Interface shape design of multi-material structures for delamination

Advance Publication by J-STAGE Mechanical Engineering Journal

DOI:10.1299/mej.15-00360

Received date : 1 July, 2015 Accepted date : 20 December, 2015 J-STAGE Advance Publication date : 4 January, 2016

© The Japan Society of Mechanical Engineers

Interface shape design of multi-material structures for delamination strength Yang LIU∗ , Daisuke MATSUNAKA∗∗ , Masatoshi SHIMODA∗∗∗ and Yoji SHIBUTANI† ∗

∗∗

∗∗∗

Department of Mechanical Engineering, Sojo University 4-22-1, Ikeda, Nishi-Ku, Kumamoto, 860-0082, Japan

E-mail: [email protected] Department of Mechanical Systems Engineering, Shinshu University

4-17-1 Wakasato, Nagano 380-8553, Japan Department of Advanced Science and Technology, Toyota Technological Institute †

2-12-1 Hisakata, Tenpaku-ku, Nagoya 468-8511, Japan Department of Mechanical Engineering, Osaka University 2-1, Yamadaoka, Suita, Osaka, 565-0871, Japan

Abstract This paper deals with interface shape optimum design of multi-material structures for the delamination strength problem. The optimum design problem is formulated as a non-parametric shape optimization problem in which the interface variation in the normal direction is considered as a design variable. The maximum value of a delamination function, an index of the delamination strength, is defined as an objective function subject to a volume constraint. The shape sensitivity, called shape gradient function, is derived by using the material derivative method and the adjoint variable method, and is applied to the H 1 gradient method to determine the optimal interface shape. With this method, the maximum value of the delamination function can be minimized while the smooth optimal interface shape can be obtained without any shape design parametrization. Several interface shape design examples are presented to verify the validity and practical utility of the proposed method. Key words : Multi-material structures, Joint strength, Delamination function, Interface shape, Optimum design

1. Introduction With remarkable increase of the automotive, exhaust gas and energy consumption have become global environmental and social problems, especially those in emerging countries. To solve this problems, lightweight design of automotive body plays a major role for reducing the fuel consumption and thus the emission of CO2 . Though a considerable progress in automotive lightweight design was made by the implementation of new materials like high-strength steels and the partial use of aluminium as well as plastics in the past years, substantial weight reduction can not be expected and new body structure design is needed. On the other hand, the multi-material design is considered as a key to achieve super lightweight design, and the significant attention to that design has been received within automotive manufacturers around the world (Sahr, 2009). The multi-material design is based on the idea of choosing light-weight materials including highstrength steel, aluminium, titanium, magnesium, and carbon-fiber-reinforced plastic (CFRP), for each part to fulfil the requirements and to minimize the weight in parallel. To determine the most capable material for each part in a multi-material structure, the topology optimization method has been adopted as a material selection methodology. There are various algorithms and their approaches were developed for the topology optimization of multi-material structures. For example, Sigmund et al. used the homogenization method to formulate three-phase topology optimization problems (Sigmund and Torquato, 1996; Sigmund and Torquato, 1999; Gibiansky and Sigmund, 2000). Wang et al. applied the variational multiple level set approach to perform multi-material topology optimization problems (Wang and Wang, 2004). Zhou et al. introduced a multi-material phase-field approach

