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Strength of Materials

, Volume 51, Issue 2, pp 271–279 | Cite as

Longitudinal Fracture Analysis of Nonlinear Elastic Circular Shafts Loaded in Torsion

  • V. I. RizovEmail author
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The longitudinal fracture of circular shafts loaded in torsion is analytically investigated. It is assumed that a longitudinal crack with a circular crack front is arbitrarily located to the radial direction of the shaft cross section. The shafts are made of nonlinear elastic material, which exhibits continuous inhomogeneity in the radial direction. The material nonlinearity is described by the Ramberg–Osgood stress–strain relation. The longitudinal fracture is studied in terms of the strain energy release rate. The developed analysis is applied to evaluate the longitudinal fracture behavior of a clamped nonlinear elastic inhomogeneous shaft loaded in torsion by a torque applied to a free end of internal crack arm. For the method verification, the strain energy release rate is also assessed by analyzing the energy balance in the clamped shaft. A parametric study is performed to assess the effects of material inhomogeneity, its nonlinearity, and crack orientation to the shaft radial direction on the longitudinal fracture behavior of the clamped shaft.

Keywords

longitudinal crack material nonlinearity circular shaft torsion 

Introduction. The methods of linear-elastic fracture mechanics are frequently used in analyses of various materials and engineering structures. The applicability of linear-elastic fracture mechanics is based upon the assumption for linear-elastic mechanical behavior of the material. However, in reality, the structural materials may exhibit nonlinear mechanical behavior. This fact emphasises the need of developing of fracture analyses with considering the material nonlinearity. Recently, several works on longitudinal fracture behavior of nonlinear elastic structural members, mainly beams of rectangular cross section, exhibiting material inhomogeneity have been published [1, 2]. The interest towards fracture analysis of inhomogeneous materials and structures is closely related to the growing application of certain kinds of inhomogeneous materials, such as functionally graded materials, in aeronautics, nuclear reactors, electronics, optics and biomedicine [3, 4, 5, 6, 7, 8, 9, 10]. One of the most important features of functionally graded materials is the continuous variation of their microstructure over the volume of the structural member. In this way, requirements for different mechanical properties in different parts of the structural member can be satisfied.

The main goal of the present paper is to develop an analysis of the longitudinal fracture behavior of nonlinear elastic circular shafts loaded in torsion. The fracture is studied in terms of the strain energy release rate assuming that the shafts exhibit smooth material inhomogeneity in radial direction. The Ramberg–Osgood stress–strain relation is applied for describing the material nonlinearity. The analysis is used for evaluating of longitudinal fracture behavior of a clamped circular nonlinear elastic shaft loaded in torsion.

1. Analysis of the Strain Energy Release Rate. Longitudinal fracture in nonlinear elastic shafts of circular cross section is studied in the present paper. The shafts exhibit smooth material inhomogeneity in radial direction. The nonlinear mechanical behavior of the inhomogeneous material is treated by the Ramberg–Osgood stress–strain relation. The shafts are loaded in pure torsion. A shaft portion with the longitudinal crack front is shown schematically in Fig. 1. The radius of the shaft cross section is r2 . The longitudinal crack represents a cylindrical surface of radius r1. Thus, the internal crack arm is a shaft of circular cross section of radius r1. The external crack arm is a shaft of ring-shaped cross section of internal radius r1 and external radius r2 . The crack front is a circle of radius r1. The torsion moment ahead of the crack front is T. The fracture behavior is studied in terms of the strain energy release rate. By using the approach developed in [11], the strain energy release rate for the longitudinal crack in Fig. 1 is expressed as
Fig. 1.

Shaft portion with the longitudinal crack front (the longitudinal crack is a cylindrical surface with radius r1).

$$ G=\frac{1}{r_1}\left(\underset{0}{\overset{r_1}{\int }}{u}_{01}^{\ast } rdr+\underset{r_1}{\overset{r_2}{\int }}{u}_{02}^{\ast } rdr-\underset{0}{\overset{r_2}{\int }}{u}_0^{\ast } rdr\right), $$
(1)

where \( {u}_{01}^{\ast },{u}_{02}^{\ast } \), and \( {u}_0^{\ast } \) are, respectively, the complementary strain energy densities in the cross section of the internal and external crack arms behind the crack front and in the shaft cross section ahead of the crack front.

