Interface profile optimization for planar stress wave attenuation in bi-layered plates

Composites Part B 82 (2015) 129e142

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Composites Part B journal homepage: www.elsevier.com/locate/compositesb

Interface profile optimization for planar stress wave attenuation in bi-layered plates R. Rafiee-Dehkharghani a, D. Bansal b, A.J. Aref a, *, G.F. Dargush c a

Department of Civil, Structural, and Environmental Engineering, University at Buffalo, State University of New York, Buffalo, NY 14260, USA Material Science and Technology Division, Physical Sciences Directorate, Oak Ridge National Lab, Oak Ridge, TN 37831, USA c Department of Mechanical and Aerospace Engineering, University at Buffalo, State University of New York, Buffalo, NY 14260, USA b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 5 February 2015 Received in revised form 19 May 2015 Accepted 6 August 2015 Available online 17 August 2015

Stress waves scatter upon entering a new medium. This occurs due to the reflection and transmission of the waves, which depends on the impedance mismatch between the two materials and the angle of incidence. For a bi-layered structure with finite dimensions and constant impedance ratio, the scattering and intensity of the stress waves may be varied by changing the interface profile between the two layers. In this paper, a methodology is proposed for optimizing the interface profile between the layers of a finite bi-layered plate for the objective of planar stress wave attenuation. The bi-layered plates are subjected at one end to highly impulsive loadings with various durations, and the geometry of the internal interface is optimized for the purpose of minimizing the amplitude of the maximum reaction force at the opposite fixed end. The optimization methodology is based on a genetic algorithm, which is coupled with a finite element method for analyzing the wave propagation behavior of the plates. It is observed that the interface profile and the amount of stress wave attenuation depend on the duration of the applied impulsive loading, with higher amounts of attenuation obtained when the wavelength associated with the impulsive load is small compared to the dimensions of the bi-layered plates. © 2015 Elsevier Ltd. All rights reserved.

Keywords: A. Plates B. Impact behavior B. Interface C. Finite element analysis (FEA) Stress wave propagation

1. Introduction Over the past few decades, layered structures have been used extensively in a broad range of engineering applications. Many of these applications are related to designing structures that are subjected to dynamic loadings. For example, in military applications, laminated composites are exploited broadly for mitigating the effects of impulsive loadings. As another example, various numbers of layered and sandwich composites have been developed for protecting structures against blast and high velocity impact. The behavior of the layered structures under dynamic loadings and their attenuation capacity depends on the material properties and impedance mismatch between the layers. Due to the existence of impedance mismatch, reflection and transmission of the incident waves take place at the interface between the two materials. This results in scattering of the waves and altering the wave propagation behavior of the layered systems. Considering the impedance mismatch phenomenon, the material properties of layered systems can be tuned for attenuating the intensity of the dynamic loadings.

* Corresponding author. Tel.: þ1 716 645 4369; fax: þ1 716 645 3733. E-mail address: [email protected] (A.J. Aref). http://dx.doi.org/10.1016/j.compositesb.2015.08.010 1359-8368/© 2015 Elsevier Ltd. All rights reserved.

The wave propagation behavior of layered structures has been explored by many researchers. In one of the early works, Lindholm and Doshi [1] studied the wave propagation in a nonhomogeneous finite elastic bar with varying modulus of elasticity, which is subjected to a transient pressure pulse. Anfinsen [2] provided a design approach for maximizing and minimizing the amplitude of a stress pulse in layered one-dimensional elastic structures. Lee et al. [3] explored the efficiency of layered plates with discontinuous and continuous changes in material properties for the objective of impact resistance. Chiu and Erdogan [4] demonstrated the onedimensional wave propagation in elastic slabs with functionally graded materials (FGMs). Sudden changes in the material properties have many disadvantages because of the high amount of stress concentration. To overcome this problem, FGMs are usually employed in engineering applications, because the material properties of these structures are gradually changing. Wave propagation in FGMs are extensively studied in the literature. Some examples include the works by Li et al. [5], Velo and Gazonas [6], Naik et al. [7], Aksoy and S¸enocak [8], Sun and Luo [9], Hui and Dutta [10], and Pandya et al. [11]. In addition, some research has been done on the optimal design of the FGMs and layered systems for the purpose of stress wave attenuation (Taha et al. [12], Luo et al. [13] and [14], Rafiee-Dehkharghani et al. [15], and Rafiee-Dehkharghani [16]).

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By reviewing the literature, it is demonstrable that most of the research on wave propagation in layered structures has focused on the effect of impedance mismatch between the layers and the way that the change in the material properties occurs, i.e., graded or abrupt. This means that the major parameter, which is explored in the literature is the material effect and it is discussed by several researchers (Liu et al. [17]; Banks-Sills et al. [18]; Nwosu et al. [19]; Hong and Lee [20]). For this reason, many of these studies (Chiu and Erdogan [4]; Bruck [21]; Samadhiya et al. [22]; Chen et al. [23]) are based on one-dimensional wave propagation, even in twodimensional structures, such as plates. In fact, in many practical applications, the structures have more than one dimension and their wave propagation behavior depends on the geometric specifications in addition to the material properties. For layered systems, the geometric properties can be attributed to the global shape of the structure and the interface profile between the layers. To the best of our knowledge, no research has been done on the effect of the interface profile between the layers. This paper describes a study that aimed at investigating the effect of the interface profile between two media in layered structures and illustrates the development of a methodology for optimizing the shape of this profile for the objective of stress wave attenuation. The developed methodology is used for optimal design of the interface profile between the layers of free-fixed rectangular bi-layered plates, which are subjected to in-plane transient dynamic loadings with different durations. It is assumed that the interface between the layers is composed of various straight line segments that are connected to each other to make a jagged path. An optimization methodology is then used to find the optimal shape of this jagged path for minimizing the intensity of the transient load, as it reaches the fixed boundary. Since the amount of attenuation and scattering of the waves depends on the inclination angle of the incident waves at the jagged interface, the theory of the reflection and transmission for an incident wave with oblique angle is explained in detail. The dynamic behavior of the bi-layered plates is analyzed using two-dimensional plane stress wave propagation theory. The finite element (FE) method is employed for this purpose due to the limitations of finding closed-form solutions for the problem of wave propagation in finite plates with a complex jagged interface. A genetic algorithm (GA) optimization approach is utilized for the problems under consideration. This optimization approach is a robust heuristic optimization methodology, which is appropriate for solving the problems stated in this paper. In particular, GAs resolve the difficulty of obtaining gradient information with respect to the design variables, and are well-suited for the highly nonlinear nature of the problem at hand. For finding the fitness value, the GA is coupled with an FE code. The coupled GA-FE

