Note on similarity solutions for viscous flow over an impermeable and non-linearly (quadratic) stretching sheet

Similarity solutions for flow over an impermeable, non-linearly (quadratic) stretching sheet were studied recently by Raptis and Perdikis (Int. J. Non-linear Mech. 41 (2006) 527–529) using a stream function of the form ψ=αxf(η)+βx2g(η). A fundamental error in their problem formulation is pointed out. On correction, it is shown that similarity solutions do not exist for this choice of ψ

Discontinuous solutions and boundary value problems for non-linearly elastic fibre-reinforced materials

En este trabajo se han analizado varios problemas en el contexto de la elasticidad no lineal basándose en modelos constitutivos representativos. En particular, se han analizado problemas relacionados con el fenómeno de perdida de estabilidad asociada con condiciones de contorno en el caso de material reforzados con fibras. Cada problema se ha formulado y se ha analizado por separado en diferentes capítulos. En primer lugar se ha mostrado el análisis del gradiente de deformación discontinuo para un material transversalmente isótropo, en particular, el modelo del material considerado consiste de una base neo-Hookeana isótropa incrustada con fibras de refuerzo direccional caracterizadas con un solo parámetro. La solución de este problema se vincula con instabilidades que dan lugar al mecanismo de fallo conocido como banda de cortante. La perdida de elipticidad de las ecuaciones diferenciales de equilibrio es una condición necesaria para que aparezca este tipo de soluciones y por tanto las inestabilidades asociadas. En segundo lugar se ha analizado una deformación combinada de extensión, inación y torsión de un tubo cilíndrico grueso donde se ha encontrado que la deformación citada anteriormente puede ser controlada solo para determinadas direcciones de las fibras refuerzo. Para entender el comportamiento elástico del tubo considerado se ha ilustrado numéricamente los resultados obtenidos para las direcciones admisibles de las fibras de refuerzo bajo la deformación considerada. En tercer lugar se ha estudiado el caso de un tubo cilíndrico grueso reforzado con dos familias de fibras sometido a cortante en la dirección azimutal para un modelo de refuerzo especial. En este problema se ha encontrado que las inestabilidades que aparecen en el material considerado están asociadas con lo que se llama soluciones múltiples de la ecuación diferencial de equilibrio. Se ha encontrado que el fenómeno de instabilidad ocurre en un estado de deformación previo al estado de deformación donde se pierde la elipticidad de la ecuación diferencial de equilibrio. También se ha demostrado que la condición de perdida de elipticidad y ^W=2 = 0 (la segunda derivada de la función de energía con respecto a la deformación) son dos condiciones necesarias para la existencia de soluciones múltiples. Finalmente, se ha analizado detalladamente en el contexto de elipticidad un problema de un tubo cilíndrico grueso sometido a una deformación combinada en las direcciones helicoidal, axial y radial para distintas geotermias de las fibras de refuerzo . In the present work four main problems have been addressed within the framework of non-linear elasticity based on representative constitutive models. Namely, problems related to the loss of stability phenomena associated with boundary value problems for fibre-reinforced materials. Each of the considered problems is formulated and analysed separately in different chapters. We first start with the analysis of discontinuous deformation gradients for a transversely isotropic material under plane deformation. In particular, the material model is an augmented neo-Hookean base with a simple unidirectional reinforcement characterised by a single parameter. The solution of this problem is related to material instabilities and it is associated with a shear band-type failure mode. The loss of ellipticity of the governing differential equations is a necessary condition for the existence of these material instabilities. The second problem involves a detailed analysis of the combined non-linear extension, inflation and torsion of a thick-walled circular cylindrical tube where it has been found that the aforementioned deformation is controllable only for certain preferred directions of transverse isotropy. Numerical results have been illustrated to understand the elastic behaviour of the tube for the admissible preferred directions under the considered deformation. The third problem deals with the analysis of a doubly fibre-reinforced thickwalled circular cylindrical tube undergoing pure azimuthal shear for a special class of the reinforcing model where multiple non-smooth solutions emerge. The associated instability phenomena are found to occur prior to the point where the nominal stress tensor changes monotonicity in a particular direction. It has been also shown that the loss of ellipticity condition that arises from the equilibrium equation and ^W=2 = 0 (the second derivative of the strain-energy function with respect to the deformation) are equivalent necessary conditions for the emergence of multiple solutions for the considered material. Finally, a detailed analysis in the basis of the loss of ellipticity of the governing differential equations for a combined helical, axial and radial elastic deformations of a fibre-reinforced circular cylindrical tube is carried out.

