Open access peer-reviewed chapter - ONLINE FIRST

A New Computerized Boundary Element Model for Three-Temperature Nonlinear Generalized Thermoelastic Stresses in Anisotropic Circular Cylindrical Plate Structures

By Mohamed Abdelsabour Fahmy

Submitted: March 15th 2019Reviewed: October 8th 2019Published: November 6th 2019

DOI: 10.5772/intechopen.90053

Downloaded: 11

Abstract

In this chapter, we propose a new theory called nonlinear generalized thermoelasticity involving three temperatures. Because of strong nonlinearity of the proposed theory, therefore, it is much more difficult to develop analytical solution for solving problems related with the proposed theory. So, we propose a new computerized boundary element model for the solution of such problems and obtaining the three-temperature nonlinear generalized thermoelastic stresses in anisotropic circular cylindrical plate structures problems which are related with the proposed theory, where we used two-dimensional three temperature nonlinear radiative heat conduction equations coupled with electron, ion and phonon temperatures. The numerical results of the current study show the temperatures effects on the thermal stresses. Also, these numerical results demonstrate the validity and accuracy of our proposed model.

Keywords

  • boundary element model
  • three-temperature radiative heat conduction
  • nonlinear generalized thermoelasticity
  • thermal stresses
  • anisotropic circular cylindrical plate structures

1. Introduction

The spiral formed tube which has been used in water transmission pipelines [1, 2] is the most common structural application of a cylindrical shell. Spiral formed pipes were initially constructed by riveting together appropriately bent plates [3] until advances in welding technology allowed for efficient tandem arc welding [1]. Recently, increasing attention has been devoted to the study of spiral welded tubes due to its many applications in water, gas and oil pipelines under both low and high pressure [4] as well as for foundation piles and primary load-bearing members in Combi-walls [5]. Spiral welded tubes provide certain benefits over traditional longitudinal and butt-welded tubes. In particular, continuous or very long tubular members may be constructed efficiently from compact coils of metal strip, eliminating the need for costly transport of long tubular members. The coil material is usually manufactured to very tight tolerances which results in a tube with consistent wall thickness [6]. Further, they exhibit a superior fatigue performance compared to longitudinal seam welded tubes [7]. They also exhibit a comparable resistance to crack growth propagation in ductile materials [8]. However, spiral welded tubes are not suitable for offshore and deep-water applications, because their diameter and wall thickness are limited to nearly 3 m and 30 mm, respectively [9] which generally makes them unsuitable for offshore and deep-water applications [10].

In recent years, great attention has been directed towards the study of generalized thermoelastic interactions in anisotropic thermoelastic models due to its many applications in physics, geophysics, astronautics, aeronautics, earthquake engineering, military technologies, plasma, robotics, mining engineering, accelerators, nuclear reactors, nuclear plants, soil dynamics, automobile industries, high-energy particle accelerators and other science and engineering applications. The main notion of photons, which are particles of light energy, has been introduced by Albert Einstein in 1905. It is difficult to interpret why temperature depends on the specific heat of the crystalline solids. So, the original notion of phonons, which are particles of heat, has also introduced by Albert Einstein in 1907 to explain this phenomenon. Our three-temperature study is essential for a wide range of low-temperature applications, such as pool and basin heating, unglazed and uninsulated flat-plate organic collectors, cold storage warehouses, outdoor applications in extreme low temperatures, cryogenic gas processing plants and frozen food processing facilities. Also, our three-temperature study is very important high temperature applications such as turbine blades, piston engine valves, turbo charger components, microwave devices, laser diodes, RF power amplifiers, tubes of steam power plant, recuperators in the metallurgical and glass industries. The proposed boundary element method (BEM) can be easily implemented for solving nonlinear generalized thermoelasticity problems. Through the present paper, the three-temperature concept introduced for the first time in the field of nonlinear generalized thermoelasticity. Duhamel [11] and Neumann [12] developed the classical thermo-elasticity (CTE) theory and obtained the strain-temperature gradients equations in an elastic body, but their theory has the following two shortcomings: First, the heat conduction equation is predicting infinite speeds of propagation. Second, the heat conduction equation does not contain elastic terms. Biot [13] developed the classical coupled thermo-elasticity (CCTE) theory to overcome the first shortcoming in CTE. Then, several generalized theories based on a modified Fourier’s law predict finite propagation speed of thermal waves such as extended thermo-elasticity (ETE) theory of Lord and Shulman (L-S) [14], temperature-rate-dependent thermo-elasticity (TRDTE) theory of Green and Lindsay (G-L) [15] and three linear generalized thermoelasticity models of Green and Naghdi (G-N) [16, 17], where Type I discusses the heat conduction theory based on Fourier’s law, type II describes the thermoelasticity theory without energy dissipation (TEWOED), and type III discusses the thermoelasticity theory with energy dissipation (TEWED). Due to the computational difficulties, inherent in solving nonlinear generalized thermoelastic problems [18], for such problems, it is very difficult to obtain the analytical solution in a general case. Instead of analytical methods, many numerical methods were developed for solving such problems approximately including the finite difference method (FDM) [19, 20], discontinuous Galerkin method (DGM) [21], finite element method (FEM) [22, 23] and boundary element method (BEM) [24, 25, 26]. The boundary element method (BEM) has been performed successfully for solving various engineering, scientific and mathematical applications due to its simplicity, efficiency, and ease of implementation [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46].

