A large deformation theory for rate-dependent elastic–plastic materials with combined isotropic and kinematic hardening

We have developed a large deformation viscoplasticity theory with combined isotropic and kinematic hardening based on the dual decompositions F=F^eF^p [Kröner, E., 1960. Allgemeine kontinuumstheorie der versetzungen und eigenspannungen. Archive for Rational Mechanics and Analysis 4, 273–334] and [Lion, A., 2000. Constitutive modelling in finite thermoviscoplasticity: a physical approach based on nonlinear rheological models. International Journal of Plasticity 16, 469–494]. The elastic distortion F^e contributes to a standard elastic free-energy ψ^(e), while , the energetic part of F^p, contributes to a defect energy ψ^(p) – these two additive contributions to the total free energy in turn lead to the standard Cauchy stress and a back-stress. Since F^e=FF^p-1 and , the evolution of the Cauchy stress and the back-stress in a deformation-driven problem is governed by evolution equations for F^p and – the two flow rules of the theory.We have also developed a simple, stable, semi-implicit time-integration procedure for the constitutive theory for implementation in displacement-based finite element programs. The procedure that we develop is “simple” in the sense that it only involves the solution of one non-linear equation, rather than a system of non-linear equations. We show that our time-integration procedure is stable for relatively large time steps, is first-order accurate, and is objective.     

Turbulent wake and vortex shedding for a stack partially immersed in a turbulent boundary layer

The effect of the jet-to-cross-flow velocity ratio, R, on the turbulent wake and Kármán vortex shedding for a cylindrical stack of aspect ratio AR=9 was investigated in a low-speed wind tunnel using thermal anemometry. The cross-flow Reynolds number was Re_D=2.3×10⁴, the jet Reynolds number ranged from Re_d=7.6×10³ to 4.7×10⁴, and R was varied from 0 to 3. The stack was partially immersed in a flat-plate turbulent boundary layer, with a boundary layer thickness-to-stack-height ratio of δ/H=0.5 at the location of the stack. From the behaviour of the turbulent wake and the vortex shedding, the flow around the stack could be classified into three regimes depending on the value of R, which were the downwash (R<0.7), cross-wind-dominated (0.7R<1.5), and jet-dominated (R1.5) flow regimes. Each flow regime had a distinct structure to the mean velocity (streamwise and wall-normal directions), turbulence intensity (streamwise and wall-normal directions), and Reynolds shear stress fields, as well as the variation of the Strouhal number and the power spectrum along the stack height.     

Fast Diffusion to Self-Similarity: Complete Spectrum,Long-Time Asymptotics,and Numerology

The complete spectrum is determined for the operator on the Sobolev space W^1,2(Rⁿ) formed by closing the smooth functions of compact support with respect to the norm Here the Barenblatt profile is the stationary attractor of the rescaled diffusion equation in the fast, supercritical regime m the same diffusion dynamics represent the steepest descent down an entropy E(u) on probability measures with respect to the Wasserstein distance d₂. Formally, the operator H=HessE is the Hessian of this entropy at its minimum , so the spectral gap H:=2–n(1–m) found below suggests the sharp rate of asymptotic convergence: from any centered initial data 0u(0,x)L¹(Rⁿ) with second moments. This bound improves various results in the literature, and suggests the conjecture that the self-similar solution u(t,x)=R(t)^–n(x/R(t)) is always slowest to converge. The higher eigenfunctions – which are polynomials with hypergeometric radial parts – and the presence of continuous spectrum yield additional insight into the relations between symmetries of Rⁿ and the flow. Thus the rate of convergence can be improved if we are willing to replace the distance to with the distance to its nearest mass-preserving dilation (or still better, affine image). The strange numerology of the spectrum is explained in terms of the number of moments of .Dedicated to Elliott H. Lieb on the occasion of his 70th birthday.     

Floquet bundles for scalar parabolic equations

For linear scalar parabolic equations such as on a finite interval 0x, with various boundary conditions, we obtain canonical Floquet solutions u _n (t, x). These solutions are characterized by the property that z(u _n (t, x))=n for all t, where z(·) denotes the zero crossing (lap) number of Matano. The coefficients a(t, x) and b(t, x) are not assumed to be periodic in t, but if they are, the solutions u _n (t, x) reduce to the standard Floquet solutions. Our results may naturally be expressed in the language of linear skew product flows. In this context, we obtain for each N1 an exponential dichotomy between the bundles span {u _n (·,·)} _n ^=1/N and .     