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based on the Cahn-Hilliard equation, which was a general method to solve multiphase structural topology optimization problems (Zhou and Wang, 2006; Zhou and Wang, 2007). However, the major challenge, and one of main cost drivers of the multi-material structure is the joining of the parts made of dissimilar materials (Lesemann and Brockerhoff, 2008). The joint strength on the interface of two dissimilar materials greatly influences static or dynamic mechanical properties of multi-material structures and decides on the implementation of multi-material concepts in future automotive generations. Unfortunately, the traditional joining solutions, such as the resistance spot welding, are difficult to provide sufficient joint strength. Though the mechanical and/or adhesive bonding techniques has proved its high potential, it still has to overcome the challenge of joint strength problem under dynamic condition (Lesemann and Brockerhoff, 2008). For achieving a maximum interface strength, this paper proposes an interface shape optimization method for multi-material design. To the best of our knowledge, this is the first research dealing with interface shape optimum design for the interface strength problem. In this paper, the maximum value of a delamination function is difined as an index of the delamination strength, and the interface shape design problem deals with a minimization of the maximum value of the delamination function (i.e., stress minmax problem) subject to a volume constraint. The difficulties for the stress minmax problem are that the socalled singularity phenomenon occurs when a local measure is used for the maximum stress, and it is theoretically difficult to determine directly the sensitivity function of the local objective functional. In this work, this difficulties are removed by employing a smooth envelope function, i.e., the Kreisselmeier-Steinhauser (KS ) function (Kreisselmeier and Steinhauser, 1979), in which a local objective functional is replaced by a smooth differentiable integral functional. The interface shape of bonded dissimilar materials is determined under the condition where the outside boundary of the multi-material structure is invariable. The optimization problem is formulated as distributed-parameter shape optimization problems and the shape gradient function is derived using the material derivative method and the adjoint variable method. The optimal interface shape is obtained by applying the derived shape gradient functions to the H 1 gradient method (Azegami, 1994; Azegami, et al., 1997; Azegami and Takeuchi, 2006; Azegami, et al., 2013). Three kinds of interface shape designs are presented to validate the proposed method and demonstrate its practical utility.

2. Interface Strength Problem of Multi-material Structure 2.1. Modelling of Multi-material Structure

ΓA f ( x)

ΩA

ΓB Γ AB

ΩB

Γ1

P ( x) nA

z y

nB nA

σ zz

x

σ zy

p ( x)n

Γ2

Fig. 1

σ zx

Multi-material structure which consists of two elastic bodies with dissimilar materials in three-dimensional space.

As shown in Fig. 1, a multi-material structure, having an initial bounded domain Ω ⊂ R3 with boundary Γ, consists of two elastic bodies with different elastic moduli in three-dimensional space. Body A occupies domain ΩA with boundary ΓA and body B occupies domain ΩB with boundary ΓB . Therefore, the interface boundary is ΓAB = ΓA ∩ ΓB . Suppose that body forces per unit volume f (x) act on Ω, concentrated loads P(x) are applied at points on boundary Γ1 and distributed pressure loads p(x)n are applied in the directions normal to the boundary Γ2 . Then, the variational governing equation for the linear elastic problem of the multi-material structure can be expressed as Eqn. (1). aA (u, w) + aB (u, w) − hA (u, w) − hB (u, w) − l(w) = 0, ∀w ∈ U,

(1)

where, u = [v1 v2 v3 ]T indicates displacement for x ∈ Ω, w = [w1 w2 w3 ]T expresses a adjoint variable and U expresses the admissible space in which the given constraint conditions is satisfied. In addition, the bilinear forms am (u, w), hm (u, w)(m = © The Japan Society of Mechanical Engineers

A, B) and the linear form l(w) are defined in Eqns. (2), (3) and (4), respectively, using tensor representation. am (u, w) = hm (u, w) = l(w) =

∫ Ω

∫ ∫

Ω

em vm wm dΩ = m i jkl k,l i, j

ΓAB

∫

m σm i j (u)n j wi dΓ,

fi wi dΩ +

∫

Γ1

m σm i j (u)ϵi j (w)dΩ,

Ωm

Pi wi dΩ +

(2) (3)

∫ Γ2

p(x)ni wi dΩ.

(4)

The infinitesimal strain tensors are then defined to be Eqn. (5) and the stress-strain relations (generalized Hook’s law) are expressed in Eqn. (6). m ϵimj (u) = (vm i, j + v j,i )/2, m m σm i j (u) = C i jkl ϵi, j (u),

x ∈ Ωm ,

(5)

x ∈ Ωm .

