A Pressure Projection Method for Nearly Incompressible Rubber Hyperelasticity, Part II: Applications

1996 ◽  
Vol 63 (4) ◽  
pp. 869-876 ◽  
Author(s):  
Jiun-Shyan Chen ◽  
Cheng-Tang Wu ◽  
Chunhui Pan
Keyword(s):  
Finite Element ◽  
Energy Density ◽  
Projection Method ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Finite Elasticity ◽  
Element Formulation ◽  
Density Functions ◽  
Hyperelastic Materials ◽  
Pressure Projection

In the first part of this paper a pressure projection method was presented for the nonlinear analysis of structures made of nearly incompressible hyperelastic materials. The main focus of the second part of the paper is to demonstrate the performance of the present method and to address some of the issues related to the analysis of engineering elastomers including the proper selection of strain energy density functions. The numerical procedures and the implementation to nonlinear finite element programs are presented. Mooney-Rivlin, Cubic, and Modified Cubic strain energy density functions are used in the numerical examples. Several classical finite elasticity problems as well as some practical engineering elastomer problems are analyzed. The need to account for the slight compressibility of rubber (finite bulk modulus) in the finite element formulation is demonstrated in the study of apparent Young’s modulus of bonded thin rubber units. The combined shear-bending deformation that commonly exists in rubber mounting systems is also analyzed and discussed.

Constitutive Equations for Hyperelastic Materials Based on the Upper Triangular Decomposition of the Deformation Gradient

2018 ◽  
Vol 24 (6) ◽  
pp. 1785-1799 ◽  
Author(s):  
Y. Q. Li ◽  
X.-L. Gao
Keyword(s):  
Energy Density ◽  
Constitutive Equations ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Deformation Gradient ◽  
Density Functions ◽  
Triangular Decomposition ◽  
Deformation Gradient Tensor ◽  
Hyperelastic Materials ◽  
Distortion Tensor

The upper triangular decomposition has recently been proposed to multiplicatively decompose the deformation gradient tensor into a product of a rotation tensor and an upper triangular tensor called the distortion tensor, whose six components can be directly related to pure stretch and simple shear deformations, which are physically measurable. In the current paper, constitutive equations for hyperelastic materials are derived using strain energy density functions in terms of the distortion tensor, which satisfy the principle of material frame indifference and the first and second laws of thermodynamics. Being expressed directly as derivatives of the strain energy density function with respect to the components of the distortion tensor, the Cauchy stress components have simpler expressions than those based on the invariants of the right Cauchy-Green deformation tensor. To illustrate the new constitutive equations, strain energy density functions in terms of the distortion tensor are provided for unconstrained and incompressible isotropic materials, incompressible transversely isotropic composite materials, and incompressible orthotropic composite materials with two families of fibers. For each type of material, example problems are solved using the newly proposed constitutive equations and strain energy density functions, both in terms of the distortion tensor. The solutions of these problems are found to be the same as those obtained by applying the polar decomposition-based invariants approach, thereby validating and supporting the newly developed, alternative method based on the upper triangular decomposition of the deformation gradient tensor.

A Pressure Projection Method for Nearly Incompressible Rubber Hyperelasticity, Part I: Theory

1996 ◽  
Vol 63 (4) ◽  
pp. 862-868 ◽  
Author(s):  
Jiun-Shyan Chen ◽  
Chunhui Pan
Keyword(s):  
Hydrostatic Pressure ◽  
Energy Density ◽  
Least Squares ◽  
Projection Method ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Least Squares Method ◽  
Finite Element Program ◽  
Reduced Integration ◽  
Pressure Projection

A least-squares-based pressure projection method is proposed for the nonlinear analysis of nearly incompressible hyperelastic materials. The strain energy density function is separated into distortional and dilatational parts by the use of Penn’s invariants such that the hydrostatic pressure is solely determined from the dilatational strain energy density. The hydrostatic pressure and hydrostatic pressure increment calculated from displacements are projected onto appropriate pressure fields through the least-squares method. The method is applicable to lower and higher order elements and the projection procedures can be implemented into the displacement based nonlinear finite element program. By the use of certain pressure interpolation functions and reduced integration rules in the pressure projection equations, this method can be degenerated to a nonlinear version of the selective reduced integration method.

