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Abstract

The behavior of embedded planar elliptical cracks within square prismatic bars that are commonly used as torque-resisting structural components motivates this work, as the literature for such stress intensity factors is evidently deficient. The stress intensity factors are evaluated using the dual boundary element program of Beasy, and studied by considering the elliptical aspect ratio and the eccentricity in the sense of an offset from the centroid. The variation of shear mode stress intensity factors for a penny crack situated near to the free surface and the square corner is also reported for the first time, complementing the available results for tensile mode.

Keywords

Stress intensity factor embedded crack square prismatic bar shear mode dual boundary element method

Introduction

Prismatic bars are ubiquitously used as structural components in mechanisms, machineries, and other engineering applications. As engineers explore limits to design products, materials defects and flaws must be examined, and fracture analysis becomes essential. Embedded cracks in solids as material flaws are invariably present as defects of material processing or as coalescence of micro-cracks due to dislocations pile up during services. The approach of examining cracks using fracture mechanics requires the stress intensity factors (SIFs); relevant solutions and off-the-shelf data for such are often sought to develop, validate, and support non-destructive techniques for evaluation of embedded defects in solids. However, very few and limited studies on embedded elliptical cracks have been reported in the literature, as this problem poses formidable challenges for both analytical and experimental solutions. The state of the art for material cutting and joining is perhaps still too limiting for creating experimental samples with embedded crack, and samples deliberately obtained by controlling metallurgical processing are often too difficult to study as crack density, size, location, and orientation almost never appeared favorably for experimental purposes. On the other hand, the analytical formulation of the boundary value problem for embedded elliptical crack is complex and challenging, and it is only amenable for special geometry and loading conditions.

Green and Sneddon¹ and Sadowsky and Sternberg² represent the earlier work on the stress field due to an elliptical crack in infinite solid under normal loading. Various results of SIFs for infinite solid under special loading can be obtained from, among others, Kassir and Sih,³ Sneddon and Lowengrub,⁴ Sih,⁵ and Keer.⁶ The handbook of Tada et al.⁷ collected the work of many authors who contributed to the advances on the treatment and the analysis of embedded cracks. Ang and Clements⁸ and Ang⁹ considered material inhomogeneity in their work. Gao and Rice¹⁰ and Gao¹¹ developed a new perturbative approach based on the technique of Rice¹² to obtain the SIFs of the nearly circular tensile and shear mode cracks. Another approach for the slightly non-circular tensile crack was formulated by Panasyuk.¹³ Livieri and Segala^14,15 presented an analytical technique based on the first-order expansion of the Oore–Burns integral to treat arbitrarily shaped embedded crack fronts obtained by homotopic transformations of a reference disk. Wang and Glinka¹⁶ reported the SIFs of embedded elliptical cracks under complex two-dimensional loading conditions using the weight function method. Based on the properties of weight functions and the available weight functions for two-dimensional cracks, they proposed new mathematical expressions using the point load weight function. Atroshchenko et al.¹⁷ analytically developed a three-dimensional unbounded-domain boundary value problem of elasticity for an elliptical crack and proposed solutions for arbitrary normal loadings composed as linear combinations of basis polynomials. Of great practical interest is the study of crack situated near the free boundary. Smith and Alavi,¹⁸ Fischer et al.,¹⁹ and Shah and Kobayashi²⁰ analytically treated the problem of a tensile crack close to the free surface of the half space; Kobayashi et al.²¹ presented a procedure for estimating the SIFs for tensile cracks near a square corner. No corresponding results for shear mode SIFs are available in the literature.

For some finite-domain problems with special geometry and loading, the results could be expressed using the Fredholm integrals, an example is the long circular cylinder.^22,23 Newman and Raju²⁴ obtained empirically the closed-form solutions for SIFs of an embedded elliptical crack located centrally within a rectangular prismatic bar subjected to tension and bending based on simulation results from the finite element method (FEM), and these closed-form solutions are the first of its kind. Le Delliou and Barthelet²⁵ elaborated the influences of crack size, shape, and the free surface on SIFs for embedded cracks in a plate using FEM. Lee²⁶ presented the solution for a pressurized elliptical crack within a long circular cylinder. Qian²⁷ reported the effects of crack aspect ratio, crack eccentricity, and effect of pipe thickness on the SIFs of an embedded elliptical crack axially oriented in a pressurized pipe using the interaction integral approach on a three-dimensional finite element crack front model. Liu et al.²⁸ investigated embedded cracks in a welded joint of pressure vessels loaded uniaxially. Imran et al.²⁹ presented the SIFs of an embedded elliptical crack in a circular cylinder with different crack aspect ratios, eccentricities, and inclinations under normal and torsional loadings.

