Philippe Rouch

The variational theory of complex rays for the calculation of medium‐frequency vibrations

A new approach called the “variational theory of complex rays” (VTCR) is developed for calculating the vibrations of weakly damped elastic structures in the medium‐frequency range. Here, the emphasis is put on the most fundamental aspects. The effective quantities (elastic energy, vibration intensity, etc.) are evaluated after solving a small system of equations which does not derive from a finite element discretization of the structure. Numerical examples related to plates show the appeal and the possibilities of the VTCR.

The variational theory of complex rays: a predictive tool for medium-frequency vibrations

Abstract Today, the main numerical modeling techniques for the analysis of medium-frequency vibrations are all based on finite element or boundary element approaches. In order to represent small-wavelength phenomena in complex structures such as car chassis, satellites or ships, these techniques require a huge number of degrees of freedom (at least seven elements per wavelength are required to represent oscillating solutions). In addition, the solution obtained is highly sensitive to material properties and boundary conditions. The use of high-frequency approaches such as the statistical energy analysis (SEA) or any of its improvements does not appear to be suitable for medium-frequency vibrations: the vibrational behavior becomes too smooth and, in general, the coupling loss factor cannot be calculated in a predictive way. In this paper, a new and multiscale approach to the calculation of the vibrations of elastic structures in the medium-frequency range, called the “variational theory of complex rays”, is presented. The feasibility and effectiveness of the method are demonstrated for structures made of plate assemblies. Several 3D examples are analyzed and compared to the results from a classical finite element approach and from a SEA code. This comparison shows that our method is able to predict the effective quantities at a very low numerical cost.

Extension of the variational theory of complex rays to shells for medium-frequency vibrations

A new approach to the calculation of vibrations of weakly damped elastic structures in the medium-frequency range, called the variational theory of complex rays (VTCR), is being developed. Here, the extension of this theory to shells of relatively small curvature is considered. Numerical examples of structures made of plates and shells demonstrate the capabilities of the VTCR.

A multiscale computational method for medium-frequency vibrations of assemblies of heterogeneous plates

Abstract A new approach, called the “variational theory of complex rays” has been developed in order to calculate the vibrations of slightly damped elastic plates in the medium-frequency range. The resolution of a small system of equations which does not result from a fine spatial discretization of the structure leads to the evaluation of effective quantities (deformation energy, vibration amplitude,…). Here, we extend this approach, which was already validated for assemblies of homogeneous substructures, to the case of heterogeneous substructures.

Inter-lamellar shear resistance confers compressive stiffness in the intervertebral disc: An image-based modelling study on the bovine caudal disc

The intervertebral disc withstands large compressive loads (up to nine times bodyweight in humans) while providing flexibility to the spinal column. At a microstructural level, the outer sheath of the disc (the annulus fibrosus) comprises 12-20 annular layers of alternately crisscrossed collagen fibres embedded in a soft ground matrix. The centre of the disc (the nucleus pulposus) consists of a hydrated gel rich in proteoglycans. The disc is the largest avascular structure in the body and is of much interest biomechanically due to the high societal burden of disc degeneration and back pain. Although the disc has been well characterized at the whole joint scale, it is not clear how the disc tissue microstructure confers its overall mechanical properties. In particular, there have been conflicting reports regarding the level of attachment between adjacent lamellae in the annulus, and the importance of these interfaces to the overall integrity of the disc is unknown. We used a polarized light micrograph of the bovine tail disc in transverse cross-section to develop an image-based finite element model incorporating sliding and separation between layers of the annulus, and subjected the model to axial compressive loading. Validation experiments were also performed on four bovine caudal discs. Interlamellar shear resistance had a strong effect on disc compressive stiffness, with a 40% drop in stiffness when the interface shear resistance was changed from fully bonded to freely sliding. By contrast, interlamellar cohesion had no appreciable effect on overall disc mechanics. We conclude that shear resistance between lamellae confers disc mechanical resistance to compression, and degradation of the interlamellar interface structure may be a precursor to macroscopic disc degeneration.

