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.

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.

A comparison of stability and bifurcation criteria for inflated spherical elastic shells

The nonlinear stability analysis introduced by Chen and Haughton [1] is employed to study the full nonlinear stability of the non-homogeneous spherically symmetric deformation of an elastic thick-walled sphere. The shell is composed of an arbitrary homogeneous, incompressible elastic material. The stability criterion ultimately requires the solution of a third-order nonlinear ordinary differential equation. Numerical calculations performed for a wide variety of well-known incompressible materials are then compared with existing bifurcation results and are found to be identical. Further analysis and comparison between stability and bifurcation are conducted for the case of thin shells and we prove by direct calculation that the two criteria are identical for all modes and all materials.

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.

Gold nanoresistors with near-constant resistivity in the cryogenic-to-room temperature range

Using a multiple plasma deposition-annealing (MDA) technique, we have fabricated an Au nanoisland-based thin film nanoresistor with a very low temperature coefficient of electrical resistivity in a cryogenic-to-room temperature range of 10 to 300 K. The nanoislanded gold film was deposited on a SiO2/Si wafer (500 nm SiO2 thickness) between two 300 nm thick Au electrodes which were separated by 100 m. A sophisticated selection of the thickness of the nanoislanded gold film, the annealing temperature, as well as the number of deposition/annealing cycles resulted in the fabrication of a nanoresistor with a temperature coefficient of electrical resistivity of 2.1 × 10-3 K-1 and the resistivity deviation not exceeding 2% in a cryogenic-to-room temperature range. We have found that the constant resistivity regime of the nanoisland-based thin film nanoresistor corresponds to a minimized nanoisland activation energy (approximately 0.3 meV). This energy can be minimized by reducing the nearest neighbor distance and increasing the size of the Au nanoislands in the optimized nanoresistor structure. It is shown that the constant resistivity nanoresistor operates in the regime where the thermally activated electron tunneling is compensated by the negative temperature dependence of the metallic-type conductivity of nanoislands. Our results are relevant to the development of commercially viable methods of nanoresistor production for various nanoelectronics-based devices. The proposed MDA technique also provides the opportunity to fabricate large arrays of metallic nanoparticles with controllable size, shapes and inter-nanoparticle gaps.

A linear-elastic model of anisotropic tumour growth

This paper examines the effect of anisotropic growth on the evolution of mechanical stresses in a linear-elastic model of a growing, avascular tumour. This represents an important improvement on previous linear-elastic models of tissue growth since it has been shown recently that spatially-varying isotropic growth of linear-elastic tissues does not afford the necessary stress-relaxation for a steady-state stress distribution upon reaching a nutrient-regulated equilibrium size. Time-dependent numerical solutions are developed using a Lax-Wendroff scheme, which show the evolution of the tissue stress distributions over a period of growth until a steady-state is reached. These results are compared with the steady-state solutions predicted by the model equations, and key parameters influencing these steady-state distributions are identified. Recommendations for further extensions and applications of this model are proposed.

A loop cancellation based active damping solution for constant power instability in vehicular power systems

In recent years, electric propulsion systems have increasingly been used in land, sea and air vehicles. The vehicular power systems are usually loaded with tightly regulated power electronic converters which tend to draw constant power. Since the constant power loads (CPLs) impose negative incremental resistance characteristics on the feeder system, they pose a potential threat to the stability of vehicular power systems. This effect becomes more significant in the presence of distribution lines between source and load in large vehicular power systems such as electric ships and more electric aircrafts. System transients such as sudden drop of converter side loads or increase of constant power requirement can cause complete system instability. Most of the existing research work focuses on the modeling and stabilization of DC vehicular power systems with CPLs. Only a few solutions are proposed to stabilize AC vehicular power systems with non-negligible distribution lines and CPLs. Therefore, this paper proposes a novel loop cancellation technique to eliminate constant power instability in AC vehicular power systems with a theoretically unbounded system stability region. Analysis is carried out on system stability with the proposed method and simulation results are presented to validate its effectiveness.

Virtual resistance based active damping solution for constant power instability in AC microgrids

With ever-increasing share of power electronic loads constant power instability is becoming a significant issue in microgrids, especially when they operate in the islanding mode. Transient conditions like resistive load-shedding or sudden increase of constant power loads (CPL) might destabilize the whole system. Modeling and stability analysis of AC microgrids with CPLs have already been discussed in literature. However, no effective solutions are provided to stabilize this kind of system. Therefore, this paper proposes a virtual resistance based active damping method to eliminate constant power instability in AC microgrids. Advantages and limitations of the proposed method are also discussed in detail. Simulation results are presented to validate the proposed active damping solution.

