scholarly journals A refined dynamic finite-strain shell theory for incompressible hyperelastic materials: equations and two-dimensional shell virtual work principle

2020 ◽  
Vol 476 (2237) ◽  
pp. 20200031 ◽  
Author(s):  
Xiang Yu ◽  
Yibin Fu ◽  
Hui-Hui Dai
Keyword(s):  
Shell Theory ◽  
Finite Strain ◽  
Virtual Work ◽  
Constitutive Relations ◽  
Strain Energy Function ◽  
Two Dimensional ◽  
Weak Formulation ◽  
Virtual Work Principle ◽  
Hyperelastic Materials ◽  
Shell Equations

Based on previous work for the static problem, in this paper, we first derive one form of dynamic finite-strain shell equations for incompressible hyperelastic materials that involve three shell constitutive relations. In order to single out the bending effect as well as to reduce the number of shell constitutive relations, a further refinement is performed, which leads to a refined dynamic finite-strain shell theory with only two shell constitutive relations (deducible from the given three-dimensional (3D) strain energy function) and some new insights are also deduced. By using the weak formulation of the shell equations and the variation of the 3D Lagrange functional, boundary conditions and the two-dimensional shell virtual work principle are derived. As a benchmark problem, we consider the extension and inflation of an arterial segment. The good agreement between the asymptotic solution based on the shell equations and that from the 3D exact one gives verification of the former. The refined shell theory is also applied to study the plane-strain vibrations of a pressurized artery, and the effects of the axial pre-stretch, pressure and fibre angle on the vibration frequencies are investigated in detail.

scholarly journals On a consistent finite-strain plate theory based on three-dimensional energy principle

2014 ◽  
Vol 470 (2171) ◽  
pp. 20140494 ◽  
Author(s):  
Hui-Hui Dai ◽  
Zilong Song
Keyword(s):  
Finite Strain ◽  
Virtual Work ◽  
Plate Theory ◽  
Three Dimensional ◽  
Force Balance ◽  
Two Dimensional ◽  
Field Equations ◽  
Order Error ◽  
Energy Principle ◽  
Balance Structure

This paper derives a finite-strain plate theory consistent with the principle of stationary three-dimensional potential energy under general loadings with a fourth-order error. Starting from the three-dimensional nonlinear elasticity (with both geometrical and material nonlinearity) and by a series expansion, we deduce a vector plate equation with three unknowns, which exhibits the local force-balance structure. The success relies on using the three-dimensional field equations and bottom traction condition to derive exact recursion relations for the coefficients. Associated weak formulations are considered, leading to a two-dimensional virtual work principle. An alternative approach based on a two-dimensional truncated energy is also provided, which is less consistent than the first plate theory but has the advantage of the existence of a two-dimensional energy function. As an example, we consider the pure bending problem of a hyperelastic block. The comparison between the analytical plate solution and available exact one shows that the plate theory gives second-order correct results. Compared with existing plate theories, it appears that the present one has a number of advantages, including the consistency, order of correctness, generality of loadings, applicability to finite-strain problems and no involvement of non-physical quantities.

On a consistent finite-strain shell theory for incompressible hyperelastic materials

2018 ◽  
Vol 24 (5) ◽  
pp. 1320-1339 ◽  
Author(s):  
Yuanyou Li ◽  
Hui-Hui Dai ◽  
Jiong Wang
Keyword(s):  
Shell Theory ◽  
Finite Strain ◽  
Circular Cylindrical Shell ◽  
Force Balance ◽  
Bottom Surface ◽  
Series Expansions ◽  
Surface Boundary ◽  
Hyperelastic Materials ◽  
Governing System ◽  
Shell Equation

In this paper, a consistent finite-strain shell theory for incompressible hyperelastic materials is formulated. First, for a shell structure made of an incompressible material, the three-dimensional (3D) governing system is derived through the variational approach, which is composed of the mechanical field equation and the constraint equation. Then, series expansions of the independent variables are conducted about the bottom surface and along the thickness direction of the shell. The recursive relations of the coefficient functions in the series expansions can be derived from the original 3D governing system. Further from the top surface boundary condition, a 2D vector shell equation is obtained, which represents the local force-balance of a shell element. The associated edge boundary conditions are also proposed. It is verified that shell equation system is consistent with the 3D variational formulation. The weak formulation of the shell equation is established for future numerical calculations. To show the validity of the shell theory, the axisymmetric deformations of a spherical and a circular cylindrical shell made of incompressible neo-Hookean materials are studied. By comparing with the exact solutions, it is shown that the asymptotic solutions obtained from the shell theory attain the accuracy of O( h2).

