Equivalent Body Forces Research Articles

Two-dimensional analysis of electrical behavior in composite semiconductor shell structures with magnetic field tuning

To precisely investigate the electric properties and reveal the intrinsic interaction mechanisms between multiple fields in the composite magneto-electric-semiconductor shell structure, two-dimensional analyses were conducted within the framework of the third-order shear deformation theory and the coupled field theory. The governing equations for the composite piezoelectric semiconductor (PS) shell structures were derived using the principle of virtual work. Field distributions of the composite PS shell structures were obtained by expanding basic field quantities into Fourier series along the width direction and using the differential quadrature method. Prior to analysis, the convergence and correctness of the adopted method were evaluated. Several numerical examples were presented to demonstrate how magnetic manipulation can tune electrical behavior. It was found that inhomogeneous shear strain induced by applied magnetic fields is the key factor affecting potential variation in the composite PS shells. The appearance of potential barriers and wells in the composite PS shell structure with an opposite c axis is explained thoroughly. Additionally, the effect of magnetic field non-uniformity on the composite PS shell structure was investigated using an equivalent body force model, offering valuable insights for designing thin-film devices.

Analytical solution of acoustic emission in soft material with cracks by using reciprocity theorem

Special property in soft materials manifests itself as slight external action may occur significant changes in structure or performance followed by crack initiation and propagation. Characterizing the transition from a crack-free to a cracked soft material remains one of the most challenging problems in structural changes. New methods are needed to predict when and how these soft materials will fail due to defects. In this context, we explored the acoustic emission phenomenon of a crack generation with micro-nanometer size in rubber seal rings. We found through the reciprocity theorem and equivalent body forces that this tiny structural change is defined by the opening crack time function, crack opening body and elastic constant. We take rubber as an example and analyze the new application of acoustic emission and propose a methodology for near surface crack detection of soft materials. Additionally, we discuss the contributions of the opening crack time function, cyclic frequency and elastic constant. The present ‘acoustic emission applications’ strategy offers an effective pathway to observe soft materials' structural changes.

Deformation induced by distributions of single forces in a layered half-space: [formula omitted

In the present paper we introduce a numerical model for the representation of displacement, strain and stress due to single forces embedded in a layered elastic half-space. The code EFGRN/EFCMP (Elastic Forces GReeN functions/Elastic Forces CoMPutation) is able to represent the mechanical effects due to pre-assigned distributions of single forces. Even if internal deformation sources can be described by distributions of equivalent body forces with vanishing resultant and moment, single forces are employed in geophysics to represent hydraulic and/or lithostatic loads, effects of internal density anomalies, and even some kind of seismic events. A distribution of single forces is also used to describe the effects of an inelastic inclusion located inside an elastic medium. In fact, the recent literature shows that poro-elastic and thermo-elastic inclusions can be represented using single forces distributed on their boundaries. EFGRN/EFCMP shares the benefits of rapid and semi-analytical calculation offered by the parent code, EDGRN/EDCMP, which is instead suitable for the representation of extended dislocation sources, as seismic faults. The present code also provides an option for computing the effects of a distribution of single forces embedded in a homogeneous half-space, by using the analytical solutions of Mindlin. Accordingly, EFGRN/EFCMP can be a valid support both for the representation of forward models of deformation sources and for the procedures of inversion of geodetic data in a layered medium. We show some applications of the code and we provide several scripts in MATLAB language which help the user to quickly start using EFGRN/EFCMP.

Def3D, a FEM simulation tool for computing deformation near active faults and volcanic centers

Abstracts Most analytical solutions of deformation near active and volcanic centers are limited to simple geometries and homogeneous media. To solve more complex problems, we have developed a FEM software package called Def3D for modeling deformation caused by earthquakes or magma chambers in laterally inhomogeneous media. Features presently available include modeling of split node technique, equivalent body forces and surface tractions. Def3D presently supports the isotropic elastic and viscoelastic materials behaving as a Maxwell body. The program has been verified in various test cases by strict comparison with analytical solutions. Displacements following the 2008 Wenchuan earthquake and magma-induced surface deformation near Changbaishan volcano are used to test the reliability of the software package.

