Archive of Applied Mechanics 70 (2000) 585±596 Ó Springer-Verlag 2000

Integrated design of graded microstructures of heterogeneous materials N. Takano, M. Zako

Summary This paper proposes a novel, integrated, computational design methodology of graded microstructures of heterogeneous materials for the emergence of macroscopic function. In the ®rst step of the design procedure, some discrete microstructures among a large number of graded microstructures are determined by the genetic algorithm as the optimization method. In this determination procedure, the homogenized modeling is adopted considering the micromacro coupling by the homogenization method. This method enables us to study complex microstructures. Homogenized properties such as elastic properties, coef®cient of thermal expansion and thermal conductivity can be calculated rigorously based on continuum mechanics. Calculated homogenized properties are stored in the micro-macro correlative database. The genetic algorithm can select the best geometrical arrangement of multiple microstructures from the pre-calculated database to construct the graded microstructure architecture. By using the database, the microscopic and macroscopic analyses are separated from each other, which reduces considerably the computational cost for the micro±macro coupled design. In the second step, continuously graded microstructures are designed using a feature-based 3D-CAD system by interpolating the discretized graded microstructures. In addition, a solid model is produced by the stereolithography technique to help in understanding the computationally designed complex microstructures. Brief descriptions of the formulation of the homogenization method for heat conduction and thermal stress problems are shown. A design problem for a plate with graded microstructures in its thickness direction is shown. The objective function for this example is the control of the wrap of the plate under the condition of temperature distribution. Key words Microstructure, heterogeneous materials, computational design, homogenization method

1 Introduction Multiscale modeling of heterogeneous materials in considering their micro-meso-macro structures is becoming a matter of interest to bridge the gap due to the length scale, [1±3]. In the ®eld of computational mechanics, there are many studies on the homogenization method. The micro±macro coupled problems are solved for heterogeneous materials in the framework of continuum mechanics. The method represents a mathematical theory ®rst developed in the early 1980s, [4±6]. In the 1990s, many pioneering works on the engineering applications of this method were published [7, 8], studying metal matrix composite materials. The advantages of the homogenization method over the classical rule of mixture are that a complex shape of the microstructure can be considered, and that both effective (or homogenized) properties and Received 19 April 1993; accepted for publication 18 January 2000 N. Takano (&), M. Zako Department of Manufacturing Science, Osaka University 2-1 Yamada-oka, Suita, Osaka 565-0871 Japan Fax: + 81-6-6879-7570; e-mail: [email protected] The authors would like to express their sincere gratitude to Denken Engineering Co., Ltd. in Japan for the help in the creation of the solid model by stereolithography. This research was supported by the Kurata Foundation, Japan. Also, the authors are grateful to Mr. Raed Rachdan for his dedicated help in writing this paper.