scholarly journals Automatic Generation of User Material Subroutines for Biomechanical Growth Analysis

2010 ◽  
Vol 132 (10) ◽  
Author(s):  
Jonathan M. Young ◽  
Jiang Yao ◽  
Ashok Ramasubramanian ◽  
Larry A. Taber ◽  
Renato Perucchio
Keyword(s):  
Code Generation ◽  
Growth Analysis ◽  
Soft Tissues ◽  
Constitutive Relations ◽  
Strain Energy Function ◽  
Automatic Generation ◽  
Passive Layer ◽  
Isotropic Layer ◽  
Element Analysis ◽  
Hyperelastic Materials

The analysis of the biomechanics of growth and remodeling in soft tissues requires the formulation of specialized pseudoelastic constitutive relations. The nonlinear finite element analysis package ABAQUS allows the user to implement such specialized material responses through the coding of a user material subroutine called UMAT. However, hand coding UMAT subroutines is a challenge even for simple pseudoelastic materials and requires substantial time to debug and test the code. To resolve this issue, we develop an automatic UMAT code generation procedure for pseudoelastic materials using the symbolic mathematics package MATHEMATICA and extend the UMAT generator to include continuum growth. The performance of the automatically coded UMAT is tested by simulating the stress-stretch response of a material defined by a Fung-orthotropic strain energy function, subject to uniaxial stretching, equibiaxial stretching, and simple shear in ABAQUS. The MATHEMATICA UMAT generator is then extended to include continuum growth by adding a growth subroutine to the automatically generated UMAT. The MATHEMATICA UMAT generator correctly derives the variables required in the UMAT code, quickly providing a ready-to-use UMAT. In turn, the UMAT accurately simulates the pseudoelastic response. In order to test the growth UMAT, we simulate the growth-based bending of a bilayered bar with differing fiber directions in a nongrowing passive layer. The anisotropic passive layer, being topologically tied to the growing isotropic layer, causes the bending bar to twist laterally. The results of simulations demonstrate the validity of the automatically coded UMAT, used in both standardized tests of hyperelastic materials and for a biomechanical growth analysis.

scholarly journals A Finite Element Procedure with Poisson Iteration Method Adopting Pattern Approach Technique for Near-Incompressible Rubber Problems

2014 ◽  
Vol 6 ◽  
pp. 272574 ◽  
Author(s):  
Young-Doo Kwon ◽  
Soon-Bum Kwon ◽  
Xiaozhe Lu ◽  
Hyun-Wook Kwon
Keyword(s):  
Finite Element ◽  
Poisson’S Ratio ◽  
Virtual Work ◽  
Poisson Ratio ◽  
Constitutive Relations ◽  
Poisson's Ratio ◽  
Lagrangian Description ◽  
Hyperelastic Materials ◽  
Finite Element Procedure ◽  
Incompressible Rubber

A finite element procedure is presented for the analysis of rubber-like hyperelastic materials. The volumetric incompressibility condition of rubber deformation is included in the formulation using the penalty method, while the principle of virtual work is used to derive a nonlinear finite element equation for the large displacement problem that is presented in a total-Lagrangian description. The behavior of rubber deformation is represented by hyperelastic constitutive relations based on a generalized Mooney-Rivlin model. The proposed finite element procedure using analytic differentiation exhibited results that matched very well with those from the well-known commercial packages NISA II and ABAQUS. Furthermore, the convergence of equilibrium iteration is quite slow or frequently fails in the case of near-incompressible rubber. To prevent such phenomenon even for the case that Poisson's ratio is very close to 0.5, Poisson's ratio of 0.49000 is used, first, to get an approximate solution without any difficulty; then the applied load is maintained and Poisson's ratio is increased to 0.49999 following a proposed pattern and adopting a technique of relaxation by monitoring the convergence rate. For a given Poisson ratio near 0.5, with this approach, we could reduce the number of substeps considerably.