Observations on the theoretical and experimental foundations of thermoplasticity

The thermodynamics of materials with internal state variables has been employed to study the properties of a class of thermoplastic materials in which the evolution equation for the internal variables is given by equation k⁽ⁱ⁾ = g⁽ⁱ⁾(σ_kl, '_kl, k⁽ⁱ⁾, θ, · '_kl) where g⁽ⁱ⁾ is homogeneous of order one in · '_kl. The most general form of the Helmholtz potential consistent with the assumption of insensitivity of the elastic relations to inelastic deformation has been derived and a geometric interpretation of the Clausius-Duhem restriction has been made employing the concept of a thermodynamic reference stress. Experimental results of one of the authors have been correlated with the theory.     

Viscoelastic sink flow in a wedge for the UCM and Oldroyd-B models

The steady planar sink flow through wedges of angle π/α with α≥1/2 of the upper convected Maxwell (UCM) and Oldroyd-B fluids is considered. The local asymptotic structure near the wedge apex is shown to comprise an outer core flow region together with thin elastic boundary layers at the wedge walls. A class of similarity solutions is described for the outer core flow in which the streamlines are straight lines giving stress and velocity singularities of O(r⁻²) and O(r⁻¹), respectively, where r1 is the distance from the wedge apex. These solutions are matched to wall boundary layer equations which recover viscometric behaviour and are subsequently also solved using a similarity solution. The boundary layers are shown to be of thickness O(r²), their size being independent of the wedge angle. The parametric solution of this structure is determined numerically in terms of the volume flux Q and the pressure coefficient p₀, both of which are assumed furnished by the flow away from the wedge apex in the r=O(1) region. The solutions as described are sufficiently general to accommodate a wide variety of external flows from the far-field r=O(1) region. Recirculating regions are implicitly assumed to be absent.     

Dynamic interaction of various beams with the underlying soil – finite and infinite, half-space and Winkler models

Various beams lying on the elastic half-space and subjected to a harmonic load are analyzed by a double numerical integration in wavenumber domain. The compliances of the beam–soil systems are presented for a wide frequency range and for a number of realistic parameter sets. Generally, the soil stiffness G has a strong influence on the low-frequency beam compliance whereas the beam parameters EI and m^′ are more important for the high-frequency compliance. An important parameter is the elastic length l=(EI/G)^1/4 of the beam–soil system. Around the corresponding frequency ω_l=v_S/l, the wave velocity of the combined beam–soil system changes from the Rayleigh wave v_R≈v_S to the bending wave velocity v_B and the combined beam–soil wave has typically a strong damping. The interaction frequency ω_l is found not far from the characteristic frequency ω₀=(G/m^′)^1/2 where an amplification compared to the static compliance is observed for special parameter constellations. In contrast, real foundation beams show no resonance effects as they are highly damped by the radiation into the soil. At medium and high frequencies, asymptotes for the compliance of the beam–soil system are found, u/P(ρv_Paiω)^−3/4 in case of the dominating damping and u/P(−m^′ω²)^−3/4 for high frequencies. The low-frequency compliance of the coupled beam–soil system can be approximated by u/P1/Gl, but it also depends weakly on the width a of the foundation. All numerical results of different beam–soil systems are evaluated to yield a unique relation u/P₀=f(a/l). The integral transform method is also applied to ballasted and slab tracks of railway lines, showing the influence of train speed on the deformation of the track beam. The presented results of infinite beams on half-space are compared with results of finite beams and with infinite beams on a Winkler support. Approximating Winkler parameters are given for realistic foundation-soil systems which are useful when vehicle-track interaction is analyzed for the prediction of railway induced vibration.     