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586

microscopic stresses can be calculated, [4±8]. Another interesting application of the homogenization method in the ®eld of material science is the design of microstructures for tailored materials. In [9, 10], a novel layout (or topology) design of a microstructure has been proposed to obtain prescribed effective properties by solving the inverse problem. A practical design of a piezoelectric device using the same technique has also been reported in [11]. However, the studies in Refs. [9±11] were based on the microstructures that have exactly the same pattern repeated periodically to form the structure. In the ®eld of material science, there have been many studies of functionally gradient materials (FGMs), [12, 13]. FGMs are made of continuously gradient heterogeneous materials, and offer desired functions which are adaptive to the environmental situations. In the conventional design of FGMs, the rule of mixture has been used. When we consider the cellular materials, however, they have very complex microstructures. Therefore, they might exhibit many interesting and important macroscopic functions such as foam materials with negative Poisson's ratio or human bone, [14, 15], possessing a layered-microstructure architecture. The same is true for arti®cial blood vessels, [15]. Obviously, the macroscopic properties of these cellular materials can not be accurately described by the rule of mixtures. Hence, a more general study is needed. Therefore, we propose a novel computational design methodology of continuously graded microstructures for the emergence of macroscopic function. The above-mentioned layered structures and FGMs are supposed to be the examples of functional composites with graded microstructures that we propose to study in this paper. In the design of graded microstructures, we have to design the optimum arrangement of a large number of microstructures. This subject is absolutely different from the previous studies on the layout design of one microstructure [9±11]. Hence, we propose a novel design methodology which consists of two steps. The ®rst step uses a discretized model and CAE techniques, including the homogenization method and the genetic algorithm (GA). Then, in the second step, the design of continuously graded microstructures is completed using a featurebased 3D-CAD. In addition, a solid model is created using a rapid prototyping (RP) technique such as the stereolithography. The solid model helps us to easily understand the designed complex graded microstructure architecture. In summary, the proposed design consists of integrated CAE/CAD/RP systems. In this paper, we will show the concept of the design methodology, however, the details are presented through an example of a plate consisting of a graded porous materials. Also, a brief description of the formulation of the homogenization method for heat conduction and thermal stress problems is shown, and the optimization procedure is brie¯y described. Now, we would like to place some remarks on the length scale that we considered in this paper. In the previous studies using the homogenization method [1±11], the microscale is de®ned in the framework of the continuum mechanics. The well-known multiscale model [16, 17] has also three scales, micro-meso-macro scales, as we wrote above. But, in Refs. [16, 17], the atoms are considered as the microscale, the grains as the mesoscale, and the structure as the macroscale. In this paper, using the conventional homogenization method for linear elasticity, the microstructure model is supposed to be described by continuum mechanics as well as the macroscopic structure. The application of the homogenization method to polycrystalline materials have not been reported as far as we know. That is, the accuracy of the homogenized properties calculated by this method has not been con®rmed, although many papers on the classical ®nite element (FE) calculations of grains have been published. To this end, the reliable length scale of the microstructures in this paper may be larger than the size of grains in the case of multiphase materials at this moment.

2 Outline of the proposed design methodology First of all, we suppose arbitrary and very complex microstructures in the framework of the continuum mechanics. The dimensions of the microstructures are supposed to be the design parameters. The topology of the microstructures are supposed to be given. In the practical applications such as the foam materials, their microstructures are very complex. The porosity can be controlled in the real manufacturing, then the topology of the microstructures can be given by observation using a microscope. Consider a structure that consists of a large number of graded microstructures. If the representative dimension of a macroscopic structure is L and that of a microstructure is `, the scale ratio between macroscale and microscale is

e ˆ `=L  1 :

…1†

In such conditions, it is impossible to design all the microstructures in the whole structure one by one. Hence, in a ®rst step, a discretized modeling is adopted to design some discrete microstructures among a large number of graded microstructures. To construct the discretized model, the homogenization method is adopted. We replace the discretized area, which consists of heterogeneous microstructures, by an equivalent macroscopic material model. Although the discretized area consists of graded microstructures, we consider one representative microstructure for one discretized area. In other words, the discretized model uses the assumption of periodic arrays of the representative microstructure in an area. In the real heterogeneous materials, the periodicity condition is not always satis®ed, and this has been considered as the limitation of the use of the homogenization method. However, from the above consideration on the representative microstructure in the discretized area, the homogenized modeling using this method is acceptable. When the design parameters, i.e. the dimensions of the microstructures are changed, the set of the homogenized coef®cients of thermal expansion and the homogenized conductivity can be calculated for each microstructure with speci®c microscopic dimensions. Then, we can construct a database of the homogenized properties associated with the design parameters. This is a micro-macro correlative database. Using this database, microscopic and macroscopic analyses can be separated. Accordingly, the computational design cost for the micro-macro coupled design can be reduced remarkably. The GA is adopted as the optimization method in this paper, because it can be applied easily to various nonlinear design problems, [18]. By using the pre-calculated database and by solving a macroscopic problem, optimum discrete microstructures are selected. Then, in the second step, continuously graded microstructures are reconstructed from the above-mentioned discretized model. As the topology is supposed to be ®xed, the feature-based 3D-CAD is well suited to this geometric design. By interpolating the discrete models, the design of continuously graded microstructures is completed. Figure 1 illustrates this design procedure. In addition, because the 3D-CAD system is used in the design procedure, the creation of a solid model of the computed design solution is easily conducted by a RP technique such as the stereolithography. The solid model can be called a 3D-plot of the computed solution. By watching and touching the 3D-plot, we can understand deeply the complex graded microstructure architecture.

