Showing papers on "Displacement field published in 1995"
TL;DR: The Element Free Galerkin method (EFG) as mentioned in this paper is a gridless gridless method for solving partial differential equations which employs moving least square interpolants for the trial and test functions.
513 citations
TL;DR: In this article, a refined theory of laminated composite plates with piezoelectric laminae is developed using an energy principle, which includes coupling between mechanical deformations and the charge equations of electrostatics.
360 citations
TL;DR: The combination of rapidly acquired parallel-tagged MR images and field-fitting analysis is a valuable tool in cardiac mechanics research and in the clinical assessment of cardiac mechanical function.
Abstract: PURPOSE: To reconstruct three-dimensional (3D) myocardial deformations from orthogonal sets of parallel-tagged magnetic resonance (MR) images. MATERIALS AND METHODS: Displacement information in the direction normal to the undeformed tag planes was obtained at points along tag lines. Three independent sets of one-dimensional displacement data were used to fit an analytical series expression to describe 3D displacement as a function of deformed position. The technique was demonstrated with computer-generated models of the deformed left ventricle with data from healthy human volunteers. RESULTS: Model deformations were reconstructed with a 3D tracking error of less than 0.3 mm. Error between estimated and observed one-dimensional displacements along the tags in 10 human subjects was 0.00 mm +/- 0.36 (mean +/- standard deviation). Robustness to noise in the tag displacement data was demonstrated by using a Monte Carlo simulation. CONCLUSION: The combination of rapidly acquired parallel-tagged MR images and fi...
263 citations
TL;DR: In this paper, a time domain model of linear viscoelasticity is developed based on a decomposition of the total displacement field into two parts: one elastic, the other anelastic.
Abstract: A time domain model of linear viscoelasticity is developed based on a decomposition of the total displacement field into two parts: one elastic, the other anelastic. The anelastic displacement field is used to describe that part of the strain that is not instantaneously proportional to stress. General coupled constitutive equations for (1) the total and (2) the anelastic stresses are developed in terms of the total and anelastic strains, and specialized to the case of isotropic materials. A key feature of the model is the absence of explicit time dependence in the constitutive equations. Apparent time-dependent behavior is described instead by differential equations that govern (1) the motion of mass particles and (2) the relaxation of the anelastic displacement field. These coupled governing equations are developed in a parallel fashion, involving the divergence of appropriate stress tensors. Boundary conditions are also treated: the anelastic displacement field is effectively an internal field, as it is driven exclusively through coupling to the total displacement, and cannot be directly affected by applied loads. In order to illustrate the use of the method, model parameters for a commonly-used high damping polymer are developed from available complex modulus data.
198 citations
TL;DR: The proposed method locally correlates images for displacements, rotations, deformations, and higher-order displacement gradient fields, and applies a global minimization procedure to insure a global consistency in the results.
Abstract: This paper focuses on the correlation of two successive scalar images for the purpose of measuring imaged fluid motions. A method is presented for deforming, or transforming, one image to another. Taylor series expansions of the Lagrangian displacement field are used, in conjunction with an integral form of the equations of motion, to approximate this transformation. The proposed method locally correlates images for displacements, rotations, deformations, and higher-order displacement gradient fields, and applies a global minimization procedure to insure a global consistency in the results. An integral form of the equations of motion is employed. No explicit spatial or temporal differentiation of the image data is required in estimating the displacement field. As a consequence, this method is appropriate for both continuous-scalar as well as discrete-particle-image data. Successive two-dimensional digital CCD images of fluid motion marked with dye, are used to verify the capabilities of the method. The utility of the method is also illustrated using a pair of Voyager 2 images of Jupiter.
174 citations
TL;DR: In this article, 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.
Abstract: 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.
142 citations
TL;DR: In this article, a nonlinear shell theory, including transverse strains perpendicular to the shell midsurface, as well as transverse shear strains, with exact description of the kinematical fields, is developed.
