Level Set function-based Functionally Graded Material for the reduction of maximum stresses around a pair of inclined unequal circular holes

Abstract

OBJECTIVE:

The aim of this work is to present the methodology for grading the Functionally Graded Material (FGM) using Level Set (LS) sign distance function around the multiple holes and parametrically analyse the maximum stresses for a pair of inclined unequal circular holes surrounded by the FGM layer in an infinite plate subjected to uniaxial tensile load using the Extended Finite Element Method (XFEM).

METHODS:

The LS method has the ability to represent the multiple geometrical boundaries with a single sign distance function which can be effectively used for grading the FGM around the multiple discontinuities such as holes, inclusions, cracks, etc. When dealing with FGM material grading around multiple discontinuities, it is important to have smooth grading to minimise the stress concentration. The grading of the material with multiple functions may result in sharp changes in the material properties at the interference region which may lead to high stresses. The LS function-based FGM material grading eliminates such sharp changes as it uses a single function.

RESULTS:

The parametric analysis shows that applying the LS function-based power law FGM layer of Titanium – Titanium Mono Boride (Ti-TiB) around the pair of inclined unequal circular holes significantly reduces the values of maximum tensile as well as compressive hoop stresses when compared with the homogeneous material case.

Keywords

Level Set Method FGM XFEM stress concentration pair of circular holes

1. Introduction

In modern engineering as well as space applications, one cannot imagine hole/cutout free mechanical component/structure, as they are necessary to put two or more components together or make a component/structure accessible for e.g. electrical/pneumatic/hydraulic lines. These holes/cutouts disturb the stress distribution and cause stress concentration which may lead to mechanical failure of component/structure. One can trace back a detailed stress distribution and concentration analysis of multiple equal size holes in the work of Howland and Knight [13]. Green [9] analysed a group of circular holes and also presented their general mathematical solution. Ling [18] solved the problem of an infinite plate with a pair of circular holes using Minslin’s bi-polar method. Haddon [11] obtained the solution for non-equal, non-intersecting pair of circular holes in an infinite 2D domain under inclined loading. Zimmerman [36] determined the stress singularity around a pair of holes. Zhang et al. [34] obtained the solution of the plain elasticity problem of stress concentration around multiple elliptical holes. Hoang and Abousleiman [12] extended Green’s solution [9] for a pair of unequal holes.

For isotropic materials, it is observed that the stress concentration basically depends on geometrical properties and independent of material properties. But it is not true for non-isotropic materials, especially for FGMs. They are new generation composite materials with varying material properties along with the physical dimension(s). Recently, it has been observed that these materials can also be used to reduce the stress concentration factor (SCF) around holes/cutouts. The first such attempt was made by Kubair and Bhanu-Chandar [17]. The SCF in FGM plate with a central circular hole under tensile loading was analysed by them using Finite Element Method (FEM). Yang et al. [28,29,31,32] performed SCF analysis on an exponential function-based FGM plate with central circular hole for different load conditions. Mohammadi et al. [19] obtained Frobenius series solution for FGM plate graded in radial direction and weakened by central circular hole. Kubair [15,16] investigated the SCF problem around circular hole in FGM plate for anti-plane shear loading. Sburlati et al. [23,24] introduced the concept of FGM layer. They used FGM around the hole and rest of the plate is of homogeneous material. Ashrafi et al. [1] obtained boundary element solution for SCF near circular hole in 3D FGM plate. Enab [5] investigated the SCF near elliptical hole in different FGM plates. Bhat and Ukadgaonker [3] presented a review on SCF around hole and their solution methodology. Gouasmi et al. [6] attempted the SCF reduction near elliptical notch using FGM layer. Yang et al. [30] used Complex Variable Theory (CVT) for obtaining the value of SCF around elliptical hole in 2D plate. Goyat et al. [7,8] investigated reduction in SCF around single and pair of circular holes using FGM layer. Chen et al. [4] enhanced the load capacity of a plate weakened by circular hole using FGM material. Guan and Li [10] solved the problem of SCF around arbitrary hole surrounded by FGM layer and subjected to far field anti-plane shear load. Nie et al. [20,21] tailored the FGM material to achieve the specific SCF near the hole. Zheng et al. [35] analysed the SCF reduction near FGM reinforced sphere. Wang et al. [26] obtained the SCF and damage factor for the FGM plate with elliptical hole using FEM. Yang et al. [27] investigated SCF near circular hole for FGM plate subjected to out-of-plane bending loading. Yesil and Atasayanlar [33] analysed the FGM plate having circular hole under bending load using FEM.