Flow and heat transfer in a stagnation-point flow over a stretching sheet with a magnetic field

The flow and heat transfer problem in the boundary layer induced by a continuous moving surface is important in many manufacturing processes in industry such as the boundary layer along material handling conveyers, the aerodynamic extrusion of plastic sheet, the cooling of an infinite metalic plate in a cooling bath (which may also be electrolyte). Glass blowing, continuous casting and spinning of fibres also involve the flow due to a stretching surface. Sakiadis [1] was the first to study the flow induced by a semi-infinite moving wall in an ambient fluid. On the other hand, Crane [2] first studied the flow over a linearly stretching sheet in an ambient fluid. Subsequently, Crane [3] also investigated the corresponding heat transfer problem. Since then several authors [4-8] have studied various aspects of this problem such as the effects of mass transfer, variable wall temperature, constant heat flux, magnetic field etc. Recently, Andersson [9] has obtained an exact solution of the Navier-Stokes equations for the MHD flow over a linearly stretching sheet in an ambient fluid. Also Chiam [10] has studied the heat transfer with variable thermal conductivity on a stretching sheet when the velocities of the sheet and the free stream are equal.

Energy scattering in weakly non-linear systems.

The Chinese Tam-Tam exhibits non-linear behavior in its vibro-acoustic response. The frequency content of the response during free, unforced vibration smoothly changes, with energy being progressively smeared out over a greater bandwidth with time. This is used as a motivating case for the general study of the phenomenon of energy cascading through weak nonlinearity. Numerical models based upon the Fermi-Pasta-Ulam system of non-linearly coupled oscillators, modified with the addition of damping, have been developed. These were used to study the response of ensembles of systems with randomized natural frequencies. Results from simulations will be presented here. For un-damped systems, individual ensemble members exhibit cyclical energy exchange between linear modes, but the ensemble average displays a steady state. For the ensemble response of damped systems, lightly damped modes can exhibit an effective damping which is higher than predicated by linear theory. The presence of a non-linearity provides a path for energy flow to other modes, increasing the apparent damping spectrum at some frequencies and reducing it at others. The target of this work is a model revealing the governing parameters of a generic system of this type and leading to predictions of the ensemble response.

A meta-analysis of latitudinal variations in life-history traits of roach, Rutilus rutilus, over its geographical range: linear or non-linear relationships?

1. We collated information from the literature on life history traits of the roach (a generalist freshwater fish), and analysed variation in absolute fecundity, von Bertalanffy parameters, and reproductive lifespan in relation to latitude, using both linear and non-linear regression models. We hypothesized that because most life history traits are dependent on growth rate, and growth rate is non-linearly related with temperature, it was likely that when analysed over the whole distribution range of roach, variation in key life history traits would show non-linear patterns with latitude.

A simple non-linear global-local finite element methodology for composite structures

A simple non-linear global-local finite element methodology is presented. A global coarse model, using 2-D shell elements, is solved non-linearly and the displacements and rotations around a region of interest are applied, as displacement boundary conditions, to a refined local 3-D model using Kirchhoff plate assumptions. The global elements' shape functions are used to interpolate between nodes. The local model is then solved non-linearly with an incremental scheme independent of that used for the global model.

