Uptake of water from soils by plant roots

Uptake of water by plant roots can be considered at two different Darcian scales, referred to as the mesoscopic and macroscopic scales. At the mesoscopic scale, uptake of water is represented by a flux at the soil–root interface, while at the macroscopic scale it is represented by a sink term in the volumetric mass balance. At the mesoscopic scale, uptake of water by individual plant roots can be described by a diffusion equation, describing the flow of water from soil to plant root, and appropriate initial and boundary conditions. The model involves at least two characteristic lengths describing the root–soil geometry and two characteristic times, one describing the capillary flow of water from soil to plant roots and another the ratio of supply of water in the soil and uptake by plant roots. Generally, at a certain critical time, uptake will switch from demand-driven to supply-dependent. In this paper, the solutions of some of the resulting mesoscopic linear and nonlinear problems are reviewed. The resulting expressions for the evolution of the average water content can be used as a basis for upscaling from the mesoscopic to the macroscopic scale. It will be seen that demand-driven and supply-dependent uptake also emerge at the macroscopic scale. Information about root systems needed to operationalize macroscopic models will be reviewed briefly.     

Time-harmonic P-waves engulfing a rectangular limited-permeable crack in piezoelectric medium: Energy density and energy release

The dynamic behavior of a limited-permeable rectangular crack in a transversely isotropic piezoelectric material is impinged by to a P-wave. The generalized Almansi theorem and the Schmidt method are used to determine the stress intensity factor and energy density factor as the primary fracture criterion of failure. The mixed boundary value problem entails the evaluation of the appropriate crack edge stress singularities that are characteristics of the fundamental functions. The stress and electric displacement intensity factors are also used to find the energy release rate that can be computed numerically and compared with the results corresponding to those of the stress intensity factor, and energy density factor. Graphical presentation shows that the energy release rate is always negative for the boundary conditions considered while the energy density factors always remain positive. Under certain conditions, the stress and electric displacement intensity factors can be negative and subject to physical limitations. Piezoelectric material boundary value problem solutions should therefore be qualified by the application of failure criteria by fracture of otherwise, particularly when the mechanical and electrical energy can release by creating free surface at the macroscopic and microscopic scales. Negative energy release rate found for the piezoelectric medium in this work can be a case in point.Positive definiteness of the energy density factor can be applied to mutliscale fracture. This is not true for the stress intensity factor nor the energy release rate. Hence, crack initiation behavior for the permittivity of a rectangular crack due to the wave propagation effects may be studied. In particular, the initiation of micro-cracks may be identified with certain critical stress wave frequency band. Negative stress intensity factor may not enhance macrocracking but it does not exclude microcrack initiation.     

Damage analysis of tetragonal perovskite structure ceramics implicated by asymptotic field solutions and boundary conditions

Cracking of ceramics with tetragonal perovskite grain structure is known to appear at different sites and scale level. The multiscale character of damage depends on the combined effects of electromechanical coupling, prevailing physical parameters and boundary conditions. These detail features are exhibited by application of the energy density criterion with judicious use of the mode I asymptotic and full field solution in the range of r/a=10⁻⁴ to 10⁻² where r and a are, respectively, the distance to the crack tip and half crack length. Very close to the stationary crack tip, bifurcation is predicted resembling the dislocation emission behavior invoked in the molecular dynamics model. At the macroscopic scale, crack growth is predicted to occur straight ahead with two yield zones to the sides. A multiscale feature of crack tip damage is provided for the first time. Numerical values of the relative distances and bifurcation angles are reported for the PZT-4 ceramic subjected to different electric field to applied stress ratio and boundary conditions that consist of the specification of electric field/mechanical stress, electric displacement/mechanical strain, and mixed conditions. To be emphasized is that the multiscale character of damage in piezoceramics does not appear in general. It occurs only for specific combinations of the external and internal field parameters, elastic/piezoelectric/dielectric constants and specified boundary conditions.     

