Small Deformation Conditions Research Articles

A Hybrid Interpolation Method for Geometric Nonlinear Spatial Beam Elements with Explicit Nodal Force

Based on geometrically exact beam theory, a hybrid interpolation is proposed for geometric nonlinear spatial Euler-Bernoulli beam elements. First, the Hermitian interpolation of the beam centerline was used for calculating nodal curvatures for two ends. Then, internal curvatures of the beam were interpolated with a second interpolation. At this point, C1 continuity was satisfied and nodal strain measures could be consistently derived from nodal displacement and rotation parameters. The explicit expression of nodal force without integration, as a function of global parameters, was founded by using the hybrid interpolation. Furthermore, the proposed beam element can be degenerated into linear beam element under the condition of small deformation. Objectivity of strain measures and patch tests are also discussed. Finally, four numerical examples are discussed to prove the validity and effectivity of the proposed beam element.

Rheology and microstructure characterisation of small intestinal digesta from pigs fed a red meat-containing Western-style diet

A method to characterise rheological and microstructure features of whole digesta collected from pigs fed meat-containing western style diets is described. Digesta are shown to contain suspensions of meat fibre particles up to 300 μm in size which exhibit weak gel behaviour, have a dynamic viscosity higher than steady shear viscosity indicative of structural breakdown at the initiation of flow, and shear thinning behaviour under steady shear. The power law index approach is useful for capturing the overall features of rheological behaviour under both small (n*<0.3) and large deformation (typically 0.2 < n < 0.5) conditions. Values obtained for rheological parameters represent a balance between the digestion of structural components (meat particles and starch granules in this case) with the absorption of water along the small intestine, and result in the absence of any systematic pattern as a function of small intestinal site. Both primary rheological data (G′, G″, η, η*) and power law index parameters show a high level of variability between animals fed identical diets. This study provides a baseline from which to investigate the in vivo effects of hydrocolloids on the rheology of porcine digesta from diets representative of human meals.

Reflection on ‘finite rotation problem’ in plate and shell theories and in finite element formulation – Back to basics

This paper results from a reflection on the problem of finite rotations in plates and shells and its presentation in finite element formulation, a well-attended subject with a wide diversity of treatments, often extremely complicated. For instance, the very topic attracted 3 out of total 15 chapters over 120 out of total 494 pages in a monograph on nonlinear FEA [5]. In another instance, a 71 page journal paper was published [2] specifically on this topic alone. Review papers on this matter can be found in various sources including Appl Mech Rev, (e.g. Ibrahimbegovic, 1997 [8]). No attempt is to be made on reviewing these sophisticated approaches but the present paper will bring answers to questions, if they have ever been asked, such as whether the sophistication in existing approaches is really necessary and whether there is a much simpler and conceptually more direct and accurate approach. A proper re-examination of the existing approaches would reveal a fundamental inconsistency: rigid body rotations have been assumed without reservation to describe the deformations of deformable plates and shells. After re-establishing the consistency based on the very basics of conventional plate and shell theories as a simple reflection, one can conclude surprisingly that the whole issue on ‘finite rotations’ results from a logic fallacy of faulty generalisation. The so-called ‘rotations’ should be displacement gradients instead. They can be considered as ‘rotations’ as conventionally perceived hitherto under the condition of small deformation. Somehow, the concept of ‘rotations’ got generalised regardless the magnitude of the deformation. Instead of calling the problem as ‘finite displacement gradients’ as it should be called, a falsely generalised term of ‘finite rotations’ have been used. Since they were called ‘rotations’, the definition of those of rigid body kinematics has been taken for granted. Finite displacement gradients should not present any additional problem apart from introducing nonlinear nature into the problem, which can be addressed as a geometrically nonlinear problem in a conventional manner. However, along the line of a falsely generalised concept of finite rigid body rotations, complications have been the norm. The complicated accounts on the ‘finite rotation’ problem in the literature, which might have enhanced the understanding of rigid body kinematics, are entirely unnecessary as far as the deformable plates and shells are concerned, involving infinitesimal or finite deformations.

