Micro-mechanical modelling of granular material. Part 1: Derivation of a second-gradient micro-polar constitutive theory

Summary This contribution is one in a series of two papers. In the current paper a constitutive law is developed that includes the micro-structural effects by particle displacement as well as particle rotation. Both degrees of freedom can be related to corresponding macroscopic kinematic continuum variables, where the resulting gradients of displacement are selected up to the fourth-order and the gradients of rotation up to the third order. The elastic micro-structural properties for an individual particle are used to derive the macro-level behavior for a fabric of equal-sized spherical particles, leading to a second-gradient micro-polar formulation. In this model, all coefficients are expressed in terms of particle stiffness and particle structure. It is shown that the second-gradient micro-polar model can be reduced to simpler forms, such as the classic linear elastic model, the second-gradient model and the Cosserat model. In the accompanying paper these reduced forms are treated in more detail by analyzing the corresponding dispersion relations for plane body wave propagation.     

Goal‐oriented adaptivity for linear elastic micromorphic continua based on primal and adjoint consistency analysis

Microscopic considerations are drawing increasing attention for modern simulation techniques. Micromorphic continuum theories, considering micro degrees of freedom, are usually adopted for simulation of localization effects like shear bands. The increased number of degrees of freedom clearly motivates an application of adaptive methods. In this work, the adaptive FEM is tailored for micromorphic elasticity. The proposed adaptive procedure is driven by a goal‐oriented a posteriori error estimator based on duality techniques. For efficient computation of the dual solution, a patch‐based recovery technique is proposed and compared to a reference approach. In order to theoretically ensure optimal convergence order of the proposed adaptive procedure, adjoint consistency of the FE‐discretized solution for the linear elastic micromorphic continua is shown. For illustration, numerical examples are provided. Copyright © 2017 John Wiley & Sons, Ltd.     

Theory of micropolar gyroelastic continua

This paper is devoted to the dynamic modeling of micropolar gyroelastic continua and explores some of the modeling and analysis issues related to them. It can be considered as an extension of the previous studies on equivalent continuum modeling of truss structures with or without angular momentum devices. Assuming unrestricted or large attitude changes for the axes of the gyros and utilizing the micropolar theory of elasticity, the energy expressions and equations of motion for undamped micropolar gyroelastic continua are derived. Whereas the micropolar gyroelastic continuum model with extra coefficients and degrees of freedom is primarily developed to account for the asymmetric stress–strain analysis in the gyroelastic continua, it also proves to be beneficial for a more comprehensive representation of the actual gyroelastic structure. The dynamic equations of the general gyroelastic continua are reduced to the case of one-dimensional gyroelastic beams. Simplified micropolar beam torsion and bending theories are used to derive the governing dynamic equations of micropolar gyroelastic beams from Hamilton’s principle. A finite element model corresponding to the micropolar gyrobeams is built in MATLAB\({^{\circledR}}\) and is used in numerical examples to study the spectral and modal behavior of simply supported micropolar gyroelastic beams.     

空间4R球面机构作为转动副的3RPS并联机构运动学分析

目的设计一种新型3RsPS并联机构,使得机构可以输出3个转动自由度与3个移动自由度耦合的运动。方法根据机构的几何关系,在4R空间机构连杆运动学分析的基础上,建立该机构的运动学方程,使用消元法对其运动学正、逆解进行了求解。使用空间4R机构替换3RPS并联机构的转动副,使得该机构在初始位形下的结构,与3RPS并联机构等效。结果使用Matlab计算得到了运动学方程消元后方程的最高次数,从而得到了运动学正解最多的个数。该机构具有3个自由度,3个移动输出和3个转动输出耦合,机构的运动学正解最多有16个,这与3RPS最多拥有的正解数目一致。结论得到了一种新型的3RsPS机构,该机构的特征为RS副的转动轴线是在空间变化的,该机构的输出为三维移动伴随三维转动,具有此类运动模式的并联机构在不规则形状的产品分拣筛选时,具有一定应用前景。     

On the geometry and dynamics of floating four-bar linkages

In this paper, we investigate the kinematics and dynamics of floating, planar four-bar linkages. The geometry of configuration space is analyzed in part through the classical theory of mechanisms due to Grashof. The techniques of symplectic and Poisson reduction are used to understand the dynamics of the system. Bifurcations of relative equilibria for linkages admitting symmetric shapes are studied using the techniques of singularity theory. The problem of reconstruction of the full dynamics and its relation to geometric phases is discussed through some examples. This research reveals that a coupled mechanical system with kinematic loops possesses richer and more complicated dynamical aspects in comparison with systems which have the same number of degrees of freedom, but no kinematic loops     

