Mechanical Modeling

View a selection of illustrative plots on rolling resistance.

In this work we make use of the advantage offered by model order-reduction techniques to identify the properties of non-Newtonian fluids using the vane test. A response surface depending on the parameters of the vane test is constructed to this aim. A power-law fluid is retained to illustrate this process and the vane test is simulated as a function of flow consistency, flow behavior index and rotational velocity. The simulated response surface is then leveraged to identify the non-Newtonian fluid flow parameters from experimental results.

The effective modeling of the flow of fresh concrete materials in settings such as that of the vane test is a challenging process that is the object of ongoing research. Previous works modeled concrete and cement pastes as solids subjected to yielding or as Bingham or power-law fluids, both in two or three dimensions. Of the existing models, those implementing power-law fluids in three dimensions carry the best predictive ability considering the typically heterogeneous composition of concrete suspensions and the relatively complex three-dimensional features of their flows. In this work, we model the vane test in a power-law cement paste using the Proper Generalized Decomposition (PGD). In this framework, the three-dimensional problem is solved as a sequence of 2D × 1D problems, thus alleviating the curse of dimensionality. This choice is supported by experience from previous works using the PGD to simulate Non-Newtonian behavior using iterative resolutions. It is also particularly useful in addressing the inverse problem corresponding to the identification of the material properties of cement pastes from experimental data, as this requires many direct resolutions of the forward problem. The use of the PGD is also appealing because the model parameters can be introduced as extra coordinates of the problem.

Recent works have proposed novel semi-analytical computational methods to model the viscoelastic rolling behavior of objects such as cylinders and spheres, including rolling resistance. This paper introduces and applies an extension of such methods to model the steady-state spinning behavior of a circular viscoelastic coating under the effect of gravity. The proposed modeling technique is wide-ranging in that it can accommodate any linear viscoelastic material characterized by its most general frequency-domain master-curves. The results of the model reveal that the mechanical behavior of the spinning viscoelastic coating is significantly different from that of a similar coating with purely elastic properties. The method can potentially be applied to characterize viscoelastic materials from either laboratory or field measurements. It could also be used to study the behavior of precision measurement devices and propellant grains under gravitational loads, or to model very large scale phenomena such as the tidal locking of celestial bodies.

Article featured in Advances in Engineering: "By modeling the resistance to motion on orthotropic viscoelastic layers, Dr. Gérard-Philippe Zéhil and his co-author have discovered important behavioral features that the scientific and engineering communities were previously unaware of. In a key scientific article contributing to excellence in engineering research, the authors show that the level of dissipation varies with the direction of motion, following a trend depending on speed. The moving object is furthermore subjected to a lateral force which tends to deviate it from its initial course to follow a path governed by velocity. The authors discuss potential impacts of broad significance of their findings, such as on the design of metamaterials and the engineering of systems to achieve motion control and optimize directional energy transfers in various applications."

New two-dimensional and three-dimensional boundary element formulations of compressible viscoelastic layers of arbitrary thickness are presented in this work. The formulations are derived in increasing order of complexity for: (i) compressible isotropic layers, (ii) transversely isotropic layers, and (iii) fully orthotropic layers. It is further shown that existing 2D and 3D models for incompressible isotropic layers may be regarded as particular instances of case (i). The proposed formulations are based on Fourier series and support any linear viscoelastic material model characterized by general frequency-domain master-curves. These approaches result in a compliance matrix for the layer's upper boundary, which includes the effects of steady-state motion. This characterization may be used as a component in various problem settings to generate sequences of high fidelity solutions for varying parameters. The proposed modeling techniques are applied, in combination with appropriate contact solvers, to the rolling resistance of rigid cylinders and spheres on compressible isotropic, transversely isotropic and orthotropic layers. The latter case reveals that the dissipated power varies with the direction of motion, which suggests new ways of optimizing the level of damping in various engineering applications of very high impact. Interesting lateral viscoelastic effects resulting from material asymmetry are unveiled. These phenomena could be harnessed to achieve smooth and `invisible' guides across three-dimensional viscoelastic surfaces, and hence suggest new ways of controlling trajectories, with a broad range of potential applications.

We present a novel three-dimensional boundary-element formulation that fully characterizes the mechanical behavior of the external boundary of a multi-layered viscoelastic coating attached to a hard rotating spherical core. The proposed formulation incorporates both, the viscoelastic, and the inertial effects of the steady-state rolling motion of the sphere, including the Coriolis effect. The proposed formulation is based on Fourier-domain expressions of all mechanical governing equations. It relates two-dimensional Fourier series expansions of surface displacements and stresses, which results in the formation of a compliance matrix for the outer boundary of the deformable coating, discretized into nodes. The computational cost of building such a compliance matrix is optimized, based on configurational similarities and symmetry. The proposed formulation is applied, in combination with a rolling contact solving strategy, to evaluate the viscoelastic rolling friction of a coated sphere on a rigid plane. Steady-state results generated by the proposed model are verified by comparison to those obtained from running dynamic simulations on a three-dimensional finite element model, beyond the transient. A detailed application example includes a verification of convergence and illustrates the dependence of rolling resistance on the applied load, the thickness of the coating, and the rolling velocity.

A three-dimensional boundary element formulation of an incompressible viscoelastic layer of finite thickness is proposed, in a moving frame of reference. The formulation is based on two-dimensional Fourier series expansions of relevant mechanical fields in the continuum of the layer. The linear viscoelastic material is characterized, in the most general way, by its frequency-domain master curves. The presented methodology results in a compliance matrix for the layer's upper boundary, which includes the effects of steady-state motion and can be used in any contact problem-solving strategy. The proposed formulation is used, in combination with a contact solver, to build a full three-dimensional model for the steady-state rolling/sliding resistance incurred by a rigid sphere on the layer. Energy losses include viscoelastic damping and surface friction. The model is tested and its results are found to be consistent with existing solutions in limiting cases. An example is explored and the corresponding results are used to illustrate the influence of different parameters on the rolling resistance. General aspects of previously-described dependences are confirmed.

This paper presents simple, yet robust and efficient algorithms for solving steady-state, frictional, rolling/sliding contact problems, in two and three dimensions. These are alternatives to powerful, well established, but in particular instances, possibly ‘cumbersome’ general-purpose numerical techniques, such as finite-element approaches based on constrained optimization. The cores of the solvers rely on very general principles: (i) resolving motional conflicts, and (ii) eliminating unacceptable surface tractions. The proposed algorithms are formulated in the context of small deformations and applied to the cases of a rigid cylinder and a rigid sphere rolling on a linear viscoelastic layer of finite thickness, in two and three dimensions, respectively. The underlying principles are elucidated, relevant mathematical expressions derived and details given about corresponding implementation techniques. The proposed contact algorithms can be extended to more general settings involving a deformable indenter, material nonlinearities and large deformations.

Modeling approaches yielding rolling resistance estimates for rigid spheres (and cylinders) on viscoelastic layers of finite thicknesses are introduced as lower-cost alternatives to more comprehensive solution-finding strategies. Detailed examples are provided to illustrate the capabilities of the different approaches over their respective ranges of validity.