Structural optimization and experimental study on the Ti-alloy Kagome structure formed by the superplastic forming/diffusion bonding process

Abstract

The structural optimization of Ti-alloy lattice structure formed by the superplastic forming/diffusion bonding (SPF/DB) process is a high-nonlinear problem with multiple design variables. The problem is solved in this research by introducing the modified Kriging response surface model based on structural mechanics analysis and the genetic algorithm into the optimization design of the Kagome structure. The comprehensive influence of the structural parameters on the shape and compressive strength of the structure is analyzed, and the optimized structural parameters are obtained. The SPF/DB forming and performance test of the optimized Kagome structure are carried out for verifying the accuracy of the model. The results show that the modified Kriging response surface model of the Kagome structure performs well in describing the relationships between the structural parameters and the simulation results, the average relative error of the predicted groove depth and compression strength are 2.4% and 4.5%, respectively. The specific compression strength and the specific compression modulus of the optimized Kagome structure are $5.21 \times 10^{4} N \cdot m / kg$ and $1.50 \times 10^{6} N \cdot m / kg$ . Based on the compression strength results calculated by the Kriging model, a new Ashby plot for the compressive strength as function of density is obtained.

Keywords

Sandwich structure SPF/DB structural optimization simulation Kriging model

Introduction

Metal sandwich structures have been applied in the aerospace field¹ due to their lightweight, high strength and multi-functions, such as integral panels,² hollow blades³ and aircraft wings.⁴ Based on topology morphology, various metal lattice truss structures are designed and researched, such as pyramidal structure, X-type structure, Kagome structure and so on. Among them, the Kagome structure is concluded to have excellent specific strength compared with the other lattice structures. It also exhibits exceptional load-bearing capabilities and energy absorption properties,⁵ making it suitable for applications that require high strength and impact resistance. The Ti-alloy Kagome lattice truss structure can be formed by investment casting,⁶ stamping and brazing,⁷ additive manufacture⁸ and superplastic forming/diffusion bonding (SPF/DB) process.⁹ SPF/DB process is considered an efficient and advanced manufacturing technology, and it can fabricate large-size overall structures with high design freedom.^10,11 Tan et al.¹² fabricated Ti6Al4V sandwich structure with a strong joint by SPF/DB process, and improved the shear property of this structure. Li et al.¹³ manufactured different titanium alloy topological structures by designing the shape of core sheets and analyzed the compression, shear, bending and torsion properties by the FE model and test. Therefore, the SPF/DB process is a good method for the manufacture of sandwich structures with favorable mechanical properties and dimensional precision.

The demand for lightweight structural design makes more scholars focus on optimizing structures and process parameters.^14,15 However, there are inhomogeneous deformations in the lattice structures during SPF/DB process, the relationship between structural parameters and performance was difficult to establish, so it is essential to find a suitable structural optimization method for the lattice structure formed by SPF/DB process. The surrogate model is used to characterize the relationship between structural parameters and performance for achieving performance-based lattice structure optimization. The conventional surrogate models include the polynomial-based model,¹⁶ the radial basis function model,¹⁷ the Kriging model,¹⁸ etc. The polynomial-based response surface model is widely applied because of its simplicity and ease of calculation. However, it is challenging to solve nonlinear and multivariable engineering problems. The radial basis function model has an excellent fitting degree, but the calculation time is long. The Kriging response surface provides an efficient and accurate way to solve nonlinear engineering optimization problems by considering the complex geometrical features and the spatial distribution of properties. Additionally, the Kriging model allows for the generation of surrogate models, which can significantly reduce the computational cost associated with evaluating the mechanical response of lattice structures.¹⁹ Du et al.²⁰ implemented the optimization of the high-accuracy hierarchical Kriging model by using the hybrid optimization algorithm and applied the model to the structural dynamic optimization of rocket engines.

