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.     

Crack size and speed interaction characteristics at micro-, meso- and macro-scale

The motivation to examine physical events at even smaller size scale arises from the development of use-specific materials where information transfer from one micro- or macro-element to another could be pre-assigned. There is the growing belief that the cumulated macroscopic experiences could be related to those at the lower size scales. Otherwise, there serves little purpose to examine material behavior at the different scale levels. Size scale, however, is intimately associated with time, not to mention temperature. As the size and time scales are shifted, different physical events may be identified. Dislocations with the movements of atoms, shear and rotation of clusters of molecules with inhomogeneity of polycrystals; and yielding/fracture with bulk properties of continuum specimens. Piecemeal results at the different scale levels are vulnerable to the possibility that they may be incompatible. The attention should therefore be focused on a single formulation that has the characteristics of multiscaling in size and time. The fact that the task may be overwhelmingly difficult cannot be used as an excuse for ignoring the fundamental aspects of the problem.Local nonlinearity is smeared into a small zone ahead of the crack. A “restrain stress” is introduced to also account for cracking at the meso-scale.The major emphasis is placed on developing a model that could exhibit the evolution characteristics of change in cracking behavior due to size and speed. Material inhomogeneity is assumed to favor self-similar crack growth although this may not always be the case. For relatively high restrain stress, the possible nucleation of micro-, meso- and macro-crack can be distinguished near the crack tip region. This distinction quickly disappears after a small distance after which scaling is no longer possible. This character prevails for Mode I and II cracking at different speeds. Special efforts are made to confine discussions within the framework of assumed conditions. To be kept in mind are the words of Isaac Newton in the Fourth Regula Philosophandi:

“Men are often led into error by the love of simplicity which disposes us to reduce things to few principles, and to conceive a greater simplicity in nature than there really is … We may learn something of the way in which nature operates from fact and observation; but if we conclude that it operates in such a manner, only because to our understanding that operates to be the best and simplest manner, we shall always go wrong.”––Isaac Newton

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.     

Logarithmic singularity of an elastic composite wedge subjected to out-of-the-plane extensional strain

When an elastic composite wedge is not under a plane strain deformation, an out-of-the-plane extensional strain exists. The singularity analysis for the stresses at the apex of the composite wedge reduces to a system of non-homogeneous linear equations. When the composite wedge consists of two anisotropic elastic materials, it is shown that the stresses have the (ln r) term for all combinations of wedge angles with few exceptions. The same is true when the materials are isotropic except that the (ln r) term may appear in the form of r(ln r) in the displacements only. For these isotropic composite wedges therefore the stresses are bounded, though not continuous, at the apex. However, there are isotropic composite wedges for which the stress singularity is logarithmic. Conditions are given for isotropic composite wedges for which the stresses are (a) uniform, (b) non-uniform but bounded and (c) logarithmic. Unlike the r^−λ singularity, the existence of the (ln r) term does not depend on the complete boundary conditions.     

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.     

Regularity criteria for the 3D MHD equations in terms of the pressure

In this paper we consider the regularity criteria for weak solutions to the 3D MHD equations. It is proved that under the condition b being in the Serrin's regularity class, if the pressure p belongs to L^α,γ with or the gradient field of pressure p belongs to L^α,γ with on [0,T], then the solution remains smooth on [0,T].     

A moving dislocation in a magneto–electro-elastic solid

This work considers the generalized plane problem of a moving dislocation in an anisotropic elastic medium with piezoelectric, piezomagnetic and magnetoelectric effects. The closed-form expressions for the elastic, electric and magnetic fields are obtained using the extended Stroh formalism for steady-state motion. The radial components, E_rand H_r, of the electric and magnetic fields as well as the hoop components, D_θ and B_θ, of electric displacement and magnetic flux density are found to be independent of θ in a polar coordinate system. This interesting phenomenon is proven to be is a consequence of the electric and magnetic fields, electric displacement and magnetic flux density that exhibit the singularity r⁻¹ near the dislocation core. As an illustrative example, the more explicit results for a moving dislocation in a transversely isotropic magneto–electro-elastic medium are provided and the behavior of the coupled fields is analyzed in detail.     

