Plane Waves of a Fiber-Reinforcement Magneto-thermoelastic Comparison of Three Different Theories

In this article, the coupled theory, Lord–Shulman theory, and Green–Lindsay (GL) theory are used to study the influence of a magnetic field on a fiber-reinforced thermoelastic half-space. Normal mode analysis is used to solve a thermal shock problem. Numerical results for the temperature, displacement components, and stress components are given and illustrated graphically. A comparison is made between the coupled and GL theories in the absence and presence of a magnetic field and reinforcement.     

Mode I crack in a soft ferromagnetic material

ABSTRACT In the existing magnetoelastic theories, stress is proportional to the square of magnetic intensity and the linear model developed is usually used to analyse magnetoelastic problems. For a crack problem, the perturbation of the magnetic field caused by deformation is not much less than the applied field. In this paper, complex potentials for a mode I crack with a nonlinear relation for magnetic intensity are developed. The boundary conditions on crack faces are represented in terms of the continuity of the magnetic field. A solution of the crack problem is obtained by solving the Riemann‐Hilbert problem. Making use of the solution, the effects of the boundary conditions on the crack faces on the magnetoelastic coupling are discussed.     

Multiscale continuum modeling of a crack in elastic media with microstructures

Cosserat type continuum theories have been employed by many authors to study cracks in elastic solids with microstructures. Depending on which theory was used, different crack tip stress singularities have been obtained. In this paper, a microstructure continuum theory is used to model a layered elastic medium containing a crack parallel to the layers. The crack problem is solved by means of the Fourier transform. The resulting integrodifferential equations are discretized using the Chebyshev polynomial expansion method for numerical solutions. By using the present theory, the explicit internal length effects upon the crack opening displacement and stress field can be observed. It is found that the stress field near the crack tip is not singular according to the microstructure continuum solution although the level of the opening stress shows an increasing trend until it gets very close to the crack tip. The rising portion of the near tip opening stress is used to project the stress intensity factor which agrees fairly well with that obtained using the FEM to perform stress analyses of the cracked layered medium with the exact geometry. The numerical solutions also indicate that treating the layered medium as an equivalent homogeneous classical elastic solid is not adequate if cracks are present and accurate stress intensity factors in the original layered medium is desired.     

Explicit solutions of magnetoelastic fields in a soft ferromagnetic solid with curvilinear cracks

The problem of curvilinear cracks lying on a soft ferromagnetic solid subjected to a remote uniform magnetic induction is considered. With the complex variable technique, the general solutions of both the magnetic field quantities and the magnetoelastic stresses can be obtained. In order to illustrate the effect of magnetic induction, the solutions for the problem with one arc crack and two arc cracks are presented in a closed form. The stress intensity factors in the vicinity of crack tip and the crack opening condition are also derived. Considering the magnetic stress induced by an oblique magnetic field on the crack surface, one can find that the stress intensity factors of mode-I and mode-II are related to the incident angle of magnetic induction, the crack half angle and the magnetic susceptibility as displayed with figures. It is noticed that the present work is available even for a ferromagnetic material with low susceptibility. For the limiting case of the crack half angle in the one arc crack problem approaching to zero, the stress intensity factors are also provided and analytically compared with the existing ones of the straight crack problem.     

A magneto-electro-elastic material with a penny-shaped crack subjected to temperature loading

Summary The analysis of intensity factors for a penny-shaped crack under thermal, mechanical, electrical and magnetic boundary conditions becomes a very important topic in fracture mechanics. An exact solution is derived for the problem of a penny-shaped crack in a magneto-electro-thermo-elastic material in a temperature field. The problem is analyzed within the framework of the theory of linear magneto-electro-thermo-elasticity. The coupling features of transversely isotropic magneto-electro-thermo-elastic solids are governed by a system of partial differential equations with respect to the elastic displacements, the electric potential, the magnetic potential and the temperature field. The heat conduction equation and equilibrium equations for an infinite magneto-electro-thermo-elastic media are solved by means of the Hankel integral transform. The mathematical formulations for the crack conditions are derived as a set of dual integral equations, which, in turn, are reduced to Abel's integral equation. Solution of Abel's integral equation is applied to derive the elastic, electric and magnetic fields as well as field intensity factors. The intensity factors of thermal stress, electric displacement and magnetic induction are derived explicitly for approximate (impermeable or permeable) and exact (a notch of finite thickness crack) conditions. Due to its explicitness, the solution is remarkable and should be of great interest in the magneto-electro-thermo-elastic material analysis and design.     

