Dynamics of biological soft tissue and rubber: internally pressurized spherical membranes surrounded by a fluid

The behavior of a family of dynamical systems representing the elastodynamic response of an internally pressurized, non-linearly elastic spherical membrane lying in an incompressible external fluid is governed primarily by the strain energy function for the membrane, the specific forcing function due to the internal pressure, and the viscosity of the external fluid. It is shown that such systems with an inviscid external fluid and having a constant internal pressure are integrable but not Hamiltonian. Under periodic internal loading, and for a small spherical radius and constitutive relations typical of many biological soft tissues, a periodic orbit in phase space exists near a static equilibrium. A viscous external fluid causes the periodic orbit to be an attractor. The dynamical system is robust under small loading perturbations common in normal biological systems. Rubber models, on the other hand, may admit structural catastrophes. For small initial sphere radii, a jump from one periodic orbit to another is possible for rubber models but not for the classical soft tissue models. It is dangerous, therefore, to model soft biological tissue as a rubber either mathematically or physically in experiments because the predicted instabilities may not exist in tissue.     

Indentation of pressurized viscoplastic polymer spherical shells

The indentation response of polymer spherical shells is investigated. Finite deformation analyses are carried out with the polymer characterized as a viscoelastic/viscoplastic solid. Both pressurized and unpressurized shells are considered. Attention is restricted to axisymmetric deformations with a conical indenter. The response is analyzed for various values of the shell thickness to radius ratio and various values of the internal pressure. Two sets of material parameters are considered: one set having network stiffening at a moderate strain and the other having no network stiffening until very large strains are attained. The transition from an indentation type mode of deformation to a structural mode of deformation involving bending that occurs as the indentation depth increases is studied. The results show the effects of shell thickness, internal pressure and polymer constitutive characterization on this transition and on the deformation modes in each of these regimes.     

Effect of thickness inhomogeneities in internally pressurized elastic-plastic spherical shells

The behaviour of elastic-plastic spherical shells under internal pressure is investigated numerically for thickness-to-radius ratios ranging from cases of thin shells to very thick shells. The shells under consideration are made of strain-hardening elastic-plastic material with a smooth yield-surface. Attention is restricted to axisymmetric deformations, and results are presented for initial thickness inhomogeneities in various axisymmetric shapes. For smooth thickness-variations in the shape of the critical bifurcation mode, the reduction in maximum pressure is studied together with the distribution of deformations in the final collapse mode. Also, the possibility of flow localization due to more localized, initially thin regions on a spherical shell is investigated.     

Inhomogeneous spherical configurations of inflated membranes

Based on physically meaningful choice of the strain measures, we study the equilibrium and stability of an inflated spherical membrane. First, we obtain general results deduced by global geometric properties and then we analyze the possibility of inhomogeneous configurations. The stability analysis shows that under special constitutive assumptions the global energy minimum can be attained by inhomogeneous spherical configurations that we analytically describe. We argue that these deformations can reproduce well-known experimental results.     

Lateral stability of lofty air-supported spherical membranes

The lateral stability of a lofty air supported spherical membrane subjected to an unsymmetrically applied concentrated load is treated. Simple expressions for the ‘critical’ load are obtained at which the configuration of the structure changes dramatically and part of the membrane comes into contact with the support. The value of this load depends strongly on the area of the support and decreases with increasing eccentricity of the load.     

Inflation and bifurcation of compressible spherical membranes

The inflation and bifurcation of spherical membranes is considered. The membrane material is assumed to be isotropic and hyperelastic but may be arbitrarily compressible. Qualitatively the behaviour of compressible membranes is shown to be the same as that of incompressible membranes but specific forms of strain-energy functions are chosen to illustrate possible quantitative differences.     

Effects of uni-directional geometric imperfections on vibrations of pressurized shallow spherical shells

This paper deals with the effects of initial geometric uni-directional imperfections on vibrations of a pressurized spherical shell or spherical cap. The analysis is based upon shallow shell theory. Frequency vs applied pressure interaction curves are plotted for various values of the imperfection amplitude. Imperfections are shown to have a severe effect in reducing the natural frequencies similar to that demonstrated in the buckling behavior of spherical shells.     

Elastic analysis of thick-walled pressurized spherical vessels coated with functionally graded materials

In recent years, functionally graded material (FGM) has been widely explored in coating technology amongst both academic and industry communities. FGM coatings are suitable substitutes for many typical conventional coatings which are susceptible to cracking, debonding and eventual functional failure due to the mismatch of material properties at the coating/substrate interface. In this study, a thick spherical pressure vessel with an inner FGM coating subjected to internal and external hydrostatic pressure is analyzed within the context of three-dimensional elasticity theory. Young’s modulus of the coating is assumed to vary linearly or exponentially through the thickness, while Poisson’s ratio is considered as constant. A comparative numerical study of FGM versus homogeneous coating is conducted for the case of vessel under internal pressure, and the dependence of stress and displacement fields on the type of coating is examined and discussed.     

