Pure Homogeneous Strain Research Articles

Interfacial wave propagation in initially stressed compressible hyperelastic materials

In this paper, the propagation of interfacial waves at the joint boundary of two initially stressed compressible half-spaces is discussed. The materials are subject to pure homogeneous strain and the wave is assumed to travel along one of the principle axes. For mathematical formulation of the problem, non-linear theory of elasticity and theory of invariants are used. Boundary conditions at the interface are applied which lead to an implicit secular equation governing the wave speed. A prototype strain-energy function is used for specialized theoretical results to understand the physical behavior of waves at the interface. A special case of biaxial initial stress is considered and the results are presented theoretically and graphically for representative numerical values of parameters. It is observed that the wave speed is considerably affected by intrinsic properties, i.e., material parameters as well as the amount of initial stress.

The secular equation for non-principal Rayleigh waves in deformed incompressible doubly fiber-reinforced nonlinearly elastic solids

The explicit and implicit secular equations for the speed of a (surface) Rayleigh wave propagating in a pre-stressed, doubly fiber-reinforced incompressible nonlinearly elastic half-space are obtained. Hence, the anisotropy is associated with two preferred directions, thereby modelling the effect of two families of fiber reinforcement. One of the principal planes of the primary pure homogeneous strain coincides with the free surface while the surface wave is not restricted to propagate in a principal direction. Results are illustrated with numerical examples. In particular, an isotropic material reinforced with two families of fibers is considered. Each family of fibers is characterized by defining a privileged direction. Furthermore, the fibers of each family are located throughout the half space and run parallel to each other and perpendicular to the depth direction, i.e. the free surface is a plane of symmetry of the anisotropy. The wave speed depends strongly on the anisotropic character of the material model as well as the direction of propagation.

Asymptotic analysis of an adhesive joint with mismatch strain

This paper proposes the study of the equilibrium problem, where two elastic bodies are bonded to a thin elastic film under mismatch strain conditions resulting in a state of residual stress. The asymptotic behavior of the film/adherent system is modeled as the film thickness tends to zero, using a method based on asymptotic expansions and energy minimization procedures. This method yields a family of non-local imperfect interface laws, which define a jump in the displacement and the traction vector fields. The amplitudes of the jumps turn out to be correlated with the state of residual stress and the elastic properties of the materials. As an example, the interface law is calculated at order zero in the case of a pure homogeneous mismatch strain and a thin isotropic film consisting of Blatz-Ko material.

ON SURFACE WAVES IN A FINITELY DEFORMED MAGNETOELASTIC HALF-SPACE

Rayleigh-type surface waves propagating in an incompressible isotropic half-space of nonconducting magnetoelastic material are studied for a half-space subjected to a finite pure homogeneous strain and a uniform magnetic field. First, the equations and boundary conditions governing linearized incremental motions superimposed on an initial motion and underlying electromagnetic field are derived and then specialized to the quasimagnetostatic approximation. The magnetoelastic material properties are characterized in terms of a "total" isotropic energy density function that depends on both the deformation and a Lagrangian measure of the magnetic induction. The problem of surface wave propagation is then analyzed for different directions of the initial magnetic field and for a simple constitutive model of a magnetoelastic material in order to evaluate the combined effect of the finite deformation and magnetic field on the surface wave speed. It is found that a magnetic field in the considered (sagittal) plane in general destabilizes the material compared with the situation in the absence of a magnetic field, and a magnetic field applied in the direction of wave propagation is more destabilizing than that applied perpendicular to it.

Problems in determining the elastic strain energy function for rubber

Abstract Lightly crosslinked natural rubber can be stretched by 600% or more, and recovers almost completely. It is often regarded as a model highly elastic material and characterized by a strain energy function to describe its stress–strain behavior under various types of deformation. A number of such functions have been proposed; some of them appear in current finite element programs. They are usually validated by comparison with measured stress–strain relations by Treloar [7] [L.R.G. Treloar, Stress–strain data for vulcanized rubber under various types of deformation, Trans. Faraday Soc. 40 (1944) 59–70] and Jones and Treloar [15] [D.F. Jones, L.R.G. Treloar, The properties of rubber in pure homogeneous strain, J. Phys. D Appl. Phys. 8 (1975) 1285–1304]. But Treloar pointed out that the relations at high strains became markedly irreversible, and he did not assign a strain energy function for strains greater than about 300%. Rivlin's universal relation between torsional stiffness and tensile stress [14] [R.S. Rivlin, Large elastic deformations of isotropic materials. Part V1: further results in the theory of torsion, shear and flexure, Philos. Trans. R. Soc. A 243 (1949) 251–288] is applied here to show that a typical elastic solid cannot be described by any strain energy function at strains greater than about 300%. Elastic strain energy functions for higher strains, or for other rubbery materials, are thus of doubtful value unless evidence for reversibility of stress–strain relations is adduced or the applicability of a strain energy function is demonstrated.

