Polymer and composite failure and durability Exam Questions

ALTERATION:

1. Interaction with liquids (reversible): the interaction with affine liquid that act as plasticizer with cohesive energy similar to the polymer one. It can induce complete dissolution of non-crosslinked polymers if the amount of liquid is enough. But in general liquid can always induce swelling (without dissolution) as in case of cross-linked polymers (increase of FREE VOLUME). The material results more ductile (Tg is reduced), its σ is greatly reduced and can lead to premature plastic deformation. It is usually quite reversible.

2. Environment stress cracking (ESC) (irreversible): First requirement for this kind of degradation is the presence of a stress (it can be externally applied or it can be a residual stress from production process). When exposed to certain environment, depending on the polymer-environment coupling (less viscosity=worst effect), we can have a decrease of the maximum strain for environmental stress cracking with respect to the one in air. The liquid

should have the same cohesive energy of the polymer and a low viscosity. So failure initiation and propagation will be faster at same applied stress in contact with a certain environment than in air. It is also possible to have a brittle-ductile transition. 3. Physical ageing (reversible): when a material is rapidly cooled under Tg, it will present a higher volume than the equilibrium one (due to high cooling rate, the material will present a higher fraction of amorphous material that has not been able to organize, which correspond to a higher free volume). Depending on the service temperature (which affects chain mobility and so ageing rate, higher T will correspond to a faster aging), the material will decrease this excess of free volume (reorganization of macromolecules) tending to the equilibrium value. This process leads to a ductile-brittle transition (material becomes more strong and less tough at increasing aging times, smaller time for fracture initiation and propagation). Materialcan be seen under a microscope). The shear yielding mechanism can be described by the following equation: τ = kγ^n where τ is the shear stress, γ is the shear strain, k is a material constant, and n is the strain hardening exponent. The design criteria for shear yielding is typically based on the maximum shear stress that the material can withstand before yielding. On the other hand, crazing is a brittle and diffuse mechanism of yielding. It involves the formation of microcracks within the material, which are called crazes. Crazes are thin fibrils that form in the direction perpendicular to the applied stress. The material undergoes significant volume expansion due to the formation of crazes, resulting in a decrease in density. Crazing can be described by the following equation: σ = kε^n where σ is the tensile stress, ε is the tensile strain, k is a material constant, and n is the strain hardening exponent. The design criteria for crazing is typically based on the maximum tensile stress that the material can withstand before crazing occurs. The parameters for both shear yielding and crazing mechanisms can depend on time and temperature. For example, at higher temperatures, the mobility of polymer chains increases, leading to a higher likelihood of shear yielding or crazing. Additionally, the rate of deformation and the duration of the applied stress can also affect the yielding behavior of polymers.

The criterion to describe this yielding mechanism is the "modified Von Mises" (because T<C in polymers). It needs two materials characteristics (T and C) and 2 characteristics of the state of stress (I1: first dilatoric invariant of state of stress, J2: second deviatoric invariant of state of stress). In the triaxial stress space the criterion turns out to define a safe region which is a cylinder for metals (T=C) and an ellipsoid for polymers (0,5<=T/C<=0,7). If we consider a biaxial state of stress (sigma3=0), the safe region becomes an ellipse which for polymers is centered in the pure compression region (third quadrant).

The effect of temperature or time is the same and consists in a decrease of the safety area (it reduces yielding limits of the material). Modified von Mises can be represented like

Crazing is a brittle and very localized mechanism of yielding. Under applied normal stress, material undergoes a cavitation which involves the formation of voids

the stress intensity factor (measure of the stress concentration at the crack tip). The Sternstein criterion states that crazing occurs when the product of the bias stress and the stress intensity factor exceeds a critical value. In addition to the Sternstein criterion, there are other factors that can influence the formation and growth of crazes, such as temperature, strain rate, and molecular weight of the polymer. These factors can affect the mobility of the polymer chains and the ability of the fibrils to align and rupture. Crazing is a common phenomenon in polymers and can have both positive and negative effects on the material's properties. On one hand, crazing can increase the toughness and impact resistance of the material. On the other hand, crazing can reduce the material's strength and stiffness, as well as its optical clarity. In conclusion, crazing is a unique deformation mechanism observed in polymers under tension. It involves the formation of interconnected voids called crazes, which are not considered cracks but can lead to crack initiation. The formation and growth of crazes are influenced by various factors, and their presence can have both beneficial and detrimental effects on the material's properties.

The first dilatoric invariant (chain mobility determining factor).

For triaxial state of stress

For biaxial state of stress

The 2D representation of the Sternstein criterion is:

The temperature or time increase reduce the safety area for crazing (T and t reduce yielding limit of the material.)

Another criterion to describe brittle yielding is the Argon's model:

If we put the two criteria together, we can identify different regions in which act different yielding mechanism depending on the state of stress.

If temperature or time is increased, both safety regions are reduced, but the material results to be more ductile because the region for pure crazing is greatly reduced, due to the fact that shear yielding is more sensitive to T and t variations than crazing and so the ellipse will shrink more, leading to a very small red area. This means that yielding limit for shear yielding decreases more than the yielding limit for crazing when T or t increases.

