A penny-shaped crack in transversely isotropic magneto-electro-thermo-elastic medium subjected to uniform symmetric heat flux

The problem of a penny-shaped crack subjected to symmetric uniform heat flux in an infinite transversely isotropic magneto-electro-thermo-elastic medium is investigated. The exact solution in the full space is in terms of line integrals and the exact solution in the crack plane also is obtained. Although we start our derivations with magneto-electro-thermo-elastic, the solution presented in this paper is also applicable for linear transversely isotropic thermopiezoelectric, thermomagnetoelastic,thermoelastic materials (see Appendix E). The solution in the crack plane, which shows a great agreement with the solution for a transversely isotropic medium obtained by Tsai (1983), indicates that _σx,_σy,_Dx,_Dy,_Bx${\sigma }_x\text{,}{\sigma }_y\text{,}{D}_x\text{,}{D}_y\text{,}{B}_x$, and _By$B_y$ along the crack rim are of the same singularity of the normal stress or its equivalent quantities. To illustrate how the applied symmetric heat fluxes affect the whole fields, a numerical example is also given.     

Static analysis of anisotropic functionally graded magneto-electro-elastic beams subjected to arbitrary loading

This paper considers the analytical and semi-analytical solutions for anisotropic functionally graded magneto-electro-elastic beams subjected to an arbitrary load, which can be expanded in terms of sinusoidal series. For the generalized plane stress problem, the stress function, electric displacement function and magnetic induction function are assumed to consist of two parts, respectively. One is a product of a trigonometric function of the longitudinal coordinate (x) and an undetermined function of the thickness coordinate (z), and the other a linear polynomial of x with unknown coefficients depending on z. The governing equations satisfied by these z-dependent functions are derived. The analytical expressions of stresses, electric displacements, magnetic induction, axial force, bending moment, shear force, average electric displacement, average magnetic induction, displacements, electric potential and magnetic potential are then deduced, with integral constants determinable from the boundary conditions. The analytical solution is derived for beam with material coefficients varying exponentially along the thickness, while the semi-analytical solution is sought by making use of the sub-layer approximation for beam with an arbitrary variation of material parameters along the thickness. The present analysis is applicable to beams with various boundary conditions at the two ends. Two numerical examples are presented for validation of the theory and illustration of the effects of certain parameters.     

An interface crack with contact zones in a piezoelectric/piezomagnetic bimaterial

An interface crack with a frictionless contact zone at the right crack tip between two semi-infinite piezoelectric/piezomagnetic spaces under the action of a remote mechanical loading, magnetic and electric fluxes as well as concentrated forces at the crack faces is considered. Assuming that all fields are independent on the coordinate x ₂ co-directed with the crack front, the stresses, the electrical and the magnetic fluxes as well as the derivatives of the jumps of the displacements, the electrical and magnetic potentials are presented via a set of analytic functions in the (x ₁, x ₃)-plane with a cut along the crack region. Two cases of magneto-electric conditions at the crack faces are considered. The first case assumes that the crack is electrically and magnetically permeable, and in the second case the crack is assumed electrically permeable while the open part of the crack is magnetically impermeable. For both these cases due to the above-mentioned representation the combined Dirichlet–Riemann boundary value problems have been formulated and solved exactly. Stress, electric and magnetic induction intensity factors are found in a simple analytical form. Transcendental equations and a closed form analytical formula for the determination of the real contact zone length have been derived for both cases of magnetic conditions in the crack region. For a numerical illustration of the obtained results a bimaterial BaTiO₃–CoFe₂O₄ with different volume fractions of BaTiO₃ has been used, and the influence of the mechanical loading and the intensity of the magnetic flux upon the contact zone length and the associated intensity factors as well as the energy release rate has been shown.     

On the N-wave equations and soliton interactions in two and three dimensions

Several important examples of the N-wave equations are studied. These integrable equations can be linearized by formulation of the inverse scattering as a local Riemann-Hilbert problem (RHP). Several nontrivial reductions are presented. Such reductions can be applied to the generic N-wave equations but mainly the 3- and 4-wave interactions are presented as examples. Their one and two-soliton solutions are derived and their soliton interactions are analyzed. It is shown that additional reductions may lead to new types of soliton solutions. In particular the 4-wave equations with ?₂ × ?₂ reduction group allow breather-like solitons. Finally it is demonstrated that RHP with sewing function depending on three variables t, x and y provides some special solutions of the N-wave equations in three dimensions.     

