Zihao Yang, Liang Ma, Qiang Ma, Junzhi Cui,4, Yufeng Nie, Hao Dong, Xiaohong An

1 Introduction

Periodic composite material structures are widely used in the engineering practice due to their various advantageous physical and mechanical properties. Generally, both the material coefficients and geometric configurations of periodic composites are microscopically periodic. However, influenced by preparation technology, hot and humid environment, fatigue, damage and other factors, the coefficients reflecting properties of periodic composite material structures are no longer whole-periodic, but local-periodic,i.e., quasi-periodic. In other words, the material coefficients can depend not only on themicroscale information but also on the macro location. The functionally gradient material structure is a representative structure with quasi-periodic properties [Yin, Paulino, Buttlar,and Sun (2007); Shim, Yang, Liu, and Lee (2005); Zhang, Ni, Liu (2014)]. With the appearance of complex and extreme service environments, many composite structures work under transient thermo-mechanical circumstances. And the fully coupled analysis will lead to more accurate results. Therefore, it is necessary to study the transient thermomechanical coupling responses of quasi-periodic composite structures.

Up to now, some works have been performed on thermo-mechanical problems of composite structures. Feng et al. [Feng and Cui (2004)] proposed the multiscale asymptotic expansion for the problem under the conditions of coupled thermoelasticity for the structure of periodic composite materials. In [Zhang, Zhang, Bi, and Schrefler (2007); Yu and Tang (2007)], the authors investigated the thermo-mechanical problem of periodic composites by a multiscale asymptotic homogenization approach and a variational asymptotic micromechanics model,respectively. Terada et al. [Terada, Kurumatani, Ushida, and Kikuchi (2010)] considered the scale effect and derived the formal expansions for thermo-mechanical problem with periodically oscillatory coefficients. Goupee et al. [Goupee and Vel (2010)] presented multiscale thermoelastic analysis of random heterogeneous materials. Khan et al. [Khan,Barello, Muliana, and Lé vesque (2011)] studied the coupled heat conduction and thermal stress analyses in particulate composites by introducing two micromechanical modeling approaches. Temizer et al. [Temizer and Wriggers (2011)] reported a survey of the known mathematical results of the homogenization method and the multiscale approach for the linear thermoelasticity. Guan et al. [Guan, Yu, and Tian (2016)] presented a thermo-mechanical model for strength prediction of concrete materials. However, these studies were devoted to one-way thermo-mechanical coupling problems, namely, the thermal effects affect the mechanical filed but not vice versa. As for the two-way coupling problems, Parnell [Parnell(2006)] has given the homogenized procedure for the transient thermo-mechanical problems with different periodic configurations. After that, Yang et al. [Yang, Cui, Wu, Wang, and Wan (2015)] investigated the transient thermo-mechanical coupling problems of periodic composites by second-order two-scale method. For quasi-periodic composites, Bensoussan et al. [Bensoussan, Lions, and Papanicolaou (1978)] presented the homogenization theory and Cao et al. [Cao and Cui (1999)] has given the first-order approximation and several basic estimations of the mechanical problems. After that, Su et al. [Su, Cui, Zhan, and Dong (2010)]present the multi-scale analysis of boundary value problems for second-order elliptic type equation for the quasi-periodic composites. Dong et al. [Dong, Nie, Cui (2017)] perform a second-order two-scale analysis and introduce a numerical algorithm for the damped wave equations of quasi-periodic composite materials. To our knowledge, we have not seen the study of the transient thermo-mechanical coupling problems of quasi-periodic composites in the existing literature.

The two-way coupling problem is strongly coupled by the hyperbolic and parabolic equations with nonlinear coefficients, and it is impossible to find the analytical solutions.As for numerical solutions, due to the quasi-periodic properties and oscillating rapidly in microscopic cells of material coefficients, in order to effectively capture the local fluctuation behaviors of temperature and displacement fields and their derivatives, the mesh size must be very small while employing the traditional numerical methods, which will lead to a prohibitive amount of computation time. Therefore, it is necessary to develop highly efficient numerical methods for predicting the nonlinear thermo-mechanical coupling performance of composite material structures with quasi-periodic properties.

