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R W Ogden - One of the best experts on this subject based on the ideXlab platform.
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the effect of rotation and initial stress on the propagation of waves in a transversely isotropic elastic solid
Wave Motion, 2014Co-Authors: R W Ogden, Baljeet SinghAbstract:In this paper the equations governing small amplitude motions in a rotating transversely isotropic initially stressed elastic solid are derived, both for compressible and Incompressible linearly elastic Materials. The equations are first applied to study the effects of initial stress and rotation on the speed of homogeneous plane waves propagating in a configuration with uniform initial stress. The general forms of the constitutive law, stresses and the elasticity tensor are derived within the finite deformation context and then summarized for the considered transversely isotropic Material with initial stress in terms of invariants, following which they are specialized for linear elastic response and, for an Incompressible Material, to the case of plane strain, which involves considerable simplification. The equations for two-dimensional motions in the considered plane are then applied to the study of Rayleigh waves in a rotating half-space with the initial stress parallel to its boundary and the preferred direction of transverse isotropy either parallel to or normal to the boundary within the sagittal plane. The secular equation governing the wave speed is then derived for a general strain–energy function in the plane strain specialization, which involves only two Material parameters. The results are illustrated graphically, first by showing how the wave speed depends on the Material parameters and the rotation without specifying the constitutive law and, second, for a simple Material model to highlight the effects of the rotation and initial stress on the surface wave speed.
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a modified holzapfel ogden law for a residually stressed finite strain model of the human left ventricle in diastole
Biomechanics and Modeling in Mechanobiology, 2014Co-Authors: Huiming Wang, R W Ogden, Boyce E Griffith, Colin BerryAbstract:In this work, we introduce a modified Holzapfel-Ogden hyperelastic constitutive model for ventricular myocardium that accounts for residual stresses, and we investigate the effects of residual stresses in diastole using a magnetic resonance imaging–derived model of the human left ventricle (LV). We adopt an invariant-based constitutive modelling approach and treat the left ventricular myocardium as a non-homogeneous, fibre-reinforced, Incompressible Material. Because in vivo images provide the configuration of the LV in a loaded state even in diastole, an inverse analysis is used to determine the corresponding unloaded reference configuration. The residual stress in this unloaded state is estimated by two different methods. One is based on three-dimensional strain measurements in a local region of the canine LV, and the other uses the opening angle method for a cylindrical tube. We find that including residual stress in the model changes the stress distributions across the myocardium and that whereas both methods yield qualitatively similar changes, there are quantitative differences between the two approaches. Although the effects of residual stresses are relatively small in diastole, the model can be extended to explore the full impact of residual stress on LV mechanical behaviour for the whole cardiac cycle as more experimental data become available. In addition, although not considered here, residual stresses may also play a larger role in models that account for tissue growth and remodelling.
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propagation of waves in an Incompressible transversely isotropic elastic solid with initial stress biot revisited
Journal of Mechanics of Materials and Structures, 2011Co-Authors: R W Ogden, Baljeet SinghAbstract:In this paper, the general constitutive equation for a transversely isotropic hyperelastic solid in the presence of initial stress is derived, based on the theory of invariants. In the general finite deformation case for a compressible Material this requires 18 invariants (17 for an Incompressible Material). The equations governing infinitesimal motions superimposed on a finite deformation are then used in conjunction with the constitutive law to examine the propagation of both homogeneous plane waves and, with the restriction to two dimensions, Rayleigh surface waves. For this purpose we consider Incompressible Materials and a restricted set of invariants that is sufficient to capture both the effects of initial stress and transverse isotropy. Moreover, the equations are specialized to the undeformed configuration in order to compare with the classical formulation of Biot. One feature of the general theory is that the speeds of homogeneous plane waves and surface waves depend nonlinearly on the initial stress, in contrast to the situation of the more specialized isotropic and orthotropic theories of Biot. The speeds of (homogeneous plane) shear waves and Rayleigh waves in an Incompressible Material are obtained and the significant differences from Biot's results for both isotropic and transversely isotropic Materials are highlighted with calculations based on a specific form of strain-energy function.