Strain-Dependent Rheological Model and Pressure Wave Prediction for Shut in and Restart of Waxy Oil Pipelines

Waxy oil gelation and rheology is investigated and modeled using strain-dependent viscosity correlations. Rotational rheometry shows a sharp viscosity increase upon gel formation. High creeping flow viscosities are observed at small deformation conditions prior to yielding. A new strain-dependent rheological model, following analogous formulation to the Carreau–Yasuda shear rate-dependent model, captures viscosity reduction associated with yielding. In addition, shear viscosity and extensional viscosity are investigated using a capillary rheometry method. Distinct shear-thinning behavior is observed in the shear mode of deformation, while distinct tension-thinning behavior is observed in the extensional mode of deformation for the model fluid systems. High Trouton ratios are obtained for the gelled model fluid systems, confirming strongly non-Newtonian fluid rheology. Finally, axial pressure wave profiles are computed at real pipeline dimensions for idealized moderate yield stress fluids using a computationally efficient 1D pipeline simulator. The Rønningsen time-dependent gel degradation model is used to emulate the fluid rheology in the simulator. Axial stress localization phenomena are shown to depend on the overall magnitude of gel degradation as established by the reduction in yield value. A high degree of gel degradation serves to afford flow commencement in a timely manner.

Flow behavior, thixotropy and dynamical viscoelasticity of sodium alginate aqueous solutions

Understanding of the rheological behavior of sodium alginate (SA) is fundamental either to design optimized products or to ensure stable flows. Of the SA samples tested that characterized by a M/G ratio of 0.38 and a weight–average molecular mass (Mw) of 1.180 × 105 g/mol, steady–shear flow, thixotropy and dynamical viscoelasticity tests were carried out to characterize rheological behavior of G–rich SA aqueous solutions as influenced by concentration (1.0–3.0%, w/v) and temperature (5–35 °C). These measurements were carried out by using an Anton Paar–Physica MCR 301 Rheometer. It was found that, above a critical shear rate, all G–rich SA solutions (1.0–3.0%, w/v) exhibited non–Newtonian shear–thinning behavior and the flow curves could be well described by the Cross model. The temperature–dependent behavior of 2.5% (w/v) SA had higher correlation to the Cross model and the Arrhenius relationship, while the flow activation energy tended to decrease with the increase of shear rate. The upward–downward rheograms showed that all the systems evaluated in this paper had a hysteresis loop which indicating strong thixotropic behavior, the higher the SA concentration, the stronger the thixotropic behavior. The dynamical viscoelastic properties characterized by oscillatory frequency sweep under small–deformation conditions showed a fluid–like viscoelastic behavior. The behavior of SA aqueous solutions was predominantly more viscous than elastic. Multiwave temperature ramp test for 2.5% SA aqueous solutions was consistent with the results from oscillatory frequency sweep test.

Coalescence of drops with mobile interfaces in a quiescent fluid

AbstractA study on the axisymmetric near-contact motion of drops with tangentially mobile interfaces under the action of a body force in a quiescent fluid is described. A long-time asymptotic analysis is presented for small-deformation conditions. Under these conditions the drops are nearly spherical, except in the near-contact region, where a flattened thin film forms. According to our analysis, a hydrostatic dome does not form in the near-contact region at long times, in contrast to the assumption underlying all previous analyses of this problem. Instead, the shape of the film in the near-contact region results from the absence of tangential stresses acting on it. As a result, the long-time behaviour of the system is qualitatively different than previously predicted. According to the theory presented herein, the minimum film thickness (rim region) decays with time as ${h}_{m} \sim {t}^{- 4/ 5} $, and the thickness at the centre of the film decays as ${h}_{0} \sim {t}^{- 3/ 5} $, which is a faster decay than predicted by prior analyses based on a hydrostatic dome. Numerical thin-film simulations quantitatively confirm the predictions of our small-deformation theory. Boundary-integral simulations of the full two-drop problem suggest that the theory also describes qualitatively the long-time evolution under finite-deformation conditions.

