Cosserat Model Research Articles

On prescribing the number of singular points in a Cosserat-elastic solid

Abstract In a geometrically non-linear Cosserat model for micro-polar elastic solids, we prove that critical points of the Cosserat energy functional with an arbitrary large (finite) number of singularities do exist, whereas Cosserat energy minimizers are known to be locally Hölder continuous. To reach that goal, we first develop a technique to insert dipole pairs of singularities into smooth maps while controlling the amount of Cosserat energy needed to do so. We then use this method to force an arbitrary number of singular points into (weak) Cosserat-elastic solids by prescribing smooth boundary data. The boundary data themselves are given in such a way, that they contain no topological obstruction to regularity. Throughout this paper, we often exploit connections between harmonic maps and Cosserat-elastic solids, so that we are able to adapt and incorporate ideas of R. Hardt and F.-H. Lin for harmonic maps with singularities, as well as of F. Béthuel for dipole pairs of singularities.

Generalized multiscale finite element method for a nonlinear elastic strain-limiting Cosserat model

For nonlinear Cosserat elasticity, we consider multiscale methods in this paper. In particular, we explore the generalized multiscale finite element method (GMsFEM) to solve an isotropic Cosserat problem with strain-limiting property (ensuring bounded linearized strains even under high stresses). Such strain-limiting Cosserat model can find potential applications in solids and biological fibers. However, Cosserat media with naturally rotational degrees of freedom, nonlinear constitutive relations, high contrast, and heterogeneities may produce challenging multiscale characteristics in the solution, and upscaling by multiscale methods is necessary. Therefore, we utilize the offline and residual-based online (adaptive or uniform) GMsFEM in this context while handling the nonlinearity by Picard iteration. Through various two-dimensional experiments (for perforated, composite, and stochastically heterogeneous media with small and big strain-limiting parameters), our numerical results show the approaches' convergence, efficiency, and robustness. In addition, these results demonstrate that such approaches provide good accuracy, the online GMsFEM gives more accurate solutions than the offline one, and the online adaptive strategy has similar accuracy to the uniform one but with fewer degrees of freedom.

Cosserat constitutive theory and one of its higher-order forms: A rediscussion on the mesh dependence problem

When the finite element method (FEM) is adopted for studying strain localization problems, the mesh dependence phenomenon often ensues. The occurrence of mesh dependency will reduce the reliability of FEM simulations, so it is still worth studying. Herein, a constitutive model with decent mesh stability named the multiscale Cosserat (MC) model which contains higher-order rotation variables based on the conventional Cosserat (CC) theory, was introduced. The theory derivation indicates that the MC model has an extra internal length scale vector Dq that can consider the microscopic geometrical characteristics of the simulated material and can easily regress to the conventional Cosserat model when Dq = 0. After revisiting the mesh dependence problem through numerical simulations of plane strain compression tests, the mechanisms and advantages of the CC and MC models in solving the mesh dependence problem were discussed. The analysis demonstrated that the CC theory can alleviate the mesh dependence problem but cannot eliminate it; when the divergence of the computation occurs, due to a stricter accuracy requirement for convergence, the computation result of the MC model tends to stabilize along with the refinement of the elements. The mesh advantage of the MC model is influenced by both the length scales l and Dq. This study can provide new insight into understanding the mesh dependence problem, and the MC model introduced here is a potential model for comprehensively eliminating the influence of mesh dependence problems.

A Coarsened-Shell-Based Cosserat Model for the Simulation of Hybrid Cables

The simulation of elastic slender objects like cables is essential for industrial applications in predicting elastic behaviors and life cycles. The Cosserat model and its variants are the dominant approaches due to their high efficiency and accuracy. However, these assume cables with homogeneous interiors and thus cannot simulate hybrid cables containing different materials. We address this by developing a novel coarsened-shell-based Cosserat (CSC) model. The CSC model constructs a material-aware elastic energy function along the cable’s cross-section to describe the global elastic behavior. The CSC model is specifically developed by carefully leveraging the strengths of three approaches: the Cosserat theory to model slender cables, the Kirchhoff–Love shell theory to model the cable’s cross-sectional energy, and numerical coarsening to reduce the degrees of freedom in the shell simulation via constructing a set of new types of material-aware shape/base functions. This allows the more accurate computation of the local and global deformations of hybrid cables, surpassing the classical Cosserat models in accuracy.

