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Chapter 15: Yielding

Last updated

Jan 28, 2025
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- Chapter 14: Composites
- Chapter 16: Polymer Gels

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Yielding Criterion:

So far we have just been talking about stress and strain up to the yield point. However what happens beyond the elastic limit of materials? Well we have already discussed that the yield point denotes that the deformation is now permanent or plastic and we have initiated dislocation or defect motion. But as engineers perhaps the better or more useful question to ask is can we predict whether yielding will occur for materials that are not just simply loaded uniaxially. There are multiple yielding criterion that we will discuss, specifically: Rankine, Tresca, and von Mises Yield Criterion.

Rankine:

Rankine criterion also known as the maximum normal stress criterion states the a material will fail or yield when the maximal principal (normal) stress (σ₁) reaches the value where the material yields in uniaxial tension or compression so the material will yield when:

$\sigma_{1}\geq\sigma_{y} \nonumber$

IMPORTANT NOTATION NOTE: Once again these mechanics people utilize some confusing notation. This σ₁is not equivalent to σ₁₁this σ₁is referring to the maximum principal normal stress. Rankine doesn’t accurately describe material yielding, it is missing shear stress.

Tresca:

So let’s take a look at the Tresca criterion or the max shear stress criterion which states that the material yields when the $\tau_{max}=\frac{\sigma_{1}-\sigma_{3}}{2}=\sigma_{y}$ reaches the value it does when material yields under uniaxial loading which is described below:

$\sigma_{1}-\sigma_{3}\geq\sigma_{y} \nonumber$

Again here it should be noted that σ₃is the minimum principal normal stress. This is better, but in 1913 for WWI/WWII subs because they found that the Tresca criterion was a poor predictor for yielding.

Von Mises:

So we developed the Von Mises criterion is also called the maximum shear deformation energy (SDE) criterion and states the material will yield when the SDE reaches the yield stress value under uniaxial loading which is:

$\sigma_{eff} = \sqrt{\frac{1}{2}[(\sigma_{11}-\sigma_{22})^2+(\sigma_{22}-\sigma_{33})^2+(\sigma_{33}-\sigma_{11})^2+6(\sigma^{2}_{23}+\sigma^{2}_{31}+\sigma^{2}_{12})]}\geq\sigma_{y} \nonumber$

or in terms of the principal stresses:

$\sigma_{eff} = \sqrt{\frac{1}{2}[(\sigma_{1}-\sigma_{2})^2+(\sigma_{2}-\sigma_{3})^2+(\sigma_{3}-\sigma_{1})^2}\geq\sigma_{y} \nonumber$

You can see by drawing the yield loci of these different criterion that Rankine is the most conservative yield criterion.

Figure $\PageIndex{1}$ : Yield Criterion. Rankine (blue), Tresca (green), Von Mises (red).

Polymer Yielding Mechanisms:

Upon application of a stress in excess of the yield stress the sample permanently deforms and any additional plastic strain is not recoverable once stress is removed. In metals, the origin of plastic deformation is the introduction and movement of dislocations in a lattice, resulting in shear banding on the macroscale; similar mechanism also apply for single-crystal polymer fibers.

In amorphous and semi-crystalline polymers, there are two observed yield behaviors: crazing and shear banding. Both can appear in the same material at the same time, or one can be observed and not the other. Crazes are observed as the opening of a crack with strands of polymer that extend between the crack faces, and leads to a change in volume (called dilational deformation). A shear band in polymers is essentially the same as shear banding in other materials, and is caused by a movement of material along directions of maximum shear stress (typically ± 45^ounder uniaxial tension) if the stress is above the yield point. Shear bands lead to the formation of necking and the reduction in cross-sectional area at the neck, versus in the bulk; however, volume is still conserved in the sample overall.

Shear Banding:

Shear yielding in glassy, amorphous polymers (or semi-crystalline polymers) looks at the macroscale very similar to the shear yielding in non-polymeric materials. Above a critical yield stress, the strain will begin to increase at a roughly constant plateau stress as the material undergoes necking. Upon reaching high strains, the stress for continued elongation again begins to increase in a process called strain hardening until eventually the material fails.

As the material is elongated due to strain, it undergoes a process called strain hardening, which means that the amount of stress required to continue straining the material increases. In polymers, the molecular mechanism for strain hardening is the alignment of polymer chains with the stress; at high strains we can imagine the polymers become aligned with the stress and require further straining the sample requires pulling on bonds. Regardless of the molecular mechanism, as the material strain hardens more and more stress is necessary to continue elongation. However, to maintain volume conservation, elongating the material along the direction of pulling necessarily reduces the cross-sectional area in the other dimensions; since stress is measured as a force divided by an area, the true stress felt by the sample thus increases as elongation increases even if the actual force applied to the sample is held constant. Typically, stress-strain curves are drawn to show the engineering stress applied, since this is directly controllable experimentally. Above the yielding point, the decrease in the cross-sectional area upon elongation leads to an increase in the true stress felt by the sample even if the engineering stress is held constant. However, at the same time the amount of stress necessary to induce further strain is also increasing due to strain hardening. The onset of necking is thus a competition between the strain hardening and the decrease in the cross-sectional area.