Deformation and mechanical behavior of electrohydrodynamic droplet under external electric field

Droplet deformation and mechanical behavior under external electric field are fundamental in electrohydrodynamic research. Based on the volume-of-fluid (VOF) method and adding electric force to Navier-Stokes equations as an additional source term, a simulation method considering fluid flow and electric field in two-way coupling is proposed. Using this method, the deformation/motion of leaky dielectric droplet and charged droplet under external uniform and non-uniform electric field is investigated respectively. The shape deformation coefficient calculated by the proposed method is close to the theory value of leaky dielectric droplet under uniform electric field in small deformation condition; thus the validity of the numerical method proposed in this study is verified. The results show that the charge redistribution modes of leaky dielectric droplet depend on the property difference between droplet and the around fluid, and the droplet may deform into either prolate or oblate shapes. The electric field strength only affects the droplet shape deformation coefficient. The external uniform electric field induces strong circulation motion inside the leaky dielectric droplet without moving the droplet. Because of the effect of Coulomb force, the charged droplet may deform into prolate and move along the electric field direction. However both leaky dielectric droplet and charged droplet may move along the electric field direction, resulting in different shapes simultaneously under the non-uniform electric field. The numerical method coupling fluid flow with electric field proposed in this paper could provide an effective approach to electrostatic spray, electrophoresis and other complex engineering electrohydrodynamic researches.

Material Mechanical Performance about Worm Bending and Twisting Deformation for Civil Engineering

On the condition of line elastic and small deformation, more than 80% failure of the drive shaft is due to fatigue damage. It relates to the alternating stress of the drive shaft under the combination effect of bending and twisting. When drive shaft is under the combination effect of bending and twisting for civil engineering, torsion shear stress is perpendicular to bending normal stress. They can’t be simply stacked. Combining with the belt conveyor tension device of the worm transmission, first, simplify the external force; do stress analysis in horizontal plane and vertical plane respectively. Then make corresponding bending moment figure and separately calculate the internal force, stress of each basic deformation and make comprehensive bending moment figure and torque figure. Then use the application of superposition principle, the fourth strength theory to calculate worm transmission’s strength. The method makes senses to other transmission mechanism shaft bending and twisting combination strength analysis as a reference.

Comparative rheological study of ionic semi-IPN composite hydrogels based on polyacrylamide and dextran sulphate and of polyacrylamide hydrogels

Ionic semi-interpenetrating polymer networks composite hydrogels were synthesized by free-radical polymerization using dextran sulphate (DxS), acrylamide as monomer and N,N′-methylene(bis)acrylamide as cross-linking agent. The viscoelastic properties of these composite hydrogels were investigated by oscillatory shear measurements under small deformation conditions comparative with those of polyacrylamide gels. Changes of the rheological properties of composite hydrogels have been studied in terms of polymerization temperature, cross-linker ratio, initial monomer concentration and molar mass of DxS. The results showed that the stability of the composite hydrogels obtained at room temperature (22 °C) was relatively low because the storage modulus (G′) was only eight times higher than the loss modulus (G″), while for those obtained by cryopolymerization (−18 °C), the stability was improved, the G′ values being about 30 times higher than those of G″. This behaviour indicated that, by conducting the synthesis of hydrogels below the freezing point of the reaction solutions, an enhancement of the hydrogels elasticity was achieved. The network parameters, i.e. the average molecular weight between two cross-links and the cross-link density of the composite hydrogels prepared at −18 °C, were estimated from rheological data.

Flow behaviour, linear viscoelasticity and surface properties of chitosan aqueous solutions

A study of rheological properties of chitosan solutions was carried out in terms of polymer concentration (0.05–2.5 wt%) and temperature (5–25 °C). The chitosan solutions studied exhibited Newtonian behaviour at concentrations lower than 0.25 wt%. Above this value, the rest of the systems showed shear-thinning behaviour. The viscoelastic properties of chitosan solutions above 0.25 wt% were characterized by oscillatory shear measurements under small-deformation conditions and the results showed fluid-like viscoelastic behaviour. Time-concentration and time-temperature superposition methods were carried out to obtain an overall master curve; where the rheological properties of chitosan solution at a specific temperature or concentration can be predicted over a broad time scale. The fact that the activation energy for steady shear flow was higher than for oscillatory shear promoted deviations from the Cox–Merz rule, when temperature was decreased at the highest chitosan concentration studied In addition, their dynamic surface tension behaviour and surface viscoelastic properties were explored. The time-dependent decrease in surface tension along with the limiting elasticity of the interface seems to indicate that commercial chitosan may play the role of an emulsifier as well as a stabiliser of O/W emulsions.

Oscillation frequency of simplified arterial tubes

To construct the spatial kinetic equation of an arterial tube and obtain its radial natural frequency, a linear-elastic and small deformation condition is assumed. The theoretical analysis is first presented and the finite element method is then used to numerically simulate the spatial kinematics. The results show that the first-order frequency is 15.8 Hz and the obtained exact analytical solutions agree well with numerical solutions, which proves that the theoretical analysis and numerical simulation are both correct.