A continuum mechanical theory for turbulence: a generalized Navier–Stokes-<Emphasis Type="Italic">α</Emphasis> equation with boundary conditions

We develop a continuum-mechanical formulation and generalization of the Navier–Stokes-α equation based on a recently developed framework for fluid-dynamical theories involving higher-order gradient dependencies. Our flow equation involves two length scales α and β. The first of these enters the theory through the specific free-energy α ²|D|², where D is the symmetric part of the gradient of the filtered velocity, and contributes a dispersive term to the flow equation. The remaining scale is associated with a dissipative hyperstress which depends linearly on the gradient of the filtered vorticity and which contributes a viscous term, with coefficient proportional to β ², to the flow equation. In contrast to Lagrangian averaging, our formulation delivers boundary conditions and a complete structure based on thermodynamics applied to an isothermal system. For a fixed surface without slip, the standard no-slip condition is augmented by a wall-eddy condition involving another length scale ℓ characteristic of eddies shed at the boundary and referred to as the wall-eddy length. As an application, we consider the classical problem of turbulent flow in a plane, rectangular channel of gap 2h with fixed, impermeable, slip-free walls and make comparisons with results obtained from direct numerical simulations. We find that α/β ~ Re ^0.470 and ℓ/h ~ Re ^−0.772, where Re is the Reynolds number. The first result, which arises as a consequence of identifying the specific free-energy with the specific turbulent kinetic energy, indicates that the choice β = α required to reduce our flow equation to the Navier–Stokes-α equation is likely to be problematic. The second result evinces the classical scaling relation η/L ~ Re ^−3/4 for the ratio of the Kolmogorov microscale η to the integral length scale L.      

Fatigue crack growth of cable steel wires in a suspension bridge: Multiscaling and mesoscopic fracture mechanics

Intrinsically, fatigue failure problem is a typical multiscale problem because a fatigue failure process deals with the fatigue crack growth from microscale to macroscale that passes two different scales. Both the microscopic and macroscopic effects in geometry and material property would affect the fatigue behaviors of structural components. Classical continuum mechanics has inability to treat such a multiscale problem since it excludes the scale effect from the beginning by introducing the continuity and homogeneity assumptions which blot out the discontinuity and inhomogeneity of materials at the microscopic scale. The main obstacle here is the link between the microscopic and macroscopic scale. It has to divide a continuous fatigue process into two parts which are analyzed respectively by different approaches. The first is so called as the fatigue crack initiation period and the second as the fatigue crack propagation period. Now the problem can be solved by application of the mesoscopic fracture mechanics theories developed in the recent years which focus on the link between different scales such as nano-, micro- and macro-scale.On the physical background of the problem, a restraining stress zone that can describe the material damaging process from micro to macro is then introduced and a macro/micro dual scale edge crack model is thus established. The expression of the macro/micro dual scale strain energy density factor is obtained which serves as a governing quantity for the fatigue crack growth. A multiscaling formulation for the fatigue crack growth is systematically developed. This is a main contribution to the fundamental theories for fatigue problem in this work. There prevail three basic parameters μ^∗, σ^∗ and d^∗ in the proposed approach. They can take both the microscopic and macroscopic factors in geometry and material property into account. Note that μ^∗, σ^∗ and d^∗ stand respectively for the ratio of microscopic to macroscopic shear modulus, the ratio of restraining stress to applied stress and the ratio of microvoid size ahead of crack tip to the characteristic length of material microstructure.To illustrate the proposed multiscale approach, Hangzhou Jiangdong Bridge is selected to perform the numerical computations. The bridge locates at Hangzhou, the capital of Zhejiang Province of China. It is a self-anchored suspension bridge on the Qiantang River. The cables are made of 109 parallel steel wires in the diameter of 7 mm. Cable forces are calculated by finite element method in the service period with and without traffic load. Two parameters α and β are introduced to account for the additional tightening and loosening effects of cables in two different ways. The fatigue crack growth rate coefficient C₀ is determined from the fatigue experimental result. It can be concluded from numerical results that the size of initial microscopic defects is a dominant factor for the fatigue life of steel wires. In general, the tightening effect of cables would decrease the fatigue life while the loosening effect would impede the fatigue crack growth. However, the result can be reversed in some particular conditions. Moreover, the different evolution modes of three basic parameters μ^∗, σ^∗ and d^∗ actually have the different influences on the fatigue crack growth behavior of steel wires. Finally the methodology developed in this work can apply to all cracking-induced failure problems of polycrystal materials, not only fatigue, but also creep rupture and cracking under both static and dynamic load and so on.     

