Complex Potential Functions Research Articles

Creep behaviour of undisturbed London Clay in triaxial stress space

The evolution of the creep strain component in triaxial stress space was investigated through performing a series of multistage drained compression tests on London Clay using a specially designed locally instrumented triaxial apparatus. Experiments along specifically defined stress paths showed significant rotation of the local creep strain component as the samples were sheared towards failure. The results indicate a need for a more complex plastic potential function to correctly predict incremental creep strains at different states in triaxial stress space. Creep deformations for stress path controlled drained compression tests were also found to require a reinterpretation of the classic secondary compression behaviour. Creep strain-rates were found to fall well outside the normal power decay function. Test data and previously reported drained creep test results on London Clay have been combined to provide a complete understanding of the incremental creep component. The experiments show how creep behaviour significantly depends on the stress conditions imposed and the approaching strain rate.

A new analytical approach for a shallow circular hydraulic tunnel

An analytical solution is presented for a shallow circular hydraulic tunnel excavated in an elastic rock mass with gravity by a new approach. Through the stress boundary condition on the ground surface, the relationship between the two analytic functions can be directly given. Thus, only one analytic function should be considered in the solving process. Based on the stress boundary condition on the excavated boundary, the analytic function is solved by power series method. The solving process is much simpler than ever. The excavation process and unbalanced force system acting on the tunnel boundary are considered by a complex potential function with logarithmic term. Through the conformal mapping method in complex variable, the basic equations for solving the stresses and displacements are obtained according to the stress boundary condition on the tunnel boundary. The examples are presented under the different coefficients of horizontal stress, tunnel depths and water head heights.

An analytic solution for the problem of two symmetrical edge cracks emanating from a circular hole with surface effect under antiplane shear

A theoretical study is performed on the fracture characteristics for the problem of two symmetrical edge cracks emanating from a circular hole with surface effect under far-field antiplane shear loading. Based on the theory of Gurtin–Murdoch surface model, a rigorous analytical solution of the stress intensity factor and the strain energy release rate at a crack tip is presented by using the complex potential function and a conformal mapping technique. When the cracked hole becomes nano-sized, the stress intensity factor and the strain energy release rate show significant size dependence. The interaction effects between cracks and hole on the stress intensity factor and the strain energy release rate are discussed in providing numerical examples.

The effect of mismatch strain induced by the a1–a2 domain pattern on dislocation generation of epitaxial ferroelectric films

This study theoretically models both mismatch strain and lattice dislocation generation induced by the a1–a2 domain pattern in epitaxial ferroelectric films. For the ferroelectric film/substrate structure, the precise complex potential functions for the Somigliana screw dislocation quadrupole that represents the lattice mismatch strain are presented, the free boundary condition of the film is approximately processed by an equivalent method, and the effect of mismatch strain on lattice dislocation generation is discussed. The results show that: the higher mismatch degree and domain wall energy evidently lead to interfacial dislocation multiplication; the density of the dislocations decreases with decreasing film thickness; the generated dislocations are mainly concentrated at the sites far away from domain walls.

Stress field of a functionally graded coated inclusion of arbitrary shape

Based on the theory of complex variable functions, the stress field in an infinite matrix containing an arbitrary shape inclusion with a functionally graded coating is analyzed. The elastic properties in the functionally graded coating change continuously and arbitrarily along the normal direction of the inclusion. By using the method of piecewise homogeneous layers and the technique of conformal mapping, the complex potential functions in the matrix, coating and inclusion are derived in the form of Laurent series and Faber series, respectively. The influences of different varying Young’s modulus on the interfacial stresses are discussed by numerical examples for various shape inclusions, including ellipse, triangle, square and rectangle. It is shown that the magnitude and distribution of interfacial stresses for arbitrary shape inclusions can be successfully designed and controlled by adding a functionally graded coating with proper varying elastic properties along the normal direction. The results for some special cases are compared with previous literature and found in good agreement.

Complex variable approach in studying modified polarization saturation model in two-dimensional semipermeable piezoelectric media

A modified polarization saturation model is proposed and addressed mathematically using a complex variable approach in two-dimensional (2D) semipermeable piezoelectric media. In this model, an existing polarization saturation (PS) model in 2D piezoelectric media is modified by considering a linearly varying saturated normal electric displacement load in place of a constant normal electric displacement load, applied on a saturated electric zone. A centre cracked infinite 2D piezoelectric domain subject to an arbitrary poling direction and in-plane electromechanical loadings is considered for the analytical and numerical studies. Here, the problem is mathematically modeled as a non-homogeneous Riemann-Hilbert problem in terms of unknown complex potential functions representing electric displacement and stress components. Having solved the Hilbert problem, the solutions to the saturated zone length, the crack opening displacement (COD), the crack opening potential (COP), and the local stress intensity factors (SIFs) are obtained in explicit forms. A numerical study is also presented for the proposed modified model, showing the effects of the saturation condition on the applied electrical loading, the saturation zone length, and the COP. The results of fracture parameters obtained from the proposed model are compared with the existing PS model subject to electrical loading, crack face conditions, and polarization angles.