Crack-induced guided wave motion and modal excitability in plates using elastodynamic reciprocity

This paper presents solutions and comparisons for guided wave motion (Lamb and shear horizontal) due to tensile and shear cracks in an isotropic plate using elastodynamic reciprocity. Cracks are treated as point sources, which for the ease of wave motion analysis have equivalent body force representations. The paper then outlines the use of elastodynamic reciprocity for obtaining the wave motion due to equivalent body forces for tensile (Mode I) and shear (Mode II or III) cracks. A far-field analysis is subsequently performed in order to simplify the results and facilitate a direct comparison of guided mode excitability due to the three cracking modes. A parametric study in an aluminum plate is carried out for the dependence of guided mode excitability on frequency, crack depth, and propagation direction. The study reveals the intricate frequency and depth dependence of the excitability for various fundamental and higher-order modes. Although tensile cracking exhibits complex Lamb mode radiation patterns, a simple and universal closed-form expression is derived for the low-frequency limit (<0.5 MHz-mm). Afterward, the analysis is validated against a 3D FEM model, confirming the predictions of the theory.

Numerical inversion of magma chamber pressurization in volcanic areas: A case study of Changbaishan volcano

Volcanoes are commonly associated with significant topographic relief; however, most analytical models of volcanoes consider the Earth's surface as flat and use half-space solutions. Finite element method can take into account topography; we applied the equivalent body force method to the Changbaishan volcanic area to account for the influence of topographic effects on the inversion of magma chamber pressurization and to calculate the influence of the magma-induced stresses on the stability of the surrounding faults. The influence of surface topography should not be ignored when calculating the surface deformation caused by the magma chamber pressurization in a volcanic area. The shallower the burial depth, the closer the surface maximum radial displacement ur to the center of the magma chamber. When the slope reaches 30°, the maximum vertical displacement uz is no longer directly above the magma chamber. An ellipsoidal pressurization was used to satisfactorily simulate the GPS and leveling measurements at the Changbaishan volcanic area from 2002 to 2003. The superposition of the magma-induced stresses and the regional tectonic stresses likely cause the present stress directions around the Changbaishan volcano, which are dominated by the EW direction to the west of Tianchi Lake and the NW-NNW direction to the north of Tianchi Lake. The Coulomb failure stress change caused by the magma chamber is conducive to the distribution of the NW-trending earthquake swarms that mainly occur to the southwest and northeast of Changbaishan crater.

SH wave generated Lamb wave in a plate with a localized region of material nonlinearity

AbstractHigher harmonics are sensitive to micro‐defects, which can be utilized in the development of nonlinear ultrasonic techniques (NLUTs). Thus, the higher harmonic generation by material nonlinearity has attracted considerable attention. However, few works have been done on the elastic wave propagation in a finite structure with a localized region of material nonlinearity. The propagation of the SH wave in a plate with a localized region of quadratic material nonlinearity is investigated in this paper. The nonlinear governing equations are reduced to a set of linear equations at different orders by using the perturbation method. The incident plane SH wave satisfies the zero‐order equations. The elastic waves generated by the nonlinear material region can be obtained by solving the first‐order equations, whose inhomogeneous terms can be regarded as the equivalent body forces and surface tractions. In the first‐order approximation, only the Lamb wave can be generated by the interaction of the SH wave with a region of quadratic material nonlinearity. In this paper, the analytical solution for the generated backward Lamb wave is obtained, whose amplitude varies with a material‐dependent constant linearly and with the length of the nonlinear region in the form of a sine function.

Scattering of a Rayleigh wave by a near surface crack which is normal to the free surface

In this paper, scattering of a Rayleigh surface wave by a 2D near-surface crack which is normal to the free surface of a homogenous, isotropic and linearly elastic half-space has been investigated. The scattered field is represented by the radiation from equivalent body forces, which are expressed in terms of the crack-opening volumes generated by the incident Rayleigh wave. The reciprocity theorem together with a virtual wave has been used to determine the amplitude of the far field of the scattered Rayleigh wave. A special case of low frequency, and small ratio of the crack length to the wavelength of the incident Rayleigh wave, has been considered to illustrate the method for different values of the ratio of crack length to the wavelength of the incident surface wave. The displacements of the analytical solution on the free surface at some distance from the crack are compared with the displacements obtained numerically. Analytical and numerical results show excellent agreement when the Rayleigh wavelength is sufficiently larger than the crack length. The results of this paper should be useful for the quantitative non-destructive evaluation of a near-surface crack.

