scholarly journals Plane Wave Reflection in a Compressible Half Space with Initial Stress

Open Physics ◽  
2019 ◽  
Vol 17 (1) ◽  
pp. 438-448
Author(s):  
Moniba Shams ◽  
Chaudry Masood Khalique ◽  
Taha Aziz
Keyword(s):  
Plane Wave ◽  
Initial Stress ◽  
Half Space ◽  
Strain Energy ◽  
Strain Energy Function ◽  
Homogeneous Plane ◽  
Special Strain ◽  
Theory Of Invariants ◽  
Graphical Illustration ◽  
Theoretical Results

Abstract In this paper, the problem of wave propagation in a compressible half-space with an initial stress is considered. General discussion on the speed of wave in the presence of an initial stress is presented. Furthermore, reflection of a homogeneous plane P−wave is also studied. A special strain energy function dependent on this initial stress is used to understand the response of the materials. Explicit formulas for the reflection coefficients are also presented. General nonlinear theory and the theory of invariants are used to derive theoretical results. Graphical illustration of theoretical results for various numerical values of parameters show that initial stress has considerable bearing on the behavior of a plane wave.

The effect of initial stress or residual stress on elastic energy calculations

1966 ◽  
Vol 56 (2) ◽  
pp. 421-424 ◽  
Author(s):  
R. Burridge ◽  
L. Knopoff
Keyword(s):  
Residual Stress ◽  
Initial Stress ◽  
Physical Interpretation ◽  
Elastic Energy ◽  
Energy Function ◽  
Strain Energy ◽  
Strain Energy Function ◽  
Energy Estimates ◽  
Energy Calculations ◽  
Stress Energy

abstract The first aim of this paper is to define a stress-energy function which coincides with the conventional strain-energy function in a non-prestressed medium, but which has a more natural physical interpretation than strain energy for prestressed media. The second aim is to show how consideration of prestress or residual stress may drastically change theoretical energy estimates for earthquakes.

Reflection of plane waves from the boundary of an initially stressed incompressible half-space

2017 ◽  
Vol 24 (2) ◽  
pp. 406-433 ◽  
Author(s):  
M Shams
Keyword(s):  
Initial Stress ◽  
Half Space ◽  
Plane Waves ◽  
Strain Energy Function ◽  
Prototype Strain ◽  
Incompressible Material ◽  
Elastic Half Space ◽  
Theory Of Elasticity ◽  
Plane Boundary ◽  
Infinitesimal Deformations

In this paper, nonlinear theory of elasticity is used to study the effect of initial stress on plane waves in an incompressible material. For this problem, the initial stress is not associated with a finite elastic deformation and the material is assumed to be isotropic in the absence of the initial stress. The theory of superposition of infinitesimal deformations on finite deformation is applied to a problem of plane incremental motions in an initially stressed incompressible homogeneous elastic half-space. The general formulation of the problem is presented first and then specialized using a prototype strain energy function. Homogeneous plane waves are considered and the analysis is carried out for incompressible materials in both the deformed and the undeformed reference configurations. In addition to this, problems for the reflection of small amplitude homogeneous waves from the plane boundary of an initially stressed half-space are also considered and graphical results are included, which show the effect of initial stress on reflection. It is noted that the reflection coefficients in this case behave in a similar fashion when the initial stress is a pre-stress.

Interfacial waves and deformations in pre-stressed elastic media

1991 ◽  
Vol 433 (1888) ◽  
pp. 313-328 ◽  
Keyword(s):  
Half Space ◽  
Strain Energy ◽  
Wave Speed ◽  
Principal Axis ◽  
Secular Equation ◽  
Sufficient Conditions ◽  
Strain Energy Function ◽  
Interfacial Wave ◽  
Existence Criterion ◽  
Necessary And Sufficient

The propagation of interfacial (Stoneley) waves along the boundary between two half-spaces of pre-stressed incompressible isotropic elastic material is examined. The underlying deformation in each half-space corresponds to a pure homogeneous strain with one principal axis of strain normal to the interface and the others having a common orientation. The secular equation governing the wave speed for propagation along a principal axis is obtained in respect of general strain-energy functions. Detailed analysis of the secular equation reveals general sufficient conditions for the existence of a wave and, in particular cases, necessary and sufficient conditions for the existence of a unique interfacial wave. It is also shown that when an interfacial wave exists its speed is greater than that of the least of the Rayleigh wave speeds for the separate half-spaces, paralleling a result from the linear theory. For the special case of quasi-static interfacial deformations (corresponding to vanishing wave speed) an existence criterion is found; moreover, it is shown that inequalities that exclude surface deformations in each half-space also exclude interfacial deformations. Dependence of the above results on the underlying homogeneous deformations and on material parameters is illustrated by numerical results for the neo-hookean strain-energy function.