Derive the theoretical model for fracture

initiation in viscoelastic materials, specifying the relevant assumptions adopted. Explain how stress history, temperature, and aging time influence the initiation time (please provide a detailed explanation and examples). Consider the crack tip under the opening mode of deformation. Ahead of the crack tip, there is a plastic zone called the process zone (α), in which we assume cohesive-like yielding. Inside this zone, the failure process takes place, and its mechanical behavior is highly non-linear and rate-dependent. Additionally, the scale of plasticity is small (α << a), meaning that the process zone is much smaller than the crack itself. The process zone is a highly damaged zone in which fibrils (columns of material) are formed. We assume that fracture initiation occurs when the first fibril fails (ξ = α), which happens when the crack tip opening displacement (δ) reaches a critical value (v is half of δ at ξ = α). The failure time depends on the state of the material, including factors such as stress history, temperature, and aging time. Stress history: The stress history experienced by the material affects its initiation time. If the material has been subjected to high stress levels in the past, it may have already undergone some damage or deformation, which can reduce the initiation time. On the other hand, if the material has experienced low stress levels, it may be less prone to failure and have a longer initiation time. For example, consider a viscoelastic material that has been subjected to cyclic loading in the past. Each cycle of loading and unloading causes some damage to the material, reducing its resistance to failure. As a result, the initiation time for this material would be shorter compared to a material that has not experienced cyclic loading. Temperature: The temperature of the material also plays a significant role in determining the initiation time. Higher temperatures generally accelerate the degradation and damage processes in the material, leading to a shorter initiation time. This is because elevated temperatures increase the mobility of molecules, making it easier for them to rearrange and cause damage. For instance, consider a viscoelastic material used in a high-temperature environment. The material is exposed to constant heat, which accelerates the degradation of its molecular structure. As a result, the initiation time for this material would be shorter compared to the same material used in a lower temperature environment. Aging time: The aging time refers to the duration for which the material has been in service or exposed to external conditions. Over time, materials can undergo various aging processes, such as creep, relaxation, or chemical degradation. These aging processes can lead to changes in the material's properties, including its resistance to failure. For example, consider a viscoelastic material that has been in service for a long time. During this period, the material may have experienced creep, which causes it to deform under constant stress. This deformation can lead to the formation of microcracks or other damage mechanisms, reducing the initiation time. Therefore, a material with a longer aging time would have a shorter initiation time compared to a material that is relatively new. In summary, the initiation time in viscoelastic materials is influenced by various factors, including stress history, temperature, and aging time. These factors can affect the material's resistance to failure and the extent of damage in the process zone, ultimately determining the initiation time.

The failure zone increases in time, but the thickness h of the damaged layer remains approximately constant.

We make the hypothesis that:

• h << α (h is the trace given by fibrils damage)
• Each fibril is as an individual tensile speciment
• Inside the process zone the stress is constant: σ = σ = σ (perfectly plastic behavior)

So the work of fracture can be written as ( J/m^2)

Assuming that the constant stress distribution inside the plastic zone is equal to σ, α can be written as:

yLength of the failure provided by Bareblatt (K contains: length of crack, geometry, value of applied stress)

Rate of crack propagation is equal to the growth rate of the process zone

CTOD elastic case (failure of plastic zone)

We have related the half of CTOD of the process zone to some material properties and to stress applied at boundaries

The total work of fracture done on the fibril at ξ = α that leads to crack initiation: Γ

At crack initiation,

The work required to break the fibril is equal to 2 criterion for viscoelastic fracture initiation (t is the time for fracture Γ, D* K initiation, if we know and we can calculate it)

In summary, an existing crack in a viscoelastic body is expected to propagate when the work done by the Γ applied stress intensity factor, K, equals the material's fracture toughness. If K is constant the equation is very simplified: *1/8

Since stress in case 2 is higher than in case 1, the time-to-failure t i2Γ turns out to be shorter. If fracture toughness is increased, then the time-to-failure is also increased.

There are three consequences from this model:

1. Failure threshold (i.e. the stress value below which failure never occurs) only exists if the creep compliance curve has an upper limit: this occurs only if the material is a thermoset (crosslinked) polymer.
2. Increasing the temperature shifts the creep compliance curve horizontally towards shorter times; the shortening in the time to

failure is simply proportional to the temperature shift factor.Γ3. Aging typically reduces the fracture toughness ( ) of the material, thus shortening the failure time.

Illustrate fracture propagation

The crack propagation rate can be predicted making the following assumptions:

1. The failure zone doesn’t have so much effect on crack displacement outside the failure zone itself (if we shift the Barenblatt solution by about 2α/3 we get a reasonable superposition between it and the elastic solution). When crack grows simply shifts rigidly from left to the right.
2. The rate at which the crack grows is a function “f” (da/dt=f(K )) of the applied stress intensity factor; I the function f is independent of the geometry of the body (geometry is taken into account by K only).
3. Materials at the crack tip sees only changes in the stress intensity factor, irrespectively on what is happening in the far field (loads, displacements or geometry changes).

Crack length is a function

of time and can be expressed as a series expansion:

Assuming that there are small changes in crack speed between t1and t2, we can neglect crack accelleration:

Considering that the crack speed is approximately constant over the zone length α and a, the stress intensity factor is approximately constant over the same time period, we can use the pseudo-plastic approximation Ԑ(t)=D(t)*σ to calculate the crack displacement.

In most cases p>3/2 (exact result from the Barenblatt solution) and so:

Also the stress acting on the failure zone turns out to have same power law dependance. The fracture energy during crack propagation can then be calculated:

t = time at which the crack crosses the process zone (crack covers a length equal α to σ.

If we assume that creep compliance can be represented for simplicity by a power