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Linear Elastic Fracture Mechanics

Introduction

Linear Elastic Fracture Mechanics (LEFM) is a well-known theory for predicting the crack growth response of linear elastic materials. The theory is nice and interesting, but it is not used frequently for polymers since they are rarely linear elastic in their response. However, since the theory is simple, it is an excellent stepping stone for covering the basics before venturing into Elastic-Plastic Fracture Mechanics (EPFM). Which is the topic for another article.

Linear Elastic Polymer Behavior

Polymers behave  almost linearly elastic if the temperature is low enough, or if the strain rate is high enough. In other words, the brittleness of a polymer depends on the material and the load load environment. Note that the material need to be linear elastic for LEFM to apply.

Linear elastic response.

Griffith Theory

Griffith in 1920 proposed that fracture will occur if the reduction in total strain energy due to crack growth is larger than the surface energy required to grow the crack:

\( \displaystyle – \left( \frac{dU}{dc} \right) \ge \gamma \left( \frac{dA}{dc} \right) \).

In this equation γ is the surface energy (per area), A is the crack area, c is the crack length, and U the strain energy. This condition works for brittle glass, but not polymers where other energies contribute to the crack growth resistance.

Alan Griffith (1893-1963)

Irwin Theory

Irwin extended the Griffith theory to include some limited plasticity (or other dissipating mechanisms) at the crack tip. According to Irwin, the Energy Release Rate (also called “crack driving force”) is given by:

\(G = \displaystyle \frac{dW}{dA} – \frac{dU}{dA} \)

Crack propagation is taken to occur if: \(G > G_c\). In this equation W is the work done by external forces, U is the stored elastic energy, A is the crack surface area, and \(G_c\) is the critical energy release rate.

Irwin

George Irwin (1907-1998)

Fracture Modes and Crack Growth

There are three different fracture modes in which a crack can be loaded: (I) opening, (II) in-plane shear, (III) out-of-plane shear. Note that brittle crack growth typically occurs in Mode I.

Fracture modes

The following images show the directions in which brittle cracks typically grow.

Stress Intensity Factors

The stress field in front of a mode I crack is given by:

Stress field

Here, KI is called the stress intensity factor. Note that the stress goes to infinity at the crack tip (r=0). In the crack plane (θ=0) the opening stress is \( \sigma_{yy} = K_I / \sqrt{2 \pi r}\). If we set the vertical opening stress to the yield stress, then the plastic zone size can be obtained:

\( r_y =\displaystyle \frac{1}{2\pi} \left(\frac{K_I}{\sigma_y} \right)^2\)

Outside the plastic zone there will be a region where the stress is dominated by K. In general, the stress intensity factor is given by: \( K_I = A \sigma \sqrt{\pi c}\), and the fracture condition is: \(K_I \ge K_{IC}\). In these equations A is a dimensionless geometric factor. It can also be shown that \(G_{IC} = K_{IC}^2 / E\). Note that there are handbooks that contain stress intensity factors as a function crack and specimen geometry, and it is also possible to determine the K values using FEA (I discuss more about that in my article on elastic-plastic fracture mechanics). 

Experimental LEFM Testing

ASTM D5045 contains information about how to experimentally determine the critical stress intensity factor using a single tension test using a compact tension specimen (see image below). For the experimental fracture tests to be valid the specimen dimensions need to exceed about 15 times the plastic zone size. This requirement is quite difficult to satisfy for ductile polymers, and is one of the reasons why elastic-plastic fracture mechanics is often preferred.

Summary

Linear Elastic Fracture Mechanics (LEFM) is a simple and well-known theory. In practice, however, it is not that useful for polymers due to their non-linear viscoplastic response. For these reasons I typically use Elastic-Plastic Fracture Mechanics (EPFM).

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