Mechanics of nonbuckling interconnects with prestrain for stretchable electronics

The performance of the flexibility and stretchability of flexible electronics depends on the mechanical structure design, for which a great progress has been made in past years. The use of prestrain in the substrate, causing the compression of the transferred interconnects, can provide high elastic stretchability. Recently, the nonbuckling interconnects have been designed, where thick bar replaces thin ribbon layout to yield scissor-like in-plane deformation instead of in-or out-of-plane buckling modes. The nonbuckling interconnect design achieves significantly enhanced stretchability. However,combined use of prestrain and nonbuckling interconnects has not been explored. This paper aims to study the mechanical behavior of nonbuckling interconnects bonded to the prestrained substrate analytically and numerically. It is found that larger prestrain,longer straight segment, and smaller arc radius yield smaller strain in the interconnects.On the other hand, larger prestrain can also cause larger strain in the interconnects after releasing the prestrain. Therefore, the optimization of the prestrain needs to be found to achieve favorable stretchability.     

MECHANICS ANALYSIS OF TWO-DIMENSIONALLY PRESTRAINED ELASTOMERIC THIN FILM FOR STRETCHABLE ELECTRONICS

Various methods have been developed to fabricate highly stretchable electronics. Recent studies show that over 100% two dimensional stretchability can be achieved by mesh structure of brittle functioning devices interconnected with serpentine bridges. Kim et al show that pressing down an inflated elastomeric thin film during transfer printing introduces two di- mensional prestrain, and therefore further improves the system stretchability. This paper gives a theoretical study of this process, through both analytical and numerical approaches. Simple analytical solutions are obtained for meridional and circumferential strains in the thin film, as well as the maximum strain in device islands, which all agree reasonably well with finite element analysis.     

Effect of interfacial stiffness on the stretchability of metal/elastomer bilayers under in-plane biaxial tension

Flexible electronic devices are often subjected to large and repeated deformation, so that their functional components such as metal interconnects need to sustain strains up to tens of percent, which is far beyond the intrinsic deformability of metal materials(～1%). To meet the stringent requirements of flexible electronics, metal/elastomer bilayers, a stretchable structure that consists of a metal film adhered to a stretchable elastomer substrate, have been developed to improve the stretch capability of metal interconnects. Previous studies have predicted that the metal/elastomer bilayers are much more stretchable than freestanding metal films. However, these investigations usually assume perfect bonding between the metal and elastomer layers. In this work, the effect of the metal/elastomer interface with a finite interfacial stiffness on the stretchability of bilayer structures is analyzed. The results show that the assumption of perfect interface(with infinite interfacial stiffness) may lead to an overestimation of the stretchability of bilayer structures. It is also demonstrated that increased adhesion between the metal and elastomer layers can enhance the stretchability of the metal layer.     

A generalized anisotropic hardening rule based on the Mroz multi-yield-surface model for pressure insensitive and sensitive materials

In this paper, a generalized anisotropic hardening rule based on the Mroz multi-yield-surface model for pressure insensitive and sensitive materials is derived. The evolution equation for the active yield surface with reference to the memory yield surface is obtained by considering the continuous expansion of the active yield surface during the unloading/reloading process. The incremental constitutive relation based on the associated flow rule is then derived for a general yield function for pressure insensitive and sensitive materials. Detailed incremental constitutive relations for materials based on the Mises yield function, the Hill quadratic anisotropic yield function and the Drucker–Prager yield function are derived as the special cases. The closed-form solutions for one-dimensional stress–plastic strain curves are also derived and plotted for materials under cyclic loading conditions based on the three yield functions. In addition, the closed-form solutions for one-dimensional stress–plastic strain curves for materials based on the isotropic Cazacu–Barlat yield function under cyclic loading conditions are summarized and presented. For materials based on the Mises and the Hill anisotropic yield functions, the stress–plastic strain curves show closed hysteresis loops under uniaxial cyclic loading conditions and the Masing hypothesis is applicable. For materials based on the Drucker–Prager and Cazacu–Barlat yield functions, the stress–plastic strain curves do not close and show the ratcheting effect under uniaxial cyclic loading conditions. The ratcheting effect is due to different strain ranges for a given stress range for the unloading and reloading processes. With these closed-form solutions, the important effects of the yield surface geometry on the cyclic plastic behavior due to the pressure-sensitive yielding or the unsymmetric behavior in tension and compression can be shown unambiguously. The closed form solutions for the Drucker–Prager and Cazacu–Barlat yield functions with the associated flow rule also suggest that a more general anisotropic hardening theory needs to be developed to address the ratcheting effects for a given stress range.     

