Numerical calculation the fracture toughness of square section beam under three-point bending test

. A universal equation is proposed to determine specific fracture energy based on numerical calculations by finite element methods in the ANSYS system referring to the three-point bending of a square beam arranged edgewise on supports. Numerical dependences of the specimen compliance on the crack length were represented as the 5th and 6th degree polynomials on the relative value of  l/h , where  l is the crack length and h is the half length of the square diagonal. To calculate the specific fracture energy, we used the derivative of specimen compliance by the crack length. The analysis has shown that finite value of specific fracture energy at  l →0 is obtained when the decomposition term at  l/h is equal to 0. A satisfactory agreement between the specific fracture energy values for the 5th and 6th degree decompositions is observed only at  l/h ≈ 1. As the values of  l decrease, the difference becomes increasingly significant. The averaging of coefficients at corresponding  l/h degrees of two considered decompositions allows us to find a power function that weakly depends on the  l/h ratio and leads to a consistent dependence of the specific fracture energy on the crack length. Due to the performed calculations, a universal equation is obtained to determine the specific fracture energy according to the test data of square beams by three-point bending in a wide range of geometric dimensions.


Introduction
A current problem in engineering applications is to obtain and to use the simplified analytical equations for express calculation of the mechanical properties of materials using experimental data.In particular, a prefracture stage associated with the initiation and stable crack propagation prior to spontaneous fracture of a specimen is of particular interest for mechanicians.This stage is characterized by the significant increase in the specimen compliance η, which is determined as the ratio of the beam elastic bending e to the applied force P: η = e/P.Change in the specimen compliance dη/dl is based on fundamental equation, which determines the intensity of elastic energy release as the crack propagates [1][2][3][4]: where P is the external load, a is the specimen thickness, l is the crack length.
The intensity of elastic energy release G, or specific fracture energy (SFE), is the most important fracture toughness characteristics of the material.In engineering practice, the stress intensity factor KI is often used associated with SFE by the following relation: where Е is the Young's modulus,  is the Poisson's ratio.
In terms of experimental implementation, theoretical and numerical calculation, the loading of a beam by three-point bending is the simplest [5,6].
This paper provides a derivation of the universal equation to determine SFE based on numerical calculations by finite element methods in the ANSYS system referring to threepoint bending of a square beam arranged edgewise on supports (Fig. 1).As experience shows [7], such an unconventional arrangement of a beam on supports contributes to prolongation of the plastic deformation stages and stable crack propagation.

Analytical and numerical calculations
According to the elasticity theory, the relationship between the elastic bending of the beam e and the force P, applied to the end of the rectangular beam with a cross-section of a×b, is determined by the well-known equation [8,9] (Fig. 2а): where l0 is the beam length, Е is the Young's modulus.
It is obvious that the same deflection will be realized in the case of the 3-point deflection of the beam with the a×b section, if the length between supports is L = 2l0 and the external force F = 2Р.In this regard, for a simplicity of calculation, the dependence of the deflection and the compliance on the crack length l was considered using the example of a cantilever embedded in a rigid foundation (Fig. 2).
Let us show that an elastic bending of the cantilever (2) does not depend on orientation the force direction P, acting in the section of the cantilever.
Consider the orientation of the console shown in Figure 2b. Figure 3 shows the contact area of the cantilever and a rigid foundation.A cantilever can be represented as a pack of extremely thin plates with a thickness of dx, which can be seen as mini-cantilevers.Figure 3 shows a minicantilever projection at a distance of x from the vertical diagonal of the square.In this case, according to the definition, a mini-cantilever has a length of l0, a thickness of a = dx and a width of b = 2(x -h), where h is the half length of the diagonal of square.Under the action of force P, applied at the cantilever's end, all the minicantilevers will bend by the same value of e.The equation ( 2) is valid for each minicantilever.Given that a = dx, according to equation (2), we can write the following expression: where dP is the elementary force, which bends a mini-cantilever by the value of e.
In this case, P is determined as an integral sum of all elementary forces dP, acting on all mini-cantilevers: ) . 4 4 Equation ( 4) corresponds to equation ( 2), if b = a.Consequently, elastic bending of the cantilever e (without a crack) does not depend on the square section orientation relatively to the force direction P, acting at the cantilever's end, what was required to prove.
According to Eq. ( 4), the exact compliance value of the cantilever without a crack is determined by the equation There is no exact analytical solution for the cantilever with a crack due to the problem of stress-strain state description in the crack mouth by the elasticity theory methods.However, it is possible to solve problems of this type on the basis of numerical calculations by finite element methods using the well-developed programs.Static problems of the elasticity theory are solved with sufficient accuracy using the ANSYS computer system.
Let us consider the problem of dependence of the cantilever compliance η on the crack length at the contact boundary of a cantilever and a rigid foundation (Fig. 4).To a certain extent, this corresponds to the process of crack propagation in the beam in the three-point bending mode.The crack is shown as a slit with a depth of l (Fig. 4) with a rectilinear boundary oriented perpendicular to the applied force P.This approach avoids the difficulties associated with the interpretation of boundary conditions on the contact surfaces of supports with a beam for the case of the three-point bending problem.The use of a rigid attachment prevents the free bending of the beam, which will seriously distort the results.

