# Mutual Influence of Geometric Parameters and Mechanical Properties on Thermal Stresses in Composite Laminated Plates with Rectangular Holes

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## Abstract

**:**

_{s}for composite materials are examined in relation to the thermal stress distribution. The thermal insulated state and Neumann boundary conditions at the hole edge are taken into account. It is found out that the hole rotation angles and heat flux angle play key roles in obtaining the optimum thermal stress distribution around the hole. The present analytical method can well investigate the interaction of effective parameters on symmetric multilayer composites under heat flux.

## 1. Introduction

## 2. Literature Review

_{s}.

## 3. Analytical Solution

_{θ}) is the only stress created around the hole. In the present analysis, the assumption of plane stress is used, and the deformations are considered small. According to the classical laminated-plate theory (CLPT), it can be proved for a symmetric laminated plate that only the $[{A}_{ij}]$ matrix was remained as extensional stiffness. By introducing the stress function $U(x,y)$, the compatibility relationship of an anisotropic plate is defined in terms of the stress function as Equation (1) [12]:

^{(h)}and E

^{(p)}are homogeneous and particular parts, respectively. The homogeneous solution is derived using Lekhnitskii’s approach; by presenting the four linear first-order differential operators, Equation (1) is presented as Equation (3) [12]:

^{t}(x+s

_{t}y), where s

_{t}represents the roots of the characteristic equation (Equation (13)) [37]:

## 4. Validation of the Analytical Solution

_{s}. The parameter θ determines the angular position of the edge of the hole relative to the horizontal axis. According to Figure 4, the closeness of the results of the present method and FEM confirms the accuracy of the analytical method.

## 5. Results and Discussion

_{s}are considered. The maximum and minimum values of stress in this interval are called desirable and undesirable stresses, respectively. It should be noted that the default values of the parameters in Section 5 are δ = 270°, w = 0.05, β = 0°, and c = 1, unless the parameter is varied to study its effect on the thermal stress value. Moreover, the material property of the laminated composite is presented in Table 1.

#### 5.1. Effect of Hole Rotation Angles

_{s}in various values of bluntness is shown in Figure 6. By changing the rotation angle of the hole, the location of the occurrence of the highest normalized stress for each composite laminate also changes. According to Figure 6, the desirable rotation angle for graphite/epoxy composite laminate happens at β = zero, 90°, and 180°, and the undesirable thermal stress occurs at β = 45° and 135°. Whereas, the desirable stress for E-Glass/epoxy wet laminate happens similar to the graphite/epoxy laminate, and the undesirable stress occurs in the range of 35–55° (125–145°) for the stacking sequence of [45/−45]

_{s}. Moreover, it is observed that the maximum thermal stress for graphite/epoxy material is equal to 1.58 and that of the E-glass/epoxy wet material is equal to 1.24 when w = 0.07. Therefore, the value of thermal stress is different in both materials, and this value is related to the hole geometric parameters.

#### 5.2. Effect of Heat Flux Angle

_{s}. Figure 9 shows the values of thermal stress in terms of heat flux angle in different values of bluntness (w) parameter. The results show that when the flux angle changes, the location of the maximum normalized stress around the rectangular hole for two composite laminates changes. In other words, in the corners of the hole for the graphite/epoxy composite laminate, if the angle between the normal curve of the boundary of the hole and flux angle is closer to 45° or 135°, the maximum normalized stress will be reduced. Whereas, for E-glass/epoxy wet composite laminate, if the angle between the normal curve of the boundary of the hole and flux angle is closer to zero or 90°, the maximum normalized stress created in these regions increases, and if this angle is closer to 45◦, the maximum normalized stress depends on the hole geometry. According to Figure 9, the desirable thermal stress for graphite/epoxy composite laminate is equal to 1.056 when δ = 90° and w = 0.07, and the undesirable thermal stress is equal to 1.559 when δ = 45° or 135° and w = 0.04. Whereas, the desirable thermal stress for E-glass/epoxy wet composite laminate is equal to 0.956 when δ = 0°, 90°, and 180° and w = 0.04, but the undesirable thermal stress value is depended on the value of bluntness and heat flux angle parameters. However, the undesirable thermal stress in the stacking sequence of [45/−45]

_{s}is equal to 1.168 when δ = 35–55° or 125–145° and w = 0.07.

_{s}. As it is clear, the maximum normalized thermal stress of laminate containing a rectangular hole increases with increasing the aspect ratios of the hole.

