Stress fields near the tip of a through crack in an elastic plate under bending and twisting moments are reviewed assuming both Kirchhoff and Reissner plate theories. The crack tip displacement and rotation fields based on the Reissner theory are calculated here for the first time. These results are used to calculate the J-integral (energy release rate) for both Kirchhoff and Reissner plate theories. Invoking Simmonds and Duva's [16] result that the value of the J-integral based on either theory is the same for thin plates, a universal relationship between the Kirchhoff theory stress intensity factors and the Reissner theory stress intensity factors is obtained for thin plates. Calculation of Kirchhoff theory stress intensity factors from finite elements based on energy release rate is illustrated. A small scale yielding like model of the crack tip fields is discussed, where the Kirchhoff theory fields are considered to be the far field conditions for the Reissner theory fields. It is proposed that, for thin plates, fracture toughness and crack growth rates be correlated with the Kirchhoff theory stress intensity factors.

A theory for the fracture of thin plates subjected to bending and twisting moments

The fracture mechanics of plates and shells under membrane, bending, twisting and shearing loads are reviewed, starting with the crack tip fields for plane-stress, Kirchhoff and Reissner theories. The energy release rate for each of these theories is calculated and is used to determine the relation between the Kirchhoff and Reissner theories for thin plates. For thicker plates this relationship is explored using three-dimensional finite element analysis. The validity of the application of twodimensional (plate theory) solutions to actual three-dimensional objects is analyzed and discussed. Crack tip fields in plates undergoing large deflection are analyzed using von Kármán theory. Solutions for cracked shells are discussed as well. A number of computational methods for determining stress intensity factors in plates and shells are discussed. Applications of these computational approaches to aircraft structures are reviewed. The relatively few experimental studies of fracture in plates under bending and twisting loads are reviewed.

Fracture Mechanics of Thin Plates and Shells Under Combined Membrane, Bending, and Twisting Loads

Two sets of photoelastic experiments employing stress freezing were performed in order to evaluate closure and precatastrophic extension of cracks in plates under cylindrical bending. Data were compared with the Hartranft-Sih theory in order to obtain quantitative results. It was found that crack closure on the compressive side of the plate produced somewhat larger local elastic stresses on the tensile side of the plate for straight through cracks than predicted by the mathematical model. When precatastrophic extension was allowed on the tension side of the plate, accompanying stress relaxation neutralized the stress intensification effect of closure. For short, nearly square cracks, three-dimensional effects appeared to reduce the stress intensification effects due to closure.

A photoelastic evaluation of the influence of closure and other effects upon the local bending stresses in cracked plates

Based on the classical plate theory in conjunction with the assumption of line contact at the compressive edge of a crack face, closed form solutions were presented for a through-the-thickness central crack in an infinite plate subjected to all around bending. The complete solutions were obtained by superposing the membrane components due to the contact forces at the crack face to the non-closure bending components. The distribution of the contact forces was found uniform by considering the contact condition which prevents mutual penetration of the crack faces at the compressive edges. The results showed that the closure of the crack faces tends to reduce the crack opening displacement at the tension side and, consequently, reduce the stress intensity factor. The finite element method was also used to investigate the present problem. The modified crack closure method in combination with the finite element method was used to find the stress intensity factors. Close agreement between the finite element and the analytical solutions was observed.

Influence of crack closure on the stress intensity factor in bending plates - A classical plate solution

Using the technique of stress freezing and slicing, a set of photoelastic experiments was conducted on plates, each containing a single through internal crack in a remote cylindrical bending field, where each crack suffered precatastrophic extension on the tensile side of the plate as well as closure on the compression side. Resulting stresses were compared with the Sih-Hartranft theory. Results indicate substantial stress relaxation near the tensile side of the plate due to precatastrophic extension. It is concluded that: (1) the Sih-Hartranft theory may be used to estimate local elastic stresses on the tensile side of the plate even when closure and precatastrophic extension occur; (2) complex coupling of closure and precatastrophic extension effects preclude the use of simple correction factors for existing mathematical models except for relatively small precatastrophic extension.

Ozcan Tuncel

Papers

An Experimental Investigation of the Crack Tip Stress Intensity Factors in Plates Under Cylindrical Bending