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 = \sqrt{\frac{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 As metals 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 (KIC) can be defined as the threshold value of stress intensity base on the material properties. Therefore, the crack will not propagate so long as KI is less than KIC. Since KIC is a material property it can be determined through experimental testing. ASTM D5045-99 provides a standard testing method for determining critical stress of plastics. Although KIC 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 As the needs for polymers for engineering purposes are increasing, the fatigue behavior of polymers is receiving more attentions. Polymer fatigue life is affected by multiple factors, including temperature, oxidation, crystallization and so on. Therefore, the need becomes vital for people to study and predict the mechanical performances of polymers under different environments. The experimental methods to study polymer fatigue vary, including pure shear test, simple extension test, single edge crack test and tearing test, among which the most widely used geometry people adopt is mode I cyclic tension test under pure geometry. This is due to the fact that polymers have viscoelastic behavior and poor conductivity of heat, and they are more sensitive to their cyclic loading conditions than metal. Unlike metals and other materials, polymer do not cyclic harden; rather, polymers perform cyclic softening for most of the time, and the extent of which usually depends on the loading conditions of the experimental setup. In a rare case, polymers can also remain cyclically stable under small strain deformations, during which the polymer remain linearly elastic. In reinforced polymers, crack initiation usually occurs at the interface of polymer fiber and the matrix. Fatigue performances in polymers caused by cyclical loading usually go through two stages:
crack initiation/nucleation and
crack growth. Hence, a lot of researcher design experiments to study the fatigue behaviors of polymers according to these two phases, especially for rubber fatigue.
Crack Nucleation Approach The crack nucleation approach considers that polymers will eventually crack under a history of stress and strains. Study under this proposal is first adapted by
August Wöhler in the 1860s, who aimed to study railroad axles. Two important parameters are involved in this conversation: maximum principal strain and strain energy density. in metals.
Hysteresis heating and Chain scission Fatigue in polymers, controlled by cyclical loading, is caused by two general mechanisms: 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, during loading and unloading, the polymer the stress-strain curve will act as a hysteresis loop as shown in Figure 2, creating energy on the material as discussed before. Because energy is dissipated to the crack tip, this process is different from the cyclic loading of an elastic material where the loading and unloading paths are the same and strain energy can be recovered. The work inputted into the material (area of the hysteresis loop) is 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. 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 σf is the fatigue stress, σm is mean stress, σa is the amplitude stress and σt 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. 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 Power Law model where ‘A’ is a pre-exponential factor. da/dN = A(K)^m
(Power Law Regime Equation) Recrystallization This process can be caused as a consequence of extensive movement of chain segments like in case or work hardening of materials.
Fatigue in Nylon When a
Nylon component is subjected to conditions of tensile fatigue, failure occurs when a minimum strain is reached. this means that the lifetime of nylon material is dictated by the time under load and not on the number of cycles
Fatigue of Short-Fibre-Reinforced Plastics Fatigue failure in these reinforced polymers is due to the formation of micro cracks that are easily initiated, and which coalesce into one crack, causing the final failure ==Impact Fracture==