We present the results for the SIFs of embedded elliptical cracks within square prismatic bar under torsion. No solution of SIFs for such finite domain under torsion has been reported in the literature to date. The effects of elliptical aspect ratio and eccentricity in the sense of an offset from the cross-sectional centroid are studied. An approximate variation of shear mode SIFs for a penny crack located near the free surface and the square corner is also reported. This brief graphical description represents the first report on shear mode penny crack situated near the free surface and the square corner, and may complement the work on opening mode by Smith and Alavi,¹⁸ Fischer et al.,¹⁹ Shah and Kobayashi,²⁰ and Kobayashi et al.²¹ All simulation results are performed using the fracture mechanics package of Beasy,³⁰ a relatively new program based on the dual boundary element method (DBEM). Notwithstanding the need for vigorous assessment on a broader scale, especially in terms of accuracy, convergence, and flop count, its validity has been variously confirmed for problems described in the literature.^31–34

Simulation of embedded elliptical cracks and evaluation of Beasy

Evaluation of SIFs using DBEM implemented in Beasy fracture mechanics package

Adopting the crack front coordinate system in Figure 1, the SIFs can be expressed as follows as the material point $x'$ approaches the crack border

\begin{matrix} K_{1} = lim_{x' \to x} \sqrt{2 π ‖ x - x' ‖} σ_{ξ ξ} \\ K_{2} = lim_{x' \to x} \sqrt{2 π ‖ x - x' ‖} σ_{ζ ξ} \\ K_{3} = lim_{x' \to x} \sqrt{2 π ‖ x - x' ‖} σ_{ζ η} \end{matrix}

(7)

These SIFs are evaluated in Beasy by way of the J-integral concept of Rice³⁵ and Cherepanov,³⁶ which gives, for crack opening in the $x_{i}$ direction, a path-independent energy integral of the form $J = \int_{Γ} (W n_{i} - t_{k} u_{k, i}) d Γ$ over a surface $Γ$ with outward normal n. This concept was developed for linear elastic materials, and it was further extended to HRR solutions³⁷ for materials with constitutive relationship in the form of Ramberg–Osgood. Following the presentation in Cruse,³⁸ and Mi and Aliabadi,³⁹ and using Green’s functions $U_{ij}$ for displacement and $T_{ij}$ for traction, the strain energy density $W (Γ)$ and the work-conjugate of traction t and displacement u in J-integral are calculated using the dual boundary integral equations as follows

\begin{matrix} u_{i} (x') + α_{ij} (x') u_{j} (x') + \int_{Γ} T_{ij} (x', x) u_{j} (x) d Γ (x) \\ = \int_{Γ} U_{ij} (x', x) t_{j} (x) d Γ (x) \end{matrix}

(1)

\begin{matrix} \frac{1}{2} t_{j} (x') + n_{i} (x') \int_{Γ} T_{ij, k} (x', x) u_{k} (x) d Γ (x) \\ = n_{i} (x') \int_{Γ} U_{ij, k} (x', x) t_{k} (x) d Γ (x) \end{matrix}

(2)

Leading to equations (1) and (2) is Somigliana’s identity expressed as an integral equation involving $T_{ij}$ and several other terms as its kernel. T_ij’s singularity of $O (1 / ‖ x - x' ‖^{2})$ as $x \to x'$ warrants regularization and treatment in the sense of Cauchy; $α_{ij} (x')$ in equation (1) is a term that emanates from this process. Mi and Aliabadi³⁹ claimed to have presented an effective numerical implementation of the dual boundary integrals of equations (1) and (2), and this technology is the basis of the fracture mechanics package of Beasy.

Figure 1.

Crack front coordinate system.