The Ramberg–Osgood stress–strain relation that is used in the present fracture analysis is written as

$$ \upgamma =\frac{\uptau}{B}+{\left(\frac{\uptau}{H}\right)}^{1/n}, $$
(2)

where γ is the shear strain, τ is the shear stress, and B, H, and n are material properties. It is assumed that B varies continuously in radial direction

$$ B=B(r), $$
(3)

where

$$ 0\le r\le {r}_2. $$
(4)

The complementary strain energy density in the cross section of the internal crack arm behind the crack front is written as [12]

$$ {u}_{01}^{\ast }=\frac{\uptau^2}{2B}+\frac{n{\uptau}^{\left(1+n\right)/n}}{\left(1+n\right){H}^{1/n}}. $$
(5)

Formula (2) indicates that τ cannot be determined explicitly. Therefore, τ is expanded in Taylor series by keeping the first three members

$$ \uptau (r)\approx \uptau \left({r}_a\right)+\frac{\uptau^{\prime}\left({r}_a\right)}{1!}\left(r-{r}_a\right)+\frac{\uptau^{\prime \prime}\left({r}_a\right)}{2!}{\left(r-{r}_a\right)}^2, $$
(6)

where

$$ 0\le r\le {r}_1, $$
(7)
$$ {r}_a={r}_1/2. $$
(8)

Formula (6) is rewritten as

$$ \uptau (r)\approx {\upxi}_1+{\upxi}_2\left(r-{r}_a\right)+{\upxi}_3{\left(r-{r}_a\right)}^2. $$
(9)

The coefficients ξ1, ξ2, and ξ3 are determined by using (2). For this purpose, the distribution of shear strains in the cross section of the internal crack arm is treated by applying the Bernoulli hypothesis for plane sections

$$ \upgamma =\frac{r}{r_1}{\upgamma}_s, $$
(10)

where

$$ 0\le r\le {r}_1. $$
(11)

In (10), γs is the shear strain at the periphery of the internal crack arm. It should be mentioned that the Bernoulli hypothesis can be applied since circular shafts of high length to diameter ratio are analyzed in the present paper.

By substituting of (9) and (10) in (2), one arrives at

$$ \frac{r}{r_1}{\upgamma}_s=\frac{\upxi_1+{\upxi}_2\left(r-{r}_a\right)+{\upxi}_3{\left(r-{r}_a\right)}^2}{B}+{\left[\frac{\upxi_1+{\upxi}_2\left(r-{r}_a\right)+{\upxi}_3{\left(r-{r}_a\right)}^2}{H}\right]}^{1/n}. $$
(12)

At r = ra, formula (12) transforms in

$$ \frac{r_a}{r_1}{\upgamma}_s=\frac{\upxi_1}{B}+\frac{\upxi_1^{1/n}}{H^{1/n}}. $$
(13)

By substituting of r = rain the first and second derivatives of (12) with respect to r, one obtains

$$ B^{\prime}\frac{r_a}{r_1}{\upgamma}_s+B\frac{\upgamma_s}{r_1}={\upxi}_2+\frac{B^{\prime }{\upxi}_1^{1/n}+B\frac{1}{n}{\upxi}_1^{1/n-1}{\upxi}_2}{H^{1/n}}, $$
(14)
$$ B^{\prime\prime}\frac{r_a}{r_1}{\upgamma}_s+2B^{\prime}\frac{\upgamma_s}{r_1}=2{\upxi}_3+\frac{B^{\prime\prime }{\upxi}_1^{1/n-1}{\upxi}_2+{\upxi}_2\left(\frac{B^{\prime }}{n}{\upxi}_1^{1/n-1}+B\frac{1-n}{n^2}{\upxi}_1^{\left(1-2n\right)/{\upxi}_2}\right)}{H_t^{1/n-1}}. $$
(15)

In this way, three equations, (13)–(15), with four unknowns γs, ξ1, ξ2, and ξ3 are written. Another equation is obtained by considering the equilibrium of the elementary forces in the cross section of the internal crack arm

$$ {T}_1=\underset{\left({A}_1\right)}{\iint}\uptau rdA, $$
(16)

where T1 is the torsion moment in the internal crack arm behind the crack front and A1 is the area of the internal crack arm cross section. It should be mentioned that B and its derivatives B′ and B′′, which participate in (13)–(15), are calculated at r = ra.

Equations (13)–(16) should be solved with respect to γs, ξ1, ξ2, and ξ3 by using the MatLab computer program for given shaft geometry, external loading and material properties. Then, \( {u}_{01}^{\ast } \) is obtained by substituting of (9) in (5).