Small-deformation theory for a surfactant-covered drop in linear flows

A small-deformation perturbation analysis is developed to study the effect of surfactant on drop dynamics in viscous flows. The surfactant is assumed to be insoluble in the bulk-phase fluids; the viscosity ratio and surfactant elasticity parameters are arbitrary. Under small-deformation conditions, the drop dynamics are described by a system of ordinary differential equations; the governing equations are given explicitly for the case of axisymmetric and two-dimensional imposed flows. Analytical results accurate to third order in the flow-strength parameter (capillary number) are derived (i) for the stationary drop shape and surfactant distribution in simple shear and axisymmetric straining flows, and (ii) for the rheology of a dilute emulsion in shear flow which include a shear-thinning viscosity and non-zero normal stresses. For drops with clean interfaces, the small-deformation theory presented here improves the results of Barthès-Biesel &amp; Acrivos (J. Fluid Mech., vol. 61, 1973, p. 1). Boundary integral simulations are used to test our theory and explore large-deformation conditions.

Coalescence of Drops with Tangentially Mobile Interfaces: Effects of Ambient Flow

According to the classical theory of film drainage for drops with tangentially immobile interfaces, the ambient flow field has no influence on film drainage aside from the hydrodynamic pushing force that it provides. The current theories of drop coalescence assume that this feature applies also to drops with tangentially mobile interfaces. This assumption is incorrect, except at short times. At long times, ambient flow conditions qualitatively affect film drainage, even under small-deformation conditions. The ambient flow can arrest film drainage, preventing coalescence, or exponentially enhance film drainage. Here, a new axisymmetric thin-film formulation is presented that correctly accounts for the influence of the ambient flow field. The formulation correctly recovers theoretical predictions for the stationary film thickness at long times. An analysis of the long-time behavior under accelerated drainage conditions explains why film thinning occurs exponentially. The numerical predictions of the new thin-film formulation are compared to whole-drop boundary integral simulations and are found to be in good agreement under small-deformation conditions. Thin-film simulations are used to describe offset collisions of drops, under the simplifying assumption that film drainage is unaffected by the tangential relative motion of the drops.

An Improved Model for Bending of Thin Viscoelastic Plate on Elastic Foundation

An improved model for bending of thin viscoelastic plate resting on Winkler foundation is presented. The thin plate is linear viscoelastic and subjected to normal distributed loading, the effect of normal stress along the plate thickness on the deflection and internal forces is taken into account. The basic equations for internal forces and stress distribution are derived based on the general viscoelastic theory under small deformation condition. The reduced equations for elastic case are given as well. It is shown that the proposed model reveals a larger flexural rigidity compared to that in classic models, in which the normal stress along the plate thickness is neglected.

SYNTHESIS AND CHARACTERIZATION OF AMPHOTERIC HYDROGELS BASED ON N-CARBOXYETHYLCHITOSAN

New amphoteric hydrogels based on carboxyethylchitosans (CECH) with various degrees of substitution (DS) were prepared using different amounts of epichlorohydrin (ECH) as the crosslinking agent. The equilibrium swelling ratio (SW) was determined as functions of pH and salt concentration. The hydrogels show typical amphoteric character responding to pH change of the external medium. At isoelectric point (IEP), the hydrogels shrink. The DS value has important effect on the swelling properties of the hydrogels. When the DS of N-carboxyethylchitosan increases from 0.32 to 0.72, the equilibrium swelling ratio (SW) of the hydrogel changes from 76 to 290 at pH 7.3 and from 117 to 499 at pH 11.3. A marked volume decrease was observed in hydrogels with increasing salt concentration in the surrounding solution. The viscoelastic properties of the hydrogels were studied by oscillatory shear measurements under small-deformation conditions. The elastic modulus G' of all the samples has no dependence on frequency and is one order of magnitude larger than the loss modulus G", corresponding to a strong gel behavior.