柔性电子封装结构中夹杂对延展性的影响分析

以岛-桥型柔性电子封装结构为研究对象,从封装材料中夹杂的刚度、位置、封装方式三个方面探讨了夹杂对柔性电子结构延展性的影响。有限元分析结果表明:随着夹杂刚度的增大,桥的最大主应变增大,整体结构最大延伸量可减小30%;夹杂埋藏位置越深,桥顶局部的整体应变水平越大,最大延伸量可减小20%;相对"硬"封装情形,相同的夹杂对"软"封装中桥的最大延伸量的影响更严重。本文所得结论对于柔性电子器件结构的设计和材料选用具有参考和指导意义。     

Laser-induced porous graphene on Polyimide/PDMS composites and its kirigami-inspired strain sensor

The laser-induced porous graphene(LIG) prepared in a straightforward fabrication method is presented,and its applications in stretchable strain sensors to detect the applied strain are also explored. The LIG formed on the polyimide/polydimethylsiloxane(PI/PDMS) composite exhibits a naturally high stretchability(over 30%), bypassing the transfer printing process compared to the one prepared by laser scribing on PI films. The PI/PDMS composite with LIG shows tunable mechanical and electronic performances with different PI particle concentrations in PDMS. The good cyclic stability and almost linear response of the prepared LIG's resistance with respect to tensile strain provide its access to wearable electronics. To improve the PDMS/PI composite stretchability, we designed and optimized a kirigami-inspired strain sensor with LIG on the top surface, dramatically increasing the maximum strain value that in linear response to applied strain from 3% to 79%.     

可延展结构的设计及力学研究新进展

可延展柔性电子器件克服了传统无机电子器件脆、硬的缺点,在保持优异电学性能的同时,以其优秀的可延展性极大拓展了微电子器件的应用范围,备受国内外学术界和电子产业界瞩目. 无机电子器件的可延展柔性化主要通过力学结构设计的方法实现,本文针对近两年具有代表性的三种可延展柔性结构设计,包括分形互联岛桥结构、折纸结构和剪纸结构,简要综述了这些结构的力学研究进展,彰显了力学在可延展柔性电子器件发展中的重要作用,并展望了未来的发展方向.     

RVE-based studies on the coupled effects of void size and void shape on yield behavior and void growth at micron scales

