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Mathematical model for predicting dynamic response of thin elastic shells of revolution during water impact
Linear thin shell theory in complex dependent variable form for determining fourth order partial differential equations, considering elastic shells with edge loads
Large elastic deformations of incompressible materials shells with inclusion of transverse normal strain, considering thickness change at boundaries
Understanding how to produce forces using biomolecular building blocks is essential for the development of adaptive synthetic cells and living materials. Here we ask whether a dynamic polymer system can generate deformation forces in soft shells by pure self-assembly, motivated by the fact that biological polymer networks like the cytoskeleton can exert forces, move objects, and deform membranes by simply growing, even in the absence of molecular motors. We address this question by investigating polymer force generation by varying the release rate, the structure, and the interactions of self-assembling monomers. First, we develop a toy computational model of polymerization in a soft elastic shell that reveals the emergence of spontaneous bundling which enhances shell deformation. We then extend our model to account more explicitly for monomer binding dynamics. We find that the rate at which monomers are released into the interior of the shell is a crucial parameter for achieving deformation through polymer growth. Lastly, we demonstrate that the introduction of multivalent particles that can join polymers can either improve or impede polymer performance, depending on the amount and on the structure of the multivalent particles. Our results provide guidance for the experimental realization of polymer systems that can perform work at the nanoscale, for example through rationally designed self-assembling proteins or nucleic acids.
Physical phenomena involving rapid and sudden transitions, such as snap buckling of elastic shells, explosions, and earthquakes, are characterized mathematically as a small disturbance causing a large-amplitude response. Because of this, standard asymptotic and perturbation methods are ill-suited to these problems. In the present paper, a new method of analyzing jump phenomena is proposed. The principal feature of the method is the representation of the response in terms of rational functions. For illustration, the method is applied to the snap buckling of an elastic arch and to a simple combustion problem.
The thickness of the elastic lithosphere in the Tharsis region of Mars is estimated from effects due to the surface load of Olympus Mons. Deformation (vertical displacement) and stress are calculated using elastic flexure theory for a range of possible lithospheric thicknesses (T), modeling the lithosphere as a thin elastic shell and the interior as a Newtonian fluid. For T below 150 km, displacement and stress rise rapidly with decreasing thickness. For T near 100 km, deformation of the region surrounding the volcano would be clearly visible in the topography, and resulting tensional stresses exceeding 5 kbar should produce observable fracturing at the surface. In contrast, for T near 200 km deformation is minimal and the tensional stress, being less than a kilobar, would not result in extensive fracturing. Since significant deformation and fracturing are not observed, it is concluded that the Martian elastic lithosphere is at least 150 km in thickness. Seismic, tectonic, and gravity observations all suggest a thick Martian lithosphere as well.
We develop here computational methods for the simulation of osmotic swelling phenomena relevant to microscopic vesicles containing transformable solute molecules. We introduce stochastic immersed boundary methods (SIBMs) that can capture osmotically driven fluid transport through semipermeable elastic membranes subject to thermal fluctuations. We further develop numerical methods to handle within SIBMs an elastic shell model for a neo-Hookean material. Our extended SIBM allows for capturing osmotic swelling phenomena driven by concentration changes and interactions between a discrete collection of confined particles while accounting for the thermal fluctuations of the semipermeable membrane and the hydrodynamic transport of solvent. We use our computational methods to investigate osmotic phenomena in regimes that go beyond the classical Van't Hoff theory. We develop statistical mechanics theories for osmotic swelling of vesicles when there are significant interactions between particles that can transform over time. We validate our theoretical results against detailed computational simulations. Our methods are expected to be useful for a wide class of applications allowing for the simulation of osmotically driven flows, thermally fluctuating semipermeable elastic structures, and solute interactions.