Theoretical and experimental studies in fracture mechanics of crack-like defects under biaxial loading

It is a common point of view in fracture mechanics that, for any geometry of the body with a crack and any boundary conditions for the loading acting in the body plane, the stress and displacement components near the crack tip can be approximated in the framework of the theory of elasticity by a one-parameter or one-term representation, i.e., strictly in terms of the stress intensity coefficients K _I and K _II for an arbitrary failure crack [1, 2]. The authors of [2] specified the Westergaard function of the singular solution for a central crack under the biaxial loading of a plate. This approximate two-component solution has satisfactory accuracy. It is clear from [2] that this method cannot be admitted as a general statement [1], although it has long been assumed to be correct. The cause is that one cannot reasonably justify neglecting the second term in the Williams representation of the stress components in the plane case in the form of eigenfunction series; the contribution of this term in the rectangular coordinate system x, y is independent of the distance from the crack tip. This method may result in a serious mistake, from both the qualitative and quantitative viewpoints, in the prediction of local stresses, displacements, and related variables that are of interest. Apparently, this can best be demonstrated by an example of biaxial loading of a plate with a crack [1]. The unfounded neglect of the second term (whose contribution is independent of the distance from the crack tip) in the series representing the stress components is the source of the above-mentioned difficulties. In this problem, the influence of the load applied in the direction parallel to the crack plane manifests itself only in the second term of the series [3]. Therefore, this term should be clearly determined and studied in detail in the case of technological welding defects (faulty fusions, incomplete fusions, undercuts, and slag inclusions) and crack-like defects (scratches and cuts) in the base metal. The influence of the stress σ _OX along the crack axis on the stress tensor σ _x , σ _y , τ _xy and on the displacements u _x and u _y is confirmed by experimental studies of cracks by the photoelasticity method [4].     

Three-dimensional analysis of doubly curved functionally graded magneto-electro-elastic shells

Three-dimensional (3D) solutions for the static analysis of doubly curved functionally graded (FG) magneto-electro-elastic shells are presented by an asymptotic approach. In the present formulation, the twenty-nine basic equations are firstly reduced to ten differential equations in terms of ten primary variables of elastic, electric and magnetic fields. After performing through the mathematical manipulation of nondimensionalization, asymptotic expansion and successive integration, we finally obtain recurrent sets of two-dimensional (2D) governing equations for various order problems. These 2D governing equations are merely those derived in the classical shell theory (CST) based on the extended Love–Kirchhoffs' assumptions. Hence, the CST-type governing equations are derived as a first-order approximation to the 3D magneto-electro-elasticity. The leading-order solutions and higher-order corrections can be determined by treating the CST-type governing equations in a systematic and consistent way. The 3D solutions for the static analysis of doubly curved multilayered and FG magneto-electro-elastic shells are presented to demonstrate the performance of the present asymptotic formulation. The coupling magneto-electro-elastic effect on the structural behavior of the shells is studied.     

Fracture mechanical assessment of interface cracks with contact zones in piezoelectric bimaterials under thermoelectromechanical loadings: I. Electrically permeable interface cracks

An electrically permeable interface crack with a frictionless contact zone at the right crack tip between two semi-infinite piezoelectric spaces under the action of a remote electromechanical loading and a temperature flux is considered. Assuming that all fields are independent on the coordinate x₂ co-directed with the crack front, the stresses, the electrical and the temperature fluxes as well as the derivatives of the jumps of the displacements, the electrical potential and the temperature at the interface are presented via a set of analytic functions in the (x₁,x₃)-plane with a cut along the crack. Due to this representation firstly an auxiliary problem concerning the direction of the heat flux permitting a transition from a perfect thermal contact to a separation has been solved for a piezoelectric bimaterial. Besides, an inhomogeneous combined Dirichlet–Riemann boundary value problem has been formulated and solved exactly for the above mentioned interface crack. Stress and electrical displacements intensity factors are found in a clear analytical form which is especially easier for a small contact zone length. A simple equation and a closed form analytical formula for the determination of the real contact zone length have been derived and compared with the associated equation of the classical (oscillating) interface crack model defining the zone of crack faces interpenetration. For a numerical illustration of the obtained results a bimaterial cadmium selenium/glass has been used, and the influence of the heat flux upon the contact zone length and the associated stress intensity factor has been shown.     

Limit cycle oscillations and a perturbed harmonic oscillator

This note is concerned with the behaviour of the dynamical system x = y, y = ?x?μ sin y which is important in electrical circuit theory.     