From the literature, it has been found that the power law-based radially graded FGM materials of lower value of Young’s modulus near the hole/cutout and higher value of Young’s modulus away from the hole/cutout are capable of reducing the SCF significantly. Most of the available literature is about the analysis and reduction of SCF around the single hole or cutout using FGM material for different load conditions. However, multiple or series of holes/cutouts also appears frequently in the mechanical structures. This work aims to present methodology for grading the FGM using LS function and to perform parametric analysis for the maximum stresses around pair of inclined unequal circular holes surrounded by FGM layer in an infinite plate subjected to uniaxial tensile load using XFEM. A power law-based radial Ti-TiB FGM layer around the pair of holes has been used to reduce the stress concentration in TiB plate. The present work can be helpful for material designers to model FGM around multiple discontinuities or geometries.

2. Materials and methods

2.1. Level Set Method (LSM)

LSM was introduced by Osher and Sethian [22] to represent arbitrary geometry in a higher dimension sign distance function. It is very helpful in XFEM to represent the geometrical discontinuities with the mesh. As mesh is independent and non-conformal with the geometrical discontinuities in XFEM. The beauty of this method is that one can make a single LS function for the multiple geometrical discontinuities. In this work, LSM is used to define the pair of holes in XFEM as well as to make FGM material model in normal direction to the pair of holes.

Fig. 1.

Problem geometry.

Figure 1 shows the geometry and the load condition of the present problem. The thin square panel of infinite length having two circular hole of different radius subjected to far-field uni-axial tensile stress of unit magnitude. The radius of bigger hole is R and smaller hole is r. The coordinate system is considered at the centre of panel and the centres of holes are equidistant from the origin. The distance between the centres of holes is s and the maximum distance between outmost points of holes circumference is l. The line passing through the centre of holes make an angle (𝜃) with the x-axis. The holes are reinforced with the layer of FGM of thickness (t).

Fig. 2.

Contour plot of LS function for (a) bigger hole of radius R, (b) smaller hole of radius r and (c) both holes.

To model the hole of radius R by LSM, the sign distance function for any point (x, y) in the panel can be written as: $\begin{eqnarray}\displaystyle {\phi}_{R}(x,y)=\sqrt{(x-x_{R})^{2}+(y-y_{R})^{2}}-R & & \displaystyle\end{eqnarray}$ (1) where, x _R and y _R are centre coordinates of the bigger hole. Figure 2(a) shows the contour plot of the sign distance function 𝜙_R, the values of the function show the normal distance from the hole boundary and the sign of the values define whether the point is inside, outside or at the hole. The hole boundary can be traced by the zero value contour (𝜙_R = 0), shown by white circle in Fig. 2. The sign distance function 𝜙_r for the smaller hole can be written using centre coordinates x _r and y _r as: $\begin{eqnarray} \displaystyle {\phi}_{r}(x,y)=\sqrt{(x-x_{r})^{2}+(y-y_{r})^{2}}-r & & \displaystyle \end{eqnarray}$ (2) Figure 2(b) shows the contour plot of the LS function 𝜙_r for the smaller hole of radius r, the white circle shows the hole boundary, i.e. 𝜙_r = 0. The single LS function for both holes is formed as: $\begin{eqnarray} \displaystyle {\phi}(x,y)=\min [{\phi}_{R}(x,y),{\phi}_{r}(x,y)] & & \displaystyle \end{eqnarray}$ (3) The contour plot of combined LS function is shown in Fig. 2(c). The combined LS function 𝜙(x, y) is used in this work to define the pair of holes in XFEM as well as the FGM material modelling.