Reflection of plane waves at the boundary of a pre-stressed elastic half-space subject to biaxial stretch and simple shear

This paper is concerned with the effect of finite pure homogeneous biaxial stretch together with simple shear deformation on the reflection from a plane boundary of elastic waves propagating in a half-space of incompressible isotropic elastic material. This generalizes the previous work in which, separately, pure homogeneous strain and simple shear were considered. For a special class of constitutive law, it is shown that an incident plane harmonic wave propagating in the considered plane gives rise to a surface wave in addition to a reflected wave for every angle of incidence. The amplitude of the surface wave may vanish at certain discrete angles depending on the state of stress, biaxial stretch and simple shear deformation and then specialized to recover results obtained previously. The amplitude of the reflected wave is independent of the pre-stress but does depend upon the magnitude of deformation under consideration. The dependence of the reflected and surface wave behavior on the angle of incidence, amount of shear strain, biaxial stretch and the state of stress is illustrated graphically.

Surface waves in a stretched and sheared incompressible elastic material

In this paper, we analyze the effect of a combined pure homogeneous strain and simple shear in a principal plane of the latter on the propagation of surface waves for an incompressible isotropic elastic half-space whose boundary is normal to the glide planes of the shear. This generalizes previous work in which, separately, pure homogeneous strain and simple shear were considered. For a special class of materials, the secular equation is obtained in explicit form and then specialized to recover results obtained previously for the two cases mentioned above. A method for obtaining the secular equation for a general form of strain–energy function is then outlined. In general, this is very lengthy and the result is not listed, but, for the case in which there is no normal stress on the half-space boundary, the result is given, for illustration, in respect of the so-called generalized Varga material. Numerical results are given to show how the surface wave speed depends on both the underlying pure homogeneous strain and the superimposed simple shear. Further numerical results are provided for the Gent model of limiting chain extensibility.

Surface waves in deformed Bell materials

Small amplitude inhomogeneous plane waves are studied as they propagate on the free surface of a predeformed semi-infinite body made of Bell constrained material. The predeformation corresponds to a finite static pure homogeneous strain. The surface wave propagates in a principal direction of strain and is attenuated in another principal direction, orthogonal to the free surface. For these waves, the secular equation giving the speed of propagation is established by the method of first integrals. This equation is not the same as the secular equation for incompressible half-spaces, even though the Bell constraint and the incompressibility constraint coincide in the isotropic infinitesimal limit.

The effect of pre-strain on the reflection and transmission of plane waves at an elastic interface

This paper illustrates the influence of pure homogeneous strain on the reflection and transmission of plane waves at the boundary between two half-spaces of incompressible isotropic elastic material. In general, the half-spaces consist of different material and are subjected to different deformations but with their principal axes aligned. For a certain class of constitutive laws it is shown that a homogeneous plane (SV) wave incident on the boundary from one half-space gives rise to a reflected wave (with angle of reflection equal to the angle of incidence) together with an interfacial wave in the same half-space, while in the other half-space two possibilities arise depending on the angle of incidence, the material properties and the magnitudes of the deformations in the two half-spaces. Either (a) there is a transmitted (homogeneous plane) wave accompanied by an interfacial wave, or (b) there are two interfacial waves with equal speeds of propagation but different rates of (spatial) decay away from the boundary. For a second class of constitutive laws similar behaviour is found for certain combinations of angle of incidence, material properties and deformations, but additional possibilities also arise. In particular, there may be two reflected waves instead of one reflected wave and an interfacial wave, coupled with either possibility (a) or (b) in the second half-space. Equally, there may be two transmitted waves for each of the possible combinations of reflected and interfacial waves in the first half-space. For each class of materials conditions under which the incident wave is not refracted are identified and the dependence of the amplitudes of the reflected, transmitted and interfacial waves on the angle of incidence, the states of deformation and the materials is illustrated graphically.

