A simple method for determining large deflection states of arbitrarily curved planar elastica

The paper discusses a relatively simple method for determining large deflection states of arbitrarily curved planar elastica, which is modeled by a finite set of initially straight flexible segments. The basic equations are built using Euler–Bernoulli and large displacement theory. The problem is solved numerically using Runge–Kutta–Fehlberg integration method and Newton method for solving systems of nonlinear equations. This solution technique is tested on several numerical examples. From a comparison of the results obtained and those found in the literature, it can be concluded that the developed method is efficient and gives accurate results. The solution scheme displayed can serve as reference tool to test results obtained via more complex algorithms

Study report conducted at the University of Massachusetts, Amherst (USA)

Raziskave so bile financirane s sredstvi Javnega sklada Republike Slovenije za razvoj kadrov in štipendije. Sredstva / štipendija je bila pridobljena po razpisnih pogojih "Štipendiranja oz. sofinanciranja raziskovalnega sodelovanja doktorskih študentov v tujini v letu 2012" po pogodbi s številko 11012-47/2012

Effects of nonlinearities and geometric imperfections on multistability and deformation localization in wrinkling films on planar substrates

Compressed elastic films on soft substrates release part of their strain energy by wrinkling, which represents a loss of symmetry, characterized by a pitchfork bifurcation. Its development is well understood at the onset of supercritical bifurcation, but not beyond, or in the case of subcritical bifurcation. This is mainly due to nonlinearities and the extreme imperfection sensitivity. In both types of bifurcations, the energy–displacement diagrams that can characterize an energy landscape are non-convex, which is notoriously difficult to determine numerically or experimentally, let alone analytically. To gain an elementary understanding of such potential energy landscapes, we take a thin beam theory suitable for analyzing large displacements under small strains and significantly reduce its complexity by reformulating it in terms of the tangent rotation angle. This enables a comprehensive analytical and numerical analysis of wrinkling elastic films on planar substrates, which are effective stiffening and/or softening due to either geometric or material nonlinearities. We also validate our findings experimentally. We explicitly show how effective stiffening nonlinear behavior (e.g., due to substrate or membrane deformations) leads to a supercritical post-bifurcation response, makes the energy landscape non-convex through energy barriers causing multistability, which is extremely problematic for numerical computation. Moreover, this type of nonlinearity promotes uni-modal, uniformly distributed, periodic deformation patterns. In contrast, nonlinear effective softening behavior leads to subcritical post-bifurcation behavior, similarly divides the energy landscape by energy barriers and conversely promotes localization of deformations. With our theoretical model we can thus explain an experimentally observed phenomenon that in structures with effective softening followed by an effective stiffening behavior, the symmetry is initially broken by localizing the deformation and later restored by forming periodic, distributed deformation patterns as the load is increased. Finally, we show that initial imperfections can significantly alter the local or global energy-minimizing deformation pattern and completely remove some energy barriers. We envision that this knowledge can be extrapolated and exploited to convexify extremely divergent energy landscapes of more sophisticated systems, such as wrinkling compressed films on curved substrates (e.g., on cylinders and spheres) and that it will enable elementary analysis and the development of specialized numerical tools

Fluid flow during phase transition

In this paper, we consider a pressure-driven flow of a viscoelastic fluid in a straight rectangular channel undergoing a solidification phase change due to polymerization. We treat the viscoelastic response of the fluid with a model based on the formalism of variable-order calculusmore specifically, we employ a model utilizing a variable-order Caputo-type differential operator. The order parameter present in the model is determined by the extent of polymerization induced by light irradiation. We model this physical quantity with a simple equation of kinetics, where the reaction rate is proportional to the amount of material available for polymerization and optical transmittance. We treat cases when the extent of polymerization is a function of either time alone or both position and time, and solve them using either analytical or semi-analytical methods. Results of our analysis indicate that in both cases, solutions evolve in time according to a variable-order decay law, with the solution in the first case having a hyperbolic cosine-like spatial dependence, while the spatial dependence in the second case conforms to a bell curve-like function. We infer that our treatment is physically sound and may be used to consider problems of more general viscoelastic flows during solidification, with the advantage of requiring fewer experimentally determined parameters

Using a generative adversarial network for the inverse design of soft morphing composite beams