Complex Potential Methods for a Crack and Three-phase Circular Composite in Anti-plane Elasticity

An interaction between an anti-plane crack with a three-phase circular composite by using complex potential methods is considered in this paper. The solution procedures for solving this problem consist of two parts. In the first part, based on complex potential methods in conjunction with analytical continuation theorem and alternating technique, the complex potential functions of a screw dislocation interacting with three-phase circular composites are obtained. The second part consists of the derivation of logarithmic singular integral equations by introducing the complex potential functions of screw dislocation along the crack border together with superposition technique. The logarithmic singular integral equations are then solved numerically by modeling a crack in place of several segments. Linear interpolation formulae with undetermined coefficients are applied to approximate the dislocation distribution along the elements, except at vicinity of the crack tip where the dislocation distribution preserves a square-root singularity. The mode-III stress intensity factors are then obtained numerically in terms of the values of the dislocation density functions of the logarithmic singular integral equations.

Полевая аналитическая модель магнитоэлектрического вентильного двигателя

Phenomena occurring in a switched magnetoelectric motor can be considered in its non-magnetic gap, which should also embrace the layer of a high-energy magnet. Complex periodic potential functions serve as a mathematical basis for analytically solving the Dirichlet problem in a non-magnetic gap having the shape of an infinite horizontal strip with boundaries confined by two parallel straight lines. The imaginary components of the complex potential functions at the boundaries of the above-mentioned strip represented by trigonometric Fourier series are the known scalar magnetic potentials of the stator winding and permanent magnets on the rotor, which are the magnetic field sources.A comparatively large width of the considered strip results in that the magnetic field in it is two-dimensional in nature due to the presence of permanent magnets. The boundary sources of magnetic field can be used for generating additional magnetic fields corresponding to the stator’s virtual teeth. The geometrical structure of the virtual teeth, which determines the amplitudes of tooth magnetic inductions, can be accurately taken into account using conformal transformation of a non-uniform air gap into an infinite strip or approximately based on the specific permeance method.

Аналитическая модель магнитного поля в ярме магнитоэлектрического вентильного двигателя

The processes occurring in a switched permanent magnet motor can be considered in the nonmagnetic gap. The high-energy magnet layer should also be related to this gap. Complex periodic potential functions serve as the mathematical basis for analytically solving the Dirichlet problem both in the nonmagnetic gap represented by an infinite horizontal strip with the boundaries delineated by two parallel straight lines and in similar strips with homogeneous magnetic permeability of the stator and rotor yokes. The imaginary components of the complex potential functions at the boundaries of the above-mentioned strips represented by trigonometric Fourier series are the known scalar magnetic potentials of magnetic field sources, namely, the stator winding and permanent magnets on the rotor. To fulfill the magnetic field boundary conditions in the yoke belts, a procedure for reducing these fields to the magnetic field in the air gap is implemented by means of special coefficients for each of the harmonic components in the magnetic induction radial and tangential components. The value of magnetic induction uniformly distributed in the yoke body is selected from the maximal magnetic induction inside the yoke without taking into account the induction impulse values at the boundaries with the air gap. Graphs showing the distribution of magnetic fields in the yokes corresponding to the obtained analytical correlations are given.

Anti-plane problem of four edge cracks emanating from a square hole in piezoelectric solids

By constructing a new numerical conformal mapping and using the Stroh-type formulism, an anti-plane problem of four edge cracks emanating from a square hole in piezoelectric solids is investigated. The explicit expressions of the complex potential function, field intensity factors, energy release rates and mechanical strain energy release rate near the crack tip are obtained under the assumptions that the surfaces of the cracks and hole are electrically permeable and electrically impermeable. Numerical examples are presented to show the influences of the geometrical parameters of defects and applied mechanical/electrical loads on the energy release rate and mechanical strain energy release rate under two electrical boundary conditions.