Generalized Solutions of the Capillary Problem

It was pointed out by Finn [2] that the capillary problem in zero gravity has not always a classical (smooth) solution in the case that the bounded domain Ω⊂ℝ² contains cusps or corners. Here, ω denotes the cross section of a given cylinder, in which a liquid is contained. If special energy terms could become infinite the variational formulation is not free of limitations as well. Therefore, the concept of generalized solutions, which can be traced back to Miranda [11], has been developed and could be a way out. We want to prove an existence result for such solutions under very weak preconditions. The proof is closely related to Giusti's paper [6], but we do not require full smoothness of the boundary. The major new difficulty is that we also want to consider domains with non-Lipschitz boundary. This excludes the application of some theorems. On the other hand, we use special geometric conditions in ℝ² and consequently, the proof cannot easily be generalized to a higher dimension. Furthermore, we construct some generalized solutions explicitly.     

Development of a turbulent boundary layer after a step from smooth to rough surface

The flow developing downstream of a step change from smooth to rough surface condition is studied in the light of Townsend’s wall similarity hypothesis. Previous studies seem to support the hypothesis for channel and pipe flows, but there are considerable controversies about its application to boundary layers and in particular to surface roughness formed by spanwise bars. It has been suggested that this controversy arises from insufficient separation of scales between the boundary layer thickness and the roughness length scale. An experimental investigation has therefore been undertaken where the flow evolves from a fully developed smooth wall boundary layer at high Reynolds numbers over a step in surface roughness (Re _θ = 13,400 at the step). The flow is mapped through the development of the internal layer until the flow is fully developed over the rough wall. The internal layer is found to grow as δ ∼ X ^0.73, and after about 15 boundary layer thicknesses at the step, the internal layer has reached the outer edge of the incoming layer. At the last rough wall measurement station, the Reynolds number has grown to Re _θ ≈ 32,600 and the ratio of boundary layer to roughness length scales is δ/k ≈ 140. The outer layer differences between the smooth and the rough wall data were found to be sufficiently small to conclude that for this setup the Townsend’s wall similarity hypothesis appears to hold.     

On the asymptotic boundary conditions of an anisotropic beam via virtual work principle

In this research, an efficient and effective method is proposed to derive the boundary conditions of an anisotropic beam in the asymptotic sense. We first set up the constrained virtual work by introducing the Lagrange multiplier on the displacement prescribed boundary. The macroscopic beam and microscopic cross-section equations with the boundary conditions are simultaneously obtained by taking the asymptotic expansion on the displacement vector. In this way, the three-dimensional characteristics of the beam are asymptotically smeared into the macroscopic beam equations and the beam boundary conditions. The boundary conditions obtained are then compared to those from the decay analysis method. The beam bending slope boundary condition obtained in the frame work of variational principle is different from the well-known average condition. This new boundary condition is more accurate than the average one for a sandwich beam. This is further demonstrated and discussed via the examples of a cantilever beam loaded at the end.     

A Three-Scale Model of pH-Dependent Flows and Ion Transport with Equilibrium Adsorption in Kaolinite Clays: I. Homogenization Analysis

A new three-scale model to describe the coupling between pH-dependent flows and transient ion transport including sorption phenomena in kaolinite clays is proposed. The kaolinite is characterized by three separate nano-micro and macroscopic length scales. The (micro)-scale consists of micro-pores saturated by an aqueous solution containing four monovalent ionic species (Na⁺, H⁺, Cl^?, OH^?) and charged solid particles surrounded by thin electrical double layers. The movement of the ions is governed by the Nernst-Planck equations and the influence of the double layers upon the flow is dictated by the Helmholtz–Smoluchowski slip boundary condition in the tangential velocity. In addition, sorption interface conditions for ion transport are postulated in the sense of Auriault and Lewandowska (Eur. J. Mech. A 15:681–704, 1996) to capture the immobilization of the ions in the electrical double layer and on particle surface due to protonation/deprotonation reactions. The intensity of sorption relative to diffusion effects is quantified by the Damköhler number, whose order of magnitude is estimated by invoking the nanoscopic modeling of the thin EDL based on Poisson–Boltzmann problem for the local electric potential coupled with a non-linear surface charge density with constitutive law dictated by the protonation/deprotonation reactions. The two-scale nano/micro model including sorption and slip boundary condition is homogenized to the core scale leading to a derivation of macroscopic governing equations.     