A new multiscale Cosserat model for size effect simulation in granular media

The Cosserat continuum, which contains a length scale factor l reported to link with the material’s grain size, is seemingly reasonable to describe the size effect of granular media. However, the physical meaning of l is still unclear, and its potential in describing the size effect is seldom studied. In this work, the performance of the conventional Cosserat model in capturing the size effect of granular media is firstly assessed. A new multiscale Cosserat constitutive model with an additional higher-order internal length scale vector Dq is then proposed based on the conventional Cosserat theory. The advantage of the proposed multiscale Cosserat model in size effect simulation for granular media is discussed through numerical simulations of centrifuge slope experiments and triaxial compression tests. The analysis results denote that the multiscale Cosserat continua are more sensitive to size effect than the conventional Cosserat continua, and the length scale vector Dq is like a symmetry-breaking factor that can trigger asymmetric deformation in structures; the influences of length scale factor l and Dq are seemingly opposite on stress concentration. This subtly improved constitutive model has the potential to simulate the size effect of a wider variety of materials.

A computational approach to identify the material parameters of the relaxed micromorphic model

We determine the material parameters in the relaxed micromorphic generalized continuum model for a given periodic microstructure in this work. This is achieved through a least squares fitting of the total energy of the relaxed micromorphic homogeneous continuum to the total energy of the fully-resolved heterogeneous microstructure, governed by classical linear elasticity. We avoid establishing exact micro–macro transition relations, as in classical homogenization theory, because defining a representative volume element is not feasible in the absence of scale separation, as such an element does not exist. The relaxed micromorphic model is a generalized continuum that utilizes the Curl of a micro-distortion field instead of its full gradient as in the classical micromorphic theory, leading to several advantages and differences. The most crucial advantage is that it operates between two well-defined scales. These scales are determined by linear elasticity with microscopic and macroscopic elasticity tensors, which respectively bound the stiffness of the relaxed micromorphic continuum from above and below. While the macroscopic elasticity tensor is established a priori through standard periodic first-order homogenization, the microscopic elasticity tensor remains to be determined. Additionally, the characteristic length parameter, associated with curvature measurement, controls the transition between the micro- and macro-scales. Both the microscopic elasticity tensor and the characteristic length parameter are here determined using a computational approach based on the least squares fitting of energies. This process involves the consideration of an adequate number of quadratic deformation modes and different specimen sizes. We conduct a comparative analysis between the least square fitting results of the relaxed micromorphic model, the fitting of a skew-symmetric micro-distortion field (Cosserat-micropolar model), and the fitting of the classical micromorphic model with two different formulations for the curvature; one simplified formulation involving only one single characteristic length and a simplified isotropic curvature with three parameters. The relaxed micromorphic model demonstrates good agreement with the fully-resolved heterogeneous solution after optimizing only four parameters. The “simplified” full micromorphic model, which includes isotropic curvature and involves the optimization of seven parameters, does not achieve superior results, while the Cosserat model exhibits the poorest fitting.

