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Stroh formalism in evaluation of 3D Green’s function in thermomagnetoelectroelastic anisotropic medium
Abstract
The paper presents studies on the Green’s function for thermomagnetoelectroelastic medium and its reduction to the contour integral. Based on the previous studies the thermomagnetoelectroelastic Green’s function is presented as a surface integral over a half-sphere. The latter is then reduced to the double integral, which inner integral is evaluated explicitly using the complex variable calculus and the Stroh formalism. Thus, the Green’s function is reduced to the contour integral. Since the latter is evaluated over the period of the integrand, the paper proposes to use trapezoid rule for its numerical evaluation with exponential convergence. Several numerical examples are presented, which shows efficiency of the proposed approach for evaluation of Green’s function in thermomagnetoelectroelastic anisotropic solids.
... where ( ) = √ 1 − 2 . The inner integral can be evaluated analytically, [22] and the outer integral is supposed to be evaluated using the trapezoid rule, since the outer integral is evaluated over the period of the integrand. However, the analytic integration of the inner integral in Equation (22) is related to the eigenvalue problem and determination of a matrix adjoint, which can also require a lot of computations. ...
... Evaluation of inner integral in Equation (22) is reduced to: [22] 1 eigenvalue problem for 2 × 2 matrix and + 1 computations of x adjoint matrices, where is the dimension of the problem (5 for general thermomagnetoelectroelasticity, 4 for pyroelectric and 3 for thermoelastic materials). Computational complexity of these transformations can be assessed as 3 + ( 2 ) ≈ 0.7 4 + 0.7 3 + ( 2 ) for computation of matrix adjoints. ...
... Computational complexity of these transformations can be assessed as 3 + ( 2 ) ≈ 0.7 4 + 0.7 3 + ( 2 ) for computation of matrix adjoints. Thus, overall complexity of the analytic evaluation of inner integral with the approach [22] equals 0.7 4 + 6 3 + ( 2 ). According to Equations (22), (23) numerical integration of the inner integral is reduced to computation of + 1 inverses of the matrix Γ ( ) at specific points and further matrix-vector multiplications. ...
The paper presents general boundary element approach for analysis of thermomagnetoelectroelastic solids containing shell‐like electrically conducting inclusions with high magnetic permittivity. The latter are modeled with opened surfaces with certain boundary conditions on their faces. Rigid displacement and rotation, along with constant electric and magnetic potentials of inclusions are accounted for in these boundary conditions. Formulated boundary value problem is reduced to a system of singular boundary integral equations, which is solved numerically by the boundary element method. Special attention is paid to the field singularity at the front line of a shell‐like inclusion. Special shape functions are introduced, which account for this square‐root singularity and allow accurate determination of field intensity factors. Approaches for fast and accurate numerical evaluation of anisotropic thermomagnetoelectroelastic kernels are discussed, which are crucial in derivation of fast and precise boundary element approach. Numerical examples are presented.
... Therefore, today the issue of optimizing and establishing a statement of facts of a 3D computer model for modelling spherical elements is acute [26]. And also, the ability to generate specific heuristics that correspond to the deep essence of this research will make it possible to reduce the number of time-consuming and cumbersome field experiments [27,28]. ...
The article presents modelling of spherical elements based on the developed computer model. We recorded the main combinations of spherical particles during filling, which are formed in the hopper. It was found that the most likely combination that occurs when modelling spherical elements consists of three balls. It should be noted that in the cross-section of such a combination passing through the center of the balls, an equilateral triangle is formed. And in the cross-section of the structure, which consists of four spherical balls, a rhombus is formed, if you connect the centers of these spherical elements. It is worth noting that from this formed combination of spherical elements, it can be seen that the rhombus forms two smaller equilateral triangles that fix the process of pushing the spherical balls apart. In turn, the process of pushing spherical elements apart made it possible to fix the contact between spherical elements, as well as to state the stable position of each (individual) particle. This paper also presents the main fragments of encoding the source text of a 3D computer model for modelling spherical elements, which made it possible to optimize the model parameters. It was found that from the obtained data on the distribution of coordination numbers for different volume fillings of spherical elements, it follows that the largest filling was 72 %, which corresponds to the state when 112 lobules have an average coordination number of 3,92.