The Asymptotic Expansion Load Decomposition elastoplastic beam model

A new higher‐order elastoplastic beam model is derived and implemented in this paper. The reduced kinematic approximation is based on a higher‐order elastic beam model using the asymptotic expansion method. This model introduces new degrees of freedom associated to arbitrary loads as well as eigenstrains applied to the beam. In order to capture the effect of plasticity on the structure, the present elastoplastic model considers the plastic strain as an eigenstrain imposed on the structure, and new degrees of freedom are added on the fly into the kinematics during the incremental‐iterative process. The radial return algorithm of J₂ plastic flow is used. Because of the constant evolution of beam kinematics, the Newton‐Raphson algorithm for satisfying the global equilibrium is modified. An application to a cantilever beam loaded at its free extremity is presented and compared to a three‐dimensional reference solution. The beam model shows satisfying results even at a local scale and for a significantly reduced computation time.     

Elastoviscoplastic constitutive frameworks for generalized continua

Summary.  A unifying thermomechanical constitutive framework for generalized continua including additional degrees of freedom or/and the second gradient of displacement is presented. Based on the analysis of the dissipation, state laws, flow rules and evolution equations are proposed for Cosserat, strain gradient and micromorphic continua. The case of the gradient of internal variable approach is also incorporated by regarding the nonlocal internal variable as an actual additional degree of freedom. The consistency of the continuum thermodynamical framework is ensured by the introduction of a viscoplastic pseudo–potential of dissipation, thus extending the classical class of so–called standard material models to generalized continua. Variants of the higher order and higher grade theories are also reported based on the explicit introduction of the plastic strain tensor as additional degree of freedom. Within this new class of models, called here gradient of strain models, one recognizes the fact that, in a second grade theory for instance, the plastic part of the strain gradient can be identified with the gradient of plastic strain. Simple examples dealing with bending and shearing of Cosserat or second grade media are given to illustrate two types of extensions of classical J ₂-plasticity : single-criterion and multi-mechanism generalized elastoplasticity. Finally, formulations at finite deformation of the proposed models are provided focusing on proper decompositions of Cosserat curvature, strain gradient and gradient of micromorphic deformation into elastic and viscoplastic parts. Received March 11, 2002 Published online: January 16, 2003     

2-RPU/UPR并联机构的自由度与工作空间分析

目的 针对目前快递行业需要对大量物品进行分拣、包装,需耗费大量的劳动力,提出一种2-RPU/UPR并联机构进行运动学分析,得到其自由度和工作空间,考虑是否可以取代人工操作。方法 首先利用SolidWorks软件对2-RPU/UPR并联机构进行三维建模,然后利用螺旋理论进行分析,得到其自由度;接着通过运动链参数D-H表示法和欧拉角表示结合求解位置反解;最后,借助Matlab软件求出其工作空间,并改变动定平台半径比,分析工作空间变化情况。结果 2-RPU/UPR并联机构的工作空间范围比较广,形状规则,呈对称分布,动定平台半径比增大,工作空间也会随之增大。结论 2-RPU/UPR并联机构具有xy等2个转动方向和1个移动方向z,通过添加对应的控制程序,可以代替人工进行分拣、包装工作。     

一种新型六自由度SCARA机器人设计与运动学分析

目的 针对传统四自由度SCARA机器人末端灵活性不足的问题，设计一种新型的六自由度SCARA机器人。方法 首先在传统四自由度SCARA机器人结构特点分析的基础上，通过改进其末端结构以增加机器人的自由度，从而提高其末端的灵活性。然后分别采用D–H参数法与代数法建立该机器人的运动学正解模型与逆解模型。在此基础上，采用对比法在Matlab的Robotics Toolbox工具箱中对该机器人开展仿真分析，验证机器人运动学模型的正确性。最后，基于五次多项式插值算法对机器人开展轨迹规划研究。结果 实验结果显示该机器人的运动轨迹、速度与加速度曲线光滑连续，没有任何断点与突变点。结论 表明该机器人运动平稳，具有良好的运动学性能，从而为后续该机器人的运动控制奠定了良好的基础。     

