Effect of temperature on dynamic behavior of cracked metallic and composite beam
Mainul Hossain^{1} , Jaan Lellep^{2}
^{1, 2}University of Tartu, Tartu, Estonia
^{1}Corresponding author
Vibroengineering PROCEDIA, Vol. 32, 2020, p. 172178.
https://doi.org/10.21595/vp.2020.21337
Received 14 February 2020; received in revised form 3 March 2020; accepted 10 March 2020; published 29 June 2020
JVE Conferences
EulerBernoulli beam model is used to analyze the thermal vibration of cracked metallic and laminated composite beam with fully clamped ends. An analytical solution technique is proposed to investigate the natural vibration of cracked beam subjected to axial thermal load. An open and stable crack is considered perpendicular to the beam axis. In this study, three different types of material such as steel, copper and aluminum and their composition are used to analyze the thermal effect on different modes of frequency. The effects of thermal stress on the vibration of cracked metallic and composite fully clamped beam are revealed. The results show that the effect of temperature is significant on the natural frequency of cracked metallic and composite beam.
 The effect of thermal vibration depends on length to height ratio of the beam
 The effect of temperature is more significant in first mode of frequency
 Under thermal load composite beam is very effective than metallic beam
Keywords: effect of temperature, natural frequency, crack, composite, beam.
1. Introduction
Effect of temperature is one of the important features for designing of structures efficiently. Unfortunately, thermal effect is ignored in most of the research which leads to incorrect results. A small change in temperature may cause a large variation in the behavior of beam vibration [1]. Thermal stresses are considered as mechanical stresses of the elements. It is the result of contraction or expansion of structural elements due to the applied thermal load. When an element is subjected to the rise of temperature it expands and when it loses temperature, it contracts. The expansion and contraction of the element due to the temperature is proportional to the change of temperature. This proportionality depends on the coefficient of linear thermal expansion of the material [2]. According to the theory of linear thermoelasticity, the thermal strain is linearly added with the mechanical strain. That is why the equations of equilibrium and compatibility of thermal problems are the same as the elasticity problem. According to this assumption, many techniques that have been used to solve the elasticity problems are applicable to the thermoelasticity problem as well [3].
Interest in multifunctional composite materials has increased rapidly in recent years. Laminated composite materials are also a very efficient type of composition. Laminated composite materials are formed by different layers of two or more materials that are effectively bonded together. It is a combination of a large number of fibers in a thin layer of matrix. Fibers in the lamina may be continuous or discontinuous, arranged in a specific direction or in a random orientation [4]. Lamination is a very useful technique to achieve more efficient material. Lamination can emphasize the properties of material such as strength, stiffness, low weight, corrosion resistance, thermal insulation, etc. [57].
The effect of thermal vibration on the cracked composite beam is very rare in the open literature. Therefore, thermal effects on macro mechanics also achieve substantial attention among researchers. Zhang et al. [8] studied the thermal effects on highfrequency vibration of beams. Their results showed that the spatial distributions and levels of energy density can be affected by the thermal effects. Cui and Hu [9] examined thermal buckling and natural vibration of the beam with an axial stickslipstop boundary. Lia and Zhang [10] investigated the thermal effect on vibration and buckling analysis of thin plates with cracks. Their results revealed that the dynamic behavior of structural elements is significantly affected by thermal effects.
In this paper, the effects of temperature on the dynamic behavior of cracked metallic and composite beam are investigated. Our main concern is to compare the effectiveness of a fully clamped metallic and composite beam under crack and temperature. The governing differential equations with boundary and intermediate conditions are solved using analytical techniques. The effects of crack severity, crack position and beam length to height ratio on the frequencies of the heated beam are investigated. On the other hand, the effect of temperature on the vibration of the cracked composite beam is analyzed and compared with the metallic beam in detail.
2. Settings of the problem
A diagram of a cracked beam is illustrated in Fig. 1. The cross section of the beam is rectangular. $L$ is the length and $b$ is the width of the beam respectively. Beam is isotropic and homogeneous where $\rho $ is the density of the metallic beam. Single crack is placed at the location of $x=a$ of the beam. The crack is considered as stable crack whose depth is $c$ and $s$ is the ratio of crack height and beam height. Three different types of material (Aluminium, Copper, and Steel) are considered for the metallic beam as shown in Table 1.
A composite laminated material is also considered for the beam which is formed by three different materials such as aluminum, copper, and steel in a certain combination.
Fig. 1. Geometry of the problem
Table 1. Material properties
Material

Young’s modulus (GPa)

Coefficient of thermal expansion (°C)

Density (kg/m^{3})

