Showing papers on "Transverse isotropy published in 2007"

A polyconvex hyperelastic model for fiber-reinforced materials in application to soft tissues

Abstract: In this paper a generalized anisotropic hyperelastic constitutive model for fiber-reinforced materials is proposed. Collagen fiber alignment in biological tissues is taken into account by means of structural tensors, where orthotropic and transversely isotropic material symmetries appear as special cases. The model is capable to describe the anisotropic stress response of soft tissues at large strains and is applied for example to different types of arteries. The proposed strain energy function is polyconvex and coercive. This guarantees the existence of a global minimizer of the total elastic energy, which is important in the context of a boundary value problem.

Finite deformations of fibre-reinforced elastic solids with fibre bending stiffness

Abstract: In the conventional theory of finite deformations of fibre-reinforced elastic solids it is assumed that the strain-energy is an isotropic invariant function of the deformation and a unit vector A that defines the fibre direction and is convected with the material. This leads to a constitutive equation that involves no natural length. To incorporate fibre bending stiffness into a continuum theory, we make the more general assumption that the strain-energy depends on deformation, fibre direction, and the gradients of the fibre direction in the deformed configuration. The resulting extended theory requires, in general, a non-symmetric stress and the couple-stress. The constitutive equations for stress and couple-stress are formulated in a general way, and specialized to the case in which dependence on the fibre direction gradients is restricted to dependence on their directional derivatives in the fibre direction. This is further specialized to the case of plane strain, and finite pure bending of a thick plate is solved as an example. We also formulate and develop the linearized theory in which the stress and couple-stress are linear functions of the first and second spacial derivatives of the displacement. In this case for the symmetric part of the stress we recover the standard equations of transversely isotropic linear elasticity, with five elastic moduli, and find that, in the most general case, a further seven moduli are required to characterize the couple-stress.

Shale dynamic properties and anisotropy under triaxial loading: experimental and theoretical investigations

Abstract: This paper is concerned with the experimental identification of the whole dynamic elastic stiffness tensor of a transversely isotropic clayrock from a single cylindrical sample under loading. Measurement of elastic wave velocities (pulse at 1 MHz), obtained under macroscopically undrained triaxial loading conditions are provided. Further macroscopic (laboratory scale) interpretation of the velocity measurements is performed in terms of (i) dynamic elastic parameters; and (ii) elastic anisotropy. Experiments were performed on a Callovo-Oxfordian shale, Jurassic in age, recovered from a depth of 613 m in the eastern part of Paris basin in France. Moreover, a physically-based micromechanical model is developed in order to quantify the damaged state of the shale under loading through macroscopic measurements. This model allows for the identification of the pertinent parameters for a general transversely isotropic orientational distribution of microcracks, superimposed on the intrinsic transverse isotropy of the rock. It is directly inspired from experimental observations and measurements. At this stage, second- and fourth-rank tensors αij and βijkl are identified as proper damage parameters. However, they still need to be explicited in terms of micromechanical parameters for the complex case of anisotropy. An illustration of the protocole of this microstructural data recovery is provided in the simpler case of isotropy. This microstructural insight includes cavities geometry, orientation and fluid-content.

Transversely isotropic properties of porcine liver tissue: experiments and constitutive modelling.

Abstract: Knowledge of the biomechanical properties of soft tissue, such as liver, is important in modelling computer aided surgical procedures. Liver tissue does not bear mechanical loads, and, in numerical simulation research, is typically assumed to be isotropic. Nevertheless, a typical biological soft tissue is anisotropic. In vitro uniaxial tension and compression experiments were conducted on porcine cylindrical and cubical liver tissue samples respectively assuming a simplistic architecture of liver tissue with its constituent lobule and connective tissues components. With the primary axis perpendicular to the cross sectional surface of samples, the tissue is stiffer with tensile or compressive force in the axial direction compared to that of the transverse direction. At 20% strain, about twice as much force is required to elongate a longitudinal tissue sample than that of a transverse sample. Results of the study suggest that liver tissue is transversely isotropic. A combined strain energy based constitutive equation for transversely isotropic material is proposed. The improved capability of this equation to model the experimental data compared to its previously disclosed isotropic version suggests that the assumption on the fourth invariant in the constitutive equation is probably correct and that anisotropy properties of liver tissue should be considered in surgical simulation.