Tectonic features on a planetary surface are commonly used as constraints on models to determine the state of stress at the time the features formed. Quantitative global stress models applied to understand the formation of the Tharsis province on Mars constrained by observed tectonics have calculated stresses at the surface of a thin elastic shell and have neglected the role of vertical structure in influencing the predicted pattern of surface deformation. Wrinkle ridges in the Lunae Planum region of Mars form a conentric pattern of regularly spaced features in the eastern and southeastern part of Tharsis; they are formed due to compressional stresses related to the response of the Martian lithosphere to the Tharsis bulge. As observed in the exposures of valley walls in areas such as the Kasei Valles, the surface plains unit is underlain by an unconsolidated impact-generated megaregolith that grades with depth into structurally competent lithospheric basement. The ridges have alternatively been hypothesized to reflect deformation restricted to the surface plains unit ('thin skinned deformation') and deformation that includes the surface unit, megaregolith and basement lithosphere ('thick skinned deformation'). We have adopted a finite element approach to quantify the nature of deformation associated with the development of wrinkle ridges in a vertically stratified elastic lithosphere. We used the program TECTON, which contains a slippery node capability that allowed us to explicitly take into account the presence of reverse faults believed to be associated with the ridges. In this study we focused on the strain field in the vicinity of a single ridge when slip occurs along the fault. We considered two initial model geometries. In the first, the reverse fault was assumed to be in the surface plains unit, and in the second the initial fault was located in lithospheric basement, immediately beneath the weak megaregolith. We are interested in the conditions underwhich strain in the surface layer and basement either penetrates or fails to penetrate through the megaregolith. We thus address the conditions required for an initial basement fault to propagate through the megaregolith to the surface, as well as the effect of the megareolith on the strain tensor in the vicinity of a fault that nucleates in the surface plains unit.
In the frame work of density functional theory, the biaxial strain induced phonon dispersion curves of monolayer boron pnictides (BX, X = N, P, As, and Sb) have been investigated. The electron-ion interactions have been modelled using ultrasoft pseudopotentials while exchange-correlation energies have been approximated by the method of local density approximation in the parameterization of Perdew-Zunger. The longitudinal and transverse acoustic phonon modes of boron pnictide sheets show linear dependency on wave vector k{sup →} while out of plane mode varies as k{sup 2}. The in-plane longitudinal and out of plane transverse optical modes in boron nitride displaying significant dispersion similar to graphene. We have analyzed the biaxial strain dependent behaviour of out of plane acoustic phonon mode which is linked to ripple for four BX sheets using a model equation with shell elasticity theory. The strain induces the hardening of this mode with tendency to become more linear with increase in strain percentage. The strain induced hardening of out of plane acoustic phonon mode indicates the absence of rippling in these compounds. Our band structure calculations for both unstrained and strained 2D h-BX are consistent with previous calculations.
We introduce a dielectric elastomer actuator (DEA) composed of liquid-phase Gallium-Indium (GaIn) alloy electrodes embedded between layers of poly(dimethylsiloxane) (PDMS) and examine its mechanics using a specialized elastic shell theory. Residual stresses in the dielectric and sealing layers of PDMS cause the DEA to deform into a saddle-like geometry (Gaussian curvature K<0). Applying voltage Φ to the liquid metal electrodes induces electrostatic pressure (Maxwell stress) on the dielectric and relieves some of the residual stress. This reduces the longitudinal bending curvature and corresponding angle of deflection ϑ. Treating the elastomer as an incompressible, isotropic, NeoHookean solid, we develop a theory based on the principle of minimum potential energy to predict the principal curvatures as a function of Φ. Based on this theory, we predict a dependency of ϑ on Φ that is in strong agreement with experimental measurements performed on a GaIn-PDMS composite. By accurately modeling electromechanical coupling in a soft-matter DEA, this theory can inform improvements in design and fabrication.
