Viscoelastic Spherical Shells Research Articles

Buckling of viscoelastic spherical shells

Viscoelastic spherical shells exhibit a wide range of time/rate-dependent buckling behaviors when subjected to pressure. For certain loadings, buckling can even occur after a significant time delay, termed creep buckling. To gain a thorough understanding of the nonlinear time-dependent buckling behavior of viscoelastic spherical shells, this work develops an analytical model employing the small-strain, moderate-rotation shell theory combined with a linearly viscoelastic material law. Numerical results are presented for axisymmetric spherical shells with geometric imperfections for two types of loading: a prescribed rate of volume change and a prescribed pressure that remains constant after it is applied. The first type reveals the rate-dependent behavior of viscoelastic buckling while the constant pressure loading is used to quantify creep buckling phenomena. The results show that viscoelasticity and loading rates play important roles in the load-carrying behavior of these shells, and the results for the constant pressure loading reveal an unexpected and important connection between the short-time elastic buckling limit and the long-time creep buckling limit. An imperfection sensitivity map is constructed for the constant pressure loading showing three regimes with qualitatively different behaviors: near-instantaneous buckling, creep buckling and no buckling.

Acoustic radiation torque of a Bessel vortex wave on a viscoelastic spherical shell nearby an impedance boundary

In this investigation, a comprehensive analytical formalism is presented for the acoustic radiation torque exerted on a viscoelastic spherical shell nearby an impedance boundary in a Bessel vortex wave field, whose incident direction is perpendicular to the boundary. Based on the sound scattering theory, an infinite series expansion of the axial acoustic radiation torque is derived by applying the series expansion method, the image theory and the translational addition theorem. Computations of the non-dimensional axial acoustic radiation torque for a viscoelastic polyethylene shell are considered with emphasis on the effects of the reflecting coefficient of the interface, the shell-boundary distance, the shell's relative thickness and the half-cone angle. It is shown that the high-amplitude radiation torque peaks exist resulting from different vibration modes at resonance scattering frequencies, which are significantly enhanced when the boundary becomes increasingly rigid. The periodical change of acoustic radiation torque is also observed as the shell moves along the wave axis. With the increase of wave order, the high-amplitude “islands” shift to larger half-cone values. The results of this study can improve our understanding of the acoustic radiation torque behaviors for viscoelastic particles near the boundary, which is commonly encountered in biomedical ultrasound and fluid dynamics.

Modelling of the collision of two viscoelastic spherical shells

In the present paper, the collision of two viscoelastic spherical shells is investigated using the wave theory of impact. The model developed here suggests that after the moment of impact quasi-longitudinal and quasi-transverse shock waves are generated, which then propagate along the spherical shells. The solution behind the wave fronts is constructed with the help of the theory of discontinuities. Since the local bearing of the materials of the colliding viscoelastic shells is taken into account, the solution in the contact domain is found via the modified Hertz contact theory involving the operator representation of viscoelastic analogs of Young’s modulus and Poisson’s ratio. The collision of two elastic spherical shells is considered first, and then using Volterra correspondence principle, according to which the elastic constants in the governing equations should be replaced by the corresponding viscoelastic operators, the solution obtained for elastic shells is extended over the case of viscoelastic shells.

Predicting surface dynamic topographies of stagnant lid planetary bodies

Although planetary mantles are viscoelastic media, numerical models of thermal convection in a viscoelastic spherical shell are still very challenging. Here, we examine the validity of simplified mechanical and rheological frameworks classically used to approximate viscoelastic dynamic topography. We compare three simplified approaches to a linear Maxwell viscoelastic shell with a pseudo upper free-surface, considered as the reference model. A viscous model with a free-slip boundary condition at the surface correctly reproduces the final relaxed shape of the viscoelastic body but it cannot reproduce the time evolution of the viscoelastic topography. Nevertheless, characterizing the topography development is important since it can represent a significant fraction of the history for planets having a thick and rigid lithosphere (e.g. Mars). A viscous model with a pseudo free-surface, despite its time-dependency, also systematically fails to describe correctly these transient stages. An elastic filtering of the instantaneous viscous topography is required to capture the essence of the time evolution of the topography. We show that a single effective elastic thickness is needed to correctly reproduce the constant transient viscoelastic topography obtained when the lithosphere corresponds to a step-like viscosity variation, while a time-dependence of the effective elastic thickness must be considered to take account of realistic temperature-dependent viscosity variations in the lithosphere. In this case, the appropriate thickness of the elastic shell can be evaluated, at a given instant, with a simple procedure based on the local Maxwell time. Furthermore, if the elastic filtering is performed using the thin elastic shell formulation, an unrealistic degree-dependence of the thickness of the elastic shell is needed to correctly approximate the viscoelastic topography. We show that a model that fully couples a viscous body to an elastic shell of finite thickness estimated using the local Maxwell time gives the best approximation of the viscoelastic deformation, whatever the degree of the load and the time of loading.