Effect of magnetostriction on fracture of a soft ferromagnetic medium with a crack-like flaw

The magnetic‐induction field in the vicinity of an elliptical inclusion embedded in an infinite soft ferromagnetic medium is determined based on complex potential theory. By using a constitutive relation of magnetostriction for isotropic materials, the stress field in the vicinity of an elliptical flaw is obtained. Furthermore, the stress field at the tip of a slender elliptical crack is determined for the case in which only an external magnetic field perpendicular to the major axis of the ellipse is applied at infinity. The results indicate that the stress field in the neighbourhood of the tip is governed by the magnetostriction and permeability of the soft ferromagnetic material. The induction magnetostrictive modulus is a key parameter in determining which of the two mechanisms, i.e., magnetostriction and magnetic‐force‐induced deformation, is dominant in determining the stress field in the neighbourhood of the tip of a crack‐like flaw. With regard to the influence of the magnetic field on the apparent toughness of a soft ferromagnetic body with a crack‐like flaw, soft ferromagnetic materials can be roughly divided into two categories: one possesses a large induction magnetostrictive modulus and the other has a small modulus. An approximate criterion for categorizing the materials is presented. For the benefit of engineering design, the expressions of the stress‐intensity factor for these two categories of soft ferromagnetic materials are presented. The results show that the stress‐intensity factor is affected not only by the flaw geometry, but also by the permeability of the medium inside the flaw.     

Time-domain boundary element analysis of elastic wave interaction with a crack

The mathematical formulation of the problem of transient wave interaction with a crack in a homogeneous, isotropic, linearly elastic solid has been reduced to the solution of an integral equation over the insonified crack face. The integral equation relates the unknown crack-opening displacement, which depends on time and position, to the incident wave field. The integral equation has been solved numerically by a time-stepping method in conjunction with a boundary element discretization of the crack surface. For normal incidence of a longitudinal step-stress wave on a penny-shaped crack, results as functions of time have been obtained for the crack-opening displacement, the elastodynamic Mode-I stress intensity factor and the scattered far-field.     

Impact response of a cracked soft ferromagnetic medium

A solution is given for the problem of an infinite soft ferromagnetic solid containing a central crack subjected to normal impact load. The solid is permeated by a uniform magnetostatic field normal to the crack surface. Laplace and Fourier transforms are employed to reduce the transient problem to the solution of integral equations in the Laplace transformed plane. A numerical Laplace inversion technique is used to compute the values of the dynamic stress-intensity factor, and the results are compared with the corresponding elastodynamic values to reveal the influence of magnetic field on the dynamic stress-intensity factor. The dynamic stress intensity factor is found to increase with increasing values of the magnetic field.With 4 Figures     

Effects of magnetic fields on cracks in a soft ferromagnetic material

The 2D problem of a soft ferromagnetic solid with a finite crack under a uniform magnetic field has been studied based on the linear theory of Pao and Yeh. Especially, in this work, the Maxwell stresses induced by the applied magnetic field are taken into account in the boundary conditions not only along the crack surfaces, but also at infinity. Based on these boundary conditions, the related boundary-value problem is solved by using Muskhelishvili’s complex variable method to obtain the complex potentials. Thus, it is found that the obtained complex potentials are constant, which indicates that both magnetic fields and stress are uniform in the solid. This implies that if only a pure magnetic field is applied, it has no effects on a crack in a soft ferromagnetic solid. To confirm this result, the same boundary-value problem is solved by the integral transform technique, which shows the same finding as that by using the complex variable method. This outcome is consistent with available experimental data but different to previously published theoretical results.     