Determination of interatomic potentials for diatomic molecules from low resolution spectra

Often, experimental spectra with only partially resolved rotational structure are observed. In such cases, determination of potential energy curves for the corresponding electronic states might be not as reliable as in the high resolution studies. In this paper, we estimate the uncertainties of the potential energy curve parameters, obtained from spectra with partially or not resolved rotational structure, where both frequencies and intensities are fitted. As an example, recently reported excitation spectrum of the B31(53P1)←X10+(51S0) electronic transition in CdAr van der Waals complex is used. Two types of Morse functions are used to model the excited state potential, but method can be readily extended to more complex potential functions. It is shown, that for reliable estimation of the uncertainties of given potential function one should take into account the possible variation of the transition dipole moment function.

The interface effect of a nano-inhomogeneity on the fracture behavior of a crack and the nearby edge dislocation

The interface effect of a nano-inhomogeneity on the elastic–plastic fracture behavior of a crack with the influence of nearby edge dislocation is studied by applying the surface/interface stress model. When the inhomogeneity and the crack become nano-sized, the bimaterial interface contributes considerable stresses to the crack surfaces, which are usually ignored in the traditional micro/macro crack problems. With the existence of nano-inhomogeneity, an extra dislocation near the crack may be either absorbed into the crack (so the crack grows bigger) or repelled away by the crack and the inhomogeneity. The Zener–Stroh mechanism is employed to model the nano-sized fracture behavior and the generalized Irwin plastic zone correction is taken to improve the fracture analysis at the crack tips. Using the complex potential functions and distributed dislocation method, the stress intensity factor, plastic zone size, and crack tip opening displacement are evaluated by solving the Cauchy singular integral equations. The numerical examples prove that the values of stress intensity factors, plastic zone size and crack tip opening displacement are sensitive to the interface properties of the current nano-scaled problem, especially when the inhomogeneity is small. When the extra edge dislocation is close to the sharp crack tip, the stress intensity factor increases significantly, which indicates the crack has a potential to absorb the nearby dislocation. Further investigations find that mixed mode loading conditions always induce much higher stress intensity factors, plastic zone size, and crack tip opening displacement comparing to pure mode I loading problem.

Anti-Plane Interaction of a Coated Circular Inclusion with a Crack Located in Matrix

In this study, the interaction between a coated circular inclusion with an anti-plane crack located in matrix is considered. The solution procedures for solving this problem consist of two parts. In the first part, based on the method of analytical continuation in conjunction with the alternating technique, the complex potential functions of screw dislocation interacting with multi-layered composites are obtained. The second part consists of the derivation of logarithmic singular integral equations by introducing the complex potential functions of screw dislocation along the crack border together with superposition technique. The logarithmic singular integral equations is the solved numerically by modeling a crack in place of several segments. Linear interpolation formulae with undetermined coefficients are applied to approximate the dislocation distribution along the elements, except at vicinity of crack tip where the dislocation distribution preserves a square-root singularity. The stress intensity factors are then obtained numerically in terms of the values of the dislocation density functions of the logarithmic singular integral equations.

Solutions of a crack interacting with a three-phase composite in plane elasticity

The interaction between a crack and a circularly cylindrical layered media under a remote uniform load for plane elasticity is investigated. Based on the method of analytical continuation associated with the alternation technique, the solutions to the crack problem for a three-phase composite are derived. A rapidly convergent series solution for the stress field, expressed in terms of an explicit general term of the corresponding homogeneous potential, is obtained in an elegant form. The solution procedures for solving this problem consist of two parts. In the first part, the complex potential functions of dislocation interacting with a three-phase composite are obtained. In the second part, the derivation of logarithmic singular integral equations by introducing the complex potential functions of dislocation along the crack border is made. The stress intensity factors (SIFs) are then obtained numerically in terms of the dislocation density functions of the logarithmic singular integral equations. The stress intensity factors (SIFs) as a function of the dimensionless crack length for various material properties and geometric parameters are shown in graphic form. The obtained results may provide some guidance for material and geometry selections by minimizing the SIF.

An infinite plate with a curvilinear hole having three poles with complex parameters

This paper covered the study of the boundary value problem for isotropic homogeneous perforated infinite elastic media. For this, we considered the problem of a thin infinite plate of specific thickness with a curvilinear hole where the origin lie outside the hole is conformally mapped outside a unit circle by means of a specific rational mapping . The complex variable method has been applied and it transforms the problem to the integrodifferential equation with Cauchy kernel that can be solved to find two complex potential functions which called Gaursat functions. Many special cases are discussed and established of these functions .Also, many applications and examples are considered. Moreover the components of stress , in each application , are computed.