optimization methodology is utilized for designing the bi-layered plates with jagged interfaces subjected to in-plane half-sine transient loadings with different durations. The efficiency of this coupled GA-FE optimization methodology is examined for layered systems with straight interfaces and for plates with circular inclusions in Rafiee-Dehkharghani et al. [15] and [24], respectively. The theory and background of the problem, details of the coupled GA-FE optimization methodology, and the results of the optimal designs are explained comprehensively in the following sections. 2. Theory and background Discontinuity in material and geometric properties lead to wave scattering in elastic media and thus provide the potential for stress wave attenuation. For example, for longitudinal wave propagation in one-dimensional bi-material bars with equal cross section areas, the stress amplitude ratio of a transmitted wave (st) to an incident wave (si) is given by st/si ¼ (2Z2/Z1)/(1þZ2/Z1), where Z1 and Z2 represent the impedances of the first and second layers, respectively, as the wave propagates (Graff [25]). Consequently, stress wave attenuation will occur when a wave passes from a high impedance to a low impedance material (i.e., Z1/Z2>1). However, in practice, the overall response is complicated due to multiple reflections in finite length bars and to oblique incidences. The more general case of wave scattering occurs for an incident wave at an oblique angle associated with the interface of two different materials, as shown in Fig. 1. The two materials can be solid, fluid, vacuum, or any other combination. The continuity in displacement and stress at an interface results in wave scattering through reflection and transmission in the two media at different angles. Solids can sustain both dilatational and shear waves, and each of these waves generates dilatational plus shear waves at an interface. Thus, for dilatational and shear wave incident on an interface at an oblique angle, eight new waves will be generated, as shown in Fig. 1. However, in fluids and vacuum, less number of waves will be generated because shear waves do not travel in nonviscous fluids, and longitudinal and shear waves do not propagate in vacuum. Fig. 1 shows the scattering of stress waves at the interface of two solid materials. In this figure, ID and Is represent displacement amplitudes of incident dilatational and shear waves and RDS, RDD, TDS and TDD correspond to the displacement amplitude of reflected shear, reflected dilatational, transmitted shear, and transmitted dilatational waves for a dilatation incident wave (ID), respectively. Similar notations are used for a shear incident wave (IS) by converting the first index from D to S, i.e., RSS, RSD, TSS and TSD.

Fig. 1. Reflection and transmission of waves at the interface of two solids, (a) dilatational incident wave, (b) shear incident wave.

R. Rafiee-Dehkharghani et al. / Composites Part B 82 (2015) 129e142

The direction of reflected and transmitted waves is governed by Snell's law [26], which can be expressed for incident dilatational and shear waves using Eqs. (1) and (2) as follows:

sinðqDI Þ sinðqDSR Þ sinðqDDR Þ sinðqDST Þ sinðqDDT Þ ¼ ¼ ¼ ¼ cD1 cS1 cD1 cS2 cD2

(1)

3 ðMat2Þ ðMat1Þ 2 3 ux  ux 0 6 ðMat2Þ ðMat1Þ 7 6 uy 7 607  uy 6 7¼6 7 405 6 ðMat2Þ ðMat1Þ 7  Sxx 4 Sxx 5 ðMat2Þ ðMat1Þ 0 Sxy  Sxy

131

2

(5)

ðMat1Þ

sinðqSI Þ sinðqSSR Þ sinðqSDR Þ sinðqSST Þ sinðqSDT Þ ¼ ¼ ¼ ¼ cS1 cS1 cD1 cS2 cD2

(2)

where DI and SI represent the dilatational and shear incident

2

cosðqDDR Þ

6 6 sinðqDDR Þ 6 6 6 ZD1 cosð2qDSR Þ 6 4 c ZS1 S1 sinð2qDDR Þ cD1

cosðqDDT Þ

sinðqDSR Þ

sinðqDDT Þ

cosðqDSR Þ

ZD2 cosð2qDST Þ c ZS2 S2 sinð2qDDT Þ cD2

ZS1 sinð2qDSR Þ

cosðqDDR Þ

6 6 sinðqDDR Þ 6 6 6 ZD1 cosð2qDSR Þ 6 4 c ZS1 S1 sinð2qDDR Þ cD1

ZS1 cosð2qDSR Þ

cosðqDDT Þ

sinðqDSR Þ

sinðqDDT Þ

cosðqDSR Þ

ZD2 cosð2qDST Þ c ZS2 S2 sinð2qDDT Þ cD2

ZS1 sinð2qDSR Þ

vuy vux þ ðl þ 2mÞ vy vx


vux vuy þ Sxy ¼ m vy vx

(6)

Similarly, the expansion of Eq. (5) for incident shear wave gives

3 2 3 2 3 7 RSD sinðqSI Þ 7 cosðqDST Þ 76 TSD 7 6 7 cosðqSI Þ 76 7 6 7 ZS2 sinð2qDST Þ 74 RSS 5 ¼ 4 ZS1 sinð2qSI Þ 5 7 5 TSS ZS1 cosð2qSI Þ ZS2 cosð2qDST Þ sinðqDST Þ

ZS1 cosð2qDSR Þ

SDT represent shear reflected, dilatational reflected, shear transmitted, and dilatational transmitted waves that are generated from a shear incident wave, respectively (see Fig. 1b), while cD1, cD2, cS1, and cS2 are the dilatational wave velocity in material 1, dilatational wave velocity in material 2, shear wave velocity in material 1, and shear wave velocity in material 2, respectively. For a traveling stress wave, the displacement and stress continuity at the interface of two media has to be satisfied. The interface stress Sxx and Sxy in thin plates are given by:

Sxx ¼ l

2 3 3 cosðqDI Þ 2 3 6 7 RDD 7 6 7 7 sinðqDI Þ cosðqDST Þ 76 TDD 7 6 7 76 7 7¼6 ZS2 sinð2qDST Þ 74 RDS 5 6 ZD1 cosð2qDSR Þ 7 6 7 7 4 5 TDS 5 cS1 ZS2 cosð2qDST Þ sinð2qDI Þ ZS1 cD1 sinðqDST Þ

waves. Also, DSR, DDR, DST, and DDT represent shear reflected, dilatational reflected, shear transmitted, and dilatational transmitted waves that are generated from a dilatational incident wave, respectively (see Fig. 1a). Similarly, SSR, SDR, SST, and

2

where ux is the sum of displacement due to incident, transðMat2Þ mitted, and reflected waves, and ux is the sum of displacement due to transmitted and reflected waves. The same description apðMat1Þ ðMat2Þ plies for Sxx and Sxx . Expansion of Eq. (5) for incident dilatational wave leads to the following relationship:

(7)

where ZD1, ZD2, ZS1, and ZS2 are the dilatational impedance in material 1, dilatational impedance in material 2, shear impedance in material 1, and shear impedance in material 2, respectively. The stress amplitude for reflected and transmitted dilatational and shear waves are plotted in Fig. 2 and Fig. 3 for incident

(3)

(4)

where l and m are the Lame's parameters, and are given by l þ 2m ¼ rc2D , and m ¼ rc2S with r represents the mass density of the medium. By enforcing displacement and stress continuity in thin plates at an interface, we have:

Fig. 2. Reflection and transmission stress coefficients for incident dilatational wave on AL-HDPE interface for varying incident angle.