New phenomenological behavior laws for rubbers and thermoplastic elastomers

A phenomenological study of rubber-like materials undergoing pure homogeneous strain such as uniaxial tension, pure shear and (equi-) biaxial tension, leads to a generalized strain energy density representation for hyperelastic incompressible elastomeric material behavior. Based on experimental observations, an additional decomposition of the strain energy density into two functions, each depending on one of the non-constant principal strain invariants, has been assumed. A new building process of these functions is then proposed combined with a simple and fast parameter identification. The former is progressive and requires only two sets of data. An excellent agreement between theory and experimental data has been observed on numerous experiments. The new model seems particularly well adapted to account for the state of strain response dependence for rubbers as for thermoplastic elastomers.

Surface waves propagating in a half-space of neo-Hookean elastic material subject to pure homogeneous strain

The propagation of infinitesimal surface waves along a principal axis of strain in a half-space of incompressible isotropic elastic material subject to a homogeneous deformation was discussed in detail by Dowaikh and Ogden (IMA. J. Appl. Math. 44 (1990) 261–284). Here, we consider propagation in an arbitrary direction parallel to the surface of the half-space. The cumbersome algebra involved precludes the possibility of obtaining an explicit form of the secular equation with respect to a general form of strain-energy function, but for the neo-Hookean form of constitutive law and with an arbitrary normal stress on the half-space boundary an explicit secular equation is obtained.

On the reflection of plane waves at the boundary of an elastic half-space subject to simple shear

This paper is concerned with the effect of finite strain principal axis orientation on the reflection from a plane boundary of infinitesimal plane waves propagating in a half-space of incompressible isotropic elastic material. Attention is focussed on waves propagating in a principal plane of the deformation corresponding to simple shear. For a special class of constitutive laws it is shown that an incident plane harmonic wave propagating in the considered plane gives rise to a surface wave in addition to a reflected wave for every angle of incidence although its amplitude may vanish at certain discrete angles depending on the state of stress and deformation. Unlike the situation in which the underlying deformation is a pure homogeneous strain, however, the amplitude ratio of the reflected (plane harmonic) wave does not in general have unit magnitude, but its magnitude is independent of the pre-stress. Moreover, the angle of reflection differs from the angle of incidence. For materials not in this special class, on the other hand, it is shown that two plane harmonic waves may be reflected when the angle of incidence lies within certain ranges of values (which depend on the shear deformation). Outside this range there is in general a single reflected wave, and a surface wave is generated. For each class of materials the dependence of the reflected and surface wave behaviour on the angle of incidence, the magnitude of the shear strain and the state of stress is illustrated graphically.

The effect of shear on the propagation of elastic surface waves

The propagation of surface waves on a half-space of a pre-stressed incompressible elastic material is discussed. In contrast to previous studies, in which the pre-stress is associated with a pure homogeneous strain with one principal axis normal to the surface of the half-space, this paper examines the influence of an underlying deformation which corresponds to a homogeneous simple shear, so that no principal axis of the deformation is along the surface normal. For a particular class of strain-energy functions an explicit secular equation for the wavespeed is obtained and then analysed to provide necessary and sufficient conditions for the existence and uniqueness of surface waves and quasi-static deformations. Numerical calculations are used to illustrate the dependence of the results on the amount of shear and on the hydrostatic stress.

Interfacial waves and deformations in pre-stressed elastic media

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.

A non-Gaussian theory for rubber in biaxial strain. I. Mechanical properties

The theory makes use of the inverse Langevin expression for the force-extension relation for the single chain ofnrandomly-jointed links. An affine deformatiom of chain vector lengths is assumed on subjection of the network to pure homogeneous strain. The equations for the principal stresses are solved by numerical computation, using the Gauss quadrature method for evaluation of the relevant integrals. Numerical data are given for the whole range of principal extension ratios in biaxial strain forn= 25 and 100, and also for the particular case of simple extension forn= 16, 25, 36, 64 and 100. The results account in a semi-quantitative manner for an important feature of experimental biaxial-strain data which has only recently been observed, and which is not accounted for by previous network theories.