The stress state around lined non-circular hydraulic tunnels below the water table using complex variable method

An elastic plane-strain solution for lined non-circular hydraulic tunnels, below the water table, is presented. Tunnels are assumed to be subjected to uniform ground load, and excavated in a semi-finite aquifer with fully saturated and drained geomaterial. Rock mass and concrete shows a homogenous, isotropic linear elastic behavior, and the non-seepage condition dominates the model. Muskhelishvili complex potential functions combined with conformal mapping method were used to investigate stress field around the tunnels. The solution was compared with ABAQUS finite element software through an example that showed a good agreement. Moreover, the proposed solution predicted the boundary values of the problem along the internal lining periphery much more accurately than did the ABAQUS solution. In the next part, an attempt was made to reduce the solution to be more practical and simply applicable for pressure tunnels when a large water pressure is exerted on the linings. The significance of the work is in the fact that it incorporates non-uniform distribution of water pressure for groundwater and water inside tunnels, and predicts stress components in a highly accurate and fast process. The method could also be used for the openings in half-plane medium under non-uniform ground load.

The Generalized Two Dimensional Thermal-Electro-Elastic Solution for the Cracked-Half-Elliptical-Hole Problem in a Half Plane

AbstractThe half elliptical hole with an edge crack in a thermopiezoelectric material is studied by using the complex variable method. First, the mapping function which maps the outside of the elliptical hole and the crack in the right half plane into the outside of a circular hole in a full plane is given by the method of conformal mapping. Then, the complex potential functions and the field intensity factors (FIF) are presented according to the boundary conditions, respectively. Some useful results can be found by numerical analysis: 1) The influence of the heat flux on FIF depends on the model of the crack; 2) The shape and the size of the hole possess a significant effect on the field distribution at the crack tip.

Dugdale model for three equal collinear straight cracks: An analytical approach

In the paper, solution of three collinear equal straight cracks has been investigated on the basis of Dugdale’s hypothesis. These cracks damage an infinite isotropic elastic perfectly plastic plate. Crack tips are very sensitive about loads applied at the infinite boundary of the plate. Each crack tip opens in mode-I type deformation on the application of repeated loads at the boundary of the plate, as a result yield zones develop at each crack tip. To stop further opening of cracks, the rims of the developed yield zones are subjected to yield stress distribution. Muskhelisvili’s complex variable method is used to derive analytical expressions for complex potential function, stress intensity factor (SIF), components of displacement and crack tip opening displacement (CTOD) at each crack tip. Some of the analytical expressions are validated with previously published work. Numerical results are obtained for load bearing capacity, yield zone length and CTOD. These results are analyzed and reported graphically for different cracks lengths.

Evaluation of the singularity exponents and characteristic angular functions for piezoelectric V-notches under in plane and out of plane conditions

Based on the assumption that the physical fields close to the V-notch vertex are expressed by the series asymptotic expansions, the evaluation of singularity exponents for piezoelectric V-notches is turned into solving the characteristic values of ordinary differential equations under given boundary conditions, which are a set of equations with variable coefficients and solved by the interpolating matrix method developed by part of the authors before. The singularity analysis for V-notches under in plane and out of plane conditions is taken into account. Numerical results show that all the singularity exponents and the characteristic angular functions can be evaluated synchronously without the need of solving the transcendental equations iteratively. The present method is not only suitable for the singularity analysis of V-notches, but also for cracks and interface ends, without encountering the problem of ill-posed equations when the complex potential function method for the singularity analysis of V-notches is degenerated to analyze the singularity of cracks.

Disturbed elastic fields in a circular 2D finite domain containing a circular inhomogeneity and a finite interfacial zone

In order to take into account the effect of a finite interfacial zone between the inhomogeneity and matrix in an inhomogeneous representative volume element (RVE), analytical solutions for a 3-phase inclusion problem are needed. In this study, the primary focus is placed on a 3-phase concentric configuration for a 2D RVE, for which the exterior matrix is bounded and different elastic properties are assigned to the interior inclusion, interfacial zone and exterior matrix. In addition to the inhomogeneity induced by material mismatch, arbitrary uniform eigenstrains are allowed to occur independently within the interfacial zone and inclusion, respectively. In this study, the analytical solution is pursued via the complex potential method with the aid of 2 auxiliary potential functions. Based on the Sokhotski–Plemelj theorem, the complex potentials in each phase are constructed to account for the eigenstrains existing in the inclusion and the interfacial zone, as well as the imposed boundary conditions on the exterior matrix. The identification of the coefficients in the complex potential functions gives the analytical expressions for the disturbance induced by the inhomogeneity and eigenstrains. When compared with FEM simulations, a firm agreement is achieved for 3-phase concentric 2D finite domains. Furthermore, if the exterior matrix is extended to infinity, the obtained analytical solution reproduces the results given by Luo and Weng for uniform eigenstrain within the inclusion and by Markenscoff and Dundurs for uniform eigenstrain in the interfacial zone.