3 Formulation of the homogenization method Suppose a macroscale x and microscale y as shown in Fig. 2. Using the scale ratio de®ned by Eq. (1), the relation between both scales is expressed as follows: y ˆ x=e :

…2†

As shown in Fig. 2, the structure is supposed to be a periodic assembly of unit cells.

Fig. 1. Schematic ¯ow of the design method

587

Fig. 2. De®nition of the macroscale and the microscale

588 For heat conduction and thermal stress problems the governing equations can be described considering the effects of the microstructure respectively, by

Z X

Z X

oT odT K ij dX ˆ oxj oxi

Z

Z f dT dX ‡

hdT dC 8 dT ; Ch

X

Z

ouk odui Eijkl dX ˆ oxl oxj

…3†

Z

Z

bi dui dX ‡

ti dui dC ‡ Ct

X

Eijkl akl …T X

T0 †

odui dX 8 dui ; …4† oxj

where K ij and Eijkl are the thermal conductivity and the elastic tensors, and akl are the coef®cients of thermal expansion. Let f and h denote internal heat sources and in¯ow of heat on Ch boundary, while bi and ti denote the body force and the traction on Ct boundary; here Kij and Eijkl re¯ect the microscopic heterogeneity represented by periodic functions and T0 is the initial temperature. The temperature and displacement ®elds are expressed using the asymptotic expansion method

T…x; y† ˆ T 0 …x† ‡ eT 1 …x; y† ;

…5†

ui …x; y† ˆ u0i …x† ‡ eu1i …x; y† ;

…6†

when we consider the microscopic heterogeneity as shown in Fig. 2. Here, T 0 and u0i are the macroscopic (or homogenized) temperature and displacement, and T 1 and u1i are the microscopic temperature and displacement, respectively. It has been proven that the macroscopic quantities are functions of only macroscale x, [4±7], as shown in Eqs. (5) and (6). By substituting Eqs. (5) and (6) into the governing equations, and using the following averaging principle for a periodic function W in the same way with the standard homogenization procedure, [4±7], the micro-macro coupled problem can be resolved into microscopic and macroscopic problems:

Z lim e!0

W X

Z Z x 1 dX ˆ W…y†dY dX ; e jYj

…7†

Y

X

where jYj denotes the volume of the unit cell. For a heat conduction problem, the microscopic and the macroscopic equations are derived respectively, as

Z K ip Y

Z X

KH ij

o/j odT 1 dY ˆ oyp oyi

Z

oT 0 odT 0 dX ˆ oxj oxi

K ij Y

Z X

odT 1 dY oyi

f H dT 0 dX ‡

8 dT 1 ; Z Ch

hdT 0 dC 8 dT 0 :

…8†

…9†

By solving the microscopic equation (8) using the periodic boundary condition, we obtain the characteristic function associated with heat conduction in the microstructure due to the mismatch of the thermal conductivities of the constituents. In Eq. (9), the homogenized H thermal conductivity tensor K H are rigorously ij and the homogenized internal heat sources f de®ned as follows:

KH ij ˆ fH ˆ

1 jYj 1 jYj

Z  K ij

K ip

Y

 o/j dY ; oyp

…10†

Z

f dY :

…11†

Y

For a thermal stress problem, microscopic and macroscopic equations are derived in the same way

Z Eijmn Y

Z Eijkl Y

Z

EH ijkl

X

1 ovkl m odui dY ˆ oyn oyj

ouk odu1i dY ˆ oyl oyj ou0k odu0i dX ˆ oxl oxj

Z Eijkl Y

odu1i dY oyj

Z Eijkl akl Y

Z

odu1i dY oyj

0 bH i dui dX ‡

Z

8 du1i ;

