Citation: Hang YANG, Li MA. Multimaterial Lattice Structures With Thermally Programmable Mechanical Behaviors[J]. Applied Mathematics and Mechanics , 2022, 43(5): 534-552. doi: 10.21656/1000-0887.430104 作者简介:

杨航（1993—），男，博士生（E-mail： [email protected] ）

;

马力（1975—），男，教授，博士，博士生导师（通讯作者. E-mail： [email protected] ）

中图分类号: O341; TB381

传统的点阵结构一旦制备完成，其力学性能通常在使用寿命内保持不变。设计和制造具有环境适应特性的智能点阵结构，可编程地感知和响应外界变化(例如光强、压强、溶液、温度、电磁场、电化学激励)，并在时间和空间上进行形状重构、模式转换和性能调控，仍然是人造材料研究领域重要的科学挑战。该文采用具有不同玻璃化转变温度和温度依赖性的多种聚合物材料，通过合理设计材料空间分布，提出了一类具有热可编程力学响应能力的多材料点阵结构。结合理论分析和有限元模拟，研究了组分材料相对刚度对多材料点阵结构的Poisson比、变形模式以及结构稳定性的影响。通过温度变化实现了对多材料点阵结构弹性常数、压溃响应和结构稳定性的调控，使多材料点阵结构表现出极大的热变形、超弹性和形状记忆效应。为设计和制造自适应保护装备、生物医学设备、航空航天领域的变形结构、柔性电子设备、自组装结构和可变形软体机器人等开辟了新途径。

可调Poisson比 结构稳定性 形状记忆效应 Abstract:

Traditional lattice structures usually maintain their mechanical properties throughout their lifetime. Designing and manufacturing intelligent materials with environmental adaptability, programmable sense and responses to external changes (such as light, pressure, solution, temperature, electromagnetic field and electrochemical reaction), shape transformation, mode conversion and performance regulation in space and time, are still important scientific challenges in the field of artificial materials. In this paper, multimaterial lattice structures with thermally programmable mechanical responses were proposed by means of polymer materials with disparate glass transition temperatures and temperature dependencies, and through reasonable design of the spatial distribution of the materials. By theoretical analysis combined with finite element simulation, the effects of the relative stiffnesses of constitute materials on Poisson’s ratios, deformation modes and structural stability of the multimaterial lattice structures, were studied. The elastic constants, crushing responses and structural stability of multimaterial lattice structures were regulated by temperature control, consequently the multimaterial lattice structures were endowed with giant thermal deformation, hyperelasticity and shape memory effects. This paper opens up new avenues for the design and manufacture of adaptive protection equipment, biomedical devices, aerospace morphing structures, flexible electronic devices, self-assembly structures and reconfigurable soft robots.

