Strain Hardening of Amorphous and Semi-crystalline Polymers
Strain hardening is observed as a strengthening of a material during large strain deformation. It is caused by large scale orientation of chain molecules and lamellar crystals. This phenomenon is quite often observed when plastic materials are stretched beyond their yield point. Polymers that exhibit greater strain hardening are often tougher and undergo ductile rather than brittle failure.
The degree of strain hardening depends on both intrinsic and extrinsic factors. Important intrinsic factors include molecular structure, cohesive energy, molecular weight (distribution), and morphology (degree of crystallinity and branching). Important extrinsic factors are strain rate, temperature, stress state, and specimen geometry. The morphology is one of the most important factors which also depends on the processing conditions.
A ductile amorphous material shows a characteristic yield point. At this point, the material starts to undergo plastic deformation. Materials with high plasticity show strong necking and cold drawing, that is, beyond the yield point, the cross section in the necking region will steadily reduce until the material breaks abruptly at rather large elongation. The necking is often followed by strain hardening because the chain molecules tend to orient and allign in the direction of the load which increases the strength and stiffness of the plastic in stretch direction. The opposite of strain hardening is strain softening which is a less common phenomenon. Miehe et al. postulated that true strain softening is the result of localized shear band formation which produces local voids which in turn weaken the polymer.1
Strain Hardening of Semi-Crystalline Polymers
The mechanical properties are greatly affected by the degree of crystallinity
and the lamellae structure and size. For example, it has been shown that the yield stress is proportional to the lamellar thickness, whereas the strain hardening modulus does not depend on crystallinity or lamellae thickness but is mainly controlled by the entanglement density of the amorphous phase
which is affected by the cooling rate. Slowly cooled polymers are less entangled than quickly cooled ones and therefore show less strain hardening.
Although the crystalline phase does not contribute to strain hardening, it
greatly affects the mechanical properties.
The degree of crystallinity and the lamellar thickness can be altered by changing the rate of crystallization from melt and by annealing (cold crystallization) at elevated temperatures above the glass transition temperature but below the melting point. Thus, the ultimate yield stress of semi-crystalline materials also depends on the processing conditions.
Engineering Stress-Strain Curves (Schematic)
Gaussian Treatment of Strain Hardening
A traditional interpretation of the polymeric strain hardening is the equilibrium theory and the theory of entropic elasticity which assumes that sub chains can move freely between crosslinks (Gaussian distribution).2 The true stress-strain curve after yielding can then be described by3,4
σtrue = σy + GP (λ2 - λ-1)
where GP is the strain hardening modulus, σtrue the true stress, λ = L/L0 the extension ratio, and σy is the yield stress. In this equation, the small strain due to elastic deformation has been neglected, that is yielding is assumed to start at λ ≈ 1.
In tensile testing, the engineering and not the true stress-strain curves are recorded. Dividing the equation above by λ yields this relationship:
σeng = σy / λ + GP (λ - λ-2)
The relation between stress and strain for different yield stress (σy = 20 - 35 MPa) is shown in the figures above. Yielding is assumed to occur at about 10% strain when using a constant hardening modulus of GP = 5 MPa. The engineering stress shows a steep and continuous increase at low strain followed by a decrease in stress after yield and large strain. This behavior is typical for many thermoplastics. The strong stretching and drop in engineering stress is usually accompanied by necking (localized stretching) which is known as geometric softening5, whereas a continuous increase in stress is typical for homogeneous elongation. The condition for the transition from homogeneous deformation to necking is,
It follows that the ratio of yield stress and strain hardening modulus has to exceed three to induce necking (σy / GP > 3 for λ → 1). The drop in engineering stress is follwoed by an increase in stress at larger elongations. The inversion point where dσ/dλ = 0 is given by
This condition describes stable growth of the necking (cold drawing).
References and Notes
C. Miehe, S. Goektepe and J. Mendez Diez, Int. J. Of Solids and Structures, Vol. 46, 1, pages 181-202 (2009)
This theory predicts qualitatively the right shape of the curve but the values are much higher than measured, that is GP = ρe k T is inconsistent with rheological measurements.
R. N. Haward, Polymer, Vol. 28, 1485 (1987)
R.N. Haward, Polymer, Vol. 35, 3858 (1994)
Strain hardening is sometimes erroneously called "strain softening" when the shear or tensile resistance decreases with increasing strain. The observed drop in engineering stress beyond the yield point is caused by a reduction of the cross section area (necking). The true stress, however, which is the load divided by the actual cross-sectional area, typically increases.
As has been shown, GP is indeed proportional to the entanglement density ρe but it is not proportional to temperature, T. Furthermore, it was found that the true GP is about 100 times larger than predicted by the Gaussian model. The much larger modulus might be attributed to internal frictional forces or to the the greater energy necessary to plastically deform glassy polymers. Nevertheless, when using empirical strain hardening moduli, the Gaussian model gives reasenable predictions for many uncrosslinked polymers.
It is assuemd that the polymer is incompressible. Then it follows σTrue = λ σeng