A Finite Element Implementation of Knowles Stored-Energy Function: Theory, Coding and Applications

2011 ◽  
Vol 58 (3) ◽  
pp. 319-346 ◽  
Author(s):  
Cyprian Suchocki
Keyword(s):  
Finite Element ◽  
Energy Function ◽  
Function Theory ◽  
Constitutive Models ◽  
Elasticity Tensor ◽  
Stored Energy ◽  
Energy Potential ◽  
Hyperelastic Materials ◽  
Finite Element Implementation ◽  
Living Tissues

A Finite Element Implementation of Knowles Stored-Energy Function: Theory, Coding and Applications This paper contains the full way of implementing a user-defined hyperelastic constitutive model into the finite element method (FEM) through defining an appropriate elasticity tensor. The Knowles stored-energy potential has been chosen to illustrate the implementation, as this particular potential function proved to be very effective in modeling nonlinear elasticity within moderate deformations. Thus, the Knowles stored-energy potential allows for appropriate modeling of thermoplastics, resins, polymeric composites and living tissues, such as bone for example. The decoupling of volumetric and isochoric behavior within a hyperelastic constitutive equation has been extensively discussed. An analytical elasticity tensor, corresponding to the Knowles stored-energy potential, has been derived. To the best of author's knowledge, this tensor has not been presented in the literature yet. The way of deriving analytical elasticity tensors for hyperelastic materials has been discussed in detail. The analytical elasticity tensor may be further used to develop visco-hyperelastic, nonlinear viscoelastic or viscoplastic constitutive models. A FORTRAN 77 code has been written in order to implement the Knowles hyperelastic model into a FEM system. The performance of the developed code is examined using an exemplary problem.

Implementing Stretch-Based Strain Energy Functions in Large Coupled Axial and Torsional Deformations of Functionally Graded Cylinder

2019 ◽  
Vol 11 (04) ◽  
pp. 1950039 ◽  
Author(s):  
Arash Valiollahi ◽  
Mohammad Shojaeifard ◽  
Mostafa Baghani
Keyword(s):  
Finite Element ◽  
Hoop Stress ◽  
Constitutive Models ◽  
Elastic Solid ◽  
Functionally Graded ◽  
Deformation Tensor ◽  
Element Analysis ◽  
Hyperelastic Materials ◽  
Ogden Model ◽  
Exponential Power

In this paper, coupled axial and torsional large deformation of an incompressible isotropic functionally graded nonlinearly elastic solid cylinder is investigated. Utilizing stretch-based constitutive models, where the deformation tensor is non-diagonal is complex. Hence, an analytical approach is presented for combined extension and torsion of functionally graded hyperelastic cylinder. Also, finite element analysis is carried out to verify the proposed analytical solutions. The Ogden model is employed to predict the mechanical behavior of hyperelastic materials whose material parameters are function of radius in an exponential fashion. Both finite element and analytical results are in good agreement and reveal that for positive values of exponential power in material variation function, stress decreases and the rate of stress variation intensifies near the outer surface. A transition point for the hoop stress is identified, where the distribution plots regardless of the value of stretch or twist, intersect and the hoop stress alters from compressive to tensile. For the Ogden model, the torsion induced force is always compressive which means the total axial force starts from being tensile and then eventually becomes compressive i.e., the cylinder always tends to elongate on twisting.

A Simple w′(λ) Function for The Valanis-Landel Form of Stored Energy Function

1998 ◽  
Vol 71 (2) ◽  
pp. 234-243 ◽  
Author(s):  
Robert F. Landel
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Long Range ◽  
Energy Function ◽  
Good Accuracy ◽  
Linear Range ◽  
Stored Energy ◽  
Element Analysis ◽  
Strong Curvature

Abstract In the Valanis-Landel formulation of the stored energy function W, stresses depend on the function w′(λ)(=dw/dλ). This function exhibits strong curvature, making it difficult to represent analytically with good accuracy. It is found for both SBR and NR that the function λw′(λ) is not only far less curved, it is essentially linear in λ for the range of about 0.4 < λ < 2.0. The long range of simple proportionality to strain invites examination of molecular theories of rubberlike elasticity. Above the linear range the response can be approximated by kλn. These simplifications should make it easier to convert w′(λ) to the W1 and W2 functions employed in finite element analysis.

