# Thermoelastic Analysis for Two Collinear Cracks in an Orthotropic Solid Disturbed by Antisymmetrical Linear Heat Flow.

1. Introduction

In the abovementioned works and other works [26-32], the partially insulated boundary condition is mainly used and the closed-form solutions have been obtained for the thermoelastic field around the penny-shaped crack under the loading of uniform heat flow.

Under the consideration of the antisymmetrical linear heat flow, the partially insulated boundary condition is applied to address the problem of two collinear cracks in an orthotropic solid and FEM solutions for the stress intensity factor K are used to investigate the accuracy of this methodology in this paper. Applying the Fourier transform technique and integral equation methods, the crack-tip thermoelastic fields involving the jumps of temperature, the elastic displacements across the cracks, and the mode II stress intensity factors are given in explicit forms. Numerical results show the heat flux to the crack surfaces and the mode II stress intensity factors are dependent on the geometries of cracks and the dimensionless thermal resistance. The thermal resistance in the heat conduction through the crack region is of much importance in analyzing the thermoelastic problem of a cracked material with a thermal loading. Furthermore, FEM solutions for the stress intensity factor K are used to compare with the solutions obtained using the method. It is revealed the friction in closed crack surface region has an effect on analyzing the stress intensity factor K and should be considered.

2. Statement of the Problem

As shown in Figure 1, two collinear cracks for an infinite transversely orthotropic material are embedded in an orthotropic solid. We use Cartesian coordinates system xoy and suppose that the two collinear cracks are situated at the segment of [absolute value of a] < [absolute value of x] < [absolute value of b]. With respect to the state of plane stress, the following constitute equations are

[mathematical expression not reproducible], (1)

where u and V denote the components of elastic displacement, [theta] stands for the temperature change, [v.sub.xy] and [v.sub.y]x are the Poisson ratios, [E.sub.xx] and [E.sub.yy] are Young's moduli, [c.sub.66] = [G.sub.xy] is the shear modulus, and [[alpha].sub.xx] and [[alpha].sub.yy] are the coefficients of linear expansion. Applying the following equations:

[mathematical expression not reproducible], (2)

we have

[mathematical expression not reproducible]. (3)

Furthermore, applying thermal equilibrium equation leads to

[mathematical expression not reproducible], (4)

where [[lambda].sub.x] and [[lambda].sub.y] are the thermal conductivities of a cracked orthotropic material. From the theory of Fourier heat conduction, one has

[q.sub.x] = -[[lambda].sub.x] [partial derivative][theta]/[partial derivative]x, [q.sub.y] = -[[lambda].sub.y] [partial derivative][theta]/[partial derivative]y. (5)

Due to the antisymmetry of the heat flux acting on two collinear cracks, the thermal field in the region x >0 is only studied. With application of the crack-face boundary conditions, one has

[mathematical expression not reproducible]. (6)

For the thermal loading, the antisymmetrical linear heat flow is acting on the crack surfaces; namely,

[q.sup.+.sub.y] (x, 0) = [q.sup.-.sub.y] (x, 0) = -[q.sub.0]x, a < x < b, (7)

where [q.sub.0] is the prescribed constant, where

[mathematical expression not reproducible]. (8)

Considering the thermal resistance [R.sub.c] in the heat conduction through the crack region, the thermoelastic boundary conditions on the crack surfaces are written as

[q.sup.+.sub.y] (x, 0) = [q.sup.-.sub.y] (x, 0) = - ([q.sub.0] - [q.sub.c]) x, a < x < b, (9)

where

[q.sub.c] = 1/[R.sub.c] ([[theta].sup.+] (x, 0) - [[[theta].sup.-] (x, 0)). (10)

[q.sub.c] denotes the heat flux to the crack surfaces. The limiting value [R.sub.c] [right arrow] 0 or [R.sub.c] [right arrow] [infinity] represents perfect conduction or perfect insulation on the crack surfaces. In addition, the continuity of elastic displacement and temperature on the crack-free parts of v-axis yields