Elastodynamic Potential Method for Transversely Isotropic Solid

Abstract: A theoretical formulation is presented for the determination of the displacements, strains, and stresses in a three-dimensional transversely isotropic linearly elastic medium. By means of a complete representation using two displacement potentials, it is shown that the governing equations of motion for this class of problems can be uncoupled into a fourth-order and a second-order partial differential equation in terms of the spatial and time coordinate under general conditions. Compatible with Fourier expansions and Hankel transforms in a cylindrical coordinate system, the formulation includes a complete set of transformed displacement-potential, strain-potential, and stress-potential relations that can be useful in a variety of elastodynamic as well as elastostatic problems. As an illustration of the application of the method, the solution for a half-space under the action of arbitrarily distributed, time-harmonic surface traction is derived, including its specialization to uniform patch loads and point forces. To confirm the accuracy of the numerical evaluation of the integrals involved, numerical results are also included for cases of different degree of the material anisotropy, frequency of excitation, and compared with existing solutions.

A continuum model for remodeling in living structures

Abstract: A new remodeling theory accounting for mechanically driven collagen fiber reorientation in cardiovascular tissues is proposed. The constitutive equations for the living tissues are motivated by phenomenologically based microstructural considerations on the collagen fiber level. Homogenization from this molecular microscale to the macroscale of the cardiovascular tissue is performed via the concept of chain network models. In contrast to purely invariant-based macroscopic approaches, the present approach is thus governed by a limited set of physically motivated material parameters. Its particular feature is the underlying orthotropic unit cell which inherently incorporates transverse isotropy and standard isotropy as special cases. To account for mechanically induced remodeling, the unit cell dimensions are postulated to change gradually in response to mechanical loading. From an algorithmic point of view, rather than updating vector-valued microstructural directions, as in previously suggested models, we update the scalar-valued dimensions of this orthotropic unit cell with respect to the positive eigenvalues of a tensorial driving force. This update is straightforward, experiences no singularities and leads to a stable and robust remodeling algorithm. Embedded in a finite element framework, the algorithm is applied to simulate the uniaxial loading of a cylindrical tendon and the complex multiaxial loading situation in a model artery. After investigating different material and spatial stress and strain measures as potential driving forces, we conclude that the Cauchy stress, i.e., the true stress acting on the deformed configuration, seems to be a reasonable candidate to drive the remodeling process.

The Eshelby Tensors in a Finite Spherical Domain—Part I: Theoretical Formulations

Abstract: This work is concerned with the precise characterization of the elastic fields due to a spherical inclusion embedded within a spherical representative volume element (RVE) The RVE is considered having finite size, with either a prescribed uniform displacement or a prescribed uniform traction boundary condition Based on symmetry and group theoretic arguments, we identify that the Eshelby tensor for a spherical inclusion admits a unique decomposition, which we coin the “radial transversely isotropic tensor” Based on this notion, a novel solution procedure is presented to solve the resulting Fredholm type integral equations By using this technique, exact and closed form solutions have been obtained for the elastic disturbance fields In the solution two new tensors appear, which are termed the Dirichlet‐Eshelby tensor and the Neumann‐Eshelby tensor In contrast to the classical Eshelby tensor they both are position dependent and contain information about the boundary condition of the RVE as well as the volume fraction of the inclusion The new finite Eshelby tensors have far-reaching consequences in applications such as nanotechnology, homogenization theory of composite materials, and defects mechanics DOI: 101115/12711227

Bio-inspired mechanics of reversible adhesion : Orientation-dependent adhesion strength for non-slipping adhesive contact with transversely isotropic elastic materials

Abstract: Geckos and many insects have evolved elastically anisotropic adhesive tissues with hierarchical structures that allow these animals not only to adhere robustly to rough surfaces but also to detach easily upon movement. In order to improve our understanding of the role of elastic anisotropy in reversible adhesion, here we extend the classical JKR model of adhesive contact mechanics to anisotropic materials. In particular, we consider the plane strain problem of a rigid cylinder in non-slipping adhesive contact with a transversely isotropic elastic half space with the axis of symmetry oriented at an angle inclined to the surface. The cylinder is then subjected to an arbitrarily oriented pulling force. The critical force and contact width at pull-off are calculated as a function of the pulling angle. The analysis shows that elastic anisotropy leads to an orientation-dependent adhesion strength which can vary strongly with the direction of pulling. This study may suggest possible mechanisms by which reversible adhesion devices can be designed for engineering applications.

