Singularities in solid mechanics: An engineering perspective

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Singularities in solid mechanics An engineering
perspective
Mathematisches Forschungsinstitut
Oberwolfach November 3-9, 2002

• Barna Szabó
• Center for Computational Mechanics
• Washington University, St. Louis, Missouri, USA

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Outline

• Singularities in solid mechanics
• Regularity
• Relationship to failure laws
• Linear models of failure
• Linear elastic fracture mechanics (LEFM)
• Generalizations
• Nonlinear models
• Examples

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Failure criteria

• Failure of materials is an essentially nonlinear
  phenomenon
• Very large and highly localized straining occurs,
  placing it outside of the scope of continuum
  mechanics (formation and coalescence of voids)
• Under what conditions can we depend on linear
  computations to predict failure?
• Example Linear elastic fracture mechanics
• Obstacles to generalizations

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Notation
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The L-shaped domain
The von Mises stress corresponding
to K1K21.000 Geometric mesh, p8, trunk space.
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Assumptions
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Prediction of crack propagation
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Typical crack growth data
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Spoiler of Boeing 7x7 aircraft
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Cracked panel Topology and BCs
So
So
L 30.0 S 6.0 H 0.7 W 0.7 R 0.25 t
0.25 So 10 ksi
Material
1/4-solid model
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Cracked panel Meshing
Crack in First Bay
Crack in DCF
Crack in Fillet
Crack in Second Bay
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Crack in DCF Details
(xc, yc) system of ellipse
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Crack in bay 1 a1.00
Y
Kmax20360
K1 along crack front
Convergence of K1
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Crack in fillet 1 a2.65
a10.250 c10.400 c 0.308
Kmax26370
K1 along crack front
Convergence of K1
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Results for crack in DCF a3.05
a10.650 c10.850 c 0.785
Kmax32710
K1 along crack front
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Results for crack in DCF a3.20
a10.80 c11.15 c 0.95
Kmax38480
K1 along crack front
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Seq for crack in DCF a3.35
von Mises stress
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Generalization
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Case 1 Isotropic without adhesive
ta2tp
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Case 1 Linear solution
PL1000 S_eq4.08E5
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Case 1 Nonlinear Solution
PL1000
PL2000
PL3000
PL4000
PL5000
Scale 12
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Case 1 Nonlinear solution
PL5000 S_eq8.33E5
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Case 2 Isotropic with adhesive (2D)
tp
ta
Elastic-plastic
tp
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Case 2 Nonlinear solution (2D)
PL1000
PL2000
PL3000
Scale 11
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Case 2 Adhesive stresses (2D)
PL1000
PL2000
PL3000
Note Cannot compute larger load because the
elements around the singularities become too
distorted and stiffness generation fails.
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Case 2b Alternative meshing
46-element mesh
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Case 2b Nonlinear solution (2D)
PL1000
PL2000
PL3000
PL3500
Scale 12
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Case 2b Adhesive stresses (2D)
PL1000
PL2000
PL3000
PL3500
Note Cannot take larger load because the
adhesive is fully plastic
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Case 2b Nonlinear - Extrusion
PL1000
PL2000
PL3000
PL3500
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Closing remarks

• With the notable exception of LEFM, and the
  J-integral, defined for the deformation theory of
  plasticity, the asymptotic expansions at
  singularities are not being used in predictions
  of failure initiation
• Lack of reliable experimental data
• LEFM is applicable under clearly defined
  conditions only.
• Data are generated from standard test specimen
• Generalization to 3D is tenuous, but nothing
  better is available, or likely to become
  available
• Alternative criteria are being developed, based
  on average stress on a reference area.
• Anisotropic strength consideration
• Geometric and material nonlinearities must be
  considered