[v.sup.+] (x, 0) = [v.sup.-] (x, 0), 0 < x < a, x > b, (11)

[u.sup.+] (x, 0) = [u.sup.-] (x, 0), 0 < x < a, x > b, (12)

[mathematical expression not reproducible]. (13)

3. Solution Procedure

3.1. Temperature Field. The temperature field is not dependent on the elastic strain; the temperature field can be solved firstly. By applying the Fourier transform technique, the solution of (4) can be defined as

[mathematical expression not reproducible]. (14)

[A.sup.[+ or -]]([xi]) is unknown and will be solved. [[delta].sup.+] = 1 for y > 0 and [[delta].sup.-] = -1 for y < 0. Hereafter the superscripts + and - denote the physical quantities of the upper (i.e., y > 0) and lower (i.e., y < 0) region, respectively. From (5), we obtain

[mathematical expression not reproducible]. (15)

Using (8) and the second relation in (13), we obtain

[A.sup.+] ([xi]) = -[A.sup.-] ([xi]). (16)

Using the boundary conditions (7), one has

[[integral].sup.[infinity].sub.0] [xi][A.sup.+] ([xi]) sin ([xi]x) d[xi] = - ([q.sub.0] - [q.sub.c])x/[lambda][[lambda].sub.y], a < x < b. (17)

Application of the first relation in (13) leads to

[[integral].sup.[infinity].sub.0] [xi][A.sup.+] ([xi]) sin ([xi]x) d[xi] = 0, 0 < x < a or x > b. (18)

In order to solve (17) and (18), we give a definition of the auxiliary function

[omega](x) = [partial derivative][[theta].sup.+] (x, 0)/[partial derivative]x. (9)

Applying the inverse Fourier transform, one obtains

[A.sup.+] ([xi])[xi] = 2/[pi] [[integral].sup.b.sub.a] [omega](s) cos ([xi]s) ds. (20)

Substituting (20) into (17) leads to

[mathematical expression not reproducible]. (21)

From the known result 

2 [[integral].sup.[infinity].sub.0] sin ([xi]x) cos ([xi]s) d[xi] = 1/[x + s] + 1/x - s. (22)

Equation (21) can be rewritten as

1/[pi] [[integral].sup.b.sub.a] [omega](s) 2/[s.sup.2] - [x.sup.2] ds = [q.sub.0] - [q.sub.c]/[lambda][[lambda].sub.y], a < x < b. (23)

Then, introducing [s.sup.2] = [s.sup.*], [x.sup.2] = [x.sup.*], 2s ds = d[s.sup.*], [a.sup.2] = [a.sup.*], and [b.sup.2] = [b.sup.*], (23) can be written as

[mathematical expression not reproducible]. (24)

By virtue of the singular integral equation theory, we can obtain the solution of (21); namely,

[mathematical expression not reproducible], (25)

where C is the constant determined by

[mathematical expression not reproducible]. (26)

By some computations, we can get C = 0. Furthermore, applying (19) and (25) yields

[mathematical expression not reproducible]. (27)

Substituting (25) into (27), we get the temperature change on the cracks as

[[theta].sup.+] (x, 0) = [q.sub.0] - [q.sub.c]/[lambda][[lambda].sub.y]([b.sup.2] - [a.sup.2]) [square root of (([x.sup.2] - [a.sup.2])([b.sup.2] - [x.sup.2))], a < x < b. (28)

When a = 0, the temperature change on the cracks of a single crack of length 2b is given as

[[theta].sup.+] (x, 0) = [q.sub.0] - [q.sub.c]x/[lambda][[lambda].sub.y][b.sup.2] [square root of ([b.sup.2] - [x.sup.2])], a < x < b. (29)