A unified treatment of strain gradient plasticity

A theoretical framework is presented that has potential to cover a large range of strain gradient plasticity effects in isotropic materials. Both incremental plasticity and viscoplasticity models are presented. Many of the alternative models that have been presented in the literature are included as special cases. Based on the expression for plastic dissipation, it is in accordance with Gurtin (J. Mech. Phys. Solids 48 (2000) 989; Int. J. Plast. 19 (2003) 47) argued that the plastic flow direction is governed by a microstress q_ij and not the deviatoric Cauchy stress σ_ij′ that has been assumed by many others. The structure of the governing equations is of second order in the displacements and the plastic strains which makes it comparatively easy to implement in a finite element programme. In addition, a framework for the formulation of consistent boundary conditions is presented. It is shown that there is a close connection between surface energy of an interface and boundary conditions in terms of plastic strains and moment stresses. This should make it possible to study boundary layer effects at the interface between grains or phases. Consistent boundary conditions for an expanding elastic-plastic boundary are as well formulated. As examples, biaxial tension of a thin film on a thick substrate, torsion of a thin wire and a spherical void under remote hydrostatic tension are investigated.     

BENDING OF ORTHOTROPIC SANDWICH PLATES WITH A FUNCTIONALLY GRADED CORE SUBJECTED TO DISTRIBUTED LOADINGS

Based on the Reissner assumptions, this paper is concerned with the bending analysis of simply supported sandwich plates with functionally graded core and orthotropic face sheets subjected to transverse distributed loadings. First, the expressions of the displacements, stresses and internal forces of the sandwich plate are presented according to the constitutive relations and stress states of the core and face sheets. Then, the solutions of bending equilibrium equations are derived by expanding the deflection w, transverse shearing forces Q_x and Q_y with double trigonometric series that satisfy the simply supported boundary conditions. Finally, the proposed solution is validated by comparing the results with available elasticity solutions for a square sandwich plate with an isotropic core and finite element simulations for one with functionally graded core. The Young's modulus of the functionally graded core is assumed to be graded by a power law distribution of volume fractions of the constituents, and the Poisson's ratio is held constant. And the effects of the core's top-bottom Young's modulus ratio ? and volume fraction exponent n₀ on the variation of the displacements of the functionally graded sandwich plate are also examined.     

Magneto elastic stress subjected to uniform magnetic field over a thin infinite plate containing an elliptical hole with an edge crack

Two dimensional solutions of the magnetic field and magneto elastic stress are presented for a magnetic material of a thin infinite plate containing an elliptical hole with an edge crack subjected to uniform magnetic field. Using a rational mapping function, each solution is obtained as a closed form. The linear constitutive equation is used for these analyses. According to the electro-magneto theory, only Maxwell stress is caused as a body force in a plate. In the present paper, it raises a plane stress state for a thin plate, the deformation of the plate thickness and the shear deflection. Therefore the magneto elastic stress is analyzed using Maxwell stress. No further assumption of the plane stress state that the plate is thin is made for the stress analysis, though Maxwell stress components are expressed by nonlinear terms. The rigorous boundary condition expressed by Maxwell stress components is completely satisfied without any linear assumptions on the boundary. First, magnetic field and stress analyses for soft ferromagnetic material are carried out and then those analyses for paramagnetic and diamagnetic materials are carried out. It is stated that those plane stress components are expressed by the same expressions for those materials and the difference is only the magnitude of the permeability, though the magnetic fields H_x, H_y are different each other in the plates. If the analysis of magnetic field of paramagnetic material is easier than that of soft ferromagnetic material, the stress analysis may be carried out using the magnetic field for paramagnetic material to analyze the stress field, and the results may be applied for a soft ferromagnetic material. It is stated that the stress state for the magnetic field H_x, H_y is the same as the pure shear stress state. Solutions of the magneto elastic stress are nonlinear for the direction of uniform magnetic field. Stresses in the direction of the plate thickness and shear deflection are caused and the solutions are also obtained. Figures of the magnetic field and stress distribution are shown. Stress intensity factors are also derived and investigated for the crack length.     

