This paper deals with the Lagrange multipliers corresponding to the intrinsic constraint equations of rigid multibody mechanical systems. The intrinsic constraint equations are algebraic equations that are associated with nonminimal sets of orientation parameters employed for the kinematic representation of large finite rotations. Two coordinate formulations are analyzed in this investigation, namely the reference point coordinate formulation (RPCF) with Euler parameters and the natural absolute coordinate formulation (NACF). While the RPCF with Euler parameters employs the four components of a unit quaternion as rotational coordinates, the NACF directly uses the orthonormal set of nine direction cosines for describing the orientation of a rigid body in the three-dimensional space. In the multibody approaches based on the RPCF with Euler parameters and on the NACF, the use of a nonminimal set of rotational coordinates facilitates a general and systematic formulation of the differential–algebraic equations of motion. Considering the basic equations of classical mechanics, the fundamental problem of constrained motion is formalized and solved in this paper by using a special form of the Udwadia–Kalaba method. By doing so, the Udwadia–Kalaba equations are employed for obtaining closed-form analytical solutions for the Lagrange multipliers associated with the intrinsic constraint equations that appear in the differential–algebraic dynamic equations developed by using the RPCF with Euler parameters and the NACF multibody approaches. Two simple numerical examples support the analytical results found in this paper.
The objective of this paper is to investigate the mechanical properties of samples of MS1 Maraging Steel (untreated and heat treated), which were produced by additive technology in various orientations in the working area of the building machine. MS1 steel (European 1.2709 and German X3NiCoMoTi 18-9-5) is well known for its high strength, high fracture toughness, good weldability, and dimensional stability during aging. The literature review, related to the mechanical properties and fracture of MS1 steel, found that there are no available studies of the effects of both building direction and heat treatment on the mechanical properties of MS1 steel. The authors decided to address this omission and present this entirely new research in this article. The uniaxial tensile tests to fracture were completed at two of the authors' workplaces. The results were statistically assessed using Grubbs' test for outliers, and then the data were processed using box plots to be easily comparable from the point of view of print direction, heat treatment, and the values declared by the metal powder producer or in the tables (for conventionally produced steel). Scanning electron microscopy was used to analyze the fracture surfaces obtained after tensile testing cylindrical samples. The results showed that there was an impact on the mechanical properties depending on the sample orientation within the same heat treatment type; there was also significant influence of heat treatment, while the possibility of the natural aging effect on mechanical properties was also noted.
This paper is focused on the development of a numerical procedure for solving the system identification problem of linear dynamical models that mathematically describe multibody mechanical systems. To this end, an input–output representation of the time evolution of a general mechanical system based on a sequence of matrices referred to as Markov parameters is employed. The set of Markov parameters incorporate the state-space matrices that allow for describing the dynamic behavior of a general mechanical system considering the assumption of structural linearity. The system Markov parameters are defined by means of a discretization process applied to the analytical description of a mechanical system, and therefore, they are difficult to obtain directly from observable measurements. However, a state observer can be introduced in order to define a set of observer Markov parameters that can be readily recovered from input–output experimental data. The observer Markov parameters obtained by using a least-square approach allow for computing in a recursive manner the system Markov parameters as well as another discrete sequence of matrices referred to as observer gain Markov parameters. Subsequently, the system and observer gain Markov parameters identified from observable input–output data are used for constructing a sequence of generalized Hankel matrices from which a state-space model of the mechanical system of interest can be extracted. This fundamental step of the identification procedure is performed in the algorithm elaborated in this work employing a numerical procedure which relies on the use of the Moore–Penrose pseudoinverse matrix obtained by means of the singular value decomposition. In the paper, the principal analytical and numerical aspects of the proposed identification algorithm are described in detail. Furthermore, a numerical example based on a simple vehicle model is discussed in order to verify by means of numerical experiments the effectiveness of the identification procedure developed in this work.
Periodic domain patterns in tetragonal ferroelectrics are explored using a phase field model calibrated for barium titanate. In this context, we discuss the standard periodic boundary condition and introduce the concept of reverse periodic boundary conditions. Both concepts allow the assembly of cubic cells in accordance with mechanical and electrical conditions. However, application of the reverse periodic boundary condition is due to an increased size of the RVE and enforces more complex structures compared to the standard condition. This may be of particular interest for other multiphysics simulations. Additionally, we formulate mechanical side conditions with minimal spherical (hydrostatic) stress, or conditions with controlled average strain. It is found that in sufficiently small periodic cells, only a uniform single domain, or the simplest stripe domains constitute equilibrium states. However, once the periodic cells are of order 20 domain wall widths in size, more complex, 3-dimensional patterns emerge. Some of these patterns are known from prior studies, but we also identify other domain patterns with long, ribbon-like domains threaded through them and some vortex-like structures.
