Polydomain nematic sheets can be designed for desired shape transition, and a typical example is a disc composed of congruent wedges with rank-1 connected director field. Recent theoretical study indicated that such a disc, if infinitesimally thin, tends to become a perfect pyramid upon illumination. Nonetheless, what is the influence of the finite thickness remains unexplored. In the present work, we re-examine this problem by treating the disc as an elastic plate with finite thickness. Analytical solution to the photo-actuated shape is obtained in small deformations, and the influences of the number of domains and the attenuation of photo intensity are discussed in details. The results are expected helpful to the design of related photo-responsive devices.

This paper investigates the robustness against localized impacts of elastic spherical shells pre-loaded under uniform external pressure. We subjected a pre-loaded spherical shell that is clamped at its equator to axisymmetric blast-like impacts applied to its polar region. The resulting axisymmetric dynamic response is computed for increasing amplitudes of the blast. Both perfect shells and shells with axisymmetric geometric imperfections are analyzed. The impact energy threshold causing buckling is identified and compared with the energy barrier that exists between the buckled and un-buckled static equilibrium states of the energy landscape associated with the pre-loaded pressure. The extent to which the impact energy of the threshold blast exceeds the energy barrier depends on the details of its shape and width. Targeted blasts that approximately replicate the size and shape of the energy barrier buckling mode defined in the paper have an energy threshold that is only modestly larger than the energy barrier. An extensive study is carried out for more realistic Gaussian-shaped blasts revealing that the buckling threshold energy for these blasts is typically in the range of at least ten to forty percent above the energy barrier, depending on the pressure pre-load and the blast width. The energy discrepancy between the buckling threshold and energy barrier is due to elastic waves spreading outward from the impact and dissipation associated with the numerical integration scheme..

Full three-dimensional cell models containing a small cavity are used to study the effect of plastic anisotropy on cavitation instabilities. Predictions for the Barlat-91 model (Barlat et al., 1991, “A Six-Component Yield Function for Anisotropic Materials,” Int. J. Plast. 7 , 693–712), with a non-quadratic anisotropic yield function, are compared with previous results for the classical anisotropic Hill-48 quadratic yield function (Hill, 1948, “A Theory of the Yielding and Plastic Flow of a Anisotropic Metals,” Proc. R. Soc. Lond. A193 , 281–297). The critical stress, at which the stored elastic energy will drive the cavity growth, is strongly affected by the anisotropy as compared with isotropic plasticity, but does not show much difference between the two models of anisotropy. While a cavity tends to remain nearly spherical during a cavitation instability in isotropic plasticity, the cavity shapes in an anisotropic material develop toward near-spheroidal elongated shapes, which differ for different values of the coefficients defining the anisotropy. The shapes found for the Barlat-91 model, with a non-quadratic anisotropic yield function, differ noticeably from the shapes found for the quadratic Hill-48 yield function. Computations are included for a high value of the exponent in the Barlat-91 model, where this model represents a Tresca-like yield surface with rounded corners.

We present a weak form implementation of the nonlinear axisymmetric shell equations. This implementation is suitable to study the nonlinear deformations of axisymmetric shells, with the capability of considering a general mid-surface shape, non-homogeneous (axisymmetric) mechanical properties and thickness variations. Moreover, given that the weak balance equations are arrived to naturally, any external load that can be expressed in terms of an energy potential can, therefore, be easily included and modeled. We validate our approach with existing results from the literature, in a variety of settings, including buckling of imperfect spherical shells, indentation of spherical and ellipsoidal shells, and geometry-induced rigidity (GIR) of pressurized ellipsoidal shells. Whereas the fundamental basis of our approach is classic and well established, from a methodological view point, we hope that this brief note will be of both technical and pedagogical value to the growing and dynamic community that is revisiting these canonical but still challenging class of problems in shell mechanics.

