Accepted Manuscripts

Hidenori Murakami
J. Appl. Mech   doi: 10.1115/1.4036317
Utilizing the kinematics, presented in the part-I paper, an active large-deformation beam model for slender, flexible or soft robots is developed from the d'Alembert principle of virtual work, which is derived for three-dimensional elastic solids from Hamilton's principle. This derivation is accomplished by refining the definition of the Cauchy stress tensor as a vector-valued 2-form to exploit advanced geometrical operations available for differential forms. From the three-dimensional principle of virtual work, both the beam principle of virtual work and beam equations of motion with consistent boundary conditions are derived, adopting the kinematic assumption of rigid cross-sections of a deforming beam. In the derivation of the beam model, √Člie Cartan's moving frame method is utilized. The resulting large-deformation beam equations apply to both passive and active beams. The beam equations are validated with the previously reported results expressed in vector form. To transform passive beams to active beams, constitutive relations for internal actuation are presented in rate-form. Then, the resulting three-dimensional beam models are reduced to an active planar beam model. To illustrate the deformation due to internal actuation, an active Timoshenko-beam model is derived by linearizing the nonlinear planar equations. For an active, simply-supported Timoshenko-beam, the analytical solution is presented. Finally, a linear locomotion of a soft inchworm-inspired robot is simulated by implementing active -beam elements in a nonlinear finite element code.
TOPICS: Kinematics, Deformation, Solids, Robots, Cross section (Physics), Equations of motion, D'Alembert's principle, Hamilton's principle, Virtual work principle, Constitutive equations, Finite element analysis, Boundary-value problems, Stress tensors
Lincong Chen, Jun Liu and Jian-Qiao Sun
J. Appl. Mech   doi: 10.1115/1.4036307
There has been no significant progress in developing new techniques for obtaining exact stationary probability density functions (PDFs) of nonlinear stochastic systems since the development of the method of generalized probability potential in 1990s. In this paper, a general technique is proposed for constructing approximate stationary PDF solutions of single-degree-of-freedom (SDOF) nonlinear systems under external and parametric Gaussian white noise excitations. This technique consists of two novel components. The first one is the introduction of new trial solutions for the reduced Fokker-Planck-Kolmogorov (FPK) equation. The second one is the iterative method of weighted residuals to determine the unknown parameters in the trial solution. Numerical results of four challenging examples show that the proposed technique will converge to the exact solutions if they exist, or a highly accurate solution with a relatively low computational effort. Furthermore, the proposed technique can be extended to multi-degree-of-freedom (MDOF) systems.
TOPICS: Statistical distributions, Stochastic systems, Probability, Density, Nonlinear systems, Iterative methods, White noise, Excitation
Hidenori Murakami
J. Appl. Mech   doi: 10.1115/1.4036308
In order to develop an active nonlinear beam model, the beam's kinematics is examined in this part-I paper, by employing the kinematic assumption of a rigid cross section during deformation. As a mathematical tool, the moving frame method, developed by √Člie Cartan (1869-1951) on differentiable manifolds, is utilized by treating a beam as a frame bundle on a deforming centroidal curve. As a result, three new integrability conditions are obtained, which play critical roles in the derivation of beam equations of motion. These integrability conditions enable the derivation of beam models in the part-II paper, starting from the three-dimensional Hamilton's principle and the d'Alembert principle of virtual work. To illustrate the critical role played by the integrability conditions, the variation of kinetic energy is computed. Finally, the reconstruction scheme for rotation matrices for given angular velocity at each time is presented.
TOPICS: Kinematics, Rotation, Deformation, Kinetic energy, Equations of motion, D'Alembert's principle, Hamilton's principle, Manifolds
Honglong Zhang, Zejun Yu, Yongmao Pei and Daining Fang
J. Appl. Mech   doi: 10.1115/1.4036298
The field-dependent Young's modulus shows a promising application in the design and miniaturization of phononic crystals, tunable mechanical resonators, interdigital transducers, etc. With the multi-field bulge-test instrument developed by our group, the electric field-tunable elastic modulus of ferroelectric films has been studied experimentally. A butterfly change in Young's modulus of PZT film under biaxial tensile stress state with electric field has been discovered for the first time. Based on phase field model, an electro-mechanical coupling model is constructed and a case of PZT ferroelectric film subjected to a vertical electric field and horizontal tensile strains is simulated. The numerical results show that the change in Young's modulus is proportional to the variation of volume fraction of 90o domain switching under a pure tensile strain. It is the constraint of 90o domain switching by electric field that contributes to the butterfly change in elastic modulus.