This paper describes the energy dissipation arising from microslip for an elastic shell incorporating shear and longitudinal deformation resting on a rough-rigid foundation. This phenomenon is investigated using finite element (FE) analysis and nonlinear geometrically exact shell theory. Both approaches illustrate the effect of shear within the shell and observe a reduction in the energy dissipated from microslip as compared to a similar system neglecting shear deformation. In particular, it is found that the shear deformation allows for load to be transmitted beyond the region of slip so that the entire interface contributes to the load carrying capability of the shell. The energy dissipation resulting from the shell model is shown to agree well with that arising from the FE model, and this representation can be used as a basis for reduced order models that capture the microslip phenomenon.
Here, we expand the second-order fluid–structure coupling scheme of Farhat et al. (1998, “Load and Motion Transfer Algorithms for 19 Fluid/Structure Interaction Problems With Non-Matching Discrete Interfaces: Momentum and Energy Conservation, Optimal Discretization and Application to Aeroelasticity,” Comput. Methods Appl. Mech. Eng., 157(1–2), pp. 95–114; 2006, “Provably Second-Order Time-Accurate Loosely-Coupled Solution Algorithms for Transient Nonlinear Computational Aeroelasticity,” Comput. Methods Appl. Mech. Eng., 195(17), pp. 1973–2001) to structural acoustics. The staggered structural acoustics solution method is demonstrated to be second-order accurate in time, and numerical results are compared to a monolithically coupled system. The partitioned coupling method is implemented in the Sierra Mechanics software suite, allowing for the loose coupling of time domain acoustics in SIERRA/SD to structural dynamics ( SIERRA/SD ) or solid mechanics ( SIERRA/SD ). The coupling is demonstrated to work for nonconforming meshes. Results are verified for a one-dimensional piston, and the staggered and monolithic results are compared to an exact solution. Huang, H. (1969, “Transient Interaction of Plane Acoustic Waves With a Spherical Elastic Shell,” J. Acoust. Soc. Am., 45(3), pp. 661–670) sphere scattering problem with a spherically spreading acoustic load demonstrates parallel capability on a complex problem. Our numerical results compare well for a bronze plate submerged in water and sinusoidally excited (Fahnline and Shepherd, 2017, “Transient Finite Element/Equivalent Sources Using Direct Coupling and Treating the Acoustic Coupling Matrix as Sparse,” J. Acoust. Soc. Am., 142(2), pp. 1011–1024).
Nonlinear effects of liquid motion inside rigid and elastic shells such as rocket propellant tanks
Spherical caps axisymmetric static and dynamic buckling under load, using axisymmetric nonlinear elastic shell theory approximation and finite difference equations
Spherical caps axisymmetric static and dynamic buckling under load, using axisymmetric nonlinear elastic shell theory approximation and finite difference equations
Structural elastic shell theory for axisymmetric hollow circular cylinders
Orbiting astronomical observatories have the potential for making observations far superior to those from earth-based mirrors. In order for this performance to be realized, the contour of the primary mirror must be very accurately controlled. A preliminary investigation of the use of thermally induced elastic strains for correcting axisymmetric deformations in space telescope mirrors has been presented. The relation between axial deformation and thermal inputs was determined by a finite difference solution of the equations for thin elastic shells. The use of this technique was demonstrated analytically on a beryllium paraboloid. This mirror had 10 equally spaced thermal inputs and results are presented which show the nature of the temperature distribution required to correct deformations due to an acceleration-type loading.
A series representation is presented of the dynamic behavior of incompressible non-viscous liquids contained in partially filled elastic shells of revolution. Using a series in which each term satisfies exactly Laplace's equation, the liquid portion of the problem becomes an integral equation on the wetted wall of the shell. Physical insight required in selecting the terms of the series is such that selection may be readily programed on a computer. Liquid-shell vibration interaction is investigated by modifying an existing shell of revolution vibration program with the series expansion of the liquid pressure loading on the wetted shell wall. Application of the modified program is demonstrated on a hemispherical tank and a realistic scaled model of a launch vehicle tank having a more complex geometry. Comparison of the present analysis with a NASTRAN finite element analysis for a hemispherical tank, and comparison with test results for a launch vehicle tank shows excellent agreement and rapid convergence of the series representation of the liquid to the true solution.