Elastic moduli of anisotropic clay

Abstract: Claymineralsareimportantcomponentsinshales,controlling their elastic properties and anisotropy. The elasticity of crystalline clay minerals differs significantly from that of clay in situ because of the ability of clay particles to bind water. In the majority of published works, only isotropic moduli for in situ clays are reported. However, anisotropy is inherent in the clay elasticity. We develop an inversion technique for determination of thestiffnesstensorofinsituclayfromtheshale’sstiffnesstensor. As an example, we obtain the stiffness tensor of a “water-clay” composite from the data on the water-saturated Greenhorn shale sample, whose clay composition consists of almost equal amounts of illite and smectite and comparable amounts of kaolinite and chlorite. The stiffness tensor of the water-clay composite is found for the Greenhorn shale with step-by-step inversion based upon an effective medium theory. The inversion uses a nonlinear optimization technique with bounds imposed on the estimated parameters. In the inversion, we apply different approaches of the effective medium theory using a published method referred to here as the generalized singular approximation GSA. The GSA method makes it possible to take into account the microstructure of shales. The resulting elasticity constants of the anisotropic transversely isotropic in situ clay composite are C11 = 23.7, C33 = 8.5, C44 = 0.8, C66 = 5.7, and C13 = 3.1 in GPa; and the density equals 2.17 g/cm 3 . The Thomsen parameters for the clay composite are = 0.89, = 3.10, and = 0.34. The elasticity constants found for this clay composite can be used in the theoretical analysis of shales that have a similar composition of clay but with different mineral compositions. The inversion technique developed can be used for general shale water-clay composites when the mineral composition and orientation of the clay platelets are known.

Second order homogenization of the elastic wave equation for non-periodic layered media

Abstract: In many cases, in the seismic wave propagation modelling context, scales much smaller than the minimum wavelength are present in the earth model in which we wish to compute seismograms. For many numerical methods these small scales are a challenge leading to high numerical cost. The purpose of this paper is to understand and to build the effective medium and equations allowing to average the small scales of the original medium without losing the accuracy of the wavefield computation. In this paper, only the simple layered medium case is studied, leaving the general 3-D medium case for future work. To obtain such an effective medium and equations, we use high order two scale homogenization applied to the wave equation for layered media with rapid variation of elastic properties and density compared to the smallest wavelength of the wavefield. We show that the order 0 homogenization gives the result that was obtained by Backus in 1962. Order 0 homogenized models are transversely isotropic even though the original model is isotropic. It appears that order 0 is not enough to obtain surface waves with correct group and phase velocities and higher order homogenization terms up to two are often required. In many cases, the order one and two simply require to correct the boundary conditions of the wave equation to obtain an accurate solution, even for surface waves. We show how to extend the theory from the periodic case to the non-periodic case. Examples in periodic and non-periodic media are shown. The accuracy of the results obtained by homogenization is checked against the normal mode solution computed in the original medium and shows good agreement.

Compression moulding of SMC: In situ experiments, modelling and simulation

Abstract: Compression mouldings of commercial SMC were performed with an instrumented industrial press under various process conditions. Results underline the influence of process parameters such as the initial SMC temperature, the axial punch velocity and the geometry of the mould on local normal stress levels. They also show negligible fibre-bundle segregation in the principal plane of the moulded parts. Thereby, a one-phase plug flow shell model is proposed as a direct extension of the plug flow model proposed by M.R. Barone and D.A. Caulk [J Appl Mech 53(191):1986;361–70]. In the present approach, the SMC is considered as a power-law viscous medium exhibiting transverse isotropy. The shell model is implemented into a finite element code especially developed for the simulation of compression moulding of composite materials. Simulation and experimental results are compared, emphasizing the role of the SMC rheology on the overall recorded stress levels. Despite the simplicity of the model, rather good comparisons are obtained.