Model-based changes in global annual mean surface temperature change (Delta T_g) and radiative forcing due to land ice albedo changes (Delta R_[LI]) over the last 5 Myr, supplementary material

It is still an open question how equilibrium warming in response to increasing radiative forcing - the specific equilibrium climate sensitivity S - depends on background climate. We here present palaeodata-based evidence on the state dependency of S, by using CO2 proxy data together with a 3-D ice-sheet-model-based reconstruction of land ice albedo over the last 5 million years (Myr). We find that the land ice albedo forcing depends non-linearly on the background climate, while any non-linearity of CO2 radiative forcing depends on the CO2 data set used. This non-linearity has not, so far, been accounted for in similar approaches due to previously more simplistic approximations, in which land ice albedo radiative forcing was a linear function of sea level change. The latitudinal dependency of ice-sheet area changes is important for the non-linearity between land ice albedo and sea level. In our set-up, in which the radiative forcing of CO2 and of the land ice albedo (LI) is combined, we find a state dependence in the calculated specific equilibrium climate sensitivity, S[CO2,LI], for most of the Pleistocene (last 2.1 Myr). During Pleistocene intermediate glaciated climates and interglacial periods, S[CO2,LI] is on average ~ 45 % larger than during Pleistocene full glacial conditions. In the Pliocene part of our analysis (2.6-5 Myr BP) the CO2 data uncertainties prevent a well-supported calculation for S[CO2,LI], but our analysis suggests that during times without a large land ice area in the Northern Hemisphere (e.g. before 2.82 Myr BP), the specific equilibrium climate sensitivity, S[CO2,LI], was smaller than during interglacials of the Pleistocene. We thus find support for a previously proposed state change in the climate system with the widespread appearance of northern hemispheric ice sheets. This study points for the first time to a so far overlooked non-linearity in the land ice albedo radiative forcing, which is important for similar palaeodata-based approaches to calculate climate sensitivity. However, the implications of this study for a suggested warming under CO2 doubling are not yet entirely clear since the details of necessary corrections for other slow feedbacks are not fully known and the uncertainties that exist in the ice-sheet simulations and global temperature reconstructions are large.

Numerical study of ultrasound induced non-linear shape and size bubble oscillations in viscoelastic media

The theoretical study of forced bubble oscillations is motivated by the importance of cavitation bubbles and oscillating encapsulated microbubbles (i.e. contrast agents) in medical sciences. In more details,theoretical studies on bubble dynamics addressing the sound-bubble interaction phenomenon provide the basis for understanding the dynamics of contrast agent microbubbles used in medical diagnosis and of non-linearly oscillating cavitation bubbles in the case of high-intensity ultrasound therapy. Moreover, the inclusion of viscoelasticity is of vital importance for an accurate theoretical analysis since most biological tissues and fluids exhibit non-Newtonian behavior.

Sensitivity of proximal femoral stiffness and areal bone mineral density to changes in bone geometry and density

Areal bone mineral density (aBMD) is the most common surrogate measurement for assessing the bone strength of the proximal femur associated with osteoporosis. Additional factors, however, contribute to the overall strength of the proximal femur, primarily the anatomical geometry. Finite element analysis (FEA) is an effective and widely used computerbased simulation technique for modeling mechanical loading of various engineering structures, providing predictions of displacement and induced stress distribution due to the applied load. FEA is therefore inherently dependent upon both density and anatomical geometry. FEA may be performed on both three-dimensional and two-dimensional models of the proximal femur derived from radiographic images, from which the mechanical stiffness may be redicted. It is examined whether the outcome measures of two-dimensional FEA, two-dimensional, finite element analysis of X-ray images (FEXI), and three-dimensional FEA computed stiffness of the proximal femur were more sensitive than aBMD to changes in trabecular bone density and femur geometry. It is assumed that if an outcome measure follows known trends with changes in density and geometric parameters, then an increased sensitivity will be indicative of an improved prediction of bone strength. All three outcome measures increased non-linearly with trabecular bone density, increased linearly with cortical shell thickness and neck width, decreased linearly with neck length, and were relatively insensitive to neck-shaft angle. For femoral head radius, aBMD was relatively insensitive, with two-dimensional FEXI and threedimensional FEA demonstrating a non-linear increase and decrease in sensitivity, respectively. For neck anteversion, aBMD decreased non-linearly, whereas both two-dimensional FEXI and three dimensional FEA demonstrated a parabolic-type relationship, with maximum stiffness achieved at an angle of approximately 15o. Multi-parameter analysis showed that all three outcome measures demonstrated their highest sensitivity to a change in cortical thickness. When changes in all input parameters were considered simultaneously, three and twodimensional FEA had statistically equal sensitivities (0.41±0.20 and 0.42±0.16 respectively, p = ns) that were significantly higher than the sensitivity of aBMD (0.24±0.07; p = 0.014 and 0.002 for three-dimensional and two-dimensional FEA respectively). This simulation study suggests that since mechanical integrity and FEA are inherently dependent upon anatomical geometry, FEXI stiffness, being derived from conventional two-dimensional radiographic images, may provide an improvement in the prediction of bone strength of the proximal femur than currently provided by aBMD.