Deformation of a single red blood cell in a microvessel

Red blood cells (RBCs) are the most common type of cells in human blood and they exhibit different types of motions and deformed shapes in capillary flows. The behaviour of the RBCs should be studied in order to explain the RBC motion and deformation mechanism. This article presents a numerical simulation method for RBC deformation in microvessels. A two dimensional spring network model is used to represent the RBC membrane, where the elastic stretch/compression energy and the bending energy are considered with the constraint of constant RBC surface area. The forces acting on the RBC membrane are obtained from the principle of virtual work. The whole fluid domain is discretized into a finite number of particles using smoothed particle hydrodynamics concepts and the motions of all the particles are solved using Navier--Stokes equations. Minimum energy concepts are used to simulate the deformed shape of the RBC model. To verify the model, the motion of a single RBC is simulated in a Poiseuille flow and the characteristic parachute shape of the RBC is observed. Further simulations reveal that the RBC shows a tank treading motion when it flows in a linear shear flow.

Revisiting borehole stresses in anisotropic elastic media : comparison of analytical versus numerical solutions

We present a rigorous validation of the analyticalAmadei solution for the stress concentration around arbitrarily orientated borehole in general anisotropic elastic media. First, we revisit the theoretical framework of the Amadei solution and present analytical insights that show that the solution does indeed contain all special cases of symmetry, contrary to previous understanding, provided that the reduced strain coefficients β11 and β55 are not equal. It is shown from theoretical considerations and published experimental data that the β11 and β55 are not equal for realistic rocks. Second, we develop a 3D finite-element elastic model within a hybrid analyticalnumerical workflow that circumvents the need to rebuild and remesh the model for every borehole and material orientation. Third, we show that the borehole stresses computed from the numerical model and the analytical solution match almost perfectly for different borehole orientations (vertical, deviated and horizontal) and for several cases involving isotropic and transverse isotropic symmetries. It is concluded that the analytical Amadei solution is valid with no restrictions on the borehole orientation or elastic anisotropy symmetry.

Novel non-linear elastic structural analysis with generalised transverse element loads using a refined finite element

In the finite element modelling of structural frames, external loads such as wind loads, dead loads and imposed loads usually act along the elements rather than at the nodes only. Conventionally, when an element is subjected to these general transverse element loads, they are usually converted to nodal forces acting at the ends of the elements by either lumping or consistent load approaches. In addition, it is especially important for an element subjected to the first- and second-order elastic behaviour, to which the steel structure is critically prone to; in particular the thin-walled steel structures, when the stocky element section may be generally critical to the inelastic behaviour. In this sense, the accurate first- and second-order elastic displacement solutions of element load effect along an element is vitally crucial, but cannot be simulated using neither numerical nodal nor consistent load methods alone, as long as no equilibrium condition is enforced in the finite element formulation, which can inevitably impair the structural safety of the steel structure particularly. It can be therefore regarded as a unique element load method to account for the element load nonlinearly. If accurate displacement solution is targeted for simulating the first- and second-order elastic behaviour on an element on the basis of sophisticated non-linear element stiffness formulation, the numerous prescribed stiffness matrices must indispensably be used for the plethora of specific transverse element loading patterns encountered. In order to circumvent this shortcoming, the present paper proposes a numerical technique to include the transverse element loading in the non-linear stiffness formulation without numerous prescribed stiffness matrices, and which is able to predict structural responses involving the effect of first-order element loads as well as the second-order coupling effect between the transverse load and axial force in the element. This paper shows that the principle of superposition can be applied to derive the generalized stiffness formulation for element load effect, so that the form of the stiffness matrix remains unchanged with respect to the specific loading patterns, but with only the magnitude of the loading (element load coefficients) being needed to be adjusted in the stiffness formulation, and subsequently the non-linear effect on element loadings can be commensurate by updating the magnitude of element load coefficients through the non-linear solution procedures. In principle, the element loading distribution is converted into a single loading magnitude at mid-span in order to provide the initial perturbation for triggering the member bowing effect due to its transverse element loads. This approach in turn sacrifices the effect of element loading distribution except at mid-span. Therefore, it can be foreseen that the load-deflection behaviour may not be as accurate as those at mid-span, but its discrepancy is still trivial as proved. This novelty allows for a very useful generalised stiffness formulation for a single higher-order element with arbitrary transverse loading patterns to be formulated. Moreover, another significance of this paper is placed on shifting the nodal response (system analysis) to both nodal and element response (sophisticated element formulation). For the conventional finite element method, such as the cubic element, all accurate solutions can be only found at node. It means no accurate and reliable structural safety can be ensured within an element, and as a result, it hinders the engineering applications. The results of the paper are verified using analytical stability function studies, as well as with numerical results reported by independent researchers on several simple frames.

