Physical Nonlinearity Research Articles

Soft Machines: Challenges to Computational Dynamics

The paper surveys the recent studies of authors on the dynamic modeling and simulation of soft machines in the frame of multibody system dynamics. The studies focus on the geometric nonlinearity of coupled overall motion and large deformation of a soft body, the physical nonlinearity of a soft body made of hyper-elastic or elastoplastic materials, the frictional impacts and contacts of soft bodies, and the efficient dynamic computation algorithm of soft multibody systems governed by a set of differential-algebraic equations of very high dimensions. The paper presents the validation of proposed approach through three case studies, including the locomotion of a soft quadrupedal robot, the spinning deployment of a solar sail of spacecraft, and the deployment of a mesh reflector of satellite antenna, as well as the corresponding experimental studies. Finally, the paper gives some remarks on future researches.

Учет физической нелинейности железобетонных конструкций при численных расчетах конструктивных систем

Учет физической нелинейности железобетонных конструкций при численных расчетах конструктивных систем

The development of fracture mechanics hypotheses applicable to the calculation of reinforced concrete structures for the second group of limit states

The problems of the fracture mechanics hypotheses development are formulated with reference to the calculation of reinforced concrete structures in the presence of cracks. The main aspects of fracture mechanics are given. They focus on the features of the predestruction zone, as well as the hypotheses and assumptions underlying the calculation of reinforced concrete structures for the second group of limit states. It is described the features of cutting a two-cantilever element including a crack for constructing an effective instrument of calculation for reinforced concrete with allowance for physical nonlinearity, cracking processes, bond of reinforcement with concrete, and the effect of discontinuity. It is obtained a new solution to the problem of the stressed-strained state of the reinforced concrete element in the zone immediately adjacent to the crack.

Simulation of performance of circular CFST columns under short-time and long-time load

The method of calculation of concrete-filled steel tubular (CFST) columns with consideration of physical nonlinearity of materials, geometric nonlinearity of the confinement and the effect of the gain in strength of the core is considered. The method uses a step iteration algorithm, which involves analytical dependencies and the ultimate element simulation method. Allowance for creep of concrete is based on using the generalized kinetic long-term deformation curve and phenomenological deformation development equations. Creep of concrete is controlled through new structural factors that determine the structure of cement rock layers between sand and mortar grains between chip grains. The method is validated by comparing experimental results and theoretical data. The suggested method allowed to study the stress-strain and limit state of circular concrete-filled steel tubular columns, as well as to evaluate their effectiveness with account for the time factor.

A refined model of elastic-plastic bending deformation of flexible reinforced shallow shells based on explicit “cross” scheme

An initial-boundary value problem is formulated for elastoplastic deformation of flexible shallow shells, cross reinforced by equidistant surfaces. The mechanical behavior of the component materials of the composition of curved panels is governed by the Prandtl-Reuss-Hill equations for an elastoplastic medium. The geometric nonlinearity of the problem is considered in the Karman approximation. The resulting governing equations and the corresponding initial and boundary conditions in the generalized kinematic variables allow one to calculate with different accuracy the stress-strain state in the components of the composition of flexible shallow shells and plates, taking into account their weakened resistance to the transverse shear. The relations corresponding to the traditional non-classical Reddy theory are obtained in the first approximation from the equations, initial and boundary conditions. The numerical integration of the initial-boundary value problem is carried out on the basis of the method of steps in time. The central finite differences are used to approximate derivatives with respect to time. On the basis of this approximation the explicit numerical “cross” scheme is constructed in the case of impact loads of explosive type. The properties of elastoplastic dynamic behavior is investigated for the reinforced shallow spherical shell of annular form in plan, with a perfectly rigid inner insert, also for the cylindrical panels of rectangular elongated shape in plan of different thickness under the action of the front load generated by the air blast. Shallow shells are rationally reinforced in the directions of principal stresses and strains and at the supported edges they are rigidly clamped. It is shown that in some cases the Reddy theory is absolutely inadmissible to produce adequate results of calculations of the elastic-plastic deformable composite shallow shells, even with a relatively small thickness. It is demonstrated that, due to the geometrical and physical nonlinearity of the investigated problem, the dynamic behavior of reinforced curved panels significantly depends on the fact that the form of the front surface of the shell, subjected to the external load, is convex or concave. It is established that the most probable mechanism of pre-destruction of such structures is the accumulation of damage due to low-cycle fatigue of binder that occurs in the oscillation process of a reinforced structure.

