Rejection and Correction of the Fracture Mechanics Singularity Approach with its Associated Tree Modes of Crack Separation
Authors: Put Tom Van der 
Published: May 2018

Limit Analysis is an prescribed exact approach of Wood Science, what is shown to also apply for wood Fracture Mechanics. Knowledge of the gradual elastic to plastic behavior and of the imitation by non-linear elasticity (and J-integral) is shown to be not needed. The linear – full plastic limit approach delivers an elastic lower bound, up to this full plastic boundary, the fracture- or yield criterion, where ultimate load behavior is described, by virtual work approach and “flow” by the normality rule. This delivers the possibility to look at any equilibrium system, which satisfies compatibility and boundary conditions and nowhere exceeds this “flow” criterion and is verified by test data. Because the accepted singularity approach does not deliver a right mixed mode fracture criterion, it is necessary to make comparisons with other possible Airy stress functions. Therefore, the derivation of the accepted, general applied, elementary singularity solution with its 3 failure modes, is discussed and compared with new theory. This new limit analysis theory is based on an older, forgotten, Airy stress function, and shows e.g., by the new approach and application to wood, that there is no real difference between strength theory and fracture mechanics and between linear and non-linear theory. It delivers the, empirical verified, exact mixed mode failure criterion for wood; shows that stresses in the isotropic wood matrix also have to be regarded separately, to explain the, only by isotropy, possible, extremely high triaxial hydrostatic stress, and stress increase by the stress spreading effect. Therefore the stresses and strengths of the isotropic wood matrix are derived. The transformation to total stresses, including the reinforcement is shortly given for the empirical verification and for literature reference. Therefore, only the derivation of the necessary corrections of the singularity approach, for isotropic material, is regarded. This leads to a necessary rejection of the, tree failure modes, singularity approach of Irwin and of associate equations. By the splitting in 3 modes, there is no compatibility and no mixed mode fracture criterion. Instead there are tree, each excluding, Airy stress functions, thus 3, each excluding, compatibility equations. Necessary is one mixed mode solution for the total load. Then the solution also is known for separate acting (thus non zero) loading components. This is done in § 4, necessarily in elliptic coordinates, to know failure, by the highest tangential, uniaxial tensile stress in the crack boundary. The tangential direction in polar coordinates is not tangential to the elliptic first expanded of the crack boundary. Therefore then not the right KIIc values are obtained. The expression in elliptic coordinates delivers (by the highest empirical correlation) the failure criterion of wood for every load combination. Transformation of this mixed mode solution to polar coordinates gives the corrected singularity method based on a mixed mode failure criterion and delivers also the definition of the stress intensity factor. This last also gives another interpretation of the Bazant curve, which is shown to be the initial mode I yield criterion.