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IN THIS ISSUE

### Research Papers

J. Appl. Mech. 2019;86(5):051001-051001-9. doi:10.1115/1.4042576.

The kinetic energy of a mass moving horizontally can be completely converted into potential energy using a spring as an intermediary. The spring can be used to temporarily store some of the energy of the mass and change the direction of motion of the mass from horizontal to vertical. A nondimensional framework is used to study this problem for a point mass, first with a linear spring and then with a nonlinear spring that is an elastica. Solutions to the problems with the linear spring and elastica show many similarities and some dissimilarities. The dynamics of the point mass and elastica resemble the mechanics of a pole-vault; and therefore, a nonconservative external torque is introduced to parallel the muscle work done by vaulters. For the nonconservative system, the problem is solved for complete transformation of the kinetic energy of the mass and the work done by the external torque into potential energy of the mass. The initial velocities for the two cases, with and without the nonconservative force, are quite similar; and therefore, the maximum potential energy of the mass is higher in the presence of the nonconservative force. A realistic dimensional example is considered; the solution to the problem, despite several simplifying assumptions, is found to be similar to data of elite pole vaulters presented in the literature.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2019;86(5):051002-051002-10. doi:10.1115/1.4042575.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2019;86(5):051003-051003-13. doi:10.1115/1.4042573.

We theoretically study the electromechanical behaviors of a laminated thin-film piezoelectric semiconductor (PS) composite plate with flexural deformation. The nonlinear equations for drift currents of electrons and holes are linearized for a small carrier concentration perturbation. Following the structural theory systemized by R. D. Mindlin, a system of two-dimensional (2D) equations for the laminated thin-film PS plate, including the lowest order coupled extensional and flexural motion, are presented by expanding the displacement, potential, and the incremental concentration of electrons and holes as power series of the plate thickness. Based on the derived 2D equations, the analytical expressions of the electromechanical fields and distribution of electrons in the thin-film PS plate with an n-type ZnO layer subjected to a static bending are presented. The numerical results show that the electromechanical behaviors and piezotronic effects can be effectively controlled by the external applied force and initial concentration of carriers. The derived 2D equations and numerical results in this paper are helpful for developing piezotronic devices.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2019;86(5):051004-051004-11. doi:10.1115/1.4042574.

Surface energy outside the contact zone, which is ignored in the classical Johnson–Kendall–Roberts (JKR) model, can play an essential role in adhesion mechanics of soft bodies. In this work, based on a simple elastic foundation model for a soft elastic half space with constant surface tension, an explicit expression for the change of surface energy outside the contact zone is proposed for a soft elastic substrate indented by a rigid sphere in terms of two JKR-type variables , a), where a is the radius of the contact zone and δ is the indentation depth of the rigid sphere. The derived expression is added to the classical JKR model to achieve two explicit equations for the determination of the two JKR variables , a). The results given by the present model are demonstrated with detailed comparison with known results reported in recent literature, which verified the validity and robust accuracy of the present method. In particular, the present model confirms that the change of surface energy of the substrate can play an essential role in micro/nanoscale contact of soft materials (defined by $W/(E*R)≥0.1$, where W is the adhesive energy, $E*$ is the substrate elasticity, and R is the rigid sphere radius). The present model offers a simpler analytical method for adhesion mechanics of a rigid sphere on a soft elastic substrate when compared with several existing methods proposed in recent literature that request more substantial numerical calculations.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2019;86(5):051005-051005-7. doi:10.1115/1.4042577.

We consider the maximum value of the magnitude of transformation strain for an Eshelby inclusion set by the requirement of non-negative dissipation. The general formulation for a linear elastic solid shows that the dissipation associated with a strain transformation can be calculated as an integral over the transformed inclusion. Closed-form expressions are given for the maximum transformation strain magnitude in an isotropic linear elastic solid for both cylindrical and spherical inclusions that have undergone transformations corresponding to either a pure volume (or area) change or a pure shear. Most results presented are for transformations in an infinite solid and presume uniform material properties. Examples of the effect of a finite boundary and of differing material properties inside and outside the transformed inclusion are also given. The analytical results indicate that non-negative dissipation typically limits the transformation strain to being a constant of order unity times the critical stress at transformation divided by a relevant elastic modulus.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2019;86(5):051006-051006-11. doi:10.1115/1.4042893.

