Crazing

Crazing, derived from the Middle English term "crasen" meaning "to break", has historically been used to describe a network of fine cracks in the surfaces of glasses and ceramics. This term was naturally extended to describe similar phenomena observed in transparent glassy polymers. Under tensile stress, these polymers develop what appear to be cracks on their surfaces, often very gradually or after prolonged periods. These fine cracks, or crazes, were noted for their ability to propagate across specimens without causing immediate failure. Crazing in polymers was first identified as a distinct deformation mechanism in the mid-20th century. Unlike inorganic glasses, most glassy polymers were found to be able to undergo significant plastic deformation before fracture occurs. Early observations noted the presence of crazes that propagated across specimens without causing immediate failure, indicating their load-bearing capacity and provided further insights into the nature of crazes, describing their appearance and behavior under stress. Significant advancements in the understanding of crazing were made in the 1960s and 1970s, illustrating the formation and structure of crazes in various polymers and on the stress conditions necessary for craze formation in polymers. Researchers demonstrated that crazes grow perpendicular to the principal stress and highlighted the critical stress levels required for their initiation. == Mechanisms of crazing ==

Craze nucleation and growth There is typically a delay between the application of stress and the visible appearance of crazes, indicating a barrier to craze nucleation. The time delay between the application of stress and the nucleation of crazes can be attributed to the viscoelastic nature of the process. Like other viscoelastic phenomena, this delay results from the thermally activated movements of polymer segments under mechanical stress. Crazing involves a localized or inhomogeneous plastic strain of the material. However, while plastic deformation essentially occurs at constant volume, crazing is a cavitation process that takes place with an increase in volume. The initiation of crazing normally requires the presence of a dilative component of the stress tensor and can be inhibited by applying hydrostatic pressure. From a solid mechanics perspective this means that a necessary condition for craze nucleation is having a positive value of I_1, the first stress invariant that represent the dilatational component: I_1 = \sigma_1 + \sigma_2 + \sigma_3>0 This condition is favored by the presence of triaxial tensile stresses, a condition that exist in defects of bulky samples subjected to plane strain. The cavitation involved in crazing allows the material to achieve plastic strain faster. The presence of cracks or defects in bulky samples will favor the initiation of crazing, as these defects are points of high concentration of stresses and can cause the formation of initial microvoids. Crazes grow on the plane of maximum principal stress. provides a fundamental explanation for the growth of crazes. This phenomenon is commonly observed when two flat plates with a layer of liquid between them are forced apart or when adhesive tape is peeled off from a substrate. The hypothesis concerning craze formation The meniscus formation is a result of the imbalance of surface tension forces, surface tension acts to minimize the surface area, and any disturbance can create a meniscus, a curved surface at the interface between two phases. This causes the polymer chains to pull apart and form a cavity filled with a fibrillar network. This type of instability is well documented in various classes of materials and the concepts were developed from experiments involving the interpenetration of two fluids with different densities. The breakdown of the craze starts gradually as voids coalesce to produce a cavity equal in thickness to the craze itself. Craze breakdown, which leads to crack extension, is crucial to the failure process. However, the detailed mechanisms involved remain a subject of debate among experts, despite the many models that have been suggested. In the framework of fracture mechanics, once a crack of size a is initiated due to an applied stress \sigma_\infin, its propagation can be analyzed using the stress intensity factor K: K=\sigma_\infin \sqrt{\pi a} which describes the stress state near the tip of a crack. According to linear elastic fracture mechanics (LEFM), the crack will propagate when the stress intensity factor reaches a critical value K_c, known as the fracture toughness of the material. This approach allows for the prediction of crack growth and the evaluation of the material's resistance to fracture under various loading conditions. It has been observed that for a crack growing relatively slowly in a stable manner and preceded by a craze, then the relationship between K and crack propagation speed \dot{a} can be described by an equation of the form: K \propto \dot{a}^n Where n is related to the viscoelastic processes at the crack tip that stabilize crack growth. == Craze yielding and shear yielding ==

The yield point of a material represents the maximum stress it can endure without resulting in a permanent strain after the load is removed, it refers to the stress level required to initiate plastic deformation. When analyzing the yielding behavior of polymers, it is crucial to differentiate between shear yielding and craze yielding due to their distinct microstructural characteristics. Shear yielding involves the material undergoing shear flow with minimal or no change in density. In contrast, craze yielding, is highly localized and the macroscopic behaviors of shear and craze yielding differ significantly. Crazing and shear yielding are the two principal deformation mechanisms inherent to polymers. Those two phenomenon are competitive mechanisms (although they are not mutually exclusive and can coexist Shear bands may form in a material that exhibits strain softening, hence when the conditions which favour crazing are suppressed, polymers will tend to form shear bands. ==Yielding criteria for polymers ==