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constitutive modelling of passive myocardium a structurally based framework for Material characterization
Philosophical Transactions of the Royal Society A, 2009Co-Authors: Gerhard A Holzapfel, R W OgdenAbstract:In this paper, we first of all review the morphology and structure of the myocardium and discuss the main features of the mechanical response of passive myocardium tissue, which is an orthotropic Material. Locally within the architecture of the myocardium three mutually orthogonal directions can be identified, forming planes with distinct Material responses. We treat the left ventricular myocardium as a non-homogeneous, thick-walled, nonlinearly elastic and Incompressible Material and develop a general theoretical framework based on invariants associated with the three directions. Within this framework we review existing constitutive models and then develop a structurally based model that accounts for the muscle fibre direction and the myocyte sheet structure. The model is applied to simple shear and biaxial deformations and a specific form fitted to the existing (and somewhat limited) experimental data, emphasizing the orthotropy and the limitations of biaxial tests. The need for additional data is highlighted. A brief discussion of issues of convexity of the model and related matters concludes the paper.
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Stress softening and residual strain in the azimuthal shear of a pseudo-elastic circular cylindrical tube
International Journal of Non-Linear Mechanics, 2001Co-Authors: R W OgdenAbstract:In a recent paper, Ogden and Roxburgh (Proc. R. Soc. London A 455 (1999a) 2861) developed a theory of pseudo-elasticity for the description of damage-induced stress softening in rubberlike solids (the Mullins effect), and the theory was modified to incorporate residual strains by Ogden and Roxburgh (in: A. Dorfmann, A. Muhr (Eds.), Proceedings of the First European Conference on Constitute Models for Rubber, Vienna, 1999, Balkema, Rotterdam, pp. 23–28.). In the present paper this theory is applied to a problem involving non-homogeneous deformation, namely the (plane strain) azimuthal shear of a thick-walled circular cylindrical tube of Incompressible Material. Loading, effected by application of a specified rotation of the outer surface of the tube relative to the inner one, is described by an isotropic elastic strain-energy function. Unloading, associated with reduction in the applied shearing stress on the outer boundary, is described by a different isotropic elastic strain-energy function, which is inhomogeneous and dependent locally on the extent of the initial loading. It is shown that if the maximum applied shear stress on the outer boundary is below a certain critical value, then there is no residual strain after the shearing stress is removed, while if it is greater than a second critical value then there is residual strain throughout the tube. In the intermediate situation there is residual strain only within a certain radius. Outside this radius the final state of deformation corresponds to a rigid rotation. The results are specialized in respect of a particular Material model, which allows the residual strain to be calculated explicitly. Also, the total residual (non-recoverable) energy due to the loading–unloading cycle is calculated. Numerical calculations are used to illustrate the stress softening effect by comparison of the shear stress on unloading with that on loading.
Kenichi Soga - One of the best experts on this subject based on the ideXlab platform.
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Projection Method in Material Point Method for Modeling Incompressible Materials
Procedia Engineering, 2017Co-Authors: Shyamini Kularathna, Kenichi SogaAbstract:Abstract Material point method is a variant of the finite element method (FEM) and is successfully applied in large deformation problems. Recently, Material point method has been applied in a wide range of engineering applications including solid and solid-fluid interaction problems. However, describing the behavior of Incompressible Materials is a challenging problem in MPM. The explicit formulation and the linear elements used in the standard MPM exhibit numerical instabilities such as mesh locking and artificial pressure oscillations in Material incompressibility. Further, the small time step used to obtain a reasonable numerical stability limits the application of MPM in problems particularly with long time durations. We present an implicit treatment of the pressure term in MPM to mitigate the numerical instabilities and small time steps in Incompressible Material problems. The set of velocity-pressure coupled governing equations resulted by the implicit formulation is solved using Choin's projection method. The numerical examples show that the present MPM implementation is capable of modeling Incompressible Materials without pressure oscillations using a significantly large time step.