2.2. FGM material model

The square panel is assumed to be made of Titanium mono-Boride (TiB) and the FGM layer is considered of Titanium (Ti) and TiB with varying composition along the normal direction to the hole boundaries, i.e. as per LS function. The Young’s modulus (E) and Poisson’s ratio of Ti and TiB are as follows: $\begin{eqnarray} \displaystyle \text{} & \text{} & \displaystyle E_{\text{Ti}}=107∼\text{GPa},\quad v_{\text{Ti}}=0.14\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle E_{\text{TiB}}=375∼\text{GPa},\quad v_{\text{TiB}}=0.34\nonumber \end{eqnarray}$ The Young’s modulus is considered to be varied along the normal direction (with LS sign distance function 𝜙(x, y)) to the hole boundary with the power law function as: $\begin{eqnarray}\displaystyle E(x,y)=E_{\text{Ti}}+(E_{\text{TiB}}-E_{\text{Ti}})\left(\frac{{\phi}(x,y)}{t}\right)^{n} & & \displaystyle\end{eqnarray}$ (4) where, n is power law exponent and t is taken as thickness of the FGM layer. The n = 0 represents the homogeneous material plate of E _TiB and n = 1 represents linear variation of E in FGM layer. The range of n is selected as 0 to 5 as per the literature. Figure 3 shows the variation of E for different values of n. The variation in Poisson’s ratio affects stress distribution and stress concentration insignificantly; therefore, Poisson’s ratio of FGM is considered as 0.34.

Fig. 3.

Variation of E with LS function 𝜙(x, y) for different values of n.

Figure 4 depicts the variation of E in the panel for different s/r values. The benefit of modeling of FGM using LS function can be observed clearly from Fig. 4. For the low s/r ratio where the FGM layers of one hole interfere with the FGM layer of other, Young’s modulus varies smoothly because the FGM is graded with a single Level Set (LS) function.

Fig. 4.

Variation of E in the panel for different values of normalized hole distance s/r (a) 0.5, (b) 1, (c) 2.5, (d) 5, (e) 7.5 and (f) 10.

2.3. Extended Finite Element Method (XFEM)

The XFEM was introduced by Belytschko and Black [2] to model discontinuities without conformal mesh. This method uses the concept of partition of unity and utilises the enrichment function to estimate the behaviour of field variables near the discontinuity. In the present work, XFEM is used to take advantage of its high accuracy, simple and non-conformal mesh. The LSM is used to define the hole boundaries within mesh of quadrilateral 8 node elements. The enrichment function used in present work was initially proposed by Sukumar et al. [25]. The displacement field (u) for finite elements can be defined using nodal shape function (N _i), nodal displacement (u _i) and LSM-based enrichment functionV _i(𝜙) as: $\begin{eqnarray} \displaystyle u=\sum N_{i}u_{i}V_{i}({\phi}) & & \displaystyle \end{eqnarray}$ (5) The value of enrichment function V _i(𝜙) is decided on the following basis: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle V_{i}({\phi})=1\quad \text{for }{\phi}\geq 0\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle V_{i}({\phi})=0\quad \text{for }{\phi}\lt 0.\end{eqnarray}$ (6) The following constitutive relation has been used to calculate the stress field (𝜎) in the considered 2D plane stress elasto-static FGM plate shown in Fig. 1: $\begin{eqnarray}\displaystyle {\sigma}=D(x,y):{\varepsilon} & & \displaystyle\end{eqnarray}$ (7) where, 𝜀 is strain field and D (x, y) is constitutive relation matrix. The expanded form of Eq. (7) can be written as: $\begin{eqnarray} \displaystyle \left\{ \begin{array}{@{} c@{}}{\sigma}_{xx}\\ {\sigma}_{yy}\\ {\tau}_{xy} \end{array} \right\}=\frac{E(x,y)}{1-v^{2}}\left[ \begin{array}{@{} ccc@{}}1 & v & 0\\ v & 1 & 0\\ 0 & 0 & {\displaystyle \frac{1-v}{2}} \end{array} \right]\left\{ \begin{array}{@{} c@{}}{\varepsilon}_{xx}\\ {\varepsilon}_{yy}\\ {\gamma}_{xy} \end{array} \right\} & & \displaystyle \end{eqnarray}$ (8) The Young’s modulus within the element has been calculated using following graded finite element relation which was initially proposed by Kim et al. [14]: $\begin{eqnarray} \displaystyle E(x,y)=\sum N_{i}E_{i} & & \displaystyle \end{eqnarray}$ (9) For the enriched elements, i.e. these elements having the hole boundary, a different scheme of numerical integration is used. In this scheme, the element is subdivided into triangular subelements and these triangles must be conformal with the boundary of hole as well as element. The triangular gauss quadrature is applied in each subelement. Further, for non-enriched, i.e. simple finite elements, the 3-point gauss quadrature is used.

Fig. 5.

Comparison of the present work with (a) Haddon [11] for R∕r = 1, (b) Haddon [11] for 𝜃 = 0°, (c) Haddon [11] for 𝜃 = 90° and (d) Sburlati et al. [24].

