Impact of elastic modulus of rock on the fracture toughness take a look at
For the reason that NSCB configuration beneficial by ISRM for testing OkayIc has no express requirement for materials properties of rock, the NSCB configuration might be utilized to any rock supplies by default. Apart from, the existence of friction between the specimen floor and the curler help isn’t thought-about within the advice. Subsequently, it’s needed to debate the affect of the change of fabric properties on the fracture toughness take a look at. On this examine, 4 sorts of rock supplies with giant variations in elastic modulus are chosen for evaluation. Desk 2 reveals the fundamental materials parameters of the 4 rocks and their values within the numerical mannequin. For the reason that distinction in Poisson’s ratio of the 4 rock supplies isn’t important, the affect of Poisson’s ratio on the fracture toughness take a look at of the rock mass isn’t thought-about within the presence of friction.
As talked about above, the friction coefficient of the three placement strategies of the curler help for rock specimens with the NSCB configuration is inside 0.327, and the friction coefficient is managed inside this vary within the later fashions and evaluation. The load utilized by the numerical mannequin is 2000 N. Along with the above materials parameters and hundreds, different parameters enter within the numerical mannequin (comparable to specimen measurement and grid sort) are the identical as these of the numerical mannequin of rock specimen with the NSCB configuration for mannequin verification in Methods Part. It’s price noting that the impact of thickness or crack entrance size can have an effect on the stress state as acknowledged in some earlier research51,52. Nevertheless, the variation legislation of stress depth issue with friction coefficient is just centered on this examine, and the two-dimensional evaluation is just carried out. As proven in Fig. 4, the OkayI decreases considerably with the rise of friction coefficient (as noticed in Methods Part). On the similar time, even when there’s friction between the specimen floor and the curler help, the change of elastic modulus has little impact on the OkayI. That is impartial of the small load utilized within the mannequin; actually, the load utilized within the mannequin is way bigger than the failure power of the specimen. Rock specimens with the NSCB configuration are simply broken below small load, and the crack opening displacement of the specimen with totally different materials properties modifications little, and the load utilized at specimen is remodeled into the vertical response pressure of the curler help on the specimen floor. In different phrases, the vertical response pressure of the curler help on the specimen floor hardly ever modifications with the numerous modifications in materials properties. Subsequently, the change of rock materials properties might be ignored when contemplating the affect of friction on the take a look at outcomes of OkayIc for rock specimens with the NSCB configuration.
Impact of various load ranges on the fracture toughness take a look at
Usually, the friction pressure exerted by the curler help on the specimen is expounded to the friction coefficient and the vertical response pressure of the curler help on the specimen floor. The vertical response pressure of the curler help on the specimen floor is intently associated to the load degree utilized to the specimen. On this examine, 4 load ranges are set (i.e. 500 N, 1000 N, 1500 N and 2000 N) to discover the consequences of various load ranges on the OkayI of rock specimens with the NSCB configuration below the motion of friction. The diorite is used on this take a look at and the fabric parameters of the diorite in “Part Effect of elastic modulus of rock on the fracture toughness test” are chosen to determine the mannequin. The diorite is chosen for a number of causes: (1) the fabric properties are impartial of the dedication of OkayI; (2) diorite has a big elasticity modulus and small deformation within the mannequin, the collection of diorite can keep away from the troublesome convergence of the mannequin after the incidence of enormous deformation. Different parameters enter into the numerical mannequin are the identical as these of the numerical mannequin of rock specimen with the NSCB configuration established in Methods Part.
As a result of friction between the curler help and the specimen floor, the crack opening is proscribed, and the OkayI of the specimen is smaller than that of the specimen within the friction-free situation. As proven in Fig. 5, the OkayI in friction-free situation (Okay0I) is taken as a reference, and the distinction between the Okay0I and the OkayI (ΔOkay) below totally different load ranges modifications with the friction coefficient. On the similar load degree, the ΔOkay will increase linearly with the rise of friction coefficient, which is according to the ends in the earlier examine. On the similar time, below the identical friction coefficient, the ΔOkay will increase with the rise of load degree. That is primarily as a result of the vertical response pressure exerted on the floor of the pattern by curler help will increase with the rise of load degree, which results in the rise of the friction pressure exerted by the curler help on the pattern.
