Simple Waves and Shock Waves Generated by an Incident Shock Wave in Two-Dimensional Hyperelastic Materials

The reflection of an oblique plane shock wave from a boundary in a two-dimensional isotropic hyperelastic material is studied. For plane strain deformations, the strain energy function W is a function of two invariants p and q of the deformation gradient. There are, in general, two reflected waves each of which can be a simple wave or a shock wave. For a special class of materials for which the strain energy function W(p, q) represents a developable surface (of which harmonic materials are particular examples), one of the reflected waves is always a shock wave. It is shown that there are materials other than harmonic materials for which the wave speeds are independent of the direction of propagation. Illustrative examples are presented to show how one can determine the reflected waves from a rigid boundary. It is also shown that for certain incident shock waves, there exists only one reflected wave.

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On the stability of plane shock waves

Journal of Fluid Mechanics ◽

10.1017/s0022112057000208 ◽

1957 ◽

Vol 2 (4) ◽

pp. 397-411 ◽

Cited By ~ 25

Author(s):

N. C. Freeman

Keyword(s):

Shock Wave ◽

Mach Number ◽

Shock Waves ◽

Plane Shock ◽

Two Dimensional ◽

Small Perturbations ◽

Diverging Channel ◽

Maximum Stability ◽

The Stability ◽

Fourier Type

The decay of small perturbations on a plane shock wave propagating along a two-dimensional channel into a fluid at rest is investigated mathematically. The perturbations arise from small departures of the walls from uniform parallel shape or, physically, by placing small obstacles on the otherwise plane parallel walls. An expression for the pressure on a shock wave entering a uniformly, but slowly, diverging channel already exists (given by Chester 1953) as a deduction from the Lighthill (1949) linearized small disturbance theory of flow behind nearly plane shock waves. Using this result, an expression for the pressure distribution produced by the obstacles upon the shock wave is built up as an integral of Fourier type. From this, the shock shape, ξ, is deduced and the decay of the perturbations obtained from an expansion (valid after the disturbances have been reflected many times between the walls) for ξ in descending power of the distance, ζ, travelled by the shock wave. It is shown that the stability properties of the shock wave are qualitatively similar to those discussed in a previous paper (Freeman 1955); the perturbations dying out in an oscillatory manner like ζ−3/2. As before, a Mach number of maximum stability (1·15) exists, the disturbances to the shock wave decaying most rapidly at this Mach number. A modified, but more complicated, expansion for the perturbations, for use when the shock wave Mach number is large, is given in §4.In particular, the results are derived for the case of symmetrical ‘roof top’ obstacles. These predictions are compared with data obtained from experiments with similar obstacles on the walls of a shock tube.

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Features of spatial shock-wave flows in vapor-gas-liquid mixtures

Proceedings of the Mavlyutov Institute of Mechanics ◽

10.21662/uim2014.1.005 ◽

2014 ◽

Vol 10 ◽

pp. 27-31

Author(s):

R.Kh. Bolotnova ◽

U.O. Agisheva ◽

V.A. Buzina

Keyword(s):

Shock Wave ◽

High Pressure ◽

Shock Waves ◽

Liquid Mixture ◽

Liquid Mixtures ◽

Pressure Vessels ◽

Water Mixture ◽

Two Dimensional ◽

Nonstationary Processes ◽

Two Phase

The two-phase model of vapor-gas-liquid medium in axisymmetric two-dimensional formulation, taking into account vaporization is constructed. The nonstationary processes of boiling vapor-water mixture outflow from high-pressure vessels as a result of depressurization are studied. The problems of shock waves action on filled by gas-liquid mixture volumes are solved.

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Mechanical power and hyperelasticity

10.1093/oso/9780198567783.003.0003 ◽

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.

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

Journal of Elasticity ◽

10.1007/s10659-021-09823-x ◽

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.

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

Journal of Composite Materials ◽

10.1177/00219983211011528 ◽

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.

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