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Fig. 3. Reflection and transmission stress coefficients for incident shear wave on ALHDPE interface for varying incident angle.

dilatational and shear waves, respectively, for a particular Aluminum (AL)-High-density polyethylene (HDPE) bi-material interface. The material properties of AL and HDPE are given in Table 1. It should be noted that Eqs. (6) and (7) are written for the displacement amplitudes; however in Figs. 2 and 3, the displacement amplitudes are converted to stress amplitudes using appropriate transformations. These figures show that for different incident wave angles, the ratio of shear and dilatational stress amplitude varies significantly. For incident shear wave, at an angle of 35 , the reflected dilatational wave becomes evanescent; however other waves will continue to propagate. Thus, for incident angles greater than 35 , Eq. (7) should be modified by removing the reflected dilatational wave, and enforcing displacement and stress continuity at the interface. These results evidently show that the incident angle has a noticeable effect in mitigating or amplifying the stress amplitudes. This means that the amplitude of the stress waves can be altered by converting the straight interface between two materials to a jagged path. This idea in fact is the impetus of the present research and will be pursued in the remainder of this paper for optimizing the interface profile of the layers in bi-layered rectangular plates with finite dimensions. It should be noted that the analytical solutions in this section are only applicable to semi-infinite media, and there is no closed form solution of this kind for finite structures. Accordingly, the FE numerical method is utilized for analyzing the proposed rectangular bi-layered plates in this study. 3. Concept of interface profile optimization In the previous section, it was observed that the angle of the incident pulse at the boundary of the two solid materials can change the characteristics of the reflected and transmitted waves significantly. Considering this fact, an optimization problem can be defined for minimizing the amplitude of the stress waves in a bi-layered plate by converting the interface profile between the two layers from a vertical straight line to a jagged shape. In this case, the angle of the incident wave at the intersection of the two media will no longer be zero (see Fig. 1). Consequently, the stress waves will be scattered and a number of reflected and transmitted waves will be generated within the structure. By employing an appropriate optimization methodology, the profile

Table 1 Mechanical properties of the plate materials. Material

E (GPa)

r (kg/m3)

n

cD (m/s)

cS (m/s)

cR (m/s)

Aluminum HDPE

68.9 1.2

2700 950

0.33 0.42

5351 1238

3097 667

2887 630

of the jagged path can therefore be optimized for the objective of stress wave attenuation at the clamped boundary of the bilayered plate. In order to introduce the concept of interface profile optimization, consider a general bi-layered plate shown in Fig. 4. A transient in-plane dynamic load is applied to the left side of the plate (Line AF) and the plate is clamped at the right side (line JE). The top and bottom parts of the plate (lines FJ and AE) are traction-free boundaries. The horizontal and vertical dimensions of the plate are Lx and Ly, respectively, and the plate is divided into two parts: ACHF and CEJH with lengths L1 and L2, respectively. The objective of the problem is minimizing the peak amplitude of the transient reaction force (on line AF), as it reaches the clamped boundary (line JE), by changing the interface profile between the two layers of the plate. To do so, an optimization zone (box) can be defined around the boundary of the two media, which can be divided into nx ¼ n1 and ny ¼ m1 segments in the horizontal and vertical directions, respectively, to generate an n  m grid, as shown in Fig. 4. Thereafter, the optimization algorithm can select the points within the generated grid to develop a jagged or smooth interface between the two media. In this paper, without loss of generality, it is assumed that only one point can be selected by the optimization algorithm at each horizontal line (a number between 1 and n); therefore, the size of the solution space is nm. The results of the optimization can be shown with an array of numbers. The length of this array is equal to the number of horizontal lines in the vertical direction (ny), and each number within the array is bounded between 1 and the number of the vertical lines in the horizontal direction (nx). To clarify further, an example is provided in Fig. 5. In this example, the optimization zone is divided into 7 and 5 sections in the horizontal and vertical directions, respectively, which generates an 8  6 grid. Therefore, there are ny ¼ 6 horizontal lines in the optimization zone and each contains nx ¼ 8 points. This means that the optimization array is composed of 6 numbers, which can have a value between 1 and 8. For instance, the optimization array for the jagged boundary in Fig. 5 is “3 8 1 4 5 2”. 4. Problem definition In the previous section, the concept of interface profile optimization for a bi-layered plate with jagged interface was introduced. This concept is utilized in this section to define a quantitative optimization problem for minimizing the amplitude of the stress waves in a bi-layered plate with finite dimensions. Consider the bi-layered plates shown in Fig. 6 with the horizontal and vertical dimensions of Lx ¼ 25.2 cm and Ly ¼ 12.6 cm, respectively. These plates are divided into two parts with equal thickness of 0.5Lx ¼ 12.6 cm, and the mid-section of the optimization zone is located at the mid-section of the plates. The thickness of the optimization zone (Lopt) is assumed to be 0.3Lx ¼ 7.56 cm, 0.6Lx ¼ 15.12 cm, and 0.9Lx ¼ 22.68 cm for the plates in Fig. 6aec, respectively. The first and second layers of the plates are made of Aluminum (AL) and High-density polyethylene (HDPE), respectively. These materials are selected because of their significant impedance mismatch, which results in higher amount of wave scattering. According to wave propagation theories, the amplitude of the transmitted stress waves in the interface of two media mitigates as the wave passes from a high to a low impedance medium (This is the other way round for the magnitude of the transmitted displacement waves). Therefore, the AL layers in Fig. 6 are placed on the left, before the HDPE layers. The mechanical properties of these materials are presented in Table 1. In this table, E, r, n, cD, cS, and cR represent Young's modulus, mass density, Poisson's ratio, dilatational wave velocity (for plane stress condition), shear wave velocity, and Rayleigh wave velocity, respectively, of the two

R. Rafiee-Dehkharghani et al. / Composites Part B 82 (2015) 129e142

L1

133

L2

F

G

H

Optimization Zone

I

Mat 1

m

J

A

B

C

D

j

Ly

Mat 2

E 1 2

Lopt

i

n

ny

2 1

nx

Lx Fig. 4. General bi-layered rectangular plate and its optimization zone.