8 du1i ;

ti du0i dC ‡

Ct

X

…12†

…13† Z

H 0 EH ijkl akl …T

X

T0 †

odu0i dX 8 du0i : oxj …14†

Microscopic Eq. (12) provides the characteristic displacement due to the mismatch of the elastic tensor of the constituents, and microscopic Eq. (13) provides the characteristic displacement due to the mismatch of the coef®cient of thermal expansion. The homogenized elastic tensor EH ijkl ; H coef®cients of thermal expansion aH ij and the body force bi are derived as follows:

EH ijkl aH ij

1 ˆ jYj

ˆ

bH i ˆ

Z  Eijkl

H Cijpq

1 jYj

Y

1 jyj

Z

Z

 ovkl m dY ; Eijmn oyn

 Epqkl akl

Y

…15†

 ouk dY ; oyl

…16†

bi dY ;

…17†

Y

H where Cijpq denotes the homogenized compliance tensor, which is the inverse of the homogenized elastic tensor. The above partial differential equations can be solved numerically by the ®nite element method. Microscopic equations are solved with the periodic boundary condition that is put on the unit cell. Not only the homogenized properties but also the microscopic quantities such as the following microscopic stress can be calculated by the homogenization method. The microscopic quantities can be considered as one of the objective functions in the optimization procedure. In the design example in this paper, however, they are not used, and only macroscopic de¯ection is considered as the objective function

 rij ˆ Eijkl

 0 ovkl m ouk Eijmn oyn oxl

 Eijkl akl

 ouk …T 0 oyl

T0 † :

…18†

589

590

4 Micro±macro correlative database The homogenized properties can be calculated by the above formulation for arbitrary complex microstructures in the framework of the continuum mechanics. When the topology of the microstructure is given, we ®rst calculate the homogenized properties by changing the dimensions of the microstructure. These pre-calculated properties are stored in the micro±macro correlative database. The database is referred to in the optimization procedure. Then, the micro-macro coupled design, which generally may require a large amount of computational cost, is separated into microscopic and macroscopic analyses. Figure 3 illustrates the ¯owchart of the microscopic and macroscopic analyses. Basically, the database of the homogenized properties associated with the design parameters consists of discrete information. Some interpolation functions can be applied to compensate the lack of data. For instance, when we consider a foam material with graded microstructures that is produced by changing of porosity, the discrete database is generated by the observation of the actual microstructures with some representative porosity ratios. Then, in the design procedure, a smoother database of the homogenized properties for various porosity ratios is generated with the help of the interpolation. Once the database is generated, it can be used for various designs for the emergence of various macroscopic functions. 5 Optimization method As shown in Fig. 1 only some discrete microstructures among a large number of graded microstructures are designed in the ®rst step of the design procedure. It seems to be quite natural to approximate the distribution of a physical quantity by some representative and discrete points. The optimization problem is equivalent to the combinatorial problem where an optimum set of microstructures is selected from the pre-calculated database. Accordingly, the GA is adopted as the optimization method. For instance, in the following design example, an optimum solution is searched for from 4010 choices. For this kind of optimization problem, the GA is effective. Moreover, and in most cases, the micro-macro coupled-optimum design problem is dif®cult to be described mathematically because of nonlinearity. The GA is easily

Fig. 3. Flowchart of the micro±macro coupled-design procedure

applied to highly nonlinear problems, and it has been recognized that it works well for solving nonlinear problems, [18]. From a manufacturing viewpoint, a constrained optimum problem should also be considered in the practical design of microstructures. Such problems are also dif®cult to be described mathematically. By using the GA, such constraints may be expressed by some rules denoting the neighboring microstructures in the selection of the candidates from the database. From the above reasons, the GA has been adopted. There have been many studies on the enhancement of the GA and on its application to practical engineering problems. After these previous works, [18±21], in this study, a conventional algorithm using reproduction, crossover and mutation is adopted. Also, the standard setting of the GA parameters are used. That is, the population size is 100, the mutation rate is 0.05, the crossover rate is between 0.6 and 0.8, [20, 21]. In the design of graded microstructures, the dimensions of the microstructures are the design parameters, and the micro-macro correlative database is generated by changing the dimensions of the microstructure. In the generation of the database, the calculated microstructures are numbered sequentially, and assigned to the chromosome of the GA. The binary coding has been used as the chromosome so far. However, in this study, the integer coding is adopted after Ref. [21]. The advantages of the integer coding come from the fact that less computational memory and population size is required to reach the global optimum, [21]. In the following example, the chromosome has an integer value between 0 and 39.

