Isotropic and Kinematic Hardening Material

108

Table 7.1 The optimal layout of structure based on the bilinear elastoplastic model.

Model Weight Filtering

Factor Final Layout Stress Distribution of Final Layout

A

Iteration 138 Max. stress 592 MPa

B

Iteration 123 Max. stress 519 MPa

C

Iteration 134 Max. stress 557 MPa

-109

Figure 7.12 Iterative material distribution on isotropic and kinematic hardening material with cyclic loading of Model A.

Figure 7.13 Iterative stress distribution on isotropic and kinematic hardening material with cyclic loading of Model A.

Iteration 187 Iteration 181

Iteration 169

Iteration 153 Iteration 146

Iteration 132

Iteration 118 Iteration 97

Iteration 36

Iteration 187 Iteration 181

Iteration 169

Iteration 153 Iteration 146

Iteration 132

Iteration 118 Iteration 97

Iteration 36

110

Figure 7.14 Iterative material distribution on isotropic and kinematic hardening material with cyclic loading of Model B.

Figure 7.15 Iterative stress distribution on isotropic and kinematic hardening material with cyclic loading of Model B.

Iteration 164 Iteration 160

Iteration 152

Iteration 141 Iteration 127

Iteration 121

Iteration 118 Iteration 71

Iteration 27

Iteration 164 Iteration 160

Iteration 152

Iteration 141 Iteration 127

Iteration 121

Iteration 118 Iteration 71

Iteration 27

111

Figure 7.16 Iterative material distribution on isotropic and kinematic hardening material with cyclic loading of Model C.

Iteration 5 Iteration 13 Iteration 20

Iteration 34 Iteration 68 Iteration 118

Iteration 121 Iteration 126 Iteration 141

Iteration 146 Iteration 152 Iteration 157

Iteration 161 Iteration 165 Iteration 172

Iteration 184 Iteration 181

Iteration 177

112

Figure 7.17 Iterative stress distribution on isotropic and kinematic hardening material with cyclic loading of Model C.

Iteration 5 Iteration 13 Iteration 20

Iteration 34 Iteration 68 Iteration 118

Iteration 121 Iteration 126 Iteration 141

Iteration 146 Iteration 152 Iteration 157

Iteration 161 Iteration 165 Iteration 172

Iteration 184 Iteration 181

Iteration 177

113

The internal energy at each iteration for all three models was plotted as figure 7.18 to investigate the convergence of the optimization problem. A tendency of all three models had increased the internal energy when the optimization iteration was increased. Only the proposed Model C was increased the internal energy smoothly, while Model A and C caused a fluctuation at iteration instance 160 due to the effect of unloading behavior. From these results, the optimization process for three models was preliminary confirmed the convergence under cyclic loading.

Figure 7.18 Internal energy of the three models during the optimization process based on the isotropic and kinematic hardening material.

Table 7.2 showed a comparison on the final layout of three optimization models with the definition of each model being specified as the same as the bilinear elastoplastic material case. The weight filtering factor, which applied to Model B, repeatedly obtained the final layout in the checkerboard pattern. Since the value of weight filtering factor in Model B usually indicates zero. On the other hand, the weight filtering factor of Model A and Model C are positive values and higher that zero. Therefore, the equation of Model B was not beneficial in optimizing the structure under the cyclic load and the nonlinear material

0 100 200 300 400 500 600

0 20 40 60 80 100 120 140 160 180 200

Internal Energy (mJ)

Iteration Model A

Model B Model C

114

properties. The final layouts of Model A and Model C obtained were quite similar lay-outs, and both layouts were not complicated to conjecture the real structure. Thus, the stress constraint of each model during the optimization process was crucial with further considerations to decide on a more effective weight filtering factor.

Table 7.2 The optimal layout of structure based on the isotropic and kinematic hardening model.

Model Weight Filtering

Factor Final Layout Stress Distribution of Final Layout

A

Iteration 187 Max. stress 492 MPa

B

Iteration 164 Max. stress 414 MPa

C

Iteration 184 Max. stress 499 MPa

The maximum stress during the optimization process, which measured maximum elemental stress at each iteration, was displayed in figure 7.19 for comparing all three weight filtering factors. The stress constraint of the three models was the similar tendency at the beginning of the optimization process. Moreover, the stress histories were divergent from the bilinear elastoplastic material model due to the effect of cyclic loading and unloading point of the isotropic and kinematic hardening material. These effects caused a high interval of stress during the optimization process.

(

)

ij r r

w =max0, ₀

-÷÷ ø ö çç

è æ

-=max 1 ⁰,0

ij

ij r

w r

(

)

max _{0 ij}

ij r r

w = -

-115

Figure 7.19 Maximum stress of three models during optimization process based on isotropic and kinematic hardening material.

The fluctuation of the stress occurred at the iteration instance 150, and the stress histories were plotted in figure 7.20 until the termination process. Model A and Model B occurred the swing effect of stress during the optimization process due to the cyclic load that forces the problem into tension and compression loads, and the unloading point of kinematic hardening material. The unloading point of the external load appeared when the load factor was equal to zero. The stress fluctuation of Model A and B obviously showed the high-stress interval between the maximum and minimum peaks, while Model C was optimized without the swing effect. The final layout of Model A and Model C occurred the maximum stress of 429 MPa and 414 MPa, respectively, which was lower than the maximum peaks of the optimization constraints. Model C caused the maximum stress of 499 MPa on the final layout, which is the maximum peak of the optimization constraints because Model C did not fluctuate. Moreover, the stress distribution on the final layout of Model C clearly demonstrated the fully stressed distribution when compared with the other two models. Therefore, the proposed weight filtering factor (Model C) was suitable for nonlinear topology optimization under the cyclic loading. Finally, 54% of the material amount can be removed from the initial design domain on Model C.

250 300 350 400 450 500 550 600 650

0 20 40 60 80 100 120 140 160 180 200

Stress (MPa)

Iteration Model A

Model B Model C

Stress Limit

116

Figure 7.20 Stress fluctuation during the optimization process based isotropic and kinematic hardening material.