L1-0.5Lopt

Lopt

L2-0.5Lopt

Optimization Zone 1.08 cm

6

7 6

5

FL

4

Mat 1

1.80 cm

8

Mat 2

3

AL

T/2

Ly

5

12.6 cm

HDPE

4 3 2 1

2

1 2 3 4 5 6 7 8

(a)

1

8.82 cm

7.56 cm

8.82 cm

1 2 3 4 5 6 7 8

25.2 cm

Lx

Optimization Zone 1.08 cm

1.80 cm

8

Fig. 5. Example for the optimized interface between the two layers.

7 6

materials. The velocity of the different types of waves can be found using the following formulas:

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E   cD ¼ r 1  n2

cS ¼

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E 2rð1 þ nÞ

FL

AL

T/2

lmin 2L

12.6 cm

3

(8)

1

(b)

5.04 cm

15.12 cm

5.04 cm

25.2 cm

(9) Optimization Zone 1.08 cm

1.80 cm

HDPE

8

(10)

7 6

The bi-layered plates are subjected to an in-plane transient halfsine loading with the duration of T/2, as shown in Fig. 6. The ratio of the wavelength associated with this pulse to the total horizontal length of the structure is called the wavelength ratio and can be designated as:

Rl ¼

HDPE

4

2

AL

0:87 þ 1:12n cR ¼ cS 1þn

5

FL

5

T/2

12.6 cm

4 3

(c)

2 1 1.26 cm

22.68 cm

1.26 cm

(11)

where lmin is the associated wavelength, which is the product of the minimum wave speed within the structure (cmin) and the duration of a complete sine pulse T. The slowest wave in the plate structures is the Rayleigh wave; thus, for the problem at hand, cmin is equal to cR of HDPE, which is 630 m/s. In order to investigate the effect of Lopt on the attenuation capacity of bi-layered plates, the structures in Fig. 6 are subjected to an in-plane transient loading with a constant wavelength ratio of

25.2 cm Fig. 6. Bi-layered plates with (a) Lopt ¼ 0.3Lx, (b) Lopt ¼ 0.6Lx, (c) Lopt ¼ 0.9Lx.

Rl ¼ 0.05, and their optimal interface profiles are then identified. It is observed that the structure with Lopt ¼ 0.3Lx (see Fig. 6a) has higher attenuation capacity compared to the other structures (the details are presented later in Section 8). Therefore, this structure is selected for studying the effect of other parameters such as the wavelength ratio of the transient loading and the grid dimensions.

134

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The wave propagation behavior and attenuation capacity of any structure depends markedly on the duration (wavelength) of the transient loading. To examine this effect, the structure identified with Lopt ¼ 0.3Lx (Fig. 6a) is subjected to four different wavelength ratios, i.e., Rl ¼ 0.05, 0.1, 0.2, and 0.4. Considering the Rayleigh wave speed in HDPE (cR ¼ 630 m/s), the duration of the half-sine pulses (T/2) for each wavelength ratio (Rl) are calculated and presented in Table 2. The objective of the defined optimization problem is to minimize the peak amplitude of the total reaction force at any instant at the clamped boundary. To provide better insight, the reaction force history at the clamped boundary can be normalized by the amplitude of the applied force, and the optimization problem can be determined with the following formula:

Objective function : Minimize RF

(12)

where, RF represents the normalized force history at the boundary and can be found as:


  F  RF ¼ max  B  F

(13)

L

with FB and FL representing the force history at the clamped boundary and amplitude of applied transient loading, respectively. Since the main purpose of this paper is showing the effectiveness of interface profile optimization, the attenuation capacity of the optimized structures is determined by comparing the performance of these structures to bi-layered plates having a straight vertical layer interface. Therefore, the attenuation capacity of each optimal design is defined using the following formula:


Attenuation ð%Þ ¼

1

ðRF ÞJ ðRF ÞS

  100

(14)

where (RF)J and (RF)S denote the normalized force history, Eq. (13), at the boundary of the bi-layered plates with jagged and straight vertical interface, respectively. It should be noted that for each optimal design, (RF)S is found by analyzing a bi-layered plate with straight vertical interface that is subjected to a transient loading with similar duration used for the corresponding optimal design. All of the properties of these plates are similar to the structures in Fig. 6, except the interface between the layers is a straight vertical line and the thickness of each layer (AL and HDPE layers) is 0.5  25.2 ¼ 12.6 cm. In layered elastic systems, the number of reflections and transmissions of the stress waves increases as the analysis is performed for a longer duration of time. Therefore, one of the important factors in characterization of these systems is the duration of analysis. For practical problems, it is impossible to analyze the structures for infinite time and a stopping time should be selected based on the duration of the applied transient loading and the wave speed within the system. In this paper, the duration of analysis (TA) has been set to be 20 times the summation of the duration of the half-sine loading (T/2) and the maximum amount of time required

Table 2 Duration of half-sine loadings (T/2) and duration of analysis (TA) for different values of wavelength ratios. Rl

T/2 (sec)

TA(sec)

0.05 0.1 0.2 0.4

2.0E-05 4.0E-05 8.0E-05 1.60E-04

8.4E-3 8.8E-3 9.6E-3 1.12E-2

for the slowest wave to reach the clamped boundary (treach). Therefore, TA is obtained by using:


