梯度多胞材料耐撞性设计的简化模型和渐近解
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摘要
多胞材料具有很强的吸能特性,并被广泛应用于结构耐撞性设计领域,其力学响应与密度分布息息相关.因此可以通过对多胞材料密度分布的设计达到对材料的宏观力学响应的有效控制,以满足实际应用中的耐撞性要求.基于非线性塑性冲击波模型可以实现对密度梯度的反向设计,但其求解较为困难.本文利用级数法获得了耐撞性反向设计理论的渐近解,针对常冲击力的耐撞性要求给出了一个具有足够精度的近似解.利用二维随机Voronoi技术构建了细观有限元模型,并运用有限元软件ABAQUS/Explicit进行了数值验证,结果表明密度分布的反向设计理论的渐近解对于指导梯度多胞材料的耐撞性设计是有效的,且二阶渐近解已提供足够的设计精度.基于冲击波模型的反向设计方法为主动设计梯度多胞材料提供了重要依据.
Cellular materials have strong energy absorption capacity and have been extensively used in crashworthiness design of structures. The mechanical responses of cellular materials are related to their relative density. Introducing a density gradient to cellular materials may further improve their crashworthiness. A backward design method based on a nonlinear plastic shock model was developed to meet the crashworthiness requirement of protecting a mass when impinging a graded cellular rod. For the case of constant impact force, an asymptotic solution to the density distribution of graded cellular rod is obtained in this paper. The density distributions between the explicit asymptotic solution and the exact solution numerically obtained with a fourth-order Runge-Kutta scheme have been compared. The first-order approximate solution is a linear approximation, and the second-order approximate prediction is basically consistent with the numerical solution. For the impact force history, the first-order approximate prediction has a large deviation, and the second-order approximate prediction has a slight deviation. For practical application the second-order approximate solution seems to be accurate enough. Cell-based finite element models are employed to verify the asymptotic solution. The finite element models have been constructed and computed via ABAQUS/Explicit code. Samples of graded cellular rods designed from the asymptotic solution are tested numerically. The predictions using the second-order approximate solution of density distribution is very close to the finite element results. The design using the first-order approximate solution is not accurate enough, but in fact it is not so bad. Compared the uniform density distribution, the second-order approximate prediction is good for the crashworthiness optimization. The design of density distribution for a specific impact scenario is found to be inappropriate for much high velocity impact. In summary, a case study shows that the second-order approximate solution is very close to the exact solution and it is good enough for the design. The backward design method and the asymptotic solution of density distribution are helpful to guide the crashworthiness design of graded cellular materials.
Cellular materials have strong energy absorption capacity and have been extensively used in crashworthiness design of structures. The mechanical responses of cellular materials are related to their relative density. Introducing a density gradient to cellular materials may further improve their crashworthiness. A backward design method based on a nonlinear plastic shock model was developed to meet the crashworthiness requirement of protecting a mass when impinging a graded cellular rod. For the case of constant impact force, an asymptotic solution to the density distribution of graded cellular rod is obtained in this paper. The density distributions between the explicit asymptotic solution and the exact solution numerically obtained with a fourth-order Runge-Kutta scheme have been compared. The first-order approximate solution is a linear approximation, and the second-order approximate prediction is basically consistent with the numerical solution. For the impact force history, the first-order approximate prediction has a large deviation, and the second-order approximate prediction has a slight deviation. For practical application the second-order approximate solution seems to be accurate enough. Cell-based finite element models are employed to verify the asymptotic solution. The finite element models have been constructed and computed via ABAQUS/Explicit code. Samples of graded cellular rods designed from the asymptotic solution are tested numerically. The predictions using the second-order approximate solution of density distribution is very close to the finite element results. The design using the first-order approximate solution is not accurate enough, but in fact it is not so bad. Compared the uniform density distribution, the second-order approximate prediction is good for the crashworthiness optimization. The design of density distribution for a specific impact scenario is found to be inappropriate for much high velocity impact. In summary, a case study shows that the second-order approximate solution is very close to the exact solution and it is good enough for the design. The backward design method and the asymptotic solution of density distribution are helpful to guide the crashworthiness design of graded cellular materials.