In this paper, the Kagome lattice structure with the different structural parameters were studied, the corresponding shape size and compressive strength were obtained by finite element simulation. Then, based on the simulation results, a modified Kriging response surface based on structural mechanics analysis was constructed, the comprehensive influence of the structural parameters on the shape and compressive strength was analyzed, and the optimized structural parameters were obtained by the genetic algorithm. Finally, the SPF/DB forming and performance test of the different lattice structures were carried out. This study provided a structural optimization method for the lattice structure formed by the SPF/DB process. It aimed to promote the application of high-temperature titanium alloy sandwich structures in aerospace.

Lattice truss structure and Structural optimization method

The SPF/DB forming principle and three lattice truss structures

Figure 1 shows the schematic diagram of the SPF/DB process for fabricating lattice structures. In Figure 1(a), the face sheets and core sheets were bonded at a specific DB process parameter. After bonding, the argon was blown into the sandwich panel, and the face sheets and the core sheets would move together to approach the dies, resulting in the formation of the structure. Figure 1(c) shows the core sheet and 3D model of Kagome lattice structures, where the green areas are the bonding areas and the blue areas are the non-welding areas.

Design variables and objective functions

Structural optimization design needs to determine design variables, constraints and objective functions.²¹ For the Kagome structure formed by SPF/DB process, design variables are the rib width $b$ , the hexagonal side length $a$ and the weldspot diameter $d$ . Based on the calculation method of relative density and compressive strength in the equations (1) and (3), when the relative density of the core layer is less than 0.02 and the strength is greater than 2MPa, the range of the rib width and the hexagonal side length can be obtained: $b \in [3, 6]$ , $a \in [8, 12]$ . Acconding to the previous diffusion bonding test, if the weldspot diameter is too small, the welding quality is poor, and the lattice structure is difficult to form. If the diameter of the solder joint is too large, the relative density of the structure increases, so the range of the weldspot diameter is determined by initial experimental investigations: $d \in [3, 6]$ . The corresponding objective function and constraint conditions are defined as²²

{\begin{cases} x = [b, a, d] \\ \max f (x) = σ_{p} \\ g (x) = h \leq 0.2 m m \\ \bar{ρ} = \frac{4 a b + b^{2}}{30 {(2 a + b)}^{2}} \\ 3 \leq b \leq 6 \\ 8 \leq a \leq 12 \\ 3 \leq d \leq 6 \end{cases}

(1)

where

f (x)

is objective function,

g (x)

is inequality constraint function,

σ_{p}

is the compression strength,

h

is the groove depth,

\bar{ρ}

is the relative density. The homogenization design method was used to establish the sample points.

Kriging response surface model

The Kriging response surface model is suitable for highly nonlinear complex engineering optimization problems. The Kriging model is composed of two terms, the output result is equal to the sum of global design value and local deviation,²³ it can be expressed as follows

y (x) = β \cdot μ (x) + z (x)

(2)

where

μ (x)

is a function about the design variable, which represents the global model,

β

is the regression coefficient of the basis function,

z (x)

is the Gaussian random function with an expectation of 0 and a variance of

σ^{2}

, which means the local deviation. The structural mechanics analysis of the three structures is carried out respectively, and the structural mechanics model is taken as the global model

μ (x)

. The corresponding model about

σ_{p}

and

h

are as follows

{\begin{cases} μ_{1} (x) = σ_{p 1} = \frac{π^{2} \cdot κ^{2} \cdot t^{2} \cdot E_{t}}{12 (a^{2} + 100)} \cdot \frac{20 \bar{ρ}}{(2 a + b) \sqrt{a^{2} + 100}} \\ μ_{2} (x) = h_{1} = q_{1} b^{2} + q_{2} a^{2} + q_{3} d^{2} + q_{4} b d + q_{5} b a + q_{6} a d + q_{7} b + q_{8} a + q_{9} d + q_{10} \end{cases}

(3)

where

κ = 1

, the

E_{t}

is tangent modulus of the material equal to 61GPa,

q_{m}

are constant coefficients. The calculation equation of Kriging covariance

z (x)

is as follows,

cov (z (x_{i}), z (x_{j})) = σ^{2} R (x_{i}, x_{j})

