Nonlinear eigenvalue problems arising from nonlinear fracture mechanics analysis E.M. Adulina, L.V. Stepanova, Samara State University
Transcript
Slide 1
Nonlinear eigenvalue problems arising from nonlinear fracture
mechanics analysis E.M. Adulina, L.V. Stepanova, Samara State
University
Slide 2
J. W. Hutchinson, Singular behavior at the end of tensile crack
in a hardening material, J. Mech. Phys. Solids 16 (1968) 13-31. J.
R. Rice, G. F. Rosengren, Plane strain deformation near a crack tip
in a power-law hardening material, J. Mech. Phys. Solids 16 (1968)
1-12. F.G. Yuan, S. Yang, Analytical solutions of fully plastic
crack-tip higher order fields under antiplane shear, Int. J. of
Fracture 69, (1994) 1-26. G.P. Nikishkov, An algorithm and a
computer program for the three-term asymptotic expansion of
elastic-plastic crack tip stress and displacement fields,
Engineering Fracture Mechanics 50 (1995) 65-83. B.N. Nguyen, P.R.
Onck, E. Van Der Giessen, On higher-order crack-tip fields in
creeping solids, Transaction of the ASME 67 (2000) 372-382. I.
Jeon, S. Im, The role of higher order eigenfields in
elastic-plastic cracks, J. Mech. Phys. Solids 49 (2001) 2789-2818.
C.Y. Hui, A. Ruina, Why K? High order singularities and small scale
yielding, Int. J. of Fracture 72 (1995) 97-120. L. Meng, S.B. Lee,
Eigenspectra and orders of singularity at a crack tip for a
power-law creeping medium, Int. J. of Fracture 92 (1998) 55-70. M.
Anheuser, D. Gross, Higher order fields at crack and notch tips in
power-law materials under longitudinal shear, Archive of Applied
Mechanics 64 (1994) 509-518.
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Hutchinson-Rice-Rosengren solution
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Crack tip geometry and coordinate systems
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Mode I crack. Basic equations
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The Airy stress potential The asymptotic behavior of the Airy
function The asymptotic behavior of the stress field
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Nonlinear eigenvalue problem
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Perturbation theory approach
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Mode III crack. Basic equations
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The stress function The asymptotic behavior of the stress
function Nonlinear eigenvalue problem
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Eigenvalues The set of linear differential equations
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The solvability condition The set of boundary value problems
The solvability condition
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Closed form analytical solution
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Mode I crack. Nonlinear eigenvalue problem
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The perturbation method The undisturbed linear problem
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The set of boundary value problems
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The solvability condition
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The linear differential equation for
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The three-term asymptotic expansions of the hardening
exponent
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The three-term asymptotic expansions for the hardening
exponent
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Eigenspectra at a Mode II crack under plane strain conditions
The nine-term asymptotic expansion of the hardening exponent
Slide 24
Eigenspectra at a Mode II crack under plane stress conditions
The nine-term asymptotic expansion of the hardening exponent
Slide 25
The asymptotic study of fatigue crack growth based on continuum
damage mechanics 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.
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The asymptotic study of fatigue crack growth based on continuum
damage mechanics Consider a fatigue growing crack lying on the
x-axis with the coordinate origin located at the moving crack
tip
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The asymptotic study of fatigue crack growth based on continuum
damage mechanics 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 The
kinetic law of damage evolution The equilibrium equations
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The asymptotic study of fatigue crack growth based on continuum
damage mechanics The compatibility equation The constitutive
equations The constitutive equations for plane stress conditions
The constitutive equations for plane strain conditions
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The asymptotic study of fatigue crack growth based on continuum
damage mechanics The traditional traction free conditions on crack
surfaces The Airy stress function The Airy stress function can be
presented in the form The Mode I stress field components at the
crack tip behave as follows
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The asymptotic study of fatigue crack growth based on continuum
damage mechanics The damage field around the crack tip can be
presented as The strain components are given as From the
compatibility equation one can obtain (for plane stress) one can
obtain (for plane strain)
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The asymptotic study of fatigue crack growth based on continuum
damage mechanics The damage evolution law allows to obtain The
symmetry of the stress and damage fields around the crack tip The
normalizing condition is chosen as The regularity requirement The
traction free conditions
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The asymptotic study of fatigue crack growth based on continuum
damage mechanics The totally damaged zone needs to be modeled in
the vicinity of the crack tip The analytical result The stress and
damage fields around the crack tip
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The asymptotic study of fatigue crack growth based on continuum
damage mechanics
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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.