On kinematic method in shakedown theory: I. Duality of extremum problems

A kinematic method for determining the safety factor in shakedown problems is developed. An upper bound kinematic functional is defined on a set of kinematically admissible time-independent velocity fields. Every value of the functional is an upper bound for the safety factor. Using convex analysis methods, conditions are established under which the infimum of the kinematic upper bounds equals the safety factor, in particular, conditions under which it is sufficient to consider only smooth velocity fields for the safety factor calculation. The method generalizes that recently proposed for the case of spherical yield surfaces by Kamenjarzh and Weichert. The extension covers a wide class of yield surfaces and inhomogeneous bodies. A shakedown problem for a beam subjected to a concentrated load is considered as an example.     

Extension of the virtual fields method to elasto-plastic material identification with cyclic loads and kinematic hardening

The virtual fields method (VFM) has been specifically developed for solving inverse problems from dense full-field data. This paper explores recent improvements regarding the identification of elasto-plastic models. The procedure has been extended to cyclic loads and combined kinematic/isotropic hardening. A specific attention has also been given to the effect of noise in the data. Indeed, noise in experimental data may significantly affect the robustness of the VFM for solving such inverse problems. The concept of optimized virtual fields that minimize the noise effects, previously developed for linear elasticity, is extended to plasticity in this study. Numerical examples with models combining isotropic and kinematic hardening have been considered for the validation. Different load paths (tension, compression, notched specimen) have shown that this new procedure is robust when applied to elasto-plastic material identification. Finally, the procedure is validated on experimental data.     

Performance of the MLPG method for static shakedown analysis for bounded kinematic hardening structures

Shakedown analysis is an extension of plastic limit analysis to the case of variable repeated loads and plays a significant role in safety assessment and structural design. This paper presents a solution procedure based on the meshless local Petrov–Galerkin (MLPG) method for lower-bound shakedown analysis of bounded kinematic hardening structures. The numerical implementation is very simple and convenient because it is only necessary to construct an array of nodes in the targeted domain. Moreover, the natural neighbour interpolation (NNI) is employed to construct trial functions for simplifying the imposition of essential boundary conditions. The kinematic hardening behaviour is simulated by an overlay model and the numerical difficulties caused by the time parameter are overcome by introducing the conception of load corner. The reduced-basis technique is applied to solve the mathematical programming iteratively through a sequence of reduced residual stress subspaces with very low dimensions and the resulting non-linear programming sub-problems are solved via the Complex method. Numerical examples demonstrate that the proposed solution procedure is feasible and effective to determine the shakedown loads of bounded kinematic hardening structures as well as unbounded kinematic hardening structures.     

Deformation gradient based kinematic hardening model

A kinematic hardening model applicable to finite strains is presented. The kinematic hardening concept is based on the residual stresses that evolve due to different obstacles that are present in a polycrystalline material, such as grain boundaries, cross slips, etc. Since these residual stresses are a manifestation of the distortion of the crystal lattice a corresponding deformation gradient is introduced to represent this distortion. The residual stresses are interpreted in terms of the form of a back-stress tensor, i.e. the kinematic hardening model is based on a deformation gradient which determines the back-stress tensor. A set of evolution equations is used to describe the evolution of the deformation gradient. Non-dissipative quantities are allowed in the model and the implications of these are discussed. Von Mises plasticity for which the uniaxial stress–strain relation can be obtained in closed form serves as a model problem. For uniaxial loading, this model yields a kinematic hardening identical to the hardening produced by isotropic exponential hardening. The numerical implementation of the model is discussed. Finite element simulations showing the capabilities of the model are presented.     

Singularity and kinematic bifurcation analysis of pin-bar mechanisms using analogous stiffness method

Based on the analogy between bifurcation of equilibrium paths in structures and kinematic bifurcation of mechanisms, this paper proposes an analogous stiffness method to detect the singularity and kinematic bifurcation of mechanisms. The analogous stiffness in mechanisms is first defined as the derivative of the state variable with respect to the controlling variable. By investigating the value of analogous stiffness, the singularity can be classified into output singularity, input singularity and architectural singularity. And the kinematic characteristics of free joints at corresponding singularity configurations are expounded. The singular and kinematic bifurcation points of mechanisms can then be determined by solving analogous stiffness equations and compatibility equations simultaneously. Following that, the analytical criterion for finite motion of corresponding free joints at singularity configurations is derived from the second-order analysis of compatibility equations. The efficiency of the proposed method is finally illustrated by three typical examples.     