(4)

where

R (x_{i}, x_{j})

represents the correlation function of

x_{i}

and

x_{j}

, it is determined by equation (4),

R (x_{i}, x_{j}) = \exp {(\sum_{k = 1}^{n} - θ_{k} | x_{i}^{k} - x_{j}^{k} |)}^{2}

(5)

where

n

is the number of variables,

θ_{k}

is obtained by maximizing the likelihood function:

\max L (θ_{k}) = - \frac{1}{2} [n \ln (σ^{2}) + \ln (\det (R))]

(6)

R = [\begin{array}{l} R (x_{1}, x_{1}) R (x_{1}, x_{2}) R (x_{1}, x_{3}) \\ R (x_{2}, x_{1}) R (x_{2}, x_{2}) R (x_{2}, x_{3}) \\ R (x_{3}, x_{1}) R (x_{3}, x_{2}) R (x_{3}, x_{3}) \end{array}]

(7)

The estimated value $β$ can be obtained by the least squares estimation method

β = {(U^{T} R^{- 1} U)}^{- 1} U^{T} R^{- 1} Y

(8)

where

U = {[μ (x^{1}), \cdot \cdot \cdot, μ (x^{n})]}^{T}

Y

is the response value matrix, and the variance

σ^{2}

can be represented as follows

σ^{2} = \frac{1}{n} {(Y - β U)}^{T} R^{- 1} (Y - β U)

(9)

The predicted value and variance of the Kriging response surface model can be expressed by the following equation²⁴

\hat{y} (x) = β \cdot \hat{μ} (x) + r^{T} (x) R^{- 1} (Y - β U)

(10)

σ_{y (x)}^{2} = σ^{2} [1 + \frac{{(μ (x) - U^{T} R^{- 1} r)}^{2}}{U^{T} R^{- 1} U} - r^{T} R^{- 1} r]

(11)

where

r (x)

can be represented

r (x) = {(R (x, x^{1}), \cdot \cdot \cdot, R (x, x^{n}))}^{T}

(12)

Finally, based on the established Kriging response surface model, the genetic algorithm toolbox in Matlab is used to optimize the optimal solution of the model and achieve the optimization of structural parameters.

Material and experimental methods

Material

TC31 titanium alloy sheet was used for the manufacture of the Kagome structure, and the thickness of the face sheet and core sheets were 1.2 mm and 0.8 mm, respectively. The high-temperature performance of the TC31 titanium alloy was studied in a previous article,²⁵ which indicated that the material had an excellent superplastic performance at 920°C.

The simulation model

The SPF process of the Kagome structure is simulated by using Abaqus 6.14, and the corresponding simulation schemes are shown in Table 1. The dies were defined as rigid, the face sheets and the core sheets were defined as deformable entities, and the stress-strain curves of the materials were chosen as the materials model.²⁵ The assembled model is shown in Figure 2(a), the dies are fixed. The corresponding diffusion bonding areas were tied²⁶ because the good diffusion bonding joint had high tear strength. The coefficient of friction between the sheets was 0.1.²⁷ The adaptive optimized pressure curve depending on the target strain rate (0.001 s⁻¹) is set in the model, and the final pressure load shown in Figure 2(b) is applied on the inner surface of the face sheet.

Table 1.

The simulation results of the different Kagome structure.

NO.	$b$ (mm)	$a$ (mm)	$d$ (mm)	$h$ (mm)	$σ_{p}$ (MPa)	$h^{'}$ (mm)	$σ_{p}^{'}$ (MPa)
1	3.0	8.4	3.6	0.34	2.05	0.34	2.02
2	3.3	9.2	4.5	0.33	1.95	0.33	1.99
3	3.6	10.0	5.4	0.25	1.87	0.26	1.85
4	3.9	10.8	3.0	0.48	1.16	0.5	1.16
5	4.2	11.6	3.9	0.45	1.06	0.44	1.06
6	4.5	8.0	4.8	0.49	3.91	0.51	3.91
7	4.8	8.8	5.7	0.46	3.70	0.45	3.70
8	5.1	9.6	3.3	0.69	2.09	0.69	2.20
9	5.4	10.4	4.2	0.64	2.35	0.63	2.03
10	5.7	11.2	5.1	0.51	1.46	0.53	1.76
11	6.0	12.0	6.0	0.38	1.47	0.38	1.38