_{s}, the desirable thermal stress value is obtained at β = 90° and δ = zero, 90°, and 180°. Whereas, the undesirable thermal stress value is obtained at β = 45° and δ = 45°or 135°. It can be seen at different rotation and flux angles that the graphite/epoxy plate has higher thermal stress than the E-glass/epoxy wet plate. Therefore, by selecting the appropriate values of effective parameters, the thermal stress distribution in the symmetric laminated composite containing a hole can be decreased remarkably.

## 6. Conclusions

_{s}were investigated on the thermal stress distribution. The results showed that the geometric parameters and material can influence the thermal stresses. According to the results, the hole angular position was found to be an important parameter influencing the thermal stress distribution around the hole. The results showed that the hole angular position of 90° at δ = 0° for the graphite/epoxy plate and hole angular position of 180° at δ = 0° for the E-glass/epoxy wet plate resulted in the minimum stress concentration for the laminate with a stacking sequence of [45/−45]s. Moreover, the desirable thermal stress distribution for the graphite/epoxy plate was obtained for c = 1 at δ = 0° and 180°, and for the E-glass/epoxy wet plate, it was achieved for c = 1 at δ = 90°. So, the heat flux angle and the hole orientation play key roles in obtaining a desirable thermal stress distribution around a rectangular hole. According to the obtained results the graphite/epoxy plate can tolerate higher thermal stress than the E-glass/epoxy wet plate. Unlike the isotropic material, the mechanical properties have a significant effect on the thermal stress distribution. Finite element analyses were employed for validating the analytical solutions. Reasonable agreement was obtained between the finite element and analytical results.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Bitsadze, L. Explicit solution of one boundary value problem of thermoelasticity for a circle with diffusion, microtemperatures, and microconcentrations. Acta Mech.
**2020**, 231, 3551–3563. [Google Scholar] [CrossRef] - Sarkar, N. Thermoelastic responses of a finite rod due to nonlocal heat conduction. Acta Mech.
**2020**, 231, 947–955. [Google Scholar] [CrossRef] - Alshaya, A.; Lin, S.J. Hybrid stress analysis of a near-surface circular hole in finite structures. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci.
**2019**, 234, 1366–1381. [Google Scholar] [CrossRef] - Song, H.P.; Song, K.; Schiavone, P.; Gao, C.F. Design of a neutral elastic inhomogeneity via thermal expansion. Acta Mech.
**2020**, 231, 2867–2876. [Google Scholar] [CrossRef] - Jafari, M.; Chaleshtari, M.H.B.; Abdolalian, H.; Craciun, E.-M.; Feo, L. Determination of forces and moments per unit length in symmetric exponential FG plates with a quasi-triangular hole. Symmetry
**2020**, 12, 834. [Google Scholar] [CrossRef] - Wang, X.; Schiavone, P. Uniformity of stresses inside a non-elliptical inhomogeneity near an irregularly shaped hole in plane elasticity. Mech. Mater.
**2020**, 145, 103389. [Google Scholar] [CrossRef] - Blesa Gracia, J.; Rammerstorfer, F.G. Increase in buckling loads of plates by introduction of cutouts. Acta Mech.
**2019**, 230, 2873–2889. [Google Scholar] [CrossRef] [Green Version] - Wang, S.R.; Wang, Y.H.; Gong, J.; Wang, Z.L.; Huang, Q.X.; Kong, F.L. Failure mechanism and constitutive relation for an anchorage segment of an anchor cable under pull-out loading. Acta Mech.
**2020**, 231, 3305–3317. [Google Scholar] [CrossRef] - Jafari, M.; Hoseyni, S.A.M.; Altenbach, H.; Craciun, E.-M. Optimum design of infinite perforated orthotropic and isotropic plates. Mathematics
**2020**, 8, 569. [Google Scholar] [CrossRef] [Green Version] - Groza, G.; Jianu, M.; Pop, N. Infinitely differentiable functions represented into Newton interpolating series. Carpathian J. Math.
**2014**, 30, 309–316. [Google Scholar] - Jafari, M.; Bayati Chaleshtari, M.H. Using dragonfly algorithm for optimization of orthotropic infinite plates with a quasi-triangular cut-out. Eur. J. Mech. A/Solids
**2017**, 66, 1–14. [Google Scholar] [CrossRef] - Lekhnitskii, S.G. Anisotropic Plates, 2nd ed.; Gordon and Breach Science: New York, NY, USA, 1968. [Google Scholar]
- Florence, A.L.; Goodier, J.N. Thermal stress at spherical cavities and circular holes in uniform heat flow. J. Appl. Mech.