The homogenization method is developed to give the overall behavior of the composite by incorporating the fluctuations due to the heterogeneities of composites. However,numerous numerical results [Feng and Cui (2004); Bensoussan, Lions, and Papanicolaou(1978); Dong, Nie, Cui (2017)] have shown that the numerical accuracy of the standard homogenization method may not be satisfactory. And then, based on homogenization methods [Bensoussan, Lions, and Papanicolaou (1978); Marchenko and Khruslov (2008)],various multi-scale methods have been proposed [Efendiev and Hou (2009); Juanes(2005); E, Engquist, Li, Ren, and Vanden-Eijnden (2007)]. However, they only considered the first-order asymptotic expansions, which are not enough to describe the local fluctuation in many physical and mechanical problems. Hence, it is necessary to seek the more effective methods. This is the motivation for higher-order multiscale asymptotic methods and associated numerical algorithms. In recent years, Cui et al. [Cui and Yu (2006); Yang, Cui, Nie (2012); Zhang, Nie, Wu (2014)] introduced the secondorder multiscale analysis method to predict different physical and mechanical behaviors of composites. By the second-order correctors, the microscopic fluctuation behaviors inside the composite materials can be captured more accurately [Yang, Cui, Wu, Wang,and Wan (2015); Cao and Cui (1999); Su, Cui, Zhan, and Dong (2010); Dong, Nie, Cui(2017)]. However, the previous multiscale asymptotic expansions and algorithms cannot be directly employed to the thermo-mechanical problems due to the nonlinearity and twoway coupling. The aim of this paper is to establish a novel high-order multiscale method with less effort and computational cost to give a better approximation of the temperature and displacement fields in the transient thermo-mechanical coupling problems.

The remainder of this paper is outlined as follows. The formulation of the multiscale asymptotic expansions for the transient thermo-mechanical coupling problems of composite structures with quasi-periodic properties and associated error estimation in nearly pointwise sense are presented in section 2. In section 3, a finite element-difference algorithm based on the multiscale method is given in details. Some numerical results are shown to verify the validity of the multiscale algorithms in section 4. Finally, the conclusions are summarized in Section 5.

For convenience, the vector or matrix functions are denoted by bold letters like,..., and the Einstein summation convention on repeated indices is used in this paper.Besides, we do not give the definitions of the associated Sobolev spaces in this paper, and we refer the reader to the book [Leoni (2009)].

2 Multiscale asymptotic expansion

Obviously, they are 1-periodic functions in, respectively.

Figure 1: Macroscopic structure and microscopic unit cell

Now we derive the multiscale computation formulas for the transient thermo-mechanical coupling problem of quasi-periodic composite material structures. For convenience, we represent problem (1) as the operator equations. Let

and problem (1) can be rewritten as

in which □ denotes the functions.

Enlightened by the work in [Yang, Cui, Wu, Wang, and Wan (2015)],andcan be expanded into following forms

Taking into account of the chain rule

From above expansions (11), we can writeas follows

Further, we can define

Inserting (8) and (17) into (3) and equating the coefficients of the same powers, we have

and we will study these equations successively and define the homogenized problems,homogenized coefficients and corresponding cell functions.

From (19) and considering (18), (14), (12) and (9), we have

According to the theory of partial differential equations, we can acquire thatandare independent of the microscale, namely

Taking (24) into (20) and considering (18), (14), (15), (12) and (9), it can be obtained

Taking (27) and (28) into (25) and (26), it can be obtained that,andare the solutions of following cell problems

Existence and uniqueness of the cell problems (31) - (33) can be established based on

suppositions (S1) and (S2), Lax-Milgram lemma and Korn's Inequalities [Bensoussan,Lions, and Papanicolaou (1978)].

