Crack Growth in Polymers

Stress Concentration and
Stress Intensity Factors

The fracture strength of structural materials is often described with the Griffith model.¹ This model is in excellent agreement with the observed fracture strength of brittle materials like glass and ceramics. However, for polymers and metals that undergo extensive plastic deformation it gives unrealistically low fracture energies, or in other words, the observed fracture energy is several orders of magnitude greater than the fracture surface energy of brittle materials.

In ductile or not ideal elastic materials a plastic zone develops at the tip of the crack. This plastic zone increases in size until the stress behind the tip of the crack unloads. The plastic deformation behind the crack tip results in dissipation of energy as heat. This energy has to be added to the (elastic) fracture surface energy as has been proposed by Orowan and Irwin independently (1948):

G = 2 (Γ_e + Γ_p)

where G  represents the total strain energy, and Γ_e and Γ_p are the elastic and nonelastic contributions to the fracture surface energy, respectively. Assuming Griffith's energy criterion is still valid, the critical stress for crack propagation can be written as

σ_cr² = G E / (π a)

where 2a is the length of the crack and E is the Youngs or tensile modulus.
Another approach is the analysis of stress distribution around areas of constrain such as notches, holes, and circumferential grooves. For example, for an elliptical hole in an infinitely wide plate subjected to a normal stress, σ_n, the stress at the tip of the hole, σ_t, is given by

σ_t = σ_n  (1 + 2 a / b)

or

σ_t = σ_n  [1 + 2 (a / ρ)^1/2]

where 2a and 2b are the ellipse diameter in a direction perpendicular and parallel to the external load and ρ = b²/a is the radius of curvature of the tip of the hole. For the special case of a spherical hole, the maximum stress at the hole is three time the normal stress, σ_t = 3 σ_n. Thus, a geometric discontinuity causes localized stress that is well above the average or far-field stress of the part.

Elliptical Notch in a Plate

The ratio of the maximum stress around a geometric constraint and the nominal stress is called the stress concentration factor:

K_t = σ_max / σ   for normal stress (tension)

K_ts = τ_max / τ   for shear stress (torsion)

In these equations, σ_max, τ_max represent the maximum stress at the edge of the defect and σ, τ are the applied farfield or nominal tensile and shear stress. In the case of an elliptical hole or elliptical grove, the stress concentration factor is simply 1 + 2 (a / ρ)^1/2. In general, stress concentration factors are either obtained analytically from the elasticity theory or finite element method, or experimentally using strain gages and other devices. They can be found in many handbooks for the most common geometries.

The stress concentration factor describes only local stress around macroscopic "defects" and changes in cross section (notches). The sharper the radius at the cross section change, the higher the stress concentration. Typically, these factors range from one to three. For microscopic defects, i.e. cracks and flaws, the radius of curvature can be extremely small and approaches zero for very sharp corners. Thus, the stress concentration factor becomes very large and strong plastic deformation around the crack tip can be expected. In this case, the allowable stress has to be calculated with the so called stress intensity factor. This factor is often used for fatigue calculations. It is an inherent property of the material and has to be  determined experimentally for each part geometry and load condition. For a sharp crack in an infinitely wide and elastic plate the stress intensity factor is given by

K_I = σ  (π a)^1/2

and for the criitcal stress that causes crack propagation (fracture) the condition reads

K_Ic = σ_c (π a)^1/2

where K_Ic is the critical stress intensity factor which is a measure for the materials resistance to fracture. This parameter is also known as the fracture toughness of the material.

By comparing the equation of stress intensity with Orowan's modification of the Griffith equation we find

K_I = (G E)^1/2                for plane stress   

K_I = [G E / (1 - ν²)^1/2]   for plane strain    

In general, stress intensity factors depend on the crack and sample geometry, and may be written in the form

K_I = β σ  (π a)^1/2

where β is the geometry correction factor which is a function of extrinsic factors such as sample width, crack size, and load.

References and Notes

1. A.A. Griffith, Philos. Trans. Roy. Soc. A, Vol. 221, Issue 582, p. 163 (1921)
2. G.R. Irwin in Handbuch der Physik, Band VI, Elastizitaet und Plastizitaet, 1958
3. M.D Lechner, K. Gehrke, E.H. Nordmeier, Makromolekulare Chemie, 1993