scholarly journals Alternative derivation of the higher-order constitutive model for six-parameter elastic shells

2021 ◽  
Vol 72 (2) ◽  
Author(s):  
Mircea Bîrsan
Keyword(s):  
Energy Density ◽  
Constitutive Model ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Shell Theory ◽  
Constitutive Relations ◽  
Three Dimensional ◽  
Classical Shell Theory ◽  
Three Dimensional Elasticity ◽  
General Method

AbstractIn this paper, we present a general method to derive the explicit constitutive relations for isotropic elastic 6-parameter shells made from a Cosserat material. The dimensional reduction procedure extends the methods of the classical shell theory to the case of Cosserat shells. Starting from the three-dimensional Cosserat parent model, we perform the integration over the thickness and obtain a consistent shell model of order $$ O(h^5) $$ O ( h 5 ) with respect to the shell thickness h. We derive the explicit form of the strain energy density for 6-parameter (Cosserat) shells, in which the constitutive coefficients are expressed in terms of the three-dimensional elasticity constants and depend on the initial curvature of the shell. The obtained form of the shell strain energy density is compared with other previous variants from the literature, and the advantages of our constitutive model are discussed.

A PD Computed Torque Control Method with Online Self-gain Tuning for a 3UPS-PS Parallel Robot

Robotica ◽  
2021 ◽  
pp. 1-13
Author(s):  
Xiaogang Song ◽  
Yongjie Zhao ◽  
Chengwei Chen ◽  
Liang’an Zhang ◽  
Xinjian Lu
Keyword(s):  
Feedback Loop ◽  
Control Method ◽  
Virtual Work ◽  
Parallel Robot ◽  
Torque Control ◽  
Virtual Work Principle ◽  
Tuning Method ◽  
Computed Torque Control ◽  
Control Feedback ◽  
Computed Torque

SUMMARY In this paper, an online self-gain tuning method of a PD computed torque control (CTC) is used for a 3UPS-PS parallel robot. The CTC is applied to the 3UPS-PS parallel robot based on the robot dynamic model which is established via a virtual work principle. The control system of the robot comprises a nonlinear feed-forward loop and a PD control feedback loop. To implement real-time online self-gain tuning, an adjustment method based on the genetic algorithm (GA) is proposed. Compared with the traditional CTC, the simulation results indicate that the control algorithm proposed in this study can not only enhance the anti-interference ability of the system but also improve the trajectory tracking speed and the accuracy of the 3UPS-PS parallel robot.

scholarly journals Application of the virtual work principle to compute magnetic forces with a volume integral method

2013 ◽  
Vol 27 (3) ◽  
pp. 418-432 ◽  
Author(s):  
Anthony Carpentier ◽  
Nicolas Galopin ◽  
Olivier Chadebec ◽  
Gérard Meunier ◽  
Christophe Guérin
Keyword(s):  
Integral Method ◽  
Virtual Work ◽  
Volume Integral ◽  
Virtual Work Principle ◽  
Magnetic Forces

Study on Drive Characteristic of Multi–Rope Friction Winder Using Functional Virtual Prototyping

2009 ◽  
Vol 628-629 ◽  
pp. 305-310
Author(s):  
Yi Liu ◽  
Guo Ding Chen ◽  
J.S. Li ◽  
Y.J. Xue
Keyword(s):  
Software Package ◽  
3D Model ◽  
Virtual Work ◽  
Virtual Prototyping ◽  
Research Work ◽  
Three Dimensional ◽  
Virtual Work Principle ◽  
Nodal Method ◽  
Hoisting System ◽  
Dynamics Characteristic

The main objective of this study was to model and simulate a reduced three-dimensional (3D) model for researching the hoisting system of a Multi – rope Friction Winder. By introducing the relative nodal method, the simplified dynamic equations have been derived via the virtual work principle and validated on a virtual prototype with the RecurDyn software package. Kinematics and dynamics characteristic date are obtained by computer-aided dynamic simulation of virtual Multi – rope friction winder. The result is in accord with theoretical analysis. The research work will provide a powerful tool and useful method for the design of economic and credible elevator system. The approach can be generalized to analysis other flexible drive fields.

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