Nonlinear eigenvalue problems arising from nonlinear fracture mechanics

$Nonlinear eigenvalue problems arising from nonlinear fracture mechanics analysis E. M. Adulina, L. V.$

Nonlinear eigenvalue problems arising from nonlinear fracture mechanics analysis E. M. Adulina, L. V. Stepanova, Samara State University

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Hutchinson-Rice-Rosengren solution

Crack tip geometry and coordinate systems

Mode I crack. Basic equations

The Airy stress potential The asymptotic behavior of the Airy function The asymptotic behavior of the stress field

Nonlinear eigenvalue problem

Perturbation theory approach

Mode III crack. Basic equations

The stress function The asymptotic behavior of the stress function Nonlinear eigenvalue problem

Eigenvalues The set of linear differential equations

The solvability condition The set of boundary value problems The solvability condition

Closed form analytical solution

Mode I crack. Nonlinear eigenvalue problem

The perturbation method The undisturbed linear problem

The set of boundary value problems

The solvability condition

The linear differential equation for

The three-term asymptotic expansions of the hardening exponent

The three-term asymptotic expansions for the hardening exponent

Eigenspectra at a Mode II crack under plane strain conditions The nine-term asymptotic expansion of the hardening exponent

Eigenspectra at a Mode II crack under plane stress conditions The nine-term asymptotic expansion of the hardening exponent

The asymptotic study of fatigue crack growth based on continuum damage mechanics o o o Zhao J. , Zhang X. The asymptotic study of fatigue crack growth based on damage mechanics// Engn. Fracture Mechanics. 1995. V. 50. 1. P. 131 -141. Li J. , Recho N. Methodes asymptotiques en mecanique de la rupture. Paris: Hermes Science Publications, 2002. 262 p. Astafiev V. I. , Radayev Y. N. , Stepanova L. V. Nonlinear fracture mechanics. Samara: Samara State University, 2001. 632 p. Stepanova L. V. Mathematical methods of fracture mechanics. Moscow: Fizmatlit, 2009. 332 p. Astafjev V. I. , Grigorova T. V. , Pastuchov V. A. Influence of continuum damage on stress distribution near a tip of a growing crack under creep conditions/ procedings of the International Colloquium «Mechanics of creep brittle materials 2» , University of Leicester, UK, 1991. P. 49 -61.

The asymptotic study of fatigue crack growth based on continuum damage mechanics o Consider a fatigue growing crack lying on the x-axis with the coordinate origin located at the moving crack tip

The asymptotic study of fatigue crack growth based on continuum damage mechanics o The essence of continuum damage mechanics is characterized by material deterioration coupled constitutive equations. Under the assumption of linear elasticity a stiffness reduction based stress-strain relationship is applied as o The kinetic law of damage evolution o The equilibrium equations

The asymptotic study of fatigue crack growth based on continuum damage mechanics o The compatibility equation o The constitutive equations for plane stress conditions o The constitutive equations for plane strain conditions

The asymptotic study of fatigue crack growth based on continuum damage mechanics o The traditional traction free conditions on crack surfaces o The Airy stress function can be presented in the form o The Mode I stress field components at the crack tip behave as follows

The asymptotic study of fatigue crack growth based on continuum damage mechanics o The damage field around the crack tip can be presented as o The strain components are given as o From the compatibility equation o one can obtain (for plane stress) o one can obtain (for plane strain)

The asymptotic study of fatigue crack growth based on continuum damage mechanics o The damage evolution law allows to obtain o The symmetry of the stress and damage fields around the crack tip o The normalizing condition is chosen as o The regularity requirement o The traction free conditions

The asymptotic study of fatigue crack growth based on continuum damage mechanics o The totally damaged zone needs to be modeled in the vicinity of the crack tip o The analytical result o The stress and damage fields around the crack tip

The asymptotic study of fatigue crack growth based on continuum damage mechanics

The asymptotic study of fatigue crack growth based on continuum damage mechanics • The new analytical presentation of stress, strain and continuity fields both for plane strain and plane stress conditions is given. • The results obtained differ from Zhao and Zhang's solution where the original formulation of the problem for plane stress conditions has been proposed. • An analytical solution of the nonlinear eigenvalue problem arising from the fatigue crack growth problem in a damaged medium in coupled formulation is obtained. • The perturbation technique is used. The method allows to find the analytical formula expressing the eigenvalue as the function of parameters of the damage evolution law.