In this paper, the nonlinear vibrations of a Rectangular hyperelastic membrane resting on a nonlinear elastic Winkler-Pasternak foundation subjected to uniformly distributed hydrostatic pressure are investigated. The membrane is composed of an incompressible, homogeneous, and isotropic material. The elastic foundation includes two Winkler and Pasternak linear terms and a Winkler term with cubic nonlinearity. Using the theory of thin hyperelastic membrane, Hamilton’s principle, and assuming the finite deformations, the governing equations are obtained. Also, the kinetic energy, the work of uniform distributed force and pressure, and the effects of damping are determined, according to the strain energy function for neo-Hookean hyperelastic constitutive law. By applying Galerkin’s method, the nonlinear partial differential equation of motion in the transversal direction is transformed to the ordinary differential equations. Then, utilizing the method of multiple scales, the superharmonic and subharmonic resonances including the 1:3 superharmonic and 3:1 subharmonic, 1:5 superharmonic, and 5:1 subharmonic, 1:7 superharmonic, and 7:1 subharmonic are analyzed. Also, the analytical results are compared with those presented by other researchers. Finally, the effect of the Winkler and Pasternak stiffness, the material properties, and various geometrical characteristics on the superharmonic and subharmonic resonances of the vibration behavior of a Rectangular hyperelastic membrane is investigated.

The aim of this paper is the formulation and numerical simulation of the growth phenomenon in skin under mechanical loading. The main feature and the novelty of the present research is that it models the skin as a membrane that obeys the constitutive equations of hyperelastic materials. Moreover, the membrane is not necassrily flat, and ca have arbitrary initial curvature in its reference configuration. At first, kinematics of membranes under large deformations is formulated and the essential tensors to be used in the next sections are introduced. Afterwards, fundamentals of the formulation of growth mechanics and its specialization for membranes are presented. In this work, growth phenomenon is characterized as an transeversely isotropic growth which accurs through a single scalar-valued growth multiplier which is defined in the surface where the growth phenomenon takes place. Growth parameter is considerd to be an internal varable that obeys a n evolution equation, which is a first-order differential equation of time. In addition, to solve the evolution equation for growth mltiplier, an unconditionally stable Euler backward method is employed. The compressible neo-Hookean strain energy density function is used to derive the expressions for the stress and the fourth-order elasticity tensors. For numerical solution of governing equations, a Total Lagrangian nonlinear finite element formulation is developed. Finally, as numerical examples, growth and large deformation of skin considering initially flat with three square, circular and Rectangular geometries, as well as an initially curved cylindrical sector under external pressure loading is investigated. Even though the presented model in this paper is much simpler than the preceding ones, the obtained results are in agreement with those available in the literature. Moreover, numerical calculations and storage space are remarkably reduced by the present formulation, so that the number of membrane elements used in the present work is one percent of that of three-dimensional elements employed in the literature.

A finite-element formulation based on triangular membranes of any order is proposed to analyze problems involving highly deformable hyperelastic materials under plane-stress conditions. The element kinematics is based on positional description and the degrees of freedom are the current plane coordinates of the nodes. Two isotropic and nonlinear hyperelastic models have been selected: the compressible neo-Hookean model and the incompressible Rivlin– Saunders model. The constitutive relations and the consistent tangent operator are condensed to the compact 2D forms imposing plane-stress conditions. The resultant algorithm is implemented in a computer code. Three benchmark problems are numerically solved to assess the formulation proposed: the Cook’ s membrane, involving bending, shear, and a singularity point; a partially loaded membrane, which presents severe mesh distortion and large compression levels; and a rubber sealing, which is a more realistic problem. Convergence analysis in terms of displacements, applied forces, and stresses is performed for each problem. It is demonstrated that mesh refinement avoids locking problems associated with incompressibility condition, bending-dominated problems, stress concentration, and mesh distortion. The processing times are relatively small even for fifth-order elements.

In this paper, numerical spline-based differential quadrature is presented for solving the boundary andinitial value problems, and its application is used to solve the fixed Rectangular membrane vibrationequation. For the time integration of the problem, the Runge– Kutta and spline-based differential quadraturemethods have been applied. The Runge– Kutta method was unstable for solving the problem, with largeerrors in its results, but the spline-based differential quadrature method obtained results that agree with theexact solution. The relative errors were calculated and investigated for different values of time and spatialnodes of discretisation. It seems that the spline-based differential quadrature method is proper for the fullsimulation of membrane vibration in both spatial and temporal solutions. For the time solving of themembrane vibration, conventional methods, such as the Runge– Kutta method, are not useful even if thetime steps are considered too small.