(6)

2.2. Quadratic Stress Criterion for Initiation of Delamination Under the three-dimensional stress state in the interface of the multi-material structure, delamination initiation generally can be predicted by a quadratic stress criterion proposed by John C. Brewer and Paul A. Lagace (Brewer and Lagace, 1988). Since the delamination initiation is mainly attributed to interface stress effects, this model takes into account the interaction of three interface stress components in a quadratic equation. The three stress components are the interface shear stresses τzx , τzy and the normal stress σzz in a local coordinate system shown in Fig. 1. Then, the delamination initiation model can be presented in the following form:  ( t )2 ( c )2 (    σzz σzz τzx )2 ( τzy )2 2 e < 1 no failure (7) + = e + +    Zt Zc Z s1 Z s2 e ≥ 1 failure, where Z t and Z c indicate the tensile strength for positive interlaminar normal stress σtzz and the compressive strength for negative σczz , respectively. Z s1 and Z s2 are the interlaminar shear strength for stress τzx and τzy . Moreover, it is assumed for simplicity that constant values of Z t , Z c , Z s1 and Z s2 can represent the interlaminar strengths throughout the interface boundary ΓAB . From the delamination initiation criterion expressed in Eqn. (7), the delamination function for multi-material structure can be defined as below. ( t )2 ( c )2 ( σzz σzz τzx )2 ( τzy )2 F(σ) = + s2 − 1. (8) + + Zt Zc Z s1 Z The definition suggests that the smaller the value of the delamination function is, the more it is resistant to the delamination initiation, and the higher the interface strength becomes to be.

3. Interface Shape Optimization Problem for Maximizing Delamination Strength 3.1. Domain Variation As shown in Fig. 2, when the interface with initial domain ΓAB undergoes domain variation {V(x), x ∈ ΓAB }, ie., s design velocity field, such that its domain becomes ΓAB . It is assumed that outside boundary of the multi-material structure remains invariable under the domain variation, ie., {V(x) = 0, x ∈ Γ}. The domain variation at this time can be expressed s s by a mapping from ΓAB to ΓAB , which is denoted as T s : X ∈ ΓAB 7→ X s (X) ∈ ΓAB , 0 ≤ s ≤ ε (ε is a small integer) given s by X s = T s (X), ΓAB = T s (ΓAB ) (Haug, et al., 1986; Choi and Kim, 2005). The subscript s expresses the iteration history of the domain variation. Assuming a shape constraint is acting on the variation in the domain, the infinitesimal variation of the domain can be expressed by T s+∆s (X) = T s (X) + ∆sV, (9) where the design velocity field V(X s ) = ∂T s (X)/∂s. The non parametric optimization method explained later is a method for determining the optimal domain variation V of interface shape. 3.2. Formulation of Interface Shape optimization Let us consider the formulation of the interface shape optimization problem. Aiming at maximizing the interface strength, the delamination function defined in Eqn. (8) is used to be the objective functional to be minimized. Letting

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V(x)

ΓB

nB

ΩB

ΩA

Γ AB

nA

ΓA

s Γ AB

Fig. 2

Shape variation of interface shape by V.

the state equation in Eqn. (1), the volume of material A be the constraint conditions, a non-parametric shape optimization s problem for finding the optimal design velocity field V, or ΓAB = ΓAB + ∆sV can be formulated as shown below: ΓAB ,

Given

V(or

find

(10) S ΓAB ),

(11)

minimize

Fmax (σ),

(12)

subject to

Eq. (1) and

(13)

∫

MA (=

ΩAS

ˆ A, dΩ) ≤ M

(14)