As for (21), using (18), (14), (15), (16) and (12), we get

Introducing (27) and (28) into (34) and (35), integrating over both sides of equations (34),(35) in and respecting (31) - (33), following equations are obtained

According to (32) - (33) and definitions (40) - (41), it is easy to prove thatis equivalent to. And according to supposition (S2) and [Bensoussan, Lions, and Papanicolaou (1978)], it followsand positive definite. Thus, the homogenized problem associated with the original problem (1) can be defined as follows

Existence and uniqueness of the cell problems (49)-(57) can be established based on suppositions (S1) and (S2), Lax-Milgram lemma and Korn's Inequalities [Bensoussan,Lions, and Papanicolaou (1978)].

In summary, the multiscale approximate solutions of problem (1) are defined as follows

Remark 2.1According to above detailed mathematical, it can be noted from (63) that the residual between the first-order multiscale approximate solutions and the solutions of original problem (1) is of orderthat does not equal to 0. In the practical engineering computation, it cannot be omitted for a constant, so engineers conclude that the firstorder multiscale approximate solutions cannot be acceptted and the microscale fluctuation of the temperature and displacement are far from being captured. This is the reason why it is necessary to seek the higher order expansions. It can be concluded from(64) that the second-order multiscale solutions are equivalent to the solutions of original problem (1) with orderin nearly pointwise sense. Moreover, the numerical results presented in Section 4 clearly show that it is important to include the second-order corrector terms.

Summing up, one obtains following results

Theorem 2.1The temperature and displacement fields for the transient thermomechanical coupling problem (1) of quasi-periodic composite materials have the multiscale asymptotic expansions as follows

And then the temperature gradient, strains and stresses can be evaluated based on the chain rule (10) and the multiscale asymptotic expansions of temperature and displacement fields (65) and (66).

3 Numerical implementation of multiscale method

In this section, the multiscale algorithms based on the finite difference method in time direction and finite element method in spatial region for predicting the transient thermomechanical coupling behaviors of quasi-periodic composite materials is presented.

Note that all the cell problems (31) - (33) and (49) - (57) are associated with macroscopic coordinates, which brings lots of complexities in numerical computation since we have to solve these cell problems at every point. In practical applications, such as the damage analysis of composite materials, engineers often take the single cell as a unit to evaluate the damage degree of composite materials, which leads to a scale separation of material coefficients. The specific meaning of scale separation is written as follows

and it is a solution of following problem

Besides, the homogenized coefficient can be rewritten as

3.1 Multiscale numerical formulations

3.1.1 Finite element computation of cell functions and homogenized parameters

3.1.2 Finite element-difference computation of homogenized equation

The homogenized equations (44) are dynamic problem coupled by hyperbolic and parabolic equations. Thus, the spatial regionis divided by using the finite element mesh first, and then the temporal domain）is divided by using the finite difference.The semi-discrete scheme for solving homogenized equations is given as follows

They are expressed as the following forms

3.1.3 Multiscale numerical solutions

According to (58) - (61), the multiscale approximation solutions based on global structurecan be evaluated by

3.2 Algorithm procedures for the multiscale method

The algorithm procedure for the multiscale method to predict the transient thermomechanical coupling performance is stated as follows

1) Determine the geometrical constructions of the macroscopic structureand cell domain, and verify the material parameters of various constituents.

2) Solve the cell problems (31) - (33) to get the finite element solutions of,and, respectively. Furthermore, the homogenized parameters,,,,andare evaluated by formulas (38)-(43).

3) With the homogenized parameters obtained in previous step, compute the homogenized solutionsandby solving the homogenized problem (44).

5) Solve the derivatives of the homogenized solutionsandwith respect to spatial and temporal variables. The derivatives with respect to spatial variable are evaluated by the average technique on relative elements [Thomas (2013)] and the derivatives with respect to temporal variable are evaluated using the difference schemes in step 3.

6) Compute the temperature and displacement fields using formulas (65) and (66),respectively.

4 Numerical examples

Figure 2: (a) Unit cell; (b) domain

Since it is difficult to find the exact solutions of above problem, we have to takeandto be their finite element (FE) solutionsandin the very fine mesh for comparison with different order approximate solutions. The triangulation partition is implemented, and the information of the FE meshes is listed in Table 2. Set

Table 1: Material properties

For convenience, we introduce the following notation

And we consider two cases

Table 2: Comparison of computational cost

Table 3: Comparison of computing results for temperature increment in -norm

?

