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### Research Papers

J. Appl. Mech. 2017;85(3):031001-031001-10. doi:10.1115/1.4038718.

Guided by the experimental observations in the literature, this paper discusses two possible modes of defect growth in soft solids for which the size-dependent fracture mechanics is not always applicable. One is omni-directional growth, in which the cavity expands irreversibly in all directions; and the other is localized cracking along a plane. A characteristic material length is introduced, which may shed light on the dominant growth mode for defects of different sizes. To help determine the associated material properties from experimental measurement, the driving force of defect growth as a function of the remote load is calculated for both modes accordingly. Consequently, one may relate the measured critical load to the critical driving force and eventually to the associated material parameters. For comprehensiveness, the calculations here cover a class of hyperelastic materials. As an application of the proposed hypothesis, the experimental results (Cristiano et al., 2010, “An Experimental Investigation of Fracture by Cavitation of Model Elastomeric Networks,” J. Polym. Sci. Part B: Polym. Phys., 48(13), pp. 1409–1422) from two polymers with long and short chain elastomeric network are examined. The two polymers seem to be susceptible to either of the two dominating modes, respectively. The results are interpreted, and the material characteristic length and other growth parameters are determined.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2018;85(3):031002-031002-10. doi:10.1115/1.4038749.

A comprehensive study is reported herein for the evaluation of Lagrangian functions for linear systems possessing symmetric or nonsymmetric coefficient matrices. Contrary to popular beliefs, it is shown that many coupled linear systems do not admit Lagrangian functions. In addition, Lagrangian functions generally cannot be determined by system decoupling unless further restriction such as classical damping is assumed. However, a scalar function that plays the role of a Lagrangian function can be determined for any linear system by decoupling. This generalized Lagrangian function produces the equations of motion and it contains information on system properties, yet it satisfies a modified version of the Euler–Lagrange equations. Subject to this interpretation, a solution to the inverse problem of linear Lagrangian dynamics is provided.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2018;85(3):031003-031003-10. doi:10.1115/1.4038719.

Penny-shaped fluid-driven cracks are often detected in many fluid–solid interaction problems. We study the combined effect of pressure and shear stress on the crack propagation in an impermeable elastic full space. Boundary integral equations are presented, by using the integral transform method, for a penny-shaped crack under normal and shear stresses. The crack propagation criterion of stress intensity factor is examined with the strain energy release rate. Dominant regimes are obtained by using a scaling analysis. Asymptotic solution of the toughness-dominant regime is derived to show the effect of shear stress on the crack opening, crack length, and pressure distribution. The results indicate that a singular shear stress can dominate the asymptotic property of the stress field near the crack tip, and the stress intensity factor cannot be calculated even though the energy release rate is finite. Shear stress leads to a smaller crack opening, a longer crack, and a slightly larger wellbore pressure. A novel dominant-regime transition between shear stress and pressure is found. Unstable crack propagation occurs in the shear stress-dominant regime. This study may help in understanding crack problems under symmetrical loads and modeling fluid–solid interactions at the crack surfaces.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2018;85(3):031004-031004-16. doi:10.1115/1.4038770.