Abstract: A nonlinear shell theory, including transverse strains perpendicular to the shell midsurface, as well as transverse shear strains, with exact description of the kinematical fields, is developed. The strain measures are derived by considering theGreen strain tensor of the three-dimensional shell body. A quadratic displacement field over the shell thickness is considered. Altogether seven kinematical fields are incorporated in the formulation. The kinematics of the shell normal is described by means of a difference vector, avoiding the use of a rotation tensor and resulting in a configuration space, where the structure of a linear vector space is preserved. In the case of linear constitutive equations, a possible consistent reduction to six degrees of freedom is discussed. The finite element formulation is based on a hybrid variational principle. The accuracy of the theory and its wide range of applicability is demonstrated by several examples. Comparison with results based on shell theories formulated by means of a rotation tensor are included.
131 citations
TL;DR: A method for estimating a dense displacement field from sparse displacement measurements based on a multidimensional stochastic model for the smoothness and divergence of the displacement field and the Fisher estimation framework for in vivo heart data is proposed.
Abstract: Magnetic resonance (MR) tagging has shown great potential for noninvasive measurement of the motion of a beating heart. In MR tagged images, the heart appears with a spatially encoded pattern that moves with the tissue. The position of the tag pattern in each frame of the image sequence can be used to obtain a measurement of the 3-D displacement field of the myocardium. The measurements are sparse, however, and interpolation is required to reconstruct a dense displacement field from which measures of local contractile performance such as strain can be computed. Here, the authors propose a method for estimating a dense displacement field from sparse displacement measurements. Their approach is based on a multidimensional stochastic model for the smoothness and divergence of the displacement field and the Fisher estimation framework. The main feature of this method is that both the displacement field model and the resulting estimate equation are defined only on the irregular domain of the myocardium. The authors' methods are validated on both simulated and in vivo heart data.
128 citations
TL;DR: In this article, an embedded representation of fracture for finite element analysis of concrete structures is presented, where the three-field Hu-Washizu variational statement is extended to bodies with internal discontinuities.
Abstract: As an alternative to the smeared and discrete crack representations, an embedded representation of fracture for finite element analysis of concrete structures is presented. The three-field Hu–Washizu variational statement is extended to bodies with internal discontinuities. The extended variational statement is then utilized for formulating elements with a discontinuous displacement field. The new elements are capable of modelling different deformation modes of an internal discontinuity at the element level. The satisfactory performance of the embedded crack representation is verified by several case studies on concrete fracture.
121 citations
TL;DR: In this paper, the authors presented the results of a combined theoretical and experimental investigation of local bending effects in a clamped circular foam-cored sandwich plate subjected to a central point load, where the basic approach is to consider the deflection of the loaded face as being governed by a two-parameter elastic foundation model.
106 citations
TL;DR: In this paper, the concept of displacement-based design is extended to complex multi-degree-of-freedom (MDOF) bridge structures, and a procedure based on the assumption of a displaced shape for the structure, and the subsequent reduction of the system to an equivalent single-degree of freedom (SDOF) system is presented.
Abstract: The concept of displacement-based design is attractive for seismic design, primarily because it places the focus of design directly on displacement demand, and hence damage, rather than on force-reduction or behaviour factors. A procedure is presented which extends the simple concept of displacement-based design to complex multi-degree-of-freedom (MDOF) bridge structures. The procedure is based on the assumption of a displaced shape for the structure, and the subsequent reduction of the system to an equivalent single-degree-of-freedom (SDOF) system. The process is shown to work well for the design of a symmetrical bridge, while suffering some shortcomings when applied to a highly irregular bridge. The topic of design-oriented displacement response spectra is also briefly addressed.
TL;DR: In this article, a non-linear stress resultant four node shell finite element is presented and the underlying shell theory is developed from the three dimensional continuum theory via standard assumptions on the displacement field.
