Love waves in compressible elastic materials with a homogeneous initial stress

2018 ◽  
Vol 24 (8) ◽  
pp. 2576-2590
Author(s):  
K Ejaz ◽  
M Shams
Keyword(s):  
Initial Stress ◽  
Finite Deformation ◽  
Theory Of Elasticity ◽  
Love Waves ◽  
Deformation Tensor ◽  
Prototype Model ◽  
Composite Effect ◽  
Hyperelastic Materials ◽  
Incompressible Materials ◽  
Theoretical Results

In this paper, the motion of Love waves is considered in hyperelastic materials with an initially stressed reference configuration. Here, the Love wave is directed by a compressible layer on a compressible half-space and both are considered to be initially stressed. For the basic formulation of the problem, we make use of the nonlinear theory of elasticity and invariants of the stress tensor and deformation tensor. The equations governing a finite deformation superimposed by infinitesimal motions are used to the study the composite effect of finite deformation and initial stress on wave speed. Graphical illustrations are presented for theoretical results for a prototype model of material and also compared with the results already obtained for incompressible materials.

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 note on the finite plane-strain equations for isotropic incompressible materials

1955 ◽  
Vol 51 (2) ◽  
pp. 363-367 ◽  
Author(s):  
J. E. Adkins
Keyword(s):  
Differential Equations ◽  
Plane Strain ◽  
Energy Function ◽  
Strain Energy ◽  
Strain Energy Function ◽  
Theory Of Elasticity ◽  
Finite Plane ◽  
Incompressible Materials ◽  
Stress Components ◽  
Strain Invariants

For elastic deformations beyond the range of the classical infinitesimal theory of elasticity, the governing differential equations are non-linear in form, and orthodox methods of solution are not usually applicable. Simplifying features appear, however, when a restriction is imposed either upon the form of the deformation, or upon the form of strain-energy function employed to define the elastic properties of the material. Thus in the problems of torsion and flexure considered by Rivlin (4, 5, 6) it is possible to avoid introducing partial differential equations into the analysis, while in the theory of finite plane strain developed by Adkins, Green and Shield (1) the reduction in the number of dependent and independent variables involved introduces some measure of simplicity. Some further simplification is achieved when the strain-energy function can be considered as a linear function of the strain invariants as postulated by Mooney(2) for incompressible materials. In the present paper the plane-strain equations for a Mooney material are reduced to symmetrical forms which do not involve the stress components, and some special solutions of these equations are derived.

scholarly journals A Strut Finite Element for Exact Incompressible Isotropic Hyperelastic Analysis

2018 ◽  
Vol 26 (1) ◽  
pp. 1-9 ◽  
Author(s):  
Vinicius F. Arcaro ◽  
Pietro C. Ferrazzo
Keyword(s):  
Mathematical Model ◽  
Finite Element ◽  
Total Potential Energy ◽  
Deformation Tensor ◽  
Analysis Software ◽  
Element Analysis ◽  
Hyperelastic Materials ◽  
Total Potential ◽  
Nodal Displacements ◽  
The Right

Abstract This text describes a mathematical model of a strut finite element for isotropic incompressible hyperelastic materials. The invariants of the Right Cauchy-Green deformation tensor are written in terms of nodal displacements. The equilibrium problem is formulated as an unconstrained nonlinear programming problem, where the objective function is the total potential energy of the structure and the nodal displacements are the unknowns. The constraint for incompressibility is satisfied exactly, thereby eliminating the need for a penalty function. The results of the examples calculated by the proposed mathematical model show five significant digits in agreement when compared with commercial finite element analysis software.

scholarly journals Comparative analysis of different variants of the Uzawa algorithm in problems of the theory of elasticity for incompressible materials

2016 ◽  
Vol 7 (5) ◽  
pp. 703-707
Author(s):  
Nikita E. Styopin ◽  
Anatoly V. Vershinin ◽  
Konstantin M. Zingerman ◽  
Vladimir A. Levin
Keyword(s):  
Comparative Analysis ◽  
Theory Of Elasticity ◽  
Uzawa Algorithm ◽  
Incompressible Materials

Finite deformation of compressible bonded rubber mounts

1992 ◽  
Vol 27 (3) ◽  
pp. 157-169 ◽  
Author(s):  
C J S Petrie ◽  
M H B M Shariff
Keyword(s):  
Explicit Form ◽  
Finite Deformation ◽  
Small Deformation ◽  
Steel Plates ◽  
Qualitative Behaviour ◽  
Linear Deformation ◽  
Vehicle Suspensions ◽  
Bulk Compressibility ◽  
Simplifying Assumptions ◽  
Theoretical Results

Bearings for bridges, earthquake isolation bearings for buildings, and elements in vehicle suspensions may be made from one or more layers of elastomer between rigid steel plates to which the elastomer is bonded. Using simplifying assumptions based on previous work, simple and explicit forms of solution are developed for finite (non-linear) deformation of bonded blocks, in plane or axisymmetric strain, with special reference to compressible elastic materials. The qualitative behaviour of the explicit form of solution is reasonable and on linearizing it agrees with published experimental and theoretical results for small deformation. The explicit form of solution also compares well with such experimental results as are available for finite deformation and with finite element calculations. Application of the results to laminated blocks, where compressibility is expected to be important is illustrated, and an assessment of the effect of the bulk compressibility of the rubber is offered.

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