Damage assessment in reinforced concrete flexural members using modal strain energy based method

Damage assessment (damage detection, localization and quantification) in structures and appropriate retrofitting will enable the safe and efficient function of the structures. In this context, many Vibration Based Damage Identification Techniques (VBDIT) have emerged with potential for accurate damage assessment. VBDITs have achieved significant research interest in recent years, mainly due to their non-destructive nature and ability to assess inaccessible and invisible damage locations. Damage Index (DI) methods are also vibration based, but they are not based on the structural model. DI methods are fast and inexpensive compared to the model-based methods and have the ability to automate the damage detection process. DI method analyses the change in vibration response of the structure between two states so that the damage can be identified. Extensive research has been carried out to apply the DI method to assess damage in steel structures. Comparatively, there has been very little research interest in the use of DI methods to assess damage in Reinforced Concrete (RC) structures due to the complexity of simulating the predominant damage type, the flexural crack. Flexural cracks in RC beams distribute non- linearly and propagate along all directions. Secondary cracks extend more rapidly along the longitudinal and transverse directions of a RC structure than propagation of existing cracks in the depth direction due to stress distribution caused by the tensile reinforcement. Simplified damage simulation techniques (such as reductions in the modulus or section depth or use of rotational spring elements) that have been extensively used with research on steel structures, cannot be applied to simulate flexural cracks in RC elements. This highlights a big gap in knowledge and as a consequence VBDITs have not been successfully applied to damage assessment in RC structures. This research will address the above gap in knowledge and will develop and apply a modal strain energy based DI method to assess damage in RC flexural members. Firstly, this research evaluated different damage simulation techniques and recommended an appropriate technique to simulate the post cracking behaviour of RC structures. The ABAQUS finite element package was used throughout the study with properly validated material models. The damaged plasticity model was recommended as the method which can correctly simulate the post cracking behaviour of RC structures and was used in the rest of this study. Four different forms of Modal Strain Energy based Damage Indices (MSEDIs) were proposed to improve the damage assessment capability by minimising the numbers and intensities of false alarms. The developed MSEDIs were then used to automate the damage detection process by incorporating programmable algorithms. The developed algorithms have the ability to identify common issues associated with the vibration properties such as mode shifting and phase change. To minimise the effect of noise on the DI calculation process, this research proposed a sequential order of curve fitting technique. Finally, a statistical based damage assessment scheme was proposed to enhance the reliability of the damage assessment results. The proposed techniques were applied to locate damage in RC beams and slabs on girder bridge model to demonstrate their accuracy and efficiency. The outcomes of this research will make a significant contribution to the technical knowledge of VBDIT and will enhance the accuracy of damage assessment in RC structures. The application of the research findings to RC flexural members will enable their safe and efficient performance.