Novel material tailoring method for internally pressurized FG spherical and cylindrical vessels

A new material tailoring method for spherical and cylindrical vessels made of functionally graded materials (FGMs) is presented. It is assumed that the FG material is composed of an Al–SiC metallic-matrix composite. A uniform ratio of in-plane shear stress to yield strength [\(\varphi \left( r \right) \)] is used as the design criterion to utilize the maximum capacity of the vessel. The aim is to find a distribution of SiC particles in the radial direction, i.e., \(f\left( r \right) \), that achieves a uniform index \(\varphi \left( r \right) =\hbox {const}.\) through the wall thickness of the internally pressurized spherical or cylindrical vessel. Both the Mori–Tanaka and rule-of-mixtures homogenization schemes are used to express the effective elastic module and Poisson’s ratio. Moreover, the strength of the composite is expressed based on the rule of mixtures. Besides, finite element simulation is carried out to verify the accuracy of the analytical solution. The effects of input parameters such as the internal pressure, strength of the SiC particles, ratio of in-plane shear stress to effective yield strength, and choice of homogenization scheme on the tailored distribution of the SiC volume fraction in the radial direction are also investigated.     

Post-bifurcation of perfect and imperfect spherical elastic membranes

When a spherical elastic membrane is inflated it is well known that it may bifurcate into an aspherical mode after the pressure maximum is reached. Upon further inflation the spherical configuration is regained. Here we follow the developing aspherical solution path, for specific forms of strain-energy function, using a simple numerical method. For a realistic strain-energy function it is shown that the post-bifurcation solution curve connects the two bifurcation points. We also consider the inflation of imperfect spherical membranes and show that bifurcation may still occur. For the class of Ogden materials we investigate the asymptotic shape of arbitrary axisymmetric membranes.     

Ponding instability of air-supported spherical membranes with initial imperfections

The stability of a spherical, air-supported membrane subjected to a concentrated load and an accumulating ponding fluid is the subject of the present paper which consists of two parts: in the first part a spherical elastic membrane with an initial imperfection at the apex is considered. The effect of the ponding fluid accumulating in the initial depression reduces the value of the critical load significantly. Simple formulae for the critical load are obtained. In the second part, the case of a non-symmetric load applied to the membrane is considered. Results indicate that the eccentricity causes a significant increase in the value of the critical load thereby making the axisymmetric loading the governing configuration.     

On the dynamic snap-out instability of inflated non-linear spherical membranes

The dynamic behavior of a spherical membrane, made of Mooney material and subjected to a uniform inflating step-pressure, is studied. Its phase-planes and nondimensional radius versus time curves are plotted for different values of the material constants. The conditions for dynamic snapout instability are discussed. The relations between the static behavior of the membrane and the dynamic instability are pointed out.     

Necking of elasto-plastic rods under tension

This work is concerned with the modeling of the necking phenomenon in metallic tension members using a non-linear theory of elasto-plastic rods with deformable cross section. The study of the coupling between the axial deformation and the cross section deformation in tensile specimens is a basic step to understand the strain-softening and localization behaviors in more complex metallic structures. The main features of the model are illustrated through the simulation of the necking of an ASTM 6351 aluminium bar under tension.     

光滑圆柱拉伸试件的颈缩分析

对Gurson本构方程作了初步的研究，并对圆柱光滑拉伸试件在颈缩阶段用Gurson本构方程做了大应变弹塑性有限元分析。讨论了颈缩区空穴形核、扩张、静水应力以及材料软化的问题，初步揭示了空穴的演化过程和材料的破坏机理。有限元分析的结果表明，颈缩阶段空穴长大聚合机理非常显著，而形核作用相对较弱。     

Plasticity and creep of pressurized membranes: A new look at the small-deflection theory

The Mechanics of the bulge test at small deflections has not been adequately investigated in the past, especially when the stress dependence on strain and/or strain-rate is represented by a power law. For this type of material response, and for die apertures of any shape, we prove new theorems relating to the extremal characterization and uniqueness of the solution, and to a general correspondence between the respective solutions for viscoplasticily or creep on the one hand and time-independent plasticity on the other. The basic problem is here solved exactly for a circular membrane, and to a close approximation for elliptical ones; the results of extensive numerical calculations are presented in full. With this background the influence of material response and tool geometry on the inherent inhomogeneity of the test can be clearly seen. A proper appreciation of these effects is of course essential when interpreting observational data.     