Mesofracture mechanics: a necessary link

Classical fracture mechanics is based on the premise that small scale features could be averaged to give a larger scale property such that the assumption of material homogeneity would hold. Involvement of the material microstructure, however, necessitates different characteristic lengths for describing different geometric features. Macroscopic parameters could not be freely exchanged with those at the microscopic scale level. Such a practice could cause misinterpretation of test data. Ambiguities arising from the lack of a more precise range of limitations for the definitions of physical parameters are discussed in connection with material length scales. Physical events overlooked between the macroscopic and microscopic scale could be the link that is needed to bridge the gap. The classical models for the creation of free surface for a liquid and solid are oversimplified. They consider only the translational motion of individual atoms. Movements of groups or clusters of molecules deserve attention. Multiscale cracking behavior also requires the distinction of material damage involving at least two different scales in a single simulation. In this connection, special attention should be given to the use of asymptotic solution in contrast to the full field solution when applying fracture criteria. The former may leave out detail features that would have otherwise been included by the latter. Illustrations are provided for predicting the crack initiation sites of piezoceramics. No definite conclusions can be drawn from the atomistic simulation models such as those used in molecular dynamics until the non-equilibrium boundary conditions can be better understood. The specification of strain rates and temperatures should be synchronized as the specimen size is reduced to microns. Many of the results obtained at the atomic scale should be first identified with those at the mesoscale before they are assumed to be connected with macroscopic observations. Hopefully, “mesofracture mechanics” could serve as the link to bring macrofracture mechanics closer to microfracture mechanics.     

Tractions, Balances, and Boundary Conditions for Nonsimple Materials with Application to Liquid Flow at Small-Length Scales

Using a nonstandard version of the principle of virtual power, we develop general balance equations and boundary conditions for second-grade materials. Our results apply to both solids and fluids as they are independent of constitutive equations. As an application of our results, we discuss flows of incompressible fluids at small-length scales. In addition to giving a generalization of the Navier–Stokes equations involving higher-order spatial derivatives, our theory provides conditions on free and fixed boundaries. The free boundary conditions involve the curvature of the free surface; among the conditions for a fixed boundary are generalized adherence and slip conditions, each of which involves a material length scale. We reconsider the classical problem of plane Poiseuille flow for generalized adherence and slip conditions.     

The average stress tensor in systems with interacting particles

The derivation of an expression of the macroscopic stress tensor in terms of microscopic variables in systems of finite interacting particles is discussed from different points of view. It is shown that in volume averaging the introduction of a fictitious “interaction stress field”T ^I with special boundary conditions on the boundary of the averaging volume is needed. In ensemble averaging similar results are obtained by using a multipole expansion of the local stress and force fields. In the appropriate limiting cases, the obtained results are shown to be consistent with the results of kinetic theories of polymer solutions. Paper, presented at the First Conference of European Rheologists at Graz, April 14 – 16, 1982.     

On a framework for small-deformation viscoplasticity: free energy,microforces, strain gradients

This study develops a general theory for small-deformation viscoplasticity based on a system of microforces consistent with its own balance; a mechanical version of the second law that includes, via the microforces, work performed during viscoplastic flow; a constitutive theory that allows for dependences on plastic strain-gradients. The microforce balance and the constitutive equations—suitably restricted by the second law—are shown to be together equivalent to a flow rule that accounts for variations in free energy due to flow. When this energy is the sum of an elastic strain energy and a defect energy quadratic, isotropic, and positive definite in the plastic-strain gradients, the flow rule takes the form of a second-order parabolic PDE for the plastic strain coupled to the usual PDE arising from the standard macroscopic force balance and the elastic stress-strain relation. The classical macroscopic boundary conditions are supplemented by nonstandard boundary conditions associated with viscoplastic flow. As an aid to solution, a weak (virtual power) formulation of the nonlocal flow rule is derived.     