Design for 1:2 Internal Resonances in In-Plane Vibrations of Plates With Hyperelastic Materials

2013 ◽  
Author(s):  
Astitva Tripathi ◽  
Anil K. Bajaj
Keyword(s):  
Finite Element ◽  
Internal Resonance ◽  
Strain Energy ◽  
Quadratic Function ◽  
The Body ◽  
Energy Potential ◽  
Internal Resonances ◽  
Element Analysis ◽  
Hyperelastic Materials ◽  
Aerial Vehicle

With advances in technology, hyperelastic materials are seeing increased use in varied applications ranging from microfluidic pumps, artificial muscles to deformable robots. They have also been proposed as materials of choice in the construction of components undergoing dynamic excitation such as the wings of a micro-unmanned aerial vehicle or the body of a serpentine robot. Since the strain energy potentials of various hyperelastic materials are more complex than quadratic, exploration of their nonlinear dynamic response lends itself to some interesting consequences. In this work, a structure made of a Mooney-Rivlin hyperelastic material and undergoing planar vibrations is considered. Since the stresses developed in a Mooney-Rivlin material are at least a quadratic function of strain, a possibility of 1:2 internal resonance is explored. A Finite Element Method (FEM) formulation implemented in Matlab is used to iteratively modify a base structure to get its first two natural frequencies close to the ratio 1:2. Once a topology of the structure is achieved, the linear modes of the structure can be extracted from the finite element analysis, and a more complete Lagrangian formulation of the hyperelastic structure can be used to develop a nonlinear two-mode model of the structure. The nonlinear response of the structure can be obtained by application of perturbation methods such as averaging on the two-mode model. It is shown that the strain energy potential for the Mooney-Rivlin material makes it possible for internal resonance to occur in such structures.

scholarly journals The finite element implementation of 3D fractional viscoelastic constitutive models

2018 ◽  
Vol 146 ◽  
pp. 28-41 ◽  
Author(s):  
Gioacchino Alotta ◽  
Olga Barrera ◽  
Alan Cocks ◽  
Mario Di Paola
Keyword(s):  
Finite Element ◽  
Constitutive Models ◽  
Finite Element Implementation

Extending Marlow’s general first-invariant constitutive model to compressible, isotropic hyperelastic materials

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Florian Hüter ◽  
Frank Rieg
Keyword(s):  
Finite Element ◽  
Constitutive Model ◽  
Test Data ◽  
Curve Fitting ◽  
Energy Function ◽  
Strain Energy ◽  
Strain Energy Function ◽  
Content Type ◽  
Hyperelastic Materials ◽  
Fitting Procedures

Purpose A general first-invariant constitutive model has been derived in literature for incompressible, isotropic hyperelastic materials, known as Marlow model, which reproduces test data exactly without the need of curve-fitting procedures. This paper aims to describe how to extend Marlow’s constitutive model to the more general case of compressible hyperelastic materials. Design/methodology/approach The isotropic constitutive model is based on a strain energy function, whose isochoric part is solely dependent on the first modified strain invariant. Based on Marlow’s idea, a principle of energetically equivalent deformation states is derived for the compressible case, which is used to determine the underlying strain energy function directly from measured test data. No particular functional of the strain energy function is assumed. It is shown how to calibrate the volumetric and isochoric strain energy functions uniquely with uniaxial or biaxial test data only. The constitutive model is implemented into a finite element program to demonstrate its applicability. Findings The model is well suited for use in finite element analysis. Only one set of test data is required for calibration without any need for curve-fitting procedures. These test data are reproduced exactly, and the model prediction is reasonable for other deformation modes. Originality/value Marlow’s basic concept is extended to the compressible case and applied to both the volumetric and isochoric part of the compressible strain energy function. Moreover, a novel approach is described on how both compressive and tensile test data can be used simultaneously to calibrate the model.

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