scholarly journals Evaluation of the Use of the Yeoh and Mooney-Rivlin Functions as Strain Energy Density Functions for the Ground Substance Material of the Annulus Fibrosus

2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
Héctor E. Jaramillo
Keyword(s):  
Finite Element ◽  
Energy Density ◽  
Intervertebral Disc ◽  
Mechanical Behavior ◽  
Range Of Motion ◽  
Strain Energy Density ◽  
Annulus Fibrosus ◽  
Strain Energy ◽  
Density Functions ◽  
Human Spine

Due to the importance of the intervertebral disc in the mechanical behavior of the human spine, special attention has been paid to it during the development of finite element models of the human spine. The mechanical behavior of the intervertebral disc is nonlinear, heterogeneous, and anisotropic and, due to the low permeability, is usually represented as a hyperelastic model. The intervertebral disc is composed of the nucleus pulposus, the endplates, and the annulus fibrosus. The annulus fibrosus is modeled as a hyperelastic matrix reinforced with several fiber families, and researchers have used different strain energy density functions to represent it. This paper presents a comparative study between the strain energy density functions most frequently used to represent the mechanical behavior of the annulus fibrosus: the Yeoh and Mooney-Rivlin functions. A finite element model of the annulus fibrosus of the L4-L5 segment under the action of three independent and orthogonal moments of 8 N-m was used, employing Abaqus software. A structured mesh with eight divisions along the height and the radial direction of annulus fibrosus and tetrahedron elements for the endplates were used, and an exponential energy function was employed to represent the mechanical behavior of the fibers. A total of 16 families were used; the fiber orientation varied with the radial coordinate from 25° on the outer boundary to 46° on the inner boundary, measuring it with respect to the transverse plane. The mechanical constants were taken from the reported literature. The range of motion was obtained by finite element analysis using different values of the mechanical constants and these results were compared with the reported experimental data. It was found that the Yeoh function showed a better fit to the experimental range of motion than the Mooney-Rivlin function, especially in the nonlinear region.

scholarly journals Mechanical-Geometrical Modeling Of The Hyperelastic Materials At Uniaxial Stretching

2021 ◽  
Vol 2131 (5) ◽  
pp. 052017
Author(s):  
Daniil Azarov
Keyword(s):  
Energy Density ◽  
Strain Energy Density ◽  
Geometric Modeling ◽  
Strain Energy ◽  
Theory Of Elasticity ◽  
Density Functions ◽  
Hyperelastic Materials ◽  
Weakly Compressible ◽  
Nonlinear Elastic ◽  
Nonlinear Theory Of Elasticity

Abstract Hyperelastic materials, such as rubber, occupy an important place in the design and operation of various technological equipment and machines. The article analyzed the deformation behavior of hyperelastic materials using a mechanical-geometric model. The method of mechanical-geometric modeling is a new method for obtaining constitutive relations and strain energy density functions for nonlinear elastic solids. It is based on physically and geometrically consistent prerequisites. The resulting models can describe broad classes of nonlinear elastic materials (both isotropic and anisotropic) depending on the mechanical and geometric properties “embedded” in them at the first stages of design. This paper discusses two basic types of models based on different initial geometry. The mechanical parameters of the models are constants, and the models themselves are considered in a statement corresponding to isotropic hyperelastic materials. The article presents the most common diagrams of deformation of artificial and natural rubbers, as well as steel. Hyperelastic materials, depending on the task, can be described in the nonlinear theory of elasticity as ideal incompressible, or as weakly compressible. Parameters of expressions of strain energy density functions of mechanical-geometric models obtained for cases of incompressible and weakly compressible continuous solids were identified. Stretch diagrams and diagrams of the transverse deformation function of the obtained mechanical-geometric models for the two cases mentioned above are plotted. The extension diagram for the model with parameters corresponding to the classic structural material of the steel type is also shown. Comments are given on the possibility of further paths of developing the method of mechanical-geometric modeling to obtain results not only in the field of nonlinear theory of elasticity, but also viscoelasticity.