Results of the numerical calculations
To calculate SFE, first of all, we need to transform Eq. ( 1) with regard to the edgewise square beam tests (Fig. 1a).The а parameter in Eq. ( 1), as applied to the beam with a support on the plane, is equal to the notch length across the specimen.In the case of an edgewise specimen, it is equal to 2l (Fig. 4).Thereby Eq. ( 1) takes the form of 2 .4Δ As a rule, the power-law dependence of fracture toughness characteristics on the relative value of the l/Wtype, where l is the crack length and W is the specimen width [10,11] is used in standard equations.In our case, we used the relative value of l/h, where h = a/√2 is the half length of the square diagonal (Fig. 3).
Let us consider the compliance dependences on the crack length for two cantilevers differing in size, 20 and 24 mm long and 4×4 and 5×5 mm, respectively.
Figure 5 presents the compliance dependences  on the relative value of the crack length l/h, for material with Young's modulus E = 2•10 11 GPa, obtained using the ANSYS software.Curve 1 corresponds to calculations for a cantilever with the size of a = 4 mm, l0 = 20 mm, and curve 2, with the size of a = 5 mm, l0 = 24 mm.
In the ORIGIN 9.1 window, the compliance dependences  on l/h can be written as the n-th degree polynomial in the following way: ( )  is the numerical value of cantilever compliance without a crack.
Let us analyze the peculiarities of the polynomial dependence (7) by the example of a cantilever with a cross-section of 44 mm.We shall write down the compliance dependence  on l/h as the 6-th degree polynomial in expanded form: The value of 0 = 63566 m/N determines the numerical value of the cantilever compliance without a crack (l = 0).The exact value, calculated by Eq. ( 5), is equal to 0.625 m/N.The comparison shows that the numerical value differs slightly from the exact compliance value by only 1.6%.Neglecting an insignificant difference, Eq. ( 8) for the cantilever generally can be written as follows: The compliance derivative by the crack length will be equal to Ea h Ea h (10) where fn=6(l/h) = −0.0269+1.05(l/h) −3.3067(l/h) 2 +13.3778(l/h) 3 −18.429(l/h) 4 +10.4687(l/h) 5 .Substituting expression (10) into Eq.( 6), and considering that the compliance of the sample in the 3-point bending mode is two times less than the compliance of the console, we obtain the following equation for calculating SFE: Ea lh (11) At l→0, the function fn=6(l/h) is −0.0269, i.e. is not equal to zero.Therefore, at l→0, the function Gn=6 goes into the negative region and infinitely grows by the absolute value of (Gn=6→ −∞).The peculiarities of dependence (10) at P = 400 N are shown by curve 1 in Fig. 6a.It is seen that at l→0, the function Gn=6→ −∞, that has no physical sense.To obtain the final result, it is logical to assume that in Eq. ( 9) the coefficient  is equal to 0. The result of such a simplification is shown by curve 2 in Fig. 6a.A decrease in the relative value of l/h results in a greater divergence of curves 1 and 2. In this case, a minimum is observed on Curve 2.
Therefore, the presentation of specimen compliance as the 6 th degree polynomial does not give a reliable result for Gn=6 in the range of small values of l/h.
In the case of the specimen compliance representation by the 5-th degree polynomial, similar calculations also lead to the divergence of the curves with a decrease in the relative value of l/h (Fig. 6b).However, when l→0, the function Gn=5 grows infinitely in the positive region.In contrast to the previous approximation, there is no minimum on curve 2 in a simplified version.On the contrary, with a decrease in l, the rate of the Gn=5 function decrease grows.Let us assume that the averaging of the simplified versions represented by G1 = (Gn=6+Gn=5)/2 will lead to a satisfactory result.In the final version, the following equation can be written for SFE:
In the presented form, Eq. ( 12), apparently, can be used in calculating SFE for a cantilever of a different length and/or cross-section.In this case, dimensional changes of the cantilever should not significantly affect the dependence of the f(l/h)-type function on l/h.
To verify this assumption, we carried out calculations for another case, when the cantilever has a cross-section of a×a = 5 mm and a length of l0 = 24 mm.A function f2(l/h) was obtained, similar to the function f1(l/h), which is as follows: f2(l/h) = 0.268 + 1.603(l/h) + 1.065(l/h) 2 −4.925(l/h) 3 + 5.147(l/h) 4 .
The dependences of f1(l/h) and f2(l/h) are shown in Fig. 8.It can be seen that despite some difference in the binomial coefficients of these functions, a good agreement between the curves of the given dependences is observed.Apparently, the averaged function f(l/h) = [f1(l/h) + f2(l/h)]/2 can be used in fracture toughness calculation of materials using the universal equation: It should be noted that formula (13) is valid for calculating the specific fracture energy of a console embedded in a rigid base.The specific fracture energy of the beam in the 3-point mode will be two times less, since the compliance of the beam is two times less than the compliance of the console (Fig. 2 and 3).Turning to the beam under the 3-point deflection mode (Fig. 1), where L = 2l0 and F = 2P, we will obtain the following equation for calculating the specific energy of beam failure under the action of force F: The SFE dependences of G (1) and stress intensity factor KI (2) on the crack length.
Consider an example.In Ref. [7] the test data for steel 60Х24АГ16 are obtained by the three-point bending of a square beam with a cross-section of 4.874.87mm and a length of 50 mm with a notch at a span length between the supports of L = 2l0 = 47mm.The specimen fractured at an external applied force of F = 2P = 1360 N. It was considered that the Young's modulus of steel is E = 200 GPa and the Poisson's ratio is  = 0.3.Figure 9 shows the dependences of fracture toughness characteristics of G and KI on the crack length corresponding to these data.The crack length at the moment of fracture was ≈1 mm.According equation ( 14), the critical fracture toughness parameters of the beam without notch are Gс = 1.651 kJ/m 2 and KIc = 19.05MPa•m 1/2 for steel 60Х24АГ16.