Table 5: Comparison of computing results for displacement in -norm

Table 5: Comparison of computing results for displacement in -norm

?

Table 4: Comparison of computing results for temperature increment in -norm

Table 4: Comparison of computing results for temperature increment in -norm

?

Table 6: Comparison of computing results for displacement in -norm

Table 6: Comparison of computing results for displacement in -norm

?

From Table 2, we can see that the mesh partition numbers of second-order multiscale approximate solutions are much less than that of refined FE solutions. Both the secondorder multiscale method and the direct FE numerical computations are performed on the same computer. And the approximate running times for the multiscale algorithm and classical FE computation with refined mesh are 471.2 seconds and 1928.7 seconds,respectively. We cannot easily solve the problem (1) directly by the classical numerical methods because it would require very fine meshes and the convergence of the FE method based on fine meshes for the nonlinear coupled problem is not very easy.Moreover, the proposed second-order multiscale method is suitable for the composite materials with a great number of cells, which can greatly save computer memory and CPU time without losing precision. And it is very important in engineering computations.From Fig.3-Fig.6 and Tables 3-6, it can be found that the newly second-order multiscale approximate solutions are in good agreement with the FE solutions in a refined mesh. But the homogenized solutions and first-order multiscale solutions have less effect approaching the refined-mesh FE solutions. The homogenized solutions give the original problem an asymptotic behavior, which is not enough forthat is not so small. So, the correctors are necessary, and the results show that the second-order correctors give much better approximation of the displacement, strain, temperature increment and its gradient.Furthermore, numerical results also show that only second-order multiscale solutions can accurately capture the microscale oscillating information of the multiscale problem.Besides, the relative errors between different approximate solutions and FE solutions obtained on refined mesh are also exhibited in Figure 7. It is worth to note that the relative errors are not growing significantly as time increases. This indicates that the multiscale method is a very good method for treating a long-time problem in some cases.Consequently, all the results demonstrate that the multiscale method is effective and efficient to predict the transient thermo-mechanical coupling behaviors of quasi-periodic composite materials.

Figure 3: Temperature increment gradient

Figure 4: Strain

Figure 5: Comparison of different solutions on the lineatfor Case 1: (a)temperature increment; (b) displacement

Figure 6: Comparison of different solutions on the line at for Case 1: (a)temperature increment gradient; (b) strain

Figure 7: The evolution of relative errors with of Case 1 for (a) temperature increment gradient and (b) strain

5 Conclusion

In this paper, the multiscale analysis method and related numerical algorithms are presented to predict the transient thermo-mechanical coupling behaviors of quasi-periodic composite structures. The multiscale formulations for the nonlinear and coupling problem are obtained, including the local cell problems, effective thermal and mechanical parameters, homogenized equations and second-order multiscale asymptotic expansions of temperature and displacement fields. The error analysis is given to indicate that the second-order multiscale approximate solutions have a much better approximation to the solutions of the original problem. Numerical results demonstrate that the local steep variations of the temperature, displacement and their gradient can be captured more precisely by adding the second-order correctors. And it can be also concluded that the multiscale method is not only feasible, but also accurate and efficient to predict the transient thermo-mechanical coupling behaviors of quasi-periodic composite structures.The high quality of the results encourage the application of proposed multiscale model and related numerical technique to deal with thermo-mechanical analysis of heterogeneous medias with much more complicated multiscale structures. And it is very helpful to the design and optimization of the composite structures.

Acknowledgement:This research was financially supported by the National Natural Science Foundation of China (11501449), the Fundamental Research Funds for the Central Universities (3102017zy043), the China Postdoctoral Science Foundation (2016T91019), the fund of the State Key Laboratory of Solidification Processing in NWPU (SKLSP201628) and the Scientific Research Program Funded by Shaanxi Provincial Education Department(14JK1353)..

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