strains are applied to the integration procedure in Nonlinear increments to decrease the errors arising from the linearization of plastic equations. Two deformation vectors are used to achieve this. The first vector is based on the deformations obtained by the first iteration of the equilibrium step, and the second is acquired from the sum of the succeeding iterations. By applying these vectors and using sub-increments, the total strain increment can vary Nonlinearly during the integration of the flow rule. Four individual variation schemes are presented for this purpose. In this paper, the strain space formulation is investigated. Numerical examples are analyzed using the traditional linear method and the suggested schemes. The examples are solved using the von Mises yield criterion and Prager's linear hardening rule. Results indicate that all Nonlinear techniques increase the convergence rate of plastic analysis. In addition, such integration methods are shown to increase the stability of incremental-iterative analyses.

In this study, Nonlinear vibration of a composite cable is investigated by considering Nonlinear stress-strain relations. The composite cable is composed of an aluminum wire as reinforcement and a rubber coating as matrix. The Nonlinear governing equations of motion are derived about to an initial curve and based on the fundamentals of continuum mechanics and the Nonlinear Green-Lagrangian strain, using the Hamilton's principle. The equations of motion of the system are reduced into the ordinary differential equations using the Galerkin method, and solved by the perturbation method (multiple scales). Time-response diagrams are presented for the composite cable with different volume fractions of the matrix and the reinforcement. It is predicted that the more volume fraction of the matrix is, the more Nonlinear quadratic and cubic terms in the governing equations of motion affects the vibration of the cable. Results indicate that increasing the length of the cable would decrease the amplitude of the time response of the system.

This study was conducted to compare some Nonlinear functions to describe the broiler growth curve of the Cobb500 strain. A flock of fifty one-day-old chicks were randomly selected from a henhouse of 2500 chicks. Our goal was to establish a growth curve using weighting data using mathematical solutions of time-dependent differential functions. In total, six equations were subjected to a statistical calibration by a sequential quadratic programming under the non-linear regression procedure of the SPSS program. The results showed that the heterogeneity rate between individuals of the same batch increases with the age of the chicks, from more than 10% an early age to less than 30% at the slaughter age. The goodness of fit for six dynamic models showed that the number of iterations required increases with the number of parameters of the model. However, the three parameter models were the best model for describing growth curve (the greatest efficiencies and the lowest error components). The asymptomatic values (3500g to 7500g) and their estimation errors (2% to 12%) are relatively acceptable for the three-parameter models compared to those of four parameters (more than 8000g and up to 100% error). Finally, the comparison between actual and predicted values by models shows that the Gompertz model was the most suitable till up to the four weeks of age. After 1 month of age, the Gompertz has a lower precision and the logistics, Von Bertalonffy and WLS models accurately described the growth curve.

A viscoelastic microcantilever beam is analytically analyzed based on the modi ed strain gradient theory. Kelvin-Voigt scheme is used to model beam viscoelasticity. By applying Euler-Bernoulli inextensibility of the centerline condition based on Hamilton's principle, the Nonlinear equation of motion and the related boundary conditions are derived from shortening e ect theory and discretized by Galerkin method. Inner damping, Nonlinear curvature e ect, and Nonlinear inertia terms are also taken into account. In the present study, the generalized derived formulation allows modeling any Nonlinear combination such as Nonlinear terms that arise due to inertia, damping, and sti ness, as well as modeling the size e ect using modi ed coupled stress or modi ed strain gradient theories. First-mode Nonlinear frequency and time response of the viscoelastic microcantilever beam are analytically evaluated using multiple time scale method and then, validated through numerical  ndings. The obtained results indicate that Nonlinear terms have an appreciable e ect on natural frequency and time response of a viscoelastic microcantilever. Moreover, further investigations suggest that due to the size e ects, natural frequency would drastically increase, especially when the thickness of the beam and the length scale parameter are comparable. The  ndings elaborate the signi cance of size e ects in analyzing the mechanical behavior of small-scale structures.

Electrical energy can be harvested from the vibrations of piezoelectric plates. The behavior of the piezoelectric plate is simulated using electromechanical coupling. Given the large strain experienced by the flexible plate, the linear theory is inadequate; therefore, the effect of Von Karman strain must be considered. This paper investigates and validates the vibrational behavior of a piezoelectric Nonlinear plate. Specifically, it employs coupled equations for a multilayered plate, incorporates Von Karman’s Nonlinear strain, and applies Mindlin’s first-order shear deformation theory. The electrical response is obtained through finite element analysis of the piezoelectric Nonlinear plate in Matlab. Different boundary conditions are considered to verify the results, including clamped edges, simply supported edges, and a combination of two simply supported and two clamped edges. The electrical response of the open-circuit case under harmonic excitation is presented. Furthermore, the phenomenon of voltage cancellation during plate vibrations is studied, and a method for enhancing energy harvesting performance using separated electrodes is proposed. The results indicate that, across all cases, the voltage response with a continuous electrode is lower than with segmented electrodes at the first natural frequency.

In present research, Nonlinear vibration of functionally graded nano-beams subjected to uniform temperature rise and resting on Nonlinear foundation is comprehensively studied. The elastic center can be defined to remove stretching and bending couplings caused by the FG material variation. The small-size effect, playing essential role in the dynamical behavior of nano-beams, is considered here applying strain-inertia gradient and non-local elasticity theory. The governing partial differential equations have been derived based on the Euler-Bernoulli beam theory utilizing the von Karman strain-displacement relations. Subsequently, using the Galerkin method, the governing equations is reduced to a Nonlinear ordinary differential equation. The closed form analytical solution of the Nonlinear natural frequency is then established using the homotopy analysis method. Finally, the effects of different parameters such as length, Nonlinear elastic foundation parameter, thermal loading, non-local parameter and gradient parameters are comprehensively investigated on the FG nano-beams vibration using the homotopy analysis method. As the main results, it is observed that by increasing the non-local parameter, the frequency ratio for strain-inertia gradient theory has an increasing trend while it has decreasing trend for non-local elasticity theory. Also, the Nonlinear natural frequencies obtained using strain-inertia gradient theory are greater than the results of non-local elasticity and classical theory.

In this research, the Nonlinear dynamics of an electrostatically actuated non-uniform microbeam equipped with a damping film and a piezoelectric layer have been studied. The Nonlinear behaviour of the system was modelled using the von Karman geometrical strain terms. In addition, the strain gradient theory was utilized and the Hamilton principle was applied to obtain equations of motion and boundary conditions, respectively. The obtained equations were reduced using the Galerkin method, and the reduced equations were solved with the multiple scale method. The size-dependent responses were then investigated for primary, super-harmonic, and sub-harmonic resonances. The influence of beam width, beam thickness, and distance between electrodes on the resonant frequency response was studied along with Nonlinearity of the system. The results showed that the static and forced vibration behaviours of microbeams strongly depended on the size of the electrodes.