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1. An existence theorem for a class of wrinkling models for highly stretched elastic sheets

   https://doi.org/10.1007/s00033-023-02111-9  

   Healey, Timothy J (December 2023, Zeitschrift für angewandte Mathematik und Physik)

   NA (Ed.)

   We consider a class of models motivated by previous numerical studies of wrinkling in highly stretched, thin rectangular elastomer sheets. The model is characterized by a finite-strain hyperelastic membrane energy perturbed by small bending energy. In the absence of the latter, the membrane energy density is not rank-one convex for general spatial deformations but reduces to a polyconvex function when restricted to planar deformations, i.e., two-dimensional hyperelasticity. In addition, it grows unbounded as the local area ratio approaches zero. The small bending component of the model is the same as that in the classical von K´arm´an model. The latter penalizes arbitrarily fine-scale wrinkling, resolving both the amplitude and wavelength of wrinkles. Here, we prove the existence of energy minima for a general class of such models. 

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2. Impact of nanometer-thin stiff layer on adhesion to rough surfaces

   https://doi.org/10.1103/PhysRevResearch.6.033015  

   Kumar, Shubhendu; Gaire, Babu; Persson, B_N J; Dhinojwala, Ali (July 2024, Physical Review Research)

   Adhesives require molecular contact, which is governed by roughness, modulus, and load. Here, we measured adhesion for stiff glassy polymer layers of varying thickness on top of a soft elastomer with rough substrates. We found that a 90-nm-thick PMMA layer on a softer elastic block was sufficient to drop macroscopic adhesion to almost zero during the loading cycle. This drop in adhesion for bilayers follows the modified Persson-Tosatti model, where the elastic energy for conformal contact depends on the thickness and modulus of the bilayer. In contrast, we observed no dependence on thickness of the PMMA layer on the work of adhesion calculated using the pull-off forces. Understanding how mechanical gradients (like bilayers) affect adhesion is critical for areas such as adhesion, friction, and colloidal and granular physics. Published by the American Physical Society2024 

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3. A comparative study of two constitutive models within an inverse approach to determine the spatial stiffness distribution in soft materials

   Mei, Y.; Stover, B.; Kazerooni, N. A.; Srinivasa, A.; Hajhashemkhani, M.; Hematiyan, M. R.; Goenezen, S. (April 2018, International journal of mechanical sciences)

   A comparative study is presented to solve the inverse problem in elasticity for the shear modulus (stiffness) distribution utilizing two constitutive equations: (1) linear elasticity assuming small strain theory, and (2) finite elasticity with a hyperelastic neo-Hookean material model. Assuming that a material undergoes large deformations and material nonlinearity is assumed negligible, the inverse solution using (2) is anticipated to yield better results than (1). Given the fact that solving a linear elastic model is significantly faster than a nonlinear model and more robust numerically, we posed the following question: How accurately could we map the shear modulus distribution with a linear elastic model using small strain theory for a specimen undergoing large deformations? To this end, experimental displacement data of a silicone composite sample containing two stiff inclusions of different sizes under uniaxial displacement controlled extension were acquired using a digital image correlation system. The silicone based composite was modeled both as a linear elastic solid under infinitesimal strains and as a neo-Hookean hyperelastic solid that takes into account geometrically nonlinear finite deformations. We observed that the mapped shear modulus contrast, determined by solving an inverse problem, between inclusion and background was higher for the linear elastic model as compared to that of the hyperelastic one. A similar trend was observed for simulated experiments, where synthetically computed displacement data were produced and the inverse problem solved using both, the linear elastic model and the neo-Hookean material model. In addition, it was observed that the inverse problem solution was inclusion size-sensitive. Consequently, an 1-D model was introduced to broaden our understanding of this issue. This 1-D analysis revealed that by using a linear elastic approach, the overestimation of the shear modulus contrast between inclusion and background increases with the increase of external loads and target shear modulus contrast. Finally, this investigation provides valuable information on the validity of the assumption for utilizing linear elasticity in solving inverse problems for the spatial distribution of shear modulus associated with soft solids undergoing large deformations. Thus, this work could be of importance to characterize mechanical property variations of polymer based materials such as rubbers or in elasticity imaging of tissues for pathology. 

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4. Shapes of a filament on the surface of a bubble

   https://doi.org/10.1098/rspa.2021.0353  

   Prasath, S. Ganga; Marthelot, Joel; Govindarajan, Rama; Menon, Narayanan (September 2021, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   The shape assumed by a slender elastic structure is a function both of the geometry of the space in which it exists and the forces it experiences. We explore, by experiments and theoretical analysis, the morphological phase space of a filament confined to the surface of a spherical bubble. The morphology is controlled by varying bending stiffness and weight of the filament, and its length relative to the bubble radius. When the dominant considerations are the geometry of confinement and elastic energy, the filament lies along a geodesic and when gravitational energy becomes significant, a bifurcation occurs, with a part of the filament occupying a longitude and the rest along a curve approximated by a latitude. Far beyond the transition, when the filament is much longer than the diameter, it coils around the selected latitudinal region. A simple model with filament shape as a composite of two arcs captures the transition well. For better quantitative agreement with the subcritical nature of bifurcation, we study the morphology by numerical energy minimization. Our analysis of the filament’s morphological space spanned by one geometric parameter, and one parameter that compares elastic energy with body forces, may provide guidance for packing slender structures on complex surfaces. 

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5. Roughness-dependent scaling of the contact area and separation gap with pressure for glassy polymers

   https://doi.org/10.1073/pnas.2503087122  

   Patil, Utkarsh; Kumar, Shubhendu; Merriman, Stephen; Dhinojwala, Ali (June 2025, Proceedings of the National Academy of Sciences)

   The contact between two rough surfaces has been a topic of significant interest since early studies on Coulombic friction and remains crucial for numerous technological applications. However, theoretical progress has outpaced experiments due to the challenges in measuring contact areas across scales ranging from subnanometers to macroscopic dimensions. Here, we demonstrate the use of commonly available infrared-based (IR) spectroscopy in combination with finite-difference time-domain (FDTD) optical simulations to measure separation gaps and contact areas for glassy polymers ranging in roughness over two orders in magnitude. With the combined IR and FDTD simulations, we can overcome the optical diffraction limits and take advantage of the chemical specificity of IR spectroscopy to overcome limitations due to scattering. The scaling of the contact area ratio as a function of pressure illustrated the limitations of using pure elastic or plastic deformation in explaining the results. At both low and high pressures, the contact area ratios scale linearly with pressure as expected for purely elastic deformations at low pressures or plastic deformations at high pressures. However, if analyzed over a broad range of pressure, the power laws we observe are much larger than 1, exemplifying the need to consider elastoplastic models in explaining results for softer polymer contacts compared to other brittle, glassy materials. In comparison, the separation gaps scale exponentially with pressure, as expected. These results have important implications for the interpretation of properties such as friction, adhesion, and conductivity for softer, glassy contact interfaces. 

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