Abstract: A simple non-linear stress resultant four node shell finite element is presented. The underlying shell theory is developed from the three dimensional continuum theory via standard assumptions on the displacement field. A model for thin shells is obtained by approximating terms describing the shell geometry. In this work the rotation of the shell director is parameterized by the two Euler angles, although other approaches can be easily accomodated. A procedure is provided to extend the presented approach by including the through-thickness variable material properties. These may include a general non-linear elastic material with varied degree of orthotropy, which is typical for fibre reinforced composites. Thus a simple and efficient model suitable for analysis of multilayered composite shells is attained. Shell kinematics is consistently linearized, leading to the Newton-Raphson numerical procedure, which preserves quadratic rate of asymptotic convergence. A range of linear and non-linear tests is provided and compared with available solutions to illustrate the approach.
TL;DR: In this paper, a finite element formulation for the analysis of singular stress states at material and geometric discontinuities in anisotropic materials loaded inplane is developed for the problem of finite element simulation.
TL;DR: In this article, a generalized micropolar continuum framework is proposed to model softening behavior by damage evolution and regularize the impending loss of ellipticity which results in strong mesh dependence of localization computations.
Abstract: The aim of this work is to extend an isotropic elastoplastic damage concept for ductile materials within a generalized micropolar continuum framework. The underlying motivation is on the one hand to model softening behaviour by damage evolution and on the other hand to regularize the impending loss of ellipticity which results in strong mesh dependence of localization computations. To this end, the classical displacement field is supplemented by an independent rotation field to yield an enhanced continuum. For the model originally proposed by Lemaitre the damage evolution follows from a dissipation potential and the hypothesis of general associativity. Special emphasis is directed towards the numerical implementation of the constitutive model within the framework of finite element analysis of inelastic boundary value problems. To compare damage evolution and standard strain softening the results are contrasted to the outcome of an analysis with the micropolar version of the classical von Mises model. Thereby, the intriguing correspondence of strain softening and damage evolution is highlighted. Additionally, several planar micropolar finite element formulations are derived within mixed variational principles to enhance the poor performance of the standard bilinear displacement and rotation expansions used so far in the literature. The merits of these new element developments are demonstrated for the example of localization within a compression problem.
TL;DR: In this article, an isotropic augmented Hooke's law with both dilatational and shearing damping has been implemented and tested using a 20-node volume element in the finite element code ASKA Acoustics.
TL;DR: In this paper, a three layer elastic-gravitational fault displacement model was developed using elastic dislocation and employed to examine the effects of rigidity layering, gravity, and stress relaxation on surface and subsurface displacements fields for a dip-slip normal fault.
Abstract: A three layer elastic-gravitational fault displacement model has been developed using elastic dislocation and employed to examine the effects of rigidity layering, gravity, and stress relaxation on surface and subsurface displacements fields for a dip-slip normal fault. Within our three layer model, layer 1 represents the upper crust, layer 2 the lower crust, and layer 3 the upper mantle. The fault is embedded in the upper crust. Horizontal as well as vertical displacement components have been determined. Displacement field changes due to postseismic stress relaxation have been calculated using both the correspondence principle and relaxed rigidity moduli, and results are in close agreement. The relaxed rigidity method provides an accurate and computationally efficient method of examining postseismic relaxation. Postseismic relaxation within the lower two layers only, gives surface uplift, increasing footwall uplift and decreasing hangingwall subsidence, and also increases the wavelength of surface vertical displacement. Postseismic stress relaxation within all three layers (i.e., uniform half-space) produces a shorter wavelength of surface vertical deformation with respect to the coseismic response. Moho topography created during coseismic deformation is initially amplified during postseismic relaxation. The relaxed Moho topography is dependent on the strength of the upper crust as well as the strength of the lower crust and mantle. Gravity has a significant influence on displacements only at the postseismic stage when the effective rigidity of the lower layers is small. Coseismic and postseismic normal strains associated with dip-slip normal faulting have been examined. For a large normal basement fault intersecting the free surface, the coseismic horizontal surface strain, perpendicular to fault strike, is compressive adjacent to the fault and in the footwall, and tensile in the hangingwall away from the fault. Coseismic stress redistribution may generate significant tensile brittle failure of the upper crust adjacent to large basement faults.