Analysis of two colliding fractionally damped spherical shells in modelling blunt human head impacts

AbstractThe collision of two elastic or viscoelastic spherical shells is investigated as a model for the dynamic response of a human head impacted by another head or by some spherical object. Determination of the impact force that is actually being transmitted to bone will require the model for the shock interaction of the impactor and human head. This model is indended to be used in simulating crash scenarios in frontal impacts, and provide an effective tool to estimate the severity of effect on the human head and to estimate brain injury risks. The model developed here suggests that after the moment of impact quasi-longitudinal and quasi-transverse shock waves are generated, which then propagate along the spherical shells. The solution behind the wave fronts is constructed with the help of the theory of discontinuities. It is assumed that the viscoelastic features of the shells are exhibited only in the contact domain, while the remaining parts retain their elastic properties. In this case, the contact spot is assumed to be a plane disk with constant radius, and the viscoelastic features of the shells are described by the fractional derivative standard linear solid model. In the case under consideration, the governing differential equations are solved analytically by the Laplace transform technique. It is shown that the fractional parameter of the fractional derivative model plays very important role, since its variation allows one to take into account the age-related changes in the mechanical properties of bone.

Generalized theory of resonance excitation by sound scattering from an elastic spherical shell in a nonviscous fluid

This work presents the general theory of resonance scattering (GTRS) by an elastic spherical shell immersed in a nonviscous fluid and placed arbitrarily in an acoustic beam. The GTRS formulation is valid for a spherical shell of any size and material regardless of its location relative to the incident beam. It is shown here that the scattering coefficients derived for a spherical shell immersed in water and placed in an arbitrary beam equal those obtained for plane wave incidence. Numerical examples for an elastic shell placed in the field of acoustical Bessel beams of different types, namely, a zero-order Bessel beam and first-order Bessel vortex and trigonometric (nonvortex) beams are provided. The scattered pressure is expressed using a generalized partial-wave series expansion involving the beam-shape coefficients (BSCs), the scattering coefficients of the spherical shell, and the half-cone angle of the beam. The BSCs are evaluated using the numerical discrete spherical harmonics transform (DSHT). The far-field acoustic resonance scattering directivity diagrams are calculated for an albuminoidal shell immersed in water and filled with perfluoropropane gas, by subtracting an appropriate background from the total far-field form function. The properties related to the arbitrary scattering are analyzed and discussed. The results are of particular importance in acoustical scattering applications involving imaging and beam-forming for transducer design. Moreover, the GTRS method can be applied to investigate the scattering of any beam of arbitrary shape that satisfies the source-free Helmholtz equation, and the method can be readily adapted to viscoelastic spherical shells or spheres.

Nanorheology of viscoelastic shells: Application to viral capsids

We study the microrheology of nanoparticle shells [A.D. Dinsmore et al., Science 298, 1006 (2002)] and viral capsids [I.L. Ivanovska et al., Proc. Natl. Acad. Sci. U.S.A. 101, 7600 (2004)] by computing the mechanical response function and thermal fluctuation spectrum of a viscoelastic spherical shell that is permeable to the surrounding solvent.We determine analytically the coupled dynamics of bending and compression modes of the viscoelastic shell coupled to solvent flows inside, outside, and through the porous viral capsid. From this calculation, we identify fundamental length and time scales in the system, and compute the finite-frequency response of the shell to pinching forces applied at antipodal points on it.This calculation determines the mechanical response of a viral capsid to an AFM-based nanoindentation study. Using these calculations, one can interpret such studies in terms of elastic and dissipative response of the viral capsid to bending and compressional deformations and thereby gain new information regarding the internal deformations of individual capsomers under stress, providing insight into their intermolecular and intramolecular interactions. As an example we study the mechanical effect of a simple allosteric transition in the capsomers.The frequency-dependent mechanics of complex, viscoelastic and possibly porous spherical shells is not only application to viruses, but also may lead to new measurements of colloidosomes, nanoparticle shells, and lipid vesicles including those that contain pore-forming proteins.