Dynamics of surfactant‐covered drops in linear flows

AbstractSurfactants modify interfacial properties and significantly affect drop behavior in flow. We study the dynamics of a drop, which is covered with a monolayer of insoluble surfactant, in linear viscous flows, both unbounded and in the presence of a wall. The effect of viscosity contrast is included. Analytical results in a form of expansions for weak flows and high viscosity drops are developed. Numerical results with three‐dimensional boundary integral simulations are used to explore large deformations. The results show that surfactant generally enhances drop deformation, certainly under small‐deformation conditions. The steady‐state drop shape and surfactant distribution are independent of viscosity contrast in straining flows (e.g. hyperbolic, axisymmetric strain). The drop shape and surfactant distribution are insensitive to viscosity contrast under small deformation conditions for any linear flow; the effect of the rotational component at higher‐order. The theory quantifies the effect of surfactant on drop migration away from a bounding wall as well as the rheology of dilute emulsions. It predicts non‐Newtonian features such as shear thinning viscosity and normal stress. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Constitutive modeling of porous viscoelastic materials

The effect of porosity on the constitutive response of an isotropic linearly viscoelastic solid that obeys a constitutive law of the standard differential form is investigated under small strain deformation conditions. The correspondence principle of linear viscoelasticity is used to solve the viscoelastic boundary value problem at a unit cell containing a spherical void and loaded axisymmetrically by macroscopic stresses. The results are used to devise a constitutive potential for the description of the porous material for any arbitrary combination of hydrostatic and deviatoric loadings, and the associated 3-D constitutive relationship is determined in the Laplace transform domain. Inversion to the time domain yields the constitutive law of the porous material as a function of porosity in the standard form of convolution integrals. The presence of porosity establishes relaxation time scales for the porous body that differ from the relaxation time of the pure matrix material and brings about a viscous character to the overall hydrostatic response. The numerical implementation of the model in a general purpose finite element code is outlined. The model is used to predict the response of a porous solid propellant material in uniaxial tension and cyclic loading at room temperature.

Elastic moduli of normal and pathological human breast tissues: an inversion-technique-based investigation of 169 samples

Understanding and quantifying the mechanical properties of breast tissues has been a subject of interest for the past two decades. This has been motivated in part by interest in modelling soft tissue response for surgery planning and virtual-reality-based surgical training. Interpreting elastography images for diagnostic purposes also requires a sound understanding of normal and pathological tissue mechanical properties. Reliable data on tissue elastic properties are very limited and those which are available tend to be inconsistent, in part as a result of measurement methodology. We have developed specialized techniques to measure tissue elasticity of breast normal tissues and tumour specimens and applied them to 169 fresh ex vivo breast tissue samples including fat and fibroglandular tissue as well as a range of benign and malignant breast tumour types. Results show that, under small deformation conditions, the elastic modulus of normal breast fat and fibroglandular tissues are similar while fibroadenomas were approximately twice the stiffness. Fibrocystic disease and malignant tumours exhibited a 3–6-fold increased stiffness with high-grade invasive ductal carcinoma exhibiting up to a 13-fold increase in stiffness compared to fibrogalndular tissue. A statistical analysis showed that differences between the elastic modulus of the majority of those tissues were statistically significant. Implications for the specificity advantages of elastography are reviewed.For more information on this article, see medicalphysicsweb.org

Viscoelastic properties of dispersed chitosan/xanthan hydrogels

In this work a gel was formed by complexation of two natural polyelectrolytes, chitosan and xanthan. Changes in the hydrogels rheological properties have been studied in terms of hydrogel concentration (7–10% w/w), chemical media used for the hydrogel dispersion, and ‘test lag time’; i.e., the time between hydrogel dispersion in the chemical media and the start of the rheological test (up to 390 min). The viscoelastic properties of this polysaccharide system were characterized by oscillatory shear measurements under small-deformation conditions and the results show that chitosan/xanthan hydrogels behave like weak gels. The shear modulus increased almost linearly with frequency in the range studied (0.1–65 s −1). The effects of hydrogel concentration and dispersion medium have been related to electrostatic equilibrium and by the presence of counter-ions modifying the internal structure of the hydrogel.

A nonlinear wave equation and exact periodic solutions in circular-rod waveguide

Under the condition of small deformation, a new nonlinear wave equation is derived to describe nonlinear wave evolution in a nonlinear elastic circular rod by means of Hamilton principle. The nonlinear constitutive relationship proposed by Cox and transverse Possion effects are simultaneously taken into account. Nonlinear wave equation and truncated nonlinear wave equation are solved by the Jacobi elliptic cosine function expansion method. The exact periodic solutions of these nonlinear equations are obtained. The limiting conditions of these solutions are also given.