TL;DR: Basic integral representations that connect the solid spherical Navier eigenvectors to the vector spherical harmonics over the unit sphere are utilized to prove the fundamental representation theorem for the elastic Herglotz solutions in full space.
Abstract: Solutions of the spectral Navier equation in the linearized theory of elasticity that satisfy the Herglotz boundness condition at infinity are introduced. The leading asymptotic terms in a neighborhood of infinity provide the far-field patterns, and the Herglotz norm is expressed as the sum of the $L^2 $-norms of these patterns over the unit sphere. Basic integral representations that connect the solid spherical Navier eigenvectors to the vector spherical harmonics over the unit sphere are utilized to prove the fundamental representation theorem for the elastic Herglotz solutions in full space. It is shown that the longitudinal and the transverse Herglotz kernels are exactly the corresponding far-field patterns of the irrotational and the solenoidal parts of the displacement field. Particular methods to obtain the displacement field from the far-field patterns, and vice versa, are also described.
TL;DR: In this article, a finite-element modeling approach for the automatic mixed-mode two-dimensional (2D) linear elastic fracture mechanics (LEFM) crack-propagation analysis has been developed.
Abstract: Based on an energy principle and a virtual crack extension technique, a practical finite-element modeling approach for the automatic mixed-mode two-dimensional (2D) linear elastic fracture mechanics (LEFM crack-propagation analysis has been developed. By decomposing the displacement field obtained from a conventional finite-element analysis into symmetric and antisymmetric displacement fields with respect to the crack, the Mode-I and Mode-II energy release rates can be determined using a generalization of the stiffness derivative method, and the corresponding stress intensity factors can then be calculated. The load at which a crack propagates and the propagation direction can be determined using one of the well-established mixed-mode crack-propagation criteria. Unlike the previously developed approaches to crack propagation, this approach does not require the use of symmetric crack-tip mesh, crack-tip singular elements, which greatly simplifies the subsequent remeshing process to allow for crack propagation. The automatic crack-propagation process is achieved by using a simple and efficient local 2D remeshing algorithm that can be easily programmed. Convergence studies, mesh sensitivity studies, and the practical use of the new approach are presented through various example problems.
TL;DR: In this article, closed form solutions for the stress and displacement fields in the soil during any stage of the unloading process are derived, and large strains are taken into account by adopting an appropriate strain definition within the plastically deforming region.
Abstract: An infinite dilatant elastic-plastic soil mass contains a single cylindrical or spherical cavity within which a slowly increasing pressure is applied. The removal of the cavity pressure takes place after a partly plastic state of the soil has been reached. Closed form solutions for the stress and displacement fields in the soil during any stage of the unloading process are derived. The non-associated Mohr-Coulomb yield criterion is used to account for dilation of the soil during shearing. Large strains are taken into account by adopting an appropriate strain definition within the plastically deforming region.
TL;DR: In this paper, the effects of a viscoelastic adhesive layer on the dynamic response and structural damping of sandwich structures were studied by employing a newly developed sandwich beam theory.
Abstract: The effects of a viscoelastic adhesive layer on the dynamic response and structural damping of sandwich structures are studied by employing a newly developed sandwich beam theory. The two face layers are considered as ordinary beams (Euler beams) with both axial and bending resistance. The core material is considered to be viscoelastic. A newly developed high-order displacement field assumption is used in order to achieve a more accurate kinematics of the flexible viscoelastic core than would be possible under the classical assumptions for sandwich beams. Imperfect interface conditions between faces and core are defined as linear relations between longitudinal displacement discontinuities and the transverse shear stress at the adhesive layer. The viscoelastic properties of the adhesive layer and the core material are assumed in a complex modulus formula that is a function of frequency for a given temperature. The linear equations of motion that describe the vibration of the sandwich finite beam a...
TL;DR: The iterative dissipative method (IDM) is based on the simple fact that current induced in a conductive medium is in some sense smaller than the external current generating the field as discussed by the authors.
TL;DR: In this article, an analytical approach to calculate the relationship between the axial curvature of a bent tube and the resulting deformation of the cross-section was developed, which accounts for both geometrical and material nonlinearities.