ˆ A denotes constraint value of the volume of material A. where M In order to avoid the ”singularity” and the “non-differentiability” phenomenon of the stress minmax problem as mentioned in Sec. 1, KS function is used to transform the local objective functional expressed in Eqn. (12) into the following smooth differentiable integral functional. { } 1 1 ∫ minimize Fmax (σ) → minimize ln ∫ exp(F(σ)ρ)dΓ . (15) ΓAB ρ ΓAB dγ 3.3. Derivation of the Shape Gradient Function (Sensitivity Function) The shape design sensitivity analysis in this paper is based on the material derivative idea of continuum mechanics and the adjoint variable method, which were introduced by Haug et al. in 1986 and Choi et al. in 2005. When the objective functional J is given as a domain integral of the distributed function Ψ s , J=

∫ Ωs

Ψ s dΩ.

(16)

˙ is given by the following expression (Choi and Kim, 2005). The material derivative J, J˙ =

∫ Ω

Ψ ′ dΩ +

∫ Γ

ΨVn dΓ,

(17)

where Vn = ni Vi . The vector n is an outward normal unit vector to the surface. (·)′ indicates a shape derivative (Lagrange derivative). When the objective functional J is given as a surface integral of the distributed function Ψ s , J= the material derivative J˙ is given by J˙ =

∫ Γ

∫

Ψ s dΓ,

(18)

{Ψ ′ + (Ψ,i ni + Ψκ)Vn }dΓ,

(19)

Γs

where κ expresses twice the mean curvature distributed on Γ in R3 .

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Then, letting w and ΛA denote the Lagrange multipliers for the state equation and volume constraint MA , respectively, the Lagrange functional L associated with the interface shape optimization problem can be expressed as } { 1 ∫ 1 exp(F(σ)ρ)dΓ L = ln ∫ ΓAB ρ ΓAB dγ − {aA (u, w) − hA (u, w) − aB (u, w) + hB (u, w) − l(w)} ˆ A ). + ΛA (MA − M

(20)

For the sake of simplicity, it is assumed that the forces P(x), p(x) and f (x) applied on the multi-material structure ˙ p˙ = f˙=0). Then, the material derivative L˙ of the do not vary with regard to the space and the iteration history s (i.e., P= Lagrange functional can be derived as shown in Eqn. (21) below using the formula of material derivative as mentioned above (Haug, et al., 1986; Choi and Kim, 2005). ∫ ΓAB

L˙ =

∂σi j ′ ∂vk vk dΓ

∂F exp(F(σ)ρ) ∂σ ij

ρ

∫

ΓAB ′

exp(F(σ)ρ)dΓ

− aA (u , w) − aA (u, w′ ) + hA (u′ , w) + hA (u, w′ ) − aB (u′ , w) − aB (u, w′ ) + hB (u′ , w) + hB (u, w′ ) + l(w′ ) ˆ A ) + ⟨Gn, V⟩ , V ∈ CΘ, + Λ˙A (MA − M ⟨Gn, V⟩ =

(21)

∫

( fi wi − ei jkl vk,l wi, j + Λ)Vn dΓ ] [ ∫ exp(F(σ)ρ)(ρF,i ni + κ) κ ∫ + ∫ VA + ΓAB ρ ΓAB n ΓAB exp(F(σ)ρ)dΓ Γ

+ [{−σiAj (u)ϵiAj (w) + (σiAj (u)nAj wi ),m nmA + (σiAj (u)nAj wi )κ A }VnA + {−σiBj (u)ϵiBj (w) + (σiBj (u)nBj wi ),m nmB + (σiBj (u)nBj wi )κ B }VnB ]dΓ + +

∫

Γ1 ∫ Γ2

(Pi, j n j wi + Pi wi, j n j + κPi wi )Vn dΓ div(pwi )Vn dΓ,

(22)

m where Vn = Vi ni , Vnm = Vi nm i (m = A, B). n and n denote an outward unit normal vector on Γ and ΓAB , respectively. The notation Gn denotes the shape gradient function (i.e., shape sensitivity function), and G is called the shape gradient density function. Additionally, CΘ expresses the admissible function space that satisfies the constraints of shape variation. The ˙ are the shape derivative and the material derivative with respect to the domain variation, respectively notation (·)′ and (·) (Haug, et al., 1986; Choi and Kim, 2005). The optimality conditions of the Lagrangian function L with respect to the state variable u and the adjoint variable w are expressed as shown below.