Effect of irrigation amounts applied with subsurface drip irrigation on corn evapotranspiration, yield, water use efficiency, and dry matter production in a semiarid climate

Quantifying the local crop response to irrigation is important for establishing adequate irrigation management strategies. This study evaluated the effect of irrigation applied with subsurface drip irrigation on field corn (Zea mays L.) evapotranspiration (ETc), yield, water use efficiencies (WUE = yield/ETc, and IWUE = yield/irrigation), and dry matter production in the semiarid climate of west central Nebraska. Eight treatments were imposed with irrigation amounts ranging from 53 to 356 mm in 2005 and from 22 to 226 mm in 2006. A soil water balance approach (based on FAO-56) was used to estimate daily soil water and ETc. Treatments resulted in seasonal ETc of 580-663 mm and 466-656 mm in 2005 and 2006, respectively. Yields among treatments differed by as much as 22% in 2005 and 52% in 2006. In both seasons, irrigation significantly affected yields, which increased with irrigation up to a point where irrigation became excessive. Distinct relationships were obtained each season. Yields increased linearly with seasonal ETc (R 2 = 0.89) and ETc/ETp (R 2 = 0.87) (ETp = ETc with no water stress). The yield response factor (ky), which indicates the relative reduction in yield to relative reduction in ETc, averaged 1.58 over the two seasons. WUE increased non-linearly with seasonal ETc and with yield. WUE was more sensitive to irrigation during the drier 2006 season, compared with 2005. Both seasons, IWUE decreased sharply with irrigation. Irrigation significantly affected dry matter production and partitioning into the different plant components (grain, cob, and stover). On average, the grain accounted for the majority of the above-ground plant dry mass (≈59%), followed by the stover (≈33%) and the cob (≈8%). The dry mass of the plant and that of each plant component tended to increase with seasonal ETc. The good relationships obtained in the study between crop performance indicators and seasonal ETc demonstrate that accurate estimates of ETc on a daily and seasonal basis can be valuable for making tactical in-season irrigation management decisions and for strategic irrigation planning and management.

Water activity of poultry litter: Relationship to moisture content during a grow-out

Poultry grown on litter floors are in contact with their own waste products. The waste material needs to be carefully managed to reduce food safety risks and to provide conditions that are comfortable and safe for the birds. Water activity (Aw) is an important thermodynamic property that has been shown to be more closely related to microbial, chemical and physical properties of natural products than moisture content. In poultry litter, Aw is relevant for understanding microbial activity; litter handling and rheological properties; and relationships between in-shed relative humidity and litter moisture content. We measured the Aw of poultry litter collected throughout a meat chicken grow-out (from fresh pine shavings bedding material to day 52) and over a range of litter moisture content (10–60%). The Aw increased non-linearly from 0.71 to 1.0, and reached a value of 0.95 when litter moisture content was only 22–33%. Accumulation of manure during the grow-out reduced Aw for the same moisture content. These results are relevant for making decisions regarding litter re-use in multiple grow-outs as well as setting targets for litter moisture content to minimise odour, microbial risks and to ensure necessary litter physical conditions are maintained during a grow-out. Methods to predict Aw in poultry litter from moisture content are proposed.

Use of the Nasicon/Na2SO4 couple in a solid state sensor for SOx (x=2,3)

The e.m.f. of a concentration cell for SO x (x=2,3)-O2 incorporating Nasicon as the main solid electrolyte has been measured in the temperature range 720 to 1080 K. The cell arrangement can be represented as,$$Pt, O'_2 + SO'_2 + SO'_3 \left| {Na_2 SO_4 \left\| {\left. {Nasicon} \right\|} \right.} \right.\left. {Na_2 SO_4 } \right|SO''_3 + SO''_2 + O''_2 , Pt$$ The Na2SO4 acts both as an auxiliary electrode, converting chemical potentials of SO x and O2 to equivalent sodium potentials, and as an electrolyte. The presence of Na2SO4 provides partial protection of Nasicon from chemical reaction with gas mixtures containing SO x . The open circuit e.m.f. of the cell is in close agreement with values given by the Nernst equation. For certain fixed inlet gas compositions of SO2+O2, the e.m.f. varies non-linearly with temperature. The intrinsic response time of the cell to step changes in gas composition is estimated to vary from sim2.0 ksec at 723K to sim 0.2 ksec at 1077K. The cell functions well for large differences in partial pressures of SO3(pPrimeSO 3/pprimeSO 3ap104) at the electrodes.