T þ treach TA ¼ 20 2

(15)

lx Cmin

(16)

treach ¼

where lx and Cmin are the horizontal length and the slowest wave speed within the system. For the problem at hand, Cmin is equal to the Rayleigh wave speed of HDPE. The duration of analysis (TA) for different values of Rl is presented in Table 2. As mentioned in the previous section, the optimization zone can be divided into a grid defined by any integer number in the horizontal and vertical directions for generating the required interface for the geometry optimization. The optimization zones of the bilayered plates in Fig. 6aec are divided into 7, 15, and 22 segments in the horizontal direction, respectively, while in the vertical direction, all of the structures are divided into 7 segments. Based on these divisions, the structures in Fig. 6aec have 8  8, 15  8, and 22  8 grids, respectively. Considering the horizontal and vertical dimensions of the optimization zones, the width and height of each cell within the generated grids is 1.08 cm and 1.80 cm, respectively. It is assumed that the optimization method can select one point for each horizontal line of the grid. Therefore, the solution spaces for the structures with 8  8, 15  8, and 22  8 grids have 88 ¼ 1.68E7, 158 ¼ 2.56E9, and 228 ¼ 5.49E10 combinations, respectively. In order to investigate the effect of grid dimensions, the optimization zone of the structure with Lopt ¼ 0.3Lx is divided into 1, 3, 5, and 7 cells in the horizontal and vertical directions to generate 2  2, 4  4, 6  6, and 8  8 grids as shown in Fig. 7. The sensitivity of the attenuation capacity of the bi-layered plates with different grid sizes is discussed in Section 8. 5. Optimization method The solution space for the defined problems is very immense and it is obvious that an exhaustive search described earlier cannot be performed for finding the best solution. Therefore, an appropriate optimization procedure should be utilized for this purpose. In this paper, GA is used for geometry optimization of the potentially jagged interface. Genetic Algorithms are stochastic methods of optimization that are inspired by Darwin's theory of evolution. The reason for employing this method lies in its capability in handling the optimization problems without the need to have gradient information, which is the case for the problems defined in Fig. 6. An extensive amount of research about GA can be found in the literature and, for the sake of brevity, an exhaustive review of this topic is not presented in this paper. An interested reader is referred to Tsypkin and Nikolic [27], Holland [28], Goldberg [29], and Mitchell [30] for more information. As mentioned in Section 2, there is no closed-form solution for the wave propagation behavior of finite bi-layered plates (with jagged interface profiles) subjected to transient dynamic loadings and FE analysis is utilized for this purpose. In the following, FE modeling and the coupling of GA and FE are explained in detail. 6. FE modeling The validity of using an FE method for wave propagation in plate structures is investigated by Moser et al. [31]. For an elastic system without damping, the matrix form of the equation of motion can be written as:

R. Rafiee-Dehkharghani et al. / Composites Part B 82 (2015) 129e142

12.6 cm

nx=ny=2

nx=ny=6

2

135

1.512 cm

2.52 cm

6 5 4

7.56 cm

AL

HDPE

AL

HDPE

3 2

1 1

nx=ny=4

1

2

2.52 cm

1

nx=ny=8

4.2 cm

2

3

4

1.08 cm

5

6

1.80 cm

8

4

7 6

3

AL

AL

HDPE 2

5

HDPE

4 3 2 1

1 1

2

3

1 2 3 4 5 6 7 8

4

Fig. 7. Bi-layered plates with Lopt ¼ 0.3Lx and nx ¼ ny ¼ 2, 4, 6 and 8.

€ þ Ku ¼ P Mu

(17)

where M and K represent the mass and stiffness matrices, respectively, and P is the applied dynamic force. The dynamic equilibrium equation, Eq. (17), can be solved using two general methods: Explicit and Implicit. In this paper, explicit dynamic analysis is used for the wave propagation analysis of the bi-layered plates, as this method is computationally efficient for dynamic modeling of the structures subjected to short dynamic loadings with an impulsive nature. The FE explicit analyses of the structures are performed using the commercial software Abaqus 6.12 (Simulia [32]). Abaqus uses a central-difference time integration rule for explicit time integration of the equation of motion. The accuracy of the wave propagation analysis using FE depends, significantly, on the temporal (time step) and spatial (element size) resolution of the FE model (Moser et al. [31]). The explicit integration method is conditionally stable, and it is required that the time step of the integration be smaller than a critical value. For a system without damping, the critical time step can be found using (Bathe [33]):

Dt < Dtcr ¼

Tn p

(18)

where Tn is the smallest period of the FE assemblage. Generally, it is difficult to find the smallest period of the FE system and the following approximate formula is used for defining the value of the time step (Simulia [32]):

L Dtz min cD

(19)

where Lmin and cD represent, respectively, the minimum length of the FE mesh and dilatational wave speed. The dilatational wave speed can be found using thepLame's constants and the mass ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi density of the materials (cD ¼ ðl þ 2mÞ=r). Generally, a smaller time step than Eq. (19) is used in the explicit integration in order to assure the stability of the analysis. For example, Abaqus uses a time step within the following range for two-dimensional (2D) analysis (Simulia [32]):

1 L L pffiffiffi min < Dt < min cd 2 cd

(20)

In order to ensure the stability of the explicit FE analyses and obtain reliable results, Eq. (20) is used for selecting the time step for all of the FE analysis of the bi-layered plates in this paper. In addition to the time step, special attention should be given to the element size of the FE mesh in order to obtain reliable analysis results. Generally, in wave propagation problems, the frequency of the applied loadings is high, and thus, their wavelength is very small. In order to resolve the spatial features, the element size of the FE mesh should be small enough. According to Alleyne and Cawley [34], it is required to have more than 10 nodes per wavelength of the applied loading. Moser et al. [31] suggest the element mesh of the FE model be smaller than the following value:

le ¼ lmin =20

(21)

where lmin is the shortest wavelength within the system. The shortest wavelength can be found using Eq. (11). The minimum required dimensions of the elements (le) for the four different values of Rl are presented in Table 3. Considering these values, the minimum size of the elements in the FE mesh of the bi-layered

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Table 3 Minimum element size of the mesh for different values of Rl. Rl

lmin (m)

le (m)

0.05 0.1 0.2 0.4

0.0252 0.0504 0.1008 0.2016

1.26E-3 2.52E-3 5.04E-3 1.008E-2

plates is chosen to be 1, 2, 4, and 8 mm for Rl ¼ 0.05, 0.1, 0.2, and 0.4, respectively. It should be noted that the bi-layered plates in this paper are modeled using a 3-node linear plane stress triangular element from the Abaqus element library (Element CPS3). In addition, it should be emphasized that detailed mesh sensitivity analyses were performed to validate element size and time step for all of the finite element models of the bi-layered plates. An example of these analyses is provided in Fig. 8 for a plate subjected to the transient loading with Rl ¼ 0.1. This figure shows that the selected mesh size of 2 mm for the loading with Rl ¼ 0.1 provides quite accurate solutions. One of the important characteristics in FE analysis of the bilayered plates is meshing of the FE model. Due to the existence of the potentially jagged interface, the meshing procedure is not straightforward, and the FE model should be partitioned appropriately. Based on the geometry of the jagged interface, the FE models are partitioned in a consistent way to produce appropriate mesh, and all of the partitioning procedures are performed using a developed script (Rafiee-Dehkharghani [16]), which is explained in the next section. 7. Coupled GA-FE methodology In order to minimize the amplitude of the stress waves in bilayered plates, it is required to find the stress history at the boundary of the structures and evaluate its peak value within each run of the GA. Since the target of the problem is optimizing the geometry of the jagged interface between the boundaries, the features of the FE models change as the optimization procedure advances. Therefore, new FE models should be consistently generated within each run of GA, while more importantly retaining the objectivity of the meshing. This is done using the Abaqus Scripting Interface. Using this interface, a comprehensive Python script is written that is capable of building all of the features of the FE model without using the GUI. The key steps of finite element modeling using the python scripts are outlined in Fig. 9. The developed scripts create the crude geometry of the plate with the jagged interface (Fig. 9a) and partition it for producing a high