The effects of two different pitching frequencies (that is, Strouhal number, St ) on the wake structure generated by two foils of aspect ratio 1.0 are examined numerically at a Reynolds number of 10,000. Strouhal numbers of 0.5 and 0.2 were studied, the first corresponding approximately to the peak in efficiency and the second corresponding to the point where the thrust is equal to the drag (the free-swimming condition). The two foils have either a square trailing edge or a convex trailing edge that mimics the shape of the caudal fin exhibited by certain species of fish. In previous works, the convex trailing edge panel was found to have higher thrust and efficiency compared with the square panel trailing edge. Here, these differences are related to their characteristic vortex formation and detachment processes leading to differences in wake coherence and extension. The wake of the square panel at St = 0.2 transitions slowly from a reverse von Kármán street (2S) pattern to a paired (2P) system as the wake develops downstream, whereas at St = 0.5 , the wake almost immediately takes on a 2P form with an attendant split in the wake structure. For the convex panel, the transition from a 2S to a 2P structure at St = 0.2 is slower than that seen for the square panel, and for St = 0.5 , the wake undergoes an abrupt transition leading to two distinct vortex streets that evolve at a considerably slower rate than seen for the square panel.

A small ball resting on a curve in a gravitational field offers a simple and compelling example of potential energy. The force required to move the ball, or to maintain it in a given position on a slope, is the negative of the vector gradient of the potential field: the steeper the curve, the greater the force required to push the ball up the hill (or keep it from rolling down). We thus observe the turning points (horizontal tangency) of the potential energy shape as positions of equilibrium (in which case the “restoring force” drops to zero). In this paper, we appeal directly to this type of system using both one- and two-dimensional shapes: curves and surfaces. The shapes are produced to a desired mathematical form generally using additive manufacturing, and we use a combination of load cells to measure the forces acting on a small steel ball-bearing subject to gravity. The measured forces, as a function of location, are then subject to integration to recover the potential energy function. The utility of this approach, in addition to pedagogical clarity, concerns extension and applications to more complex systems in which the potential energy would not be typically known a priori, for example, in nonlinear structural mechanics in which the potential energy changes under the influence of a control parameter, but there is the possibility of force probing the configuration space. A brief example of applying this approach to a simple elastic structure is presented.

To determine the impact of cohesive law shapes on the modeling of interfacial debonding in lithium-ion battery electrodes, analytical methods based on different cohesive models for the debonding process have been developed individually. Three different cohesive laws, namely, triangular, trapezoidal, and rectangular laws, have been employed. To ensure comparability, the cohesive strength and the fracture toughness have been set to be identical for different cohesive laws. The evaluation of debonding onset has suggested that the cohesive law shape affects the modeling results only when the interface is ductile. The largest possible difference for the triangular law and the rectangular law on the debonding onset has been estimated. A discussion for specific electrodes has also been provided.

Experimental data have made it abundantly clear that the strength of polycrystalline silicon (poly-Si) microelectromechanical systems (MEMS) structures exhibits significant variability, which arises from the random distribution of the size and shape of sidewall defects created by the manufacturing process. Test data also indicated that the strength statistics of MEMS structures depends strongly on the structure size. Understanding the size effect on the strength distribution is of paramount importance if experimental data obtained using specimens of one size are to be used with confidence to predict the strength statistics of MEMS devices of other sizes. In this paper, we present a renewal weakest-link statistical model for the failure strength of poly-Si MEMS structures. The model takes into account the detailed statistical information of randomly distributed sidewall defects, including their geometry and spacing, in addition to the local random material strength. The large-size asymptotic behavior of the model is derived based on the stability postulate. Through the comparison with the measured strength distributions of MEMS specimens of different sizes, we show that the model is capable of capturing the size dependence of strength distribution. Based on the properties of simulated random stress field and random number of sidewall defects, a simplified method is developed for efficient computation of strength distribution of MEMS structures.

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.

Irradiation-induced oxidation of lipid membranes is implicated in diseases and has been harnessed in medical treatments. Irradiation induces the formation of oxidative free radicals, which attack double bonds in the hydrocarbon chains of lipids. Studies of the kinetics of this reaction suggest that the result of the first stage of oxidation is a structural change in the lipid that causes an increase in the area per molecule in a vesicle. Since area changes are directly connected to membrane tension, irradiation-induced oxidation affects the mechanical behavior of a vesicle. Here, we analyze shape changes of axisymmetric vesicles that are under simultaneous influence of adhesion, micropipette aspiration, and irradiation. We study both the equilibrium and kinetics of shape changes and compare our results with experiments. The tension–area relation of a membrane, which is derived by accounting for thermal fluctuations, and the time variation of the mechanical properties due to oxidation play important roles in our analysis. Our model is an example of the coupling of mechanics and chemistry, which is ubiquitous in biology.