Formula (5) is used also to calculate the complementary strain energy density in the cross section of the external crack arm behind the crack front. For this purpose, τ is replaced with τe , where τe is the shear stress in the external crack arm cross section behind the crack front. The shear stress τe is expanded in series of Taylor by keeping the first three members

$$ {\uptau}_e(r)\approx {\upxi}_{e1}+{\upxi}_{e2}\left(r-{r}_e\right)+{\upxi}_{e3}{\left(r-{r}_e\right)}^2, $$
(17)

where

$$ {r}_1\le r\le {r}_2, $$
(18)
$$ {r}_e=\frac{r_1+{r}_2}{2}. $$
(19)

The coefficients ξe1, ξe2, and ξe3 are determined by using Eqs. (13)–(16). For this purpose, γs, ξ1, ξ2, ξ3, ra, T1, τ, and A1 are replaced, respectively, with γes, ξe1, ξe2, ξ3, re, T2, τe, t2 , τe , and A2 , where γes is the shear strain at the periphery of the external crack arm, T2 is the torsion moment in the external crack arm behind the crack front, and A2 is the area of the external crack arm cross section. The complementary strain energy density \( {u}_{02}^{\ast } \) is calculated by substituting of (17) in (5).

The complementary strain energy density in the cross section of the shaft ahead of the crack front is obtained by replacing of τ with τu in formula (5), where τu is the shear stress in the shaft cross section ahead of the crack front. Also, γs, ξ1, ξ2, ξ3, ra, T1, τ, and A1 are replaced, respectively, with γus, ξu1, ξu2, ξu3, ru, T, τu, and A in formulae (9) and (13)–(16). Here, γus is the shear strain at the periphery of the shaft in the cross section ahead of the crack front, T is the torque moment in the shaft ahead of the crack front, A is the area of shaft cross section, and ru = r2/2.

The strain energy release rate is calculated by substituting of \( {u}_{01}^{\ast },{u}_{02}^{\ast } \), and \( {u}_0^{\ast } \) in (1). The integration in (1) should be performed the MatLab computer program for particular shaft configuration, loading conditions and material properties.

2. Longitudinal Fracture in a Clamped Shaft. Longitudinal fracture behavior of a clamped circular shaft loaded in torsion (Fig. 2) is studied in terms of the strain energy release rate by applying the analysis developed in Section 2 of the present paper.
Fig. 2.

Clamped circular shaft with longitudinal cylindrical crack loaded in torsion.

There is a cylindrical longitudinal crack of length a in the shaft. The radius of the internal crack arm is r1. The loading consists of a torsion moment T applied at the free end of the internal crack arm. Thus, the external crack arm is free of stresses. The shaft length is denoted by l. It is assumed that the material property B [refer to the Ramberg–Osgood stress–strain relation (2)] is distributed continuously along the radius of the shaft cross section according to the following logarithmic law:

$$ B={B}_0\ln \left(e+b\frac{r}{r_2}\right), $$
(20)

where

$$ 0\le r\le {r}_2. $$
(21)

In (20), B0 is the value of B in the center of the shat cross section and b is a material property that controls the material inhomogeneity in radial direction.

The complementary strain energy density in the internal crack arm behind the crack front is calculated by formula (5). For this purpose, first, γs, ξ1, ξ2, and ξ3 are determined from Eqs. (13)–(16). Formula (20) is substituted in (13)–(15). Also, (9) is substituted in (16). In this way, one derives