The OkayI can be utilized to explain the stress discipline close to the crack tip, which is the idea of linear elastic fracture mechanics (LEFM). Just for a super elastic materials, when the OkayI of the specimen is bigger than the OkayIc of the fabric, the specimen fails instantly after the crack initiation. For elastic–plastic supplies comparable to rock and concrete, plastic deformation is simple to happen close to the crack tip, however the failure of the specimen is attributable to the subcritical crack propagation after a time frame. Subsequently, the stress state close to the preliminary crack tip can not consequence within the fracture of elastic–plastic supplies immediately. The modified most tangential stress fracture criterion proposed by Smith et al.53 can completely describe and predict the fracture drawback of elastic–plastic supplies. In keeping with this criterion, when the tensile stress on the boundary of the fracture course of zone is bigger than the tensile power of the fabric, the specimen will fracture. It needs to be famous that the elastic–plastic materials has a big measurement of fracture course of zone. Subsequently, in accordance with the utmost tangential stress fracture criterion, the stress distribution within the far-field on the entrance of the crack tip needs to be thought-about within the fracture evaluation of elastic–plastic supplies. Consequently, the variation of the stress discipline distribution on the entrance of the crack tip with the friction coefficient can be price discussing.
Determine 6 reveals the altering pattern of tensile stress on the entrance of the crack tip below totally different friction coefficients and cargo ranges. On the similar time, the tensile stress discipline decided by Okay0I can be drawn on this determine. The tensile stress extracted from the numerical mannequin established below the friction-free situation is sort of equal to the tensile stress decided by Okay0I within the area close to the crack tip. Nevertheless, with the growing distance from the crack tip, the stress discipline distribution decided by Okay0I progressively deviates from that extracted from the numerical mannequin. On the similar load degree, with the rise of friction coefficient, the tensile stress on the entrance of the crack tip decreases progressively; with the rise of the load degree, the lower of the tensile stress on the entrance of the crack tip will increase. In different phrases, in accordance with the utmost tangential stress fracture criterion, a larger load on the specimen might be generated to the failure of the specimen within the fracture take a look at below the situation of a giant friction coefficient, reasonably than below the friction-free situation. At the moment, if the OkayIc is calculated by the components of OkayI derived below the friction-free situation, the take a look at worth of OkayIc will likely be larger than the inherent fracture toughness of the fabric. Consequently, the distinction between the examined OkayIc and the inherent OkayIc will likely be larger for the fabric with giant fracture toughness. As well as, the tensile stress within the space removed from the crack tip doesn’t change considerably because of the change of the friction coefficient. This reveals that the friction pressure between the curler help and the specimen floor has little impact on the high-order time period coefficient of Williams growth39.
Impact of various spans of curler help on the fracture toughness take a look at
The span of curler help is the important thing issue affecting the take a look at outcomes of OkayIc for rock specimens with the NSCB configuration. If the span of curler help is just too small, mode I fracture hardly happens within the specimen. When there’s friction between the curler help and the specimen floor, the affect of the span of curler help on the take a look at outcomes of OkayIc turns into extra complicated. Subsequently, 4 numerical fashions with totally different spans of curler help are established, that’s, the ratio of the half span of curler help to the specimen radius is S/R = 0.2, 0.4, 0.6 and 0.8 respectively. To keep away from the big deformation of the specimen, a load of 1000 N is utilized to the mannequin. Different parameters of the mannequin are the identical as these established in Methods Part.
