Fracture mechanics

Fracture mechanicsLecture 11 – Fracture mechanics: Irwin theory

CDM - N.Bonora 2016

Fracture mechanics

Introduction

• In 1967, G. R. Irwin following the approach proposed by N. Muskhelishvili, formulated the problem for the stress-field estimation at crack-tip into an infinite plate in biaxial loading condition

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N. MuskhelishviliG. R. Irwin

Fracture mechanics

Introduction

• Two-dimensional planar state

• Writing of equilibrium and consistency equations

• Airy’s function formulation:

• The solution is a series expansion of a complex variable

• Irwin: truncation at the first term of the series

• Displacement field -> strain -> stress

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Fracture mechanics

Irwin-Williams solution

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ij

r, k

r

f

ij A

mr

m / 2g

ij

m

m0

“S” “B”

Index i,j = 1,2,3 (or x,y,z)The functions fij are trigonometric functions in polar coordinate system: radius r is from the crack-tip and the same for

Coefficients k and Am are constant

r

Fracture mechanics

Irwin-Williams solution

CDM - N.Bonora 2016

Fracture mechanics

Irwin-Williams solution

CDM - N.Bonora 2016

Fracture mechanics

Irwin-Williams solution

CDM - N.Bonora 2016

Fracture mechanics

Irwin-Williams solution

CDM - N.Bonora 2016

Fracture mechanics

Irwin-Williams solution

CDM - N.Bonora 2016

• The stress field at the crack tip is singular as for r-½

• The field is univocally defined by only one term: stress intensity factor, SIF, KI

• SIF depends on geometry and applied load: increasing the stress the field is self-similar

• SIF unit length is [MPa m½]

• SIF as failure criterion: «In a material that has a defect failure happens if SIF is equal to the critical value»

• Critical value of SIF is called fracture toughness

Ln()

Ln(2pr)

1/2

ln 𝜎𝑦𝑦 = ln(𝐾𝐼) −1

2ln(2𝜋𝑟)

KI1< KI2 <KIc

Fracture mechanics

Irwin-Williams solution: comments

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• SIF expression for an infinite plate in biaxial loading condition is:

• Essential hypothesis of fracture mechanics: for geometries different than the reference (infinite plate in biaxial loading condition) the solution is formally the same, only the intensity of the K is properly scaled

𝐾𝐼 = 𝜎0 𝜋𝑎

𝐾𝐼 = 𝑌𝑎

𝑊𝜎0 𝜋𝑎

Fracture mechanics

Irwin-Williams solution: comments

CDM - N.Bonora 2016

• SIF as design criterion:

• For an assigned load, to consider materials with higher toughness leads to a better tolerance on the maximum value of the allowable defect’s length

𝜎 =𝐾𝐼

𝜋𝑎

Fracture mechanics

Irwin-Williams solution: comments

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• SIF as design criterion:

• For an assigned defect’s size, to consider materials with higher toughness leads to an higher allowable load

𝜎 =𝐾𝐼

𝜋𝑎

Fracture mechanics

Estimation of the fracture toughness

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• Fracture toughness is a materials’ property

• It is expressed as: «the capability of a material to resist to a crack advance»

• To estimate its value is necessary to guarantee the validity of the linear elastic fracture mechanics assumption:

• Absence of plastic deformations

• Planar strain stress state

• Fracture toughness is measured experimentally, and being a material property it can be used to evaluate the resistance of a component (transferability)

Typical ASTM standard plane-strain fracture toughness test specimens. (a) Compact tension. (b) Bending. (c) Photograph of specimens of various sizes. Charpy and tensile specimens are also shown, for comparison purposes. (Courtesy of MPA, Stuttgart.)

Fracture mechanics

Estimation of the fracture toughness

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• REFERENCE STANDARDS

• ASTM E 399-90: Standard Test Methods for Plane-StrainFracture Toughness of MetallicMaterials, 1990.

• ASTM E 1820-01: Standard Test Method for Measurement of Fracture Toughness, 2001.

• ASTM E 561-98: Standard Practice for R-Curve determination, 1998.

B = thickness; a = crack length; W = width (often W = 2B and a/W=0.5 are fixed)

Fracture mechanics

Estimation of the fracture toughness

CDM - N.Bonora 2016

• REFERENCE STANDARDS

• ASTM E 399-90: Standard Test Methods for Plane-StrainFracture Toughness of MetallicMaterials, 1990.

• ASTM E 1820-01: Standard Test Method for Measurement of Fracture Toughness, 2001.

• ASTM E 561-98: Standard Practice for R-Curve determination, 1998.

Fracture mechanics

Estimation of the fracture toughness

CDM - N.Bonora 2016

• REFERENCE STANDARDS

• ASTM E 399-90: Standard Test Methods for Plane-StrainFracture Toughness of MetallicMaterials, 1990.

• ASTM E 1820-01: Standard Test Method for Measurement of Fracture Toughness, 2001.

• ASTM E 561-98: Standard Practice for R-Curve determination, 1998.

Fracture mechanics

Estimation of the fracture toughness

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• In the test, the specimen response is measured in terms of applied load vs displacement under the load line (VLL or crack mouth opening displacement, CMOD)

• As the load for the crack propagation is determined, the KQ (Conditional toughness) is calculated

• KQ has to be verified: to guarantee the conditions for MFLE assumption

Fracture mechanics

Estimation of the fracture toughness

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• ASTM E399

𝑎 ≥ 2.5𝐾𝐼𝑐𝜎𝑌

2

Fracture mechanics

Estimation of the fracture toughness

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• Fracture toughness dependence on the specimen thickness

Fracture mechanics

Condition for K-dominance

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Fracture mechanics

G and KQ equivalence

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• Into Griffith’s approach, G is the elastic energy release rate related to the crack advance

• KI represents the field intensity at the crack-tip

• For a purely elastic material is possible to define an equivalence

• Therefore: there is a correlation one-by-one between released energy and field intensity at the crack-tip

IK a p2

a

GE

p

IK EG

21I

EGK

Fracture mechanics

Irwin’s solution for real material

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• Irwin’s solution for an elastic material establishes that the stress field tends to infinity for r → 0

• That leads to an absurd: for a very low remote load, the stress intensity at the crack tip will cause an immediate failure

• Conclusion: into real materials the stress field near the crack-tip is limited by the development of plastic deformation and a crack-tip radius not equal to 0

x

Plastic Zone

ry

Y

x0