When the material’s cross-section decreases by a greater amount than the material strain hardens, the true stress will be sufficient to drive further elongation of the sample even without an increase in the engineering stress. As a result, the material will continuously elongate at a plateau in the engineering stress until it strain hardens enough to prevent this unstable elongation and force other parts of the material to harden instead. This unstable propagation is called necking because physically the strain will concentrate in one section of the sample (typically where there is some fluctuation in the microstructure that leads to a stress concentrator) which will then preferentially elongate and form the neck. In polymers, because the stiffness of an aligned polymer is highly anisotropic, there is a huge amount of strain hardening at high elongations once the sample becomes aligned with the applied engineering stress. This leads to stable necking because upon neck formation the neck will unstably reduce to a small cross-sectional area consistent with full alignment, at which point the highly strain-hardened neck region will stop elongating, and the other softer regions will deform instead to increase the volume of the necked region. Hence, the polymers tend to form very stable necks. Again, this entire process appears to occur at constant stress because the stress in a stress-strain curve is the engineering stress, but the true stress felt in the material increases as the cross-sectional area decreases.

Finally, it is possible to observe the morphological transitions behind strain hardening. As necking occurs, x-ray scattering patterns can be used to observe the alignment of chains. Prior to necking, there is some isotropy to the polymer sample, which is then oriented by the application of stress. Upon elongation of a long neck, almost all chains tend to be elongated, leading to the high degree of strain hardening as just noted.

Shear bands are zones of material alignment that are 1000s of nm wide and nucleate at stresses that are approximately (theoretical yield stress) and continue to grow as stress exceeds. Shear band deformation is highly nonuniform. They are also not regions of high defect density but instead they are regions of chain alignment. Shear banding nucleation is also highly temperature dependent. Typically for uniaxial tension shear bands will form at 45^orelative to the applied stress at the locations of maximum shear stress.

Crazing:

Crazes regions of highly localized deformation and they are distinct from a crack. Fibrils will span the craze. Crazes will form perpendicular to the loading axis with fibrils aligned with the loading axis. Crazing will not occur under compressive stress. You can modify the von Mises criterion when considering shear banding and Tresca when considering crazing.

Crazes are also regions of highly localized deformation which is distinct from both a crack and a void but very similar to both. A craze occurs when a void is formed between two faces of a polymer and some polymer chains, fibrils/tendrils, will span the void and connect the two faces. Crazes occur in tension only, and look and act very similarly to cracks, though they are not identical. A crack is essentially the same but without the polymer tendrils connecting faces; crazes are thus unique to polymer samples because they have the long molecules necessary to cross across the void volume. Note that metals cannot form crazes because they do not have long chains in their microstructure that form these tendrils. Due to the formation of the void area between the two faces, the volume of the polymer increases, unlike the case of shear yielding where volume is conserved. Typically the fibrils are on the order of 20-50˚A, and about 50% of the void is filled with fibrils.

Crazes are very visible in samples because they scatter light due to a large change in the index of refraction between the bulk sample and the void, and the void is sufficiently wide that this scattering is visible. The craze will open such that the void is oriented perpendicular to the applied tensile stress, and will continue to grow/open in the perpendicular direction, essentially growing in width. Craze formation is favored by the inclusion of agents that lower the surface energy of the faces, essentially leading to nucleation points in the bulk where crazes can first grow.

We can understand the formation of crazes, and especially the formation of fibrils, by thinking about the surface free energy of the exposed craze face. As the craze grows, the polymer at the edge of the growing craze will have a large surface area, which is unfavorable given some positive surface energy (since polymer prefers to be in bulk, and the lack of neighboring polymers when facing a void effectively is an energetic penalty). At some point, it is more favorable for fibrils to break off from the bulk in order to exist independently in order to minimize their surface area. This is called the Taylor meniscus instability, and explains why fibrils form as the craze grows. As the craze continues to grow, the fibrils will elongate (since the faces essentially move farther apart), leading to some crystallization which will oppose the further growth of the craze.

The maximum stretching we can get is the length of polymer between entanglements, assuming entanglements are fixed over the timescale of craze formation. We can approximate the contour length between entanglements in terms of the entanglement molecular weight divided by the molecular weight of a monomer.

Asymmetry in Yield Criterion:

Unlike metals, polymers have a significant response to hydrostatic pressure, which changes the yield criteria considerably. The influence of pressure can be thought of as packing the sample together if the pressure is compressive. In metals the sample typically may not respond because the influence of pressure should not significantly change the metallic bonding in metals, which develop from the sharing of electrons, not favorable packing. In polymers, however, non-specific, relatively weak van der Waals interactions stabilize the crystalline state, interactions that are highly dependent on the distance in between polymer molecules. These interactions are weak enough that pressure can significantly influence the local packing of polymer chains. Hence, applying a compressive pressure improves the packing of the polymer which then requires more stress to yield the polymer crystal, while an outward pressure decreases the amount of energy required to deform the material. This means that the yield criterion for polymers is not symmetric as is the case for metals!