aA (u, w′ ) + aB (u, w′ ) − hA (u, w′ ) − hB (u, w′ ) − l(w′ ) = 0, ∀w′ ∈ U, ∫ ′

′

′

′

aA (u , w) + aB (u , w) − hA (u , w) − hB (u , w) −

ΓAB

∂F exp(F(σ)ρ) ∂σ ij

ρ

∫

ΓAB

ˆ A ) = 0. Λ˙A (MA − M

∂σi j ′ ∂vk vk dΓ

exp(F(σ)ρ)dΓ

= 0,

(23) ∀u′ ∈ U,

(24) (25)

Equation (23) is the governing equation for the state variable u and coincides with the state equation (1), and Eqn. (24) is the adjoint equation for the adjoint variable w. As mentioned in Sec. 3.1, the outside boundary of the multi-material structure is assumed to be invariable, ie., Vn = 0. Then, taking the following relationships on the interface of the multi-material structure into account, nA = −nB ,

(26)

B

κ = −κ ,

(27)

wA = w B ,

(28)

A

σiAj nAj

=

−σiBj nBj

,

(29) © The Japan Society of Mechanical Engineers

⟨Gn, V⟩) can be expressed as below: ⟨Gn, V⟩ =

∫ ΓAB

[

{ } exp(F(σ)ρ)(ρF,i ni + κ) κ ∫ + ∫ − σiAj (u)ϵiAj (w) − σiBj (u)ϵiBj (w) ρ ΓAB ΓAB exp(F(σ)ρ)dΓ

{ } A B + σiAj (u)nAj ϵim (w) − ϵim (w) nmA ]VnA dΓ. (30)

When the optimality conditions are satisfied, Eqn. (21) becomes L˙ = ⟨Gn, V⟩ , V ∈ CΘ .

(31)

Then, the sensitivity function (i.e., the shape gradient function Gn) for the this problem is derived in Eqn. (32) by using the state variable u calculated in Eqn. (23) and the adjoint variable w calculated in Eqn. (24). { } { } exp(F(σ)ρ)(ρF,i ni + κ) κ A B + ∫ − σiAj (u)ϵiAj (w) − σiBj (u)ϵiBj (w) + σiAj (u)nAj ϵim (w) − ϵim (w) nmA . (32) G= ∫ ρ ΓAB ΓAB exp(F(σ)ρ)dΓ The derived shape gradient function is then applied to the H 1 gradient method to determine the optimal shape varias tion V or the optimal interface shape ΓAB .

4. Calculation of Optimal Shape Variation V by Applying the H1 Gradient Method The non-parametric shape optimization method described here for the interface design of the multi-material structures is based on the H 1 gradient method, which is also called the traction method and is a type of gradient method in a Hilbert space. It is a node-based shape optimization method that can treat all nodes as design variables and does not require any shape design parametrization. The original H 1 gradient method was proposed by Azegami in 1994, and was modified by Shimoda and Liu to perform the free-form shell optimization (Shimoda and Liu, 2014; Liu and Shimoda, 2014a; Liu and Shimoda, 2015b), and the shape optimum design of stiffeners on thin-walled structures (Liu and Shimoda, 2014a; Liu and Shimoda, 2015b). In this work, we apply the H 1 gradient method to obtain the optimal shape variation V of the interface on the multimaterial structure. As shown in Fig. 3, the Dirichlet conditions are defined for a pseudo-elastic body composed of body A and body B in the case of interface shape optimization. Here, it should be noted that the elastic moduli for ΩA and ΩB are set to the same value. A distributed force proportional to the shape gradient function −Gn is applied in the normal direction of the interface. The analysis for the optimal shape variation V variation is called the velocity analysis and the governing equation is expressed as Eqn. (33). The shape gradient function is not applied directly to the shape variation but rather is replaced by a force, to vary shape of interface. This makes it possible both to reduce the objective functional and to maintain the smoothness, i.e., mesh regularity.