3.2. Elastic Field. In what follows, we pay an attention to the solutions of the partial differential equations (3). [u.sup.[+ or -]](x, y) and [v.sup.[+ or -]](x, y) are written as two parts

[u.sup.[+ or -]] (x, y) = [u.sup.[+ or -].sub.1] (x, y) + [u.sup.[+ or -].sub.2] (x, y), [v.sup.[+ or -]] (x, y) = [v.sup.[+ or -].sub.1] (x, y) + [v.sup.[+ or -].sub.2] (x, y). (30)

[u.sup.[+ or -].sub.1](x, y) and [v.sup.[+ or -].sub.1](x, y) correspond to the general solutions under [[theta].sup.[+ or -]](x, y) = 0. [u.sup.[+ or -].sub.2](x, y) and [v.sup.[+ or -].sub.2](x, y) are the particular solutions under a thermal flux loading. With application of the Fourier transform, [u.sup.[+ or -].sub.1](x, y) and [v.sup.[+ or - ].sub.1](x, y) can be expressed as

[mathematical expression not reproducible]. (31)

[A.sup.[+ or -].sub.j]([xi]) (j = 1, 2) are unknown to be solved. [[gamma].sub.j] (j = 1, 2) are the roots of equation as follows: (Re [gamma](j) > 0)

[C.sub.22][C.sub.66][[gamma].sup.4] + ([c.sup.2.sub.12] + 2[c.sub.12][c.sub.66] - [c.sub.12][c.sub.22]) [[gamma].sup.2] + [c.sub.11][c.sub.66] = 0, (32)

where [[eta].sub.j] (j = 1, 2) can be calculated as follows:

[[eta].sub.j] = [c.sub.11] - [c.sub.66][[gamma].sup.2.sub.j]/([c.sub.12] + [c.sub.66]) [[gamma].sub.j]. (33)

In addition, [u.sup.[+ or -].sub.2](x, y) and [v.sup.[+ or -].sub.2](x, y) are chosen as

[mathematical expression not reproducible]. (34)

Substituting (34) into (3) leads to

[mathematical expression not reproducible]. (35)

From (31) and (34), the components of stress can be written as

[mathematical expression not reproducible], (36)

[mathematical expression not reproducible], (37)

[mathematical expression not reproducible]. (38)

It is seen that the elastic field was only induced by the loading -[q.sub.0]x. We only consider the antisymmetrical linear heat flow -[q.sub.0]x in this paper, and it follows that

[[sigma].sup.[+ or -].sub.yy] (x, 0) = 0, x > 0. (39)

From (37) and (39), one arrives at

[mathematical expression not reproducible]. (40)

Utilizing the second relation of (6) and (12) yields

[mathematical expression not reproducible], (41)

[mathematical expression not reproducible]. (42)

To solve the dual integral equations, we introduce a new auxiliary function [phi](x).

[phi](x) = [partial derivative][[u.sup.+] (x, 0) - [u.sup.-] (x, 0)]/[partial derivative]x. (43)

By virtue of theory of the inverse Fourier transform, one gets

[2.summation over (j=1)] [A.sup.+.sub.j] ([xi]) [xi] + [M.sub.1][A.sup.+] (x) = -2/[pi] [phi](s) sin ([xi]s) ds. (44)

Making use of (40) and (44) yields

[mathematical expression not reproducible]. (45)

Application of (42) and (45) leads to

[mathematical expression not reproducible], (46)

with

[mathematical expression not reproducible]. (47)

With application of the inverse Fourier transform and (28), one obtains

[mathematical expression not reproducible]. (48)

Recalling the known result 

[[integral].sup.b.sub.a] [square root of ((s - a)(b - s)/x - s)] ds = [pi]/2 {2x - a - b). (49)

Inserting (28) into (48), one gets

[mathematical expression not reproducible], (50)

where

p = K ([q.sub.0] - [q.sub.c])/[lambda][[lambda].sub.y] ([b.sup.2] - [a.sup.2]). (51)