An elastomer is a polymer with the property of viscoelasticity, generally having notably low Young's modulus and high yield strain compared with other materials. Elastomers, in particular rubbers, are used in a wide variety of products ranging from rubber hoses, isolation bearings, and shock absorbers to tires. Rubber has good properties and is thermal and electrical resistant. We used carbon nanotube in rubber and modeled this composite with ABAQUS software. Because of hyper elastic behavior of rubber we had to use a strain energy function for nanocomposites modeling. A sample of rubber was tested and gained uniaxial, biaxial and planar test data and then the data used to get a good strain energy function. Mooney-Rivlin form, Neo-Hookean form, Ogden form, Polynomial form, reduced polynomial form, Van der Waals form etc, are some methods to get strain function energy. Modulus of elasticity and Poisson ratio and some other mechanical properties gained for a representative volume element (RVE) of composite in this work. We also considered rubber as an elastic material and gained mechanical properties of composite and then compared result for elastic and hyperelastic rubber matrix together.

In this research, in order to investigate the effect of the piezoelectric patch which is used as a sensor or actuator in rotating flexible structures such as a helicopter blade, the free vibrations of the rotating Rectangular sheet with and without the piezoelectric patch have been presented. First-order shear deformation theory is considered for plate displacement and piezoelectric field. Considering the effect of Coriolis acceleration, centrifugal acceleration and centrifugal in-plane forces, the equations of motion are derived from Hamilton's principle and the electromechanical couple equation is obtained from Maxwell's equation. For piezoelectric, two electrical conditions, open circuit and closed circuit, which are used in sensors and actuators, respectively, have been considered. The equations are discretized with the help of the numerical method of generalized differential squares and the matrices of inertia mass, eccentricity, Coriolis and stiffness matrix are obtained. Natural frequency values for beam and rotating plate have been compared in Abaqus software. Also, the values obtained from the numerical solution in MATLAB have been verified with articles and ABAQUS, which have high accuracy. The effect of parameters such as hub radius, rotation speed, sheet thickness, aspect ratio, piezoelectric patch thickness and applied voltage on the natural frequency of the system has also been investigated.

Biologic tissues modeling play an important role in understanding the tissue behavior and development of synthetic materials for medical applications. It is also a vital action to develop the predictive models for a wide range of uses including medical and tissue engineering. Various strain energy functions have been introduced to model arteries to date. The newest introduced strain energy function is the Nolan strain energy function. Two-layer arterial modeling using this strain energy function has not been performed so far. In this paper, modeling the arteries was carried out in the form of double layers including media and adventitia and hyperelastic material assumption. At first, governing equations were driven based on continuum mechanics. Boundary conditions including inner pressure of artery, axial load and torque as well as static equilibrium were applied. Moreover, Cauchy stress components were gotten by using the continuum mechanics relations. Then, the equilibrium equations in cylindrical coordinate were obtained by using the Cauchy stress. Stress distribution through the artery wall was specified by solving the resulting nonlinear partial differential equations based on generalized differential quadrature method. In the beginning, the artery modeling was conducted in the form of monolayer including the media layer and the results were compared with experimental ones, comparison between stresses in the artery wall and experimental data showed that the volcanic energy function of Nolan is suitable for modeling. After that, the stress distribution was obtained by artery modeling in the form of double layers including the media and adventitia layers.

This research investigates the creep phenomenon of a polymeric rotating disk using the generalized Maxwell’s visco-hyperelastic model. After extracting the Lagrangian partial differential equation of equilibrium governing the problem, the rotating disk was analyzed by scripting in FlexPDE. The disk modelling in ANSYS with coding in the APDL environment showed that the radial displacement and Von-Mises stress are in excellent agreement with the FlexPDE results. The advantages of FlexPDE over ANSYS include one-dimensional analysis of axisymmetric plane stress instead of two-dimensional analysis, reduction of computational cost, possibility of defining variable thickness (without additional coding) and need for fewer elements in the radial direction to achieve acceptable accuracy (necessity of using 20 elements in FlexPDE compared to 100 elements in ANSYS). It was shown that with the passage of time and the increase in angular velocity, the radial displacement and Von- Mises stress of the rotating disk due to the creep phenomenon increase. It was shown that by increasing the angular velocity and decreasing the power in the thickness profile n_h, the displacement and Von-Mises stress at a specific time increase, but the change in angular velocity (as the applied load) and the change in parameter n_h (as a geometric characteristic) do not have much effect on the relaxation time of the rotating disk.