As proven in Fig. 7, the ΔOkay will increase with the rise of friction coefficient, which is according to the earlier outcomes. In contrast with different spans of curler help, when S/R = 0.2, the change of OkayI with friction coefficient isn’t important. On the similar time, below the identical friction coefficient, the ΔOkay will increase considerably with the rise of S/R from 0.2 to 0.4. Nevertheless, when S/R modifications from 0.4 to 0.8, the ΔOkay doesn’t change. When the identical load is utilized to rock specimens with the NSCB configuration, the smaller the span of curler help, the smaller the angle between the underside finish face of the specimen and the vertical course, the smaller the friction pressure appearing on the specimen floor. Thus, when the span of curler help is small, the friction has little impact on the OkayI, and the ΔOkay hardly modifications when S/R is 0.2. When S/R will increase from 0.4 to 0.8, the angle between the underside finish face of the specimen and the vertical course hardly ever modifications, so the magnitude of the friction pressure is similar. When S/R will increase from 0.2 to 0.4, the friction pressure modifications significantly. In consequence, the discount of OkayI reveals totally different traits with the modifications within the span of curler help.
Determine 8 reveals the change of tensile stress on the entrance of the crack tip with friction coefficient below totally different spans of curler help. When S/R is 0.2, the tensile stress decided by Okay0I could be very near that extracted from the numerical mannequin. With the rise of the span of curler help, the distinction between the 2 will increase progressively. It signifies that the OkayIc examined below the situation of the small span of curler help can higher mirror the inherent fracture traits of the fabric. With the rise of friction coefficient, the tensile stress on the entrance of the crack tip decreases progressively. Totally different from different spans of curler help, below the loading situation of S/R = 0.2, the tensile stress on the crack tip decreases considerably when the friction-free situation is transitioned to the friction situation, however doesn’t change considerably with the rise of friction coefficient. Underneath totally different spans of curler help, the tensile stress within the space removed from the crack tip is much less affected by the friction coefficient. This additional reveals that the friction pressure has little impact on the coefficients of the higher-order phrases of the Williams growth.
The span of curler help of the NSCB configuration beneficial by ISRM for the fracture toughness take a look at is from S/R = 0.5 to S/R = 0.84. It may be seen from Fig. 7 that inside this span of curler help, the affect of friction on the stress depth issue is similar on the similar load degree. In keeping with the speculation of linear elastic fracture mechanics, for the loading situation with a big span of curler help, the rock will fracture at a small load degree, and the affect of friction on the stress depth issue will likely be comparatively small. Subsequently, a bigger span needs to be chosen to hold out the experiment. Nevertheless, contemplating the distinction between the true tensile stress distribution on the crack tip entrance of the specimen and that decided by the stress depth issue, it may be discovered that the smaller the span, the smaller the distinction. Primarily based on the above two features, for rock supplies with excessive power, the big span of curler help needs to be chosen because of the small fracture course of zone size; for rock supplies with low power, the small span of curler help needs to be chosen in accordance with the precise loading situations.
Impact of various friction coefficients on measurement impact of fracture toughness
The scale impact of fracture toughness in rock supplies has all the time been emphasised in rock fracture mechanics54,55,56. As a result of rock specimens with the NSCB configuration are handy in pattern processing and loading take a look at, and the transformation from mode I loading to mode II loading situations might be realized by adjusting the place of the curler help, the NSCB configuration has been broadly used within the examine of measurement impact of the fracture toughness of mode I and mode II in rock. Nevertheless, the impact of friction on the fracture toughness take a look at in rock specimens with the NSCB configuration is never thought-about29,30,31. From the evaluation of the earlier sections, it may be discovered that friction has an important affect on the fracture toughness take a look at. For a similar materials, the OkayIc worth can differ a number of instances below totally different experimental situations of friction coefficient. At current, many research have tried to determine the mathematical relationship between OkayIc and the specimen measurement of the rock29,30,31,57,58,59,60. Nevertheless, it’s not clear how the existence of friction will have an effect on the institution of the mathematical relationship between them. Subsequently, numerical fashions with totally different specimen sizes and totally different friction coefficients are established on this examine.