Comparison of isotropic hardening and kinematic hardening in thermoplasticity

A coupled thermo-mechanical problem is presented in this paper. The constitutive model is based on thermoplastic model for large strains where both kinematic and isotropic hardening are included. It is shown that a non-associated plasticity formulation enables thermodynamic consistent heat generation to be modeled, which can be fitted accurately to experimental data. In the numerical examples the effect of heat generation is investigated and both thermal softening and temperature-dependent thermal material parameters are considered. The constitutive model is formulated such that pure isotropic and pure kinematic hardening yield identical uniaxial mechanical response and mechanical dissipation. Thus, differences in response due to hardening during non-proportional loading can be studied. Thermally triggered necking is studied, as well as cyclic loading of Cook’s membrane. The numerical examples are solved using the finite element method, and the coupled problem that arises is solved using a staggered method where an isothermal split is adopted.     

A modified basis reduction method for limited kinematic hardening shakedown analysis under complex loads

A novel extension of the basis reduction method for kinematic hardening shakedown problem is presented. Firstly, the basis reduction method is implemented based on the modified Newton–Raphson (N-R) method. Then a new technique for the construction of back stress field is introduced, where the simultaneous influence of multiple load corners in shakedown is taken into consideration. Finally, two typical numerical examples are investigated. The results compared with previous works in literatures demonstrated that the proposed method is accurate and the performance in reducing of computation time is significant.     

On kinematic coupling equations within the scaled boundary finite-element method

The scaled boundary finite element method is a semi-analytical analysis technique, which combines the advantages of the finite element method and the boundary-element method. Assuming that the geometry of the governing structure can be represented by mapping its boundary with respect to the so-called scaling coordinate, the problem can be handled in a closed-form analytical manner in the scaling direction and by a finite-element approximation in the other directions. Thus, a discretization of the boundary is sufficient and the nodal degrees of freedom are functions of the scaling coordinate. In some situations, such as the analysis of the free-edge effect in laminated plates, it is useful to introduce kinematic coupling equations, which are valid not only on the boundary, but also within the domain. The implementation of linear kinematic coupling equations within the method is presented for the case of a three-dimensional structure with scaling in a fixed Cartesian direction. Rigid-body modes are handled by using the concept of generalized inverse matrices. In some benchmark examples the efficiency of the approach is demonstrated and comparison with the results of the finite-element method shows good accordance.     

316L不锈钢随动强化模型改进研究

本文对316L不锈钢进行了单轴与多轴非比例路径下的应力控制棘轮试验,考察了应力幅值、平均应力和加载历程对棘轮特性的影响。同时进行了应变控制循环试验以研究材料的应力松弛特性。试验结果表明轴向棘轮效应在对称剪切荷载下效果明显,同时棘轮应变随应力幅值和平均应力的增加而增加。研究了Chen-Jiao随动强化模型与Jiang-Sehitoglu随动强化模型采用的单轴与多轴参数对背应力分量增量方向的影响,将Chen-Jiao模型中的多轴系数替换为界面饱和率,并在此基础上引入新的参数对塑性模量系数进行修正,计算结果表明修正后的模型能提升应力控制下多轴棘轮的预测精度,并能很好的预测应力松弛现象,表明了新模型的正确性与有效性。     

A nonlinear programming approach to kinematic shakedown analysis of frictional materials

This paper develops a novel nonlinear numerical method to perform shakedown analysis of structures subjected to variable loads by means of nonlinear programming techniques and the displacement-based finite element method. The analysis is based on a general yield function which can take the form of most soil yield criteria (e.g. the Mohr–Coulomb or Drucker–Prager criterion). Using an associated flow rule, a general yield criterion can be directly introduced into the kinematic theorem of shakedown analysis without linearization. The plastic dissipation power can then be expressed in terms of the kinematically admissible velocity and a nonlinear formulation is obtained. By means of nonlinear mathematical programming techniques and the finite element method, a numerical model for kinematic shakedown analysis is developed as a nonlinear mathematical programming problem subject to only a small number of equality constraints. The objective function corresponds to the plastic dissipation power which is to be minimized and an upper bound to the shakedown load can be calculated. An effective, direct iterative algorithm is then proposed to solve the resulting nonlinear programming problem. The calculation is based on the kinematically admissible velocity with one-step calculation of the elastic stress field. Only a small number of equality constraints are introduced and the computational effort is very modest. The effectiveness and efficiency of the proposed numerical method have been validated by several numerical examples.     