Crack repair using an elastic filler

The effect of repairing a crack in an elastic body using an elastic filler is examined in terms of the stress intensity levels generated at the crack tip. The effect of the filler is to change the stress field singularity from order 1/r^1/2 to 1/r^(1-λ) where r is the distance from the crack tip, and λ is the solution to a simple transcendental equation. The singularity power (1-λ) varies from (the unfilled crack limit) to 1 (the fully repaired crack), depending primarily on the scaled shear modulus ratio γ_r defined by G₂/G₁=γ_rε, where 2πε is the (small) crack angle, and the indices (1, 2) refer to base and filler material properties, respectively. The fully repaired limit is effectively reached for γ_r≈10, so that fillers with surprisingly small shear modulus ratios can be effectively used to repair cracks. This fits in with observations in the mining industry, where materials with G₂/G₁ of the order of 10^-3 have been found to be effective for stabilizing the walls of tunnels. The results are also relevant for the repair of cracks in thin elastic sheets.     

Linear Stability Analysis of Resonant Periodic Motions in the Restricted Three-Body Problem

The equations of the restricted three-body problem describe the motion of a massless particle under the influence of two primaries of masses 1 −μ and μ, 0≤μ≤ 1/2, that circle each other with period equal to 2π. When μ=0, the problem admits orbits for the massless particle that are ellipses of eccentricity e with the primary of mass 1 located at one of the focii. If the period is a rational multiple of 2π, denoted 2π p/q, some of these orbits perturb to periodic motions for μ > 0. For typical values of e and p/q, two resonant periodic motions are obtained for μ > 0. We show that the characteristic multipliers of both these motions are given by expressions of the form in the limit μ→ 0. The coefficient C(e,p,q) is analytic in e at e=0 and C(e,p,q)=O(e^|p-q|). The coefficients in front of e^|p-q|, obtained when C(e,p,q) is expanded in powers of e for the two resonant periodic motions, sum to zero. Typically, if one of the two resonant periodic motions is of elliptic type the other is of hyperbolic type. We give similar results for retrograde periodic motions and discuss periodic motions that nearly collide with the primary of mass 1 −μ.     

Determination of all stress components from measurements of the stress invariant by the thermoelastic stress method

A method for the resolution of all stress components from the first invariant J₁ measured by thermoelastic stress analyzer is described. This method may be used to determine, not only surface stress, but also internal stress and stress on the underside.The method is based on the following procedure:

1. (1) Pick an arbitrary domain Ω, within the structure, for which the stresses are required.

2. (2) Measure J₁ on the surface of Ω.

3. (3) Determine the optimum traction along the boundary Γ, which is a part of Ω, by the least squares method such that the difference between the measured J₁ and the calculated J₁ is at a minimum. Either FEM or BEM may be used for this calculation.

Examples of stress resolution for a two-dimensional stress concentration problem and a three-dimensional stress concentration problem are shown. The accuracy of the stress resolution is discussed.     

Effect of material nonhomogeneities on the HRR dominance

This work studies the asymptotic stress and displacement fields near the tip of a stationary crack in an elastic–plastic nonhomogeneous material with the emphasis on the effect of material nonhomogeneities on the dominance of the crack tip field. While the HRR singular field still prevails near the crack tip if the material properties are continuous and piecewise continuously differentiable, a simple asymptotic analysis shows that the size of the HRR dominance zone decreases with increasing magnitude of material property gradients. The HRR field dominates at points that satisfy |α⁻¹ ∂α/∂x_δ|1/r, |α⁻¹ ∂²α/(∂x_δ ∂x_γ)|1/r², |n⁻¹ ∂n/∂x_δ|1/[r|ln(r/A)|] and |n⁻¹ ∂²n/(∂x_δ ∂x_γ)|1/[r²|ln(r/A)|], in addition to other general requirements for asymptotic solutions, where α is a material property in the Ramberg–Osgood model, n is the strain hardening exponent, r is the distance from the crack tip, x_δ are Cartesian coordinates, and A is a length parameter. For linear hardening materials, the crack tip field dominates at points that satisfy |E_tan⁻¹ ∂E_tan/∂x_δ|1/r, |E_tan⁻¹ ∂²E_tan/(∂x_δ ∂x_γ)|1/r², |E⁻¹ ∂E/∂x_δ|1/r, and |E⁻¹ ∂²E/(∂x_δ ∂x_γ)|1/r², where E_tan is the tangent modulus and E is Young’s modulus.     