In this work, the collapse load of a masonry arch subjected to an actual horizontal displacement of the supports is assessed, in the deformed configuration of the structure, by means of limit analysis theory. Masonry is modelled as a rigid in compression, no-tension material and avoiding the occurrence of sliding failures. A numerical tool, based on the approach of the kinematical theorem of the collapse state, is proposed for the collapse computation of circular arches subjected to dead loads and to incremental concentrated load applied at their crown. A parametric study has been carried out in order to develop a deeper understanding of the influence of the involved parameters. It is shown that the existence of a kinematically admissible collapse mechanism is correlated to a thickness/mean radius ratio dependent on the value of the actual horizontal displacement of the supports. In the paper, a relationship between these two last parameters is proposed.
This paper presents an analytical analysis and optimization of vibration-induced fatigue in a generalized, linear two-degree-of-freedom inerter-based vibration isolation system. The system consists of a source body and a receiving body, coupled through an isolator. The isolator consists of a spring, a damper, and an inerter. A broadband frequency force excitation of the source body is assumed throughout the investigation. Optimized system, in which the kinetic energy of the receiving body is minimized, is compared with sub-optimal systems by contrasting the fatigue life of a receiving body helical spring with several alternative isolator setup cases. The optimization is based on minimizing specific kinetic energy, but it also increases the number of cycles to fatigue failure of the considered helical spring. A significant portion of this improvement is due to the inclusion of an optimally tuned inerter in the isolator. Various helical spring deflection and stress correction factors from referent literature are discussed. Most convenient spring stress and deflection correction factors are adopted and employed in conjunction with pure shear governed proportional stress in the context of high-cycle fatigue.
This paper evaluates the longitudinal wave speed through a plate in which its two opposing sides are elastically restrained in the width direction, taking into consideration the material auxeticity and strain as well as changes to the density and cross-sectional area. Apart from the known role of Young’s modulus and density, the present results reveal that the wave speed can be enhanced by increasing the width elastic restraint. In the case of high elastic restraint, the speed of both tensile and compressive waves can be minimized by selecting plate materials with Poisson’s ratio of low magnitude. In the case of low elastic restraint, the speed of tensile and compressive waves can be greatly reduced by selecting plate materials with large positive and large negative Poisson’s ratio, respectively. For the special case of negligible strain, the longitudinal wave speed reduces to the elementary wave speed in prismatic rods and in plates of infinite width when the width elastic restraint stiffness approaches zero and infinity, respectively. The obtained results not only avail more parameters for adjusting the longitudinal waves in plates, but also identify the differing methods of effectively controlling the wave speed between tensile and compressive waves when the strain magnitude is non-negligible.
The combination of materials with either pronounced ferroelectric or ferromagnetic effect characterizes multiferroic heterostructures, whereby the different materials can be arranged in layers, columns or inclusions. The magnetization can be controlled by the application of electrical fields through a purely mechanical coupling at the interfaces between the different materials. Thus, a magneto-electric coupling effect is obtained. Within a continuum mechanics formulation, a phase field is used to describe the polarization and the magnetization in the ferroelectric and ferromagnetic layers, respectively. The coupling between polarization/magnetization and strains within the layers, in combination with the mechanical coupling at the sharp layer interfaces, yields the magneto-electric coupling within the heterostructure. The continuum formulations for both layers are discretized in order to make the differential equations amenable to a numerical solution with the finite element method. A state-of-the-art approach is used for the ferroelectric layer. The material behavior of the ferromagnetic layer is described by a continuum formulation from the literature, which is discretized using a newly proposed approach for the consistent interpolation of the magnetization vector. Four numerical examples are presented which show the applicability of the newly proposed approach for the ferromagnetic layer as well as the possibility to simulate magneto-electric coupling in multiferroic heterostructures.
The problem of optimal energy harvesting for a piezoelectric element driven by mechanical vibrations is stated in terms of an ODE system with hysteresis under the time derivative coupling a mechanical oscillator with an electric circuit with or without inductance. In the piezoelectric constitutive law, both the self-similar piezoelectric butterfly character of the hysteresis curves and feedback effects are taken into account in a thermodynamically consistent way. The physical parameters of the harvester are chosen to be the control variable, and the goal is to maximize the harvested energy for a given mechanical load and a given time interval. If hysteresis is modeled by the Preisach operator, the system is shown to be well-posed with continuous data dependence. For the special case of the play operator, we derive first-order necessary optimality conditions and an explicit form of the gradient of the total harvested energy functional in terms of solutions to the adjoint system.