Model geometry and embedded crack shape

This simulation for the SIFs uses a prismatic square bar with the cross section of 10 × 10 mm² and length of 40 mm. It is twisted with a torque $M_{t}$ that corresponds to the maximum shear stress of $τ_{\max} = 100 MPa$ . An isotropic linear elastic material with Young’s modulus of 210 GPa and Poisson’s ratio of $ν = 0.29$ is used; such stiffness moduli are typical for steel alloys. The parameters and notations used for the embedded crack are depicted in Figure 2. Embedded cracks with fixed b = 0.5 mm and aspect ratio of b/a are introduced in the square bar. Fine mesh with a typical nodal distance along the crack front of approximately 0.02 mm is used all cracks studied, and this corresponds to roughly 120 nodal points along the elliptical crack front. The built-in quadratic discontinuous quadrilateral element as described in Mi and Aliabadi³⁹ is used for all cases simulated in this article. All SIFs are normalized by $K_{0} = τ_{\max} \sqrt{π b}$ unless stated otherwise.

Figure 2.

The model of an embedded crack in a square bar under torsion and the parameters and notations used in this study. $ϕ$ is the parametric angle of the ellipse and s is the normalized arc length. e_n and e_t are the unit tangent and normal vectors, respectively.

Evaluation of Beasy with some available solutions

The problem of embedded crack contained in bounded domain of general geometry is not amenable to analytical solution. For some special geometry, it can be treated using the dual integral equations technique with solution presented as integral equations of the Fredholm type.^22,23 To assess the accuracy of the solutions obtained from the DBEM fracture mechanics package of Beasy, some of the available solutions for penny cracks within infinite solid under shear loading provided in Kassir and Sih,³ Sneddon and Lowengrub,⁴ and others are used. In addition, the Newman–Raju (NR) empirical solution²⁴ for embedded elliptical crack in a square bar loaded normally is used to benchmark the results of the same obtained from Beasy. In these simulations for the evaluation of Beasy, fine mesh of a similar typical size with that for the study for embedded cracks under torsional shear stress is used.

Embedded penny crack under axisymmetric torsion

The infinite solid containing a penny crack is subjected to an axisymmetric torsion (see Figure 3), where the shear stress varies linearly in the radial direction r in accordance with $τ_{z ϕ} = τ (r / a)$ . This is equivalent to a torque of $T = π τ a^{3} / 2$ . Sneddon and Lowengrub⁴ and Kassir and Sih³ give the SIFs as $K_{1}^{\infty} = K_{2}^{\infty} = 0$ and

K_{3}^{\infty} = \frac{4 τ \sqrt{a}}{3 \sqrt{π}}

(3)

Figure 3.

Penny crack of radius a in an infinite solid under axisymmetric torsion T.

The SIFs due to axisymmetric torsion for a penny crack of radius a embedded centrally in circular cylinders of radius R and length L (see Figure 4) are compared with the solutions for infinite solid, that is, equation (3). The axisymmetric torque T is loaded in such a way that the shear stress is linearly varying in the radial direction and assumes the same value as that in Figure 3 at the radial distance of r = a. A length of L = 6R is used. This bounded-domain boundary value problem could be treated using the dual integral equations technique with the results presented using the Fredholm integral equations.^5,22,23 The results for crack radius of a = 0.5 mm contained in cylinder of various ratios of R/a are shown in Figure 5. Normalized SIF $K_{3} / K_{3}^{\infty}$ is seen to begin to level off around R/a = 10 and converge to approximately 1.045 as the size of the physical domain increases. The convergence of SIF and a deviation of about 4.5% from $K_{3}^{\infty}$ confirm the accuracy of the results from Beasy.

Figure 4.

Penny crack of radius a in a circular cylinder of radius R and length L under axisymmetric torsion T.

Figure 5.

Normalized stress intensity factor as a function of R/a.

Penny crack under uniform shear

A penny crack in an infinite solid is subjected to a uniform shear on one of its crack surface as depicted in Figure 6. The other crack surface is subjected to the same load, but in the opposite direction. Kassir and Sih³ give $K_{1}^{\infty} = 0$ and

K_{2}^{\infty} = \frac{4 τ \sqrt{a}}{(2 - ν) \sqrt{π}} \cos (ϕ - β)

(4)

K_{3}^{\infty} = \frac{4 (1 - ν) τ \sqrt{a}}{(2 - ν) \sqrt{π}} \sin (ϕ + β)

(5)

Figure 6.