Flexible elastic beams can function as dexterous manipulators at multiple length-scales and in various niche applications. As a step toward achieving controlled manipulation with flexible structures, we introduce the problem of approximating desired quasi-static deformations of a flexible beam, modeled as an elastica, by optimizing the loads applied. We presume the loads to be concentrated, with the number and nature of their application prescribed based on design considerations and operational constraints. For each desired deformation, we pose the problem of computing the requisite set of loads to mimic the target shape as one of optimal approximations. In the process, we introduce a novel generalization of the forward problem by considering the inclinations of the loads applied to be functionals of the solution. This turns out to be especially beneficial when analyzing tendon-driven manipulators. We demonstrate the shape control realizable through load optimization using a diverse set of experiments.

Void coalescence is known to be the last microscopic event of ductile fracture in metal alloys and corresponds to the localization of plastic flow in between voids. Limit-analysis has been used to provide coalescence criteria that have been subsequently recast into effective macroscopic yield criteria, leading to models for porous materials valid for high porosities. Such coalescence models have remained up to now restricted to cubic or hexagonal lattices of spheroidal voids. Based on the limit-analysis kinematic approach, a methodology is first proposed to get upper-bound estimates of coalescence stress for arbitrary void shapes and lattices. Semi-analytical coalescence criteria are derived for elliptic cylinder voids in elliptic cylinder unit cells for an isotropic matrix material, and validated through comparisons to numerical limit-analysis simulations. The physical application of these criteria for realistic void shapes and lattices is finally assessed numerically.

The elastic interaction energy between several precipitates is of interest since it may induce ordering of precipitates in many metallurgical systems. Most of the works on this subject assumed homogeneous systems, namely, the elastic constants of the matrix and the precipitates are identical. In this study, the elastic fields, and self and interaction energies of inhomogeneous anisotropic precipitates have been solved and assessed, based on a new iterative approach using the quasi-analytic Fourier transform method. This approach allows good approximation for problems of several inhomogeneous precipitates in solid matrix. We illustrate the calculation approach on γ ′ -Ni 3 Ti precipitates in A-286 steel and demonstrate that the influence of elastic inhomogeneity is in some incidences only quantitative, while in others it has essential effect. Assuming homogeneous system, disk shape precipitate is associated with minimum elastic energy. Only by taking into account different elastic constants of the precipitate, the minimum self-energy is found to be associated with spherical shape, and indeed, this is the observed shape of the precipitates in A-286 steel. The elastic interaction energy between two precipitates was calculated for several configurations. Significant differences between the interactions in homogeneous and inhomogeneous were found for disk shape morphologies. Only quantitative differences (9% higher interaction between inhomogeneous precipitates) were found between two spherical precipitates, which are the actual shape.

Void growth in an anisotropic ductile solid is studied by numerical analyses for three-dimensional (3D) unit cells initially containing a void. The effect of plastic anisotropy on void growth is the main focus, but the studies include the effects of different void shapes, including oblate, prolate, or general ellipsoidal voids. Also, other 3D effects such as those of different spacings of voids in different material directions and the effects of different macroscopic principal stresses in three directions are accounted for. It is found that the presence of plastic anisotropy amplifies the differences between predictions obtained for different initial void shapes. Also, differences between principal transverse stresses show a strong interaction with the plastic anisotropy, such that the response is very different for different anisotropies. The studies are carried out for one particular choice of void volume fraction and stress triaxiality.