Soft electroactive materials can undergo large deformation subjected to either mechanical or electrical stimulus, and hence, they can be excellent candidates for designing extremely flexible and adaptive structures and devices. This paper proposes a simple one-dimensional soft phononic crystal (PC) cylinder made of dielectric elastomer (DE) to show how large deformation and electric field can be used jointly to tune the longitudinal waves propagating in the PC. A series of soft electrodes, which are mechanically negligible, are placed periodically along the DE cylinder, and hence, the material can be regarded as uniform in the undeformed state. This is also the case for the uniformly prestretched state induced by a static axial force only. The effective periodicity of the structure is then achieved through two loading paths, i.e., by maintaining the longitudinal stretch and applying an electric voltage over any two neighboring electrodes or by holding the axial force and applying the voltage. All physical field variables for both configurations can be determined exactly based on the nonlinear theory of electroelasticity. An infinitesimal wave motion is further superimposed on the predeformed configurations, and the corresponding dispersion equations are derived analytically by invoking the linearized theory for incremental motions. Numerical examples are finally considered to show the tunability of wave propagation behavior in the soft PC cylinder. The outstanding performance regarding the band gap (BG) property of the proposed soft dielectric PC is clearly demonstrated by comparing with the conventional design adopting the hard piezoelectric material. One particular point that should be emphasized is that soft dielectric PCs are susceptible to various kinds of failure (buckling, electromechanical instability (EMI), electric breakdown (EB), etc.), imposing corresponding limits on the external stimuli. This has been carefully examined for the present soft PC cylinder such that the applied electric voltage is always assumed to be less than the critical voltage except for one case, in which we illustrate that the snap-through instability of the axially free PC cylinder made of a generalized Gent material may be used to efficiently trigger a sharp transition in the BGs.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2018;85(3):031005-031005-5. doi:10.1115/1.4038884.

Knowledge of the ideal shear strength of solid single crystals is of fundamental importance. However, it is very hard to determine this quantity at finite temperatures. In this work, a theoretical model for the temperature-dependent ideal shear strength of solid single crystals is established in the view of energy. To test the drawn model, the ideal shear properties of Al, Cu, and Ni single crystals are calculated and compared with that existing in the literature. The study shows that the ideal shear strength first remains approximately constant and then decreases almost linearly as temperature changes from absolute zero to melting point. As an example of application, the “brittleness parameter” of solids at elevated temperatures is quantitatively characterized for the first time.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2018;85(3):031006-031006-6. doi:10.1115/1.4038809.

Scar structures of natural animals can reinforce the wounds both mechanically and biologically to maintain the functions of the injured muscle and skin. Inspired by the scar structure, we present a dielectric elastomer (DE) with silver nanowire electrodes possessing the scar-like ability. This DE membrane can tolerate the failures by both electric breakdown and mechanical rupture. The DE actuator (DEA) can maintain their performances of force and displacement output after multiple failures. Scanning electronic microscope (SEM) images show that the scar-like structures accumulate around the electromechanical failure locations on the DE membrane as the stiffened and insulated regions, which prevent further short current and membrane rupture. J-integrals and stress distribution around the failure location have been calculated by finite element analysis to verify the mechanical reinforcements of the scar-like structures over crack propagation.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2018;85(3):031007-031007-11. doi:10.1115/1.4038990.

This paper discusses the dynamic interaction between a monoatomic chain of solid particles and a thin-walled spherical pressure vessel. The objective is to find a relationship between the highly nonlinear solitary waves (HNSWs) propagating within the chain and the internal pressure of the vessel. The paper introduces first a general finite element model to predict the abovementioned interaction, and then a specific application to tennis balls. The scope is to demonstrate a new nondestructive testing (NDT) method to infer the internal pressure of the balls. The overarching idea is that a mechanically induced solitary pulse propagating within the chain interacts with the thin-walled ball to be probed. At the chain–ball interface, the acoustic pulse is partially reflected back to the chain and partially deforms the rubber giving rise to secondary pulses. The research hypothesis is that one or more features of the reflected waves are monotonically dependent on the internal pressure. Both numerical and experimental results demonstrate a monotonic relationship between the time of flight (TOF) of the solitary waves and the internal pressure of the tennis balls. In addition, the pressure inferred nondestructively with the HNSWs matches very well the pressure measured destructively with an ad hoc pressure gauge needle. In the future, the results presented in this study could be used to develop a portable device to infer anytime anywhere the internal pressure of deformable systems (including biological systems) for which conventional pressure gages cannot be used noninvasively.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2018;85(3):031008-031008-11. doi:10.1115/1.4038883.