For amorphous polymers, the yield surface has two additional effects

Crazing Yield Line: Onset of crazing, which can only occur under tension, leads to a separate yield line above which crazing is preferred to shear banding
Asymmetry in Yield Surface Under Compression: Yield occurs at higher stress when a polymer is under compression than under tension. The yield surface is distorted from the typical ellipse as a result.

Figure $\PageIndex{3}$ : Shear Banding Yield Curve

Finally, it should again be noted that the onset of yield will be temperature and time dependent, with the yield stress dropping to 0 at either T_Mor T_Gdepending on the type of polymer.

Atomistic Perspective of Polymer Yielding:

At the atomistic level, plastic flow for semi-crystalline polymers can be explained by the slip of polymer chains past each other, where the aligned lamella deform as chains move past each other. This process is qualitatively analogous to the slip of metallic materials in a crystalline lattice, which makes sense since crystalline regions of polymer are also characterized by a crystalline lattice. However, how does plastic deformation occur for glassy polymers?

Well in a glassy state, we can think of polymer molecules as frozen in place in an amorphous state. This frozen state is due to a low temperature and minimal free volume, meaning that molecules have insufficient thermal energy to jump past near-neighbors due to a large barrier to this motion. We can think of the application of stress, then, as modifying the energy barrier to jumping, and can use an effective activated jumping model, which in this case is called the Eyring Theory of viscous flow. In Eyring theory, we imagine the molecules of the glass as largely confined to some lattice, where there is an energy barrier associated with hopping from one site on the lattice to the other. Under stress-free conditions, this energy barrier is prohibitively high, but we can imagine that applying a stress reduces the barrier for hopping in the direction at which the stress is applied. Under compression, then, we can imagine squeezing the polymer sample by reducing the barrier for flow in the directions orthogonal to the compression. Since we are assuming that all molecular motion is due to a series of jumps, the total strain rate should thus be related to the effective jumping rate after the application of stress. Let’s call the jumping rate in the absence of stress ν₀, and assume it is given by an Arrhenius type equation:

$\nu_0 = B \exp ( -\Delta G^* / kT ) \nonumber$

Figure $\PageIndex{4}$ : Shear reduces energy barrier for hopping.

This expression gives the jump rate for a glassy molecule in the absence of stress, where B is some constant and ∆G^∗is the energy barrier for jumping. Now consider applying some stress σ across and area A acting over a jump distance of x. The application of this stress changes the energy barrier by an amount of σAx, since this is a work term. We expect the barrier to be lowered in the direction in which the stress is applied, giving a new forward jumping rate ν_f

$\nu_f = B \exp[ ( -\Delta G^* - \sigma A x) / kT ] = \nu_0 \exp ( \sigma A x/kT) \nonumber$

Note that we rewrote the final forward jumping rate in terms of the initial jumping rate. Similarly, the backwards jumping rate, that is, the rate at which molecules jump in a direction opposed to the stress, should be significantly lower since the energy barrier to jumping backwards is higher by the same factor of σAx. The backwards jumping rate ν_bis thus

$\nu_b = \nu_0 \exp (- \sigma A x/kT) \nonumber$

Note the sign change as the only difference - this rate is much lower than the initial jumping rate ν₀. We can thus express the strain rate as proportional to the difference between the forward and backwards jump rates, since this difference is essentially some net flow in the forward direction. We thus write the strain rate as

$\frac{d\epsilon}{dt} = \nu_f - \nu_b = \nu_0 \left [ \exp ( \sigma A x/kT) - \exp (- \sigma A x/kT) \right ] = K \sinh \left ( \frac{\sigma V}{kT} \right ) \nonumber$

Note that the term on the left is equivalent to the sinh function with new prefactor K as a lumped constant, and replacing Ax = V , reflecting the fact that the cross sectional area A times the distance x is a volume, called the activation volume. Physically, the activation volume is the volume of polymer material that must move to yield shear flow.

If σV is comparable to ∆G^∗, then the jump rate starts to become very high, and we would expect very high strain rates, which we can think of as essentially a liquid state. This would then mark the onset of yield, since now it is much easier for the glassy molecules to move past each other.

Plastic Strain Recovery in Polymeric Materials:

One extremely unique property of polymers is the ability of polymers to recover plastic deformation at long times and elevated temperatures. Polymers that undergo plastic deformation, can recover plastic strain due to the ability of polymers to flow at long times/elevated temperatures. This behavior is also techically accessible to metals but the conditions are very extreme to achieve this behavior. For polymers heating to near T_gcan result in significant recovery depending on the material, processing history, and loading history.

Polymer Self-Healing:

Polymers also have the unique ability to exhibit self-healing behavior. Certain polymers when damaged can intrinsically self heal and repair cracks via cross-linking or some type of triggered polymerization. Also for some materials like silly putty which are above T_gat room temperature this material can self heal at reasonable time scales due to polymer interdiffusion. Alternatively you can make extrinsic selfhealing polymers that will self heal when a crack triggers some type of microcapsules that will polymerize and fill the crack gap.