Two-phase geothermal flows with conduction and the connection with Buckley-Leverett theory

Two-phase flows of boiling water and steam in geothermal reservoirs satisfy a pair of conservation equations for mass and energy which can be combined to yield a hyperbolic wave equation for liquid saturation changes. Recent work has established that in the absence of conduction, the geothermal saturation equation is, under certain conditions, asymptotically identical with the Buckley-Leverett equation of oil recovery theory. Here we summarise this work and show that it may be extended to include conduction. In addition we show that the geothermal saturation wave speed is under all conditions formally identical with the Buckley-Leverett wave speed when the latter is written as the saturation derivative of a volumetric flow.Roman Letters C(P, S,q) geothermal saturation wave speed [ms^–1] (14) - c _t (P, S) two-phase compressibility [Pa^–1] (10) - D(P, S) diffusivity [m s^–2] (8) - E(P, S) energy density accumulation [J m^–3] (3) - g gravitational acceleration (positive downwards) [ms^–2] - h _w (P),h _w (P) specific enthalpies [J kg^–1] - J _M (P, S,P) mass flow [kg m^–2 s^–1] (5) - J _E (P, S,P) energy flow [J m^–2s^–1] (5) - k absolute permeability (constant) [m²] - k _w (S),k _s (S) relative permeabilities of liquid and vapour phases - K formation thermal conductivity (constant) [Wm^–1 K^–1] - L lower sheetC<0 in flow plane - m, c gradient and intercept - M(P, S) mass density accumulation [kg m^–3] (3) - O flow plane origin - P(x,t) pressure (primary dependent variable) [Pa] - q volume flow [ms^–1] (6) - S(x, t) liquid saturation (primary dependent variable) - S ^*(x,t) normalised saturation (Appendix) - t time (primary independent variable) [s] - T temperature (degrees Kelvin) [K] - T _sat(P) saturation line temperature [K] - TdT _sat/dP saturation line temperature derivative [K Pa^–1] (4) - T _c ,T _D convective and diffusive time constants [s] - u _w (P),u _s (P),u _r (P) specific internal energies [J kg^–1] - U upper sheetC > 0 in flow plane - U(x,t) shock velocity [m s^–1] - x spatial position (primary independent variable) [m] - X representative length - x, y flow plane coordinates - z depth variable (+z vertically downwards) [m] Greek Letters _P , _S remainder terms [Pa s^–1], [s^–1] - double-valued saturation region in the flow plane - h =h _s –h _w latent heat [J kg^–1] - = _w – _s density difference [kg m^–3] - line envelope - =D _K /D ₀ diffusivity ratio - porosity (constant) - _w (P), _s (P), _t (P, S) dynamic viscosities [Pa s] - v _w (P),v _s (P) kinematic viscosities [m²s^–1] - v ₀ =kh/KT kinematic viscosity constant [m² s^–1] - ₀ =v ₀ dynamic viscosity constant [m² s^–1] - _w (P), _s (P) density [kg m^–3] Suffixes r rock matrix - s steam (vapour) - w water (liquid) - t total - av average - 0 without conduction - K with conduction     

Symmetry and secondary bifurcation of elastic structures

We consider non-linear bifurcation problems for elastic structures modeled by the operator equation F[w;α]=0 where F:X×R^k→Y,X,Y are Banach spaces and XY. We focus attention on problems whose bifurcation equations are of the form

f_i(α₁,α₂;λ,μ)=(a_iμ+b_iλ)α_i+p_iα_i³+q_iα_i∑_j=1,j≠i^kα_j+1²+α_ih_i(λ,μ;α₁,α₂,…α_k) i=1,2,…k

which emanates from bifurcation problems for which the linearization of F is Fredholm operators of index 0. Under the assumption of F being odd we prove an important theorem of existence of secondary bifurcation. Under this same assumption we prove a symmetry condition for the reduced equations and consequently we got an existence result for secondary bifurcation. We also include a stability analysis of the bifurcating solutions.     