Cosserat-phase-field modeling of grain nucleation in plastically deformed single crystals

Thermomechanical processing of crystalline materials induces microstructural evolution such as grain nucleation and growth. In the numerical simulation of these processes, grain nucleation is generally treated as an additional ad hoc step in which circular or spherical grains are added in regions where a critical dislocation density, stress or strain are reached. In this paper, systematic finite element simulations are performed showing that the Kobayashi–Warren–Carter (KWC) phase field model and its coupling with Cosserat crystal plasticity predict spontaneous nucleation of new grains in single crystals in the presence of lattice orientation/rotation gradients. The numerical analysis of the stability of gradients of lattice rotation and dislocation-based stored energy indicates that a gradient of stored energy alone is not sufficient to trigger grain formation. As an application, the KWC–Cosserat model is used to simulate the torsion and annealing of a copper single crystal bar with a circular cross section. This mechanical loading produces a large, fairly uniform axial rotation gradient which induces nucleation in the form of a stack of cylindrical grains. Plastic strain gradients in cross-sections predicted by the 3D finite element simulation, are not strong enough to compete with the longitudinal nucleation process, as confirmed by experimental observations from the literature.

Green’s functions for the isotropic planar relaxed micromorphic model — Concentrated force and concentrated couple

We derive the Green’s functions (concentrated force and couple in an infinite space) for the isotropic planar relaxed micromorphic model. Since the relaxed micromorphic model particularizes into the micro-stretch, Cosserat (micropolar), couple-stress, and linear elasticity model for certain choices of material parameters, we recover the fundamental solutions in all these cases.

Closed-loop Shape Control of Deformable Linear Objects Based on Cosserat Model

Closed-loop Shape Control of Deformable Linear Objects Based on Cosserat Model

Cosserat Rod-Based Dynamic Modeling of a Hybrid-Actuated Soft Robot for Robot-Assisted Cardiac Ablation

Soft robotics has emerged as a promising field due to the unique characteristics offered by compliant and flexible structures. Overcoming the challenge of precise position control is crucial in the development of such systems that require accurate modeling of soft robots. In response, a hybrid-actuated soft robot employing both air pressure and tendons was proposed, modeled, and validated using the dynamic Cosserat rod theory. This approach comprehensively addresses various aspects of deformation, including bending, torsion, shear, and extension. The designed robot was intended for robot-assisted cardiac ablation, a minimally invasive procedure that is used to treat cardiac arrhythmias. Within the framework of the Cosserat model, dynamic equations were discretized over time, and ordinary differential equations (ODEs) were solved at each time step. These equations of motion facilitated the prediction of the robot’s response to different control inputs, such as the air pressure and tension applied to the tendons. Experimental studies were conducted on a physical prototype to examine the accuracy of the model. The experiments covered a tension range of 0 to 3 N for each tendon and an air pressure range of 0 to 40 kPa for the central chamber. The results confirmed the accuracy of the model, demonstrating that the dynamic equations successfully predicted the robot’s motion in response to diverse control inputs.

A Soft Variable Stiffness Hand With an Integrated Actuating–Cooling System

In this paper, a new soft finger is presented to achieve a fast variable stiffness mechanism using polycaprolactone based on an integrated actuating–cooling system. First, the variable stiffness finger demonstrates a dramatic 15-fold stiffness variation from <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$10.2 \times {10}^4{\bm{\ }}$</tex-math></inline-formula> to 0.67 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$ \times {10}^4$</tex-math></inline-formula> Nmm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> when the temperature is changed in a range from 20 °C to 60 °C. The cycle time of the variable stiffness based on the integrated actuating–cooling hydraulic system is approximately 505 s, which is much shorter than that of 3870 s based on the pneumatic method. Second, due to the varied weight of water in the chamber, Cosserat rod theory is used to model the bending shape of the soft finger instead of the constant curvature model. A feasible method to control the position and stiffness simultaneously based on the Cosserat model is presented. Finally, a three-fingered soft hand is fabricated based on the variable stiffness and self-locking method, which can hold objects up to 3560 g successfully, which is more than 5.2 times its own weight of 680 g.