Physical modeling and analysis of P-wave attenuation anisotropy in transversely isotropic media

Abstract: Anisotropic attenuation can provide sensitive attributes for fracture detection and lithology discrimination. This paper analyzes measurements of the P-wave attenuation coefficient in a transversely isotropic sample made of phenolic material. Using the spectral-ratio method, we estimate the group effective attenuation coefficient of P-waves transmitted through the sample for a wide range of propagation angles from 0° to 90° with the symmetry axis. Correction for the difference between the group and phase angles and for the angular velocity variation help us to obtain the normalized phase attenuation coefficient A governed by the Thomsenstyle attenuation-anisotropy parameters Q and Q. Whereas the symmetry axis of the angle-dependent coefficient A practically coincides with that of the velocity function, the magnitude of the attenuation anisotropy far exceeds that of the velocity anisotropy. The quality factor Q increases more than tenfold from the symmetry axisslow direction to the isotropy planefast direction. Inversion of the coefficient A using the Christoffel equation yields large negative values of the parameters Q and Q. The robustness of our results critically depends on several factors, such as the availability of an accurate anisotropic velocity model and adequacy of the homogeneous concept of wave propagation, as well as the choice of the frequency band. The methodology discussed here can be extended to field measurements of anisotropic attenuation needed for AVO amplitude-variation-with-offset analysis, amplitude-preserving migration, and seismic fracture detection.

Noninvasive Determination of Ligament Strain with Deformable Image Registration

Abstract: Ligament function and propensity for injury are directly related to regional stresses and strains. However, noninvasive techniques for measurement of strain are currently limited. This study validated the use of Hyperelastic Warping, a deformable image registration technique, for noninvasive strain measurement in the human medial collateral ligament using direct comparisons with optical measurements. Hyperelastic Warping determines the deformation map that aligns consecutive images of a deforming material, allowing calculation of strain. Diffeomorphic deformations are ensured by representing the deformable image as a hyperelastic material. Ten cadaveric knees were subjected to six loading scenarios each. Tissue deformation was documented with magnetic resonance imaging (MRI) and video-based experimental measurements. MRI datasets were analyzed using Hyperelastic Warping, representing the medial collateral ligament (MCL) with a hexahedral finite element (FE) model projected to a manually segmented ligament surface. The material behavior was transversely isotropic hyperelastic. Warping predictions of fiber stretch were strongly correlated with experimentally measured strains (R (2) = 0.81). Both sets of measurements were in agreement with previous ex vivo studies. Warping predictions of fiber stretch were insensitive to bulk:shear modulus ratio, fiber stiffness, and shear modulus in the range of +2.5SD to -1.0SD. Correlations degraded when the shear modulus was decreased to 2.5SD below the mean (R (2) = 0.56), and when an isotropic constitutive model was substituted for the transversely isotropic model (R (2) = 0.65). MCL strains in the transitional region near the joint line, where the material behavior and material symmetry are more complex, showed the most sensitivity to changes in shear modulus. These results demonstrate that Hyperelastic Warping requires the use of a constitutive model that reflects the material symmetry, but not subject-specific material properties for accurate strain predictions for this application. Hyperelastic Warping represents a powerful technique for noninvasive strain measurement of musculoskeletal tissues and has many advantages over other image-based strain measurement techniques.

Large deformation response of a hyperelastic fibre reinforced composite: Theoretical model and numerical validation

Abstract: The mechanical behaviour of an incompressible neo-Hookean material directionally reinforced with a generalised neo-Hookean fibre is examined in the finite deformation regime. To consider the interaction between the fibre and the matrix, we use a composite model for this transversely isotropic material based on a multiplicative decomposition of deformation, which decouples the uniaxial deformation along the fibre direction from the remaining shear deformation. The model is then verified numerically by a unit cell model with periodic boundary conditions. The strain energy stored in the unit cell is compared with the energy predicted by the proposed theoretical model and excellent agreement is reported.

