Uniaxial compliance

Note 3 Uniaxial compliance may be evaluated using tensile or compressive uniaxial deformation. If determined using tensile deformation it may be termed tensile compliance. Note 4 D = l/E, where E is Young s modulus. 

A general description of the fundamental relationships governing the dynamic response of linear viscoelastic materials may be found in several sources (28, 37, 93). In general, sinusoidally applied strains (stresses) result in sinusoidal stresses (strains) that are out of phase. Measurements may be made under uniaxial, shear, or dilational loading conditions, and the resultant complex moduli or compliance and loss-phase angle are computed. Rotating radius vectors are usually taken to represent the... 

We note that, in principle, the main physical discussions related to filler networking in this paper do not change if a sinusoidal tensile or uniaxial compres-sional stress (amplitude 0) is imposed on the rubber material. In some examples the complex dynamic modulus is then denoted with E = E + iE" and the compliance with C = C - iC". All theoretical considerations use the shearing modulus G. ... 

The Poisson ratio, like the bulk, tensile, and shear creep compliance, is an increasing function of time because the lateral contraction cannot develop instantaneously in uniaxial tension but takes an infinite time to reach its ultimate value. In response to a sinusoidal uniaxial stretch, the complete Poison ratio is obtained ... 

Strictly speaking, there are no static viscoelastic properties as viscoelastic properties are always time-dependent. However, creep and stress relaxation experiments can be considered quasi-static experiments from which the creep compliance and the modulus can be obtained (4). Such tests are commonly applied in uniaxial conditions for simphcity. The usual time range of quasi-static transient measurements is limited to times not less than 10 s. The reasons for this is that in actual experiments it takes a short period of time to apply the force or the deformation to the sample, and a transitory dynamic response overlaps the idealized creep or relaxation experiment. There is no limitation on the maximum time, but usually it is restricted to a maximum of 10" s. In fact, this range of times is complementary, in the corresponding frequency scale, to that of dynamic experiments. Accordingly, to compare these two complementary techniques, procedures of interconversion of data (time frequency or its inverse) are needed. Some of these procedures are discussed in Chapters 6 and 9. 

Uniaxial compressive creep tests were carried out for granite sampled from Three Gorges Project (TGP) site in China at seven temperatures from 20°C to 300°C in order to determine the parameters C , Ci, Cr and Cj in eq. (25). The curves of creep compliance vs. time in logarithmic coordinates... 

The relationship between the compliance matrix and the technical constants such as Young s modulus ( z) shear modulus (Gi) and Poisson s ratio (vy) measured in mechanical tests such as uniaxial or pure shear is expressed in Equation 47.6. 

As an example, it can be shown that the Reuss average compliance 53333 for a uniaxial sample consisting of an aggregate of transversely isotropic units is given in terms of the compliances Sy/y of the unit by... 

A material, which can be described as an SLS, was found to have unrelaxed and relaxed Young s modulus values of 70 and 50 GPa, respectively. Determine the relaxation and retardation times. Plot graphically the compliance of the material as a function of time, under the action of a constant uniaxial tensile stress. 

The first two terms on the right-hand side of equation [12.6] are viscoelastic terms proposed by Schapery, where e represents uniaxial kinematic (or total) strain at time t, o is the Cauchy stress at time t, is the instantaneous compliance and AD(r[i ) is a transient creep compliance function. The factor g defines stress and temperature effects on the instantaneous elastic compliance and is a measure of state dependent reduction (or increase) in stiffness. Transient compliance factor gi has a similar meaning, operating on the creep compliance component. The factor gj accounts for the influence of loading rate on creep. The function i ) represents a reduced timescale parameter defined by ... 

Studies of mechanical anisotropy in polymers have been made on specimens of two distinct types. Uniaxially drawn filaments or films have fibre symmetry, with isotropy in the plane perpendicular to the draw direction. Films drawn at constant width or films drawn uniaxially and subsequently rolled and annealed under closely controlled conditions, show orthorhombic symmetry. For fibre symmetry (also called transverse isotropy) the number of independent elastic constants reduces to five and the compliance matrix is... 

The most common technique employed to date has been that of creep in uniaxial tension. It was shown above that with the inclusion of lateral strain measurements this is a powerful technique giving access to up to 6 independent creep compliance functions. This is more than for any other known method. It further has the overwhelming advantage over many methods, such as say torsional or flexural creep, that the stress is sensibly uniform over the working volume of the specimen. This advantage is paramount in studies of materials displaying non-linear behaviour in creep since analysis of the non-uniform stress situation in non-linear systems is not well developed. Attempts to overcome the non-uniform stress situation in torsion, by recourse to, say, torsion of thin walled tubes, lead to severe difSculties in specimen preparation in oriented materials, when anisotropy of behaviour is to be studied. 

The law of linear proportionality between uniaxial strain and uniaxial stress discovered by Hooke in 1676 can be generalized to a linear connection relating all nine elements of the strain tensor and all nine elements of the stress tensor, implying the existence of 81 constants of proportionality, or elastic compliances, Sijki, relating generically the strain tensor component sy to the stress tensor component in an expression of the type... 

Special specimen preparation as with tensile testing. However, the extraction of intrinsic mechanical parameters from creep indentation data is analytically complex [3, 4]. Confined compression or unconfined compression tests require preparation of cylindrical cored specimens of tissue and underlying bone. With unconfined compression, the free draining tissue edges and low aspect ratio, layered nature of the test specimen may introduce error. Compression of a laterally confined specimen by a porous plunger produces uniaxial deformation and fluid flow. Confined compression creep data has been analyzed to yield an aggregate equilibrium compressive modulus and permeability coefficient [5] and uniaxial creep compliance [6]. 

A fibre or a uniaxially drawn film will usually show no preferred orientation in the plane normal to the fibre axis or the draw direction (see Figure 7.2). The compliance matrix then reduces to a form with only five independent constants ... 

The present section deals with the review and extension of Schapery s single integral constitutive law to two dimensions. First, a stress operator that defines uniaxial strain as a function of current and past stress is developed. Extension to multiaxial stress state is accomplished by incorporating Poisson s effects, resulting in a constitutive matrix that consists of instantaneous compliance, Poisson s ratio, and a vector of hereditary strains. The constitutive equations thus obtained are suitable for nonlinear viscoelastic finite-element analysis. 

Two test cases are used to validate the linear viscoelastic analysis capability implemented in the present finite-element program named NOVA. In the first case, the tensile creep strain in a single eight-noded quadrilateral element was computed for both the plane-stress and plane-strain cases using the program NOVA. The results were then compared to the analytical solution for the plane-strain case presented in Reference 49. A uniform uniaxial tensile load of 13.79 MPa was applied on the test specimen. A three-parameter solid model was used to represent the tensile compliance of the adhesive. The Poisson s ratio was assumed to remain constant with time. The following time-dependent functions were used in Reference 49 to represent the tensile compliance for FM-73M at 72 °C ... 

For one-dimensional deformation, the body undergoes uniaxial tension or compression. For a three-dimensional deformation, the body undergoes the shear force. That is, for a three-dimensional deformation, we have shear stress (labeled CTs) and shear strain (labeled y). The shear modulus G and the shear compliance J are defined and related by... 

Unique relationships among the compliance constants and stiffness constants for cubic crystals can be derived by considering simple states of stress and deformation of the crystal. For example, consider a uniaxial tensile stress [Pg.171]

Fig. 77. Elastic compliance of Tmo jjSe under uniaxial pressure p in the x direction. (After Boppart et al. 1980a.)...

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