Yield-Craze Behavior of Amorphous Polymers

Molecular Criterion for Failure Mode

Many fractures of plastics are caused by cavitation and crazing or by yielding. Which mechanism dominates depends on the properties of the plastic and on the test conditions. Important plastic properties that affect the fracture behavior include free volume, entanglement density, molecular weight, as well as number, size and shape of defects (voids) in the plastic. The fracture is also affected by the test method and test conditions such as rate of deformation, stress state, temperature, age of sample, and specimen geometry. The effect of these extrinsic and intrinsic variables on the brittle-ductile behavior is as follows

• Aging of polymers in the glassy state reduces the ductility but improves load-bearing properties such as mechanical strength, modulus, and creep resistance. These changes can be explained by a decrease of the initial porosity as aging proceeds, meaning the plastic becomes denser, which in turn, increases the stress at craze or yield initiation.

• Temperature increase leads to a decrease in the critical yield and craze stress because both the modulus and density decrease with increasing temperature. These effects can be explained in terms of chain relaxation; with increasing temperature the time for chain relaxation (disentanglement) increases relative to the time scale of the deformation process. An increase in temperature also increases the free volume which leads to weaker intermolecular interactions which further lowers the modulus.

• Strain rate (test speed) has a similar effect as temperature; if the strain rate is high, the time for chain disentanglement becomes long with respect to the time scale of the experiment so that more micro voids are formed as more chains tend to break. An increase in strain rate also increases the critical stress at craze or yield initiation and reduces the area under the stress-strain curve, indicating a more brittle (less tough) behavior under these conditions.

• Entanglements behave like physical cross-links which increase the resistance to void formation and crack propagation. Thus, high molecular weight polymers with low entanglement density tend to craze whereas polymers with high entanglement density tend to yield. The entanglement density is directly related to the chain stiffness which is often expressed in terms of characteristic ratio (C_∞). Very flexible chains reach the limiting value of C_∞ = 2.

• Molecular weight (MW) is another important intrinsic factor. Lowering the MW increases the mobility of the polymer chains.⁴ Therefore, a lower MW polymer has a greater tendency to yield than a higher MW polymer, or in other words, with decreasing MW, the brittle-ductile transition shifts to lower temperatures. A dramatic change in the brittle-ductile behavior occurs when the MW reaches the critical entanglement weight, M_c.⁵ Below M_c no continuous network of entanglements exists. Thus, the polymer is much more ductile.

Because of their viscoelastic properties, the fracture behavior of polymeric materials varries considerably with the temperature. At low temperatures and/or high strain rates, polymers tend to fail via cavitation and crazing whereas at high temperatures and/or low strain rates yielding is more likely to occur. Thus the two failure modes compete with each other.

The intrinsic ductility, i.e. the propensity for yielding, increases as the characteristic ratio (C_∞) decreases. It has been postulated¹ that the yield stress σ_y is directly proportional to C_∞ and the cohesive energy density (or square of solubility parameter), e_coh = δ_h²:

σ_y ≅ C_∞ δ_h²

Crazing, on the other hand, is initiated by chain scission, and thus the energy to initiate cavitation and crazing, E_cav, should be proportional to the entanglement density ν_e. Since the crazing or cavitation stress, σ_z, is proportional to e_cav^½, we find:^(2,3)

σ_z ≅ ν_e^1/2

Combining these two equation gives the molecular condition for the main failure mode of the craze-yield behavior:

σ_z / σ_y ≅ ν_e^1/2 / C_∞ δ_h²

With

ν_e = ρ_a / (3 M_r C_∞²)

the expression above can be rewritten as

σ_z / σ_y ≅ (ρ_a / 3M_r)^1/2 / [C_∞ δ_h]²

or

σ_z / σ_y ≅ (3M_r / ρ_a)^1/2 ν_e / δ_h²

The factor (ρ_a / 3M_r)^1/2 is more or less constant for the majority of polymers, except for those with high packing density, ρ_a, and/or with very large repeat units, M_r^2,3. Thus the competition between crazing and yielding is determined by the entanglement density ν_e (or characteristic ratio C_∞) and by the cohesive energy e_coh or by the solubility parameter δ_h, respectively. Polymers having a low entanglement density ν_e (large characteristic ratio C_∞) and/or high cohesive energy density e_coh have a low σ_z / σ_y ratio, and thus tend to craze whereas polymers with small C_∞ (high ν_e) and low e_coh have a high σ_z / σ_y ratio, and thus tend to yield rather than craze.⁶

References & Notes

1. R. P. Kambour, Polymer Communications, 24 (1983) 292.

2. Souheng Wu, Poly. Engineering & Science, Vol 30, No. 13 (1990)

3. Souheng Wu, Polymer International, 29, 229-247 (1992)

4. Shorter polymers have more end groups per unit volume which have more degrees of
   freedom, which increases the free volume and thus the chain mobility.

5. The critcal entanglement weight M_c is about twice the weight of an entanglement strand,
   M_c ≈ 2M_e

6. Both the entanglement density and cohesive energy depend on the temperature. Thus,
   when the temperature changes, the ratio σ_z / σ_y changes as well. Since a decrease in
   cohesive energy density and packing density increases chain mobility, polymers tend to
   yield at higher temperatures.