An empirical equation of the relative viscosity of polymer melts filled with various inorganic fillers

Summary Based on Maron-Pierce's equation, an empirical equation was suggested, which relates the relative viscosity ( _r ) of the polymer melt filled with various inorganic filler, such as glass fiber, carbon fiber, talc, precipitated- and natural-calcium carbonate powder, and glassy small sphere, to the volume fraction () of the filler. The equation is _r = (1 –/A)^–2, whereA is a parameter relating to the packing geometry of the filler, which is similar to the parameter ₀ in Maron-Pierce's equation. In the equation _r is defined as the ratio of the viscosity of the filledsystem to that of the medium at the same shear stress not the shear rate. The applicability of the equation is above the shear stress about 10⁴ dyne/cm². The equation has a simple form and is considered to have a practical utility for filled-polymer melt systems.With 2 figures and 1 table     

Dynamic fracture of a kane-mindlin plate

We study dynamic crack problems for an elastic plate by using Kane-Mindlin's kinematic assumptions. The general solutions of the Laplace transformed displacements and stresses are first derived. Path independent integrals for stationary cracks subjected to transient loads and steadily growing cracks are deduced. For a stationary crack in a very thin plate subjected to impact loads, the crack tip dynamic stress intensity factor (DSIF), K₁(t), is related to the far field plane stress one, K₁⁰(t), by where ν is Poisson's ratio. For a crack steadily growing with speed V, the crack tip DSIF, K₁(V), is given by where K₁⁰(V) is the plane stress DSIF and A(V) and B(V) are known functions of V. These results are applied to compute the DSIF for a semi-infinite stationary crack in an unbounded plate subjected to impact pressure on the crack faces. The results of DSIF for a finite crack in an infinite plate under uniform impact pressure on the crack surfaces show that for each plate thickness, the maximum DSIF is higher than that for the plane stress case.     

On the propagation of elastic waves through composite media

Summary With the aid of an ultrasonic pulse technique, the propagation of elastic waves (longitudinal as well as transverse) through polyurethane rubbers filled with different amounts of sodium chloride particles was studied. The velocity of both longitudinal and transverse waves was found to increase with filler content. From the measured wave velocities, the effective modulus for longitudinal waves,L, bulk modulus,K, and shear modulus,G, were calculated according to the relations for a homogeneous isotropic material. All three moduli appear to be monotonously increasing functions of the filler content over the whole experimentally accessible temperature range (–70 °C to + 70 °C forL andK;}-70 °C to about –20 °C forG) and they, moreover, reflect the glassrubber transition of the binder.Poisson's ratio,, was found to decrease with increasing filler content and show a rise at the high temperature side of the experimentally accessible temperature range (about –20 °C) as a result of the approach of the glass-rubber transition.In addition to the velocities, the attenuation of both longitudinal and transverse waves was measured in the temperature ranges mentioned. It was found that in the hard region tan _L as well as tan _G are independent of the filler content within the accuracy of the measurements. In the rubbery region, however, tan _L, increases with increasing filler content.Finally, the experimental data are compared with a simple macroscopic theory on the elastic properties of composite media.     

Pseudo plane stress mode II crack growth in plastic materials

The near tip field of mode II crack that grows in thin bodies with power hardening or perfectly plastic behavior is analyzed. It is shown that for power hardening behavior, the pseudo plane stress field possesses the logarithm singularity, i.e. σ (ln r)²/(n−1), (ln r)^{2n/(n − 1)}, where r is the distance from the crack tip, n the hardening exponent is σⁿ. When n → ∞ the solution reduced to that for the perfectly plastic case.     