On Asymptotics of Solutions of Nonlinear Differential Equations of the Second Order

We study the problem of asymptotics of unbounded solutions of differential equations of the form y″ = α₀ p(t)ϕ(y), where α₀ ∈ {−1, 1}, p: [a, ω[→]0, +∞[, −∞ < a < ω ≤ +∞, is a continuous function, and ϕ: [y ₀, +∞[→]0, +∞[ is a twice continuously differentiable function close to a power function in a certain sense.__________Translated from Neliniini Kolyvannya, Vol. 8, No. 1, pp. 18–28, January–March, 2005.     

Axisymmetric resonant disturbances in a homogeneous wave guide

The initial boundary-value linear stability problem for small localised axisymmetric disturbances in a homogeneous elastic wave guide, with the free upper surface and the lower surface being rigidly attached to a half-space, is formally solved by applying the Laplace transform in time and the Hankel transforms of zero and first orders in space. An asymptotic evaluation of the solution, expressed as a sum of inverse Laplace-Hankel integrals, is carried out by using the approach of the mathematical formalism of absolute and convective instabilities. It is shown that the dispersion-relation function of the problem D₀ (κ, ω), where the Hankel parameter κ is substituted by a wave number (and the Fourier parameter) κ, coincides with the dispersion-relation function D₀ (k, ω) for two-dimensional (2-D) disturbances in a homogeneous wave guide, where ω is the frequency (and the Laplace parameter) in both cases. An analysis for localised 2-D disturbances in a homogeneous wave guide is then applied. We obtain asymptotic expressions for wave packets, triggered by axisymmetric perturbations localised in space and finite in time, as well as for responses to axisymmetric sources localised in space, with the time dependence satisfying e^−iω₀t + O(e^−εt) for t → ∞, where Im ω₀ = 0, ε > 0, and t denotes time, i.e. for signalling with frequency ω₀. We demonstrate that, for certain combinations of physical parameters, axisymmetric wave packets with an algebraic temporal decay and axisymmetric signalling with an algebraic temporal growth, as √t, i.e., axisymmetric temporal resonances, are present in a neutrally stable homogeneous wave guide. The set of physically relevant wave guides having axisymmetric resonances is shown to be fairly wide. Furthermore, since an axisymmetric part of any source is L²-orthogonal to its non-axisymmetric part, a 3-D signalling with a non-vanishing axisymmetric component at an axisymmetric resonant frequency will generally grow algebraically in time. These results support our hypothesis concerning a possible resonant triggering mechanism of certain earthquakes, see Brevdo, 1998, J. Elasticity, 49, 201–237.     

Collinear Central Configurations and Singular Surfaces in the Mass Space

For a given m=(m₁,...,m_n)(R⁺)ⁿ, let p and q(R³)ⁿ be two central configurations for m. Then we call p and q equivalent and write pq if they differ by an SO(3) rotation followed by a scalar multiplication as well as by a permutation of bodies. Denote by L(n,m) the set of equivalent classes of n-body collinear central configurations in R³ for any given mass vector m=(m₁,...,m_n)(R⁺)ⁿ. The main discovery in this paper is the existence of a union H₃ of three non-empty algebraic surfaces in the mass half space (m₁,m₂–m₁,m₃–m₂)R⁺×R² besides the planes generated by equal masses, which decreases the number of collinear central configurations. The union H₃ in R⁺×R ² is explicitly constructed by three 6-degree homogeneous polynomials in three variables such that, for any mass vector m=(m₁,m₂,m₃)(R⁺)³, ^# L(3,m)=3, if m₁, m₂, and m₃ are mutually distinct and (m₁,m₂–m₁,m₃–m₂)H₃, ^# L(3,m)=2, if m₁, m₂, and m₃ are mutually distinct and (m₁,m₂–m₁,m₃–m₂)H₃, ^# L(3,m)=2, if two of m₁, m₂, and m₃ are equal but not the third, ^# L(3,m)=1, if m₁=m₂=m₃. We give also a sharp upper bound on ^#L(n,m) for any positive mass vector m(R⁺)ⁿ.     