The compression simulation model of the lattice truss structure is shown in Figure 2(c). The bottom of the model was fixed, and the velocity of the punch was 0.5 mm/min. The direct cracking stress after cracking and direct cracking strain was set as 1200 MPa and 0.075.²⁸ The compressive strength curve was obtained by extracting the reaction force on the fixed point.

SPF/DB experiment and compression experiment

The SPF/DB experiment was conducted on a vacuum SPF/DB machine. At first, all sheets were polished with emery paper and corrosive fluid (HF: HNO₃: H₂O = 1: 3: 16 mixed corrosive liquid). Then, the face sheets and the core sheets were put into a vacuum furnace. The diffusion bonding experiment shown in Figure 3(a)

was conducted in a vacuum of 2.0 × 10⁻² Pa at 920°C/3 MPa/60 min.²⁹ As shown in Figure 3(b), the bonded multilayer sheet and the superplastic dies were set into the machine. The argon was blown into the lattice structure at 920°C according to the pre-designed loading curve of gas pressure (shown in Figure 2(b)). Sufficient argon was supplied to enable the contact of face sheets and dies to facilitate the formation of the final structure.

The compression experiment shown in Figure 3(c) was performed on the UTM5504X testing machine according to GB/T 1453-2005, and the size of compressive sample was 90 mm × 80 mm × 20 mm. The compressive speed of punch was 0.5 mm/min. The compression force-displacement curves of the X-type lattice structures were obtained by the compression experiment, and two samples were tested to verify the accuracy of the results.

Results and discussion

Optimized result of the Kriging response model

Table 1 shows the simulation results of the different Kagome lattice structures. The Kriging response surface model is established according to the obtained test points, the average relative error (AARE) is used to evaluate the accuracy of the model, and it can be expressed as follows:

A R R E = \frac{1}{n} \sum_{i = 1}^{n} \frac{| c_{i} - c_{i}^{'} |}{c_{i}} \times 100 %

(13)

where

c_{i}

denotes the simulation result, and

c_{i}^{'}

denotes the computed result by the Kriging response model. The AARE of the groove depth and compression strength is 2.4% and 4.5%, respectively, which indicates that the Kriging response surface model has good prediction accuracy and can be used for optimization in the subsequent multi-objective genetic algorithm.

Figure 4 shows the Kriging response result of the Kagome structure. It can be seen from Figure 4(a) that with the increase of the weldspot diameter and the hexagon side length, the groove depth increases. However, the effect of ribs width is opposite to that of the other variables. In Figure 4(b), with the increase of the weldspot diameter and the hexagon side length, the compression strength decreases. It can also be seen that the hexagonal side length has a more significant impact on the compressive strength than the ribs width and the weldspot diameter. Finally, the optimized structural parameters obtained by the genetic algorithm are $b = 4.4 mm$ , $a = 8.2 mm$ , $d = 6 mm$ .