Non-local theory solution of a mode-I crack in a piezoelectric/piezomagnetic composite material plane

The non-local theory solution of a mode-I permeable crack in a piezoelectric/piezomagnetic composite material plane was given by using the generalized Almansi’s theorem and the Schmidt method in this paper. The problem was formulated through Fourier transform into two pairs of dual integral equations, in which the unknown variables are the displacement jumps across the crack surfaces. To solve the dual integral equations, the displacement jumps across the crack surfaces were directly expanded as a series of Jacobi polynomials. Numerical examples were provided to show the effects of the crack length and the lattice parameter on the stress field, the electric displacement field and the magnetic flux field near the crack tips. Unlike the classical elasticity solutions, it is found that no stress, electric displacement and magnetic flux singularities are present at the crack tips in piezoelectric/piezomagnetic composite materials. The non-local elastic solution yields a finite hoop stress at the crack tip, thus allowing us to use the maximum stress as a fracture criterion.     

Effect of defective holes on crack using the boundary element method

Boundary value problem of a plate with crack and defect such as the circular and/or elliptical holes is a multi-connected domain problem, this kind of problem is suitable for solving by boundary element method with its higher precision. The sub-region method is used in the paper, a center cracked plate subjected to remote tensile and shear loading is studied numerically. The effect of the circular hole on Mode-I and Mode-II stress intensity factors is studied assuming the plane strain condition, the results are more precise than that of using FEM. The effect of the elliptical hole on Mode-I stress intensity factors is studied as well and some significant results are obtained.     

Magnetic stress intensity factor for an edge crack in a soft ferromagnetic elastic half-plane under tension

Summary This paper applies the theory for magnetoelasticity to solve the plane problem of an edge crack in a soft ferromagnetic half-plane subjected to a far-field tension and a uniform magnetic field. Fourier transform techniques are used to formulate the mixed boundary value problem as a singular integral equation. The stress intensity factor is calculated and is shown graphically. Tensile tests are also performed on a cracked ferromagnetic plate with strain gage technique, and the numerical results are compared with the test data.     

Magneto-rotation-fibre-reinforced thermoelastic with gravity and energy dissipation

In this article, two theories of the generalized thermoelasticity Green-Naghdi theory (of type II and III) are applied, as well as the coupled theory to study the effect of magnetic field and rotation under influence of gravity on 2D problem of a fibre-reinforced thermoelastic. The normal mode analysis is used to obtain the expressions for the temperature, displacement components and the thermal stresses distributions. The resulting formulation is applied for two different concrete problems. The first concerns the case of a punch moving across the surface of semi-infinite thermoelastic half-space subjected to appropriate boundary conditions. The second deals with a thick plate subjected to a time-dependent heat source on each face. Numerical results are illustrated graphically for each problem considered. A comparison is made with the results predicted obtained by the two theories in the presence and absence of magnetic field, rotation and gravity field.     

The scattering of oblique flexural waves of a cracked conducting plate in a uniform magnetic field

Summary Following a classical plate bending theory for magneto-elastic interactions under quasistatic electromagnetic field, we consider the scattering of time harmonic flexural waves by a through crack in a conducting plate under a uniform magnetic field normal to the crack surface. It is assumed that the plate has the finite electric conductivity, and the electric and magnetic permeabilities of the free space. An incident wave giving rise to moments symmetric about the crack plane is applied in an arbitrary direction. Fourier transform method is used to solve the mixed boundary value problem which reduces to a pair of dual integral equations. These dual integral equations are further reduced to a Fredholm integral equation of the second kind. The dynamic moment intensity factor versus frequency for several values of incident angle is computed and the influence of the magnetic field on the normalized values is displayed graphically.     