A solid-shell based finite element model for thin-walled soft structures with a growing mass

This paper presents a solid-shell based nonlinear finite element model for the numerical analysis of the growth of thin-walled soft structures. A multiplicative decomposition of the deformation gradient tensor is employed to describe the total shape change induced by the mass growth and the elastic deformation. Then a solid-shell model with only displacement degree of freedom is developed and the shell kinematics of deformation considering the growth effect are constructed. The enhanced assumed strain and assumed natural strain methods are employed in the finite element algorithm, so that numerical difficulties arising from the Poisson-thickness locking, volumetric locking and shearing locking phenomena can be avoided. In the finite element formula, an additional term related to the growth in the tangent modulus emerges and an equivalent body force that acts as an additional driving force for the deformation of the materials from the numerical aspect arises. Several representative two- and three-dimensional examples with in-plane and volumetric growth modes are presented to demonstrate the efficiency and accuracy of the proposed model. The model is also proved to be a versatile tool for the modeling of many fascinating growth-induced shape changes and actuating behaviors observed in nature and engineering.

Transient elastodynamic analysis with a combination of convolution quadrature method and pseudo-initial condition method

Purpose The Convolution Quadrature Method (CQM) has been widely applied to solve transient elastodynamic problems because of its stability and generality. However, the CQM suffers from the problems of huge memory requirement in case of direct implementation in time domain or CPU time in case of its reformulation in Laplace domain. The purpose of this paper is to combine the CQM with the pseudo-initial condition method (PICM) to achieve a good balance between memory requirement and CPU time. Design/methodology/approach The combined methods first subdivide the whole analysis into a few sub-analyses, which is dealt with the PICM, namely, the results obtained by previous sub-analysis are used as the initial conditions for the next sub-analysis. In each sub-analysis, the time interval is further discretized into a number of sub-steps and dealt with the CQM. For non-zero initial conditions, the pseudo-force method is used to transform them into equivalent body forces. The boundary face method is employed in the numerical implementation. Three examples are analyzed. Results are compared with analytical solutions or FEM results and the results of reformulated CQM. Findings Results demonstrate that the computation time and the storage requirement can be reduced significantly as compared to the CQM, by using the combined approach. Originality/value The combined methods can be successfully applied to the problems of long-time dynamic response, which requires a large amount of computer memory when CQM is applied, while preserving the CQM stability. If the number of time steps is high, then the accuracy of the proposed approach can be deteriorated because of the pseudo-force method.

A stochastic boundary element method for piezoelectric problems

In this paper, the stochastic boundary element method of piezoelectric problems with randomness of material parameters and randomness of applied loads is proposed. By using the method of first-order Taylor expansion, each random quantity is written into the sum of the mean and deviation. The boundary integral equations corresponding to the means and deviations of the displacements and electric potential are derived, respectively. It is demonstrated that the randomness of material parameters can be transformed into the equivalent random body forces and equivalent random charge density, so that the fundamental solutions of deterministic piezoelectric problems can be used in boundary integral equations of the means or deviations due to the similarity between the governing equations with randomness and those of deterministic piezoelectric problems. Finally, several numerical examples are performed to verify the validity of the proposed stochastic boundary element method.

Reflection of ultrasound from a region of cubic material nonlinearity due to harmonic generation

Two models are proposed to obtain information on the material nonlinearity of an inclusion in a solid body. Material nonlinearity is usually generated by the development of material microscale damage. When the region of nonlinear material is large, incidence of ultrasound on the interface between the perfectly joined regions of linear and nonlinear material behavior produces very useful information. Using the continuity condition of stress and displacement at the interface, the harmonics in the nonlinear region, together with the compensatory waves, yield a reflected wave whose amplitude contains the defining constant of the material nonlinearity near the interface. The compensatory waves are introduced to ensure the continuity conditions at the interface. When the nonlinear region is an inclusion, the equivalent body force induced by the material nonlinearity generates a backscattered wave. The backscattered wave is determined in a simple manner by the use of the reciprocity theorem of elastodynamics. The backscattered wave obtained in this manner yields information on the nonlinear material properties and the size of the inclusion. In addition, a model based on the superposition of back-propagated compensatory waves from the two interfaces of the nonlinear region reveals the physical mechanism of wave scattering from the nonlinear inclusion.