Nanorheology of viscoelastic shells: Applications to viral capsids

We study the microrheology of nanoparticle shells [A. D. Dinsmore, Science 298, 1006 (2002)] and viral capsids [I. L. Ivanovska, Proc. Natl. Acad. Sci. U.S.A. 101, 7600 (2004)] by computing the mechanical response function and thermal fluctuation spectrum of a viscoelastic spherical shell that is permeable to the surrounding solvent. We determine analytically the damped dynamics of bend and compression modes of the shell coupled to the solvent both inside and outside the sphere in the zero Reynolds number limit. We identify fundamental length and time scales in the system, and compute the thermal correlation function of displacements of antipodal points on the sphere and the mechanical response to pinching forces applied at these points. We describe how such a frequency-dependent antipodal correlation and/or response function, which should be measurable in new AFM-based microrheology experiments, can probe the viscoelasticity of these synthetic and biological shells constructed of nanoparticles.

Amplitude-modulated acoustic radiation force experienced by elastic and viscoelastic spherical shells in progressive waves

The dynamic acoustic radiation force resulting from a dual-frequency beam incident on spherical shells immersed in an inviscid fluid is examined theoretically in relation to their thickness and the contents of their interior hollow regions. The theory is modified to include a hysteresis type of absorption inside the shells’ material. The results of numerical calculations are presented for stainless steel and absorbing lucite (PolyMethyMethacrylAte) shells with the hollow region filled with water or air. Significant differences occur when the interior fluid inside the hollow region is changed from water to air. It is shown that the dynamic radiation force function Y d deviates from the static radiation force function Y p when the modulation size parameter δ x = ∣ x 2 − x 1∣ ( x 1 = k 1 a, x 2 = k 2 a, k 1 and k 2 are the wave vectors of the incident ultrasound waves, and a is the outer radius of the shell) starts to exceed the width of the resonance peaks in the Y p curves.

Acoustic radiation force acting on elastic and viscoelastic spherical shells placed in a plane standing wave field

The theory of the acoustic radiation force acting on elastic spherical shells suspended in a plane standing wave field is developed in relation to their thickness and the content of their hollow regions. The theory is modified to include the effect of a hysteresis type of absorption of compressional and shear waves in the material. The fluid-loading effect on the acoustic radiation force function Y st is analyzed as well. Results of numerical calculations are presented for a number of elastic and viscoelastic materials, with the hollow region filled with water or air. These results show how the damping due to absorption, the change of the interior fluid inside the shells’ hollow regions, and the exterior fluid surrounding their structures, affect the acoustic radiation force.

Dynamics of a viscoelastic spherical shell with a nonconvex strain energy function

We study the radial motion of an incompressible viscoelastic spherical shell with a nonconvex strain energy function that models a material that can undergo a phase transition. In addition to the classical Newtonian viscosity for viscoelastic materials, we consider a material with two microstructural coefficients that are supposed to sense local configurational changes that take place during a deformation. Conditions necessary to show the effect of the nonconvexity of the strain energy function during a phase transition of the material, are determined, and the resulting dynamics is analyzed. It is shown that, though small periodic vibrations are possible, the system can easily revert into a mode of large amplitude motion as a result of small external excitation. Such motion may be transient to periodic motion or to chaotic motion. Boundaries in parameter space for the occurrence of this type of motion are determined and examples are given.

Dynamics of viscoelastic spherical shells during internal explosive loading

The dynamics of a thick-walled closed spherical shell was investigated theoretically and experimentally for determining the dynamic viscoelastic characteristics of steel St. 3. A liquid explosive charge placed in a capsule discharge in the center of a flask was used for uniform loading. The value of the logarithmic decrement was obtained from the results of measuring the meridional deformation by means of a strain-gauge complex.