Two-dimensional bow and stern flows in a finite-depth fluid with constant vorticity

This thesis is concerned with two-dimensional free surface flows past semi-infinite surface-piercing bodies in a fluid of finite-depth. Throughout the study, it is assumed that the fluid in question is incompressible, and that the effects of viscosity and surface tension are negligible. The problems considered are physically important, since they can be used to model the flow of water near the bow or stern of a wide, blunt ship. Alternatively, the solutions can be interpreted as describing the flow into, or out of, a horizontal slot. In the past, all research conducted on this topic has been dedicated to the situation where the flow is irrotational. The results from such studies are extended here, by allowing the fluid to have constant vorticity throughout the flow domain. In addition, new results for irrotational flow are also presented. When studying the flow of a fluid past a surface-piercing body, it is important to stipulate in advance the nature of the free surface as it intersects the body. Three different possibilities are considered in this thesis. In the first of these possibilities, it is assumed that the free surface rises up and meets the body at a stagnation point. For this configuration, the nonlinear problem is solved numerically with the use of a boundary integral method in the physical plane. Here the semi-infinite body is assumed to be rectangular in shape, with a rounded corner. Supercritical solutions which satisfy the radiation condition are found for various values of the Froude number and the dimensionless vorticity. Subcritical solutions are also found; however these solutions violate the radiation condition and are characterised by a train of waves upstream. In the limit that the height of the body above the horizontal bottom vanishes, the flow approaches that due to a submerged line sink in a $90^\circ$ corner. This limiting problem is also examined as a special case. The second configuration considered in this thesis involves the free surface attaching smoothly to the front face of the rectangular shaped body. For this configuration, nonlinear solutions are computed using a similar numerical scheme to that used in the stagnant attachment case. It is found that these solution exist for all supercritical Froude numbers. The related problem of the cusp-like flow due to a submerged sink in a corner is also considered. Finally, the flow of a fluid emerging from beneath a semi-infinite flat plate is examined. Here the free surface is assumed to detach from the trailing edge of the plate horizontally. A linear problem is formulated under the assumption that the elevation of the plate is close to the undisturbed free surface level. This problem is solved exactly using the Wiener-Hopf technique, and subcritical solutions are found which are characterised by a train of sinusoidal waves in the far field. The nonlinear problem is also considered. Exact relations between certain parameters for supercritical flow are derived using conservation of mass and momentum arguments, and these are confirmed numerically. Nonlinear subcritical solutions are computed, and the results are compared to those predicted by the linear theory.

Behaviour and strength of hollow flange channel sections under torsion and bending

Hollow flange channel section is a cold-formed high-strength and thin-walled steel section with a unique shape including two rectangular hollow flanges and a slender web. Due to its mono-symmetric characteristics, it will also be subjected to torsion when subjected to transverse loads in practical applications. Past research on steel beams subject to torsion has concentrated on open sections while very few steel design standards give suitable design rules for torsion design. Since the hollow flange channel section is different from conventional open sections, its torsional behaviour remains unknown to researchers. Therefore the elastic behaviour of hollow flange channel sections subject to uniform and non-uniform torsion, and combined torsion and bending was investigated using the solutions of appropriate differential equilibrium equations. The section torsion shear flow, warping normal stress distribution, and section constants including torsion constant and warping constant were obtained. The results were compared with those from finite element analyses that verified the accuracy of analytical solutions. Parametric studies were undertaken for simply supported beams subject to a uniformly distributed torque and a uniformly distributed transverse load applied away from the shear centre. This paper presents the details of this research into the elastic behaviour and strength of hollow flange channel sections subject to torsion and bending and the results.

Multimodal imaging of the collagen and elastic fibre networks in the bovine intervertebral disc

INTRODUCTION. The intervertebral disc is the largest avascular structure in the human body, withstanding transient loads of up to nine times body weight during rigorous physical activity. The key structural elements of the disc are a gel-like nucleus pulposus surrounded by concentric lamellar rings containing criss-crossed collagen fibres. The disc also contains an elastic fiber network which has been suggested to play a structural role, but to date the relationship between the collagen and elastic fiber networks is unclear. CONCLUSION. The multimodal transmitted and reflected polarized light microscopy technique developed here allows clear differentiation between the collagen and elastic fiber networks of the intervertebral disc. The ability to image unstained specimens avoids concerns with uneven stain penetration or specificity of staining. In bovine tail discs, the elastic fiber network is intimately associated with the collagen network.

Semi-analytical finite element analysis of a strip footing on an elastic reinforced soil

A plane strain elastic interaction analysis of a strip footing resting on a reinforced soil bed has been made by using a combined analytical and finite element method (FEM). In this approach the stiffness matrix for the footing has been obtained using the FEM, For the reinforced soil bed (halfplane) the stiffness matrix has been obtained using an analytical solution. For the latter, the reinforced zone has been idealised as (i) an equivalent orthotropic infinite strip (composite approach) and (ii) a multilayered system (discrete approach). In the analysis, the interface between the strip footing and reinforced halfplane has been assumed as (i) frictionless and (ii) fully bonded. The contact pressure distribution and the settlement reduction have been given for different depths of footing and scheme of reinforcement in soil. The load-deformation behaviour of the reinforced soil obtained using the above modelling has been compared with some available analytical and model test results. The equivalent orthotropic approach proposed in this paper is easy to program and is shown to predict the reinforcing effects reasonably well.