Key words: lattice structure temperature control tunable Poisson’s ratio structural stability thermal deformation shape memory effect Figure 2. The influence of relative stiffness ${E_1}/{E_2}$ of the constituent materials on elastic constants of bimaterial double-V structures: (a) equivalent Poisson’s ratio ${\nu _{xy}}$ as functions of relative stiffness ${E_1}/{E_2}$ ; (b) relative Young’s modulus ${E_y}/{E_2}$ as functions of relative stiffness ${E_1}/{E_2}$ ; (c) equivalent Poisson’s ratio ${\nu _{yx}}$ as functions of relative stiffness ${E_1}/{E_2}$ ; (d) relative Young’s modulus ${E_x}/{E_2}$ as functions of relative stiffness ${E_1}/{E_2}$ Figure 3. The influences of temperatures on elastic constants of bimaterial double-V structures: (a) simulated lateral deformation for y- direction load at low temperatures; (b) simulated lateral deformation for y- direction load at high temperatures; (c) theoretical, numerical and experimental results of Poisson’s ratio ${\nu _{xy}}$ as functions of temperature T ; (d) theoretical, numerical and experimental results of Young’s modulus ${E_y}$ as functions of temperature T Figure 6. Bimaterial double-U snapping structures: (a) schematic diagram of bimaterial double-U structures and geometric parameters of the unit cell; (b) storage modulus E _b vs. temperature T for PLA and TPU; (c) force-displacement curves of the monostable and bistable unit cells; (d) potential energy-displacement curves of the monostable and bistable unit cells Figure 7. System stability analysis of von Mises truss models: (a) the symmetric bistable system with complete constraints at both ends; (b) the symmetric bistable system with incomplete constraints at both ends; (c) the asymmetric bistable or monostable system with a vertical spring and complete constraints at both ends; (d) the asymmetric bistable or monostable system with a vertical spring and incomplete constraints at both ends Figure 8. The force-displacement responses of von Mises truss models: (a) the influence of ${K_2}/{K_1}$ on the force-displacement curves of the system for ${K_3}/{K_1} \to \infty $ and $H/L = 1$ ; (b) the influence of ${K_3}/{K_1}$ on the force-displacement curves of the system for ${K_2}/{K_1} = 0$ and $H/L = 1$ ; (c) the influence of ${K_3}/{K_1}$ on the force-displacement curves of the system for ${K_2}/{K_1} = 0.1$ and $H/L = 1$ ; (d) the influence of $H/L$ on the force-displacement curves of the system for ${K_2}/{K_1} = 0.1$ and ${K_3}/{K_1} = 1$ Figure 10. Thermal deformations and thermal expansion coefficients of bimaterial double-U structures during heating recovery: (a) the positive thermal expansion of an initial convex unit cell during heating recovery; (b) the negative thermal expansion of an initial concave unit cell during heating recovery; (c) the thermal expansion coefficients of bimaterial double-U structures during heating recovery compared with the experimental results of materials and structures reported previously 不同材料和结构的热力循环 $\varepsilon {\text{-}} T {\text{-}} \sigma$ 示意图：(a) 形状记忆聚合物 $\varepsilon {\text{-}} T {\text{-}} \sigma$ 示意图 ^{[ 90 - 94 ]} ；(b) 超弹性材料 $\varepsilon {\text{-}} T {\text{-}} \sigma$ 示意图；(c) 热塑性材料 $\varepsilon {\text{-}} T {\text{-}} \sigma$ 示意图；(d) 稳定性转换双U结构单元 $\varepsilon {\text{-}} T {\text{-}} \sigma$ 示意图；(e) 模式切换结构 $\varepsilon {\text{-}} T {\text{-}} \sigma$ 示意图 ^{[ 48 ]} ；(f) 预应力装配体 $\varepsilon {\text{-}} T {\text{-}} \sigma$ 示意图 ^{[ 49 ]} Figure 11. The $\varepsilon {\text{-}} T {\text{-}} \sigma$ diagrams of the thermomechanical cycle for different materials and structures: (a) the $\varepsilon {\text{-}} T {\text{-}} \sigma$ diagram of shape memory polymers ^{[ 90 - 94 ]} ; (b) the $\varepsilon {\text{-}} T {\text{-}} \sigma$ diagram of hyperelastic materials; (c) the $\varepsilon {\text{-}} T {\text{-}} \sigma$ diagram of themoplastic materials; (d) the $\varepsilon {\text{-}} T {\text{-}} \sigma$ diagram of the unit cell of stability transforming double-U structures; (e) the $\varepsilon {\text{-}} T {\text{-}} \sigma$ diagram of the pattern switching structures ^{[ 48 ]} ; (f) the $\varepsilon {\text{-}} T {\text{-}} \sigma$ diagram of the prestressed assemblies ^{[ 49 ]} Figure 12. The shape reconfiguration and recovery mechanism of the stability transforming double-U structures: (a) the shape reconfiguration and recovery mechanism of initial structures with fully expanded configurations; (b) the shape reconfiguration and recovery mechanism of initial structures with fully contracted configurations; (c) the shape reconfiguration and recovery mechanism of shape memory polymers Figure 1.

Bimaterial concave double-V structures: (a) schematic diagram of bimaterial concave double-V structures; (b) boundary conditions for loads along y and x directions; (c) storage modulus E _b vs. temperature T for PLA and PC; (d) schematic diagram of quasi-static uniaxial compression

Figure 2.

The influence of relative stiffness ${E_1}/{E_2}$ of the constituent materials on elastic constants of bimaterial double-V structures: (a) equivalent Poisson’s ratio ${\nu _{xy}}$ as functions of relative stiffness ${E_1}/{E_2}$ ; (b) relative Young’s modulus ${E_y}/{E_2}$ as functions of relative stiffness ${E_1}/{E_2}$ ; (c) equivalent Poisson’s ratio ${\nu _{yx}}$ as functions of relative stiffness ${E_1}/{E_2}$ ; (d) relative Young’s modulus ${E_x}/{E_2}$ as functions of relative stiffness ${E_1}/{E_2}$

Figure 3.