The effective permeability of a heterogeneous porous medium

The effective (single-phase) permeability of an (infinite) heterogeneous porous medium is studied using a formalism of Green's functions. We give formal expressions for it in the form of a series expansion involving the microscopic random-permeability field many-body correlation functions of higher and higher order.The particular case of a log-normal medium of infinite extent is studied using field-theoretical methods. Using partial series resummation techniques, we derivea formula up to all orders in the local correlations which was first reckoned by many authors by means of a first-order calculation. The formula — which remains an approximation — works whatever the dimensionality of the space, and gives the following simple estimate for the effective permeability in 3 D:K _eff=k ^1/33. The method is general and the approximations can be systematically improved on when more complex situations are studied.Roman Letters D number of dimensions of the space in which the flow takes place - f(r) body force field,N - f(q) Fourier-transformed body-force field, Nm³ - G ₀(r, r) Green's function of the Laplace operator, m^–1 - g(k,r, r) velocity propagator before averaging, m^–1 - G(r, r) velocity propagator after averaging, m^–1 - j(r) a scalar dimensionless field - k(r) local value of the permeability at point r, m² - K _eff effective permeability - K _g geometric average of the local permeability, m² - l typical size of the averaging volume, m - L characteristic length of the porous medium or of the reservoir, m - L(r, r) projection operator, m^–2 - M(r, r) scattering operator, m^–3 - p(r) local value of the pressure, Nm^–2 - p(k,r, r) pressure propagator before averaging, m^–1 - P(r, r) pressure propagator after averaging, m^–1 - r position vector, m - r modulus of vectorr, m - unit vector pointing in the direction ofr - q Fourier wave vector, m^–1 - q modulus of the Fourier wave-vectorq, m^–1 - unit vector pointing in the direction ofq - projector over vector - 1 unit tensor - X(r) a local random variable - ¯X(r) volume averaged local random variable - X (r) ensemble averaged local random variable - V large-scale averaging volume, m³ - Z(j) generating functional of a random field - Z(r,j) modified generating functional of a random field - Z normalization factor Greek Letters ₀ average value of the logarithm of the permeability - (r) fluctuation of the logarithm of permeability at pointr - viscosity of the fluid, Nt/m² - (r–r) two-point correlation function of the fluctuations of the logarithm of the permeability - _k correlation length of the permeability correlation function, m - _u correlation length of the velocity correlation function, m     

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.     

The wedge subjected to tractions: a paradox resolved

The classical two-dimensional solution provided by Lévy for the stress distribution in an elastic wedge, loaded by a uniform pressure on one face, becomes infinite when the opening angle 2 of the wedge satisfies the equation tan 2 = 2. Such pathological behavior prompted the investigation in this paper of the stresses and displacements that are induced by tractions of O(r ^–) as r0. The key point is to choose an Airy stress function which generates stresses capable of accommodating unrestricted loading. Fortunately conditions can be derived which pre-determine the form of the necessary Airy stress function. The results show that inhomogeneous boundary conditions can induce stresses of O(r ^–), O(r ^– ln r), or O(r ^– ln² r) as r0, depending on which conditions are satisfied. The stress function used by Williams is sufficient only if the induced stress and displacement behavior is of the power type. The wedge loaded by uniform antisymmetric shear tractions is shown in this paper to exhibit stresses of O(ln r) as r0 for the half-plane or crack geometry. At the critical opening angle 2, uniform antisymmetric normal and symmetric shear tractions also induce the above type of stress singularity. No anticipating such stresses, Lévy used an insufficiently general Airy stress function that led to the observed pathological behavior at 2.     

Rheological properties of skim milk gels at various temperatures; interrelation between the dynamic moduli and the relaxation modulus

The rheological properties of rennet-induced skim milk gels were determined by two methods, i.e., via stress relaxation and dynamic tests. The stress relaxation modulusG _c (t) was calculated from the dynamic moduliG andG by using a simple approximation formula and by means of a more complex procedure, via calculation of the relaxation spectrum. Either calculation method gave the same results forG _c (t). The magnitude of the relaxation modulus obtained from the stress relaxation experiments was 10% to 20% lower than that calculated from the dynamic tests.Rennet-induced skim milk gels did not show an equilibrium modulus. An increase in temperature in the range from 20° to 35 °C resulted in lower moduli at a given time scale and faster relaxation. Dynamic measurements were also performed on acid-induced skim milk gels at various temperatures andG _c (t) was calculated. The moduli of the acid-induced gels were higher than those of the rennet-induced gels and a kind of permanent network seemed to exist, also at higher temperatures. G storage shear modulus,N·m^–2; - G loss shear modulus,N·m^–2; - G _c calculated storage shear modulus,N·m^–2; - G _c calculated loss shear modulus,N·m^–2; - G _e equilibrium shear modulus,N·m^–2; - G _ec calculated equilibrium shear modulus,N·m^–2; - G(t) relaxation shear modulus,N·m^–2; - G _c (t) calculated relaxation shear modulus,N·m^–2; - G ^*(t) pseudo relaxation shear modulus,N·m^–2; - H relaxation spectrum,N·m^–2; - t time,s; - relaxation time,s; - angular frequency, rad·s^–1. Partly presented at the Conference on Rheology of Food, Pharmaceutical and Biological Materials, Warwick, UK, September 13–15, 1989 [33].     

