Large deformation stress-strain relation

This consists of experimental measurements of stress-strain relations and analysis of the data in terms of the mathematical theory of elastic continua. Rivlin7-10 was the first to pply the finite (or large) deformation theory to the phenomenologic analysis of rubber elasticity. He correctly pointed out the above-mentioned restrictions on W, and proposed an empirical form... 

At moderate strains, the value of J2 is often large enough for terms involving W2 to be neglected. Some stress-strain relations are now derived using this approximation to illustrate how such calculations are carried out and to deduce under what conditions the deformations become unstable. Instabilities are... 

Because the strain energy function for rubber is valid at large strains, and yields stress-strain relations which are nonlinear in character, the stresses depend on the square and higher powers of strain, rather than the simple proportionality expected at small strains. A striking example of this feature of large elastic deformations is afforded by the normal stresses tn,(22,133 that are necessary... 

Figure 5 demonstrates the different behaviour resulting from Eqs. (49) and (54) in the case of uniaxial compression. We also tested the elastic potential (Eq. (44)) in the two cases v = 1/2 and v = —1/4 by comparing the corresponding stress-strain relations with biaxial extension experiments which cover relatively small as well as large deformation regions for an isoprene rubber vulcanizate. In the rectan-... 

The general factorable single integral constitutive equation is an equation that describes well the viscoelastic properties of a large class of crosslinked rubbers > and the elasticoviscous properties of many polymer melt8 under various types of deformation An appropriate way to compare the stress-strain relation for cured elastomers and polymer melts therefore is to calculate the strain-dependent function contained in this constitutive equation from experimental results and to compare the strain measures so obtained. 

A full discussion of stress-strain relations at equilibrium for large deformations in rubbery cross-linked polymers is beyond the scope of this chapter there have been many investigations of uniaxial deformations (simple extension, compression,) torsion, and biaxial deformations, which have been critically reviewed elsewhere. A few comments will introduce the discussion of nonlinear viscoelastic behavior. 

The finite element procedures for the analysis of elastic>plastic solids at large strain have been given by Lee [9] and implemented by Chiou [11] and Chiou et al. [10]. In this work, only a few comments on the finite element procedures will be made. Equation (16), which links the Truesdell stress rate tensor and the deformation rate tensor, may be regarded as the stress-strain relation in rate fom with a being the "slope" at a particular point in stress space. However, in nonlinear finite element analysis, one has to have a stress-strain relation in incremental form which enables the increments in displacements, strains, and stresses not to be infinitesimally small. Therefore, it is proposed to adopt the following incremental stress-strain relation... 

It is consistent with the approximations of small strain theory made in Section A.7 to neglect the higher-order terms, and to consider the elastic moduli to be constant. Stated in another way, it would be inconsistent with the use of the small deformation strain tensor to consider the stress relation to be nonlinear. The previous theory has included such nonlinearity because the theory will later be generalized to large deformations, where variable moduli are the rule. 

An external pressure (stress) that is exerted on a material will cause its thickness to decrease. A shear stress is applied parallel to the surface of a material, and may cause the sliding of atomic layers over one another. The resultant deformation in the size/shape of the material is referred to as strain, related to the bonding scheme of the atoms comprising the solid. For example, a rubbery material will exhibit a greater strain than a covalently bound solid such as diamond. Since steels contain similar atoms, most will behave similarly as a result of an applied stress. If a stress causes a material to bend, the resultant flex is referred to as shear strain. For small shear stresses, steel deforms elastically, involving no permanent displacement of atoms. The deformation vanishes when shear stress is removed. However, for a large shear stress, steel will deform plastically, involving the permanent displacement of atoms, known as slip. 

Dynamic mechanical measurements are performed at very small strains in order to ensure that linear viscoelasticity relations can be applied to the data. Stress-strain data involve large strain behavior and are accumulated in the nonlinear region. In other words, the tensile test itself alters the structure of the test specimen, which usually cannot be cycled back to its initial state. (Similarly, dynamic deformations at large strains test the fatigue resistance of the material.)... 

The question of whether microhardness is a property related to the elastic modulus E or the yield stress T is a problem which has been commented on by Bowman Bevis (1977). These authors found an experimental relationship between microhardness and modulus and/or yield stress for injection-moulded semicrystalline plastics. According to the classical theory of plasticity the expected microindentation hardness value for a Vickers indenter is approximately equal to three times the yield stress (Tabor s relation). This assumption is only valid for an ideally plastic solid showing sufficiently large deformation with no elastic strains. PE, as we have seen, can be considered to be a two-phase material. Therefore, one might anticipate a certain variation of the H/ T 3 ratio depending on the proportion of the compliant to the stiff phase. 

In practice, large deformation properties are far more useful, and determination of the full relation between stress and strain gives the best information. Materials vary widely in their linear region, i.e., the strain range over which stress and strain remain proportional. Deforming it much farther, the material may eventually break. Relevant parameters then are fracture stress or strength, fracture strain or shortness, and work of fracture or toughness. The correlation between fracture parameters and the modulus is often poor. Since many soft solids exhibit viscoelastic behavior, the values of these parameters can depend, often markedly, on the strain rate. 

In general, SMPF is perceived as a two-phase composite material with a crystalline phase mixed with an amorphous phase. A multiscale viscoplasticity theory is developed. The amorphous phase is modeled using the Boyce model, while the crystalline phase is modeled using the Hutchinson model. Under an isostrain assumption, the micromechanics approach is used to assemble the microscale RVE. The kinematic relation is used to link the micro-mechanics constitutive relation to the macroscopic constitutive law. The proposed theory takes into account the stress induced crystallization process and the initial morphological texture, while the polymeric texture is updated based on the apphed stresses. The related computational issue is discussed. The predictabihty of the model is vahdated by comparison wifli test results. It is expected that more accurate measurement of the stress and strain in the SMPF with large deformation may further enhance the predictability of the developed model. It is also desired to reduce the number of material parameters in the model. In other words, a deeper understanding and physics based theoretical modeling are needed. 

Slip lines in polycrystaUine MgO can be seen in Fig. 4.33. The slip observation was on polished specimens following the deformation on specimens shown as stress-strain curves in Fig. 4.34. A pressure-dependent BDT is observed. At higher pressures, large strains may be achieved without fracture, since the stress required for fracturing increases. The level of the stress-strain curves increases in both the brittle and ductile ranges. Furthermore, the type of jacket influences stress-strain curves. The curves in Fig. 4.34 (and also those in Fig. 4.33) relate to coarse-grained [henceforth CG] MgO. 

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