... In the works [27], mainly informational calculation methods and basic physical and chemical characteristics of powder materials are considered [28]. The peculiarity of these works is that the analysis of only the chemical composition of components or their aggregate state of the source material is studied [29,30]. ...
This article examines the main problems of modelling spherical (circular) particles. The main method of the initial process of filling lobules using the Cauchy and Reynolds problem is substantiated. An image of an object-oriented complex of free fall of a spherical particle and their many non-collision spheres is presented. Based on the obtained research results, the main parameters of the process of filling particles of heterogeneous materials. An example of visualization of the developed software product for filling material particles is given, taking into account a number of cross-sections of a cylindrical hopper in height. A histogram of the distribution of material particles from porosity over the volume of a cylindrical hopper is also constructed.
... Thus, current trends in the development of automation systems in mechanical engineering technology are an urgent task of our time, which require constant improvement and significantly new research results. partly related to the desolate production technology [21,22], that is, scientists are faced with the task of replacing manual labour of human labour with technical and automated research systems at all stages of development [23,24]. Scientific results [25] reveal the basic principles of constructing ABC technological processes, the principles of creating automated production systems, and other subtleties of production automation [26,27]. ...
In this scientific study, the problem of automation of machine-building production is justified. A 3D model of the lathe is presented and its design is improved. Standard layout schemes based on the upgraded spindle assembly have been developed, which make it possible to increase the speed of this type of machine. The results obtained make it possible to achieve the desired cutting speed, which has significantly increased by 2-2,5 times. The constructed dependence of the deviation on the roundness of samples by the finite element method allows predicting the main indicators: feed rate, spindle speed, cutting depth, static imbalance, initial and final pressure. Also, the obtained analytical results allow us to establish the main regularities of forming the accuracy of this lathe.
... Yue (1999) developed a layered Green's function, and Yuan et al. (2003) obtained three-dimensional Green's functions for composite laminates. Recently, Pasternak et al. (2017) have studied the Green's function for thermomagnetoelectroelastic medium and its reduction to the contour integral. So far, no one has used Stroh formalism to solve the problem of anisotropic poroelasticity. ...
The dynamic responses of an anisotropic multilayered poroelastic half-space to a point load or a fluid source are studied based on Stroh formalism and Fourier transforms. Taking the boundary conditions and the continuity of the materials into consideration, the three-dimensional Green’s functions of generalized concentrated forces (force and fluid source) applied at the free surface, interface and in the interior of a layer are derived in the Fourier transformed domain, respectively. The actual solutions in the frequency domain can further be acquired by inverting the Fourier transform. Finally, numerical examples are carried out to verify the presented theory and discuss the Green’s fields due to three cases of a concentrated force or a fluid source applied at three different locations for an anisotropic multilayered poroelastic half-space.
The paper presents a novel explicit expression of three-dimensional Green’s function of magnetoelectroelastic solids with general anisotropy. The novel explicit expression of Green’s function is an alternative of the simple explicit expression in the non-degenerate case. The most impressive property of the novel explicit expression is its applicability in degenerate cases. The explicit expression is assumed to be unified since it is valid in both degenerate and non-degenerate cases. The numerical results of Green’s function of a transversely isotropic magentoelectroelastic material by using the unified explicit expression show the high accuracy of the numerical evaluation of unified explicit expression in degenerate, nearly-degenerate and non-degenerate cases.
A new expression for the fundamental solution is introduced, presenting three relevant characteristics: (i) it is explicit in terms of the Stroh’s eigenvalues, (ii) it remains well-defined when some Stroh’s eigenvalues are repeated, and (iii) it is exact. A fast and robust numerical scheme for the evaluation of the fundamental solution and its derivatives developed from double Fourier series representations is presented. The Fourier series representation is possible due to the periodic nature of the solution. The attractiveness of this series solution is that the information of the material properties is contained only in the Fourier coefficients, while the information of the dependence of the evaluation point is contained in simple trigonometric functions. This implies that any order derivatives can be determined by spatial differentiation of the trigonometric functions. Moreover, Fourier coefficients need to be obtained only once for a given material, leading to an efficient methodology. The robustness of the scheme arises from the properties (i) and (ii) of the new expression for the fundamental solution, which is used to compute the Fourier coefficients. The proposed approach combines the clean structure of the Stroh formalism with the simplicity of Fourier expansions, addressing the old drawbacks of anisotropic fundamental solutions.