ΓB

-G nB

ΩB

nA

ΩA Γ AB

ΓA Fig. 3

Schematic of the non-parametric interface shape optimization method.

a(V, w) = −⟨Gn, w⟩,

V ∈ CΘ , ∀w ∈ CΘ .

(33)

Next, we confirm that the domain variation V determined by the velocity analysis can certainly reduce the Lagrangian functional L. When the state equation, the adjoint equation and the constraints are satisfied, the perturbation expansion of the Lagrangian functional L can be written as ∆L = ⟨Gn, ∆sV⟩ + O(|∆s|2 ).

(34) © The Japan Society of Mechanical Engineers

Substituting Eqn. (33) into Eqn. (34) and taking into account the positive definitiveness of a(u, w), based on the positive definitiveness of the elastic tensor ei jkl , ∃α > 0 : a(ξ, ξ) ≥ α∥ξ∥2 , ∀ξ ∈ U,

(35)

the following relationship is obtained when ∆s is sufficiently small: ∆L = −a(V, ∆sV) < 0.

(36)

In problems where convexity is assured (Sinha, 2006), this relationship definitely reduces the Lagrangian functional in the process of changing the domain using the optimal domain variation V determined by Eqn. (33). For the regularity of the shape optimization problems, it was theoretically discussed in a previous paper of Azegami (Azegami, et al., 1997) by using the regularity theorem for elliptic boundary value problems (Pironneau, 1984; Ladyzhenskaya and Ural’tseva, 1968). It is confirmed that a design domain reshaped by the domain variation V obtained in Eqn. (33) has a boundary that agrees with the original smoothness, and has a smoother boundary for one-time differentiability than that obtained by the direct gradient method in which the boundary is moved in proportion to the shape gradient. Therefore, in the case of initial strict smoothness, the boundary smoothness can be maintained in the iterations by our proposed method. It suggests that the stiffness tensor in the governing Eqn. (33) serves as a smoother for maintaining mesh regularity.

5. Flowchart of the Interface Shape Optimization System Flowcharts of the interface shape optimization systems developed for the delamination strength maximization problem is schematized in Fig. 4. Firstly, the stress analysis expressed in Eqn. (1) or Eqn. (23) and the adjoint analysis expressed in Eqn. (24) are done using a standard commercial FEM code, in which the element quality of the model is inspected in order to guarantee the mesh regularity. Subsequently, outputs of the analyses are utilized to calculate the shape gradient function expressed in Eqn. (32). After that, the velocity analysis expressed in Eqn. (33) is implemented, where a distributed force proportional to the negative shape gradient function is applied in the normal direction on ΓAB to determine the optimal shape variation V. Then the shape is updated using the obtained shape variation V. Finally, a distributed force proportional to the Lagrange multiplier ΛA is applied in the normal direction of the interface to satisfy the volume constraint by adjusting the magnitude of ΛA repeatedly. This process is repeated until the repeat count reaches the prescribed value and the changing rate of the objective function is lower than a threshold value. Start

By standard commercial FEM code

Stress analysis By self-made program Adjoint analysis Calculation of shape gradient function G Velocity analysis using -G for V (H1 gradient method)

Shape updating Volume constraint analysis Satisfy volume constraint ?

No

Yes No

Convergence ? Yes End

Fig. 4

Flowchart of the interface shape optimization system.