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implicit formulation of Material point method for analysis of Incompressible Materials
Computer Methods in Applied Mechanics and Engineering, 2017Co-Authors: Shyamini Kularathna, Kenichi SogaAbstract:Abstract Material point method (MPM) is a particle based numerical technique with application to solid mechanics problems involving large deformations. Recently, MPM has been successfully applied in many engineering applications including solid and solid–fluid interaction problems due to its capability to model complex history dependent Material behaviours. However, the conventional explicit MPM approach exhibits numerical limitations in Incompressible or near Incompressible Material behaviour problems. We study an implicit treatment of the pressure in MPM algorithm to simulate Material incompressibility avoiding artificial pressure oscillations and significantly small time steps present in the explicit MPM approach. Applying Chorin’s projection method to the velocity–pressure coupled equations, an elliptic equation is solved for the pressure which is then used to solve the divergence free velocity field. The results of dam break problem show no spurious pressure oscillations and large time steps compared to the traditional explicit MPM. The ability of the present implicit MPM formulation to model Incompressible Materials will be an additional advantage in simulating Materials with complex constitutive equations which exhibit a time dependent memory.
Yanping Lian - One of the best experts on this subject based on the ideXlab platform.
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Incompressible Material point method for free surface flow
Journal of Computational Physics, 2017Co-Authors: Fan Zhang, Xiong Zhang, Yanping LianAbstract:To overcome the shortcomings of the weakly compressible Material point method (WCMPM) for modeling the free surface flow problems, an Incompressible Material point method (iMPM) is proposed based on operator splitting technique which splits the solution of momentum equation into two steps. An intermediate velocity field is first obtained by solving the momentum equations ignoring the pressure gradient term, and then the intermediate velocity field is corrected by the pressure term to obtain a divergence-free velocity field. A level set function which represents the signed distance to free surface is used to track the free surface and apply the pressure boundary conditions. Moreover, an hourglass damping is introduced to suppress the spurious velocity modes which are caused by the discretization of the cell center velocity divergence from the grid vertexes velocities when solving pressure Poisson equations. Numerical examples including dam break, oscillation of a cubic liquid drop and a droplet impact into deep pool show that the proposed Incompressible Material point method is much more accurate and efficient than the weakly compressible Material point method in solving free surface flow problems. An Incompressible MPM is proposed to simulate free surface flow.A scheme for calculating pressure gradients at nodes in semi-staggered grid is proposed.Hourglass damping is employed to suppress spurious velocity modes.
Fan Zhang - One of the best experts on this subject based on the ideXlab platform.
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an augmented Incompressible Material point method for modeling liquid sloshing problems
International Journal of Mechanics and Materials in Design, 2018Co-Authors: Fan Zhang, Xiong Zhang, Yan LiuAbstract:The Incompressible Material point method was proposed for modeling the free surface flow problems based on the operator splitting technique which decouples the solution of the velocity and the pressure in our previous work. To further model the coupling problems between the Incompressible fluid and the moving irregular solid bodies, an augmented Incompressible Material point method is proposed in this paper based on the energy minimization form of operator splitting technique. The interaction between the fluid and the solid is taken into account via the work done by the fluid pressure on the solid bodies. By minimizing the total work done by the fluid pressure, volume-weighted pressure Poisson equations are obtained. The proposed method is validated with liquid sloshing in a rectangular tank subjected to various base-excitations, and is then used to study the optimal height of baffles mounted on the bottom of the tank to mitigate the sloshing wave.
