Linear fracture mechanics

The analysis was first carried out by Griffith in a treatment of the brittle fracture of metals. Its transfer to poljmieric solids may look questionable at first, as these are neither ideally elastic nor linear in the response at strains near to failure. Actually, Griffith s considerations are of general nature and can also be applied to polymers, after introducing some physically important but formally simple modifications. 

We consider the plate shown in Fig. 10.41, which contains a crack of length 2c. The plate has a thickness d, a Young s modulus E, and its surface area is assumed as infinite. If a tensile force is applied perpendicular to the crack direction, one finds a uniform uniaxial stress far from the crack, denoted cr j. It is possible to calculate the drop of the elastic free energy of the plate, which results if the length of the crack is increased by A2c. The solution of the problem is given by the expression 

The increase of the crack length produces additional surfaces on both sides and this requires a work AW, which is proportional to their area 

The equation includes w as proportionality constant. Treating perfectly brittle solids, Griffith identified w with the surface free energy. For polymers, the meaning of w has to be modified. As will be discussed, other contributions appear and even take control. The condition for fracture follows from a comparison of the two quantities AE and AW, which independent of the contributions to w. A crack grows if the strain energy release rate Gj, given by 

Equation (10.63) is known as Griffith s fracture criterion. It includes a critical value of the strain energy release rate, denoted Gjc and the latter is determined by the surface parameter w. The subscript F is used in the literature to indicate reference to the opening mode I of Fig. 10.42 and to 

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Reiterer, A. and Sinn, G. (2002). Fracture behaviour of modified spruce wood a smdy using linear and non-linear fracture mechanics. Holzforschung, 56(2), 191-198. 

According to Eqs. (13.145) and (13.148) the fracture stress in plane strain is a factor 1 /(1-v2) 1 /0.84 1.2 higher than in plane stress. Experimentally, however, the difference is much bigger. The reason for this discrepancy is that Griffith s equations were developed in linear fracture mechanics, which is based on the results of linear elasticity theory where the strains are supposed to be infinitesimal and proportional to the stress. 

Specimen dimensions B, W agree with the Non Linear Fracture Mechanics criteria, and J data can be considered as reliable B, W-ao > 25 x Jg/cTy [17]. Experimental data are independent of specimen geometry (B, W) the scatter in fracture criteria results is very small. Two fracture... 

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