Finite Element Modeling of Grain Aspect Ratio and Strain Energy Density in a Textured Copper Thin Film

1996 ◽  
Vol 436 ◽  
Author(s):  
R. P. Vinci ◽  
J. C. Bravman
Keyword(s):  
Finite Element ◽  
Energy Density ◽  
Aspect Ratio ◽  
Finite Element Modeling ◽  
Strain Energy Density ◽  
Driving Force ◽  
Strain Energy ◽  
Interface Energy ◽  
Element Modeling ◽  
Grain Aspect Ratio

AbstractWe have modeled the effects of grain aspect ratio on strain energy density in (100)-oriented grains in a (111)-textured Cu film on a Si substrate. Minimization of surface energy, interface energy, and strain energy density (SED) drives preferential growth of grains of certain crystallographic orientations in thin films. Under conditions in which the SED driving force exceeds the surface- and interface-energy driving forces, Cu films develop abnormally large (100) oriented grains during annealing. In the elastic regime the SED differences between the (100) grains and the film average arise from elastic anisotropy. Previous analyses indicate that several factors (e.g. elimination of grain boundaries during grain growth) may alter the magnitude of the SED driving force. We demonstrate, using finite element modeling of a single columnar (100) grain in a (111) film, that changes in grain aspect ratio can significantly affect the SED driving force. A minimum SED driving force is found for (100) Cu grains with diameters on the order of the film thickness. In the absence of other stagnation mechanisms, such behavior could cause small grains to grow abnormally and then stagnate while large grains continue to grow. This would lead to a bimodal grain size distribution in the (100) grains preferred by the SED minimization.

Large Deformation Analysis of the Arterial Cross Section

1971 ◽  
Vol 93 (2) ◽  
pp. 138-145 ◽  
Author(s):  
B. R. Simon ◽  
A. S. Kobayashi ◽  
D. E. Strandness ◽  
C. A. Wiederhielm
Keyword(s):  
Finite Element ◽  
Energy Density ◽  
Numerical Integration ◽  
Density Function ◽  
Cross Section ◽  
Finite Deformation ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Exponential Form ◽  
Strain Energy Density Function

Possible relations between arterial wall stresses and deformations and mechanisms contributing to atherosclerosis are discussed. Necessary material properties are determined experimentally and from available data in the literature by assuming the arterial response to be a static finite deformation of a thick-walled cylinder constrained in a state of plane strain and composed of an incompressible, nonlinear elastic, transversely isotropic material. Experimental justification from the literature and supporting theoretical considerations are presented for each assumption. The partial derivative of the strain energy density function δW1/δI , necessary for in-plane stress calculation, is determined to be of exponential form using in situ biaxial test results from the canine abdominal aorta. An axisymmetric numerical integration solution is developed and used as a check for finite element results. The large deformation finite element theory of Oden is modified to include aortic material nonlinearity and directional properties and is used for a structural analysis of the aortic cross section. Results of this investigation are: (a) Fung’s exponential form for the strain energy density function of soft tissues is found to be valid for the aorta in the biaxial states considered; (b) finite deformation analyses by the finite element method and numerical integration solution reveal that significant tangential stress gradients are present in arteries commonly assumed to be “thin-walled” tubes using linear theory.

Creep Analysis and Thermal-Fatigue Life Prediction of the Lead-Free Solder Sealing Ring of a Photonic Switch

2002 ◽  
Vol 124 (4) ◽  
pp. 403-410 ◽  
Author(s):  
J. Lau ◽  
Z. Mei ◽  
S. Pang ◽  
C. Amsden ◽  
J. Rayner ◽  
...  
Keyword(s):  
Finite Element ◽  
Fatigue Life ◽  
Energy Density ◽  
Thermal Fatigue ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Constitutive Law ◽  
Isothermal Fatigue ◽  
Thermal Fatigue Life ◽  
Sealing Ring

Thermal reliability of the solder sealing ring of Agilent Technologies’ bubble-actuated photonic cross-connect switches has been investigated in this paper. Emphasis is placed on the determination of the thermal-fatigue life of the solder sealing ring under shipping/storing/handling conditions. The solder ring is assumed to obey the Garofalo-Arrhenius creep constitutive law. The nonlinear responses such as the deflections, stresses, creep strains, and creep strain energy density of the 3-D photonic package have been determined with a commercial finite element code. In addition, isothermal fatigue tests have been performed to obtain the relationship between the number of cycle-to-failure and the strain energy density. Thus, by combining the finite element results and the isothermal fatigue test results, the average thermal-fatigue life of the solder sealing ring is readily determined and is found to be more than adequate for shipping/storing/handling the photonic switches.

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