Some qualitative properties of incompressible hyperelastic spherical membranes under dynamic loads

Some nonlinear dynamic properties of axisymmetric deformation are ex- amined for a spherical membrane composed of a transversely isotropic incompressible Rivlin-Saunders material. The membrane is subjected to periodic step loads at its inner and outer surfaces. A second-order nonlinear ordinary differential equation approximately describing radially symmetric motion of the membrane is obtained by setting the thick- ness of the spherical structure close to one. The qualitative properties of the solutions are discussed in detail. In particular, the conditions that control the nonlinear periodic oscillation of the spherical membrane are proposed. In certain cases, it is proved that the oscillating form of the spherical membrane would present a homoclinic orbit of type ＂∞＂, and the amplitude growth of the periodic oscillation is discontinuous. Numerical results are provided.     

Necking of an anisotropic plastic material

The two conditions for stable necking, treated as a bifurcation of stress path, are derived. The anisotropic material considered is one for which the plastic work increment = $\overline{\sigma }d\overline{e}$, where both the generalized yield stress $\overline{\sigma }$ and generalized plastic strain increment $d\overline{e}$ are invariant functions. The method is applied to the necking of a thin cylinder under internal pressure.     

Necking of biaxially stretched elastic-plastic circular plates

The necking of an elastic-plastic circular plate under uniform radial tensile loading is investigated both within the framework of the three-dimensional theory and within the context of the plane-stress approximation. Attention is restricted to axisymmetric deformations of the plate. The material behavior is described by two different constitutive laws. One is a finite-strain version of the simplest flow-theory of plasticity and the other is a finite-strain generalization of the simplest deformationtheory, which is employed as a simple model of a solid with a vertex on its yield surface. For an initially uniform plate made of an incompressible material, bifurcation from the uniformly stretched state is studied analytically. The regimes of stress and moduli where the governing axisymmetric three-dimensional equations are elliptic, parabolic or hyperbolic are identified. The plane-stress local-necking mode emerges as the appropriate limiting mode from the bifurcation modes available in the elliptic regime. In the elliptic regime, the main qualitative features of the bifurcation behavior are revealed by the plane-stress analysis, although three-dimensional effects delay the onset of necking somewhat. For the deformation theory employed here, the first bifurcation modes are encountered in the parabolic regime if the hardening-rate is sufficiently high. These bifurcations are not revealed by a plane-stress analysis. For a plate with an initial inhomogeneity, the growth of an imperfection is studied by a perturbation method, by a plane-stress analysis of localized necking, and by numerical computations within the framework of the three-dimensional theory. When bifurcation of the corresponding perfect plate takes place in the elliptic regime, the finite element results show that the plane-stress analysis gives reasonably good agreement with the numerical results. When bifurcation of the corresponding perfect plate first occurs in the parabolic regime, then a bifurcation of the imperfect plate is encountered, that is, the finite element stiffness matrix ceases to be positive definite.     

Necking of anisotropic micro-films with strain-gradient effects

Necking of stubby micro-films of aluminum is investigated numerically by considering tension of a specimen with an initial imperfection used to onset localisation. Plastic anisotropy is represented by two different yield criteria and strain-gradient effects are accounted for using the visco-plastic finite strain model. Furthermore, the model is extended to isotropic anisotropic hardening （evolving anisotropy）. For isotropic hardening plastic anisotropy affects the predicted overall nominal stress level, while the peak stress remains at an overall logarithmic strain corresponding to the hardening exponent. This holds true for both local and nonlocal materials. Anisotropic hardening delays the point of maximum overall nominal stress.     

Strain gradient plasticity solution for an internally pressurized thick-walled spherical shell of an elastic–plastic material

An analytical solution is presented for an internally pressurized thick-walled spherical shell of an elastic strain-hardening plastic material. A strain gradient plasticity theory is used to describe the constitutive behavior of the material undergoing plastic deformations, whereas the generalized Hooke’s law is invoked to represent the material response in the elastic region. The solution gives explicit expressions for the stress, strain and displacement components. The inner radius of the shell enters these expressions not only in non-dimensional forms but also with its own dimensional identity, unlike classical plasticity-based solutions. As a result, the current solution can capture the size effect. The classical plasticity-based solution of the same problem is shown to be a special case of the present solution. Numerical results for the maximum effective stress in the shell wall are also provided to illustrate applications of the newly derived solution. The new solution can be used to construct improved expanding cavity models in indentation mechanics that incorporate both the strain-hardening and indentation size effects.