Stabilized finite element formulation applied to the kinematic Ponomarenko dynamo problem

A stabilized finite element (B, q) formulation is developed to solve the kinematic dynamo problem. As a test case, we solve the induction equation for a given solid body helical flow, embedded in a cylindrical conducting shell. This problem corresponds to the well-known Ponomarenko dynamo. It has the interesting property to have an exact dispersion relation giving the magnetic growth rate as a function of the flow properties. Therefore, it is a good benchmark to test our kinematic dynamo code. We calculated the dynamo threshold and plotted the geometry of the generated magnetic field. We also evaluated the residual error due to our stabilized formulation.     

A strain space nonlinear kinematic hardening/softening plasticity model

A strain space plasticity theory based on the nonlinear kinematic hardening and softening rule is developed in order to accommodate work-hardening, work-softening, and elastic-perfectly plastic materials with one set of constitutive equations, and to facilitate strain controlled calculations. A generalized hardening/softening parameter is proposed, and the potential of linking the parameter to micro-mechanical material changes is discussed. The theory is used to investigate work-softening materials numerically and highlights a need for additional experimental results in this area.     

Crack growth resistance in non-perfect plasticity: Isotropic versus kinematic hardening

The Strain Energy Density Theory is applied for analyzing energy dissipation and crack growth in the three-point bending specimen when the material behavior follows a multilinear strain-hardening stress-strain relationship. The problem is solved through the application of incremental theory of plasticity and finite element method.The rate of change of the strain energy density factor S with crack length a is verified to be governed by the relation . Results are obtained for isotropic and kinematic hardening. Moreover, the effects of loading step and specimen size are pointed out.     

Dimensional synthesis of the optimal RSSR mechanism for a set of variable design parameters

This paper deals with the dimensional synthesis of the RSSR mechanism, also known as spatial four-bar linkage (R and S stand for revolute and spherical kinematic pairs respectively). To univocally describe the geometry of the RSSR mechanism a specific set of geometry parameters is necessary. Generally speaking, in a synthesis problem a subset of these parameters, defined as design parameters, is usually considered as assigned whereas the remaining ones, defined as design variables, have to be found by the synthesis procedure. That is, the goal of the synthesis procedure is to find the values of the design variables, while satisfying both functional requirements of the mechanism and constraints on the design parameters. In this paper each design parameter is assigned as variable within a given range rather than being assigned as a single value. In general, varying a design parameter means obtaining a different mechanism which has different functional performances as a consequence. This feature gives raise to a novel synthesis problem, which has not been treated in the literature yet. In particular, the RSSR mechanism synthesis presented in this paper takes the optimization of the force transmission as an objective function, while referring to a given range of values of each design parameter. In addition, prescribed constraints on given extreme angular positions for both the input and the output links, on the upper and lower bounds for the transmission ratio, and on the upper and lower bounds for the design variable values have to be satisfied. The synthesis problem, set as a constrained minimization problem, is solved numerically in two steps by means of a Genetic algorithm followed by a quasi-Newton algorithm. As an example of application, a dimensional synthesis of an RSSR mechanism used in a hand exoskeleton is reported.     

Trajectory planning in parallel kinematic manipulators using a constrained multi-objective evolutionary algorithm

Generating manipulator trajectories considering multiple objectives with kinematics and dynamics constraints is a non-trivial optimization. In this paper, a constrained multi-objective genetic algorithm (MOGA) based technique is proposed to address this problem for a general motor-driven parallel kinematic manipulator. The planning process is composed of searching for a motion ensuring the accomplishment of the assigned task, minimizing the traverse time, and expended energy subject to various constraints imposed by the associated kinematics and dynamics of the manipulator. This problem is treated via an adequate parametric path representation in the task space of the moving platform, and then the use of the constrained MOGA for solving the resulted nonlinear multi-objective optimization problem. Simulation results are presented for the trajectories of the parallel kinematic manipulator, and a subsequent comparison with the weighted sum method is also carried out.     