Strain-energy density function for rubberlike materials

T he strain-energy density function surface for the rubber tested by L. R.G. T reloak (1944a) is determined from bis stress-strain data. The data were given for the three pure homogeneous strain paths of simple elongation, pure shear, and equi-biaxial extension of a thin sheet. The surface is formed by plotting calculated points of the strain-energy function above a plane having the first and second strain invariants as rectangular cartesian coordinates. The strain-energy function is expressed as a double power series in the invariants expanded about the zero energy state which is the origin of coordinates. An analysis of this experimentally derived surface provides the information required for the rational selection of terms and the determination of the coefficients in the series expansion, thus defining a function within the Rivlin-type formulation. The function so determined is tested by employing it in the appropriate constitutive formulae to compute stresses for comparison with experimental values. Another test is made by utilizing the function to predict shapes of an inflated membrane for comparison with experimentally observed shapes. Excellent agreement between prediction and experiment is found. A second demonstration is given for another rubber tested by D.F. J ones and L.R.G. T reloar (1975). Again, excellent results are obtained.

The properties of rubber in pure homogeneous strain

Measurements are reported on the relationship between the principal stresses t1 and t2 and the corresponding extension ratios lambda 1 and lambda 2 in the general biaxial strain of a rubber sheet. The resulting data are shown to be consistent with the Valanis-Landel hypothesis, according to which the strain energy W is a separable function w( lambda ) of the principal extension ratios. The formulation of the series for lambda (dw/d lambda ) in terms of strain invariants I1 and I2 enables delta W/ delta I1 and delta W/ delta I2 to be accurately calculated for any values of I1 and I2. It is concluded that both delta W/ delta I1 and delta W/ delta I2 are functions of both I1 and I2. The results confirm the conclusions of earlier workers that the empirical Mooney constants derived from uniaxial extension or compression bear no direct relation to the values of delta W/ delta I1 and delta W/ delta I2.

Strain energy functions of rubber. II. The characterization of filled vulcanizates

In part I it was shown that if the strain energy function was expanded to a sufficient degree and the coefficients Cij were found by regression to pure homogeneous strain data, then stress–strain equations could be derived to give accurate solutions to design problems even at relatively high extensions. The problems of applying this theory to filed vulcanizates are discussed and a way of obtaining pure homogeneous strain data for for such materials is suggested. Stress–strain equations fitted to the data are found to be general within the range of experimental strains and in some cases will extrapolate outside this range. The equations can be used in design applications where strains are greater than would be experienced in normal engineering practice.

Toward a turbulent constitutive relation

In an attempt to explain the failure of the various pure homogeneous strain experiments to reach equilibrium (and consequently to support the contention of Townsend of an equilibrium structure of the Reynolds stress dependent only on geometry), the nature of the general Reynolds stress-mean velocity relation is examined. It is shown that if homogeneous flows become asymptotically independent of initial conditions, and if the Reynolds stress bearing structure can be characterized by a single time scale (i.e.–at sufficiently high Reynolds number) then these flows behave like classical non-linear viscoelastic media, with the Reynolds stress structure dependent on the (strain-rate) (time scale) product. Thus, the existence of an equilibrium structure implies the existence of an equilibrium time scale and a universal value of the product. The ideas permitting Reynolds stress and mean velocity to be related are applied to the dissipative structure in homogeneous flows, and it is found that in such flow the time scale never ceases to grow, so that these flows can never reach an equilibrium structure. With the aid of an ad-hoc assumption these flows are examined in some detail, and the results of experiments are predicted with considerable accuracy. It is suggested that (inhomogeneous) flows having an equilibrium time scale may, in the homogeneous limit, be expected to display a universal structure. The small departure from universality induced by the large eddies associated with inhomogeneity may be adequately predicted by this same ad-hoc model.

Volume changes and mechanical anisotropy of strained rubbers

The Gaussian network theory, suitably modified to take account of the finite compressibility of an actual rubber, is used to derive the relation between the change of volume and the applied stresses in a pure homogeneous strain of the most general type. The anisotropy of compressibility is also worked out for the case of a pure homogeneous strain maintained by fixed applied forces. The formulae derived can be reduced to those previously given by Flory for the particular case of simple extension. It is further shown that the anisotropy of compressibility for a rubber in equilibrium with a swelling liquid is identical with that for a slightly compressible dry rubber.