Modeling the interface between two adherents in a co-cured composite joint for a delamination analysis is always a challenge since properties and thickness of the material forming the interface are not clearly defined or well characterized. In a conventional finite element (FE) analysis using virtual crack closure technique (VCCT) based on a linear elastic fracture mechanics (LEFM) theory, adherents are assigned to share the same common nodes along their intact interface. On the other hand, an FE analysis using cohesive elements or analytical methods based on an adhesive joint model for a delamination analysis of a co-cured joint will require modeling of the interface as well as the appropriate selection of its thickness and properties. The purpose of this paper is to establish the applicability and limitation of the adhesive joint model for a delamination analysis of a co-cured composite joint. In particular, it will show that when certain requirements are met, the strain energy release rates (SERR) become independent or nearly independent of the adhesive stiffness and thickness, and thus, SERR of an adhesive joint will be the same as that for a co-cured joint. These requirements are determined from a theoretical consideration, and they can be expressed explicitly in terms of joint characteristic (or load transfer) lengths and joint physical lengths. The established requirements are further validated by numerical results for various cracked joint geometries. Finally, implication of a mode ratio obtained by the proposed adhesive joint model for a corresponding delamination crack is discussed.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2019;86(5):051007-051007-11. doi:10.1115/1.4042567.

The objective of the present work is to investigate the possibility of improving both stiffness and energy absorption in interlocking, architectured, brittle polymer blocks through hierarchical design. The interlocking mechanism allows load transfer between two different material blocks by means of contact at the mating surfaces. The contacting surfaces further act as weak interfaces that allow the polymer blocks to fail gradually under different loading conditions. Such controlled failure enhances the energy absorption of the polymer blocks but with a penalty in stiffness. Incorporating hierarchy in the form of another degree of interlocking at the weak interfaces improves stress transfer between contacting material blocks; thereby, improvement in terms of stiffness and energy absorption can be achieved. In the present work, the effects of hierarchy on the mechanical responses of a single interlocking geometry have been investigated systematically using finite element analysis (FEA) and results are validated with experiments. From finite element (FE) predictions and experiments, presence of two competing failure mechanisms have been observed in the interlock: the pullout of the interlock and brittle fracture of the polymer blocks. It is observed that the hierarchical interface improves the stiffness by restricting sliding between the contacting surfaces. However, such restriction can lead to premature fracture of the polymer blocks that eventually reduces energy absorption of the interlocking mechanism during pullout deformation. It is concluded that the combination of stiffness and energy absorption is optimal when fracture of the polymer blocks is delayed by allowing sufficient sliding at the interfaces.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2019;86(5):051008-051008-11. doi:10.1115/1.4042894.

This work examines elastic wave propagation phenomena in open-cell foams with the use of the Bloch wave method and finite element analysis. Random foam topologies are generated with the Surface Evolver and subsequently meshed with Timoshenko beam elements, creating open-cell foam models. Convergence studies on band diagrams of different domain sizes indicate that a representative volume element (RVE) consists of at least 83 cells. Wave directionality and energy flow features are investigated by extracting phase and group velocity plots. Explicit dynamic simulations are performed on finite size domains of the considered foam structure to validate the RVE results. The effect of topological disorder is studied in detail, and excellent agreement is found between the wave behavior of the random foam and that of both the regular and perturbed Kelvin foams in the low-frequency regime. In higher modes and frequencies, however, as the wavelengths become smaller, disorder has a significant effect and the deviation between regular and random foam increases significantly.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2019;86(5):051009-051009-6. doi:10.1115/1.4042919.