Understanding the Piche evaporimeter as a simple integrating mass transfer meter

Results are presented of a comparison of measured and calculated evaporation rates of the Piche evaporimeter under indoor and outdoor (within a meteorological screen) conditions. In both cases, application of mass transfer formulae in use for horizontal (turbulent) flow to the evaporating blotting paper of the instrument yield very good results under pure forced convection conditions. For mixed convection regimes, comparisons using either pure free (combined heat and mass transfer) or pure forced convection equations give as expected too low calculated values. Reasons for such differences with measured values are reviewed. Our forced convection results confirm that main stream turbulence is only of influence on mass transfer to a zero incidence flow in combination with pressure gradient (bluff body) effects, which under our conditions appear to be absent around the Piche surfaces. The same results prove absence of any influence of the particular temperature distribution over the blotting paper on the mass transfer. The understanding and importance of these conclusions in relation to the use of the Piche evaporimeter as a simple integrating mass transfer meter under actual farming conditions are discussed. The importance to obtain such mass transfer data is explained in the introduction.Nomenclature A Numerical constant in free convection Sherwood number - Coefficient of thermal expansion (K^–1) - C _{(s, b)} Water vapour concentration average at the evaporating surface (s) and in the bulk air (b) (g m^–3) - D Coefficient of molecular diffusion of water vapour in air (m² s^–1) - d Characteristic dimension of the paper disc in the direction of flow (m) - E _{(c, m)} Evaporation rates of the Piche evaporimeter, calculated (c) and measured (m) (units in text) - e _{(s, b)} Partial water vapour pressure average at the evaporating surface (s) and in the bulk air (b) (mbar) - Gr Grashof number - g Acceleration of gravity (m s^–2) - m Number of measuring periods - n Numerical constant in free convection Sherwood number - Coefficient of kinematic viscosity of air (m² s^–1) - P Atmospheric pressure (mbar) - Re Reynolds number - _{(s, b)} Air density average at the evaporating surface (s) and of the bulk air (b) (g m^–3) - Sh Sherwood number - T _{(s, b)} Temperature average of the evaporating surface (s) and in the bulk air (b) (K) - T _{(vs, vb)} Virtual temperature average at the evaporating surface (s) and in the bulk air (b) (K) - U Wind speed (air movement) average of the bulk air (m s^–1)     

Limiting analytical solutions for complex saturated and unsaturated transport problems

Non-linear diffusion and velocity-dependent dispersion problems are under consideration. The necessary and sufficient conditions allowing the comparison of solutions to the two dimensional convection-dispersion equations with different coefficients are obtained. These conditions provide a framework within which solutions to the complex non-linear problems mentioned above can be estimated by solutions to the problems possessing analytical solvability.Nomenclature c(x, y, t) concentration of solute in solution,ML ^–3 - C(h)=d/dh moisture capacity function - D,D _ij hydrodynamic dispersion coefficient, a second order tensor,L ² T ^–1 - D _L longitudinal hydrodynamic dispersion coefficient,L ² T ^–1 - D _m molecular diffusion coefficient,L ² T ^–1 - D _T transverse hydrodynamic coefficient,L ² T ^–1 - G flow domain for the unsaturated flow problem - G _z , G _w flow domain and complex potential domain, respectively, for the hydrodynamic dispersion problem - h piezometric head,L - I _n given mass flux normal to the boundary,MLT ^–1 - k hydraulic conductivity,LT ^–1 - K(h) unsaturated hydraulic conductivity,LT ^–1 - L continuously differentiable function with respect to all arguments - m porosity - n(x,t) outer normal vector to the boundary - t time,T - V(x, y, t) seepage velocity vector withV=V,LT ^–1 - x Cartesian coordinate system - x horizontal coordinate,L - y vertical coordinate (elevation),L - (x),(x,t) given functions in initial and boundary conditions (3), (4) - ₁(,) angle between vectors ₁c andV - boundary of the flow domain - _L , _T longitudinal and transverse dispersivities, respectively,L - water mass density,ML ^–3 - v _i components of a unit vector in the direction of the outward normal to the boundary - =–kh velocity potential - =/m - stream function defined such thatw=+i is the complex potential - =/m     

Third-order systems: Periodicity conditions

Recently a third-order existence theorem has been proven to establish the sufficient conditions of periodicity for the most general third-order ordinary differential equation

x+f(t,x,x^′,x^″)=0