Deformation mechanisms in negative Poisson's ratio materials: structural aspects

Poisson's ratio in materials is governed by the following aspects of the microstructure: the presence of rotational degrees of freedom, non-affine deformation kinematics, or anisotropic structure. Several structural models are examined. The non-affine kinematics are seen to be essential for the production of negative Poisson's ratios for isotropic materials containing central force linkages of positive stiffness. Non-central forces combined with pre-load can also give rise to a negative Poisson's ratio in isotropic materials. A chiral microstructure with noncentral force interaction or non-affine deformation can also exhibit a negative Poisson's ratio. Toughness and damage resistance in these materials may be affected by the Poisson's ratio itself, as well as by generalized continuum aspects associated with the microstructure.     

Algebraic geometry and group theory in geometric constraint satisfaction for computer-aided design and assembly planning

Mechanical design and assembly planning inherently involve geometric constraint satisfaction or scene feasibility (GCS/SF) problems. Such problems imply the satisfaction of proposed relations placed between undefined geometric entities in a given scenario. If the degrees of freedom remaining in the scene are compatible with the proposed relations or constraints, a set of entities is produced that populate the scenario satisfying the relations. Otherwise, a diagnostic of inconsistency of the problem is emitted. This problem appears in various forms in assembly planning (assembly model generation), process planning, constraint driven design, computer vision, etc. Previous attempts at solution using separate numerical, symbolic or procedural approaches suffer serious shortcomings in characterizing the solution space, in dealing simultaneously with geometric (dimensional) and topological (relational) inconsistencies, and in completely covering the possible physical variations of the problem. This investigation starts by formulating the problem as one of characterizing the solution space of a set of polynomials. By using theories developed in the area of algebraic geometry, properties of Grobner Bases are used to assess the consistency and ambiguity of the given problem and the dimension of its solution space. This method allows for die integration of geometric and topological reasoning. The high computational cost of Grobner Basis construction and the need for a compact and physically meaningful set of variables lead to the integration of known results on group theory. These results allow the characterization of geometric constraints in terms of the subgroups of the Special Group of Euclidean displacements in E³, SE(3). Several examples arc developed which were solved with computer algebra systems (MAPLE and Mathematica). They are presented to illustrate the use of the Euclidean group-based variables, and to demonstrate the theoretical completeness of the algebraic geometry analysis over the domain of constraints expressible as polynomials.     

基于P-Rob六自由度机械臂运动学建模与仿真

目的为实现后续机械臂控制算法研究,检验机械臂运动学模型构建的正确性,基于PersonalRobotics,对六自由度机械臂进行运动学模型构建和轨迹仿真。方法通过标准D-H法建立运动学模型,实现机械臂的正、逆运动学方程求解,根据机械臂的结构特性,对传统逆向运动学求解的解析法进行改进。结果使用仿真软件Matlab验证了运动学模型建立的准确性,改进的逆向运动学求解算法降低了传统求解算法的复杂度。使用仿真软件Matlab验证了运动学模型建立的准确性,并通过Matlab对改进的逆行运动学求解方法进行了验证,结果表明,改进的逆向运动学求解速度是传统逆向运动学求解速度的一半。结论根据六自由度机械臂的运动学研究,对实际机械臂的运动控制具有一定的参考价值。在实际的P-Rob机械臂上进行了仿真数据的测试,再次验证了运动学模型建立的准确性,仿真数据可应用于实际的机械臂控制中。实验现象表明针对此机械结构的机械臂,使用改进解析法求解逆解的方法计算简单、误差小、可行性强。     

基于线结构光传感器的工业机器人运动学参数标定

针对工业机器人绝对定位精度较低问题,提出一种基于线结构光传感器的工业机器人运动学参数标定方法.首先,将线结构光传感器固定安装在机器人末端,建立传感器测量模型,然后建立机器人运动学模型,通过手眼关系将传感器模型与机器人模型连接组成完整的标定系统模型.其次,使线结构光传感器在不同位姿下对某一固定点进行测量,得到该点在机器人...     