The electromechanical loading situation at cracks in ferroelectric ceramics is essentially affected by domain switching. Under high electrical and/or mechanical external fields, the state of polarization and remanent strains is substantially changed at the crack tip. These irreversible dissipative processes influence the fracture toughness of the cracked ferroelectric material. In the present paper, the micromechanical domain switching processes at the crack tip are studied by numerical simulation and compared with the in situ experimental results obtained by Jones et al. (Acta Mater 55(16):5538-5548, 2007) using X-ray diffraction analyses in synchrotron. Main attention is payed to the spatial distribution of preferred domain orientation in a mechanically loaded compact tension specimen made of a soft tetragonal lead zirconate titanate ceramics. It is found that the mechanically induced favored domain orientation distribution depends on position within the plane of the CT specimen and correlates with projected deviatoric stresses and strains. Some issues concerning shortcomings in the experimental and simulation results are raised and discussed. The outcome of this type of simulations forms the basis for more realistic fracture mechanical evaluations in future.
The paper deals with evaluating the mean stress effect in multiaxial criteria for fatigue limit estimation, with special emphasis on the mean shear stress effect. The usual practice of accepting the mean normal stress effect and neglecting the effect of static torsion is scrutinized. Two methods—two critical plane criteria, PCr (Papuga Criterion) and QCP (Quadratic parameter on the Critical Plane)—are described, and additional local stress parameters representing the mean torsion effect are implemented. The efficiency of the new implementations is evaluated on a large data set of 407 fatigue limits. Additionally, outputs of two other well-known methods—the Crossland method and the Dang Van method—are provided for comparison. The positive outcome of including the mean shear stress effect is evident not only in cases of applied mean torsion load, but also in cases with purely axial loading or with biaxial configurations.
The present paper deals with a new holistic closed-form analytical model for the local buckling load of thin-walled composite beams with I-, Z-, C-, L- and T-cross sections under axial compressive load. The beam is simply supported at both ends (Euler case II), and the plate behaviour of web and flanges is described by the Classical Laminated Plate Theory. Furthermore, symmetric and orthotropic laminates are considered. In previous investigations on composite beams under compression, the web and flange plates are considered as separate composite plates. The present analysis is performed using the Ritz method in which an approach for the entire cross section is realized. The individual webs and flanges of the beam are assembled by suitable continuity conditions into one system. In order to achieve that, new displacement shape functions for web and flange that fulfil all boundary conditions have been developed. The present closed-form analytical method enables the explicit representation of the buckling load for the entire composite beam under axial compression. The comparison between the present approach and comparative finite element simulations shows a very satisfactory agreement. The present method is ideal for pre-designing such structures, highly efficient in terms of computational effort and very suitable for practical engineering work.
This paper focuses on development of a new mathematical model and its analytical solution for the buckling analysis of elastic longitudinally cracked columns with finite axial adhesion between the cracked sections. Consequently, the analytical solution for buckling loads is derived for the first time. The critical buckling loads are calculated for different crack lengths and various degrees of the contact adhesion. It is shown that the critical buckling loads can be greatly affected by the crack length and degree of the connection between the cracked sections. Finally, the presented results can be used as a benchmark solution.
This work presents the results of a numerical sensitivity analysis of material density on crack propagation in nonlinear viscoelastic softening material models. Two basic material models are analyzed: Maxwell and Kelvin material models, under the loading with changing rate (slow or fast impact loading). The material is discretized as a lattice model, and the analysis is performed on simple examples consisting of several lattice bars. The mathematical description is based on the theory of dynamical systems, i.e., it is a system of nonlinear differential equations for Kelvin, and a system of nonlinear differential-algebraic equations for Maxwell material model. Sensitivity of displacements on mass and load rate accompanies the analysis.