The present paper extends the Gurson and GLD models [Gurson, A.L., 1977. Continuum theory of ductile rupture by void nucleation and growth, Part I—yield criteria and flow rules for porous ductile media. J. Mech. Phys. Solids 99, 2–15; Gologanu, M., Leblond, J.B., Devaux, J., 1993. Approximate models for ductile metals containing non-spherical voids—case of axisymmetric prolate ellipsoidal cavities. J. Mech. Phys. Solids 41, 1723–1754; Gologanu, M., Leblond, J.B., Devaux, J., 1994. Approximate models for ductile metals containing non-spherical voids—case of axisymmetric oblate ellipsoidal cavities. J. Eng. Mater. Technol. 116, 290–297] to involve the coupled effects of void size and void shape on the macroscopic yield behavior of non-linear porous materials and on the void growth. A spheroidal representative volume element (RVE) under a remote axisymmetric homogenous strain boundary condition is carefully analyzed. A wide range of void aspect ratios covering the oblate spheroidal, spherical and prolate spheroidal void are taken into account to reflect the shape effect. The size effect is captured by the Fleck–Hutchinson phenomenological strain gradient plasticity theory [Fleck, N.A., Hutchinson, J.W., 1997. Strain gradient plasticity. In: Hutchinson, J.W., Wu, T.Y. (Eds.), Advance in Applied Mechanics, vol. 33, Academic Press, New York, pp. 295–361]. A new size-dependent damage model like the Gurson and GLD models is developed based on the traditional minimum plasticity potential principle. Consequently, the coupled effects of void size and void shape on yield behavior of porous materials and void growth are discussed in detail. The results indicate that the void shape effect on the yield behavior of porous materials and on the void growth can be modified dramatically by the void size effect and vice versa. The applied stress triaxiality plays an important role in these coupled effects. Moreover, there exists a cut-off void radius r_c, which depends only on the intrinsic length l₁ associated with the stretch strain gradient. Voids of effective radius smaller than the critical radius r_c are less susceptible to grow. These findings are helpful to our further understanding to some impenetrable micrographs of the ductile fracture surfaces.     

Controlled buckling of thin film on elastomeric substrate in large deformation

Electronic systems with large stretchability have many applications. A precisely controlled buckling strategy to increase the stretchability has been demonstrated by combining lithographically patterned surface bonding chemistry and a buckling process. The buckled geometry was assumed to have a sinusoidal form, which may result in errors to determine the strains in the film. A theoretical model is presented in this letter to study the mechanics of this type of thin film/substrate system by discarding the assumption of sinusoidal buckling geometry. It is shown that the previous model overestimates the deflection and curvature in the thin film. The results from the model agree well with finite element simulations and therefore provide design guidelines in many applications ranging from stretchable electronics to micro/nano scale surface patterning and precision metrology.     

A nonlinear mechanics model of bio-inspired hierarchical lattice materials consisting of horseshoe microstructures

Development of advanced synthetic materials that can mimic the mechanical properties of non-mineralized soft biological materials has important implications in a wide range of technologies. Hierarchical lattice materials constructed with horseshoe microstructures belong to this class of bio-inspired synthetic materials, where the mechanical responses can be tailored to match the nonlinear J-shaped stress–strain curves of human skins. The underlying relations between the J-shaped stress–strain curves and their microstructure geometry are essential in designing such systems for targeted applications. Here, a theoretical model of this type of hierarchical lattice material is developed by combining a finite deformation constitutive relation of the building block (i.e., horseshoe microstructure), with the analyses of equilibrium and deformation compatibility in the periodical lattices. The nonlinear J-shaped stress–strain curves and Poisson ratios predicted by this model agree very well with results of finite element analyses (FEA) and experiment. Based on this model, analytic solutions were obtained for some key mechanical quantities, e.g., elastic modulus, Poisson ratio, peak modulus, and critical strain around which the tangent modulus increases rapidly. A negative Poisson effect is revealed in the hierarchical lattice with triangular topology, as opposed to a positive Poisson effect in hierarchical lattices with Kagome and honeycomb topologies. The lattice topology is also found to have a strong influence on the stress–strain curve. For the three isotropic lattice topologies (triangular, Kagome and honeycomb), the hierarchical triangular lattice material renders the sharpest transition in the stress–strain curve and relative high stretchability, given the same porosity and arc angle of horseshoe microstructure. Furthermore, a demonstrative example illustrates the utility of the developed model in the rapid optimization of hierarchical lattice materials for reproducing the desired stress–strain curves of human skins. This study provides theoretical guidelines for future designs of soft bio-mimetic materials with hierarchical lattice constructions.     