Penny crack of radius a in an infinite solid. The crack surfaces are subjected to a uniform inclined shear stress $τ$ . The shear stress on the top and bottom surfaces are equal and opposite. $β$ indicates the direction of the shear stress.

We compare the SIFs for infinite solid, that is, equations (4) and (5), with that of a penny crack of radius a embedded centrally in circular cylinders of radius R and length L under equal and opposite uniform shear (see Figure 7). Similar to the previous problem, this bounded-domain boundary value problem could be treated using the dual integral equations technique with the results presented by the Fredholm integral equations.^5,22,23 The long cylinder has a radius of R = 15 mm, length of L = 120 mm, and a crack radius of a = 0.5 mm. It is well known that under such shear loading at the ends of the cylinder, the shear stress would vary quadratically on the cross section (assuming crack free) of the long cylinder, with the maximum value at the neutral axis which equals to 1.5 times the uniform shear stress applied at the cylinder ends. Furthermore, over a domain with the size of the crack surface radius of a = 0.5 mm, that is, with the range of 0 < r/R < a/R = 0.033, the shear stress is almost constant as depicted in Figure 8. This maximum shear stress will be applied in equations (4) and (5) for assessment of the results of Beasy. Figure 9 depicts the comparison of the results from Beasy and the analytical results. The distributions are observed to be correct, and excellent accuracies of $K_{2} / K_{2}^{\infty} = 1.021$ and $K_{3} / K_{3}^{\infty} = 0.979$ at the location of maximum SIFs are obtained.

Figure 7.

Penny crack of radius a in a circular cylinder of radius R and length L under equal and opposite uniform shear $τ$ in the direction of $β$ .

Figure 8.

The shear stress distribution on the circular cross section of radius R. r/R is the normalized radial position.

Figure 9.

Normalized stress intensity factors along the crack front for uniform shear loading.

Comparison with NR’s solution for square prismatic bar under normal loading

In the absence of any kind of solutions for SIFs of elliptical cracks embedded within square prismatic bar under torsional loading, we compare the results of Beasy using normal loading with NR empirical solution.²⁴ The results of K₁ for aspect ratios b/a = 0.5, 1, and 2 subjected to far-field normal stress $σ$ are presented in Figure 10. These results are normalized with the SIF of a penny crack in infinite solid under normal loading; Kassir and Shi³ give this normalization factor as $K_{1}^{\infty} = (2 σ \sqrt{a}) / \sqrt{π}$ . Good agreement with NR solution is observed—the results overlap each other for most part, and there is a small difference for maximum values, albeit that for minimum seems to be sizable. We remark that NR closed-form solution is approximate.

Figure 10.

K ₁ due to normal loading on an embedded elliptical crack with aspect ratios b/a = 0.5, 1, and 2 within a square bar of dimensions 20 mm × 20 mm × 80 mm.

Numerical results and discussion

Prelude: results from elasticity

The theory of elasticity (see, for example, Timoshenko and Goodier⁴⁰) gives the xz and yz components of the shear stress of a rectangular in Figure 11(a) twisted with torque $M_{t}$ as follows

\begin{matrix} τ_{yz} = \frac{8}{k π^{2}} \sum_{n = 1, 3, 5, \dots}^{\infty} \frac{1}{n^{2}} {(- 1)}^{(n - 1) / 2} (1 - \frac{\cosh \frac{n π y}{2 A}}{\cosh \frac{n π B}{2 A}}) \sin \frac{n π x}{2 A} \\ τ_{xz} = - \frac{8}{k π^{2}} \sum_{n = 1, 3, 5, \dots}^{\infty} \frac{1}{n^{2}} {(- 1)}^{(n - 1) / 2} \frac{\sinh \frac{n π y}{2 A} \cos \frac{n π x}{2 A}}{\cosh \frac{n π B}{2 A}} \end{matrix}

(5)

The above are normalized by $τ_{\max} = M_{t} / (k_{2} (2 A)^{2} (2 B))$ where $B / A \geq 1$ . For square bars, $k_{2} = 0.208$ and $k = 0.675$ . The total shear stress is the vectorial sum, $τ = \sqrt{τ_{xz}^{2} + τ_{yz}^{2}}$ . Figure 11(b) shows the shear stress distributions along $X_{0}$ and $X_{π / 4}$ , and these results would be useful to understand the behavior of SIFs for cracks with offsets from the centroid.