Abstract: The objective of this study is to develop an analytical approach to calculate the relationship between the axial curvature of a bent tube and the resulting deformation of the cross-section. The model accounts for both geometrical and material nonlinearities. An approximate expression in trigonometric form is introduced for the displacement field, which reflects the change of wall thickness and neutral axis shift during bending. The total deformation theory is employed as a constitutive relation. The solution is found using a minimization approach and the energy principle. A better approach for springback prediction might be obtained from the deformation model, which predicts a more accurate moment of inertia change during bending.
TL;DR: In this paper, a new total Lagrangian finite-element formulation for general laminated composite shells undergoing large displacement, large rotation, and large strain motion is presented, which fully accounts for geometric nonlinearities (large rotations), large in-plane strains, general initial curvatures, transverse shear deformations, and interlaminar normal stresses by using Jaumann stress and strain measures.
TL;DR: In this paper, a finite element formulation is developed to determine the order and angular variation of singular stress states at material and geometric discontinuities in anisotropic materials subject to antiplane shear loading.
Abstract: A finite element formulation is developed to determine the order and angular variation of singular stress states at material and geometric discontinuities in anisotropic materials subject to antiplane shear loading. The displacement field of the sectorial element is quadratic in the angular co-ordinate direction and asymptotic in the radial direction measured from the singular point. The formulation of Yamada and Okumura14 for in-plane problems is adapted for this purpose. The simplicity and accuracy of the formulation are demonstrated by comparison to several analytical antiplane shear solutions for both isotropic and anisotropic multi-material wedges and junctions with and without disbonds. The nature and speed of convergence of the eigensolution suggests that the solution presented here could be used in developing enriched elements for accurate and computationally efficient evaluation of stress intensity factors in problems having complex global geometries.
TL;DR: In this article, a phenomenological theory for long-wavelength polar optical oscillations to mesoscopic layered semiconductor structures is applied to calculate the normal modes of a quantum wire and of a free-standing wire, the cylindrical geometry is adopted with circular cross-section of radius r 0.
Abstract: By applying a phenomenological theory for long-wavelength polar optical oscillations to mesoscopic layered semiconductor structures, we calculate the normal modes of a quantum wire and of a free-standing wire, the cylindrical geometry is adopted with circular cross-section of radius r0. The displacement field u and the electric potential phi are calculated for the different modes, as well as the dispersion relation curves. The case of the GaAs/AlAs structure is analysed. We limit ourselves to the study of oscillations perpendicular to the wire axis. The electron-phonon interaction Hamiltonian is derived for the present problem using the second-quantization formalism.
01 Jan 1995
TL;DR: A methodology based on Chebyshev polynomials approximation to analyze the non-linear boundary value problems in rectangular domain is developed in this paper, where the von K&man equations governing the behavior of moderately large deformations of rectangular plates are expressed in displacement field.
Abstract: The von K&man equations governing the behavior of moderately large deformations of rectangular plates are expressed in displacement field. A methodology based on Chebyshev polynomials approximation to analyze the non-linear boundary value problems in rectangular domain is developed. These non-linear partial differential equations of motion are linearized using quadratic extrapolation techniques. The inertia and dissipative terms are evaluated by employing Houbolt implicit time-marching scheme. The spatial discretization of the differential equations generates incompatibility, viz. greater number of equations than the unknowns. The multiple linear regression analysis, based on the least-square error norm, is employed to overcome the incompatibility and a compatible solution is obtained. Convergence study has been carried out. The clamped and simply supported immovable rectangular plates subjected to static and dynamic loadings are analyzed. Results have been compared with the results obtained by other numerical and analytical methods.
TL;DR: In this paper, a formulation for the plane 4-node quadrilateral finite element based on the principle of virtual displacements for a deformable body is developed, which is suitable for nonlinear analysis.