6 Construction of continuously graded microstructures There may exist some possible techniques to complete the design of our continuously graded microstructures after the discrete microstructures are determined. For instance, a mathematical interpolation function may be applicable. However, some constraint conditions, which might be required for the real manufacturing, should be included in this step. In this paper, therefore, we propose to use the commercial feature-based 3D-CAD system. Since the topology of the microstructure is ®xed in our problem's setting, a unit cell can be de®ned as a feature, and the change of the dimensions (geometry) of the feature can be easily realized in the feature-based 3D-CAD. 7 A design example 7.1 Problem's setting Consider a plate which consists of graded microstructures in the thickness direction. The temperature on the upper and lower surfaces is supposed to be ®xed and is distributed through the thickness. If the plate is made of a homogeneous material or periodic arrays of single microstructure, then the mismatch of the thermal strains on the upper and lower surfaces warps the plate. On the contrary, the designed graded microstructures can control both temperature and thermal strain ®elds through the thickness. This will result in control of the warp. That is, if we can control the thermal strain to be constant through the thickness, the plate will not warp even under the temperature gradient. To achieve this goal, the optimum distribution of the coef®cients of the thermal conductivity and the thermal expansion is designed. This procedure can provide one solution to this problem's setting. As an example of the topology of the microstructure, a porous materials shown in Fig. 4 is studied in this paper. Three design parameters, A; B and C, are considered. This microstructure model is similar to a real foam material which exhibits a negative Poisson's ratio, [14, 15]. Both the concave line and the hinge architecture of this microstructure model can be seen as the reasons for which the material exhibits various interesting and unpredictable behaviors. The FE model with 1325 nodes and 960 elements was used for homogenization analysis. In the following calculation, the thickness of the plate is set to be 10 mm, and the number of graded microstructures in the thickness direction is set to be 30. Consequently, the scale ratio e is equal to 1/30. 7.2 Optimum design using a discrete model Firstly, the homogenized coef®cients of thermal conductivity and thermal expansion are calculated by the homogenization method for forty kinds of microstructures. The microstructures

591

592

Fig. 4. Unit cell of the microstructure

are assigned to the integer chromosome ranging from 0 to 39. These microstructures were generated by changing the microscopic dimensions. Because three design parameters are considered, it is hard to draw the correlation between the homogenized properties and the microscopic dimensions. However, it was recognized that the homogenized properties changed nonlinearly with respect to the changes of the microscopic dimensions. This implies the dif®culty of the micro-macro coupled design. Another important point is that a set of homogenized coef®cients of thermal conductivity and thermal expansion is determined when the geometry of the microstructure is changed. In other words, if we want to change the thermal conductivity, the coef®cient of thermal expansion must be also changed as a pair. Note here that the total computational time for the calculation of the homogenized coef®cients for forty microstructures was around one hour using a low level PC (MMX Pentium, 200 MHz). Next, ten among the 30 graded microstructures are picked, and the macroscopic discrete model is considered as shown in Fig. 5. In the ®rst step, one representative microstructure is considered in one discretized region in Fig. 5. Hence we have 4010 choices in this optimization procedure. The problem's setting in this paper yields a one-dimensional problem, where every macroscopic quantity distributes only in the thickness direction. When the temperatures on the upper and lower surfaces are ®xed, the distribution of the temperature and the thermal strain can be analyzed theoretically. In the following calculation, the temperature on the upper surface is supposed to be 373 K, and that on the lower surface is supposed to be 298 K. That is, the difference of the temperature through the thickness is 75 K. When ten discretized domains have the same thickness t, the macroscopic heat ¯ux q is calculated by

75 q ˆ P10 1 ; t nˆ1 jH

…19†

n

where jH n denotes the homogenized thermal conductivity to represent the discretized domain n. Using the Fourier's law, the macroscopic temperature gradient in the thickness direction z is obtained as follows:

dT 75 ˆ P10 1 : H dz jn t nˆ1 jH

…20†

n

Fig. 5. Discretized modeling

By integrating Eq. (20), the macroscopic temperature at the center of each discretized domain is calculated. Subsequently the thermal strain en is also calculated using the homogenized coef®cient of thermal expansion aH n . Then the variance of the macroscopic thermal strain distribution in the thickness direction Vstrain is calculated, which is de®ned as the objective function in the optimization procedure.