Figure 11.

(a) Geometry of a rectangular bar with torque of M_t and (b) shear stress distributions.

Center cracks of different aspect ratio

Figure 12 depicts SIFs K₂ and K₃ for center cracks with aspect ratio $e = b / a \in {1, 1.25, 1.5, 1.75, 2}$ with b = 0.5 mm. K₁ is irrelevant as it would equal to zero for normally oriented cracks. For e = 1 (i.e. a penny crack), the in-plane sliding mode K₂ along the crack front is approximately zero and the anti-plane tearing mode K₃ is uniform. These could be expected intuitively as the shear stress field without cracks appears to be nearly concentric circles for small radial distances away from the centroid. The twofold symmetry of a penny crack about the centroid with respect to the loading results in the periodic behavior is seen in Figures 12 and 13. As the crack aspect ratio increases, the maximum value of K₂ increases and K₃ decreases; it is seen to shift toward the crack front positions of 1/4 and 3/4 for Mode II and stays at these positions for Mode III. As depicted in Figure 14, $K_{2}^{\max} < K_{3}^{\max}$ is observed for smaller b/a. In addition, $K_{2}^{\max}$ is observed to be increasing at a slower rate for e > 1.5, while $K_{3}^{\max}$ decreases more or less linearly with b/a. The change of position for these maximum as e changes is slight.

Figure 12.

K ₂ for embedded center cracks with different aspect ratios e (b = 0.5 mm).

Figure 13.

K ₃ for embedded center cracks with different aspect ratios e (b = 0.5 mm).

Figure 14.

The magnitude and position of $K_{2}^{\max}$ and $K_{3}^{\max}$ .

Eccentric cracks

We define the eccentricity $α$ in the sense of an offset from the centroid as $α \cdot A / (5 \cos θ) = true$ distance from centroid, and sample the locations of $α \in {0, 1, 2, 3, 4}$ on $X_{θ}$ (see Figures 11 and 15). Nearly symmetrical or anti-symmetrical profile about the major or minor elliptic axes is observed in Figure 16(a)–(d). $K_{2}^{\max}$ and $K_{3}^{\max}$ are found to be, respectively, at locations which are very close to the major and minor axes. The maximum SIFs as a function of the eccentricity, as depicted in Figure 16(e) and (f), are qualitatively similar to Figure 11(b); for all cases, $K_{2}^{\max} > K_{3}^{\max}$ is observed.

Figure 15.

Eccentric embedded crack with aspect ratio b/a = 1.

Figure 16.

SIFs for eccentric cracks of $α \in {0, 1, 2, 3, 4}$ along $X_{θ}$ : (a) K₂ along $X_{0}$ axis, (b) K₃ along $X_{0}$ axis, (c) K₂ along $X_{π / 4}$ axis, (d) K₃ along $X_{π / 4}$ axis, (e) $K_{2}^{\max}$ , and (f) $K_{3}^{\max}$ .

Penny crack located near to the free surface and corner under shear mode

Of practical interest is the variation of SIFs for penny crack situated close to the free surface and corner as illustrated in Figure 17. Smith and Alavi,¹⁸ Fischer et al.,¹⁹ and Shah and Kobayashi²⁰ presented some results for crack under opening mode near the free surface; Kobayashi et al.²¹ evaluated the tensile crack near a square corner. No results of any kind have been reported to date in the literature for penny cracks situated near free surface or corner loaded by shear stress.

Figure 17.

Penny crack situated close to the free surface and corner.