Experiment and numerical modeling suspended ceiling with identification of working diagram material

The aim of this topic is to describe creating of working diagrams of structural materials. These will be used as input parameters for nonlinear analysis of real structures. Working diagrams of construction material are derived from experimental tensile tests and numerical analysis. We will use construction steel in this case. Derived working diagrams are used as an input parameter for numerical modelling of real structures. They are also used as a comparison with data from realized physical tests. Numerical modelling is based on 3D model of structure and it takes into account physical and geometrical nonlinearities.

Mechanical model of carbon dioxide vibrational spectrum

Classical dynamics methods have been used to study the nonlinear vibrations of a CO2 molecule. Consideration includes not only the anharmonicity valence angle, which enables one to explain the Fermi resonance, but also the physical nonlinearity of the force field (stiffness and softness of springs). In the farthest neighbor approximation (with regard to oxygen–oxygen interaction), a set of nonlinear differential equations in the Lagrangian form has been derived. Their analytical solution has been derived using the method of invariant normalization. The occurrence of a strange attractor has been discovered by numerical simulation. Recommendations for the selection of initial conditions are given that take into account the possibility of regular beatings that change into to chaotic beatings.

Calculation and Designing of Reinforced-Concrete Non-Pressure Pipes with Inner Lining

This article presents the results in investigations of the load-carrying ability of reinforced-concrete non-pressure pipes with the inner lining intended to protect concrete against corrosion. The lining (covers) are mounted as a stay-in-place formwork when manufacturing the pipes. To manufacture the covers, polyethylene sheets with the thickness of 3-5 mm are used. To be fastened in the pipe wall concrete, the linings are provided with special anchoring elements. Two lining types are considered: 1 – with anchoring elements in the form of solid longitudinal ribs; 2 – with anchoring elements distributed equidistantly over the shell surface. The investigations have been performed by the numerical simulation method using a volumetric finite-element model. The load was applied according to a trilinear pattern which is used to test the pipes for strength (including that specified in ЕN 1916). The calculation was performed by the iteration method taking into account the physical non-linearity of concrete. It has been determined that the load-carrying ability of the pipes with the first-type protective linings is lower than that of the pipes with the second-type linings. The maximum (up to 25%) reduction of the load-carrying ability is observed in the pipes reinforced with a single cylindrical cage. To protect the interior faces of reinforced-concrete non-pressure pipes, it is recommended to use the lining covers with the V-type discrete anchoring members ensuring the reliable mechanical fastening of the linings in the pipe wall concrete. This type of the anchoring elements causes practically no effect upon the load-carrying ability of the pipes.DOI: http://dx.doi.org/10.5755/j01.sace.15.2.15218

Influence of geometric and physical nonlinearities on the internal resonances of a finite continuous rod with a microstructure

In this work, nonlinear longitudinal vibrations of a finite composite rod are studied including geometric and physical nonlinearities. An original boundary value problem for a heterogeneous rod yielded by the macroscopic approximation obtained earlier by the higher-order asymptotic homogenization method is used. The effects of internal resonances and modes coupling are predicted, validated and analyzed. The defined novel continuous problem governed by PDEs is solved using space-discretization and the method of multiple time scales. We are aimed at understanding and analyzing how the presence of the microstructure influences the processes of mode interaction. It is shown that, depending on a scaling relation between the amplitude of the vibrations and the size of the unit cell, different scenarios of the modes coupling can be realized. Additionally to the asymptotic solution, numerical simulation of the modes coupling is performed by means of the Runge–Kutta fourth-order method. The obtained numerical and analytical results demonstrate good qualitative agreement.

Limit Dependences in Stability Calculations With Account for Physical Nonlinearity

AbstractStability of bars, plates, shells, and other thin-walled structures in conditions of small physical nonlinearity is considered, when stresses exceed the proportionality limit, the amount of deformations being limited. Shanley's concept is used. The critical state is determined by means of some limit dependences. In a large number of cases, when creating efficient highly-stressed constructions, limited plastic deformations are allowed in them. When analysing stability in the critical state, the calculated stresses turn out to exceed the proportionality limit and the Young's modulus of elasticity turns out to be greater than the tangent modulus corresponding to the calculated stress on the diagram “deformation-stress”. The objective of this work is to show that stability calculation beyond the proportionality limit is reduced to the analysis of some limit dependences as well as to develop a general solution algorithm for similar problems.