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improved Incompressible Material point method based on particle density correction
International Journal of Computational Methods, 2017Co-Authors: Fan Zhang, Xiong Zhang, Ky Sze, Yong Liang, Yan LiuAbstract:In the Incompressible Material point method (iMPM), the momentum equations were solved at the background grid nodes while the divergence-free conditions were enforced at grid cell centers. The density of each particle was assumed to be constant but the particles could distribute nonuniformly in space over time. Therefore, the fluid density would be nonuniform and violate the Incompressible condition. In this paper, the original iMPM is improved by explicitly imposing the density-invariant condition. A new particle shifting scheme is proposed for particle density correction. Particles are shifted along their density gradient to guarantee that the density field of the fluid is constant and the momentum is conserved. The proposed method has been implemented in our MPM code, and validated by simulating a dam breaking inside a tank, another dam breaking with an obstacle and a sloshing problem.
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Incompressible Material point method for free surface flow
Journal of Computational Physics, 2017Co-Authors: Fan Zhang, Xiong Zhang, Yanping LianAbstract:To overcome the shortcomings of the weakly compressible Material point method (WCMPM) for modeling the free surface flow problems, an Incompressible Material point method (iMPM) is proposed based on operator splitting technique which splits the solution of momentum equation into two steps. An intermediate velocity field is first obtained by solving the momentum equations ignoring the pressure gradient term, and then the intermediate velocity field is corrected by the pressure term to obtain a divergence-free velocity field. A level set function which represents the signed distance to free surface is used to track the free surface and apply the pressure boundary conditions. Moreover, an hourglass damping is introduced to suppress the spurious velocity modes which are caused by the discretization of the cell center velocity divergence from the grid vertexes velocities when solving pressure Poisson equations. Numerical examples including dam break, oscillation of a cubic liquid drop and a droplet impact into deep pool show that the proposed Incompressible Material point method is much more accurate and efficient than the weakly compressible Material point method in solving free surface flow problems. An Incompressible MPM is proposed to simulate free surface flow.A scheme for calculating pressure gradients at nodes in semi-staggered grid is proposed.Hourglass damping is employed to suppress spurious velocity modes.
Graca Ana - One of the best experts on this subject based on the ideXlab platform.
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Subspace analysis to alleviate the volumetric locking in the 3D solid-shell EFG method
Elsevier, 2013Co-Authors: Graca Ana, Cardoso, Rui P. R., Yoon, Jeong WhanAbstract:The main objective of this work is to reduce the volumetric locking pathology at the Element Free Galerkin (EFG) 3D solid-shell meshless method. For this purpose, a subspace analysis is performed in order to extract the linearly independent Incompressible deformation modes that can be reproduced by different background integration cells. Additional deformation modes reproduced by a cell configuration produces more flexibility for the meshless formulation. The Enhanced Assumed Strain (EAS) method is blended with the EFG formulation in such a way that additional (mathematical) variables are included in the formulation increasing its flexibility for nearly Incompressible Materials. Three numerical examples show the effect of the EAS formulation in alleviating the locking pathology for an almost Incompressible Material condition
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Subspace analysis to alleviate the volumetric locking in the 3D solid-shell EFG method
'Elsevier BV', 2013Co-Authors: Cardoso, Rui P.r., Yoon, Jeong Whan, Graca AnaAbstract:© 2012 Elsevier B.V. All rights reserved. The main objective of this work is to reduce the volumetric locking pathology at the Element Free Galerkin (EFG) 3D solid-shell meshless method. For this purpose, a subspace analysis is performed in order to extract the linearly independent Incompressible deformation modes that can be reproduced by different background integration cells. Additional deformation modes reproduced by a cell configuration produces more flexibility for the meshless formulation. The Enhanced Assumed Strain (EAS) method is blended with the EFG formulation in such a way that additional (mathematical) variables are included in the formulation increasing its flexibility for nearly Incompressible Materials. Three numerical examples show the effect of the EAS formulation in alleviating the locking pathology for an almost Incompressible Material condition