A rigid inclusion is embedded at a finite depth in a soft layer resting on a rigid substrate. A spherical indenter presses vertically onto the surface, deforming the matrix and displacing the inclusion. A subsurface inclusion initially near the indentation axis moves primarily downward, until an unstable lateral jump occurs to minimize the energy stored in the elastic medium. Such an instability is unique to soft materials undergoing large deformation. A two-dimensional plane-strain finite element analysis is used to simulate the 3D phenomenon.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2019;86(5):051010-051010-10. doi:10.1115/1.4042570.

Emerging stretchable piezoelectric devices have added exciting sensing and energy harvesting capabilities to wearable and implantable soft electronics. As conventional piezoelectric materials are intrinsically stiff and some are even brittle, out-of-plane wrinkled or buckled structures and in-plane serpentine ribbons have been introduced to enhance their compliance and stretchability. Among those stretchable structures, in-plane piezoelectric serpentine ribbons (PSRs) are preferred on account of their manufacturability and low profiles. To elucidate the trade-off between compliance and sensitivity of PSRs of various shapes, we herein report a theoretical framework by combining the piezoelectric plate theory with our previously developed elasticity solutions for passive serpentine ribbons without piezoelectric property. The electric displacement field and the output voltage of a freestanding but nonbuckling PSR under uniaxial stretch can be analytically solved under linear assumptions. Our analytical solutions were validated by finite element modeling (FEM) and experiments using polyvinylidene fluoride (PVDF)-based PSR. In addition to freestanding PSRs, PSRs sandwiched by polymer layers were also investigated by FEM and experiments. We found that thicker and stiffer polymers reduce the stretchability but enhance the voltage output of PSRs. When the matrix is much softer than the piezoelectric material, our analytical solutions to a freestanding PSR are also applicable to the sandwiched ones.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2019;86(5):051011-051011-8. doi:10.1115/1.4042920.

Laminated ribbons have been widely adopted for structures of flexible electronics to simultaneously achieve the electronic functions and mechanical performances. Their effective tensile stiffness and bending stiffness, which are extensively used as fundamental parameters in the mechanical analysis, are usually obtained by the plane-strain hypothesis for simplicity. However, it is found that the practical condition is usually closer to the traction free, even for the cases with a relatively large width. Here, a traction-free model is proposed to analytically obtain the effective tensile stiffness and bending stiffness of laminated ribbons, which can be used directly in the mechanical analysis of flexible electronics. The prediction of the traction-free model agrees very well with the precise result obtained by 3D finite element analysis (FEA) for the cases that are in the range of structure designs of flexible electronics. It is found that the tensile/bending stiffness of traction-free model is between the plane-stress model and plane-strain model, but is closer to the plane-stress model. The use of the plane-strain model sometimes may yield a considerable error in the mechanical analysis of flexible electronics. The parameter study shows that this model is very important for the problems with advanced materials, such as metamaterials with negative Poisson's ratio. This work provides a theoretical basis for the mechanical analysis of flexible electronics.

Commentary by Dr. Valentin Fuster

### Technical Brief

J. Appl. Mech. 2019;86(5):054501-054501-6. doi:10.1115/1.4042696.

Conventional wisdom would have it that moving mechanical systems that dissipate energy by Coulomb friction have no relationship between force and average speed. One could argue that the work done by friction is constant per unit of distance travelled, and if propulsion forces exceed friction, the net work is positive, and the system accumulates kinetic energy without bound. We present a minimalistic model for legged propulsion with slipping under Coulomb friction, scaled to parameters representative of single kilogram robots and animals. Our model, amenable to exact solutions, exhibits nearly linear (R2 > 0.96) relationships between actuator force and average speed over its entire range of parameters, and in both motion regimes, it supports. This suggests that the interactions inherent in multilegged locomotion may lead to governing equations more reminiscent of viscous friction than would be immediately obvious.

Commentary by Dr. Valentin Fuster