Observer-based sensor fault estimation in nonlinear systems

Sensor bias faults and sensor gain faults are two important types of faults in sensor. Simultaneous estimation of these sensor faults in nonlinear systems in the presence of input disturbance and measurement noise is challenging and has not been adequately addressed in literature. Hence, this article develops an observer-based sensor fault estimation method for generalized sector-bounded nonlinear systems in the presence of input disturbance and measurement noise. A generalized sector-bounded nonlinearity was chosen because it encompasses a wide range of nonlinearities including Lipschitz, positive real, and dissipative. This article presents necessary and sufficient conditions to achieve a suboptimal cost for a cost function consisting of the sum of the square integrals of the estimation errors to the square integrals of the disturbances in the form of linear matrix inequality. The linear matrix inequality can be solved offline to explicitly calculate observer gain, and the resulting observer simultaneously estimates the system states as well as both bias and gain faults in the sensors. Compared to previous literature, the proposed methodology is designed to work in the presence of both input disturbance and measurement noise. Additionally, this article considers a generalized sector-bounded nonlinearity which encompasses a variety of different physical nonlinearities. Furthermore, the observer does not require the online solution of the Riccati equation and is thus computationally less intensive compared with the methods of extended Kalman filtering. The observer design procedure is demonstrated through two illustrative examples consisting of a fourth-order double spring–mass system and a third-order wind turbine power transmission mechanism.

Chaotic Map Construction from Common Nonlinearities and Microcontroller Implementations

This work presents novel discrete-time chaotic systems with some known physical system nonlinearities. Dynamic behaviors of the models are examined with numerical methods and Arduino microcontroller-based experimental studies. Many new chaotic maps are generated in the form of [Formula: see text] and high-dimensional chaotic systems are obtained by weak coupling or cross-coupling the same or different chaotic maps. An application of the chaotic maps is realized with Arduino for chaotic pulse width modulation to drive electrical machines. It is expected that the new chaotic maps and their microcontroller implementations will facilitate practical chaos-based applications in different fields.

Nonlinear elastic effects in phase field crystal and amplitude equations: Comparison toab initiosimulations of bcc metals and graphene

We investigate non-linear elastic deformations in the phase field crystal model and derived amplitude equations formulations. Two sources of non-linearity are found, one of them based on geometric non-linearity expressed through a finite strain tensor. It reflects the Eulerian structure of the continuum models and correctly describes the strain dependence of the stiffness. In general, the relevant strain tensor is related to the left Cauchy-Green deformation tensor. In isotropic one- and two-dimensional situations the elastic energy can be expressed equivalently through the right deformation tensor. The predicted isotropic low temperature non-linear elastic effects are directly related to the Birch-Murnaghan equation of state with bulk modulus derivative $K'=4$ for bcc. A two-dimensional generalization suggests $K'_{2D}=5$. These predictions are in agreement with ab initio results for large strain bulk deformations of various bcc elements and graphene. Physical non-linearity arises if the strain dependence of the density wave amplitudes is taken into account and leads to elastic weakening. For anisotropic deformations the magnitudes of the amplitudes depend on their relative orientation to the applied strain.

On the contact interaction between two rectangular plates

A mathematical model of contact interaction between two plates is presented, considering certain types of nonlinearity of each of the plates. Stress–strain state (SSS) of the interacting structural members is analyzed by the method of variational iterations, and the theorem of convergence of this method is provided. An iterative procedure for solving contact problems is developed and its convergence is also proved. Physical nonlinearity is considered by means of the method of variable parameters of elasticity. The SSS of a two-layer system of rectangular plates, depending on a type of boundary conditions as well as distances between plates, is investigated and supplemented with stress–strain curves $$\sigma _i^{(i)} (e_i^{(i)} )$$ for each of the plates.

Modeling of crack propagation on a mesoscopic length scale

AbstractModeling of crack propagation in materials has long been a challenge in solid‐state physics and materials science. The phase‐field method is basically developed for modeling solidification processes and has now established as one of the tools for the description of crack propagation. The applied models are thermodynamically consistent and predict crack propagation in homogeneous materials under the consideration of different loading types, multiple physical fields, geometrical and physical nonlinearities. One of the open challenges is the modeling of crack propagation on a mesoscopic length scale in multiphase or multi‐grain systems. In this work, we summarize our preliminary work to get closer to the goal of describing crack propagation on a mesoscopic length scale. (© 2016 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Auxeticity from nonlinear vibrational modes