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6. Results of Numerical Analysis The proposed method was applied to three fundamental design problems in order to confirm its validity and practical utility for the multi-material structures. In each design example, multi-material structure consists of two dissimilar materials with material properties: E A = 19.8GPa, E B = 198GPa, νA =νB =0.33. In the quadratic stress criterion for initiation of delamination (Eqn. (7)) and the delamination strength function (Eqn. (8)), the interlaminar strengths are assumed to be Z t =200MPa, Z c =800MPa, Z s1 =Z s2 =400MPa. It suggests that the normal tensile stress σtzz more easily lead to initiation of delamination than the shear stresses τzx , τzy , and remarkably more easily than the normal compressive stress σczz . 6.1. Uniaxial Tensile of Rectangular Block

F Material A

Interface boundary

Representative element

Fixed Material B (a) Initial shape and boundary conditions Fig. 5

(b) Obtained shape.

Uniaxial tensile of rectangular block (Design example 1)

A multi-material block with uniform rectangular cross section was considered as the first example to minimize the delamination strength function by the proposed interface shape optimization method. The initial shape and boundary conditions in the stress analysis of this problem are shown in Fig. 5(a), where the side of material B was fixed and upward distributed tensile force was applied on the other side of material A. The total volume is invariable and the initial volume ˆ A. of material A was given as the constraint value M The obtained optimal shape is shown in Fig. 5(b), in which the interface shape was varied from plane into ”V” shape while maintaining the symmetry, and deformation at four corners is more outstanding due to stress concentration in the initial shape. Iteration convergence histories of the volume, the objective function in Eqn. (15) and the maximum value of delamination function in Eqn. (8) are shown in Fig. 6. The values were normalized to those of the initial shape. As shown in Fig. 6, the objective function was reduced by approximately 65%, and the maximum value of delamination function F(σ) was reduced about 43% while satisfying the given volume constraint. It is clear that the delamination strength was increased efficiently by the optimized shape. To investigate variation of the interfacial stress components, the normal tensile stress σtzz and a resultant shear stress τ s of a representative element shown in Fig. 5(b), were compared between the initial shape and the optimal shape. The resultant shear stress τ s was defined in Eqn. (37). Fig. 7 shows the iteration histories of σtzz and τ s from the initial to the final shape, where the values are normalized to the normal tensile stress σtzz of the initial shape. It is clear that the interfacial stress of the initial shape was dominated almost entirely by the normal tensile stress σtzz . During the optimization process, the resultant shear stress τ s increased steadily while the normal tensile stress σtzz decreased substantially, that contribute to reduce the maximum value of delamination function F(σ). These results show the effectiveness of the proposed method for designing the optimal interface shape to improve the delamination strength of the multi-material structure. √ τ s = τ2zx + τ2zy . (37)

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1.2 1.0 0.8 0.6

Volume Max. value of F(σ) Objective functional

0.4 0.2 0.0 0

10

20

30

40

50

Iteration No. Fig. 6

Iteration histories of design example 1.

σzz

σzz (τs=0)

τzy τs τzx

1.2

Normal tensile stress σzz

1.0 0.8 0.6 0.4 0.2

Shear stress τs

0.0 0

10

20

30

40

50

Iteration No. Fig. 7

Stress components variation of the representative element.

© The Japan Society of Mechanical Engineers

6.2. Rectangular Beam under Distributed Lateral Load

Fixed

Material A

Material B

Fixed

Representative element

Interface boundary (a) Initial shape and boundary conditions Fig. 8

(b) Obtained shape.

Rectangular beam under distributed lateral load (Design example 2).