From (46) and (50), we can get

[mathematical expression not reproducible], (52)

Considering the following condition:

[mathematical expression not reproducible]. (53)

By further calculations, one obtains

C = -p([a.sup.2] - [b.sup.2])/4 + 2p[b.sup.2] ([a.sup.2] + [b.sup.2])E(k)/3F(k) -4p[a.sup.2][b.sup.2]/3. (54)

With application of (43), one has

[mathematical expression not reproducible]. (55)

Substituting (52) into (55), we have

[mathematical expression not reproducible], (56)

where k = [square root of (([b.sup.2] - [a.sup.2])/[b.sup.2])] and F(*) and E(*) are the complete elliptic integrals of the first and second kind, respectively. F([alpha], k) and E([alpha], k) are the incomplete elliptic integrals of the first and the second kinds with

[alpha] = arcsin [square root of ([b.sup.2] - [x.sup.2]/[b.sup.2] - [a.sup.2]). (57)

Applying (38) and (45), the shearing stresses near the crack-tips are expressed as

[mathematical expression not reproducible]. (58)

Substituting (50) and (52) into (58), one has

[mathematical expression not reproducible]. (59)

3.3. Stress Intensity Factors. According to (10), (16), and (28), one has

[mathematical expression not reproducible]. (60)

The quantity [h.sub.c] = [[lambda].sub.y]/[R.sub.c] is the dimensionless thermal resistance. One can see from (60) that the heat flux to the crack surfaces is dependent on the geometries of the cracks, the material properties, and applied thermal loading. Furthermore, the crack opening displacement has great influences on analyzing the thermo elastic problem. From the obtained results, it is easy to determine

[mathematical expression not reproducible]. (61)

From (56), one arrives at

[mathematical expression not reproducible]. (62)

From (62), it is seen that the stress intensity factors are expressed in the closed forms. In addition, to consider the stress fields near the crack-tips, the mode II stress intensity factors are defined as

[mathematical expression not reproducible]. (63)

One has

[mathematical expression not reproducible]. (64)

It is easy to find that the mode II stress intensity factors are dependent not only on the geometries of the cracks and the heat flux to the crack surfaces [q.sub.c] but also on the material properties.

In order to verify (64), we consider a limiting case as a [right arrow] 0, corresponding a single crack with the length of 2b. It is not difficult to find

[K.sub.0] = [c.sub.66]p'[N.sub.2][b.sup.2][square root of b[pi]]/2[c.sub.22] ([[gamma].sub.2][[eta].sub.2] - [[gamma.sub.1][[eta].sub.1]) (1/2 + 2E(k)/3F(k)), (65)

where

p' = K ([q.sub.0] - [q.sub.c])/[lambda][[lambda].sub.y][b.sup.2]. (66)

4. Numerical Results and Discussion

In this section, numerical results are carried out to research the effects of the dimensionless thermal resistance of the cracks surfaces and the geometries of the cracks on the temperature change and the mode II stress intensity factors under the antisymmetrical linear heat flow. A ceramic material of Tyrannohex is chosen and the corresponding material properties are given in . That is, one has

[mathematical expression not reproducible]. (67)

Figure 2 shows the variations of [q.sub.c]/[q.sub.0] versus x/b with a/b = 0.25 and [h.sub.c] = 0, 1, 2, and 4, respectively. Obviously, when [h.sub.c] = 0, we have [q.sub.c] = 0, corresponding to fully thermally insulated cracks. When [h.sub.c] = 1, it means that if the heat conduction through the cracks region is the same as that of the orthotropic material, the cracks approximate to thermally permeable ones. Moreover, one can find it is feasible to reduce the thermal stress concentrations near the crack-tips by the way of filling the matrix material into a crack in engineering problems. When [h.sub.c] = 2 or [h.sub.c] = 4, it shows the heat conduction through the crack region is twice or 4 times as much as thermal conductivities of an orthotropic material. When [h.sub.c] [right arrow] [infinity], we have [q.sub.c] = [q.sub.0], which means fully thermally permeable cracks. It is revealed that applying the thermally insulated assumption [h.sub.c] = 0 will underestimate much the heat flux to the crack surfaces and applying the fully thermally permeable assumption [h.sub.c] [right arrow] [infinity] will overestimate much the heat flux to the crack surfaces. The observation reveals that the effects of the heat conduction through the crack region on the crack-tip thermoelastic fields cannot be disregarded.