Conclusion
This work presents a universal equation to determine SFE based on numerical calculations by finite element methods in the ANSYS system referring to the three-point bending of a square beam arranged edgewise on supports.As experience shows [6], such an unconventional beam arrangement on supports, contributes to prolongation of the plastic deformation stages and stable crack propagation.
Numerical dependences of the specimen compliance on the crack length were presented as the 5 th and 6 th degree polynomials on the relative value of l/h, where l is the crack length and h is the half length of the square diagonal.To calculate SFE, we used a derivative of the specimen compliance by the crack length.
The analysis has shown that the finite value of SFE at l→0 is obtained when the decomposition term at l/h is equal to 0. In this case, a satisfactory agreement between the SFE values for the 5-th and 6-th degree decompositions are observed only at l/h ≈ 1.As the values of l decrease, the difference becomes increasingly significant.The averaging of coefficients at corresponding l/h degrees of two considered decompositions allows us to find a power function that weakly depends on the l/h ratio and leads to a consistent dependence of SFE on the crack length.Due to the performed calculations, a universal equation is obtained to determine SFE according to the test data of square beams by threepoint bending in a wide range of geometric dimensions.

Fig. 3 .
Fig. 3. Contact area of the cantilever with a rigid foundation.

Fig. 4 .
Fig. 4. Crack at the contact boundary of a cantilever and a rigid foundation.

Fig. 5 .
Fig. 5. Dependences of compliance  on the crack length.Explanations are in the text.

Fig 6 .
Fig 6.Dependences of SFE Gn=6 (a) and Gn=5 (b) on the crack length: 1 is the original version, 2 is the simplified version.

Figure 7
Figure 7 demonstrates all three versions of the SFE dependence on the crack length in the cantilever.It is seen that the averaging led to a consistent result.The rate of elastic energy release during crack propagation first increases almost linearly (Fig. 7, curve 3), then starts increasing smoothly.