Vstrain ˆ

10 1 X …en 10 nˆ1

e†2 ;

…21†

where e is the mean value of macroscopic thermal strain distribution. Finally, the optimum design problem can be expressed as follows: H find jH n ; an …n ˆ 1; 2; . . . ; 10† to minimize Vstrain :

593

…22†

The inverse of Vstrain is used as the ®tness function for the GA search. In this optimization procedure, only a macroscopic analysis has to be carried out, which requires little computational cost. Figure 6 shows the designed discrete microstructures. In this paper, no constraint condition is considered in the selection of the microstructures. The microstructures are graded from the lower surface (#1) to the upper surface (#10). The distribution of the homogenized coef®cients of thermal conductivity and thermal expansion is shown in Fig. 7. The vertical axis denotes the normalized location in the thickness direction. The coef®cient of thermal expansion near the upper surface with high temperature is smaller. This result seems to be quite reasonable if we want to make the distribution of the thermal strain constant and control the warp of the plate. However, the obtained distribution of the thermal conductivity was unpredictable. To this end, and as shown in Fig. 8, the temperature distribution through the thickness is not linear. The linear temperature distribution, plotted also in Fig. 8, is for the case of a homogeneous material. The obtained distribution of the thermal strain was almost constant

Fig. 6. Designed discretized graded microstructures

594

Fig. 7. Distribution of homogenized properties in the thickness direction

Fig. 8. Temperature distribution in the thickness direction

through the thickness. Accordingly, the warp of the plate is expected to be controlled. To verify this, a 3D FE analysis was carried out. We found that the warp of the plate was almost zero compared to the case of the homogeneous material which has the minimum thermal expansion in the database. The computing time for the optimum design by the GA, using the pre-calculated micromacro correlative database, was only three minutes with the above-mentioned low level PC. The simpli®cation in evaluating the objective function in this example contributed much to the reduction of the computing time.

7.3 Design of continuously graded microstructures by 3D-CAD In order to complete the design of the graded microstructures, a feature-based 3D-CAD system was used, and the continuously graded microstructures were obtained by interpolating the discrete microstructures. The resulting graded microstructures in the thickness direction are shown in Fig. 9. The plate consists of periodic arrays of the set of graded microstructures. 7.4 Creation of a solid model by stereolithography Because the proposed design methodology adopts 3D-CAD to complete the design, it is natural and quite easy to create the solid model by the stereolithography. Figure 10 shows the created solid model of the unit set of the designed graded microstructures. The considered plate actually consists of a large number of arrays of this unit set of graded microstructures as shown in Fig. 5. In a future study, the solid model can be used to check the function of the computationally designed graded microstructures.

595

Fig. 9. Final designed continuously graded microstructures

Fig. 10. Solid model of graded microstructures

8 Conclusion The concept of the design of graded microstructures for the emergence of macroscopic function is proposed based on continuum mechanics. An integrated computational design methodology using the homogenization method, genetic algorithm, feature-based 3D-CAD system and stereolithography technique is presented. To design a large number of microstructures, the discrete modeling is used ®rst, and continuously graded microstructures are subsequently designed. As an example, a plate whose macroscopic de¯ection is controlled under the condition of the temperature distribution is shown. Only one-dimensionally graded problem's setting was described in this paper, but the proposed design methodology can be applied to a two-dimensionally graded problem. Not only the macroscopic quantity but also the microscopic quantities such as strain, stress and heat ¯ux can be considered as the objective functions. The proposed methodology is expected to be applied to the designs of practical functional materials such as foam materials, synergy ceramics, etc.

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