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Fracture Mechanics in Polymers

Fracture mechanics in polymers has become an increasingly concerning field as many industries transition to implementing polymers in many critical structural applications. As industries make the shift to implementing polymeric materials, a greater understanding of failure mechanisms for these polymers is needed . Polymers may exhibit some inherently different behaviors than metals when cracks are subject to loading. This is largely attributed to their tough and ductile mechanical properties. Microstructurally, metals contain grain boundaries, crystallographic planes and dislocations while polymers are made up of long molecular chains. In the same instance that fracture in metals involves breaking bonds, the covalent and van der Waals bonds need to be broken for fracture to occur. These secondary bonds (van der Waals) play an important role in the fracture deformation at crack tip. Many materials, such as metals, use linear elastic fracture mechanics to predict behavior at the crack tip. For some materials this is not always the appropriate way to characterize fracture behavior and an alternate model is used. Elastic-plastic fracture mechanics relates to materials that show a time independent and nonlinear behavior or in other words plastically deform. The initiation site for fracture in these materials can often occur at inorganic dust particles where the stress exceeds critical value.

Under standard linear elastic fracture mechanics, Griffiths law can be used to predict the amount of energy needed to create a new surface by balancing the amount of work needed to create new surfaces with the sample’s stored elastic energy. His popular equation below provides the necessary amount of fracture stress required as a function of crack length. E is the young’s modulus of the material, γ is the surface free energy per area and a is the crack length.

$\sigma =\surd (2\gamma *E/\pi *a)$

Griffith Law

While many ideas from the linear elastic fracture mechanics (LEFM) models are applicable to polymers there are certain characteristics that need to be considered when modeling behavior. Additional plastic deformation should be considered at crack tips as yielding is more likely to occur in plastics.

Yielding Mechanisms

Figure 1 - Crack-tip Craze Zone

As polymers yield through dislocation motions in the slip planes, polymer yield through either shear yielding or crazing. In shear yielding, molecules move with respect to one another as a critical shear stress is being applied to the system resembling a plastic flow in metals. Yielding through crazing is found in glassy polymers where a tensile load is applied to a highly localized region. High concentration of stress will lead to the formation of fibrils in which molecular chains form aligned sections. This also creates voids which are known as cavitation and can be seen at a macroscopic level as a stress-whitened region as shown in Figure 1.  These voids surround the aligned polymer regions. The stress in the aligned fibrils will carry majority of the stress as the covalent bonds are significantly stronger than the van der Waals bonds. The plastic like behavior of polymers leads to a greater assumed plastic deformation zone in front of the crack tip altering the failure process.

Crack Tip Behavior

Just as in metals, when the stress at the crack tip approaches infinity, a yield zone will form at this crack tip front. Craze yielding is the most common yielding method at the crack front under tension due to the high triaxial stresses being applied in this local region.  The Dugdale- Barenblatt strip-yield model is used to predict the length of the craze zone. KI represent the stress intensity factor, s is the crazing stress being applied to the system (perpendicular to the crack in this situation), and r is the crazing zone length.

$\rho =\pi /8(K_{I}/\sigma _{c})$

Dugdale-Barenblatt strip-yield model

The equation for stress intensity factor for a specimen with a single crack is given in the following equation where Y is a geometric parameter, s is the stress being applied and a is the crack length. For an edge crack ‘a’ is the total length of the crack where as a crack not on the edge has a crack length of ‘2a’.

$K=Y\sigma \surd (\pi *a)$

Stress Intensity Equation

As the fibrils in the crack begin to rupture the crack will advance in either a stable, unstable or critical growth depending on the toughness of the material. To accurately determine the stability of a crack growth and R curve plot should be constructed. A unique tip of fracture mode is called stick/slip crack growth. This occurs when an entire crack zone ruptures at some critical crack tip opening displacement (CTOD) followed by a crack arrest and then the formation of a new crack tip.