The main aim of the present chapter is to propose a new theory called nonlinear generalized thermoelasticity involving three-temperature. A new boundary element model was proposed for solving nonlinear generalized thermoelastic problems in anisotropic circular cylindrical plate structures which are associated with the proposed theory, where we used two-dimensional three-temperature (2D-3T) nonlinear time-dependent radiative heat conduction equations coupled with electron, ion and photon temperatures in the formulation of such problems. The numerical results are presented graphically to show the effects of electron, ion and photon temperatures on the thermal stress components. The validity and accuracy of our proposed BEM model were confirmed by comparing our BEM obtained results with the corresponding results of finite element method (FEM).

A brief summary of the chapter is as follows: Section 1 outlines the background and provides the readers with the necessary information to books and articles for a better understanding of mechanical behaviour of anisotropic circular cylindrical plate structures and their applications. Section 2 describes the formulation of the new theory and its related problems. Section 3 discusses the implementation of the new BEM for solving the three-temperature heat conduction equations, to obtain the temperature fields. Section 4 studies the development of new BEM and its implementation for solving the equilibrium equation based on the three-temperature fields. Section 5 presents the new numerical results that describe the temperatures effects on the thermal stresses generated in anisotropic circular cylindrical plate structures.

2. Formulation of the problem

We consider a cylindrical coordinate system rθzfor the circular cylindrical plate structure (Figure 1) within the region Rwhich bounded by boundary S. Pressure distribution over the structure’s entire surface has been shown in Figure 2. Geometry of meridional cross section of the considered structure has been shown in Figure 3, where =1r.

Figure 1.

Geometry of circular cylindrical plate structure.

Figure 2.

Pressure distribution over the structure’s entire surface.

Figure 3.

Geometry of meridional cross section of considered structure.

The equilibrium equations for anisotropic plate structures can be written as follows

σpj,j=0E1

where

σpj=Cpjkluk,lβpjTαrτE2

Three radiative heat conduction equations coupled with electron, ion and phonon temperatures can be written as follows

ceTerττ1ρKeTerτ=WeiTeTiWepTeTpE3
ciTirττ1ρKiTirτ=WeiTeTiE4
4ρcpTp3Tprττ1ρKpTprτ=WepTeTpE5

where TeTiTp,cecicpand KeKiKpare respectively temperatures, specific heat capacities and conductive coefficients of electron, ion and phonon.

The total temperature

T=Te+Ti+TpE6

3. BEM solution for three-temperature field

The nonlinear time-dependent two dimensions three temperature (2D-3T) radiative heat conduction Eqs. (3)(5) coupled by electron, ion and phonon temperatures can be written as

δ1jKα+δ2jKαTαrτW¯rτ=cαρδ1δ1jTαrττE7

where

W¯rτ=ρWeiTeTi+ρWerTeTp+W¯¯,α=e,δ1=1ρWeiTeTi+W¯¯,α=i,δ1=1ρWerTeTp+W¯¯,α=p,δ1=4ρTp3E8
W¯¯rτ=δ2jKαṪα,ab+βabTα0Åδ1ju̇a,b+τ0+δ2ju¨a,b+ρcατ0+δ1jτ2+δ2jT¨αTα,ar,Tα,ar=Tα,aaE9
Wei=ρAeiTe2/3,Wer=ρAerTe1/2,Kα=AαTα5/2,α=e,i,Kp=ApTp3+BE10

The total energy can be written as follows

P=Pe+Pi+Pp,Pe=ceTe,Pi=ciTi,Pp=1ρcpTp4E11

By applying the following conditions

Tαxy0=Tα0xy=g1xτE12
KαTαnΓ1=0,α=e,i,TrΓ1=g2xτE13
KαTαnΓ2=0,α=e,i,pE14

By using the fundamental solution that satisfies the following Eq. [46]

D2Tα+Tαn=δrpiδτrE15

where D=Kαρcand piare singular points.