The extended displacement discontinuity boundary integral–differential equation method is developed for the analysis of an interface crack of arbitrary shape in a three-dimensional (3D), transversely isotropic magnetoelectrothermoelastic bimaterial. The extended displacement discontinuities (EDDs) include conventional displacement discontinuities, electric potential discontinuity, magnetic potential discontinuity as well as temperature discontinuity across the interface crack faces, correspondingly, while the extended stresses represent conventional mechanical stresses, electric displacement, magnetic induction, heat flux, etc. By virtue of the potential functions and Hankel transformation technique, the fundamental solutions for unit-point EDDs on the interface in a 3D transversely isotropic magnetoelectrothermoelastic bimaterial are derived, then the extended displacements and stresses are all obtained in terms of EDDs. An analysis method is proposed based on the analogy with the solution in an isotropic thermoelastic bimaterial. The singular indices and the singular behaviors of the near crack-tip fields are studied. The combined extended stress intensity factors for three new fracture modes are derived in terms of the EDDs and are compared with those in magnetoelectroelastic bimaterials.
Explicit expressions of Green’s function and its derivative for three-dimensional infinite solids are presented in this paper. The medium is allowed to exhibit a fully magneto-electro-elastic (MEE) coupling and general anisotropic behaviour. In particular, new explicit expressions for the first-order derivative of Green’s function are proposed. The derivation combines extended Stroh formalism, Radon transform and Cauchy’s residue theory. In order to cover mathematical degenerate and non-degenerate materials in the Stroh formalism context, a multiple residue scheme is performed. Expressions are explicit in terms of Stroh’s eigenvalues, this being a feature of special interest in numerical applications such as boundary element methods. As a particular case, simplifications for MEE materials with transversely isotropic symmetry are derived. Details on the implementation and numerical stability of the proposed solutions for degenerate cases are studied.
This paper aims to develop three-dimensional fundamental thermo-elastic solutions for an infinite/half-infinite space of a two-dimensional hexagonal quasi-crystal, which is subjected to a point heat source. Starting from the newly developed general solution in terms of quasi-harmonic potential functions, the corresponding fundamental solutions are derived by means of the trial-and-error technique. Six appropriate potential functions involved in the general solution are observed. The present fundamental solutions are applied to construct boundary integral equations governing the contact problems. Numerical calculations are performed to show the distributions of the thermo-elastic coupling field variables in a half-space subjected to a point thermal source.
The paper derives Somigliana type boundary integral equations for 3D thermomagnetoelectroelasticity of anisotropic solids. In the absence of distributed volume heat and body forces these equations contain only boundary integrals. Besides all of the obtained terms of integral equations are to be calculated in the real domain, which is advantageous to the known equations that can contain volume integrals or whose terms should be calculated in the mapped temperature domain. All kernels of the derived integral equations and the 3D thermomagnetoelectroelastic Green׳s function for a point heat are obtained explicitly based on the Radon transform technique. Verification of the obtained equations and fundamental solutions is provided.
It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed, and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators.
A method based on the Radon transform is presented to determine the displacement field in a general anisotropic solid due to the application of a time-harmonic point force. The Radon transform reduces the system of coupled partial differential equations for the displacement components to a system of coupled ordinary differential equations. This system is reduced to an uncoupled form by the use of properties of eigenvectors and eigenvalues. The resulting simplified system can be solved easily. A back transformation to the original coordinate system and a subsequent application of the inverse Radon transform yields the displacements as a summation of a regular elastodynamic term and a singular static term. Both terms are integrals over a unit sphere. For the regular dynamic term, the surface integration can be evaluated numerically without difficulty. For the singular static term, the surface integral has been reduced to a line integral over half a unit circle. Reductions to the cases of isotropy and transverse isotropy have been worked out in detail. Examples illustrate applications of the method.
