Polymers under compressive shear stresses

Ductile Failure of Brittle Polymers under Compressive Shear Stresses... 

Tphe literature is replete with examples showing that the application of hydrostatic pressure enhances the ductile behavior of strained amorphous polymers. In this paper we present a possible explanation of this effect and two experiments demonstrating the enhanced ductility of polymers under compressive shear stresses applied orthogonally to the plane of shear. 

If the component T of an applied compressive shear stress orthogonal to the plane of fracture combines with the normal component o-yy of the local stress at the tip of a crack, then the combined higher stress will minimize (AH — U0)/ductile failure ensues. This can occur if the orthogonal compressive stress is locally inhomogeneous. Hence, a polymer can fail in a more ductile fashion under orthogonal compressive shear stresses than in their absence. 

Flow behaviour of polymer melts is still difficult to predict in detail. Here, we only mention two aspects. The viscosity of a polymer melt decreases with increasing shear rate. This phenomenon is called shear thinning [48]. Another particularity of the flow of non-Newtonian liquids is the appearance of stress nonnal to the shear direction [48]. This type of stress is responsible for the expansion of a polymer melt at the exit of a tube that it was forced tlirough. Shear thinning and nonnal stress are both due to the change of the chain confonnation under large shear. On the one hand, the compressed coil cross section leads to a smaller viscosity. On the other hand, when the stress is released, as for example at the exit of a tube, the coils fold back to their isotropic confonnation and, thus, give rise to the lateral expansion of the melt. 

Under compression or shear most polymers show qualitatively similar behaviour. However, under the application of tensile stress, two different defonnation processes after the yield point are known. Ductile polymers elongate in an irreversible process similar to flow, while brittle systems whiten due the fonnation of microvoids. These voids rapidly grow and lead to sample failure [50, 51]- The reason for these conspicuously different defonnation mechanisms are thought to be related to the local dynamics of the polymer chains and to the entanglement network density. 

The formation of shear bands under compression is found in crystalline polymers when loaded at temperatures lower than 0.75 T. Under such a condition the shear bands interact with certain morphological features such as spherulite boundaries or lamellar arrangements inside the spherulites. The band initiation stress, ct, increases and the strain at break, Cp, decreases with decreasing temperature and increasing stiffness of the tested polymer, i.e. increasing degree of crystallinity. 

For macroscopically isotropic polymers, the Tresca and von Mises yield criteria take very simple analytical forms when expressed in terms of the principal stresses cji, form surfaces in the principal stress space. The shear yield surface for the pressure-dependent von Mises criterion [Eqs (14.10) and (14.12)] is a tapering cylinder centered on the applied pressure increases. The shear yield surface of the pressure-dependent Tresca criterion [Eqs (14.8) and (14.12)] is a hexagonal pyramid. To determine which of the two criteria is the most appropriate for a particular polymer it is necessary to determine the yield behavior of the polymer under different states of stress. This is done by working in plane stress (ct3 = 0) and obtaining yield stresses for simple uniaxial tension and compression, pure shear (di = —CT2), and biaxial tension (cti, 0-2 > 0). Figure 14.9 shows the experimental results for glassy polystyrene (13), where the... 

A related process is die-drawing. In this process the sample is pulled, not pushed, through the die. The important difference here is that the type of stress applied is quite different. In extrusion the polymer is under compressive stress, whereas in die-drawing it is largely under tensile or shear stress, with plane sections normal to the draw direction often remaining almost plane, giving rise to plug flow. This type of stress is necessary for some polymers in order to allow them to draw and orient. 

It was observed empirically by Hooke that, for many materials under low strain, stress is proportional to strain. Young s modulus may then be defined as the ratio of stress to strain for a material under uniaxial tension or compression, but it should be noted that not all materials (and this includes polymers) obey Hooke s law rigorously. This is particularly so at high values of strain but this section only considers the linear portion of the stress-strain curve. Clearly, reality is more complicated than described previously because the application of stress in one direction on a body results in a strain, not only in that direction, but in the two orthogonal directions also. Thus, a sample subjected to uniaxial tension increases in length, but it also becomes narrower and thinner. This quickly leads the student into tensors and is beyond the scope of this chapter. The subject is discussed elsewhere [21-23]. There are four elastic constants usually used to describe a macroscopically isotropic material. These are Young s modulus, E, shear modulus, G, bulk modulus, K, and Poisson s ratio, v. They are defined in Figure 9.2 and they are related by Equations 9.1-9.3. 

Figure 12.21 The strain rate dependence of the octahedral shear stress r at atmospheric pressure using data from torsion (o), tension (A), and compression. (Reproduced with permission from Duckett, R.A., Coswami, B.C., Smith, LS.A. et al. (1978) Yielding and crazing behavior of polycarbonate In torsion under superposed hydrostatic-pressure. Brit. Polym. J., 10, 11. Copyright (1978) Society of Chemical Industry.)...

For which polymers and under which conditions do crazes occur Crazes form primarily in amorphous polymers, for molecular weights above the entanglement limit. There is no craze formation under compression or under pure shear. The typical situation leading to craze initiation is the imposition of an uniaxial or biaxial tensile stress. If such stresses are applied and fulfill certain threshold conditions, crazes form statistically, preferentially at first at the sample surface. The initiation rate depends on the applied stress, as is shown in Fig. 8.22. The higher the stress imposed, the shorter is the time for the observation of the first crazes. After the initial increase with time, the craze density saturates. Removing the stress, the crazes close their openings somewhat, but survive. They disappear only if the sample is annealed at temperatures above the glass transition. 

In order to determine which is the most appropriate yield criterion for a particular polymer it is necessary to follow the yield behaviour under a variety of states of stress. This is most conveniently done by working in plane stress = 0) and making measurements in pure shear (o- = -0-2) and biaxial tension (o-i, 02 > 0) as well as in the simple uniaxial cases. The results of such experiments on glassy polystyrene are shown in Fig. 5.28. The modified von Mises and Tresca envelopes are also plotted. In both cases they have been fitted to the measured uniaxial tensile and compressive yield stresses, oy, and oy. It can be seen that the von Mises... 

Dimensional stability is one of the most important properties of solid materials, but few materials are perfect in this respect. Creep is the time-dependent relative deformation under a constant force (tension, shear or compression). Hence, creep is a function of time and stress. For small stresses the strain is linear, which means that the strain increases linearly with the applied stress. For higher stresses creep becomes non-linear. In Fig. 13.44 typical creep behaviour of a glassy amorphous polymer is shown for low stresses creep seems to be linear. As long as creep is linear, time-dependence and stress-dependence are separable this is not possible at higher stresses. The two possibilities are expressed as (Haward, 1973)... 

In many applications materials are subjected to compressive stresses. The macroscopic phenomena of collapse under an axial compression are the well-known shear and kink bands. In polymers they are caused by the buckling of chains, accompanied by changes in the chain conformation. The resistance against buckling is expressed by the yield strength under axial compression, ac max. Northolt (1981) found a relationship between c,max and Tg. 

---