The corresponding dual reciprocity boundary integral equation can be written as [46]

CTα=DKαOτSTαqTαqdSdτ+DKαOτRbTαdRdτ+RTαiTατ=0dRE16

which can be expressed as

CTα=STαqTαqdSRKαDTατTαdRE17

In order to transform the domain integral into the boundary, we assume that

Tατj=1NfjrjajτE18

where fjrand ajτare known functions and unknown coefficients, respectively.

We assume that T̂αjis a solution of

2T̂αj=fjE19

Thus, from (17) we can write the following boundary integral equation

CTα=STαqTαqdS+j=1NajτD1CT̂αjSTαjqq̂jTαdSE20

where

q̂j=KαT̂αjnE21
ajτ=i=1Nfji1TriττE22
Fji=fjriE23

By using (20) and (22), we obtain

CṪα+HTα=GQE24

where

C=HT̂αGQ̂F1D1E25

and

T̂ij=T̂jxiE26
Q̂ij=q̂jxiE27

For solving (24) numerically, the functions q, Tαand its derivative with time can be written as

q=1Θqm+Θqm+1,0Θ1E28
Tα=1ΘTαm+θTαm+1,0Θ=ττmτm+1τm1E29
Ṫα=dTαdΘdΘ=Tαm+1Tαmτm+1τm=Tαm+1TαmτmE30

By substituting from Eqs. (28)(30) into Eq. (24), we obtain

Cτm+ΘHTαm+1ΘGQm+1=Cτm1ΘHTαm+1ΘGQmE31

By applying the initial and boundary conditions, we obtain

aX=bE32

This system yields the temperature in terms of the displacement field.

4. BEM solution for displacement field

The equilibrium Eqs. (1) for anisotropic plate structures can be written as follows [47]

Cijkld4wdx4Td2wdx2=p+T1rE33

where

T2=pr2E34
T1=CijklT2CijklhwrE35

By using (34) and (35), we can write (33) in the following form

Cijkld4wdx4+pr2d2wdx2+Cijklhr2w=p1Cijkl2E36

where

A=Cijklhr2E37
B=p1Cijkl2E38

By using Eqs. (37) and (38), we can write (36) as follows

Cijkld4wdx4Td2wdx2+Aw=BE39

where

β=T22Cijklk,0<β2<1E40

The general solution of (39) can be obtained as

wx=C1chδxcosγx+C2chδxsinγx+C3shδxcosγx+C4shδxsinγx+wpartE41

where

δ=α1+β;γ=α1β;α=k4Cijkl4,β=T22CijklkE42

and the particular solution can be determined as p=constantas follows

wpart=pr2Cijklh1Cijkl2E43

Thus, Eq. (41) can be written as

wx=pr2Cijklh1Cijkl2+C1chδxcosγx+C4shδxsinγxE44

By implementing the following boundary conditions.

atx=±l2dwdx=0E45
atx=l2w=2pr2Cijklhd3wdx3E46

we can write the unknown C1and C4as follows

C1=2pr2Cijklh1Cijkl2u1chu1sinu2+u2shu1cosu2u2sh2u1+u1sin2u2ε1E47
C4=2pr2Cijklh1Cijkl2u2chu1sinu2u1shu1cosu2u2sh2u1+u1sin2u2ε1E48

where

ε1=11+lhAA1u1u2E49
A1u1u2=1β2ch2u1cos2u2u2sh2u1+u1sin2u2E50
u1=δl2=u1+β,u2=γl2=u1+β,u=0.64251rhE51

If we neglected the longitudinal forces influence on the bending of the circular cylindrical shell, we can write (39) in the following form

CijklwIV+kw=qE52

Now, the approximate solution has been reduced for solving problem of bending single span beam with the following compliance

kII=2r2CijklAE53

The deflection of the considered shell in the cross section and reference section, respectively, is as follows

w0=pr2Cijklh1v211u1+B1E54
wl2=pr2Cijklh1v2B11+B1E55

Also, the bending moment in the cross section and reference section, respectively, is as follows

M10=pl2241v21χ1u1+B1E56
M1l2=pl2121v2χ2u1+B1E57

The Cauchy model with two-bed scheme can be described as follows

DvIYx+pr2vx+Cijklhr2vx=p1μ2E58
v0;φ0=v0E59
M0=Dv0Tv0E60
Q0=Dv0Tv0E61

where the characteristic equation of (58) can be defined as

Cijklk4+pr2k2+Cijklhr2=0,k2=tE62
Cijklt2+pr2t+Cijklhr2=0E63

which roots

k1,2,3,4=±pr2±pr224Cijkl2hr22CijklE64
t1,2=pr2±pr224Cijkl2hr22CijklE65

The systems (32) and (58) can be solved by using the algorithm of Fahmy [35] to obtain the three temperatures and displacements components. Then we can compute thermal stresses distributions along radial distance r. we refer the reader to recent references [48-51] for details of boundary element technique.