A comprehensive Cosserat rod drill-string model for arbitrary well geometry that includes the dynamics of the cutting and lateral contact

In this paper, a comprehensive continuous Cosserat rod model is developed to study the dynamics of a drill-string. The model presents the following features: (1) it can simulate the 3-D dynamics of the system, including lateral, axial and torsional motion; (2) it includes damping effects by adopting a Kelvin–Voigt material; (3) it uses a velocity-independent bit-rock interaction formulation in which the forces and torques at the bit are obtained by considering the dynamics of the cutting blades and the evolution of the soil profile, which is simulated with the help of an advection equation; (4) lateral contact is also included. In this paper, two application cases are analysed. The first one deals with a vertical borehole. It includes four simulations that are used to further investigate the hypothesis that low-dimensional lumped formulations are not able to capture the dynamics of a drill-string and, while doing so, evaluate the effect of the damping in the system’s response. The predictions are compared with those given by two other models: the first one considers a 2-degrees-of-freedom (DOF) formulation, with 1-axial and 1-torsional DOF; the second one is a semi-discrete approach, with 1 DOF for the axial dynamics and a continuous formulation for the torsional one. Divergences between the predictions are found, associated with the presence of higher frequencies in the signals obtained with the Cosserat model, which reassures the hypothesis that the new continuous approach can capture aspects of the dynamics that cannot be modelled with low-dimensional representations. The efficiency of the drilling is also assessed for the simulations obtained with the new model. The highest performance was observed in those simulations where only axial oscillations occurred, while a performance drop was seen when those vibrations were accompanied by torsional stick–slip, suggesting that an optimum damping value associated with the best drilling performance could be found, were damping controllable. Finally, the second application case is used to illustrate the full potential of the new model to tackle the 3-D lateral, axial, and torsional dynamics in a curved borehole.

Piecewise Linear Strain Cosserat Model for Soft Slender Manipulator

Recently soft robotics has rapidly become a novel and promising area of research with many designs and applications due to their flexible and compliant structure. However, it is more difficult to derive the nonlinear dynamic model of such soft robots. The differential kinematics and dynamics of the soft manipulator can be formulated as a set of highly nonlinear partial differential equations (PDEs) via the classic Cosserat rod theory. In this work, we propose a discrete modeling technique named piecewise linear strain (PLS) to solve the PDEs of Cosserat-based models, based on which the associated analytic models are deduced. To validate the accuracy of the proposed Cosserat model, the static model of the conical cantilever rod under gravity as a simple example is simulated by using different discretization methods. Results indicate that PLS Cosserat model is comparable to the mechanical deformation behavior of a real-world soft manipulator. Finally, a parameters identification scheme for this model is established, and the simulation as well as experimental validation demonstrate that using this method can identify the model physical parameters with high accuracy.

A Geometrically Exact Assumed Strain Modes Approach for the Geometrico- and Kinemato-Static Modelings of Continuum Parallel Robots

There is a growing interest on the study of continuum parallel robots (CPRs) due to their higher stiffness and better dynamics capacities than serial continuum robots (SCRs). Several works have focused on the computation of their geometrico- and kinemato-static models that can be sorted into two main categories. Models based on the continuous Cosserat equations are very accurate but assessing elastic stability with them is tricky, and discretized models allow easily checking the elastic stability, but they require a large number of elastic variables to be accurate. In this article, we extend an approach based on assumed strain modes developed for the dynamics of SCRs to the statics of CPRs. This method is able to predict the robot configuration with an excellent accuracy with a very limited number of elastic variables, contrary to other discretization methods. The method is also more than 100 times faster than finite differences for a better prediction accuracy. Finally, it is possible to assess the robot elastic stability by only checking the Hessian of the potential energy as for any discretization method, thus making the analysis of this property simpler than for the continuous Cosserat model. All results are validated through simulations on two case studies.