$$ \frac{r_a}{r_1}{\upgamma}_s=\frac{\upxi_1}{B_0\ln \upchi}+\frac{\upxi_1^{1/n}}{H^{1/n}}, $$
(22)
$$ \frac{B_0b}{r_2\upchi}\frac{r_a}{r_1}{\upgamma}_s+{B}_0\ln \upchi \frac{\upgamma_s}{r_1}={\upxi}_2+\frac{\frac{B_0b}{r_2\upchi}{\upxi}_1^{1/n}+{B}_0\ln \upchi \frac{1}{n}{\upxi}_1^{1/n-1}{\upxi}_2}{H^{1/n}}, $$
(23)
$$ -\frac{B_0{b}^2}{r_2^2\upeta}\frac{r_a}{r_1}{\upgamma}_s+2\frac{B_0b}{r_2\upchi}\frac{\upgamma_s}{r_1}=2{\upxi}_3+\frac{-\frac{B_0{b}^2}{r_2^2\upeta}{\upxi}_1^{1/n}+\frac{B_0b}{n{r}_2\upchi}{\upxi}_1^{1/n-1}{\upxi}_2+{\upxi}_2\left(\frac{B_0b}{n{r}_2\upchi}{\upxi}_1^{1/n-1}+{B}_0\ln \upchi \frac{1-n}{n^2}{\upxi}_1^{\left(1-2n\right)/n}{\upxi}_2\right)}{H_t^{1/{n}_t}}, $$
(24)
$$ {T}_1=2\uppi \left(\frac{\upxi_1}{3}{r}_1^3+\frac{\upxi_2}{4}{r}_1^4-\frac{\upxi_2}{3}{r}_a{r}_1^3+\frac{\upxi_3}{5}{r}_1^5-\frac{\upxi_3}{2}{r}_a{r}_1^4+\frac{\upxi_3}{3}{r}_a^2{r}_1^3\right), $$
(25)

where

$$ {T}_1=T, $$
(26)
$$ \upchi =e+{b}_{r_2}^{r_a}, $$
(27)
$$ \upeta ={\left(e+b\frac{r_a}{r_2}\right)}^2. $$
(28)

After solving of Eqs. (22)–(25) with respect to γs, ξ1, ξ2, and ξ3 by the MatLab software program, \( {u}_{01}^{\ast } \) is obtained by (5).

Since the external crack arm of the clamped shaft (Fig. 2) is free of stresses,

$$ {u}_{02}^{\ast }=0. $$
(29)

Formula (5) is applied also calculate the complementary strain energy density in the shaft cross section ahead of the crack front. For this purpose, γs, ξ1, ξ2, ξ3, ra, T1, τ, and A1 are replaced, respectively, with γus, ξu1, ξu2, ξu3, ru, T, τu, and A in (9) and (22)–(25). Then, Eqs. (22)–(25) are solved and \( {u}_0^{\ast } \)is found by substituting (9) in (5).

The strain energy release rate is obtained by performing the integration in (1) with the help of the MatLab computer program.

The solution to the strain energy release rate is verified by analyzing the cylindrical longitudinal crack in the shaft configuration shown in Fig. 3 with the help of the energy balance. By assuming a small increase δa of the crack length, the energy balance is expressed as
Fig. 3.

The strain energy release rate in nondimensional form presented as a function of b at (curve 1) nonlinear mechanical behavior of the shaft and (curve 2) linear-elastic behavior of the shaft.

$$ T\updelta \upvarphi =\frac{\partial U}{\partial a}\updelta a+G{l}_{cr}\updelta a, $$
(30)

where φ is the angle of twist of the free end of the internal crack arm, lcr is the length of the crack front, and U is the strain energy cumulated in the shaft. After substituting of

$$ {l}_{cr}=2\uppi {r}_1 $$
(31)

in (30), the strain energy release rate is derived as

$$ G=\frac{T}{2\uppi {r}_1}\frac{\mathrm{\partial \upvarphi }}{\partial a}-\frac{1}{2\uppi {r}_1}\frac{\partial U}{\partial a}. $$
(32)

By using the integrals of Maxwell–Mohr, φ is obtained as

$$ \upvarphi =\frac{\upgamma_s}{r_1}a+\frac{\upgamma_{us}}{r_2}\left(l-a\right). $$
(33)

The strain energy cumulated in the shaft is written as

$$ U=a\underset{\left({A}_1\right)}{\iint }{u}_{01} dA+\left(l-a\right)\underset{(A)}{\iint }{u}_0 dA, $$
(34)

where u01 and u0 are the strain energy densities in the internal crack arm and the uncracked shaft portion, respectively.

The strain energy density in the internal crack arm is calculated by the following formula [12]:

$$ {u}_{01}=\frac{\uptau^2}{2B}+\frac{\uptau^2}{2B}+\frac{\uptau^{\left(1+n\right)/n}}{\left(1+n\right){H}^{1/n}}1, $$
(35)

where τ is found by (9).