Abstract: A formulation for the plane 4-node quadrilateral finite element is developed based on the principle of virtual displacements for a deformable body. Incompatible modes are added to the standard displacement field. Then expressions for gradient operators are obtained from an expansion of the basis functions into a second-order Taylor series in the physical co-ordinates. The internal degrees of freedom of the incompatible modes are eliminated on the element level. A modified change of variables is used to integrate the element matrices.
For a linear elastic material, the element stiffness matrix can be separated into two parts. These are equivalent to a stiffness matrix obtained from underintegration and a stabilization matrix.
The formulation includes the cases of plane stress and plane strain as well as the analysis of incompressible materials. Further, the approach is suitable for non-linear analysis. There, an application is given for the calculation of inelastic problems in physically non-linear elasticity.
The element is efficient to implement and it is frame invariant. Locking effects and zero-energy modes are avoided as well as singularities of the stiffness matrix due to geometric distortion. A high accuracy is obtained for numerical solutions in displacements and stresses.
TL;DR: An in-depth study of spatio-temporal patterns in a simplified version of a mechanical model for pattern formation in mesenchymal morphogenesis and shows that when a two-dimensional mode number admits two or more degenerate mode pairs, the solution of the full nonlinear system of partial differential equations is a mixed mode solution in which all the degenerate Mode pairs are represented in a frequency locked oscillation.
Abstract: We present an in-depth study of spatio-temporal patterns in a simplified version of a mechanical model for pattern formation in mesenchymal morphogenesis. We briefly motivate the derivation of the model and show how to choose realistic boundary conditions to make the system well-posed. We firstly consider one-dimensional patterns and carry out a nonlinear perturbation analysis for the case where the uniform steady state is linearly unstable to a single mode. In two-dimensions, we show that if the displacement field in the model is represented as a sum of orthogonal parts, then the model can be decomposed into two sub-models, only one of which is capable of generating pattern. We thus focus on this particular sub-model. We present a nonlinear analysis of spatio-temporal patterns exhibited by the sub-model on a square domain and discuss mode interaction. Our analysis shows that when a two-dimensional mode number admits two or more degenerate mode pairs, the solution of the full nonlinear system of partial differential equations is a mixed mode solution in which all the degenerate mode pairs are represented in a frequency locked oscillation.
TL;DR: In this paper, the authors used three-dimensional finite element models of center cracked plates to study the variation in the biaxiality factor β with the crack aspect ratio a/w.
Abstract: Thick three-dimensional (3-D) finite element models of centre cracked plates are used to study the variation in the biaxiality factor β with the crack aspect ratio a/w. The use of two widely accepted methods to evaluate the T-stress in two-dimensions, namely the boundary layer method and the displacement field method, to calculate the T-stress in three-dimensions is studied. It is shown that the boundary layer method gives results that compare rather well with the two-dimensional plane strain values (maximum difference of 6 percent), while the displacement field method results are about 15 percent lower. Two parameters are shown to affect the three-dimensional evaluation of the biaxiality factor, namely the material's Poisson's ratio ν and the specimen's thickness t. The biaxiality factor is directly proportional to ν and inversely proportional to t. Three-dimensional analysis is required to assess correctly the effect of ν and t on β.
TL;DR: A generalized k‐space (GKS) formulation is presented for vectorial elastodynamic scattering problems, which represents a generalization of Bojarski’s scalar k‐ space formulation.
Abstract: A generalized k‐space (GKS) formulation is presented for vectorial elastodynamic scattering problems. It represents a generalization of Bojarski’s scalar k‐space formulation. From the basic second‐order partial differential equation or its integral representation in the space‐frequency (r‐f) domain, a local equation is derived for the displacement field in the spectral‐frequency (k‐f) domain. This equation, together with the constitutive equation in the r‐f domain, reduces the original scattering problem into two simultaneous local equations with two unknowns (displacement field and the induced source), which are then solved by the conjugate‐gradient (CG) method. The connection between the k‐f domain and r‐f domain is obtained by the spatial fast Fourier transform (FFT) algorithm. The number of complex multiply‐add operations per CG iteration is O(N log2 N), and the storage requirement is only O(N), where N is the number of spatial discretization points. This is much more efficient than the conventional m...