Some solutions for a compressible isotropic elastic material

In this article we obtain closed-form solutions for the combined inflation and axial shear of an elastic tube in respect of the compressible Isotropic elastic material introduced by Levinson and Burgess. Several other boundary-value problems are also examined, including the bending of a rectangular block and straightening of a cylindrical sector, both coupled with stretching and shearing, and an axially varying twist deformation. Some of the solutions appear in closed form, others are expressible in terms of elliptic functions.

Generalized azimuthal shear deformations in compressible isotropic elastic materials

In this article we study the azimuthal shear deformations in a compressible Isotropic elastic material. This class of deformations involves an azimuthal displacement as a function of the radial and axial coordinates. The equilibrium equations are formulated in terms of the Cauchy-Green strain tensors, which form an overdetermined system of partial differential equations for which solutions do not exist in general. By means of a Legendre transformation, necessary and sufficient conditions for the material to support this deformation are obtained explicitly, in the sense that every solution to the azimuthal equilibrium equation will satisfy the remaining two equations. Additionally, we show how these conditions are sufficient to support all currently known deformations that locally reduce to simple shear. These conditions are then expressed both in terms of the invariants of the Cauchy-Green strain and stretch tensors. Several classes of strain energy functions for which this deformation can be supported are studied. For certain boundary conditions, exact solutions to the equilibrium equations are obtained. © 2005 Society for Industrial and Applied Mathematics.

On extension and torsion of a compressible elastic circular cylinder

In this paper we examine the combined extension and torsion of a compressible isotropic elastic cylinder of finite extent. The equilibrium equations are formulated in terms of the principal stretches and then applied to the special case of pure torsion superimposed on a uniform extension (an isochoric deformation). Explicit necessary and sufficient conditions on the strain-energy function for the material to support this deformation with vanishing traction on the lateral surfaces of the cylinder are obtained. Some strain-energy functions satisfying these conditions are considered, existing results are recovered as special cases and new results are obtained. We also point out how the strain-energy functions generated from the considered isochoric deformation considered (of a compressible material) can be used to generate energy functions and corresponding solutions for the incompressible theory.

Predictions of angle dependent tortuosity and elasticity effects on sound propagation in cancellous bone

The anisotropic pore structure and elasticity of cancellous bone cause wave speeds and attenuation in cancellous bone to vary with angle. Previously published predictions of the variation in wave speed with angle are reviewed. Predictions that allow tortuosity to be angle dependent but assume isotropic elasticity compare well with available data on wave speeds at large angles but less well for small angles near the normal to the trabeculae. Claims for predictions that only include angle-dependence in elasticity are found to be misleading. Audio-frequency data obtained at audio-frequencies in air-filled bone replicas are used to derive an empirical expression for the angle-and porosity-dependence of tortuosity. Predictions that allow for either angle dependent tortuosity or angle dependent elasticity or both are compared with existing data for all angles and porosities.

A node-based smoothed conforming point interpolation method (NS-CPIM) for elasticity problems

This paper formulates a node-based smoothed conforming point interpolation method (NS-CPIM) for solid mechanics. In the proposed NS-CPIM, the higher order conforming PIM shape functions (CPIM) have been constructed to produce a continuous and piecewise quadratic displacement field over the whole problem domain, whereby the smoothed strain field was obtained through smoothing operation over each smoothing domain associated with domain nodes. The smoothed Galerkin weak form was then developed to create the discretized system equations. Numerical studies have demonstrated the following good properties: NS-CPIM (1) can pass both standard and quadratic patch test; (2) provides an upper bound of strain energy; (3) avoid the volumetric locking; (4) provides the higher accuracy than those in the node-based smoothed schemes of the original PIMs.

Evaluation cortical bone elasticity in response to pulse power excitation using ultrasonic technique

This paper presents the ultrasonic velocity measurement method which investigates the possible effects of high voltage high frequency pulsed power on cortical bone material elasticity. Before applying a pulsed power signal on a live bone, it is essential to determine the safe parameters of pulsed power applied on bone non-destructively. Therefore, the possible changes in cortical bone material elasticity due to a specified pulsed power excitation have been investigated. A controllable positive buck-boost converter with adjustable output voltage and frequency has been used to generate high voltage pulses (500V magnitude at 10 KHz frequency). To determine bone elasticity, an ultrasonic velocity measurement has been conducted on two groups of control (unexposed to pulse power but in the same environmental condition) and cortical bone samples exposed to pulsed power. Young’s modulus of cortical bone samples have been determined and compared before and after applying the pulsed power signal. After applying the high voltage pulses, no significant variation in elastic property of cortical bone specimens was found compared to the control. The result shows that pulsed power with nominated parameters can be applied on cortical bone tissue without any considerable negative effect on elasticity of bone material.

A novel quadratic Edge-based Smoothed Conforming Point Interpolation Method (ES-CPIM) for elasticity problems

his paper formulates an edge-based smoothed conforming point interpolation method (ES-CPIM) for solid mechanics using the triangular background cells. In the ES-CPIM, a technique for obtaining conforming PIM shape functions (CPIM) is used to create a continuous and piecewise quadratic displacement field over the whole problem domain. The smoothed strain field is then obtained through smoothing operation over each smoothing domain associated with edges of the triangular background cells. The generalized smoothed Galerkin weak form is then used to create the discretized system equations. Numerical studies have demonstrated that the ES-CPIM possesses the following good properties: (1) ES-CPIM creates conforming quadratic PIM shape functions, and can always pass the standard patch test; (2) ES-CPIM produces a quadratic displacement field without introducing any additional degrees of freedom; (3) The results of ES-CPIM are generally of very high accuracy.