3. Results and discussion

The XFEM computer code was prepared using MATLAB language to analyse the pair of non-equal inclined holes in an infinite square panel of 40l (see Fig. 1). The quadrilateral 8 node elements were used to model the nonlinear behaviour of FGM efficiently. FGM was graded in the normal direction to the boundary of both the holes using LS function (see Eq. (4)). To validate and check the accuracy of computer code, the obtained results were compared with the work of Haddon [11] and Sburlati et al. [24]. To make the current FGM function equivalent to the work of Haddon [11], the value of power law exponent (n) was taken as 0, which represents the homogeneous material case. Figure 5(a) shows the result of present work and their comparison with Haddon [11] for a pair of equal size circular hole with inclination angle 𝜃 = 0°, 45° and 90°. A good agreement is noticed from Fig. 5(a) in the results of present work and Haddon [11]. Figures 5(b and c) show the comparative analysis of present work with Haddon [11] for radius ratio (R∕r = 1, 2, 5 and 10) with 𝜃 = 0° and 90° respectively. The results obtained by the developed code had an error in the range of 1.5%. To make the current FGM function equivalent with Sburlati et al. [24] the following changes were made in Eq. (4), R∕r = 1, s = 0, n = 0.5, 𝜙 = 𝜙 + r, t = t + r and E _Ti = 0. The obtained results of present work and that of Sburlati et al. [24 ] are depicted in Fig. 5(d), wherein it is observed that the results of the present work are quite accurate and in good agreement with Sburlati et al. [24 ].

After the validation of developed computer code, the Geometric and FGM parameter and their values were selected on the basis of literature survey for parametric analysis. These are: (a) normalized hole distance s∕r = 0.5 to 10; (b) power law coefficient n = 0, 0.25, 0.5, 1, 2 and 5; (c) normalized FGM layer thickness t∕r = 0.5, 1, 2 and 4; (d) radius ratio R∕r = 1, 2, 5 and 10; (e) inclination angle of line passing through center of hole with x-axis 𝜃 = 0°, 15°, 30°, 45°, 60°, 75° and 90°.

Fig. 6.

Maximum tensile and compressive hoop stress relations with s/r and n, for t∕r = 1 and different values of 𝜃 = (a) 0°, (b) 15°, (c) 30°, (d) 45°, (e) 60°, (f) 75° and (g) 90°.

Fig. 6 (Continued).

The variations of maximum tensile and compressive hoop stresses with s/r and n were obtained and depicted in Fig. 6(a–g) for different values of 𝜃 = 0°, 15°, 30°, 45°, 60°, 75° and 90° respectively. The normalised FGM layer thickness was taken as t∕r = 1. For all the cases of FGMs the maximum tensile and compressive stresses are observed to be less than that the homogeneous material case, i.e. n = 0. The trends of different FGM cases are found to be nearly the same as that of homogeneous case with reduced values stresses for 𝜃 = 0° to 45° and for a particular value of n.

However, for 𝜃 > 45°, a sharp rise in maximum tensile stresses around s∕r = 2t∕r is noticed. The sharp rise in the stresses is due to the interaction of the FGM layers and it is observed that this effect is higher for higher values of n therefore, it suggests that lower value of power law coefficient n is preferable. On the other hand, there are no such changes in the trends of the maximum compressive stresses.

For a particular s∕r and 𝜃 value, it is noticed that the maximum tensile stress first decreases and then increases with increase in n. Further, the maximum compressive stresses first decrease and then remains almost constant with increase in n.

One can choose n = 0.75, as it shows least values of maximum tensile and compressive stresses in most of the cases. For low values of s/r, the percentage reduction in maximum tensile stresses for n = 0.75 (in comparison with homogeneous material case) is found to be first increase and then decreases with increase in 𝜃. The maximum reduction is observed as 56% at 𝜃 = 45° and minimum reduction as 25% at 𝜃 = 90°. But, for higher values of s/r the reduction in maximum tensile stress is almost 51%. However, for maximum compressive stresses the reduction is nearly constant, about 57.5 ± 2.5%.

Figure 7(a–g) shows the variations of maximum tensile and compressive hoop stresses with s/r and t/r for n = 0.75. It is observed from Fig. 7 that for a particular value of s/r, the maximum tensile stress decreases with increase in t/r from 0 to 2 and further increase in t/r shows mixed response. However, the maximum compressive stresses are nearly same for all values of t/r and particular value of s/r and 𝜃.

Fig. 7.