A finite‐strain quadrilateral shell element based on discrete Kirchhoff–Love constraints

This paper improves the 16 degrees‐of‐freedom quadrilateral shell element based on pointwise Kirchhoff–Love constraints and introduces a consistent large strain formulation for this element. The model is based on classical shell kinematics combined with continuum constitutive laws. The resulting element is valid for large rotations and displacements. The degrees‐of‐freedom are the displacements at the corner nodes and one rotation at each mid‐side node. The formulation is free of enhancements, it is almost fully integrated and is found to be immune to locking or unstable modes. The patch test is satisfied. In addition, the formulation is simple and amenable to efficient incorporation in large‐scale codes as no internal degrees‐of‐freedom are employed, and the overall calculations are very efficient. Results are presented for linear and non‐linear problems. Copyright © 2005 John Wiley & Sons, Ltd.     

Finite-rotation theories for composite laminates

Summary For the finite-element analysis of arbitrary composite laminates various finite-rotation theories are presented in a single formulation. First a refined theory is derived which allows a quadratic shear deformation distribution across the thickness. The so-called difference vector appearing in the kinematic relations is expressed in terms of rotational degrees of freedom permitting a clear determination of the deformed normal vector in every nonlinear range. The constitutive relations derived are applicable to orthotropic material properties varying arbitrarily across the thickness and to curvilinear laminate coordinates as well. This refined theory is then transformed into simplified theories of Kirchhoff-Love and Mindlin-Reissner types. Finally, the last formulation is used to formulate a layer-wise theory able to grasp the shear deformations very accurately. Kinematic relations and the corresponding constraints are presented in two alternative forms suitable for the application of isoparametric and classical finite-element formulations.     

串联式球面剪叉机构的运动学建模与轨迹规划

目的研究一种用于球体表面装饰的串联式球面剪叉机构的运动学性能与轨迹规划。方法运用螺旋理论和修正的Kutzbach-Grübler公式计算机构的自由度,利用球面三角运算法则和坐标变换法,建立串联式球面剪叉机构的运动学模型,并采用Denavit-Hartenberg(D-H)四元数法对机构进行正运动学和逆运动学的求解,利用Matlab软件进行机构的工作空间求解,利用SolidWorks动态仿真法对机构进行轨迹规划,并进行运动性能分析。结果通过实例得出了机构在球面上按给定轨迹运行时的运动规律与实际规划轨迹的运动规律性能相符。结论串联式球面剪叉机构具有工作空间大、运动灵活、运动性能良好等优点,适用于在球体外包装表面上雕刻、喷绘以及其他相关应用领域。     

Non‐reflecting boundary conditions for atomistic,continuum and coupled atomistic/continuum simulations

We present a method to numerically calculate a non‐reflecting boundary condition which is applicable to atomistic, continuum and coupled multiscale atomistic/continuum simulations. The method is based on the assumption that the forces near the domain boundary can be well represented as a linear function of the displacements, and utilizes standard Laplace and Fourier transform techniques to eliminate the unnecessary degrees of freedom. The eliminated degrees of freedom are accounted for in a time‐history kernel that can be calculated for arbitrary crystal lattices and interatomic potentials, or regular finite element meshes using an automated numerical procedure. The new theoretical developments presented in this work allow the application of the method to non‐nearest neighbour atomic interactions; it is also demonstrated that the identical procedure can be used for finite element and mesh‐free simulations. We illustrate the effectiveness of the method on a one‐dimensional model problem, and calculate the time‐history kernel for FCC gold using the embedded atom method (EAM). Copyright © 2005 John Wiley & Sons, Ltd.     

The uncertainty of parallel model coordinate measuring machine

The Stewart platform is one of 6-DOF (6 degrees of freedom) parallel mechanism. A parallel mechanism is a mechanism to connect six actuators in parallel between the base and the stage. It is necessary to calibrate the kinematic parameters that compose the mechanism in order to use the parallel mechanism as the coordinate measuring machine. Additionally, it is necessary to estimate the uncertainty about the measurement results. In this study, the kinematics parameters are estimated by using the calibration point in one point, the uncertainty is estimated at several measurement coordinates from kinematic parameters and postures angle. The verification method is shown following; i) The amount of the actuator expansion and contraction in the parallel mechanism is assumed to be a value obtained with the sensor and, the kinematic parameters are estimated from the values by using the least squares method. ii)The coordinates of the stylus are calculated from the estimated kinematic parameters by forward kinematics. The uncertainty is estimated from the coordinates of the stylus. As a result, the influence of the uncertainty is reported when the target coordinates are measured.