Assessment of Anisotropic Tissue Elasticity of Cortical Bone from High-Resolution, Angular Acoustic Measurements

Abstract: Assessment of anisotropic elastic properties at the tissue level is still one of the major challenges in bone research. In previous studies, bone sections were cut in different directions relative to a principle axis of symmetry. This causes a high preparation and measurement effort. We have developed a new acoustic scanning procedure that allows one to measure the angular dependence of the acoustic impedance of cylindrically shaped samples (diameter: 4.4 mm) with a single measurement. Our scanning acoustic microscope was equipped with a rotational stage, and a scanning procedure was developed that measures the surface reflection of the rotating cylinder. It was shown in a previous study that the acoustic impedance derived from the reflection coefficient is highly correlated with the elastic coefficient in the probing direction. From the angular reflection, the independent elastic coefficients were derived using assumptions of transverse isotropy and continuum micromechanical model constraints. This method was applied to the inspection of human femoral bone samples. Four cylinders were prepared from the anterior, posterior, medial, and lateral regions. The measurements were performed with a 50 MHz transducer, providing a lateral resolution of 23 mum. Remarkable structural and elastic variations were observed between the four samples. The means and standard deviations of the derived elastic coefficients were: c33 = 29.9 plusmn 5.0 GPa, c11 - 21.9 plusmn 2.1 GPa, C12 = 9.2 plusmn 1.5 GPa, c13 = 9.7 plusmn 1.6 GPa, and c44 = 6.7 plusmn 1.2 GPa. The results demonstrate that microstructural and anisotropic elastic tissue parameters can be assessed by ultrasound in very small bone samples.

Plane-wave attenuation anisotropy in orthorhombic media

Abstract: Orthorhombic models are often used in the interpretation of azimuthally varying seismic signatures recorded over fractured reservoirs. Here, we develop an analytic framework for describing the attenuation coefficients in orthorhombic media with orthorhombic attenuationi.e., the symmetry of both the real and imaginary parts of the stiffness tensor is identical under the assumption of homogeneous wave propagation. The analogous form of the Christoffel equation in the symmetry planes of orthorhombic and VTI transversely isotropic with a vertical symmetry axis media helps to obtain the symmetry-plane attenuation coefficients by adapting the existing VTI equations. To take full advantage of this equivalence with transverse isotropy, we introduce a parameter set similar to the VTI attenuation-anisotropy parametersQ,Q, andQ. This notation, based on the same principle as Tsvankin’s velocity-anisotropy parameters for orthorhombic media, leads to concise linearized equations for the symmetry-plane attenuation coefficients of all three modes P, S1, and S2.The attenuation-anisotropy parameters also allow us to simplify the P-wave attenuation coefficient AP outside the symmetry planes under the assumptions of small attenuation and weak velocity and attenuation anisotropy. The approximate coefficient AP has the same form as the linearized P-wave phase-velocity function, with the velocity parameters

Anisotropic reverse-time migration for tilted TI media

Abstract: Seismic anisotropy in dipping shales results in imaging and positioning problems for underlying structures. We develop an anisotropic reverse-time depth migration approach for P-wave and SV-wave seismic data in transversely isotropic (TI) media with a tilted axis of symmetry normal to bedding. Based on an accurate phase velocity formula and dispersion relationships for weak anisotropy, we derive the wave equation for P-wave and SV-wave propagation in tilted transversely isotropic (TTI) media. The accuracy of the P-wave equation and the SV-wave equation is analyzed and compared with other acoustic wave equations for TTI media. Using this analysis and the pseudo-spectral method, we apply reverse-time migration to numerical and physical-model data. According to the comparison between the isotropic and anisotropic migration results, the anisotropic reverse-time depth migration offers significant improvements in positioning and reflector continuity over those obtained using isotropic algorithms.

A horizontal shear surface wave in magnetoelectroelastic materials

Abstract: The constitutive equations that involve the interaction of elastic, electric and magnetic fields in magnetoelectroelastic media relate stresses, strains, and electric and magnetic fields. The solution of the governing equations under appropriate boundary conditions describes a wave that decays exponentially from the surface and is a surface wave. The existence of this surface wave is investigated in transversely isotropic magnetoelectroelastic materials. It is noted that it does not arise in an elastic homogeneous material of the same elastic symmetry as the magnetoelectroelastic material. An analysis is carried out on the free surface of a magnetoelectroelastic half-space. Expressions for the velocity of propagation of the surface waves are obtained. The theoretical results can be used in the design of high-frequency surface wave devices.