Deformation mode

Different types of deformation can also be distinguished in another way, for they can be either time-dependent or time-independent. A deformation is time-dependent if the material responds with a delay to changes of the load. If - in contrast - the deformation coincides with the change of the load, the deformation is time-independent. Time-dependent deformations are denoted by the prefix visco-. Altogether, four different deformation types exist since elastic as well as plastic deformations can be time-dependent or time-independent. 

In this chapter, we will start by discussing how external forces and the resulting material deformations can be described. Subsequently, the time-independent elastic behaviour of materials will be described. Often, it is simply called the elastic behaviour , although this is not completely correct. 

Time-independent plastic deformation will be described in chapters 3, 6, and 8, the time-dependent plastic behaviour is subject of chapters 8 and 11. Time-dependent elastic behaviour is mainly observed in polymers, described in chapter 8. 

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Fig. XV-10. Illustration of a bilayer membrane and two of its deformation modes (a) twist (b) splay. (From Ref. 75.)...

The situation is not so simple when these various parameters are time dependent. In the latter case, the moduli, designated by E(t)and G(t), are evaluated by examining the (time dependent) value of o needed to maintain a constant strain 7o- By constrast, the time-dependent compliances D(t) and J(t)are determined by measuring the time-dependent strain associated with a constant stress Oq. Thus whether the deformation mode is tension or shear, the modulus is a measure of the stress required to produce a unit strain. Likewise, the compliance is a measure of the strain associated with a unit stress. As required by these definitions, the units of compliance are the reciprocals of the units of the moduli m in the SI system. 

The simplest dynamic system to analyse is one in which the stress and strain are changing in a sinusoidal fashion. Fortunately this is probably the most common type of loading which occurs in practice and it is also the basic deformation mode used in dynamic mechanical testing of plastics. 

Each of these will contribute to the behaviour of the fluid although since each results from a different deformation mode, one effect may dominate depending on the geometry of the situation. This will be seen more clearly later when speciflc design problems are tackled. 

Accordingly, we have supposedly found the shear modulus G.,2. However, a relationship such as Equation (2.107) does not exist for strengths because strengths do not transform like stiffnesses. Thus, this experiment cannot be relied upon to determine S, the ultimate shear stress (shear strength), because a pure shear deformation mode has not been excited with accompanying failure in shear. Accordingly, other approaches to obtain S must be used. 

As an indication of the changes in deformation modes that can be produced in ionomers by increase of ion content, consider poly(styrene-co-sodium methacrylate). In ionomers of low ion content, the only observed deformation mode in strained thin films cast from tetra hydrofuran (THF), a nonpolar solvent, is localized crazing. But for ion contents near to or above the critical value of about 6 mol%, both crazing and shear deformation bands have been observed. This is demonstrated in the transmission electron microscope (TEM) scan of Fig. 3 for an ionomer of 8.2 mol% ion content. Somewhat similar deformation patterns have also been observed in a Na-SPS ionomer having an ion content of 7.5 mol%. Clearly, in both of these ionomers, the presence of a... 

Thermal treatment and the nature of the casting solvent can also affect the deformation modes achieved in strained films of ionomers. For example, in films cast from polar dimethylformamide (DMF), the solvent interacts with ion-rich clusters and essentially destroys them, as is evident form absence of a second, higher temperature loss peak in such samples. As a result, even in a cast DMF sample of Na-SPS ionomer of high ion content (8.5 mol%), the only deformation mode observed in tensile straining is crazing. However, when these films are given an additional heat treatment (41 h at 210°C), shear... 

Studies of PMMA-based ionomers also demonstrate the influence of thermal treatment on deformation modes (16). For Na salts of PMMA-based ionomers of 6 and 12 mol% that were cast from DMF, only crazes were observed on straining. However, after an additional heat treatment (48 h at 160°C), which also removes any DMF solvent that is present, shear deformation zones are induced. Hence, the ionic cluster phase, which was destroyed by the polar solvent, has been restored by the heat treatment. 

The combined effects of a divalent Ca counterion and thermal treatment can be seen from studies of PMMA-based ionomers [16]. In thin films of Ca-salts of this ionomer cast from methylene chloride, and having an ion content of only 0.8 mol%, the only observed deformation was a series of long, localized crazes, similar to those seen in the PMMA homopolymer. When the ionomer samples were subject to an additional heat treatment (8 h at 100°C), the induced crazes were shorter in length and shear deformation zones were present. This behavior implies that the heat treatment enhanced the formation of ionic aggregates and increased the entanglement strand density. The deformation pattern attained is rather similar to that of Na salts having an ion content of about 6 mol% hence, substitution of divalent Ca for monovalent Na permits comparable deformation modes, including some shear, to be obtained at much lower ion contents. 

DEFORMATION MODE AND CRITICAL RESOLVED SHEAR STRESS... 

Figure 4. Temperature dependence of CRSS for the six different deformation modes observed in Ti-56 at.%Al single crystals.

The strongest mode observed near 800 cm 1 is polarized along c and is a totally symmetrical vibration mode (Ai) corresponding to the niobium-oxygen vibrations vs (NbO) of infinite chains (NbOF4 )n running along the c -axis. The mode observed at 615 cm 1 is polarized perpendicular to c and corresponds to the NbF vibrations of the octahedrons of the same chains. The mode at 626 cm 1 is attributed to NbF vibrations of isolated complex ions - NbF 2 . The lines at 377, 390 and 272 cm 1 correspond to deformation modes 8(FNbF) of the two polyhedrons. 

As the most notable contribution of ab initio studies, it was revealed that the different modes of molecular deformation (i.e. bond stretching, valence angle bending and internal rotation) are excited simultaneously and not sequentially at different levels of stress. Intuitive arguments, implied by molecular mechanics and other semi-empirical procedures, lead to the erroneous assumption that the relative extent of deformation under stress of covalent bonds, valence angles and internal rotation angles (Ar A0 AO) should be inversely proportional to the relative stiffness of the deformation modes which, for a typical polyolefin, are 100 10 1 [15]. A completly different picture emerged from the Hartree-Fock calculations where the determined values of Ar A0 AO actually vary in the ratio of 1 2.4 9 [91]. 

Fig. 5. Log-iog plot illustrating the hardness dependence of density for PE samples crystallized from the melt. The plot yields two straight sections which can be ascribed to two preferential deformation modes a crystal destruction and b compression of amorphous domains...

For density values g > 0.92 g/cm3 the deformation modes of the crystals predominate. The hard elements are the lamellae. The mechanical properties are primarily determined by the large anisotropy of molecular forces. The mosaic structure of blocks introduces a specific weakness element which permits chain slip to proceed faster at the block boundaries than inside the blocks. The weakest element of the solid is the surface layer between adjacent lamellae, containing chain folds, free chain ends, tie molecules, etc. 

From the morphology of the fibrous structure of the deformed polymer one concludes that the dominant deformation modes of the drawn polymer under the stress field of the indenter involve ... 

In particular it can be shown that the dynamic flocculation model of stress softening and hysteresis fulfils a plausibility criterion, important, e.g., for finite element (FE) apphcations. Accordingly, any deformation mode can be predicted based solely on uniaxial stress-strain measurements, which can be carried out relatively easily. From the simulations of stress-strain cycles at medium and large strain it can be concluded that the model of cluster breakdown and reaggregation for prestrained samples represents a fundamental micromechanical basis for the description of nonlinear viscoelasticity of filler-reinforced rubbers. Thereby, the mechanisms of energy storage and dissipation are traced back to the elastic response of tender but fragile filler clusters [24]. 

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