New kinematic parameters of finite rotation of a solid

A new family of kinematic (global and local) orientation parameters of a solid is presented and described. All kinematic parameters are obtained by the method of mapping the variables onto the corresponding oriented subspace (hyperplane). In particular, the method of stereographic projection of a point of a five-dimensional sphere S ⁶ ? R ⁶ onto the oriented hyperplane R ⁵ is presented for the classical direction cosines of the angles determining the orientation of two coordinate systems. A family of global kinematic parameters obtained by the method of mapping of five-dimensional kinematic Hopf parameters given in the space R ⁵ onto the four-dimensional oriented subspace R ⁴ is described. The theorem of the homeomorphism of two topological spaces (the four-dimensional sphere S ⁴ ?R ⁵ with one deleted point and the oriented hyperplane R ⁴) is used to establish the correspondence between five- and four-dimensional kinematic parameters defined in the corresponding spaces. It is also shown what global four-dimensional orientation parameters, quaternions defined in the subspace R ⁴ are associated with the classical local parameters, i.e., the Rodrigues and Gibbs three-dimensional finite rotation vectors. The projection method is used to obtain the kinematic differential equations (KDE) of rotation corresponding to the five- and four-dimensional orientation parameters. All above-introduced kinematic orientation parameters of a solid permit efficiently solving the classical Darboux problem by using the corresponding KDE, i.e., determining the body current angular position in the space R ³ from the known (measured) angular velocity of the object rotation and its initial position in space.     

On shakedown of three-dimensional elastoplastic strain-hardening structures

The present article considers the shakedown problem of structures made of either kinematic or mixed strain-hardening materials. Some basic and useful shakedown properties of elastoplastic strain-hardening structures are proved mathematically. It is impossible for a kinematic strain-hardening structure to be involved in incremental plastic collapse, and so its only possible failure mode is that of alternating plasticity. A time-independent self-equilibrium stress field has no influence on the shakedown of a kinematic strain-hardening structure although it contributes to the magnitude of plastic deformation. The sufficient shakedown conditions for either kinematic or mixed strain-hardening structures are deduced, from which the lower bound of shakedown load domain can be obtained via a mathematical programming problem. It should be pointed out that, to guarantee the safety of an elastoplastic strain-hardening structure, the damage analysis is also necessary to determine the maximum load factor the structure can bear. The shakedown analysis of strain-hardening structures can be simplified by the conclusions obtained in this article, as is illustrated by two simple examples.     

Biquaternion solution of the kinematic control problem for the motion of a rigid body and its application to the solution of inverse problems of robot-manipulator kinematics

The problem of reducing the body-attached coordinate system to the reference (programmed) coordinate system moving relative to the fixed coordinate system with a given instantaneous velocity screw along a given trajectory is considered in the kinematic statement. The biquaternion kinematic equations of motion of a rigid body in normalized and unnormalized finite displacement biquaternions are used as the mathematical model of motion, and the dual orthogonal projections of the instantaneous velocity screw of the body motion onto the body coordinate axes are used as the control. Various types of correction (stabilization), which are biquaternion analogs of position and integral corrections, are proposed. It is shown that the linear (obtained without linearization) and stationary biquaternion error equations that are invariant under any chosen programmed motion of the reference coordinate system can be obtained for the proposed types of correction and the use of unnormalized finite displacement biquaternions and four-dimensional dual controls allows one to construct globally regular control laws. The general solution of the error equation is constructed, and conditions for asymptotic stability of the programmed motion are obtained. The constructed theory of kinematic control of motion is used to solve inverse problems of robot-manipulator kinematics. The control problem under study is a generalization of the kinematic problem [1, 2] of reducing the body-attached coordinate system to the reference coordinate system rotating at a given (programmed) absolute angular velocity, and the presentedmethod for solving inverse problems of robotmanipulator kinematics is a development of the method proposed in [3–5].     

Dynamic shakedown and a reduced kinematic theorem

A dynamic shakedown theory is formulated, which preserves all the essentials of the classical quasistatic theory. The emphasis is given to the kinematic theorem. A reduced path-independent kinematic inequality, which does not involve time integrals, is deduced. An analytical example illustrates the application of both static and reduced kinematic theorems in the dynamic range.