Figure 3 shows the variations of K/[K.sub.0] versus x/b with [h.sub.c] = 1 and a/b = 0.25, 0.5, and 0.75, respectively. Figure 4 shows variations of K/[K.sub.0] versus x/b with a/b = 0.25 and [h.sub.c] = 1, 2, and 4, respectively. Hereafter [K.sub.0] stands for the mode II stress intensity factor for a single and fully thermally insulated crack with the length of 2b under the antisymmetrical linear heat flow. K/[K.sub.0] tends to zero as [h.sub.c] increases. Obviously, it is unrealistic to assume that the crack is fully thermally permeable. From the above results, it is revealed that the fully thermally insulated or fully thermally permeable assumption on the crack-face is only a limiting case of a real crack. It is concluded that the material properties and the thermal resistance in the heat conduction through the crack region are of much importance and should be considered in analyzing the thermoelastic problem of a cracked material with a thermal loading. EM solutions for the stress intensity factor K are used to compare with the solutions obtained using the method outlined above.

Figures 6 and 7 present the temperature profile and thermal stress analysis results, respectively. Another model is employed to check the mesh density convergence, with a local mesh four times finer than the mesh used in this analysis. The maximum difference between these meshes is less than 1%. For crack-tip A, the difference between present solution [K.sub.II] and those obtained using FEM is less than 2%. However, the present solution [K.sub.II] for crack-tip B differs from the FEM solutions by up to 20%. Upon examination, the differences in these analyses are due to the assumption of crack being open during the calculation of stress intensity factors. This assumption is not validated by FEM analysis. To overcome this shortcoming, the friction in closed crack surface region should be considered, in order to obtain more accurate values for [K.sub.II] at tip B. As this is beyond the scope of the present paper, it will be a topic to consider for future studies.

5. Conclusions

Considering the effects of the thermal resistance, the problem of two collinear cracks in an orthotropic solid under antisymmetrical linear heat flow is addressed. Applying the Fourier transform technique, the thermoelastic fields involving the jumps of temperature, the elastic displacements across the cracks, and the mode II stress intensity factors are given in closed forms. Numerical results are carried out to show the heat flux to the crack surfaces [q.sub.c] depends on the geometries of the cracks, the material properties, and applied thermal loading, and the mode II stress intensity factors are dependent on the heat flux to the crack surfaces [q.sub.c], the material properties, and the geometries of the cracks. The fully thermally permeable or fully thermally insulated cracks are the limiting case of the partially insulated cracks. The observations reveal that the thermal resistance in the heat conduction through the crack region is of much importance and should not be neglected in analyzing the thermoelastic problem of a cracked material with a thermal loading. Also, FEM solutions for the stress intensity factor K are used to compare with the solutions obtained using the method; it is revealed that the friction in closed crack surface region should be considered. As this is beyond the scope of the present paper, it will be a topic to consider for future studies.