As the second example, the proposed optimization method was applied to a rectangular beam laminated by dissimilar material A and material B. The initial shape and boundary conditions are shown in Fig. 8(a). In the stress analysis, left and right sides of the beam were clamped, and upward uniform distributed forces were applied along the upper surface. ˆ A. The total volume is invariable and the initial volume of material A was given as the constraint value M Figure 8(b) shows the obtained optimal shape, where the volume of two fixed sides of the material A was reduced, and the reduced volume shifted to the central part, that is opposite to the material B. Iteration convergence histories of the volume, the objective function in Eqn. (15) and the maximum value of delamination function in Eqn. (8) are shown in Fig. 9. The values were normalized to those of the initial shape. As shown in Fig. 9, the objective function was reduced by approximately 40%, and the maximum value of delamination function F(σ) was reduced about 70% while satisfying the given volume constraint. Similar to Sec. 6.1, variation of the normal tensile stress σtzz and the resultant shear stress τ s of a representative element were investigated during the optimization process. The iteration histories of σtzz and τ s are shown Fig. 10, where the values are normalized to the resultant shear stress τ s of the initial shape. It is clear that the interfacial stress of the initial shape was dominated almost entirely by the resultant shear stress τ s due to the distributed bending stress. During the optimization process, the normal compression stress σczz increased steadily while the resultant shear stress τ s decreased substantially, that contribute to reduce the maximum value of delamination function F(σ) and improve the delamination strength of the multi-material structure.

Max. value of F(σ)

Volume Objective functional 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0

10

20

30

40

Iteration No. Fig. 9

Iteration histories of design example 2.

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τs

σzz

σzz τs

1.2 1.0

Shear stress τs

0.8 0.6 0.4 0.2

Normal tensile stress σzz

0.0 -0.2 -0.4 0

10

20

30

40

Iteration No. Fig. 10

Stress components variation of the representative element.

6.3. Thick-walled Cylinder under A Pair of Forces

Material A

Fixed

Material B F

Interface boundary F (a) Initial shape and boundary conditions Fig. 11

(b) Obtained shape.

Thick-walled cylinder under a pair of forces (Design example 3)

The last design example is a thick-walled cylinder consisting of two jointed bodies with dissimilar materials, material A and material B. The initial shape and boundary conditions in the stress analysis are shown in Fig. 11(a), in which the right side of material A was fixed, and a pair of forces with equal magnitudes but opposite directions were applied on the left side of material B. The total volume is invariable and the initial volume of material A was given as the constraint value ˆ A. M The obtained interface shape is shown in Fig. 11(b). Iteration convergence histories of the volume, the objective function in Eqn. (15) and the maximum value of delamination function in Eqn. (8) are shown in Fig. 12. The values were normalized to those of the initial shape. As shown in Fig. 12, the objective function was reduced by approximately 48%, and the maximum value of delamination function F(σ) was reduced about 36% while satisfying the given volume constraint. It is confirmed again that the proposed method is effective to obtain the optimal interface shape with the maximum delamination strength.

7. Conclusions For obtaining the maximum interface delamination strength, this paper has proposed a non-parametric shape optimization method for designing the interface shape of the multi-material structure. The optimization problem was for© The Japan Society of Mechanical Engineers

1.2 1.0 0.8 0.6

Volume Max. value of F(σ) Objective functional

0.4 0.2 0.0 0

10

20

30

Iteration No. Fig. 12

Iteration histories of design example 3.

mulated as a non-parametric shape optimization problem under the assumptions that the interface domain varies in the normal direction and the outline of the multi-material structure remains invariable. The maximum value of the delamination function, an index of the delamination strength, was used to be the objective functional to be minimized under a volume constraint. In the proposed method, the issue of non-differentiability is avoided by transforming the local measure to an integral functional by using the KS function to transform the local objective functional into the smooth differentiable integral functional. The shape gradient function for this problem was theoretically derived and the method for the optimal design velocity using the adjoint variation was presented. The proposed method has been applied to three kinds of interface shape designs of multi-material structures, and the numerical results showed that smooth optimal interface shapes were obtained in each design problem to achieve the maximum interface delamination strength, that should contribute to the implementation of multi-material concepts in the future automotive generations.

Acknowledgment This paper is based on results obtained from a future pioneering program commissioned by the New Energy and Industrial Technology Development Organization(NEDO).

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