Parametric study of the collision modes of compound droplets in simple shear flow

Compound droplet collision has been found in various industrial and academic applications. Such colliding phenomena of two droplets in simple shear flow are numerically resolved by a front-tracking technique. Initially, each compound droplet, assumed cylindrical, contains one concentric inner droplet. They are separated at the lateral and vertical intervals denoted by Δx₀ and Δy₀. Because of the shear flow, the compound droplets interact with each other and exhibit three collision modes: passing-over, merging (i.e. coalescence) and reversing. These modes and their transition are affected by many parameters including the Reynolds Re and Capillary Ca numbers (based on the properties of the outer fluid), the viscosity ratios μ₁₃ and μ₂₃, the interfacial tension ratio σ₁₂ of the inner to outer interfaces, the ratio of the radii of the inner to outer droplets R₁₂ and the initial distance between them (in terms of Δx₀ and Δy₀). It is found that from a merging mode, decreasing Re from 2.51 to a value less than or equal to 1.0 induces a transition to a reversing mode, whereas, increasing Ca from 0.005 to a value greater than or equal to 0.04 leads to a transition to a passing-over mode. A transition from the merging to passing-over modes also appears when varying μ₂₃ in the range of 0.1–10.0 or varying R₁₂ in the range of 0.2–0.8. A transition from a passing-over mode to a reversing one is available when increasing Δx₀ or decreasing Δy₀. Three modes of collision all occur when μ₁₃ is varied in the range of 0.1–10.0. However, the variation of σ₁₂ does not induce any transition between different modes. Several phase diagrams in terms of Re versus Ca, or Δx₀ versus Δy₀ are also proposed to show the transitions between these modes.     

Stress and finite-strain analysis of a circular ring under diametral compression

Using a moiré, large-strain analysis method, a complete solution is shown in this paper of the fields of strain and stress for a circular ring subjected to diametral compression between two flat platens. The isotheticsu andv, obtained using 1000-lines-per-inch gratings, were differentiated photographically by the shifting technique (moiré-of-moiré) to determine ?u/?x, ?v/?y, ?u/?y and ?v/?x. Using the exact finite strain-displacement relationship, the Eulerian strains ? _x ^E , ? _y ^E and γ _xy ^E were computed. From these, the principal Eulerian strains were obtained. These results were verified with the isochromatics obtained from a large-deformation photoelasticity analysis. The ring was made of a polyurethane rubber which exhibits a linear relationship between natural strain and a newly introduced concept of “natural stress”. The Eulerian strains were converted to natural strains, and from these natural stresses were computed using the newly developed concept. Results are presented graphically for the whole field of the ring.     

Magnetic field and magneto elastic stress in an infinite plate containing an elliptical hole with an edge crack under uniform electric current

Two-dimensional solutions of the electric current, magnetic field and magneto elastic stress are presented for a magnetic material of a thin infinite plate containing an elliptical hole with an edge crack under uniform electric current. Using a rational mapping function, the each solution is obtained as a closed form. The linear constitutive equation is used for the magnetic field and the stress analyses. According to the electro-magneto theory, only Maxwell stress is caused as a body force in a plate which raises a plane stress state for a thin plate and the deformation of the plate thickness. Therefore the magneto elastic stress is analyzed using Maxwell stress. No further assumption of the plane stress state that the plate is thin is made for the stress analysis, though Maxwell stress components are expressed by nonlinear terms. The rigorous boundary condition expressed by Maxwell stress components is completely satisfied without any linear assumptions on the boundary. First, electric current, magnetic field and stress analyses for soft ferromagnetic material are carried out and then those analyses for paramagnetic and diamagnetic materials are carried out. It is stated that the stress components are expressed by the same expressions for those materials and the difference is only the magnitude of the permeability, though the magnetic fields H_x, H_y are different each other in the plates. If the analysis of magnetic field of paramagnetic material is easier than that of soft ferromagnetic material, the stress analysis may be carried out using the magnetic field for paramagnetic material to analyze the stress field, and the results may be applied for a soft ferromagnetic material. It is stated that the stress state for the magnetic field H_x, H_y is the same as the pure shear stress state. Solving the present magneto elastic stress problem, dislocation and rotation terms appear, which makes the present problem complicate. Solutions of the magneto elastic stress are nonlinear for the direction of electric current. Stresses in the direction of the plate thickness are caused and the solution is also obtained. Figures of the magnetic field and stress distribution are shown. Stress intensity factors are also derived and investigated for the crack length and the electric current direction.     

Correlation of turbulent flow rate-pressure drop data for non-newtonian solutions and slurries in pipes

Isothermal and non-isothermal flow rate-pressure drop data in turbulent flow through smooth pipes have been obtained for non-Newtonian fluids, including aqueous solutions of polymers and aqueous suspensions of titanium dioxide. It has been found that the friction factor, f, is a function of a new form of Reynolds number, Re_B, based on the parameters A, x and w of Bowen's correlation, viz.