引文
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18 Wu H X,Liu Y.Influences of density gradient variation on mechanical performances of density-graded honeycomb materials.Explosion Shock Waves,2013,33:163-168[吴鹤翔,刘颖.梯度变化对密度梯度蜂窝材料力学性能的影响.爆炸与冲击,2013,33:163-168]
19 Liu Y,Wu H X,Wang B.Gradient design of metal hollow sphere(MHS)foams with density gradients.Compos Part B-Eng,2012,43:1346–1352
20 Zhang J J,Wang Z H,Zhao L M.Dynamic response of functionally graded cellular materials based on the Voronoi model.Compos Part B-Eng,2015,85:176–187
21 Fan J H,Zhang J J,Wang Z H,et al.Dynamic crushing behavior of random and functionally graded metal hollow sphere foams.Mater Sci EngA,2013,561:352–361
22 Koohbor B,Kidane A.Design optimization of continuously and discretely graded foam materials for efficient energy absorption.Mater Des,2016,102:151–161
23 Karagiozova D,Alves M.Stress waves in layered cellular materials—Dynamic compaction under axial impact.Int J Mech Sci,2015,101-102:196 –213
24 Cui L,Kiernan S,Gilchrist M D.Designing the energy absorption capacity of functionally graded foam materials.Mater Sci Eng-A,2009,507:215 –225
25 Liu J G,Hou B,Lu F Y,et al.A theoretical study of shock front propagation in the density graded cellular rods.Int J Impact Eng,2015,80:133–142
26 Wang S L,Ding Y Y,Wang C F,et al.Dynamic material parameters of closed-cell foams under high-velocity impact.Int J Impact Eng,2017,99:111 –121
2 Gibson L J,Ashby M F.Cellular Solids:Structure and Properties.Cambridge University Press,1999
3 Ashby M F,Evans T,Fleck N A,et al.Metal Foams:A Design Guide.Elsevier,2000
4 Wang X K,Zheng Z J,Yu J L,et al.Impact resistance and energy absorption of functionally graded cellular structures.Appl Mech Mater,2011,69 :73–78
5 Li K,Gao X L,Subhash G.Effects of cell shape and cell wall thickness variations on the elastic properties of two-dimensional cellular solids.Int J Solids Struct,2005,42:1777–1795
6 Li K,Gao X L,Wang J.Dynamic crushing behavior of honeycomb structures with irregular cell shapes and non-uniform cell wall thickness.Int J Solids Struct,2007,44:5003–5026
7 Gibson L J.Mechanical behavior of metallic foams.Annu Rev Mater Sci,2000,30:191–227
8 Wang X K,Zheng Z J,Yu J L.Crashworthiness design of density-graded cellular metals.Theor Appl Mech Lett,2013,3:031001
9 Shen C J,Yu T X,Lu G.Double shock mode in graded cellular rod under impact.Int J Solids Struct,2013,50:217–233
10 Zheng J,Qin Q,Wang T J.Impact plastic crushing and design of density-graded cellular materials.Mech Mater,2016,94:66–78
11 Zheng Z J,Wang C F,Yu J L,et al.Dynamic stress–strain states for metal foams using a 3D cellular model.J Mech Phys Solids,2014,72:93–114
12 Cai Z Y,Ding Y Y,Wang S L,et al.Anti-blast analysis of graded cellular sacrificial cladding(in Chinese).Explosion Shock Waves,2017,37:396-404[蔡正宇,丁圆圆,王士龙,等.梯度多胞牺牲层的抗爆炸分析.爆炸与冲击,2017,37:396-404]
13 Ding Y Y,Wang S L,Zhao K,et al.Blast alleviation of cellular sacrificial cladding:A nonlinear plastic shock model.Int J Appl Mech,2016,08:1650057
14 Yang J,Wang S L,Ding Y Y,et al.Crashworthiness of graded cellular materials:A design strategy based on a nonlinear plastic shock model.Mater Sci Eng-A,2017,680:411–420
15 Zheng Z J,Yu J L,Li J R.Dynamic crushing of 2D cellular structures:A finite element study.Int J Impact Eng,2005,32:650–664
16 Ajdari A,Nayeb-Hashemi H,Vaziri A.Dynamic crushing and energy absorption of regular,irregular and functionally graded cellular structures.Int J Solids Struct,2011,48:506–516
17 Zhang X C,Liu Y.Research on the dynamic crushing of honeycombs with density gradient.Eng Mech,2012,29:372-377[张新春,刘颖.密度梯度蜂窝材料动力学性能研究.工程力学,2012,29:372-377]
18 Wu H X,Liu Y.Influences of density gradient variation on mechanical performances of density-graded honeycomb materials.Explosion Shock Waves,2013,33:163-168[吴鹤翔,刘颖.梯度变化对密度梯度蜂窝材料力学性能的影响.爆炸与冲击,2013,33:163-168]
19 Liu Y,Wu H X,Wang B.Gradient design of metal hollow sphere(MHS)foams with density gradients.Compos Part B-Eng,2012,43:1346–1352
20 Zhang J J,Wang Z H,Zhao L M.Dynamic response of functionally graded cellular materials based on the Voronoi model.Compos Part B-Eng,2015,85:176–187
21 Fan J H,Zhang J J,Wang Z H,et al.Dynamic crushing behavior of random and functionally graded metal hollow sphere foams.Mater Sci EngA,2013,561:352–361
22 Koohbor B,Kidane A.Design optimization of continuously and discretely graded foam materials for efficient energy absorption.Mater Des,2016,102:151–161
23 Karagiozova D,Alves M.Stress waves in layered cellular materials—Dynamic compaction under axial impact.Int J Mech Sci,2015,101-102:196 –213
24 Cui L,Kiernan S,Gilchrist M D.Designing the energy absorption capacity of functionally graded foam materials.Mater Sci Eng-A,2009,507:215 –225
25 Liu J G,Hou B,Lu F Y,et al.A theoretical study of shock front propagation in the density graded cellular rods.Int J Impact Eng,2015,80:133–142
26 Wang S L,Ding Y Y,Wang C F,et al.Dynamic material parameters of closed-cell foams under high-velocity impact.Int J Impact Eng,2017,99:111 –121