To extra simulate the true fracture state of affairs of rock, the mode I fracture take a look at knowledge of marble specimens with NSCB configurations in several sizes (out there within the literature57) are employed to determine the numerical mannequin. Desk 3 reveals the detailed take a look at knowledge. It’s price noting that the take a look at knowledge are assumed to be obtained below friction-free loading situations on this examine. Within the presence of friction, the load is utilized on the numerical mannequin till the OkayI is the same as the examined OkayIc in Desk 3, and the load at the moment is substituted into the calculation components of OkayI deduced below the friction-free situation. Then the calculated OkayI is the OkayIc below the corresponding friction coefficient. In keeping with the above steps, the variation legislation of OkayIc with specimen measurement below totally different friction coefficients is obtained. As proven in Fig. 9, below the friction-free situation, the OkayIc will increase with the rise of specimen measurement, and the growing pattern is progressively lowered. Nevertheless, when the specimen measurement will increase from 95 to 190 mm, the OkayIc decreases on the friction coefficient of 0.2 and 0.3. It signifies that when the specimen measurement will increase to a sure extent, the affect of the friction on the OkayIc decreases.
Many research have reported the dimensions impact of OkayIc in rock. To successfully apply the OkayIc examined within the laboratory to the engineering website, the connection between the OkayIc and the specimen measurement have to be established. The scale impact legislation proposed by Bazant et al.61,62 has been broadly used to foretell the OkayIc if the specimen measurement tends to infinity. Within the subsequent half, the influence of various friction coefficients on the institution of Bazant’s measurement impact legislation is mentioned.
Description of Bazant’s measurement impact legislation
In keeping with the LEFM, when a construction is subjected to nominal stress σN, the expression of OkayI is as follows:
$$K_{{textual content{I}}} = sigma_{{textual content{N}}} sqrt L kleft( alpha proper)$$
(1)
the place L is the attribute size (equal to the radius of the specimen with NSCB configuration), and ok(α) denotes dimensionless features. The nominal stress σN of the specimen with NSCB configuration might be outlined by:
$$sigma_{{textual content{N}}} = frac{P}{tR}$$
the place P is the fracture load, and t and R are the thickness and radius of the specimen with the NSCB configuration, respectively. Subsequently, the power launch charge G(α) per unit size of crack propagation might be written as:
$$Gleft( alpha proper) = frac{{K_{{textual content{I}}}^{{2}} }}{E} = frac{{sigma_{{textual content{N}}}^{{2}} L}}{E}gleft( alpha proper)$$
(2)
Right here g(α) = ok(α)2 represents the dimensionless power launch charge. Primarily based on the equal LEFM, when the equal crack tip is at a sure distance c forward of the preliminary crack tip, the utmost a great deal of specimens with varied sizes might be reached. Subsequently, the size of c also needs to be thought-about within the efficient crack size within the calculation of the power launch charge. When the specimen measurement tends to infinity, the corresponding worth of c in measurement impact legislation is denoted as cf., and the next equation might be established:
$$Gleft( {alpha_{0} + c_{{textual content{f}}} /L} proper) = frac{{sigma_{{{textual content{Nu}}}}^{{2}} L}}{E}gleft( {alpha_{0} + c_{{textual content{f}}} /L} proper) = G_{{textual content{f}}}$$
(3)
the place Gf is the fracture power and belongs to the fabric fixed; σNu is the nominal failure stress of the specimen and α0 is the ratio of the size of the prefabricated crack to the attribute size of the specimen. By approximating g(α0 + cf/L) with its Taylor collection growth at α0, the linear time period of the growth is just retained, Eq. (4) might be obtained:
$$sigma_{{{textual content{Nu}}}} = sqrt {frac{{E*G_{{textual content{f}}} }}{{Lgleft( {alpha_{0} } proper) + c_{{textual content{f}}} g^{^{prime}} left( {alpha_{0} } proper)}}}$$
(4)
Equation (4) is changed with the next easy variables as follows:
$$X = L,quad Y = sigma_{{textual content{N}}}^{{ – 2}}$$
(5)
$$A = frac{{gleft( {alpha_{0} } proper)}}{{E * G_{{textual content{f}}} }};quad C = frac{{c_{{textual content{f}}} g^{^{prime}} left( {alpha_{0} } proper)}}{{E * G_{{textual content{f}}} }}$$
(6)
Then the connection between failure stress and specimen measurement might be simplified as follows:
the place A and C are constants associated to E, Gf and cf. The values of A and C might be obtained by the linear becoming methodology with ample take a look at knowledge. In keeping with the connection between failure stress and specimen measurement established based mostly on Bazant’s measurement impact legislation, the OkayIc of specimens in any measurement might be predicted.