0.165 0.163

|FB/FL|

0.161 0.159 0.157 0.155 10

8

6

4

2

0

Mesh size (mm) Fig. 8. Mesh sensitivity analysis for a plate with the jagged interface under a transient loading with Rl ¼ 0.1.

quality mesh (Fig. 9b). Thereafter, the half-sine loading with the duration of TA and boundary conditions are assigned to the left and right side of the plate, respectively (Fig. 9c). Finally, the mesh will be generated by calculating the mesh size (le) and assigning element CPS3 to the all regions (Fig. 9d). This script is implemented in the fitness function calculation of the GA procedure, and performs the following tasks in summary (Rafiee-Dehkharghani et al. [24]):  Building the components of the FE model such as parts, materials, sections, and loading  Estimating TA and le using Eqs. (15) and (21)  Partitioning the FE model to generate an appropriate mesh  Running the explicit dynamic FE analysis  Extracting the stress history at the clamped boundary from the output database  Calculating the fitness value using Eq. (13) It should be noted that the optimization toolbox (version 6.2) of Matlab R2012a (MATLAB [35]) is used for solving the optimization problems presented herein. For each problem, the details of the GA runs are presented in the next section. 8. Results and discussion In this section, the interface profiles of the structures in Figs. 6 and 7 are optimized using the developed GA-FE methodology and the results are presented for the effect of different parameters separately. The following parameters are investigated: length of the optimization zone (Lopt), grid dimensions, and wavelength ratio of the transient loading (Rl). 8.1. Effect of the length of the optimization zone (Lopt) In order to investigate the effect of Lopt, the optimal designs of the bi-layered plates with Lopt ¼ 0.3, 0.6, and 0.9Lx are explored using the proposed GA-FE methodology. The dimensions of these structures and their optimization zones are shown in Fig. 6. The grid dimensions of these plates are similar and are subjected to a transient loading with the constant wavelength ratio of Rl ¼ 0.05. As the optimization algorithm can select one point for each horizontal line of the grid, the GA runs are performed with 8 integer variables with the lower and upper bounds of [1,8], [1,15], and [1,22] for the structures with Lopt ¼ 0.3, 0.6, and 0.9Lx, respectively. To choose an appropriate population size for the GA runs, the problems are run with different population sizes ranging from 20 to 300, and the optimal population size is selected by examining the stress wave attenuation versus population size diagram. An example of such a diagram for structure B (structure B is defined in Section 8.3) is shown in Fig. 10. This figure shows that by increasing the population size from 20 to 50 the amount of attenuation increases significantly. However, by increasing the population size from 50 to 100, the amount of attenuation increases only slightly. For population sizes larger than 100, the amount of attenuation will not increase notably. For the populations with 150 and 200 members, the attenuation is constant and for the population with 300 members, the amount of attenuation increases approximately one percent. Therefore, for the problems with Lopt ¼ 0.3Lx (Fig. 6a), the population size is set to be 100. Using a similar analysis, the population size for the structures with Lopt ¼ 0.6Lx (Fig. 6b) and 0.9Lx (Fig. 6c) is set to be 200. It is obvious that the structures with Lopt ¼ 0.6 and 0.9Lx require larger population sizes, because their solution spaces are much larger than the structure with Lopt ¼ 0.3Lx (compare 158 and 228 with 88).

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137

Attenuation (%)

Fig. 9. Steps of FE modeling using the Python scripts.

40 39 38 37 36 35 34 33 32 31 30

(a)

Lopt=0.3Lx

Attenuation =58%

1 1 2 4

AL

HDPE

7 8

0

50

100 150 200 Population size

250

300

350

1 7

Fig. 10. Attenuation versus population size for structure B.

For all of the optimal designs, the GA was run until the change in the average fitness value of the generations becomes less than a tolerance level, which is set herein at 1  106. In order to assure the results of the GA optimization procedure, the problems are run several times and the results of the previous runs are used as the initial population for the next runs. This process is repeated until no further improvement is observed in the results. The optimized vertical position strings (value of nx at each ny) and attenuation capacity for different values of Lopt are presented in Table 4 and the schematics of the optimal designs are depicted in Fig. 11. The results of the optimization in Table 4 and Fig. 11 show that the optimal interface profiles for the structures with Lopt ¼ 0.6 and 0.9Lx have a narrow length compared to the length of the optimization zone (Lopt). For example, the difference between the minimum and maximum values of nx for the structures with Lopt ¼ 0.6 and 0.9Lx in Fig. 11b and c is 113 ¼ 8 and 144 ¼ 10, respectively.

(b)

Lopt=0.6Lx 5

11 11 5

AL

Optimization string

(nx)min

(nx)max

Attenuation (%)

0.3 0.6 0.9

71874211 5 3 6 5 11 11 5 10 12 13 14 12 10 8 4 10

1 3 4

8 11 14

58 56 53

HDPE

6 3 5

(c)

Lopt=0.9Lx

Attenuation =53%

10

4 8 10

AL

HDPE

12 14

Table 4 Optimization string and attenuation capacity for different values of Lopt. Lopt

Attenuation =56%

10

13 12 Fig. 11. Optimal designs of the bi-layered plates in Fig. 6 for Rl ¼ 0.05, (a) Lopt ¼ 0.3Lx, (b) Lopt ¼ 0.6Lx, (c) Lopt ¼ 0.9Lx.