Thermal fracture of bonded dissimilar media containing a penny-shaped crack

The problem of uniform heat flow disturbed by an insulated penny-shaped crack along the common plane between two semi-infinite elastic media with different thermo-mechanical properties is formulated in terms of two potential functions in a half-space which in turn is reduced to a plane problem solvable by Muskhelishvili's method in complex function theory. Explicit expressions for the stress-intensity factors and the local stress field are derived and used in conjunction with Griffith's energy criterion to obtain the critical temperature gradient which motivates and produces initial crack extension along the bonding surface. The stress analysis involved is also applicable to a penny-shaped crack between two dissimilar solids under shear loadings.

     

Thermo-mechanical Deformation in Magneto-micropolar Elastic Medium

The present investigation is concerned with a two-dimensional problem in electromagnetic micropolar elasticity for a half-space whose surface is subjected to distributed (concentrated or continuous) thermo-mechanical sources in the presence of a transverse magnetic field. As an application of the approach, the sources are taken as uniformly or linearly distributed. Laplace and Fourier transform techniques are used to solve the problem. The integral transforms have been inverted by using a numerical technique to obtain the components of normal strain, normal stress, tangential couple stress, and temperature distribution in the physical domain. Magnetic effects on the components of normal strain, normal stress, tangential couple stress, and temperature distribution have been depicted graphically for two different theories of generalized thermoelasticity, Lord and Shulman (L–S) theory and Green and Lindsay (G–L) theory. A particular case of interest is also deduced from the present investigation.     

An alternative complex variable formulation for an inclined crack in an orthotropic medium

Of concern in this paper is the study of the elastostatic fracture response of an orthotropic plate with an inclined crack and subjected at infinity to a biaxial uniform load. To this end an unconventional approach to the derivation of the complex variable expressions of the elastic fields is proposed. The above formulation has been used to solve the boundary value problem as superposition of Mode-I and Mode-II crack problems and it is shown that the near tip asymptotic expressions of stress and displacement fields are affected by non-singular terms originated by load biaxiality. The maximum tensile stress criterion is applied in order to investigate the effects of non-singular terms on the angle of crack extension.     

压电-压磁板条中反平面裂纹的电-磁-弹性分析

基于线性电磁弹性理论,获得了压电-压磁板条中反平面裂纹尖端附近的奇异应力、电场和磁场。假设裂纹位于和板条边界平行的中心位置,并且裂纹是电磁渗透型的。利用Fourier变换,将裂纹面的混合边值问题化为对偶积分方程,即而归结为第二类Fredholm积分方程。通过渐近分析,得到了裂纹尖端附近应力、应变、电位移、电场、磁场和磁感的封闭表达式。结果表明,对于电磁渗透裂纹,电场强度因子和磁场强度因子总为0;板条的宽度对应力强度因子有显著的影响;能量释放率总为正值。     

Stress–strain state of a piecewise homogeneous ferromagnetic body with an interfacial penny-shaped crack

This paper considers the stress–strain state of a body consisting of two bonded uniform ferromagnetic half-spaces, with an interfacial penny-shaped crack. The body is assumed to be acted on by a uniform magnetic field in a direction normal to the crack and interface. The problem is reduced to the solution of singular integral equations with respect to two unknown functions on a finite interval. These functions are defined analytically, allowing us to obtain formulae for the magnetoelastic stresses and coefficient of stress intensity. From these formulae, the influence of various physico-mechanical parameters and the magnetic field on the stress strain state of the body in the vicinity of crack is studied.     

Singular stress and electric fields of a piezoelectric ceramic strip with a finite crack under longitudinal shear

Summary Following the theory of linear piezoelectricity, we consider the problem of determining the singular stress and electric fields in an orthotropic piezoelectric ceramic strip containing a Griffith crack under longitudinal shear. The crack is situated symmetrically and oriented in a direction parallel to the edges of the strip. Fourier transforms are used to reduce the problem to the solution of a pair of dual integral equations. The solution of the dual integral equations is then expressed in terms of a Fredholm integral equation of the second kind. Numerical values on the stress intensity factor and the energy release rate for piezoelectric ceramics are obtained, and the results are graphed to display the influence of the electric field.