The influences of temperatures on elastic constants of bimaterial double-V structures: (a) simulated lateral deformation for y- direction load at low temperatures; (b) simulated lateral deformation for y- direction load at high temperatures; (c) theoretical, numerical and experimental results of Poisson’s ratio ${\nu _{xy}}$ as functions of temperature T ; (d) theoretical, numerical and experimental results of Young’s modulus ${E_y}$ as functions of temperature T

Figure 4.

The influences of temperatures on deformation modes of bimaterial double-V structures: (a) simulated crushing process for y- direction load at a low temperature; (b) simulated crushing process for y- direction load at a high temperature

Figure 5.

The influence of temperature on stress-strain response of bimaterial double-V structures: (a) numerical results of stress-strain curve for y -direction load at low temperature; (b) numerical results of stress-strain curve for y -direction load at high temperature

Figure 6.

Bimaterial double-U snapping structures: (a) schematic diagram of bimaterial double-U structures and geometric parameters of the unit cell; (b) storage modulus E _b vs. temperature T for PLA and TPU; (c) force-displacement curves of the monostable and bistable unit cells; (d) potential energy-displacement curves of the monostable and bistable unit cells

Figure 7.

System stability analysis of von Mises truss models: (a) the symmetric bistable system with complete constraints at both ends; (b) the symmetric bistable system with incomplete constraints at both ends; (c) the asymmetric bistable or monostable system with a vertical spring and complete constraints at both ends; (d) the asymmetric bistable or monostable system with a vertical spring and incomplete constraints at both ends

Figure 8.

The force-displacement responses of von Mises truss models: (a) the influence of ${K_2}/{K_1}$ on the force-displacement curves of the system for ${K_3}/{K_1} \to \infty $ and $H/L = 1$ ; (b) the influence of ${K_3}/{K_1}$ on the force-displacement curves of the system for ${K_2}/{K_1} = 0$ and $H/L = 1$ ; (c) the influence of ${K_3}/{K_1}$ on the force-displacement curves of the system for ${K_2}/{K_1} = 0.1$ and $H/L = 1$ ; (d) the influence of $H/L$ on the force-displacement curves of the system for ${K_2}/{K_1} = 0.1$ and ${K_3}/{K_1} = 1$

Figure 9.

Experimental processes of shape reconfiguration and recovery of double-U structures: (a) shape reconfiguration and recovery of initial double-U structure with fully expanded configuration; (b) shape reconfiguration and recovery of initial double-U structure with fully contracted configuration

Figure 10.

Thermal deformations and thermal expansion coefficients of bimaterial double-U structures during heating recovery: (a) the positive thermal expansion of an initial convex unit cell during heating recovery; (b) the negative thermal expansion of an initial concave unit cell during heating recovery; (c) the thermal expansion coefficients of bimaterial double-U structures during heating recovery compared with the experimental results of materials and structures reported previously

Figure 11.

The $\varepsilon {\text{-}} T {\text{-}} \sigma$ diagrams of the thermomechanical cycle for different materials and structures: (a) the $\varepsilon {\text{-}} T {\text{-}} \sigma$ diagram of shape memory polymers ^{[ 90 - 94 ]} ; (b) the $\varepsilon {\text{-}} T {\text{-}} \sigma$ diagram of hyperelastic materials; (c) the $\varepsilon {\text{-}} T {\text{-}} \sigma$ diagram of themoplastic materials; (d) the $\varepsilon {\text{-}} T {\text{-}} \sigma$ diagram of the unit cell of stability transforming double-U structures; (e) the $\varepsilon {\text{-}} T {\text{-}} \sigma$ diagram of the pattern switching structures ^{[ 48 ]} ; (f) the $\varepsilon {\text{-}} T {\text{-}} \sigma$ diagram of the prestressed assemblies ^{[ 49 ]}

Figure 12.

The shape reconfiguration and recovery mechanism of the stability transforming double-U structures: (a) the shape reconfiguration and recovery mechanism of initial structures with fully expanded configurations; (b) the shape reconfiguration and recovery mechanism of initial structures with fully contracted configurations; (c) the shape reconfiguration and recovery mechanism of shape memory polymers