Adaption and use of a compressible flow code for turbomachinery design

The present paper presents and discusses the use of dierent codes regarding the numerical simulation of a radial-in ow turbine. A radial-in ow turbine test case was selected from published literature [1] and commercial codes (Fluent and CFX) were used to perform the steady-state numerical simulations. An in-house compressible- ow simulation code, Eilmer3 [2] was also adapted in order to make it suitable to perform turbomachinery simulations and preliminary results are presented and discussed. The code itself as well as its adaptation, comprising the addition of terms for the rotating frame of reference, programmable boundary conditions for periodic boundaries and a mixing plane interface between the rotating and non-rotating blocks are also discussed. Several cases with dierent orders of complexity in terms of geometry were considered and the results were compared across the dierent codes. The agreement between these results and published data is also discussed.

The modulus of elasticity of steel - is it 200 GPa?

In cold-formed steel construction, the use of a range of thin, high strength steels (0.35 mm thickness and 550 MPa yield stress) has increased significantly in recent times. A good knowledge of the basic mechanical properties of these steels is needed for a satisfactory use of them. In relation to the modulus of elasticity, the current practice is to assume it to be about 200 GPa for all steel grades. However, tensile tests of these steels have consistently shown that the modulus of elasticity varies with grade of steel and thickness. It was found that it increases to values as high as 240 GPa for smaller thicknesses and higher grades of steel. This paper discusses this topic, presents the tensile test results for a number of steel grades and thicknesses, and attempts to develop a relationship between modulus of elasticity, yield stress and thickness for the steel grades considered in this investigation.

Fine tuning of elasticity via crosslinking collagen-based materials to mediate mechanotransduction and stability using corona treatment

The common goal of tissue engineering is to develop substitutes that can closely mimic the structure of extracellular matrix (ECM). However, similarly important is the intensive material properties which have often been overlooked, in particular, for soft tissues that are not to bear load assumingly. The mechanostructural properties determine not only the structural stability of biomaterials but also their physiological functionality by directing cellular activity and regulating cell fate decision. The aim here is to emphasize that cells could sense intensive material properties like elasticity and reside, proliferate, migrate and differentiate accordinglyno matter if the construct is from a natural source like cartilage, skin etc. or of synthetic one. Meanwhile, the very objective of this work is to provide a tunable scheme for manipulating the elasticity of collagen-based constructs to be used to demonstrate how to engineer cell behavior and regulate mechanotransduction. Articular cartilage was chosen as it represents one of the most complex hierarchical arrangements of collagen meshwork in both connective tissues and ECM-like biomaterials. Corona discharge treatment was used to produce constructs with varying density of crosslinked collagen and stiffness accordingly. The results demonstrated that elastic modulus increased up to 33% for samples treated up to one minute as crosslink density was found to increase with exposure time. According to the thermal analysis, longer exposure to corona increased crosslink density as the denaturation enthalpy increased. However the spectroscopy results suggested that despite the stabilization of the collagen structure the integrity of the triple helical structure remained intact. The in vitro superficial culture of heterologous chondrocytes also determined that the corona treatment can modulate migration with increased focal adhesion of cells due to enhanced stiffness, without cytotoxicity effects, and providing the basis for reinforcing three-dimensional collagen-based biomaterials in order to direct cell function and mediate mechanotransduction.

High-throughput indentational elasticity measurements of hydrogel extracellular matrix substrates

Much interest surrounds the effect of extracellular matrix (ECM) elasticity on cell behavior. Here we present a rapid method for measuring the elasticity of synthetic ECM substrates based on indentation of the substrate with a ferromagnetic sphere and optical tracking of the resulting deformation. We find that this method yields order-of-magnitude agreement with atomic force microscopy elasticity measurements, but that the degree of this agreement depends strongly on sphere density and gel elasticity. In its regime of greatest accuracy, we envision that this method may be used for high-throughput characterization of ECM substrates in cell biological studies.

Swelling induced finite strain flexure in a rectangular block of an isotropic elastic material

The deformation of a rectangular block into an annular wedge is studied with respect to the state of swelling interior to the block. Nonuniform swelling fields are shown to generate these flexure deformations in the absence of resultant forces and bending moments. Analytical expressions for the deformation fields demonstrate these effects for both incompressible and compressible generalizations of conventional hyperelastic materials. Existing results in the absence of a swelling agent are recovered as special cases.

On helical shear of a compressible elastic circular cylindrical tube

In this paper we examine the combined azimuthal and axial shear of a compressible isotropic elastic circular cylindrical tube of finite extent, otherwise referred to as helical shear (which is an isochoric deformation). The equilibrium equations are formulated in terms of the principal stretches, and explicit necessary and sufficient conditions on the strain-energy function for the material to support this deformation are obtained and compared with those obtained previously for this problem. Several classes of strain-energy functions are derived and in some general cases complete solutions of the equilibrium equations are obtained. Existing results are recovered as special cases and some new results for the strain-energy functions derived are determined and discussed.