Some dynamic properties of a prestressed incompressible hyperelastic material

1973 ◽  
Vol 8 (1) ◽  
pp. 61-73 ◽  
Author(s):  
J.A. Belward
Keyword(s):  
Wave Propagation ◽  
Plane Wave ◽  
Energy Function ◽  
Strain Energy ◽  
Dynamic Properties ◽  
Strain Energy Function ◽  
Hyperelastic Material ◽  
Elastic Materials ◽  
Incompressible Hyperelastic Material ◽  
Plane Wave Propagation

The work continues some earlier investigations into dynamic properties of prestressed incompressible elastic materials. Whereas the material was previously assumed to be a Mooney material, it is here allowed to have any strain energy function. Plane wave propagation and the motions caused by an impulsive line of traction are examined. The results obtained are compared with the earlier work.

Mechanical power and hyperelasticity

2017 ◽  
Author(s):  
David J. Steigmann
Keyword(s):  
Energy Function ◽  
Strain Energy ◽  
Strain Energy Function ◽  
Mechanical Power ◽  
Elastic Materials ◽  
Cyclic Motion

This chapter covers the notion of hyperelasticity—the concept that stress is derived from a strain—energy function–by invoking an analogy between elastic materials and springs. Alternatively, it can be derived by invoking a work inequality; the notion that work is required to effect a cyclic motion of the material.

scholarly journals Modelling the Inflation and Elastic Instabilities of Rubber-Like Spherical and Cylindrical Shells Using a New Generalised Neo-Hookean Strain Energy Function

2021 ◽  
Author(s):  
Afshin Anssari-Benam ◽  
Andrea Bucchi ◽  
Giuseppe Saccomandi
Keyword(s):  
Experimental Data ◽  
Energy Function ◽  
Limit Point ◽  
Strain Energy ◽  
Cylindrical Shells ◽  
Strain Energy Function ◽  
Model Parameters ◽  
Wide Range ◽  
Cylindrical Tubes ◽  
Gent Model

AbstractThe application of a newly proposed generalised neo-Hookean strain energy function to the inflation of incompressible rubber-like spherical and cylindrical shells is demonstrated in this paper. The pressure ($P$ P ) – inflation ($\lambda $ λ or $v$ v ) relationships are derived and presented for four shells: thin- and thick-walled spherical balloons, and thin- and thick-walled cylindrical tubes. Characteristics of the inflation curves predicted by the model for the four considered shells are analysed and the critical values of the model parameters for exhibiting the limit-point instability are established. The application of the model to extant experimental datasets procured from studies across 19th to 21st century will be demonstrated, showing favourable agreement between the model and the experimental data. The capability of the model to capture the two characteristic instability phenomena in the inflation of rubber-like materials, namely the limit-point and inflation-jump instabilities, will be made evident from both the theoretical analysis and curve-fitting approaches presented in this study. A comparison with the predictions of the Gent model for the considered data is also demonstrated and is shown that our presented model provides improved fits. Given the simplicity of the model, its ability to fit a wide range of experimental data and capture both limit-point and inflation-jump instabilities, we propose the application of our model to the inflation of rubber-like materials.