Shear vibration modes of granular structures: Continuous and discrete approaches

AbstractThis work is dedicated to the study of shear vibration modes of one‐dimensional granular microstructured beams using a discrete Cosserat model. The dynamic response of the one‐dimensional granular beam (discrete beam) is investigated for various boundary conditions, through the analytical resolution of an exact discrete eigenvalue problem. For an infinite number of grains, the dynamic behavior of the discrete problem converges towards a Bresse–Timoshenko continuum beam model. First, the effective modeling parameters, the constitutive equations and the governing equations of motion are derived in a difference form. For simply supported granular beams, the appearance of pure shear modes of vibration is highlighted (vibration modes without deflection), a phenomenon which is already known in the free vibration of a continuous Bresse–Timoshenko beam problem. The eigenfrequencies of these pure shear modes for the discrete granular beam coincide with the critical frequencies of the discrete system. These pure shear modes may appear for granular beams composed of few grains, which give a physical support of a phenomenon well studied in the literature from a continuum beam model. The eigenfrequencies of the discrete granular beam are also calculated for various boundary conditions, including clamped, hinge and free end boundary conditions. Based on the discrete element method (DEM), several numerical simulations of dynamic tests in some representative examples have been performed. The responses of the numerical approach are accurately compared against the exact results of the analytical solutions based on the difference eigenvalue problem, which illustrates the relevance of the proposed approach.

Mechanics of fiber-reinforced soft manipulators based on inhomogeneous Cosserat rod theory

In this study, an inhomogeneous Cosserat rod theory is introduced and compared to the conventional homogeneous rod for modeling soft manipulators. The inhomogeneity is addressed by considering the pressure actuation as part of the rod’s constitutive laws, resulting in shifting the neutral axis. This shift is investigated for a soft manipulator with three parallel fiber-reinforced actuators. Furthermore, a fiber-reinforced actuator is modeled using nonlinear continuum mechanics to extract the effect of radial pressure on axial deformation and is combined with Cosserat model. Finally, several numerical methods are employed to solve the proposed model and validated by a series of experiments.

Higher-order and Weil generalization of Grassmannian

We give a simple mechanical motivation for a generalization of the classical Grassmannian considered as a space of m-dimensional linear subspaces of ℝk to higher-orders cases. Our efforts are prolongated to the Weil functor theory, motivated by non-holonomic and semi-holonomic jets, applicable in the description of the Cosserat model.

ON THE EQUILIBRIUM EQUATIONS OF LINEAR 6-PARAMETER ELASTIC SHELLS

We consider the linearized theory of 6-parameter elastic shells with general anisotropy. We derive the equilibrium equations from the virtual power statement and formulate the corresponding variational problem in the suitable functional framework. Then, using a Korn-type inequality for the linearized strain measures we prove the existence and uniqueness of weak solutions. Finally, we show that our general theo­rem can be applied to obtain existence results in the case of isotropic elastic shells. We illustrate this procedure by investigating three differ­ent linear shell models established previously in the literature, namely the simplified isotropic 6-parameter shell, the Cosserat isotropic model, and the higher-order 6-parameter Cosserat model.

Convexity and Quasiconvexity in a Cosserat Model for Fiber-Reinforced Elastic Solids

Convexity and Quasiconvexity in a Cosserat Model for Fiber-Reinforced Elastic Solids

Computer-assisted assembly process planning for the installation of flexible cables modeled according to a viscoelastic Cosserat rod model

Available computer-aided assembly process planning methods fail to provide the necessary level of accuracy required by modern complex planning processes owing to the neglect of the internal friction arising in the process of cable movement and large cable deformation conditions. The present study addresses this issue by developing a computer-aided assembly process planning method that inherently considers the tensile, bending, and torsional deformations of cables based on a dynamic viscoelastic Cosserat model describing cable motion. The simulation module of the virtual cable assembly process integrates and optimizes an efficient and robust algorithm for rapidly solving the dynamic cable model based on a semi-implicit Euler method. The performance of the proposed method is verified by demonstrating the rationality and accuracy of the assembly path planning and multi-cable assembly sequence obtained by the virtual assembly simulation module for a modeled prototype of a complex electromechanical product produced in the aerospace industry.