.(30) are shown in Table 4. Table 4 Initial mode ratio and initial normalized mode ratio ID INITIAL MODE RATIO (GII/G) (%) INITIAL NORMALIZED MODE RATIO (GII0) (%) Test 1 96 51 Test 2 90 45 Test 3 74 37 Test 4 66 30 Test 5 59 22 The test with the highest initial mode ratio and the one with the lowest mode ratio have been plotted in Fig. 14 in terms of the normalized mode ratios. 0246810 0 0.2 0.4 0.6 0.8 1 GI0 Test 1 (96%) GII0 Test 1 (96%) GI0 Test 5 (59%) GII0 Test 5 (59%) Crack advance(mm) GI0 and GII0 Fig. 14 Experimental normalized mode ratios GI0and GII0 When the mode II is clearly predominant the GII0 curve is above the GI0 curve, as in Fig. 12. Nevertheless, when the initial mode ratio is 59%, the curve of GI0 is above the curve of GII0 during almost all the propagation, in contrast to what happened with GII and GI. As linear criterion is adopted, the sum of the two normalized mode ratios GI0+GII0 depicted in Fig. 15, should be close to 1. - 23 - 0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1 Test 1 (96%) Test 2 (90%) Test 3 (74%) Test 4 (66%) Test 5 (59%) Crack advance(mm) GI0 + GII0 Fig. 15 Sum of experimental normalized mode ratios The mean value of the curves during propagation is always higher than 0.85, and in the case of tests 3 and 4 is very close to 1. The maximum deviation occurs in the tests with mode ratio near to pure mode II, tests 1 and 2, and the maximum relative error is 15% with respect to the linear criterion. 0 2 4 6 8 10 100 200 300 400 500 600 Test 1 (96%) Test 2 (90%) Test 3 (74%) Test 4 (66%) Test 5 (59%) Crack advance (mm) Geq (J/m2) Fig. 16 Equivalent ERR, experimental Geq The proposed equivalent energy release rate of Eq. (15) has been applied to the experimental data and the results are presented in Fig. 16. In all the cases the equivalent ERR tends to a constant value. The mean values of the curves during propagation are between 505 and 555 (J/m2). The value of the plateau agrees with the theoretical value of GCeq=550 (J/m2) of Table 1. The maximum deviation corresponds to the tests 1 and 2 and in those cases the maximum relative error is 8.5% with respect to the theoretical - 24 - value of GCeq.Consequently, according to the results shown in Fig. 16, the proposed equivalent ERR approach can be considered suitable for defining crack propagation condition of the ENFR test. 6.3.3 Stability evaluation During the testing phase, it has been assumed that unstable crack growth occurs when there is a jump in the load-displacement curve. In Fig. 17 the load-displacement curves of the tests presented in Table 3 are presented. The circular marker included in figures indicates the crack inititation point or the NL point. 0 1 2 3 4 5 0 50 100 150 200 250 300 350 Displacement (mm) Load (N) Test 1 (96%) Test 2 (90%) Test 3 (74%) Test 4 (66%) Test 5 (59%) Fig. 17 Experimental load-displacement curves When the initial mode ratio is close to pure mode II, tests 1 and 2, when the crack starts to propagate the load continues increasing, which means that it is behaving in a stable manner. After the load reaches its maximum value it decreases suddenly. Besides, there are jumps in both curves. Therefore the tests 1 and 2 can be considered stable during the most part of the propagation, and at the final stage they become unstable. In the rest of the cases, there are not jumps or sudden load drops during crack propagation. It can be concluded then, that they are all stable. The definition of stability of section 4.1, which defines a test as stable if an infinitesimal change of displacement, does not cause an infinite change in the crack length, has been taken into account to analyze the experimental data. In Fig. 18 the relationship between crack length and the displacement is depicted. - 25 - 012345 25 30 35 40 45 50 55 60 Displacement (mm) a (mm) Test 1 (96%) Test 2 (90%) Test 3 (74%) Test 4 (66%) Test 5 (59%) Fig. 18 Crack length versus displacement In the curves of tests 1 and 2 at the first stage of the propagation, a finite change in displacement causes a finite change in the crack length and thus they can be considered stable. However, at the final stage of the propagation a small change in displacement causes a much greater change in crack length, and there are even jumps in the curve. Therefore, they become unstable. For the rest of the tests, the crack propagation starts earlier, and the slope of the curve is similar, which means that a finite change in the displacement is needed to have a finite change in the crack length. Consequently, they are all stable according to this definition. This is the same conclusion drawn by analyzing P-  curves. Since the experimental tests have been carried out at a constant displacement rate which means under displacement control, theoretical stability condition based on the derivative of Geq,of Eq. (26) is applied to the experimental data.