Mechanical characterization of a woven multi-layered hyperelastic composite laminate under uniaxial loading

2021 ◽  
pp. 002199832110115
Author(s):  
Shaikbepari Mohmmed Khajamoinuddin ◽  
Aritra Chatterjee ◽  
MR Bhat ◽  
Dineshkumar Harursampath ◽  
Namrata Gundiah
Keyword(s):  
Composite Materials ◽  
Energy Function ◽  
Strain Energy ◽  
Strain Energy Function ◽  
Mechanical Tests ◽  
Stress Strain ◽  
Aerospace Applications ◽  
Envelope Material ◽  
The Individual ◽  
High Dielectric

We characterize the material properties of a woven, multi-layered, hyperelastic composite that is useful as an envelope material for high-altitude stratospheric airships and in the design of other large structures. The composite was fabricated by sandwiching a polyaramid Nomex® core, with good tensile strength, between polyimide Kapton® films with high dielectric constant, and cured with epoxy using a vacuum bagging technique. Uniaxial mechanical tests were used to stretch the individual materials and the composite to failure in the longitudinal and transverse directions respectively. The experimental data for Kapton® were fit to a five-parameter Yeoh form of nonlinear, hyperelastic and isotropic constitutive model. Image analysis of the Nomex® sheets, obtained using scanning electron microscopy, demonstrate two families of symmetrically oriented fibers at 69.3°± 7.4° and 129°± 5.3°. Stress-strain results for Nomex® were fit to a nonlinear and orthotropic Holzapfel-Gasser-Ogden (HGO) hyperelastic model with two fiber families. We used a linear decomposition of the strain energy function for the composite, based on the individual strain energy functions for Kapton® and Nomex®, obtained using experimental results. A rule of mixtures approach, using volume fractions of individual constituents present in the composite during specimen fabrication, was used to formulate the strain energy function for the composite. Model results for the composite were in good agreement with experimental stress-strain data. Constitutive properties for woven composite materials, combining nonlinear elastic properties within a composite materials framework, are required in the design of laminated pretensioned structures for civil engineering and in aerospace applications.

Three-Dimensional Stress Distribution in Arteries

1983 ◽  
Vol 105 (3) ◽  
pp. 268-274 ◽  
Author(s):  
C. J. Chuong ◽  
Y. C. Fung
Keyword(s):  
Experimental Data ◽  
Stress Distribution ◽  
Vessel Wall ◽  
Strain Energy ◽  
Arterial Wall ◽  
Three Dimensional ◽  
Strain Energy Function ◽  
Exponential Form ◽  
Strain Relationship ◽  
Longitudinal Stretch

A three-dimensional stress-strain relationship derived from a strain energy function of the exponential form is proposed for the arterial wall. The material constants are identified from experimental data on rabbit arteries subjected to inflation and longitudinal stretch in the physiological range. The objectives are: 1) to show that such a procedure is feasible and practical, and 2) to call attention to the very large variations in stresses and strains across the vessel wall under the assumptions that the tissue is incompressible and stress-free when all external load is removed.

Bifurcation of rotating circular cylindrical elastic membranes

1980 ◽  
Vol 87 (2) ◽  
pp. 357-376 ◽  
Author(s):  
D. M. Haughton ◽  
R. W. Ogden
Keyword(s):  
Axial Force ◽  
Strain Energy ◽  
Elastic Strain Energy ◽  
Strain Energy Function ◽  
Angular Speed ◽  
Elastic Membrane ◽  
Axial Extension ◽  
End Conditions ◽  
Elastic Membranes ◽  
Realistic Choice

SummaryBifurcation from a finitely deformed circular cylindrical configuration of a rotating circular cylindrical elastic membrane is examined. It is found (for a physically realistic choice of elastic strain-energy function) that the angular speed attains a maximum followed by a minimum relative to the increasing radius of the cylinder for either a fixed axial extension or fixed axial force.At fixed axial extension (a) a prismatic mode of bifurcation (in which the cross-section of the cylinder becomes uniformly non-circular) may occur at a maximum of the angular speed provided the end conditions on the cylinder allow this; (b) axisyim-metric modes may occur before, at or after the angular speed maximum depending on the length of the cylinder and the magnitude of the axial extension; (c) an asymmetric or ‘wobble’ mode is always possible before either (a) or (b) as the angular speed increases from zero for any length of cylinder or axial extension. Moreover, ‘wobble’ occurs at lower angular speeds for longer cylinders.At fixed axial force the results are similar to (a), (b) and (c) except that an axisym-metric mode necessarily occurs between the turning points of the angular speed.

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