Critical Stress Intensity Factor

The critical stress intensity factor (K_IC) can be defined as the threshold value of stress intensity base on the material properties. Therefore, the crack will not propagate so long as K_I is less than K_IC. Since K_IC is a material property it can be determined through experimental testing. ASTM D20 provides a standard testing method for determining critical stress of plastics. Although K_IC is material dependent it can also be a function of thickness. Where plane stress is dominant in low thickness samples increasing the critical stress intensity. As your thickness increases the critical stress intensity will decrease and eventually plateau. This behavior is caused by the transitioning from the plane stress to plain strain conditions as the thickness increases. Fracture morphology is also dependent on conditions at the located at the crack tip.

Fatigue

Figure 2 - Hysteresis Stress Strain Curve

Fatigue in polymers, caused by cyclical loading, is controlled by two general methods; hysteresis heating and chain scission. If the polymer is relatively brittle it will exhibit fatigue crack growth through chain scission. In this mechanism the crack-tip yielding is limited by the brittle material properties and each loading cycle breaks a specific amount of bonds allowing the crack front to advance. Polymers with viscoelastic behavior fatigue by the hysteresis heating mechanism. In this mechanism, when loading and unloading the polymer the stress-strain curve will act as a hysteresis loop as shown in Figure 2, creating a significant amount of work on the material. This is different than an elastic material where the loading and unloading paths are the same and strain energy is able to be recovered. The work inputted into the material (area of the hysteresis loop) will be converted to heat raising the temperature of the material possibly above the glass transition temperature. This creates a localized melting at the crack tip allowing the crack to advance. The magnitude at which the crack front will advance is largely dependent on the amount/magnitude of cycles, glass transition temperature of the material and the thermal conductivity of the polymer. A polymer that has a high thermal conductivity will dissipate the heat much fast than a material with a low coefficient.

Figure 3 - Cumulative Loading S-N Curve

Cyclic loading of samples without a notch will be the most resistant to fatigue but can still fail based on loading conditions and cycles. A S-N curve represents the amount of cycles being applied along with the stress amplitude and can derived from the Goodman relationship.

$\sigma _{a}=\sigma _{f}(1-\sigma _{m}/\sigma _{t})$

Goodman Relationship

Where σ_f is the fatigue stress, σ_m is mean stress, σ_a is the amplitude stress and  σ_t is the tensile stress of the sample being tested. In certain applications of polymers, materials will experience cyclic loading at different stress levels. Figure 3 gives an S-N diagram of cumulative cycles being applied at different stress amplitudes. The variable n represents the number of cycles being applied at the designated stress level and N is the fatigue life at the same stress level.

Figure 4 - Paris Law Regime

Many times, polymeric materials containing a crack are subject to cyclic loading in service. This decreases the life expectancy of the sample drastically and should be taken into consideration. In cases where polymers such as PVC follow the rules of linear elastic fracture mechanics, Paris law may be used to relate the fatigue crack propagation rate to the magnitude of stress intensity being applied. Below a certain stress intensity, crack propagation slowly increases until stable crack propagation is reached from higher levels of stress intensity. Higher levels of stress intensity leads to an unstable crack rate as shown in Figure 4. This figure is a log plot of the crack propagation rate versus the max stress intensity example. The stable crack growth regime represents the linear region of the red curve which is described using the Paris Law where ‘A’ is a pre-exponential factor.

$da/dN=A(K)^{m}$

Paris Law

References

Ideas for Project:

1. Fracture Mechanics in Polymers. Go further in depth about crack initiation and crack growth within a polymer at different locations. Particularly what happens at the crack tip in polymers should be unique when compared to metals.Fracture in polymers
2. Welding of liquid-crystal polymer (LCP) such as Kevlar. This polymer has superior properties such high strength and extremely intert. Welding these polymers can degrade these desirable properties. I would expand on how welding could be conducted on these polymers to preserve properties. Liquid-crystal polymer
3. Welding of composite polymers. This topic can consider both advanced composites and reinforced plastics (or one or the other). The addition of the fibers is implemented to reinforce the composite giving it more desirable properties.Polymer matrix composite

Fracture Mechanics in Polymers (Rough Draft):

Instructor Comments

All three topics are good. For welding of polymeric composites, that topic can be very broad, so you may want to limit yourself to a particular group of composites (e.g. advanced composites, random fiber composites, particulate reinforced composites, etc.).

Please let me know which topic you choose. Thanks!