Modeling and simulation of a compact push–pull rubber actuator energized by an outer coil of shape memory wire

This paper introduces the concept and develops the theory for a push–pull actuator made by winding a thin shape memory wire on a solid rubber cylinder. The incompressibility of the rubber converts the thermomechanical contraction of the heated wire in twice as much axial strain available in the rubber core. The intrinsic elastic backup provided by the core allows push–pull action of the device. Based on the assumption of both material and geometric linearity, a simple theoretical model is developed, which culminates in a simple closed-form equation for the output stroke of the actuator. The theoretical predictions closely agree with refined finite element simulations, anticipating a net stroke of about 5–6% of the overall actuator length, depending on the outer load applied. The working principle of the actuator and the accuracy of the model are validated by tests on a proof-of-concept prototype.     

A study of microbend test by strain gradient plasticity

Metallic materials display strong size effect when the characteristic length associated with plastic deformation is on the order of microns. This size effect cannot be explained by classical plasticity theories since their constitutive relations do not have an intrinsic material length. Strain gradient plasticity has been developed to extend continuum plasticity to the micron or submicron regime. One major issue in strain gradient plasticity is the determination of the intrinsic material length that scales with strain gradients, and several microbend test specimens have been designed for this purpose. We have studied different microbend test specimens using the theory of strain gradient plasticity. The pure bending specimen, cantilever beam, and the microbend test specimen developed by Stolken and Evans (Stolken, J.S., Evans, A.G., 1998. A microbend test method for measuring the plasticity length scale Acta Mater. 46, 5109–5115) are found suitable for the determination of intrinsic material length in strain gradient plasticity. However, the double cantilever beam (both ends clamped) is unsuitable since its deformation is dominated by axial stretching. The strain gradient effects significantly increase the bending stiffness of a microbend test specimen. The deflection of a 10-μm thick beam is only a few percent of that estimated by classical plasticity.     

Spherical indentation of freestanding circular thin films in the membrane regime

We present theoretical and experimental results to describe the mechanics of indentation of a clamped circular membrane with a frictionless spherical indenter. Analytical expressions and numerical simulations are presented for the relationships between contact radius, finite indentation strains (and stresses), pre-stretch, loads and deflection. These closed-form solutions are contrasted with point-load models that neglect the contact size (i.e. classical Schwerin-type solutions), and lead to important differences in the indentation strain and load-deflection response. The accuracy of these closed form expressions is illustrated by comparisons with detailed numerical results and experiments on thin elastomer films. We show that the closed-form solutions can be used to extract mechanical properties from indentation testing of freestanding films, with important implications for developing new tests on nanoscale films and/or compliant materials such as polymers and biological substances.     

DAMAGE MODEL FOR HARDENING AND SOFTENING MATERIAL IN GENERAL STATE OF STRESS

The present paper is aimed to simulate progression of damage,hardeningand softening response in brittle materials such as concrete or rock in general state ofstress.Similar shape of surfaces for yield,failure and damage progressing areavailable,and softening strain is treated as plasticity.Then,the proposed model isapplied to solving several boundary value problems.     

任意形状热夹杂位移场的三角形单元离散算法

平面夹杂模型在纤维增强型复合材料中有广泛应用.复合材料内部通常含有不规则形状夹杂,而夹杂物的存在能严重影响材料的机械力学性能,往往导致应力集中及裂纹萌生等失效先兆.先前关于多边形夹杂的研究大多数关注受均匀本征应变下的应力/应变解,而对位移的分析较少. 基于格林函数方法和围道积分,本文给出了平面热夹杂边界线单元的封闭解析解,可方便应用于受任意分布本征应变的任意形状平面热夹杂位移场的数值计算.当夹杂受均匀本征应变时, 只需将该夹杂边界进行一维离散,因而本文方法可直接得出受均匀分布热本征应变的任意多边形夹杂位移场的封闭解析解.当夹杂区域存在非均匀分布本征应变时,可将该区域划分为足够小的三角形单元进行数值计算. 众所周知,应力应变场在多边形夹杂顶点处具有奇异性,容易导致数值计算上的处理困难及相应的数值稳定性问题; 然而本文工作表明,在多边形顶点处位移场是连续有界的, 因而数值稳定性较好.本文算法可以便捷高效地通过计算机编程实现. 文中给出的验证算例,均体现了本文离散方法的高精度、以及计算编程的鲁棒性.      