A cylindrical coaxial MHD generator

A MHD generator with a novel geometry is analyzed as a possible dc power source. The generator channel consists of two coaxial cylinders with a smooth annular space between them through which pressure driven ionized gas flows axially. Magnetic poles and electrodes separated by insulators are embedded in both the inner and outer cylinders. A one-dimensional steady state analysis is presented. It is shown that the internal impedance of the generator is a very sensitive function of the ratio of areas of the charge collecting electrodes to that of the magnetic poles. The generator efficiency analysis, on the other hand, indicates that there is an optimum area ratio corresponding to the maximum conversion efficiency. A comparison of the performance characteristics of this generator with those of a generator of rectangular cross section is presented. The average gas temperature and velocity, the magnetic flux density at the poles, and the volume displacement rate, etc., are assumed identical for the two cases in comparison. It is inferred that the novel channel analyzed herein is, in general, superior to the simple rectangular channel in the energy conversion scheme.Nomenclature a _n - 2a width of the rectangular channel - a _1n , a _2n , b _1n , b _2n constants - B magnetic flux density, both induced and applied - B _r0 maximum value of radial component of B at r=r _i - B ₀ applied magnetic field in the rectangular generator = B _r0 - 2b height of the rectangular channel - C _n r _i r _o ⁿ +r _o r _i ⁿ - C _–n r _i r _o ⁿ +r _o r _i ^–n - c integration constant - D _n - E electric field strength - maximum value of azimuthal component of E at r=r _i - G _n C _–n r _n +C _n r ^– _n - G _–n C _–n r _n –C _n r ^– _n - H _n G _n r ^–1 - H _–n G _–n r ^–1 - I _r total radial current between a pair of opposite electrodes - j electric current density - p pressure of the ionized gas - P number of magnetic poles in each cylinder of the generator - P _HT power loss due to heat transfer to the walls - P _i power input - P _o power output - R _ic internal impedance of the coaxial channel MHD generator consisting of an opposite pair of electrodes associated with the magnetic poles, insulators, and the channel in between, for a unit length of the channel - R _ir internal impedance of the rectangular generator for a unit length of the channel = a/b - R ₀ external load connected to the MHD generator - r radial coordinate of the cylindrical coordinate system - r _i, r _o radii of the inner and outer cylinders, respectively - V fluid velocity - z axial coordinate of the cylindrical coordinate system - _n nP/2 - azimuthal coordinate of the cylindrical coordinate system - _e electrode angular width - _pi pole-insulator angular width - electrical conductivity of the ionized gas - permeability of the medium - _v coefficient of viscosity - (r, ) electric potential - (r _i, )–(r _o, ) potential difference between an opposite pair of electrodes - conversion efficiency of a MHD generator A paper based on some of this material was presented at the International Electron Devices Meeting, Washington (D.C.) October 1967.     

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.     

Stress singularities at crack corners

In this paper the stress and displacement fields near an embedded crack corner in a linear elastic medium are analytically computed. The conical-spherical coordinate system is introduced to solve this problem. It is observed that the strength of the stress singularity depends on the angle of the crack corner. The singularity becomes weaker, varying from r ^-1 to r ⁰, as the angle of the crack corner varies from 360° to 0°. Both symmetric and skew-symmetric loadings give the same variation of the behavior of the stress singularity. It is also found that the order of the singularity is independent of the Poisson's ratio, unlike the corner cracks at a free surface where Poisson's ratio affects the results.