5. Numerical results and discussion

The BEM that has been used in the current chapter can be applicable to a wide variety of plate structures problems associated with the proposed theory of three temperatures nonlinear generalized thermoelasticity. In order to evaluate temperatures effects on the thermal stresses, the numerical results are carried out and depicted graphically for electron, ion and phonon temperatures.

Figure 4 shows the distributions of the three temperatures Te,Ti,Tpand total temperature TT=Te+Ti+Tpalong the radial distance r. It was shown from this figure that the three temperatures are different and they may have great effects on the connected fields.

Figure 4.

Variation of the temperatures Te, Ti, Tp and T along the radial distance r.

Figures 57 show the distributions of the thermal stresses σ11,σ12andσ22respectively, with the radial distance r for the three temperatures Te,Ti,Tpand total temperature . It was noticed from these figures that the three temperatures have great effects on the thermal stresses.

Figure 5.

Variation of the thermal stress σ11 with radial distance r.

Figure 6.

Variation of the thermal stress σ12 with radial distance r.

Figure 7.

Variation of the thermal stress σ22 with radial distance r.

Figure 8 shows the distributions of the thermal stresses σ11,σ12,σ22and total temperature Twith the radial distance r for BEM results and finite element method (FEM) results of COMSOL Multiphysics software version 5.4 to demonstrate the validity and accuracy of our proposed model based on replacing heat conduction with three-temperature heat conduction.

Figure 8.

Thermal stresses and total temperature variations with r.

6. Conclusion

The main objective of this chapter is to propose a new theory called nonlinear generalized thermoelasticity involving three-temperature and new BEM model for the solution of problems which are associated with the proposed nonlinear theory, where we used the three-temperature radiative heat conduction equations coupled with electron, ion and phonon temperatures to describe the thermal stresses in anisotropic circular cylindrical plate structures. It can be concluded from numerical results of our proposed model that the generalized theories of thermoelasticity can be connected with the three-temperature radiative heat conduction to describe the deformation of anisotropic circular cylindrical plate structures. The validity and accuracy of the proposed model was examined and confirmed by comparing the obtained results with those known previously. Because there are no available data to confirm the validity and accuracy of our results, we replace the three-temperature radiative heat conduction results with one-temperature heat conduction results as a special case from results of our current general model for circular cylindrical plate structures. In the special case under consideration, the results obtained with the BEM have been compared graphically with the FEM results of COMSOL Multiphysics software version 5.4. Excellent agreement is obtained between BEM results and FEM results. Understanding the behaviour of the three-temperature thermal stresses in anisotropic circular cylindrical plate structures should be a key for extending the application of these behaviors to a wide range of structures. The numerical results for our general model which is associated with our proposed theory may provide interesting information for computer scientists and engineers, geotechnical and geothermal engineers, researchers who will industrialize the thermoelastic devices using additive manufacturing and the materials designers and developers, etc.

Nomenclature

βij

stress-temperature coefficients

δij

Kronecker delta ij=12

εij

strain tensor

θ

thermodynamic temperature

λ

tractions

μ0

magnetic permeability

ϑ0

viscoelastic relaxation time

ϖ

weights of control points

ρ

density

σij

force stress tensor

c

specific heat capacity

Cijkl

constant elastic moduli

elij

piezoelectric tensor

Fi

mass force vector

Kα

conductive coefficients

M1

bending moment

P

total energy of unit mass

Tα

temperature functions

ui

displacement vector

wx

general solution

Wei

electron-ion energy coefficient

Wep

electron-photon energy coefficient

Download

chapter PDF

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

How to cite and reference

Link to this chapter Copy to clipboard

Cite this chapter Copy to clipboard

Mohamed Abdelsabour Fahmy (November 6th 2019). A New Computerized Boundary Element Model for Three-Temperature Nonlinear Generalized Thermoelastic Stresses in Anisotropic Circular Cylindrical Plate Structures [Online First], IntechOpen, DOI: 10.5772/intechopen.90053. Available from:

chapter statistics

11total chapter downloads

More statistics for editors and authors

Login to your personal dashboard for more detailed statistics on your publications.

Access personal reporting

We are IntechOpen, the world's leading publisher of Open Access books. Built by scientists, for scientists. Our readership spans scientists, professors, researchers, librarians, and students, as well as business professionals. We share our knowledge and peer-reveiwed research papers with libraries, scientific and engineering societies, and also work with corporate R&D departments and government entities.

More About Us