The response of brain to transient elevations in intraventricular pressure

A mathematical theory has been developed to evaluate the change in size of the cerebral ventricles in response to a square-wave pressure pulse within the ventricles. Specifically, the theory utilizes a lumped compartment viscoelastic spherical shell model of the brain and takes into account available creep compliance data. The pressure change in the cortical subarachnoid space plays a critical role in determining the net displacement of the brain. The rise in cortical subarachnoid pressure is governed by a rate constant γ which is nearly proportional to the absorption coefficient when absorption is rate-limiting, inversely proportional to the compliance of the spinal subarachnoid space, and also depends upon the resistance to flow through CSF pathways. A rapid rise in pressure within the cortical subarachnoid space results in an inward displacement of the ventricles immediately after the ventricular pressure pulse has ceased. Thereafter, the ventricles return toward their initial size as the pressure in the cortical subarachnoid space decays. A slow rise, on the other hand, favors permanent ventricular enlargement. Decreased absorption of CSF, long duration pulses, and bulk flow of CSF into brain, which are observed in hydrocephalus, all favor the development of ventricular enlargement.

Large Deflections of a Linearly Viscoelastic Shallow Spherical Shell

Numerical solutions for the time history of motion of a shallow viscoelastic spherical shell are presented under large deformations. Axisymmetric motion is considered due to uniform pressure with arbitrary time variation. The dynamic viscoelastic buckling load is established by comparing long-time equilibrium states due to small changes in the applied pressure. The problem is formulated as a coupled pair of nonlinear integro-differential equations which is solved by Newton’s method.

Transmission of Acoustic Waves through Submerged Viscoelastic Spherical Shells

A pulsating sphere is located inside a thick concentric viscoelastic spherical shell, which is surrounded by an infinite acoustic medium. The space between the sphere and the shell is filled with the same acoustic medium. A shaped beam of axisymmetric spherical harmonic wave, emitted from the sphere, is transmitted through the shell to the outside medium. The shell is analyzed by use of elastic wave potentials. The emitted acoustic waves are transmitted to the external medium by coupling to the shell through continuity and boundary conditions at the inner and outer surfaces of the shell. The investigation shows that the radiated farfield pressure can be attenuated by as much as 10% through the use of stiff polymers.

Attenuation of Acoustic Waves by Submerged Viscoelastic Spherical Shells

A pulsating sphere is located inside a thick concentric viscoelastic spherical shell, which is surrounded by an infinite acoustic medium. The space between the sphere and the shell is filled with the same acoustic medium. A shaped beam of axisymmetric spherical harmonic wave, emitted from the sphere, is transmitted through the shell to the outside medium. The shell is analyzed by use of elastic wave potentials. The emitted acoustic waves are transmitted to the external medium by coupling to the shell through continuity and boundary conditions at the inner and outer surfaces of the shell. The investigation shows that the radiated farfield pressure can be attenuated by as much as 10% through the use of stiff polymers.

Uniform Spherical Radiation through Thick Shells

Wave propagation from a spherical transducer through a fluid medium and a concentric viscoelastic spherical shell (not necessarily thin) to a fluid medium of infinite extent has been analyzed. Formulas for the acoustic field when the transducer is driven with constant-amplitude sinusoidal radial velocity, have been derived. Numerical results show that the insertion loss due to the presence of the shell is an oscillatory function of frequency, indicating the existence of regions of favorable and unfavorable transmission, which are primarily controlled by the transducer-shell spacing. The magnitudes of the maxima and minima of the insertion loss are determined primarily by the mass per unit area of the shell. The insertion-loss function does not depend strongly upon the elastic constants or the material dissipation of the shell. A simple model consisting of a fluid-filled tube with a piston and a spring-supported “masslike” wall exposed to ambient fluid has been found to be a useful tool for interpreting the results. The analysis has been extended to the case of uniform radiation in the presence of a multilayered viscoelastic spherical shell. Numerical results show that by properly choosing the parameters, a low-pass filter effect can be obtained.

Response of shallow viscoelastic spherical shells to time-dependent axisymmetric loads

Response of shallow viscoelastic spherical shells to time-dependent axisymmetric loads