For reducing the sticking or stick–slip phenomenon, this paper proposed a new torsional oscillator which could produce a circumferential torque vibration in the drilling process. Based on the working principle and fundamental structure, the generation mechanism of torsional vibration and the relationships of the mechanical parts are derived. Then, the dynamic model is established according to the drilling operation condition. With this, the further analysis of changing relationship between the inlet flow and torque is studied. The angular displacement and velocity of the valve body are analyzed according to the dynamics theory models. With the combination of the theoretical analysis model, the parameter studies are carried out by numerical example calculation. Finally, experiment test is conducted to identify the accuracy and reliability of the calculation method under the corresponding working conditions. The results show that with the constant pressure the average torque is proportional to the inlet flow. However, while the inlet pressure is invariable, the collision period is inversely proportional to the inlet flow, and the angular displacement and velocity are proportional to the inlet flow. The research results can provide references for the design and application of the torsional vibration tools, which is potential and significant in drilling engineering.
For engineering structures, the limits of internal stresses, nodal displacements and fundamental frequencies must be simultaneously considered. This had been paid attention in the theory of structural optimization. Actually, most examples of only considering static constraints or only considering dynamic constraints were presented for simultaneously considering static and dynamic constraints. A few examples of considering both static and dynamic constraints were presented, but the advantage could not be presented. It is the reason that the singularity of structural optimization for considering dynamic constraints has not been discussed. To discover the singularity, an optimization model to simultaneously consider static and dynamic constraints is used for the truss size optimization. And according to the extremum conditions of the optimization problem, Ratio-Extremum method is proposed to solve the optimization problems of considering both static and dynamic constraints and only considering dynamic constraints, in which a new searching direction of design variables is to be discussed. Particularly, the step-size factors can be determined by formulas to iteratively solve Lagrangian multipliers and design variables. Numerical examples of 15-bar planar and 72-bar spatial trusses are used to show the singular solutions. On the convergent points, the optimization weights of only considering dynamic constraints are about 66.17% and 71.14% more than the weights of considering both static and dynamic constraints, respectively. The convergent solutions of only considering dynamic constraints are not the best results. However, additional static constraints can be helpful to obtain better results for considering dynamic constraints.
Microstructure evolution in magnetic materials is typically a non-local effect, in the sense that the behaviour at a material point depends on the magnetostatic energy stored within the demagnetisation field in the entire domain. To account for this, we propose a finite element framework in which the internal state variables parameterising the magnetic and crystallographic microstructure are treated as global fields, optimising a global potential. Contrary to conventional micromagnetics, however, the microscale is not spatially resolved and exchange energy terms are neglected in this approach. The influence of microstructure evolution is rather incorporated in an effective manner, which allows the computation of meso- and macroscale problems. This approach necessitates the development and implementation of novel mixed finite element formulations. It further requires the enforcement of inequality constraints at the global level. To handle the latter, we employ Fischer-Burmeister complementarity functions and introduce the associated Lagrange multipliers as additional nodal degrees-of-freedom. As a particular application of this general methodology, a recently established energy-relaxation-based model for magnetic shape memory behaviour is implemented and tested. Special casesincluding ellipsoidal specimen geometriesare used to verify the magnetisation and field-induced strain responses obtained from finite element simulations by comparison to calculations based on the demagnetisation factor concept.
This contribution deals with investigations on enhanced Fischer-Burmeister nonlinear complementarity problem (NCP) functions applied to a rate-dependent laminate-based material model for ferroelectrics. The framework is based on the modelling and parametrisation of the material's microstructure via laminates together with the respective volume fractions. These volume fractions are treated as internal-state variables and are subject to several inequality constraints which can be treated in terms of Karush-Kuhn-Tucker conditions. The Fischer-Burmeister NCP function provides a sophisticated scheme to incorporate Karush-Kuhn-Tucker-type conditions into calculations of internal-state variables. However, these functions are prone to numerical instabilities in their original form. Therefore, some enhanced formulations of the Fischer-Burmeister ansatz are discussed and compared to each other in this contribution.
A novel phase field formulation implemented within a material point method setting is developed to address brittle fracture simulation in anisotropic media. The case of strong anisotropy in the crack surface energy is treated by considering an appropriate variational, i.e. phase field approach. Material point method is utilized to efficiently treat the resulting coupled governing equations. The brittle fracture governing equations are defined at a set of Lagrangian material points and subsequently interpolated at the nodes of a fixed Eulerian mesh where solution is performed. As a result, the quality of the solution does not depend on the quality of the underlying finite element mesh and is relieved from mesh distortion errors. The efficiency and validity of the proposed method are assessed through a set of benchmark problems.
A closed finite-dimensional system of dynamical equations for an unbounded periodic set of edge dislocations obtained previously from homogenization reasoning (Berdichevsky in J Mech Phys Solids 106:95–132, 2017) is rederived in this paper using some elementary means.