Post-buckling analysis for the precisely controlled buckling of thin film encapsulated by elastomeric substrates

The precisely controlled buckling of stiff thin films (e.g., Si or GaAs nano ribbons) on the patterned surface of elastomeric substrate (e.g., poly(dimethylsiloxane) (PDMS)) with periodic inactivated and activated regions was designed by Sun et al. [Sun, Y., Choi, W.M., Jiang, H., Huang, Y.Y., Rogers, J.A., 2006. Controlled buckling of semiconductor nanoribbons for stretchable electronics. Nature Nanotechnology 1, 201–207] for important applications of stretchable electronics. We have developed a post-buckling model based on the energy method for the precisely controlled buckling to study the system stretchability. The results agree with Sun et al.’s (2006) experiments without any parameter fitting, and the system can reach 120% stretchability.     

Reprint of “Post-buckling analysis for the precisely controlled buckling of thin film encapsulated by elastomeric substrates” [In. J. Solids Struct. 45 (2008) 2014–2023]

The precisely controlled buckling of stiff thin films (e.g., Si or GaAs nano ribbons) on the patterned surface of elastomeric substrate (e.g., poly(dimethylsiloxane) (PDMS)) with periodic inactivated and activated regions was designed by Sun et al. [Sun, Y., Choi, W.M., Jiang, H., Huang, Y.Y., Rogers, J.A., 2006. Controlled buckling of semiconductor nanoribbons for stretchable electronics. Nature Nanotechnology 1, 201–207] for important applications of stretchable electronics. We have developed a post-buckling model based on the energy method for the precisely controlled buckling to study the system stretchability. The results agree with Sun et al.’s (2006) experiments without any parameter fitting, and the system can reach 120% stretchability.     

An endochronic theory accounted for deformation induced anisotropy

An anisotropic quadratic form of plastic strain increment is used to define the intrinsic time in the endochronic theory of plasticity. Based on this new definition, a yield function can be derived. This new version of endochronic theory can describe the expansion, translation, rotation, and distortion of the yield surface. While the initial yielding is in the form of the Mises yield criterion, the distortion of subsequent yield surfaces is expressed by the compression or stretching of the Mises yield surface. The effect of sharp front and blunt rear of the yield surface is considered to be of secondary importance and neglected in the interest of keeping the equations simple. This idealization will not much affect the prediction power of the model, because the plastic strain increment is in the radial direction emanating from the center of the current yield surface and is not normal to the current yield surface. In this theory, the plastic deformation is thus not sensitive to the exact shape of the yield surface. It has been shown that the proposed theory is capable of describing the experimental results of three different metals considered. The test series investigated include several different paths of prestress.     

Conductive hydrogels incorporating carbon nanoparticles: A review of synthesis,performance and applications

As one of the most rapidly expanding materials, hydrogels have gained increasing attention in a variety of fields due to their biocompatibility, degradability and hydrophilic properties, as well as their remarkable adhesion and stretchability to adapt to different surfaces. Hydrogels combined with carbon-based materials possess enhanced properties and new functionalities, in particular, conductive hydrogels have become a new area of research in the field of materials science. This review aims to provide a comprehensive overview and up-to-date examination of recent developments in the synthesis, properties and applications of conductive hydrogels incorporating several typical carbon nanoparticles such as carbon nanotubes, graphene, carbon dots and carbon nanofibers. We summarize key techniques and mechanisms for synthesizing various composite hydrogels with exceptional properties, and represented applications such as wearable sensors, temperature sensors, supercapacitors and human-computer interaction reported recently. The mechanical, electrical and sensing properties of carbon nanoparticles conductive hydrogels are thoroughly analyzed to disclose the role of carbon nanoparticles in these hydrogels and key factors in the microstructure. Finally, future development of conductive hydrogels based on carbon nanoparticles is discussed including the challenges and possible solutions in terms of microstructure optimization, mechanical and other properties, and promising applications in wearable electronics and multifunctional materials.