Institution of Bazant’s measurement impact legislation below varied friction coefficients
The failure load of the specimen below every friction coefficient might be obtained from the numerical mannequin and substituted into Eq. (5) to calculate the values of X and Y. Then the values of X and Y are drawn in Fig. 10. It may be seen that whether or not there’s friction or not, the variation legislation of OkayIc with specimen measurement obtained from the take a look at conforms to Bazant’s measurement impact legislation. Nevertheless, it’s troublesome to tell apart whether or not the fracture take a look at is affected by the friction from the becoming goodness of the information. On this examine, the linear becoming on the take a look at knowledge of every group is carried out to acquire the slope A and intercept C of the straight line, as proven in Desk 4. In keeping with Eq. (6), the fracture power and the fracture course of zone size might be obtained when the specimen measurement tends to infinity; nevertheless, the modifications of the fracture power and the fracture course of zone size with the friction coefficient are solely mentioned on this examine. As described in Eq. (6), the fracture power is just associated to the worth of A, and the 2 are inversely proportional. The fracture course of zone size is just proportional to the ratio of C and A. Subsequently, the values of A and C below totally different friction coefficients are in comparison with acquire the connection among the many fracture power and the fracture course of zone size and the friction coefficient. It may be seen that with the rise of friction coefficient, the worth of A progressively decreases, indicating that the fracture power progressively will increase. The friction coefficient will increase from 0 to 0.3, and the fracture power will increase by 42%. On the similar time, with the rise of friction coefficient, the ratio of C to A progressively decreases, indicating that the fracture course of zone size is progressively decreased. When the friction coefficient will increase from 0 to 0.3, the fracture course of zone size decreases by about 90%. In abstract, though the existence of friction doesn’t have an effect on the institution of Bazant’s measurement impact legislation, it has a big influence on the prediction of the fracture power and the fracture course of zone size when the specimen measurement tends to infinity.
Along with the prediction of fracture power when the specimen measurement tends to infinity, the correct dedication of the fracture course of zone size of varied specimen sizes can be essential for the prediction of failure load of specimen14,15,16,57,58,59. Many research imagine that the boundary of the fracture course of zone is the vary the place the tensile stress of the crack tip is bigger than or equal to the tensile power of the fabric16,57,63,64,65. As talked about earlier, there are two strategies to find out the tensile stress on the entrance of the crack tip. Within the first methodology, the tensile stress on the entrance of the crack tip is immediately decided by the OkayI alone; within the second methodology, the higher-order time period of Williams growth is taken into account, the tensile stress within the far-field of the crack tip might be described precisely. Clearly, the fracture course of zone size calculated by the 2 strategies of figuring out the tensile stress is totally different. The tensile power of marble used for the fracture take a look at is 5.37 MPa, then the fracture course of zone size of the specimen below every friction coefficient might be obtained. As proven in Fig. 11, the fracture course of zone size decided by the OkayI is way bigger than that decided by the numerical mannequin. In keeping with the dedication standards of the fracture course of zone talked about earlier, the fracture course of zone size will likely be giant as a result of the tensile power of marble on this take a look at is small. Nevertheless, the OkayI can solely precisely describe the stress discipline close to the crack tip. Subsequently, making use of the OkayI to find out the fracture course of zone size will trigger an important error. As well as, with the rise of friction coefficient, the fracture course of zone size additionally will increase. That is totally different from the earlier ends in that the fracture course of zone size of the specimen with an infinite measurement decreases with the rise of the friction coefficient. This reveals that the estimation error of fracture course of zone size will increase with the rise of the friction coefficient. As well as, with the rise of specimen measurement, the impact of friction coefficient on the fracture course of zone size will increase progressively. It signifies that for small-size specimens, the impact of friction on the estimation of fracture course of zone size is small.