**1959**, 26, 293–294. [Google Scholar] - Florence, A.L.; Goodier, J.N. Thermal stresses due to disturbance of uniform heat flow by an insulated ovaloid hole. J. Appl. Mech.
**1960**, 27, 635–639. [Google Scholar] [CrossRef] - Hasebe, N.; Wang, X. Complex variable method for thermal stress problem. J. Therm. Stresses
**2005**, 28, 595–648. [Google Scholar] [CrossRef] - Tarn, J.Q.; Wang, Y.M. Thermal stresses in anisotropic bodies with a hole or a rigid inclusion. J. Therm. Stresses
**1993**, 16, 455–471. [Google Scholar] [CrossRef] - Chao, C.K.; Chang, K.W. Contact stresses in an infinite plate containing a rigid circular inclusion under thermal loads. Acta Mech.
**2001**, 152, 95–108. [Google Scholar] [CrossRef] - Wang, X.; Dong, M.C.; Lu, G. Thermal elastic-plastic stress analysis of an anisotropic structure. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci.
**2003**, 217, 723–733. [Google Scholar] [CrossRef] - Ukadgaonker, V.G.; Rao, D.K.N. A general solution for stresses around holes in symmetric laminates under inplane loading. Compos. Struct.
**2000**, 49, 339–354. [Google Scholar] [CrossRef] - Sharma, D.S. Stresses around polygonal hole in an infinite laminated composite plate. Eur. J. Mech. A/Solids
**2015**, 54, 44–52. [Google Scholar] [CrossRef] - Goyat, V.; Verma, S.; Garg, R.K. Reduction of stress concentration for a rounded rectangular hole by using a functionally graded material layer. Acta Mech.
**2017**, 228, 3695–3707. [Google Scholar] [CrossRef] - Dai, M.; Yang, H.-B.; Schiavone, P. Stress concentration around an elliptical hole with surface tension based on the original Gurtin–Murdoch model. Mech. Mater.
**2019**, 135, 144–148. [Google Scholar] [CrossRef] - Damghani, M.; Harrison, C.; Kennedy, D. The effects of composite laminate stiffness and loading on stress resultant concentration factor around a hole. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci.
**2018**, 232, 1033–1049. [Google Scholar] [CrossRef] - Choi, H.J. Thermal stresses due to a uniform heat flow disturbed by a pair of offset parallel cracks in an infinite plane with orthotropy. Eur. J. Mech. A/Solids
**2017**, 63, 1–13. [Google Scholar] [CrossRef] - Xiao, J.; Xu, Y.; Zhang, F. An analytic solution for the problem of two symmetrical edge cracks emanating from a circular hole with surface effect under antiplane shear. Acta Mech.
**2018**, 229, 4915–4925. [Google Scholar] [CrossRef] - Guo, L.-C.; Noda, N. An analytical method for thermal stresses of a functionally graded material cylindrical shell under a thermal shock. Acta Mech.
**2010**, 214, 71–78. [Google Scholar] [CrossRef] - Khan, K.A.; Barello, R.; Muliana, A.H.; Lévesque, M. Coupled heat conduction and thermal stress analyses in particulate composites. Mech. Mater.
**2011**, 43, 608–625. [Google Scholar] [CrossRef] - Zhang, Z.; Demir, K.; Gu, G.X. Computational analysis of thermally induced stress concentration in structures with geometric constraints. Mech. Mater.
**2019**, 133, 102–110. [Google Scholar] [CrossRef] - Li, L.; Jia, P.; Pan, W. Temperature effect on the tensile behaviors of carbon/polyimide composite laminate. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci.
**2016**, 231, 4592–4602. [Google Scholar] [CrossRef] - Mahmoudi, H.; Atefi, G. Analytical solution for thermal stresses in a hollow cylinder under periodic thermal loading. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci.
**2011**, 226, 1705–1724. [Google Scholar] [CrossRef] - Jafari, M.; Nazari, M.B.; Taherinasab, A. Thermal stress analysis in metallic plates with a non-circular hole subjected to uniform heat flux. Eur. J. Mech. A/Solids
**2016**, 59, 356–363. [Google Scholar] [CrossRef] - Jafari, M.; Bagher Nazari, M.; Taheri Nasab, A. Study of the effective parameters on stress distribution around triangular hole in metallic plates subjected to uniform heat flux. J. Therm. Stresses
**2016**, 39, 333–344. [Google Scholar] [CrossRef] - Rasouli, M.; Jafari, M. Thermal stress analysis of infinite anisotropic plate with elliptical hole under uniform heat flux. J. Therm. Stresses
**2016**, 39, 1341–1355. [Google Scholar] [CrossRef] - Chao, C.K.; Chen, F.M.; Lin, T.H. Thermal stresses induced by a remote uniform heat flow interacting with two circular inclusions. J. Therm. Stresses
**2017**, 40, 564–582. [Google Scholar] [CrossRef] - Chao, C.-K.; Gao, B. Mixed boundary-value problems of two-dimensional anisotropic thermoelasticity with elliptic boundaries. Int. J. Solids Struct.
**2001**, 38, 5975–5994. [Google Scholar] [CrossRef] - Chiang, C.-R. Thermal stress at a circular hole in cubic crystals under uniform heat flow. Acta Mech.
**2018**, 229, 3963–3969. [Google Scholar] [CrossRef] - Jafari, M.; Jafari, M. Thermal stress analysis of orthotropic plate containing a rectangular hole using complex variable method. Eur. J. Mech. A/Solids
**2019**, 73, 212–223. [Google Scholar] [CrossRef] - Jafari, M.; Jafari, M. Effect of hole geometry on the thermal stress analysis of perforated composite plate under uniform heat flux. J. Compos. Mater.
**2018**, 53, 1079–1095. [Google Scholar] [CrossRef] - Hahn, D.W.; Özişik, M.N. Heat Conduction Fundamentals. In Heat Conduction, 3rd ed.; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 2012; pp. 1–39. [Google Scholar]