TOPICS: Electric fields, Design, Instrumentation, Transducers, Elastic moduli, Phononic crystals, Tension, Young's modulus
Yilun Liu, Mengjie Li, Jingran Liu and Xi Chen
J. Appl. Mech   doi: 10.1115/1.4036256
In this work, the surface wrinkle modulation of the film/substrate system caused by eigenstrain in the film is studied. A theoretical model is proposed which shows the change of the wrinkle amplitude is completely determined by four dimensionless parameters, i.e. the eigenstrain in the film, plane strain modulus ratio between the film and substrate, film thickness to wrinkle wavelength ratio, and initial wrinkle amplitude to wavelength ratio. The surface wrinkle amplitude becomes smaller (even almost flat) for the contraction eigenstrain in the film, while for the expansion eigenstrain it becomes larger. If the expansion eigenstrain exceeds a critical value, secondary wrinkling on top of the existing one is observed for some cases. In general, the deformation diagram of the wrinkled film/substrate system can be divided into three regions, i.e. the change of surface wrinkle amplitude, the irregular wrinkling, and secondary wrinkling, governed by the four parameters above. Parallel finite element method (FEM) simulations are carried out which have good agreement with the theoretical predictions. The findings may be useful to guide the design and performance of stretchable electronics, cosmetic products, soft materials and devices.
TOPICS: Deformation, Wavelength, Simulation, Finite element methods, Design, Engineering simulation, Film thickness, Plane strain, Electronics
Technical Brief  
Yuyan Gao, Yuhang Li, Rui Li and Jizhou Song
J. Appl. Mech   doi: 10.1115/1.4036257
A recently developed transfer printing technique: laser-driven non-contact micro-transfer printing, which involves laser-induced heating to initiate the separation at the interface between the elastomeric stamp (e.g., PDMS) and hard micro-/nano-materials (e.g., Si chip), is valuable to develop advanced engineering systems such as stretchable and curvilinear electronics. The previous thermo-mechanical model has identified the delamination mechanism successfully. However, that model is not valid for small-size Si chip because the size effect is ignored for simplification in the derivation of the crack tip energy release rate. This paper establishes an accurate interfacial fracture mechanics model accounting for the size effect of the Si chip. The analytical predictions agree well with finite element analysis. This accurate model may serve as the theoretical basis for system optimization, especially for determining the optimal condition in the laser-driven non-contact micro-transfer printing.
TOPICS: Lasers, Printing, Silicon chips, Size effect, Heating, Delamination, Electronics, Accounting, Fracture mechanics, Separation (Technology), Plasma desorption mass spectrometry, Fracture (Materials), Engineering systems and industry applications, Finite element analysis, Nanomaterials, Optimization
Jianmin Long, Yue Ding, Weike Yuan, Wen Chen and Gangfeng Wang
J. Appl. Mech   doi: 10.1115/1.4036214
The conventional contact mechanics does not account for surface tension, however which gets important for micro- or nano-sized contacts. In the present paper, the influences of surface tension on the indentations of an elastic half space by a rigid sphere, a cone and a flat-ended cylinder are investigated respectively, and the corresponding singular integral equations are formulated. Due to the complicated structure of the integral kernel, it is difficult to obtain their analytical solutions. By using the Gauss-Chebyshev quadrature formula, the integral equations are solved numerically firstly. Then, for each indenter, the analytical solutions of two limit cases considering only the bulk elasticity or surface tension are presented. It is interesting to find that, through a simple combination of the solutions of two limit cases, the dependence of load on contact radius or indent depth for general case can be given explicitly. The advanced analytical relations agree well with the numerical results by solving the singular integral equations. The results incorporate the contribution of surface tension in contact mechanics, and are helpful to understand contact phenomena at micro- and nano-scale.
TOPICS: Surface tension, Solids, Integral equations, Contact mechanics, Nanoscale phenomena, Cylinders, Elastic half space, Stress, Elasticity
Yue Gao, Zhanli Liu, Zhuo Zhuang and Keh-Chih Hwang
J. Appl. Mech   doi: 10.1115/1.4036194
The anisotropic poroelastic constitutive model is reexamined in this article. The assumptions and conclusions of previous works, i.e. Thompson and Willis, and Cheng are compared and clarified. The micromechanics of poroelasticity is discussed by dividing the medium into connected fluid part and solid skeleton part. And the latter includes, in turn, solid part and, possibly, disconnected fluid part, i.e. fluid islands, therefore the solid skeleton part is inhomogeneous. The constitutive model is complicated both in the whole medium and in the solid skeleton because of their inhomogeneity, but the formulations are simplified successfully by introducing a new material constant which is defined differently by Cheng and by Thompson and Willis. All the unmeasurable micromechanical material constants are lumped together in this constant. Four levels of assumptions used in poroelasticity are demonstrated, and with the least assumptions, the constitutive model is formulated. The number of independent material constants are discussed, and the procedures in laboratory tests to obtain the constants are suggested.
TOPICS: Anisotropy, Fluids, Constitutive equations, Micromechanics (Engineering)

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