Landau–De Gennes Theory of Nematic Liquid Crystals: the Oseen–Frank Limit and Beyond

We study global minimizers of a continuum Landau–De Gennes energy functional for nematic liquid crystals, in three-dimensional domains, subject to uniaxial boundary conditions. We analyze the physically relevant limit of small elastic constant and show that global minimizers converge strongly, in W ^1,2, to a global minimizer predicted by the Oseen–Frank theory for uniaxial nematic liquid crystals with constant order parameter. Moreover, the convergence is uniform in the interior of the domain, away from the singularities of the limiting Oseen–Frank global minimizer. We obtain results on the rate of convergence of the eigenvalues and the regularity of the eigenvectors of the Landau–De Gennes global minimizer. We also study the interplay between biaxiality and uniaxiality in Landau–De Gennes global energy minimizers and obtain estimates for various related quantities such as the biaxiality parameter and the size of admissible strongly biaxial regions.     

Influence of material parameters and crystallography on the size effects describable by means of strain gradient plasticity

In the context of single-crystal strain gradient plasticity, we focus on the simple shear of a constrained strip in order to study the effects of the material parameters possibly involved in the modelling. The model consists of a deformation theory suggested and left undeveloped by Bardella [(2007). Some remarks on the strain gradient crystal plasticity modelling, with particular reference to the material length scales involved. Int. J. Plasticity 23, 296–322] in which, for each glide, three dissipative length scales are considered; they enter the model through the definition of an effective slip which brings into the isotropic hardening function the relevant plastic strain gradients, averaged by means of a p-norm. By means of the defect energy (i.e., a function of Nye's dislocation density tensor added to the free energy; see, e.g., Gurtin [2002. A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations. J. Mech. Phys. Solids 50, 5–32]), the model further involves an energetic material length scale. The application suggests that two dissipative length scales may be enough to qualitatively describe the size effect of metals at the microscale, and they are chosen in such a way that the higher-order state variables of the model be the dislocation densities. Moreover, we show that, depending on the crystallography, the size effect governed by the defect energy may be different from what expected (based on the findings of [Bardella, L., 2006. A deformation theory of strain gradient crystal plasticity that accounts for geometrically necessary dislocations. J. Mech. Phys. Solids 54, 128–160] and [Gurtin et al. 2007. Gradient single-crystal plasticity with free energy dependent on dislocation densities. J. Mech. Phys. Solids 55, 1853–1878]), leading mostly to some strengthening. In order to investigate the model capability, we also exploit a Γ-convergence technique to find closed-form solutions in the “isotropic limit”. Finally, we analytically show that in the “perfect plasticity” case, should the dissipative length scales be set to zero, the presence of the sole energetic length scale may lead, as in standard plasticity, to non-uniqueness of solutions.     

Asymptotic Analysis for a Mixed Boundary-Value Contact Problem

The variational solution of the nonlinear Signorini contact problem determines also the active contact zone Γ ^c . If the latter is known, then the elastic field is a solution of a linear mixed boundary value problem in which on Γ ^c the normal displacement and tangential traction are given, while on the non-contact part the total traction is zero. Such mixed boundary conditions in general generate singularities of the solution's stress field at the points P ^{( k )} where the boundary conditions change. For smooth data, however, the variational solution of the Signorini contact problem actually belongs to H ²(Ω)², which implies the disappearance of these singularities, i.e., that the corresponding stress intensity factors vanish. This paper is devoted to the characterization of the active contact zone Γ ^c by the vanishing stress intensity factors including their sensitivity with respect to varying Γ ^c for two-dimensional problems provided that Γ ^c consists of a finite number of intervals. We use the method of asymptotic expansions and derive an explicit formula for the sensitivity, which is rigorously justified by employing weighted Sobolev spaces with detached asymptotics. These results can be used to determine the points P ^{( k )} with a corresponding Newton iteration. Accepted July 6, 2000?Published online January 22, 2001     

Time evolution of structures in rotating and stratified turbulence

Rotating and stably stratified turbulence exhibit not only significant anisotropies but also dynamics, which are qualitatively different from purely rotating or stratified turbulence. Furthermore, the different time scales due to rotation, stratification and the turbulence one open up a wide field of possibilities for the temporal evolution of rotating and stratified turbulence.We analyze results from DNS with different parameters α = f/N by visualizing iso-enstrophy surfaces, the temporal evolution of velocity correlation length scales and angular energy spectra.We retrieve standard results, such as a large anisotropy for small scales in rotating turbulence and a large anisotropy for intermediate scales in the vortex mode of stratified turbulence. Furthermore, at large times we find qualitatively different phenomena for cases α = 10 and α = 0.1 such as modified cascades due to the existence of potential energy or small scale vorticity production respectively.