Yielding criteria 's hexagonal yield surface. A yield criterion is a general condition that must be satisfied by the applied stress tensor for yield to occur. A yield criterion expressed in terms of stress can be visualized as a surface encompassing the origin in principal stress space. Yielding does not occur until the stress increases from zero (the origin) to some point on this surface. For isotropic elastic materials with a ductile failure mode, the most used criteria are the Tresca criterion of maximum tangential stress and von Mises yield criterion based on maximum distortion energy. The latter is the most used and states that yielding of a ductile material begins when the second invariant of deviatoric stress I_2 reaches a critical value. I_2=\frac{1}{6}\left[(\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2 \right] The criterion assumes that for yield to not occur the stress coordinate must be contained within the cylindrical surface described by the following equation: (\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2 = 9\tau_{oct}^2 where \tau_{oct} is the octahedral shear stress: \tau_{oct} = \frac{1}{3} \sqrt{(\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2} This criterion is observed quite well by most metals however it cannot be used to describe shear yielding in polymers since in those materials the hydrostatic component of the stress tensor affects the yield stress. The modified von Mises criterion for shear yielding Experiments have shown that nelther the Tresca nor the von Mises criterion adequately describes the shear yield behaviour of polymers, because for example the true yield stress is invariably higher in uniaxial compression than in tension, and uniaxial-tensile tests conducted in a pressure chamber show that yield stresses of polymers increase significantly with hydrostatic pressure. The von Mises criterion can be modified to incorporate the effect of pressure on the state of the material by substituting in its original formulation: (\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2 = 9\tau_{oct}^2 a value of \tau_{oct} that is linearly dependent on the hydrostatic component of the stress tensor: \tau_{oct} = \tau_0-\mu \sigma_m while \sigma_m represents the hydrostatic component: \sigma_m = \frac{\sigma_1 + \sigma_2 + \sigma_3}{3} \tau_{0} and \mu are material parameters that depend on loading rate and temperature. The constant \tau_{0} is the yield stress in pure shear, since under this stress state the value of \sigma_m is zero. In plane stress the modified von Mises criterion is an ellipse on the principal axis space, but differently from the standard criterion it is shifted with respect to the origin, due to the different behavior of the polymeric materials depending on the hydrostatic component of the stress tensor. Yielding criteria for crazing An effective crazing criterion has been proposed in the early 70's by Sternstein and coworkers. attempted to create a more comprehensive relationship valid for a general triaxial state of stress. Their assumption is that crazing occurs when the strain in any direction reaches a critical value (\epsilon_c) that depends on the hydrostatic component of the stress tensor: \epsilon_c = Y_1(t,T) + \frac{X_1(t,T)}{I_1} Where X_1 and Y_1 are again time and temperature dependent parameters. The maximum tensile strain in an isotropic body under a general state of stress defined by the principal stresses is always in the direction of the maximum principal stress (\sigma_1>\sigma_2>\sigma_3) and is given by: \epsilon_1 = \frac{1}{E}(\sigma_1-\nu\sigma_2-\nu\sigma_3) where E is Young's modulus and \nu is Poisson's ratio. So the previous equation can be re-written as to define the criterion in terms of the principal stresses: \sigma_1-\nu\sigma_2-\nu\sigma_3 = Y_1E + \frac{X_1E}{\sigma_1+\sigma_2+\sigma_3} for plane stress this equation is very similar to the one proposed by Sternstein and coworkers. proposed an alternative crazing criterion based on a molecular theory for distortional plasticity, he described the process of crazing as a micromechanical problem of elastic-plastic expansion of initially stable micropores produced by a thermally activated mechanism under stress to form a craze nucleus. With his analysis of the condition of craze nucleation he provided a derivation of crazing. This model provides an elegant criterion that can be easily applied for any stress state and it is not based on strain, which is a poor parameter of state: \tau_{oct}=\frac{C_1}{C_2+\sigma_m} where \tau_{oct} is the octahedral shear stress, while \sigma_m represents the hydrostatic component: \sigma_m = \frac{\sigma_1 + \sigma_2 + \sigma_3}{3} and C_1 and C_2 are time-temperature constants. General yielding criterion By combining the criterion for shear yielding and crazing a region can be found in which no yielding can occur. This can be easily seen in the \sigma_1-\sigma_2 plane (considering plane stress condition), where the two criteria intersect a transition between the two mechanisms is expected. Considering that polymers have a viscoelastic behaviour an effect of loading rates and temperatures on shear yield stress and on crazing yield is observed. When the loading conditions (T,\dot{\epsilon}) are such that the tensile stress for shear yielding is lower than the crazing stress no crazing will be observed on the material and a brittle to ductile transition can be expected. In order to have a comprehensive yielding criterion both yielding phenomenon must be taken into account and their dependence on external parameters has to be determined. Only if these conditions are known a proper yield criterion expressed in terms of stress can obtained as a surface encompassing the origin in principal stress space. ==See also==