The inverse design of structures having tailored properties is challenging mainly due to the multiple design solutions that can satisfy the prescribed conditions. For example, in the inverse design of morphing composite beams, different fabrication solutions exist because the material, geometry and actuation can be varied. On the other hand, the problem can be highly nonlinear due to the large deformations present in such problems. For this reason, we present a generative adversarial network-based inverse design method for constructing soft composite beams that morph into target shapes and can carry out complex prescribed motions. Our approach makes use of composites with passive and active layers that deform into prescribed shapes due to the strain mismatch induced by the non-homogeneous geometric and material properties as well as temperature actuation. To test the proposed method and explore the parametric space much faster than with heating and cooling, we established a mechanical analog (a toy model) that exploits the mechanical stretching of highly elastic, active layers. Experiments and numerical examples demonstrate the effectiveness of our simple toy model, for which the generator network takes the target shapes as inputs and generates the corresponding design parameters for the fabrication of composite beams that self-deploy into prescribed shapes when released. We extended our method for generating the design parameters for forming soft, morphing composite beams that exhibit complex targeted motions when actuated by temperature. Our data-driven method is simple, yet robust enough to provide solutions to complex problems and aid in the future design of soft robots and smart-deployable structures

Spherical harmonics-based pseudo-spectral method for quantitative analysis of symmetry breaking in wrinkling of shells with soft cores

A complete understanding of the wrinkling of compressed films on curved substrates remains illusive due to the limitations of both analytical and current numerical methods. The difficulties arise from the fact that the energetically minimal distribution of deformation localizations is primarily influenced by the inherent nonlinearities and that the deformation patterns on curved surfaces are additionally constrained by the topology. The combination of two factors – the need for dense meshes to mitigate the topological limitations of discretization in domains such as spheres where there is no spherically-symmetric discretizations, and the intensive search for minima in a highly non-convex energy landscape due to nonlinearity – makes existing numerical methods computationally impractical without oversimplifying assumptions to reduce computational costs or introducing artificial parameters to ensure numerical stability. To solve these issues, we have developed a novel (less) reduced version of shell theory for shells subjected to membrane loads, such as during wrinkling. It incorporates the linear contributions of the usually excluded tangential displacements in the membrane strain energy and thus retains the computational efficiency of reduced state-of-the-art methods while nearly achieving the accuracy of the full Kirchhoff–Love shell theory. We introduce a Galerkin-type pseudo-spectral method to further reduce computational costs, prevent non-physical deformation distribution due to mesh-induced nucleation points, and avoid singularities at the poles of the sphere. The method uses spherical harmonic functions to represent functions on the surface of a sphere and is integrated into the framework of minimizing the total potential energy subject to constraints. This robust approach effectively solves the resulting non-convex potential energy problem. Our method accurately predicts the transition between deformation modes based solely on the material and geometric parameters determined in our experiments, without the need to introduce artificial parameters for numerical stability and/or additional fitting of the experimental data

Digital image correlation based internal friction characterization in granular materials

Based on the realization that Newtonian fluids have the unique property to redirect the forces applied to them in a perpendicular direction, a new apparatus, called the Granular Friction Analyzer (GFA), and the related GFA index, were proposed for characterizing the internal friction and related flow behavior of granular materials under uniaxial compression loading. The calculation of the GFA index is based on the integration of the internal pressure distribution along the cylinder wall, within which the granular material is being uniaxially compressed by a piston. In this paper an optical granular friction analyzer (O-GFA) is presented, where a digital image correlation (DIC) method is utilized to assess the cylinder strains used to calculate the internal pressure distribution. The main advantage of using the DIC method is that the starting point (piston-powder contact point) and the length of the integration considering the edge effects can be defined. By using the DIC full-field, instead of a few points strain measurements, a 2% improvement of the GFA indexʼs accuracy has been achieved and its robustness with respect to the number of points has been demonstrated. Using the parametric error analysis it has been shown that most of the observed total error (7.5%) arises from the DIC-method-based measurements of the strains, which can be improved by higher-resolution cameras and DIC algorithms for the strain evaluation. Additionally, it was shown that the GFA index can be used for determining the well-known Janssen model parameters. The latter was demonstrated experimentally, by testing three SS 316 L granular material samples with different mean particle sizes. The results confirm that the mean particle size regulates the internal friction of granular materials