Simulation and experimental result of the optimized Kagome structure

Figure 5 shows the simulation and experimental results of the optimized Kagome structure. It can be seen that the Kagome structure with good dimension accuracy can be manufactured by SPF/DB process. The simulation results show that the area with maximum strain is located at the transition fillet between the ribs and the bonding areas, where the maximum strain is 0.99 and the minimum thickness is 0.52 mm. To compare the difference between simulation and experiment, the contour and thinning rate of region A and region B in Figure 5(b) are shown in Figure 6. In Figure 6(a), the edge of the structure fits the die surface. The maximum groove depth of the experimental result is 0.18 mm, while the simulation result is 0.23 mm, and the AARE is 1.9%. The outline and thinning rate of ribs at area B are shown in Figure 6(b) and (c). The rib outline of the experimental results is almost consistent with the simulation, and the AARE is 3.0%. The included angle between the rib and the horizontal plane is about 55°. The maximum thinning area is the same as that shown in Figure 6(b). The maximum thinning rate of the rib is 37.8%, the simulation value is 38.0%, and the error is about 0.5%. Figure 6(d) shows the compression curve of the Kagome structure. The compression strength of the test sample is 4.36 MPa, the simulation result is 4.56 MPa, and the error is about 4.4%. The material properties of the formed structure are reduced because of the thermal cycle, resulting in the compressive strength obtained by the experiment is smaller than that of the simulation.³⁰ The results show that the simulation model can accurately predict the shape size and compression performance of the Kagome structure. It also can be seen from Figure 6(d) that the ribs buckling during the initial deformation, and the cracks begin to initiate and expand. When the compression stress reaches the peak stress, the ribs break and the stress decreases rapidly, and the other ribs break in succession until the structure failed. The compressive failure mode of the Kagome structure is ductile fracture. In order to study the compression fracture mode of Kagome structure, the fracture morphology is shown in the Figure 6(e). It can be seen that there are a large number of dimples with different shapes on the section. Under the action of compressive stress, dimples are generated at the defects with microcracks, grow and aggregate at the crack tip, and finally form hemispherical cavities when fracture occurs.

According to the equation (1), the relative density $\bar{ρ}$ of the optimized Kagome structure is 0.0188, and the optimized Kagome structure has the high specific compression strength $σ_{p} / ρ$ and the high specific compression modulus $E_{p} / ρ$ , the corresponding values are $5.21 \times 10^{4} N \cdot m / kg$ and $1.50 \times 10^{6} N \cdot m / kg$ , respectively. Based on the compression strength results calculated by the Kriging model, an Ashby plot for the compressive strength as function of density for different structures is shown in Figure 7.³¹ It can be seen that within the constraints of structural parameters in this paper, the density range of the Kagome structure formed by 0.8 mm TC31 titanium alloy sheet is 73.78∼107.96 kg/m³ and the maximum compressive strength is 9.59 MPa. Furthermore, the Ti-alloy Kagome structure locates in the left region of the figure, which indicates it is lighter than most of metal structures, the density is even one order of magnitude lower than some metal materials,³¹displaying the significant advantages of lightweight. The compressive strength of this structure fabricated by SPF/DB process is higher the other lattice structures, such as the Ti pyramidal structures, the Ti X-type structures, the metal foams and polymer foams.

Conclusions

A modified Kriging response surface model based on structural mechanics analysis and genetic algorithm were introduced into the optimization design of Ti-alloy Kagome lattice structure formed by the SPF/DB process. With the validation of numerical results, the modified Kriging response surface model demonstrated the capability to capture the relationships between the structural parameters and the simulation results, such as the groove depth and the compression strength. The AARE of the predicted groove depth and compression strength were 2.4% and 4.5%, respectively. The optimized structural parameters of the Kagome structure were $b = 4.4 mm$ , $a = 8.2 mm$ , $d = 6 mm$ .

Based on the optimized structural parameter, the corresponding Kagome lattice structure with good dimension accuracy could be manufactured by SPF/DB process. The area of formed structures with maximum strain was located at the transition fillet between the ribs and the bonding areas, in which the strain and thinning rate of the Kagome structure were the largest with the value of 0.99 and 38.0%. The compression strength of the test piece is 4.36 MPa, the simulation result is 4.56 MPa, and the error is about 4.4%. The compressive failure mode of the Kagome structure is ductile fracture. Based on the compression strength results calculated by the Kriging response surface model, an Ashby plot for the compressive strength as function of density for different structures was completed to compare the compressive properties of different structures.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (No. 51805256), the Fundamental Research Funds for the Central Universities (NO. NS2023026) and Postdoctoral Research Foundation of China (NO. 2020M670792).

ORCID iDs

Dipeng Wu

Yong Wu

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