Appendix

To investigate the accuracy of this methodology, FEM solutions for the stress intensity factor K are used to compare with the solutions obtained using the method outlined above. For this problem, an orthotropic plate (thickness = 1 mm) with cracks is considered to be H long and W wide. As shown in Figure 5, the dimensions H, W, a(a = a'), and b (b = b') are chosen to be 20 mm, 40 mm, 1 mm, and 4 mm, respectively. The ambient temperature and the heat flux are set at 20.0[degrees]C and 400 J, respectively. The resultant mesh has 1,472 elements (using an 8-noded isoparametric element model) and 4,711 nodes by using software FEMAP . To model the singularity effect we use a fine mesh in the vicinity of the crack-tip region. Gap elements are inserted between crack surfaces to prevent crack top and bottom surface penetrating each other. The "quarter point element" method  is used in this section. The midpoint node is moving to the 1/4 point for the elements surrounding the crack-tip. This method uses the displacement field on the crack-face. This has an advantage over the stress correlation method because displacements are primary solution variables in the finite element method and can be obtained with higher degree precision than the stress field. Williams et al.  propose an equation to calculate SIF from FEM to include the effect of the crack length on the element size at the crack-tip:

[mathematical expression not reproducible], (A.1)

where [[delta].sub.x] and [[delta].sub.y] are the crack opening displacement in y and x direction and l is the distance from the crack-tip. The stress intensity factors obtained using FEM are given as follows:

At crack-tip A, [K.sub.II] = 0.974 Mpa[(m).sup.1/2] and [K.sub.I] = 0.0342 Mpa[(m).sup.1]2].

At crack-tip B, [K.sub.II] = 0.534 Mpa[(m).sup.1/2] and [K.sub.I] = 0 Mpa[(m).sup.1/2] (as the crack surface near crack-tip B has closed, see Figure 7).

https://doi.org/10.1155/2017/5093404

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by National Key R&D Program of China (2017YFC0404800), by the Fundamental Research Funds for the Central Universities (no. 2015B35814, no. 2015B41914, and 2017B20614), by the China Scholarship Council (no. 201606710018), and by the National Natural Science Foundation of China (no. 51479052).

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Bing Wu, (1,2,3) Jun-gao Zhu, (1,2) Daren Peng, (3) Rhys Jones, (3) Shi-hu Gao, (1,2) and Yang-yang Lu (1,2)

(1) Key Laboratory of Ministry of Education for Geomechanics and Embankment Engineering, Hohai University, Nanjing, Jiangsu 210098, China

(2) Jiangsu Research Center of Geotechnical Engineering Technology, Hohai University, Nanjing, Jiangsu 210098, China

(3) Department of Mechanical and Aerospace Engineering, Monash University, Clayton, VIC 3800, Australia

Correspondence should be addressed to Jun-gao Zhu; 2841453079@qq.com

Received 2 May 2017; Revised 19 July 2017; Accepted 31 July 2017; Published 19 November 2017

Caption: Figure 1: Two collinear cracks in an orthotropic material under antisymmetrical linear heat flow.

Caption: Figure 2: The variations of [q.sub.c]/[q.sub.0] versus x/b with a/b = 0.25 and [h.sub.c] = 0, 1, 2, and 4, respectively.

Caption: Figure 3: The variations of K/[K.sub.0] versus x/b with [h.sub.c] = 1 and a/b = 0.25, 0.5, and 0.75, respectively.

Caption: Figure 4: The variations of K/[K.sub.0] versus x/b with a/b = 0.25 and [h.sub.c] = 1, 2, and 4, respectively.

Caption: Figure 5: The plate with two collinear cracks is modelled.

Caption: Figure 6: Temperature profile in an orthotropic plate with two collinear cracks under antisymmetrical linear heat flow.

Caption: Figure 7: Thermal stress analysis results and crack local detail.
Title Annotation: Printer friendly Cite/link Email Feedback Research Article Wu, Bing; Zhu, Jun-gao; Peng, Daren; Jones, Rhys; Gao, Shi-hu; Lu, Yang-yang Mathematical Problems in Engineering Report Jan 1, 2017 5303 Distributed Robust [H.sub.[infinity]] Consensus Control for Uncertain Multiagent Systems with State and Input Delays. Electric Vehicle Routing Problem with Charging Time and Variable Travel Time. Cracking (Materials) Mathematical models Thermal conductivity Thermal stresses