Maximum tensile and compressive hoop stress relations with s/r and t/r, for n = 0.75 and different values of 𝜃 = (a) 0°, (b) 15°, (c) 30°, (d) 45°, (e) 60°, (f) 75° and (g) 90°.

Fig. 7 (Continued).

The interaction effect of FGM layers can be observed in Fig. 7(f and g) for 𝜃 = 75° and 90° respectively. The interaction effect is higher in the cases of low values of t/r, i.e. t∕r = 0.5 and 1. From Fig. 7, the normalised FGM layer thickness t∕r = 2 can be chosen as it shows least stresses for most of the cases. The reduction in maximum tensile stresses for t∕r = 2 case, when compared with t∕r = 0 i.e. homogeneous material case, is observed as maximum of 58% at s∕r = 10 and 𝜃 = 0° and minimum of 38% at s∕r = 0.75 and 𝜃 = 90°. However, for maximum compressive stresses, the reduction is around 57% in all FGM cases.

To make a comparison, the homogeneous and best FGM case (n = 0.75, t∕r = 2) of R∕r = 1, 2, 5 and 10 are depicted in Fig. 8(a–g) for different values of 𝜃 = 0°, 15°, 30°, 45°, 60°, 75° and 90° respectively. The FGM layer of t∕r = 2 and n = 0.75 shows the reduced maximum stress values (both tensile and compressive) as compared with the homogeneous material cases for all values of R/r. For a particular s/r value, the change in R/r shows less variation in the values of maximum tensile stress in homogeneous material cases of 𝜃 = 0° to 30° but, for FGM cases significant changes have been noticed, the maximum tensile stress increases with increase in R/r. The homogeneous material cases of 𝜃 > 45° show sharp increase in maximum tensile stress with increase in R/r, however, by applying FGM ring the maximum tensile stress reduced significantly. The reduction in maximum tensile stress, as compared with homogeneous material case, is observed as max 58% at s∕r = 10, R∕r = 1 and 𝜃 = 0° and min 24% at s∕r = 0.5–10, R∕r = 10 and 𝜃 = 0°. Further, for maximum compressive stresses the reduction is noticed as maximum of 61% at s∕r = 10, R∕r = 1 and 𝜃 = 0° and minimum of 25% at s∕r = 2, R∕r = 10 and 𝜃 = 0°.

Fig. 8.

Maximum tensile and compressive hoop stress relations with s/r and R/r, for n = 0.75, t∕r = 2 and different values of 𝜃 = (a) 0°, (b) 15°, (c) 30°, (d) 45°, (e) 60°, (f) 75° and (g) 90°.

Fig. 8 (Continued).

The results of FGM cases show that the maximum stresses around the pair of inclined unequal holes can be controlled by the FGM parameters. It is observed that the FGM layer of n = 0.75 and t∕r = 2 gives the least values of maximum stresses for most of the cases of R∕r = 1. Further, the FGM layer of n = 0.75 and t∕r = 2 significantly reduce the values of maximum stresses for R∕r > 1. However, it is also observed that the optimisation analysis is required to be done to get the least values of stresses for individual cases.

4. Conclusions

The methodology for grading the FGM with LS sign distance function around the multiple holes and parametric analysis for maximum stresses for the pair of inclined unequal circular holes surrounded by the FGM layer of Ti-TiB in an infinite plate of TiB subjected to unity uniaxial tensile load using XFEM are presented in this work. The conclusions from the present work are as follows:

The maximum tensile and compressive hoop stresses around the pair of inclined unequal circular holes can be reduced significantly by using the LS function-based power law FGM layer.

The power law coefficient n plays a significant role in the reduction of maximum tensile and compressive hoop stresses. The interaction effect of FGM rings can be controlled by selecting the n < 1. The n = 0.75 shows least values of maximum tensile and compressive hoop stresses for most of the cases considered in the study.

The normalised FGM layer thickness t∕r = 2 is found to be suitable for most of the cases. But it is also observed that case to case basis optimisation is required to be done to get least values of stresses.

The radius ratio of holes R/r greatly affected the values of maximum hoop stresses, especially for 𝜃 > 45°.

The maximum compressive hoop stress is not a serious concern, if the FGM layer is used around the hole, as its value can be controlled to bring down below 1.

The present work may help the material engineers in efficiently grading the FGM around multiple holes or other structural discontinuities with a single LS function in order to get optimised values of maximum hoop stresses.

Footnotes

Conflict of interest

None to report.

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