**Figure 1.**Symmetric composite laminated containing a rectangular hole subjected to steady heat flux [37].

**Figure 4.**FEM and analytic solution comparison for different composite materials in various stacking sequence.

**Figure 8.**Effect of hole rotation angle on the maximum normalized stress in different values of $\delta $.

**Figure 11.**Effect of heat flux angle on the maximum normalized stress in different values of $\beta $.

**Table 1.**Mechanical properties of the studied materials [33].

Material | E_{11} (GPa) | E_{22} (GPa) | G_{12} (GPa) | υ_{12} | α_{11} (K^{−1}) | α_{22} (K^{−1}) | K_{11} (Wm^{−1} K^{−1}) | K_{22} (Wm^{−1} K^{−1}) |
---|---|---|---|---|---|---|---|---|

Graphite/epoxy | 144.8 | 9.7 | 4.1 | 0.3 | −3 × 10^{−6} | 2.8 × 10^{−5} | 4.62 | 0.72 |

E-Glass/epoxy wet | 35 | 9 | 4.7 | 0.28 | 5.5 × 10^{−6} | 2.5 × 10^{−5} | 2.2 | 1.1 |

${E}_{33}={E}_{22}$$,{G}_{12}={G}_{13}={G}_{23}$, ${\upsilon}_{12}={\upsilon}_{13}={\upsilon}_{23}$$,{\alpha}_{33}={\alpha}_{22}$, ${K}_{33}={K}_{22}$$,{K}_{12}=0$ |

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**MDPI and ACS Style**

Chaleshtari, M.H.B.; Jafari, M.; Khoramishad, H.; Craciun, E.-M.
Mutual Influence of Geometric Parameters and Mechanical Properties on Thermal Stresses in Composite Laminated Plates with Rectangular Holes. *Mathematics* **2021**, *9*, 311.
https://doi.org/10.3390/math9040311

**AMA Style**

Chaleshtari MHB, Jafari M, Khoramishad H, Craciun E-M.
Mutual Influence of Geometric Parameters and Mechanical Properties on Thermal Stresses in Composite Laminated Plates with Rectangular Holes. *Mathematics*. 2021; 9(4):311.
https://doi.org/10.3390/math9040311

**Chicago/Turabian Style**

Chaleshtari, Mohammad Hossein Bayati, Mohammad Jafari, Hadi Khoramishad, and Eduard-Marius Craciun.
2021. "Mutual Influence of Geometric Parameters and Mechanical Properties on Thermal Stresses in Composite Laminated Plates with Rectangular Holes" *Mathematics* 9, no. 4: 311.
https://doi.org/10.3390/math9040311