AbstractAbstractauthoren It is demonstrated that the large‐amplitude, short‐wavelength vibrational modes excited in the lattice can change its elastic properties due to physical and/or geometric nonlinearity of the lattice bonds. Depending on the symmetry of the vibrational mode the symmetry of the elastic properties of the lattice can also change. Using as an example the two‐dimensional honeycomb structure with ‐FPU pairwise interparticle interactions, we demonstrate that the excitation of a large‐amplitude vibrational mode in combination with equiaxial tensile strain can change the sign of the Poisson's ratio from positive to negative, thus leading to the auxetic property of the lattice. It is shown that the considered lattice supports discrete breathers, i.e., spatially localized nonlinear vibrational modes. The excitation of the discrete breathers as a result of the modulational instability of the extended short‐wavelength modes is analyzed. Our results contribute to the understanding of the relation between the elastic properties and nonlinear dynamics of the lattices of interacting particles.Extended vibrational modes in the 2D honeycomb lattice presented by the stroboscopic pictures of the particle motion. Excitation of such modes with sufficiently large amplitude in combination with equiaxial tension changes the sign of the Poisson's ratio of the lattice from positive to negative.

Failure Surfaces for Common Brickwork Settings Used in Historical Masonry Structures

ABSTRACTThe need to preserve and protect the historical masonry structures has been the emergence of researches directed towards the analytical modeling of failure in masonry. However, since masonry is a composite material made of units (brick, blocks, etc.) and mortar, the fracturing mechanism in masonry is a complex phenomenon due to its components’ distinct material properties. Masonry is also a material which exhibits distinct directional properties because the mortar joints act as planes of weakness. An accurate analysis of masonry structures in a macro-modeling (or composite) perspective requires a material description for all stress states. Uniaxial behavior of masonry is dictated by the tensile cracking and compressive crushing mechanisms, while in-plane mechanical behavior of masonry panels under biaxial loading is a more complex phenomenon. Due to the orthotropic nature of masonry, its failure cannot be defined simply in terms of a criterion based on the principal stresses at any point; therefore, the influence of a third variable, must be also considered. One suitable graphical representation of the failure surface for in-plane behavior of masonry panels could be in terms of the full stress components (σx, σy, τxy) whereas the material axes are assumed to be defined by the bed joints direction (y direction). Another possible representation can be obtained in terms of principal stresses and an angle, θ, which measures the angle between the principal stress axes and the material axes. Clearly, different diagrams in the principal stress planes are found according to different values of θ. This study focusing on a series of masonry models with the roman brick setting (Wright 2009) aimed to develop a complete (σ1, σ2, θ) failure surface for all principal stress combinations. Employing micro-modeling approach, the Distinct Element Method was used to numerically model the brickwork unit discrete medium, in which both geometric and physical nonlinearities of the intact bricks and brick-mortar discontinuities are taken into account. For two other types of brickwork settings used in ancient buildings (such as historical vaulted constructions in Tabriz Bazaar) so called Herringbone arrangements (Brunskill 1997), a graphical representation of the failure surface in terms of the full stress vector (σx, σy, τxy) was also obtained.

Numerical Analysis of Spatial Structural Node Bearing Capacity in the View of the Geometrical and Physical Nonlinearity

The authors set the task to calculate and design the connection node of spatial rod constructions; the calculations were carried out in view of the geometrical and physical nonlinearity. To that effect, the solid-state node model was created and a numerical simulation was carried out. Based on calculation results, the authors obtained values of node deformations and equivalent stress, and completed their analysis. It was found that the node clamp disk, one of the node elements, undergoes a plastic deformation when subject to loadings. The node stiffness was enhanced by entering additional elements. Parameters of the enhanced node in the stress-strain state were re-calculated in a nonlinear formulation. The upgraded node construction can be recommended for the efficient industrial use.The proposed design technique allows the authors to evaluate structural behavior at all stages of the life cycle: model creation, preparation of design documentation, manufacture, operation in all modes and conditions.

Load Carrying Capacity of Steel Arch Reinforcement Taking into Account the Geometrical and Physical Nonlinearity

The paper deals with the calculation of the load carrying capacity of the steel arch reinforcements of underground and mine works with respect to the resulting large displacement and physical nonlinearities. Solution is based on the application of the so-called effective bending stiffness, which is defined as a function of the axial force and bending moment. The numerical model was verified using the values of the load carrying capacity, which have been experimentally obtained using strain-stress test, and implemented into the software that allows very effectively calculate load carrying capacity of steel arch reinforcements.

An approximate analytical method for stress analysis in a viscoelastic body with a circular inclusion considering the geometric and physical nonlinearity

An approximate analytical method for stress analysis in a viscoelastic body with a circular inclusion considering the geometric and physical nonlinearity