Aluminium

69

24x10^{6}

2780

Copper

117

17x10^{6}

8960

Steel

200

13x10^{6}

8050

3. Mathematical formulation
A beam is considered along the $X$ axis where transverse deflection $W\left(X,t\right)$ is described along the $Y$ axis. According to the theory of EulerBernoulli for small deflection, the equation of motion expressed by the transverse deflection $W\left(X,t\right)$ can be obtained as:
For separating the variable, this function $W\left(X,t\right)=\stackrel{}{W}{e}^{i{\omega}_{c}t}$ can be used. Then the Eq. (1) becomes:
Let us assume that the beam is homogenous having same temperature over its entire length. As a result of thermal expansion, an additional axial force ${N}_{T}$ can be presented as:
$\alpha $ is the linear thermal extension coefficient, $\theta $ is the temperature difference between the actual and initial temperature. Let us introduce dimensionless variables:
The Eq. (2) becomes:
Using this function $w\left(x\right)=A{e}^{i\lambda x}$, we can write the characteristic equation as follow:
Roots of the equation can be presented as follow:
Solutions of the equation are:
Boundary conditions for the fully clamped beam as follow:
These four boundary conditions and other four intermediate conditions for crack are used to solve the Eqs. (6) and (7). For crack, the stress intensity factor for the single edge notched tension is presented as follow (see [11]):
Using the intensity factor intermediate conditions for the crack can be defined as below:
${w}_{i+1}^{\text{'}}\left({x}_{i}\right){w}_{i}^{\text{'}}\left({x}_{i}\right)=C{w}^{\text{'}\text{'}}\left({x}_{i}\right),{w}_{i}^{\text{'}\text{'}\text{'}}\left({x}_{i}\right)={w}_{i+1}^{\text{'}\text{'}\text{'}}\left({x}_{i}\right).$
In composite beam, the Young’s modulus of composite system can be written as:
Thermal expansion coefficient of composite material can be presented as:
4. Results and discussion
The temperature effect on the vibration of metallic and composite beams is significant. To develop a reliable model to analyse thermal effect on metallic and composite beam are also very essential. In this section, the outcomes of this analysis are presented with graphs and tables. Three different materials such as steel, copper, aluminium and their composite are used in this analysis. In composite, these three materials such as steel, copper, and aluminium are used as a ratio of 40 %, 30 %, and 30 % respectively. The results of this analysis show good accordance with the papers of other researcher in open literature.
Table 2. Natural frequency of solid beam with clamped ends
$n=$1

Frequency in different modes


1

2

3

4

5

6


Current work

22.095

61.295

120.49

199.44

298.09

417.19

Jiang and Wang [12]

21.736

57.967

107.74

166.61

231.10

298.71

Table 2 illustrates the dimensionless natural frequency of fully clamped solid beam. Current research data show good agreement with data of other researcher [12] in lower mode of frequency. In current work, EulerBernoulli theory is used. On the other hand, Jiang and Wang [12] used nonlocal theory of elasticity with nonlocal parameter value 0.05. That is why, there are some differences in both data sets in higher modes of frequency.
Table 3. Natural frequency of solid beam for different axial loads
Frequency in different modes


$n$

1

2

3

4

5

6

0

22.37383

61.67188

120.9063

199.8543

298.5484

417.6172

2

21.81445

60.91797

120.0859

198.9844

297.6367

416.7578

4

21.24023

60.16211

119.2523

198.1406

296.7656

415.9063

8

20.04102

58.60547

117.5859

196.3945

294.9844

414.1328

16

17.36621

55.35547

114.1641

192.8613

291.3594

410.6563

32

9.87793

48.15234

106.998

185.6094

284.0156

403.7109

Table 3 depicts the natural frequency of solid beam under dimensionless axial loads. Natural frequency decreases with increase of axial load. Axial load is very effective on the 1st mode of frequency instead of higher mode of frequency.
Table 4. Natural frequency of composite beam for different temperatures
$L/h$=20

Aluminum (30%), Copper (30%), Steel (40%)

Aluminum (40%), Copper (30%), Steel (30%)

Aluminum (30%), Copper (40%), Steel (30%)