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This means that the maximum horizontal length of the optimized interface profiles for the structures with Lopt ¼ 0.6 and 0.9Lx is 8  1.08 ¼ 8.64 cm (0.34Lx) and 10  1.08 ¼ 10.80 cm (0.43Lx), respectively. These results show that by increasing Lopt beyond some length, the optimal interface profiles do not occupy the whole length of the optimization zone. On the other hand, the optimal interface profile of the structure with Lopt ¼ 0.3Lx does occupy the entire length of the optimization zone, as shown in Fig. 11a. Notice that, according to the details in Table 4 and Fig. 11, the attenuation capacity of the structures decreases slightly for larger values of Lopt. This indicates that the GA has not converged fully for the cases with Lopt ¼ 0.6 and 0.9Lx, which can be attributed to the extremely large size of the solution spaces for these structures (158 and 228, respectively). As mentioned above, to search the solution space further, the population size of the GA optimization for the plates with Lopt ¼ 0.6 and 0.9Lx is increased to 200 (compared to 100 for Lopt ¼ 0.3); however, from Fig. 11, the resulting attenuation capacity is reduced slightly, because the solution space has not been explored completely, which is always a possibility with heuristic optimization approaches. While it is not practically feasible to increase the population size of the GA even further to achieve exactly the same profile as found for Lopt ¼ 0.3, notice that the resulting profiles for Lopt ¼ 0.6 and 0.9Lx show no tendency to fill the optimization zone. Consequently, based on these results and observations, in the remainder of this paper, the structure with Lopt ¼ 0.3Lx (see Fig. 6a) is selected for investigating the effect of grid dimensions and wavelength ratio of the transient loading.

8.2. Effect of grid dimensions To investigate the effect of grid dimensions, the optimal design of the structures in Fig. 7 are obtained using the developed optimization methodology. The length of the optimization zone (Lopt) for all of the structures is 0.3Lx and the structures are subjected to an in-plane transient loading with Rl ¼ 0.05. It should be noted that GA is not used for the optimal design of the structures with nx ¼ xy ¼ 2 and 4 (see Fig. 7) as the solution space has only 22 ¼ 4 and 44 ¼ 256 combinations, respectively. Therefore, simple exhaustive search is utilized for the optimization of these structures. However, the coupled GA-FE methodology is employed for finding the optimal design of the structures with nx ¼ xy ¼ 6 and 8. The details of the GA runs are exactly the same as the previous section and the population size is set to be 100. The optimization string and attenuation capacity of the optimal design of the structures in Fig. 7 are presented in Table 5 and the schematics of the optimal designs are depicted in Fig. 12. The results show that the attenuation capacity of the bi-layered plates increases significantly when the number of grid points varies from 2 to 4 (compare 28% and 54% for these two cases, respectively). This phenomenon (increasing the attenuation capacity) is not observed by increasing the grid points from 4 to 6 and 6 to 8. Therefore, for the problem at hand, it is not efficient to generate a very fine grid to obtain higher attenuation capacity. Due to these facts, in the next section, the effect of wavelength ratio of the transient loading on the behavior of bi-layered plates will be investigated for a structure

Table 5 Optimization string and attenuation capacity for different values of nx and ny. nx and ny

Optimization string

Attenuation (%)

2 4 6 8

1 3 1 7

28 54 55 58

2 411 12654 1874211

with Rl ¼ 0.05 and the grid dimensions of 1.08  1.80 cm2 (see the structure with nx ¼ ny ¼ 8 in Fig. 6a or Fig. 7). 8.3. Effect of wavelength ratio (Rl) In order to investigate the effects of the wavelength ratio of the applied transient loading (Rl) on the optimized solutions, the structure in Fig. 6a (with Lopt ¼ 0.3Lx and nx ¼ ny ¼ 8) is optimized for four different values of Rl as indicated in Table 2. In the remainder of this paper, the optimal designs for Rl ¼ 0.05, 0.1, 0.2 and 0.4 are identified as structures A, B, C, and D, respectively. The GA runs are performed with 8 integer variables with the lower and upper bounds of [1,8], and their population size is set to be 100. Other details of the GA runs are similar to the previous sections. The optimized vertical position strings (value of nx at each ny) for different values of Rl are presented in Table 6. The schematic of the optimal designs and the absolute value of the ratio of the force history at the boundary (jFB =FL j) of structures A, B, C, and D are presented in Figs. 13e16, respectively. In order to show the efficiency of the optimal designs, the stress history at the boundary of a similar bi-layered plate with a straight vertical interface is also depicted in these figures (with dotted lines). The results of the optimization problems can be summarized as follows:  The optimization string for Rl ¼ 0.05 (structure A e Fig. 13) is “7 1 8 7 4 2 1 1” and the maximum amount of attenuation that can be obtained by this structure is 58%. Fig. 13b shows that the stress history at the boundary of the straight structure has a very large value at the beginning; however, there is no such peak in the stress history of the jagged structure, which results in a significant amount of attenuation.  The optimization string for Rl ¼ 0.1 (structure B e Fig. 14) is “5 5 8 5 2 1 1 1” and the maximum amount of attenuation that can be obtained by this structure is 36%. Fig. 14b shows that the peak values of the force history at the boundary of the straight structure is efficiently lessened by the jagged structure; however, the amount of attenuation at Rl ¼ 0.1 is less than at Rl ¼ 0.05.  The optimization string for Rl ¼ 0.2 (structure C e Fig. 15) is “3 4 4 2 7 7 6 8” and the maximum amount of attenuation that can be obtained by this structure is 28%. Again, the amount of attenuation at Rl ¼ 0.2 is decreased compared to Rl ¼ 0.05 (structure A) and 0.1 (structure B).  The optimization string for Rl ¼ 0.4 (structure D e Fig. 16) is “5 5 8 4 2 1 1 1” and the maximum amount of attenuation that can be obtained by this structure is only 15%. This structure is very similar to structure A, except the fourth number in the optimization string is 4 instead of 5. Compared to the other structures, it is obvious that the lowest amount of attenuation takes place at Rl ¼ 0.4. These analyses show that the optimal interface profile depends, significantly, on the wavelength ratio Rl of the applied loading. Moreover, the amount of attenuation decreases significantly by increasing Rl. Considering the dimensions of the plate in Fig. 6a, the minimum wavelength for structures A, B, C, and D is 0.1 L, 0.2L, 0.4 L, and 0.8L, respectively. Since the thickness of the optimization zone is 0.3L, it can be concluded that higher amount of attenuation can be obtained if the optimization zone has larger length compared to the wavelength of the loading. It should be mentioned that there is a symmetry in the solution space, and each solution has a corresponding pair structure in this solution space. However, the pair is not easily eliminated from the solution space without adversely impacting the genetic operators.