$\text{τ}{}_{\mathrm{w}}\text{D}{}^{\mathrm{x}}\text{=A}\text{u}{}^{\mathrm{w}}$

where τ_w is the wall shear strees, ?u the mean velocity, D the pipe diameter; A, x and w are experimentally derived parameters which characterise the fluid.     

On the similarity solutions of wave propagation for a general class of non-linear dissipative materials

Deductive similarity analysis is employed to study one-dimensional wave propagation in rate dependent materials whose constitutive laws are special cases of Maxwellian materials (σ_t = φ(ε, σ)ε_t + ψ(ε, σ), ε = strain, σ = stress). The general problem is shown not to have a similar solution although many special cases have the independent similar variable (x ? c)/(t ? d)^e. These cases are studied and tabulated. Analytic similar solutions are presented for several cases and a discussion of permissable boundary conditions is given.     

Scattering of sound by two parallel semi-infinite screens

A plane sound wave is incident upon two semi-infinite rigid plates, lying along y = 0, x > 0 and y = -h, x < 0, respectively, where (x, y) are two-dimensional Cartesian coordinates. The problem is formulated into a matrix Wiener-Hopf equation which is uncoupled by the introduction of an infinite sum of poles. The exact solution is then easily obtained in terms of the coefficients of the poles, where these coefficients are shown to satisfy a linear system of algebraic equations. The far-field solution is obtained and an asymptotic approximation to the total potential is determined in the limit as h, the plate spacing, becomes small compared to the wavelength of the incident wave. The algebraic system is solved numerically in this limit and the results are shown to agree with those obtained by the method of matched asymptotic expansions.     

Stress singularity due to traction discontinuity on an anisotropic elastic half-plane

In a half-plane problem with x₁ paralleling with the straight boundary and x₂ pointing into the medium, the stress components on the boundary whose acting plane is perpendicular to x₁ direction may be denoted by t₁ = [σ₁₁, σ₁₂, σ₁₃]^T. Stress components σ₁₁ and σ₁₃ are of more interests since σ₁₂ is completely determined by the boundary conditions. For isotropic materials, it is known that under uniform normal loading σ₁₁ is constant in the loaded region and vanishes in the unloaded part. Under uniform shear loading, σ₁₁ will have a logarithmic singularity at the end points of shear loading. In this paper, the behavior of the stress components σ₁₁ and σ₁₃ induced by traction-discontinuity on general anisotropic elastic surfaces is studied. By analyzing the problem of uniform tractions applied on the half-plane boundary over a finite loaded region, exact expressions of the stress components σ₁₁ and σ₁₃ are obtained which reveal that these components consist of in general a constant term and a logarithmic term in the loaded region, while only a logarithmic term exists in unloaded region. Whether the constant term or the logarithmic term will appear or not completely depends on what values of the elements of matrices Ω and Γ will take for a material under consideration. Elements for both matrices are expressed explicitly in terms of elastic stiffness. Results for monoclinic and orthotropic materials are all deduced. The isotropic material is a special case of the present results.     

Power-law creep and residual stresses in a carbopol gel

We report on the interplay between creep and residual stresses in a carbopol microgel. When a constant shear stress σ is applied below the yield stress σ _y, the strain is shown to increase as a power law of time, γ(t) = γ ₀ + (t/τ) ^α , with an exponent α = 0.39 ± 0.04 that is strongly reminiscent of Andrade creep in hard solids. For applied shear stresses lower than some typical value σ _c ? 0.2σ _y, the microgel experiences a more complex, anomalous creep behavior, characterized by an initial decrease of the strain, that we attribute to the existence of residual stresses of the order of σ _c that persist after a rest time under a zero shear rate following preshear. The influence of gel concentration on creep and residual stresses are investigated as well as possible aging effects. We discuss our results in light of previous works on colloidal glasses and other soft glassy systems.     

Transient response of a cracked magnetoelectroelastic strip under anti-plane impact

The transient anti-plane problem of a magnetoelectroelastic strip containing a crack vertical to the boundary is considered. Singular integral equations for the impermeable crack are obtained by using Fourier and Laplace transforms. Numerical results show the effects of the relative loading parameters κ_D and κ_B, and the crack configuration on the dynamic fracture behavior. The results obtained indicate that for the impermeable crack, the electric and magnetic impacts have significant influences on the dynamic stress intensity factor and the dynamic energy density factor.