Due to the fourfold symmetry and anti-symmetry of geometry and torsional loading (Figure 11) of the square prismatic bar studied, the shear stress is orthogonal to both $X_{0}$ and $X_{π / 4}$ , that is, $β$ indicated in Figure 17 equals to $π / 2$ and $5 π / 4$ , respectively. For a small crack with small a/A as in this work, where a/A = 0.5/5 = 0.1 << 1, the variation of stress field along $X_{θ}$ over the distance of the crack size a is very small and could reasonably be approximated as constant. Therefore, the SIF results obtained using torsional loading could serve as an approximate equivalent system for a small penny crack subjected to equal and opposite uniform shear stress on its top and bottom crack surfaces. In Figure 18, the variation of SIFs for a penny crack under uniform shear obtained using torsional loading is plotted: variation along $X_{0}$ for penny crack situated near a free surface (Figure 18(a)) and variation along $X_{π / 4}$ for penny crack situated near a square corner (Figure 18(b)). The value plotted is the maximum value over the entire crack front $(K_{2}^{\max} and K_{3}^{\max})$ and normalized with $K^{\infty}$ as expressed in equations (4) and (5). The results of (a) are expected to be close to that for a half space since a/A is very small and (b) are expected to be close to that of a quarter space for the same reason. Along $X_{0}$ or for cracks near the free surface, the effect of the free surface on the in-plane sliding mode K₂ is seen to be insignificant; the tearing mode K₃ decreases as the orthogonal distance of the crack from the free surface h reduces. This trend is also observed in Figure 5. For all h/a, K₃ is larger than K₂. Along $X_{π / 4}$ or for cracks near a square corner, the effect of the corner on K₃ is similar to that along $X_{0}$ with a slightly smaller magnitude. K₂, however, is maximal near the corner and reduces as h/a increases, and the geometrical effect of the corner is observed to vanish roughly for the distance of $h / a \geq 8$ . These graphical presentations of SIFs variations under shear mode for penny crack may serve as a complement, albeit brief, to the opening mode results developed by authors mentioned earlier.^18–21

Figure 18.

Behavior of SIFs $(K_{2}^{\max} and K_{3}^{\max})$ for penny crack located near to the (a) free surface (along $X_{0}$ ) and (b) corner (along $X_{π / 4}$ ).

Conclusion

The SIFs for elliptical embedded cracks in a square prismatic bar subjected to torsion are evaluated by considering the elliptical aspect ratio and the eccentricity in the sense of an offset from the centroid. Using the results of eccentric cracks, the approximate shear mode SIFs of a penny crack situated close to the free surface and the square corner are also presented. In general, the results conform to the expectation by way of theory of elasticity. As the aspect ratio b/a increases, both in-plane and anti-plane SIF increase, and the most severe location is at or close to the semi-major axis. With offset, K₂ is maximal at a location close to the major axis while K₃ is maximal close to the minor axis. For penny crack situated near the free surface and subjected to uniform shear, $K_{2}^{\max}$ is observed to be relatively constant as the crack gets closer to the boundary; $K_{3}^{\max}$ increases as the orthogonal distance from the boundary increases. For penny crack near the square corner and subjected to uniform shear, $K_{2}^{\max}$ is maximal near the corner and the corner effect becomes insignificant roughly for distances of $h / a \geq 8$ . $K_{3}^{\max}$ , however, varies in a similar way to that for free surface.

Footnotes

Appendix 1

Handling Editor: Davood Younesian

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported, in part, by the UM-BKP Grant (UM.TNC2/IPPP/PPGP/638/BK007-2015) and Malaysia MOHE High Impact Research Grant (UM.C/625/1/HIR/MOHE/ENG/33).

ORCID iD

Haw Ling Liew

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Stress intensity factors for embedded cracks within torsionally loaded square prismatic bars

Abstract

Keywords

Introduction

Simulation of embedded elliptical cracks and evaluation of Beasy

Evaluation of SIFs using DBEM implemented in Beasy fracture mechanics package

Model geometry and embedded crack shape

Evaluation of Beasy with some available solutions

Embedded penny crack under axisymmetric torsion

Penny crack under uniform shear

Comparison with NR’s solution for square prismatic bar under normal loading

Numerical results and discussion

Prelude: results from elasticity

Center cracks of different aspect ratio

Eccentric cracks

Penny crack located near to the free surface and corner under shear mode

Conclusion

Footnotes

Appendix 1

Declaration of conflicting interests

Funding

ORCID iD

References