Using reciprocity to derive the far field displacements due to buried sources and scatterers.

It is shown that elastodynamic reciprocity provides a simpler approach for deriving the far-field displacements due to buried (sub-surface) sources in a half-space, compared with integral transform techniques. The auxiliary fields employed in this approach are the fields associated with the reflection of plane waves of the three possible polarisations, and the required far field can be expressed in terms of these well-known auxiliary fields. The crucial step in this approach is to evaluate a surface integral involving cross-work terms between an outgoing spherical wavefront and the auxiliary fields consisting of incident and reflected plane waves. This integral can be evaluated by the stationary phase approximation for the two-dimensional case, or by a generalisation of this approximation for the three-dimensional case. Although this evaluation involves several distinct contributions, the final result is shown to be very simple, and it can be interpreted as a generalisation of a known result for the one-dimensional case, whereby the net contribution arises only from counter-propagating waves of the same mode. The results derived for a buried force are extended to the case of buried cracks by exploiting the body force equivalents for displacement discontinuities across a surface.

A robust Riks-like path following method for strain-actuated snap-through phenomena in soft solids

In this paper, a practical and robust Riks-like path following method is developed to automatically trace the nonlinear equilibrium path for initial-finite-strain-induced snap-through phenomena in soft solids. The deformation gradient is decomposed into an elastic part and an initial-strain part, where the initial strain is characterized through a prescribed pattern tensor and an unknown strain factor describing the magnitude. In the non-linear finite element (FE) solution scheme, an equivalent body force proportional to the increment of the strain factor is derived, which varies in each iteration step due to its dependence on deformed configuration and stress. Both the displacement and the strain factor increments are introduced as unknowns in the linearized equation, which permits to predict the critical equilibrium states before and after the snap-through. Several examples with various configurations and strain histories are constructed, including multilayer beams and film–substrate systems, which demonstrate the effectiveness and robustness of the algorithm to capture the strain-induced snap-through phenomena.

Time domain scattering of elastic waves by a cavity, represented by radiation from equivalent body forces

A method of representing scattering in the time domain by a cavity in an elastic body of infinite extent by radiation from equivalent body forces is presented. It is shown that the equivalent body forces are a combination of double and single forces. The equivalent forces are determined in terms of properties of the incident wave, the solid's elastic constants and the volume of the scatterer, which may be of arbitrary shape. Once the equivalent forces have been determined, the scattered field is determined by the use of a Green's function. Scattering by cavities of 2D and 3D general shape is considered and analyzed. Two special cases, namely, a circular cavity and a spherical cavity, are used to illustrate the method. The results are valid for large measurement distances and waves that are long relative to a characteristic length of the cavity. The method can easily be extended to scattering by cavities in waveguides.

A model to predict the ultrasonic field radiated by magnetostrictive effects induced by EMAT in ferromagnetic parts

An Electro-Magnetic Acoustic Transducer (EMAT) is a non-contact source used in Ultrasonic Testing (UT) which generates three types of dynamic excitations into a ferromagnetic part: Lorentz force, magnetisation force, and magnetostrictive effect. This latter excitation is a strain resulting from a magnetoelastic interaction between the external magnetic field and the mechanical part. Here, a tensor model is developed to transform this effect into an equivalent body force. It assumes weak magnetoelastic coupling and a dynamic magnetic field much smaller than the static one. This approach rigorously formulates the longitudinal Joule’s magnetostriction, and makes it possible to deal with arbitrary material geometries and EMAT configurations. Transduction processes induced by an EMAT in ferromagnetic media are then modelled as equivalent body forces. But many models developed for efficiently predicting ultrasonic field radiation in solids assume source terms given as surface distributions of stress. To use these models, a mathematical method able to accurately transform these body forces into equivalent surface stresses has been developed. By combining these formalisms, the magnetostrictive strain is transformed into equivalent surface stresses, and the ultrasonic field radiated by magnetostrictive effects induced by an EMAT can be both accurately and efficiently predicted. Numerical examples are given for illustration.