The skeleton of many natural and artificial soft materials can be abstracted as networks of fibers/polymers interacting in a nonlinear fashion. Here, we present a numerical model for networks of nonlinear, elastic polymer chains with rate-dependent crosslinkers similar to what is found in gels. The model combines the worm-like chain (WLC) at the polymer level with the transition state theory for crosslinker bond dynamics. We study the damage evolution and the force—displacement response of these networks under uniaxial stretching for different loading rates, network topology, and crosslinking density. Our results suggest a complex nonmonotonic response as the loading rate or the crosslinking density increases. We discuss this in terms of the microscopic deformation mechanisms and suggest a novel framework for increasing toughness and ductility of polymer networks using a bio-inspired sacrificial bonds and hidden length (SBHL) mechanism. This work highlights the role of local network characteristics on macroscopic mechanical observables and opens new pathways for designing tough polymer networks.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2018;85(3):031009-031009-10. doi:10.1115/1.4038919.

Flexoelectric effect is a universal and size-dependent electromechanical coupling between the strain gradient and electric field. The mathematical framework for flexoelectricity, which involves higher-order gradients of field quantities, is difficult to handle using traditional finite element method (FEM). Thus, it is important to develop an effective numerical method for flexoelectricity. In this paper, we develop a three-dimensional (3D) mixed finite element considering both flexoelectricity and strain gradient elasticity. To validate the developed element, we simulate the electromechanical behavior of a flexoelectric spherical shell subjected to inner pressure and compare the numerical results to analytical results. Their excellent agreement shows the reliability of the proposed FEM. The developed finite element is also used to simulate the electromechanical behavior of a nanometer-sized flexoelectric truncated pyramid. By decreasing the sample size, we observed the increase of its effective piezoelectricity. However, due to the effects of strain gradient elasticity and the influence of flexoelectricity on stiffness, the dependency of effective piezoelectricity on the sample size is not trivial. Numerical results indicate that, when the sample size is smaller than a certain value, the increase of effective piezoelectricity slows down. This finding also shows the importance of a numerical tool for the study of flexoelectric problems.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2018;85(3):031010-031010-6. doi:10.1115/1.4038921.

In this study, the plastic deformation mechanism of a fully clamped beam under oblique loading at its free end is analyzed. Supposing the cross sections are variable along the beam length, a characteristic length $L∗≡MP/NP$, defined as the ratio between fully plastic bending moment $MP$ and fully compression force $NP$, is employed to estimate the load carrying capacity of each cross section. By finite element (FE) simulations of the conical tubes, it is validated that if the initial failure positon locates in the middle of the beam, it will not change with the total beam length. Then, based on the analytical analysis and FE simulation, a progressive deformation mechanism triggered by bending, notated as progressive bending, is proposed for the first time. From the optimization result of maximizing loading force that the unit mass can withstand, the tubes with constant thickness are found to be better than tubes with graded thickness, when they are used as supporting structures. The multi-objective optimization for tubes with varying cross sections under oblique loading with different angles is also given. Then, two methods to improve the load carrying capacity of tubes are given: (1) to design the cross section of the tube, which is corresponding to let the critical loading force of all the cross sections be equal; (2) to optimize the initial failure point, so as to produce repeated failure modes.

Commentary by Dr. Valentin Fuster

### Technical Brief

J. Appl. Mech. 2018;85(3):034501-034501-4. doi:10.1115/1.4038808.

Configurational forces acting on two-dimensional (2D) elastic line singularities are evaluated by path-independent J-, M-, and L-integrals in the framework of plane strain linear elasticity. The elastic line singularities considered in this study are the edge dislocation, the line force, the nuclei of strain, and the concentrated couple moment that are subjected to far-field loads. The interaction forces between two similar parallel elastic singularities are also calculated. Self-similar expansion force, M, evaluated for the line force shows that it is exactly the negative of the strain energy prelogarithmic factor as in the case for the well-known edge dislocation result. It is also shown that the M-integral result for the nuclei of strain and the L-integral result for the line force yield interesting nonzero expressions under certain circumstances.

Topics: Stress , Dislocations
Commentary by Dr. Valentin Fuster