Intervertebral disc characterization by shear wave elastography: An in vitro preliminary study

Patient-specific numerical simulation of the spine is a useful tool both in clinic and research. While geometrical personalization of the spine is no more an issue, thanks to recent technological advances, non-invasive personalization of soft tissue’s mechanical properties remains a challenge. Ultrasound elastography is a relatively recent measurement technique allowing the evaluation of soft tissue’s elastic modulus through the measurement of shear wave speed. The aim of this study was to determine the feasibility of elastographic measurements in intervertebral disc. An in vitro approach was chosen to test the hypothesis that shear wave speed can be used to evaluate intervertebral disc mechanical properties and to assess measurement repeatability. In total, 11 oxtail intervertebral discs were tested in compression to determine their stiffness and apparent elastic modulus at rest and at 400 N. Elastographic measurements were performed in these two conditions and compared to these mechanical parameters. The protocol was repeated six times to determine elastographic measurement repeatability. Average shear wave speed over all samples was 5.3 ± 1.0 m/s, with a repeatability of 7% at rest and 4.6% at 400 N; stiffness and apparent elastic modulus were 266.3 ± 70.5 N/mm and 5.4 ± 1.1 MPa at rest, respectively, while at 400 N they were 781.0 ± 153.8 N/mm and 13.2 ± 2.4 MPa, respectively. Correlations were found between elastographic measurements and intervertebral disc mechanical properties; these preliminary results are promising for further in vivo application.

Diffusion model to describe osteogenesis within a porous titanium scaffold

In this study, we develop a two-dimensional finite element model, which is derived from an animal experiment and allows simulating osteogenesis within a porous titanium scaffold implanted in ewes hemi-mandible during 12 weeks. The cell activity is described through diffusion equations and regulated by the stress state of the structure. We compare our model to (i) histological observations and (ii) experimental data obtained from a mechanical test done on sacrificed animal. We show that our mechano-biological approach provides consistent numerical results and constitutes a useful tool to predict osteogenesis pattern.

A substructured version of the variational theory of complex rays dedicated to the calculation of assemblies with dissipative joints in the medium-frequency range

Purpose – This paper deals with numerical techniques dedicated to the predictive calculation of complex structures undergoing medium‐frequency vibrations. This field presents challenging difficulties. The first difficulty is the development of an efficient computational method because with the traditional finite element method (FEM), as the frequency increases, it becomes more expensive to control the pollution error. The second difficulty is the availability of sufficiently realistic joint models to take into account damping phenomena because in vibration problems dissipation controls the magnitude of the response directly.Design/methodology/approach – We use the Variational Theory of Complex Rays (VTCR), an approach which effectively avoids the difficulties encountered with traditional FE techniques. Using two‐scale shape functions which verify the dynamic equation and the constitutive relation within each substructure, the VTCR can be viewed as a means of expressing the power balance at the different i...

ANALYSIS OF STRUCTURES WITH STOCHASTIC INTERFACES IN THE MEDIUM-FREQUENCY RANGE

This paper proposes efficient techniques to obtain effective quantities when dealing with complex structures (including stochastic parameters, such as interface parameters) in medium-frequency vibrations. The first ingredient is the use of a dedicated approach — the Variational Theory of Complex Rays (VTCR) — to solve the medium-frequency problem. The VTCR, which uses two-scale shape functions verifying the dynamic equation and the constitutive relation, can be viewed as a means of expressing the power balance at the different interfaces between substructures. The second ingredient is the use of the Polynomial Chaos Expansion (PCE) to calculate the random response. Since the only uncertain parameters are those which appear in the interface equations (which, in this application, are the complex connection stiffness parameters), this approach leads to very low computation costs.

A new method for the evaluation of the end-to-end distance of the knee ligaments and popliteal complex during passive knee flexion

BACKGROUND Accurate knowledge about the length variation of the knee ligaments (ACL, PCL, MCL and LCL) and the popliteal complex during knee flexion/extension is essential for modelling and clinical applications. The aim of the present study is to provide this information by using an original technique able to faithfully reproduce the continuous passive knee flexion-extension kinematics and to reliably identify each ligament/tendon attachment site. METHODS Twelve lower limbs (femur, tibia, fibula, patella) were tested and set in motion (0-120°) using an ad hoc rig. Tibio-femoral kinematics was obtained using an optoelectronic system. A 3D digital model of each bone was obtained using low-dosage stereoradiography. Knee specimens were dissected and the insertion of each ligament and popliteal complex were marked with radio opaque paint. ACL, PCL and MCL were separated into two bundles. Bone epiphyses were CT-scanned to obtain a digital model of each ligament insertion. Bones and attachment site models were registered and the end-to-end distance variation of each ligament/tendon was computed over knee flexion. RESULTS A tibial internal rotation of 18°±4° with respect to the femur was observed. The different bundles of the ACL, MCL and LCL shortened, whereas all bundles of the PCL lengthened. The popliteal complex was found to shorten until 30° of knee flexion and then to lengthen. CONCLUSION The end-to-end distance variation of the knee ligaments and popliteal complex can be estimated during knee flexion using a robust and reliable method based on marking the ligaments/tendon insertions with radiopaque paint. LEVEL OF EVIDENCE Level IV.