R. Rafiee-Dehkharghani et al. / Composites Part B 82 (2015) 129e142

nx=ny=2

Attenuation =28%

139

nx=ny=6

AL

5

HDPE

6

AL

HDPE

2 1 1

1

nx=ny=4

Attenuation =55%

4

2

Attenuation =54%

1

nx=ny=8

Attenuation =58%

1 1 2

1

AL 4

HDPE

4

AL

HDPE

7 8 1

3

7

Fig. 12. Optimal design of the bi-layered plates in Fig. 7 for Rl ¼ 0.05.

Table 6 Optimization string and attenuation capacity of the structure in Fig. 6a for different values of Rl. Rl

Structure

Optimization string

Attenuation (%)

0.05 0.1 0.2 0.4

A B C D

7 5 3 5

58 36 28 15

1 5 4 5

8 8 4 8

7 5 2 4

4 2 7 2

2 1 7 1

1 1 6 1

1 1 8 1

Fig. 14. Structure B, (a) schematic of the optimal design, (b) force history at the boundary for Rl ¼ 0.1.

Fig. 13. Structure A, (a) schematic of the optimal design, (b) force history at the boundary for Rl ¼ 0.05.

Consequently, in this paper, we have chosen to include both members of the pair in the solution space, recognizing that the GA could converge to either of the two optimal solutions. Of course, we should also recognize that with heuristic optimization algorithms, we have no guarantee that the true optimum will be found, although the preset GA is quite robust. In addition, it should be noted that the symmetry is only in the vertical direction of the optimization zone (e.g. the jagged profiles “55852111” and “11125855” give the same amount of attenuation) and there is not such a symmetry in the horizontal direction (i.e. replacing “1” with “8”, “2” with “7”, and so on). For example, Fig. 17 shows the amount

140

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Attenuation =36%

1 1 1 1 2

AL

HDPE 5 8 5

SAL=146 cm2

SHDPE=171 cm2

5

Attenuation =13% 8 8 8 7

AL

HDPE

4 1

SAL=171 cm2

4 4

SHDPE=146 cm2

Fig. 15. Structure C, (a) schematic of the optimal design, (b) force history at the boundary for Rl ¼ 0.2. Fig. 17. Attenuation in the symmetric jagged profiles in the horizontal direction, (a) jagged profile “55852111”, (b) jagged profile “44147888”.

the attenuation-Rl curve for each structure is plotted in Fig. 18, and accordingly, the attenuation capacity of each structure can be explained as follows:  Structure A has a very high attenuation capacity for Rl ¼ 0.05 (the wavelength for which it is optimized) and Rl ¼ 0.1; however, its performance is poor for larger Rl values, especially for Rl ¼ 0.4. The minimum value of attenuation for this structure over the four cases is 2%, which occurs at Rl ¼ 0.4.  Structures B and D have a very similar attenuation capacity, because their geometries are quite similar. The optimization capacities of these structures is very good for Rl ¼ 0.2 (it is only 1% less than the attenuation capacity of structure C). For Rl ¼ 0.05, the attenuation capacity of structures B and D is about 10% lower than that of structure A. The minimum values of attenuation over the four cases for structures B and D are 13% and 15%, respectively.  Structure C provides a lower amount of attenuation for Rl ¼ 0.05, 0.1 and 0.4, and it is only efficient for Rl ¼ 0.2 (the wavelength for which it is optimized). The minimum attenuation over the four cases for this structure is 9%.

Fig. 16. Structure D, (a) schematic of the optimal design, (b) force history at the boundary for Rl ¼ 0.4.

of attenuation in the plates with the jagged profiles “55852111” and “44147888”, and it is obvious that the attenuation values differ because the area of the Aluminum (SAL) and HDPE (SHDPE) sections are different in the two structures. To explore further the attenuation capacity of the optimal designs, structures A, B, C, and D are subjected to transient loadings with various Rl values and their corresponding attenuation capacity are presented in Table 7. Examining the results of this table,

Based on these observations, either structure B or D is recommended for attenuating the intensity of the transient loadings with Rl values between 0.05 and 0.4. A more comprehensive approach for finding the optimal interface profile for a plate, which is subjected to the transient loadings with different loading functions, would be to implement the loading function as a parameter into the coupled GA-FE procedure. In this case, the best solution at each generation of GA will be subjected to different types of loadings, and consequently, the final optimal solution will be the most optimal design for a wide range of loading functions. Although this method is very comprehensive, it is extremely expensive considering the computational cost because the time required for each run of GA will be multiplied by the

R. Rafiee-Dehkharghani et al. / Composites Part B 82 (2015) 129e142 Table 7 Amount of attenuation (%) in the optimized structures for different values of Rl.

is required to be performed for practical implementation and complete realization of these concepts.

Structure

Rl

0.05 0.1 0.2 0.4

A

B

C

D

58 36 19 2

49 36 27 13

47 25 28 9

48 35 27 15

70

B

50

C

40

D

30 20 10

Rλ

0

0

0.1

0.2

0.3

0.4

Acknowledgment The research described in this paper was funded, primarily, by the US National Science Foundation under Grant No. CMMI0900338 with additional support to the first and second authors by MCEER. The authors gratefully acknowledge this support. References

A

60

A enua on (%)

141

0.5

Fig. 18. Attenuation-Rl curves for the optimized structures.

number of loading functions, as the fitness value of each solution should be calculated for multiple number of loadings (instead of one). Such an approach was taken in Dargush and Sant [36] within the context of seismic design.

9. Conclusions The stress wave attenuation in bi-layered rectangular plates with a potentially jagged interface is studied in this paper. The structures are subjected to in-plane transient half-sine loading with various durations, and their interface profile is optimized for the objective of stress wave attenuation. A coupled GA-FE optimization methodology is developed for finding the optimal design of the interfaces. The effect of different parameters such as the length of the optimization zone, the dimensions of the optimization grid, and the wavelength ratio of the applied transient loading is investigated. It is observed that the attenuation capacity of the bi-layered plates with jagged interface does not increase significantly by increasing the length of the optimization zone or by decreasing the dimensions of the grid cell (making a very fine optimization grid). The results show that the interface profile has a significant effect in attenuating the stress waves, and the amount of attenuation depends directly on the associated wavelength of the applied transient loading. In addition, there is no unique interface profile for all of the transient loadings and the optimal design of the interface varies for the loads with different wavelengths. The results also show that higher attenuation can be obtained if the associated wavelength of the applied load is small compared to the dimensions of the structure and the length of the optimization zone. The results obtained in this paper suggest that the stress wave attenuation capacity of the layered structures can be increased significantly by optimizing the interface profile between the layers; however, further analytical, computational, and experimental work

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