Formula (35) is applied also to calculate u0. For this purpose, is replaced with τu . By substituting of (33) and (34) in (32), one arrives at

$$ G=\frac{T}{2\uppi {r}_1}\left(\frac{\upgamma_s}{r_1}-\frac{\upgamma_{us}}{r_2}\right)-\frac{1}{2\uppi {r}_1}\left(\underset{\left({A}_1\right)}{\iint }{u}_{01} dA-\underset{(A)}{\iint }{u}_0 dA\right). $$
(36)

The integration in (36) should be performed by using the MatLab computer program. The strain energy release rate calculated by (36) is exact match of the strain energy release rate derived by (1). This fact is a verification of the fracture analysis developed in the present paper. It should be noted that the fracture behavior is analyzed also by keeping more than three members in the Taylor series (6). The results are very close to these obtained by keeping the first three members (the difference is less than 2%).

The solution to the strain energy release rate (1) is applied to perform a parametric investigation of the longitudinal fracture behavior of the clamped shaft (Fig. 2). The calculated strain energy release rates are presented in nondimensional form by using the formula GN = G/(B0r2) It is assumed that r2 = 0.01m and T = 50 N·m. The material inhomogeneity in radial direction is characterized by b. The cylindrical crack location in radial direction of the shaft cross section is characterized by r1/r2 ratio.

In order to evaluate the influence of material inhomogeneity on the longitudinal fracture, the strain energy release rate in nondimensional form is presented as a function of b in Fig. 3 at r1/r2 = 0.2, H/B0 = 0.5, and n = 0.8. It can be observed in Fig. 3 that the strain energy release rate decreases with increasing of b. This finding is attributed to the increase of the shaft stiffness with increasing of b. The effect of nonlinear mechanical behavior of the inhomogeneous material on the longitudinal fracture is studied too. For this purpose, the strain energy release rate obtained assuming linear-elastic behavior of the shaft is presented in nondimensional form in Fig. 3 for comparison with the nonlinear solution. It should be mentioned that the linear-elastic solution for the strain energy release rate is derived by substituting of H → ∞in (1) because at H → ∞ the Ramberg–Osgood stress–strain relation (2) transforms in the Hooke law assuming that B is the shear modulus of the inhomogeneous material. The curves in Fig. 3 indicate that the material nonlinearity leads to increase of the strain energy release rate.

The influence of H/B0 ratio on the longitudinal fracture behavior of the shaft is also evaluated. For this purpose, the strain energy release rate in nondimensional form is presented as a function of H/B0 ratio in Fig. 4 at three r1/r2 ratios. One can observe in Fig. 4 that the strain energy release rate decreases with increasing of H/B0 ratio. Concerning the influence of the longitudinal cylindrical crack location in radial direction on the fracture behavior, the curves in Fig. 4 show that the strain energy release rate decreases with increasing of r1/r2 ratio (this behavior is due to the increase of the internal crack arm stiffness).
Fig. 4.

The strain energy release rate in nondimensional form presented as a function of H/B0 ratio at r1/r2 = 0.2 (curve 1), r1/r2= 0.5 (curve 2), and r1/r2= 0.8 (curve 3).

Conclusions. The longitudinal fracture behavior of nonlinear elastic circular shafts loaded in torsion is analyzed. It is assumed that the shafts exhibit smooth material inhomogeneity in radial direction. The Ramberg– Osgood stress–strain relation is applied in order to model the nonlinear mechanical behavior of the material. A longitudinal cylindrical crack (the crack front is a circle) is located arbitrary in radial direction of the shaft cross section. The internal crack arm is a shaft of circular cross section, the external crack arm is a shaft of ring-shaped cross section. The fracture behavior is studied in terms of the strain energy release rate by assuming that the material property B varies continuously in radial direction of the shaft cross section. The analysis is applied for investigating the longitudinal fracture in a clamped shaft. The external loading of the shaft consists of a torsion moment applied at free end of the internal crack arm. The longitudinal fracture behavior of the clamped shaft is studied also by considering of the energy balance for verification of the strain energy release rate. A parametric study of the longitudinal fracture behavior is carried-out. The influences of the material inhomogeneity, the crack location in radial direction and the nonlinear mechanical behavior of the material on the fracture are investigated. It is found that the strain energy release rate decreases with increasing of b (the material property b controls the material inhomogeneity in radial direction).

The investigation shows also that the strain energy release rate decreases with increasing of H/B0 ratio. It is found that increase of r1/r2 ratio leads to a decrease of the strain energy release rate (this finding is attributed to the increase of the internal crack arm stiffness).

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Authors and Affiliations

  1. 1.Department of Technical MechanicsUniversity of Architecture, Civil Engineering and GeodesySofiaBulgaria

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