Green’s functions for transversely isotropic thermoelastic biomaterials are established in the paper. We first express the compact general solutions of transversely isotropic thermoelastic material in terms of harmonic functions and introduce six new harmonic functions. The three-dimensional Green’s function having a concentrated heat source in steady state is completely solved using these new harmonic functions. The analytical results show some new phenomena of temperature and stress distributions at the interface. The temperature contours are normal to the interface for the isotropic material but not for the orthotropic one. The normal stress contours are parallel to the interface at the boundary in the isotropic region only and shear failure is most likely at the heat source due to the highly degenerated direction of shear stress contours.
We use the compact harmonic general solutions of transversely isotropic thermoelastic materials to construct the three-dimensional fundamental solutions for a steady point heat source in an infinite transversely isotropic thermoelastic material and a steady point heat source on the surface of a semi-infinite transversely isotropic thermoelastic material by three newly introduced harmonic functions, respectively. All components of coupled field are expressed in terms of elementary functions and are convenient to use. Numerical results for hexagonal zinc are given graphically by contours.
Fundamental solutions play an important role in the analyses of coupled fields in electro-magneto-thermo-elastic material. However, most works available on this topic address the case of uniform temperature. Based on the compact general solution of transversely isotropic electro-magneto-thermo-elastic material, which is expressed in harmonic functions, and employing the trial-and-error method, the three-dimensional fundamental solution for a steady point heat source in an infinite transversely isotropic electro-magneto-thermo-elastic material is presented by five newly induced harmonic functions. Numerical results are given graphically by contours.
The general solution of three-dimensional problems in transversely isotropic magnetoelectroelastic media is obtained through five newly introduced potential functions. The displacements, electric potential, magnetic potential, stresses, electric displacements and magnetic inductions can all be expressed concisely in terms of the five potential functions, all of which are harmonic. The derived general solution is then applied to find the fundamental solution for a generalized dislocation and also to derive Green's functions for a half-space magnetoelectroelastic solid.
The effective thermal‐expansion and pyroelectric coefficients of two‐phase coupled electroelastic composite materials are obtained by coupling various micromechanics theories with exact relations connecting their effective thermal and electroelastic moduli. Results are obtained for composites reinforced by ellipsoidal piezoelectric inhomogeneities and thus are applicable to a wide range of microstructural geometry including lamina, spherical particle, and continuous fiber reinforcement. The analytical framework is developed in a matrix formulation in which the electroelastic moduli are conveniently represented by a 9×9 matrix and the thermal‐expansion and pyroelectric coefficients by a 9×1 column vector. Numerical results are presented for some typical composite microstructures which show the interesting behavior of these materials.
Based on the governing equations of transversely isotropic magnetoelectroelastic media, four general solutions on the cases of distinct eigenvalues and multiple eigenvalues are given and expressed in five mono-harmonic displacement functions. Then, based on these general solutions, employing the trial-and-error method, the three-dimensional Green’s functions of infinite, two-phase and semi-infinite magnetoelectroelastic media under point forces, point charge and magnetic monopole are all presented in terms of elementary functions for all cases of distinct eigenvalues and multiple eigenvalues. Numerical results are also presented.
The behavior of cracked linear magnetoelectroelastic solids is analysed by means of the dual Boundary Element Method (BEM) approach. Media possessing fully coupled piezoelectric, piezomagnetic and magnetoelectric effects are considered. An explicit 2-D Green’s function in terms of the extended Stroh formalism for magnetoelectroelastic full-plane under static loading is implemented. Hypersingular integrals arising in the traction boundary integral equations are computed through a regularization technique. Evaluation of fracture parameters directly from computed nodal values is discussed. The stress intensity factors (SIF), the electric displacement intensity factor (EDIF), the magnetic induction intensity factor (MIIF) as well as the mechanical strain energy release rate (MSERR) are evaluated for different crack configurations in both finite and infinite solids subjected to in-plane combined magnetic–electric–mechanical loading conditions. The accuracy of the boundary element solution is confirmed by comparison with selected analytical solutions in the literature. The new results that can be of interest in the design and maintenance of novel magnetoelectroelastic devices are also discussed.