$\theta $

1

2

3

1

2

3

1

2

3

0

22.3730

61.6738

120.898

22.3744

61.6736

120.898

22.373

61.6738

120.898

5

22.2676

61.5352

120.742

22.2637

61.5273

120.742

22.2676

61.5273

120.742

10

22.1660

61.3867

120.602

22.1543

61.377

120.572

22.1582

61.3789

120.586

20

21.9551

61.1055

120.289

21.929

61.0723

120.246

21.9395

61.0898

120.258

30

21.7402

60.8242

119.977

21.7046

60.7695

119.914

21.7207

60.793

119.945

40

21.5254

60.5352

119.668

21.4746

60.4727

119.586

21.498

60.4961

119.617

50

21.3076

60.2461

119.345

21.2441

60.1602

119.254

21.2715

60.1992

119.305

Table 4 reveals the effect of temperature on natural frequency of composite beam. Three different types of materials such as steel, copper and aluminum are used in these composites. Three different composition ratios of these materials are used to create three different composites. Table data show that increase of temperature decreases frequency. On the other hand, ratio of these compositions is not very significant on natural frequency.
Fig. 2. Frequency versus change of temperature and length to height ratio
Fig. 2 illustrates the effect of temperature and length to height ratio on the natural frequency of beam for different materials. Natural frequency decreases with increase of temperature and the length to height ratio. Temperature increases axial tension that decreases natural frequency. Similarly, increase of length to height ratio decreases height and cross sectional area of beam that increases axial stress. Increase of axial stress decreases natural frequency. Composite material shows the effective results considering the behavior of other materials.
Fig. 3 displays the effect of crack depth ratio on the natural frequency presence and absence of temperature. This figure depicts that increase of crack depth ratio decreases natural frequency. On the absence of temperature, the effect of crack depth ratio is more significant than the presence of that. High temperature expands the length of the beam and reduces the width of the crack that decreases slightly the effect of crack on frequency.
Fig. 4 demonstrates the effect of temperature on the frequency of cracked beams with different crackdepth ratio. At the low crack depth ratio, frequency decreases with the increase of temperature. On the other hand, at the high crack depth ratio, frequency increases with the increase of temperature. Increase of temperature expands the length of the beam and decreases the width of the crack that enhances the frequency of the beam. Fig. 5 describes the effect of crack location on the natural frequency of beam on presence and absence of temperature. Crack position at the ends and at the mid of the beam show lower frequency. On the other hand, in between crack position shows higher frequency of the beam. Frequency of crack beam with temperature is more significant than the crack without temperature.
Fig. 3. Frequency versus crack depth ratio with and without temperature
Fig. 4. Frequency versus change of temperature for different crack depth ratios
Fig. 5. Frequency versus crack location with and without temperature
5. Conclusions
In this study, EulerBernoulli beam theory is employed to develop an analytical technique for analyzing thermal effect on dynamic behavior of cracked metallic and composite beam. The results are presented for three different types of material and their composite. The results of this investigation showed that the effect of thermal vibration depends on length to height ratio of the beam. The effect of temperature is more significant in first mode of frequency. In higher mode of frequency, the temperature effect is less significant. Under thermal load composite beam is very effective than metallic beam.
Acknowledgements
The partial support from Estonian Doctoral School of Mathematics and Statistics is acknowledged.
References
 Avsec J., Oblak M. Thermal vibrational analysis for simply supported beam and clamped beam. Journal of Sound and Vibration, Vol. 308, 2007, p. 514525. [Publisher]
 Barron R. F., Barron B. R. Design for Thermal Stresses. John Wiley and Sons, 2012. [Publisher]
 Hetnarski R. B., Eslami M. R. Thermal StressesAdvanced Theory and Applications. Springer Science+Business Media, 2009. [Search CrossRef]
 Mukhopadhyay M. Composite Materials and Structures. Universities Press (India) Private Limited, 2004. [Search CrossRef]
 Robert M. J. Mechanics of Composite Materials. Taylor and Francis, 1999. [Search CrossRef]
 Vasiliev V. V. Advanced Mechanics of Composite Materials and Structural Elements. Elsevier, 2013. [Publisher]
 Gibson R. F. Principles of Composite Material Mechanics. CRC Press, Taylor and Francis Group, 2012. [Search CrossRef]
 Zhang W., Chen H., Zhu D., Kong X. The thermal effects on highfrequency vibration of beams using energy flow analysis. Journal of Sound and Vibration, Vol. 333, 2014, p. 25882600. [Publisher]
 Cui D. F., Hu H. Y. Thermal buckling and natural vibration of the beam with an axial stickslipstop boundary. Journal of Sound and Vibration, Vol. 333, 2014, p. 22712282. [Publisher]
 Lai S. K., Zhang L. H. Thermal effect on vibration and buckling analysis of thin isotropic/orthotropic rectangular plates with crack defects. Engineering Structures, Vol. 177, 2018, p. 444458. [Publisher]
 Anderson T. L. Fracture Mechanics Fundamentals and Applications. CRC Press, Taylor and Francis Group, 2017. [Publisher]
 Jiang J., Wang L. Analytical solutions for thermal vibration of nanobeams with elastic boundary conditions. Acta Mechanica Solida Sinica, Vol. 30, 2017, p. 474483. [Publisher]