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Research Papers

J. Appl. Mech. 2017;84(5):051001-051001-15. doi:10.1115/1.4036018.

Attributed to its significance in a wide range of practical applications, the post-buckling behavior of a beam with lateral constraints has drawn much attention in the last few decades. Despite the fact that, in reality, the lateral constraints are often flexible or deformable, vast majority of studies have considered fixed and rigid lateral constraints. In this paper, we make a step toward bridging this gap by studying the post-buckling behavior of a planar beam that is laterally constrained by a deformable wall. Unfortunately, the interaction with a compliant wall prevents derivation of closed-form analytical solutions. Nevertheless, careful examination of the governing equations of a simplified model reveals general properties of the solution, and let us identify the key features that govern the behavior. Specifically, we construct universal “solution maps” that do not depend on the mode number and enable simple and easy prediction of the contact conditions and of the mode-switching force (the force at which the system undergoes instantaneous transition from one equilibrium configuration (or mode) to another). The predictions of the mathematical model are validated against finite element (FE) simulations.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2017;84(5):051002-051002-8. doi:10.1115/1.4036094.

A cantilever beam is subjected to both lateral force and compression under gravity. By taking into account the potential energy variation of the system, we develop a theoretical result that greatly simplifies the bending vibration frequency calculation in agreement with the experiments.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2017;84(5):051003-051003-11. doi:10.1115/1.4036193.

Two types of tubular dielectric elastomers (DE) torsional actuators are studied in this work, which are, respectively, reinforced by a family and two families of helical inextensible fibers. When subject to a radial electric field, torsional deformation will be induced in the DE actuators due to the constraint of inextensible fibers. By conducting finite deformation analysis with the principal axis approach and adopting appropriate constitutive equations, simple analytical solutions are obtained for the considered DE actuators. Furthermore, the effects of material parameters and the fiber angles as well as externally applied axial force and twist moment on the voltage-induced torsional behaviors of the two DE actuators are discussed in order to explore their maximum torsional actuation capability. The concept design presented here provides an effective approach for achieving large torsional deformation, and the developed model and revealed results will aid the design and fabrication of soft actuators and soft robots.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2017;84(5):051004-051004-7. doi:10.1115/1.4036192.

Hydraulic fracturing (fracking) technology in gas or oil shale engineering is highly developed last decades, but the knowledge of the actual fracking process is mostly empirical and makes mechanicians and petroleum engineers wonder: why fracking works? (Bažant et al., 2014, “Why Fracking Works,” ASME J. Appl. Mech., 81(10), p. 101010) Two crucial issues should be considered in order to answer this question, which are fracture propagation condition and multiscale fracture network formation in shale. Multiple clusters of fractures initiate from the horizontal wellbore and several major fractures propagate simultaneously during one fracking stage. The simulation-based unitary fracking condition is proposed in this paper by extended finite element method (XFEM) to drive fracture clusters growing or arresting dominated by inlet fluid flux and stress intensity factors. However, there are millions of smeared fractures in the formation, which compose a multiscale fracture network beyond the computation capacity by XFEM. So, another simulation-based multiscale self-consistent fracture network model is proposed bridging the multiscale smeared fractures. The purpose of this work is to predict pressure on mouth of well or fluid flux in the wellbore based on the required minimum fracture spacing scale, reservoir pressure, and proppant size, as well as other given conditions. Examples are provided to verify the theoretic and numerical models.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2017;84(5):051005-051005-12. doi:10.1115/1.4036113.

A theoretical model of polyelectrolyte gels is presented to study continuous and discontinuous volume phase transitions induced by changing salt concentration in the external solution. Phase diagrams are constructed in terms of the polymer–solvent interaction parameters, external salt concentration, and concentration of fixed charges. Comparisons with previous experiments for an ionized acrylamide gel in mixed water–acetone solvents are made with good quantitative agreement for a monovalent salt (NaCl) but fair qualitative agreement for a divalent salt (MgCl2), using a simple set of parameters for both cases. The effective polymer–solvent interactions vary with the volume fraction of acetone in the mixed solvent, leading to either continuous or discontinuous volume transitions. The presence of divalent ions (Mg2+) in addition to monovalent ions in the external solution reduces the critical salt concentration for the discontinuous transition by several orders of magnitude. Moreover, a secondary continuous transition is predicted between two highly swollen states for the case of a divalent salt. The present model may be further extended to study volume phase transitions of polyelectrolyte gels in response to other stimuli such as temperature, pH and electrical field.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2017;84(5):051006-051006-8. 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.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2017;84(5):051007-051007-8. doi:10.1115/1.4036214.

The conventional contact mechanics does not account for surface tension; however, it is important for micro- or nanosized contacts. In the present paper, the influences of surface tension on the indentations of an elastic half-space by a rigid sphere, cone, and flat-ended cylinder are investigated, 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 first. 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 and fitting the direct numerical results, the dependence of load on contact radius or indent depth for general case can be given explicitly. The results incorporate the contribution of surface tension in contact mechanics and are helpful to understand contact phenomena at micro- and nanoscale.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2017;84(5):051008-051008-9. 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. 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 is discussed, and the procedures in laboratory tests to obtain the constants are suggested.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2017;84(5):051009-051009-6. 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 multifield bulge-test instrument developed by our group, the electric field-tunable elastic modulus of ferroelectric films has been studied experimentally. A butterfly change in the Young's modulus of lead titanate zirconate (PZT) film under biaxial tensile stress state with electric field has been discovered for the first time. Based on the phase field model, an electromechanical 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 the Young's modulus is proportional to the variation of volume fraction of 90-deg domain switching under a pure tensile strain. It is the constraint of 90-deg domain switching by the electric field that contributes to the butterfly change in the elastic modulus.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2017;84(5):051010-051010-10. doi:10.1115/1.4036325.

Wrinkling of thin films is an easy-to-implement and low-cost technique to fabricate stretch-tunable periodic micro and nanoscale structures. However, the tunability of such structures is often limited by the emergence of an undesirable period-doubled mode at high strains. Predictively tuning the onset strain for period doubling via existing techniques requires one to have extensive knowledge about the nonlinear pattern formation behavior. Herein, a geometric prepatterning-based technique is introduced that can be implemented even with limited system knowledge to predictively delay period doubling. The technique comprises prepatterning the film/base bilayer with a sinusoidal pattern that has the same period as the natural period of the system. This technique has been verified via physical and computational experiments on the polydimethylsiloxane (PDMS)/glass bilayer system. It is observed that the onset strain can be increased from the typical value of 20% for flat films to greater than 30% with a modest prepattern aspect ratio (2·amplitude/period) of 0.15. In addition, finite element simulations reveal that (i) the onset strain increases with increasing prepattern amplitude and (ii) the delaying effect can be captured entirely by the prepattern geometry. Therefore, one can implement this technique even with limited system knowledge, such as material properties or film thickness, by simply replicating pre-existing wrinkled patterns to generate prepatterned bilayers. Thus, geometric prepatterning is a practical scheme to increase the operating range of stretch-tunable wrinkle-based devices by at least 50%.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2017;84(5):051011-051011-9. 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, the plane strain modulus ratio between the film and the substrate, the film thickness to wrinkle wavelength ratio, and the 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 the 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, biomedical engineering, soft materials, and devices.

Commentary by Dr. Valentin Fuster

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