Soft network materials that incorporate wavy filamentary microstructures have appealing applications in bio-integrated devices and tissue engineering, in part due to their bio-mimetic mechanical properties, such as “J-shaped” stress–strain curves and negative Poisson's ratios. The diversity of the microstructure geometry as well as the network topology provides access to a broad range of tunable mechanical properties, suggesting a high degree of design flexibility. The understanding of the underlying microstructure-property relationship requires the development of a general mechanics theory. Here, we introduce a theoretical model of infinitesimal deformations for the soft network materials constructed with periodic lattices of arbitrarily shaped microstructures. Taking three representative lattice topologies (triangular, honeycomb, and square) as examples, we obtain analytic solutions of Poisson's ratio and elastic modulus based on the mechanics model. These analytic solutions, as validated by systematic finite element analyses (FEA), elucidated different roles of lattice topology and microstructure geometry on Poisson's ratio of network materials with engineered zigzag microstructures. With the aid of the theoretical model, a crescent-shaped microstructure was devised to expand the accessible strain range of network materials with relative constant Poisson's ratio under large levels of stretching. This study provides theoretical guidelines for the soft network material designs to achieve desired Poisson's ratio and elastic modulus.

A systematic study is performed on the plane contact and adhesion of two elastic solids with an interface groove. The nonadhesion and Johnson–Kendall–Roberts (JKR) adhesion solutions for a typical groove shape are obtained in closed form by solving singular integral equations and using energy release rate approaches. It is found that the JKR adhesion solution depends solely on a dimensionless parameter α and the groove is predicted to be unstably flattened with no applied load when α ≥ 0.535 . Furthermore, the corresponding Maugis–Dugdale adhesion model has been revisited with three possible equilibrium states. By introducing the classical Tabor parameter μ , a complete transition between the nonadhesion and the JKR adhesion contact models is captured, which can be recovered as two limiting cases of the Maugis–Dugdale model. Depending on two nondimensional parameters α and μ , where α 2 represents the ratio of the surface energy in the groove to the elastic strain energy when the grooved surface is flattened, different transition processes among three equilibrium states are characterized by one or more jumps between partial and full contact. Larger values of α and μ tend to induce more energy loss due to adhesion hysteresis. Combination values of α and μ are also suggested to design self-healing interface grooves due to adhesion.

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.

The nonlinear response of a flexible structure, subjected to generally supported conditions with nonlinearities, is investigated for the first time. An analytical procedure is proposed first. Moreover, a simulation technique usually employed in static analysis is developed for confirmation. Generally, ordinary perturbation methods could analyze dynamics of flexible structures with linear boundary conditions. As nonlinear boundaries are taken into account, they are out of operation for the modal shape that is hardly to be obtained, which is the key to the analysis. In order to overcome this, nonlinear boundary conditions are rescaled and the technique of modal revision is employed. Consequently, each governing equation with different time-scales could be analyzed exactly according to corresponding rescaled boundary conditions. The total response of any point at the flexible structure will be composed by harmonic responses yielded by the analytical method. Furthermore, the differential quadrature element method (DQEM), a numerical simulation technique could satisfy boundary conditions strictly, is introduced to certify analytical results. The comparison shows a reasonable agreement between these two methods. In fact, the accuracy of the analytical method for nonlinear boundaries could be explained in theory. Based on the certification, boundary nonlinearities are discussed in detail analytically and found to play an important role in responses. Because of the important role played by the nonlinear factors in the vibration and control of the flexible structure, this paper will open the vibration analysis and numerical study of the flexible structure with nonlinear constraints.

The flexoelectric effect is an electromechanical phenomenon that is universally present in all dielectrics and exhibits a strong size-dependency. Through a judicious exploitation of scale effects and symmetry, flexoelectricity has been used to design novel types of structures and materials including piezoelectric materials without using piezoelectric. Flexoelectricity links electric polarization with strain gradients and is rather difficult to estimate experimentally. One well-acknowledged approach is to fabricate truncated pyramids and/or cones and examine their electrical response. A theoretical model is then used to relate the measured experimental response to estimate the flexoelectric properties. In this work, we revisit the typical model that is used in the literature and solve the problem of a truncated cone under compression or tension. We obtained closed-form analytical solutions to this problem and examine the size and shape effects of flexoelectric response of the aforementioned structure. In particular, we emphasize the regime in which the existing models are likely to incur significant error.