Transformation of body force generated by non-contact sources of ultrasound in an isotropic solid of complex shape into equivalent surface stresses

Non-contact techniques in ultrasonic nondestructive evaluation use external non-mechanical excitation (electromagnetic, heat) which interacts with the mechanical part to be tested. The part itself becomes the source of ultrasounds by transforming the non-mechanical energy into a mechanical one. This process involves the generation of dynamic body forces or of an eigenstrain that can be modeled as equivalent body forces, these forces being confined in the vicinity of the part surface. Many models developed for predicting ultrasonic field radiation in solids assume source terms given as surface distributions of stress. In order to predict ultrasonic fields radiated by non-contact sources by means of these radiation models, we developed a method to transform dynamic body forces into equivalent surface stress distributions, irrespective of the nature of the excitation. The approximate transformation relies on a second order expansion of Green’s integral formulation of the elastic wave equation. To make this transformation applicable broadly, the geometry of the surface considered herein is of complex shape, implying thorough differential and tensorial analyses to achieve our aim. Some assumptions, notably isotropic elasticity, are made in deriving the transformation method, which are discussed in detail to clearly define its applicability. Numerical examples of radiated fields are given for illustration and validation.

Use of equivalent body forces for acoustic emission from a crack in a plate

A method to determine acoustic emission of surface waves from a crack near the free edge of a plate, is presented, in terms of the function f(t), which defines the time dependence of the crack opening process, the crack opening volume per unit thickness of the plate, and the elastic constants of the plate. The determination of the time-varying displacement is based on the use of equivalent body forces, which are shown to be two double forces. The acoustic emission of the crack, or the equivalent radiation from the double forces, has been obtained by a novel use of the elastodynamic reciprocity theorem. It is of interest that the normal surface-wave displacement at a position x0 of the free edge comes out as depending on df/dt evaluated at x0 for t>x0/cR, where cR is the velocity of surface waves on the free edge.

Body Force and Fluid Source Equivalents for Dynamic Dislocations in Fluid-Saturated Porous Media

In this study, an explicit expression is derived for body forces and fluid sources equivalent to a general dynamic dislocation in a fluid-saturated porous medium. The components of the source discontinuity vector across the source plane to represent dynamic point dislocations are found by using the surface vector harmonics. Furthermore, we demonstrate that point forces, couples, simple fluid sources, and fluid dipoles can be conversely replaced by point dislocations acting across the horizontal plane. It is shown that the discontinuity in the components of the generalized stress-displacement vector for representing these point sources agrees with those found in our previous research, and thus, the validity of our results for equivalent body forces and fluid sources is examined. With these equivalent body forces and fluid sources, the wave fields in stratified fluid-saturated porous solids can be evaluated by using the reflection and transmission matrix method.

A multiplicative finite element algorithm for the inhomogeneous swelling of polymeric gels

The paper presents a nonlinear Finite Element (FE) algorithm to simulate the inhomogeneous steady swelling of polymeric gels. The algorithm is developed based on the multiplicative decomposition of the deformation gradient into an elastic part and a swelling part. The corresponding constitutive framework and linearization of the FE equation are elaborated, which leads to an equivalent body force driving the materials to swell. A staggered iterative procedure is designed to ensure the strongly coupled chemical and mechanical fields to reach the equilibrium state simultaneously. The constitutive equations are constructed in the framework of the volumetric–isochoric splitting of the free energy function, which is helpful both for a different treatment of the incompressible part in the FE equation and for a physical interpretation of the coupling effects between the chemical and mechanical fields. In the present work, an additive free energy function, coupling the contributions of the Flory–Huggins model and the non-Gauss statistical–mechanical model, is adopted and the corresponding consistent tangent modulus in the current configuration is also derived. To implement the FE algorithm, two two-dimensional (2D) elements and one three-dimensional (3D) element are developed, which are validated by some analytical solutions, such as free swelling of gels in 2D and 3D cases, squeezing a swollen gel with uniform pressure and the constrained swelling of a single-layer and a core–shell gel ring. The algorithm is also proved to be capable of handling the surface instabilities and some engineering applications such as hydrogel-based composite beams and thin films.