Fracture mechanics

Fracture mechanics

Lecture 11 – Fracture mechanics: Irwin theory

CDM - N.Bonora 2016

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

N. Muskhelishvili G. R. Irwin

• 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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Irwin-Williams solution



  r,   k r



  

 f

    A

r

g

  

m0



“S” “B”

Index i,j = 1,2,3 (or x,y,z)

The functions f_ij are trigonometric functions in polar coordinate

system: radius r is from the crack-tip and the same for  ^

r

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Irwin-Williams solution

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Irwin-Williams solution

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

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

• SIF depends on geometry and applied load:

increasing the stress the field is self-similar

• SIF unit length is [MPa m

]

• 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 𝜎_𝑦𝑦 = ln(𝐾_𝐼) −1

2ln(2𝜋𝑟)

K_I1< K_I2 <K_Ic

Irwin-Williams solution: comments

• 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

𝐾

= 𝜎

𝜋𝑎

𝐾

= 𝑌 𝑎

𝑊 𝜎

𝜋𝑎

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• 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 = 𝐾

𝜋𝑎

Irwin-Williams solution: comments

• SIF as design criterion:

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

𝜎 = 𝐾

𝜋𝑎

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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.)

Estimation of the fracture toughness

• REFERENCE STANDARDS

• ASTM E 399-90: Standard Test Methods for Plane-Strain

Fracture Toughness of Metallic Materials, 1990.

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

• ASTM E 561-98: Standard

B = thickness; a = crack length; W = width (often W

= 2B and a/W=0.5 are fixed)

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• REFERENCE STANDARDS

• ASTM E 399-90: Standard Test Methods for Plane-Strain

Fracture Toughness of Metallic Materials, 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.

Estimation of the fracture toughness

• REFERENCE STANDARDS

• ASTM E 399-90: Standard Test Methods for Plane-Strain

Fracture Toughness of Metallic Materials, 1990.

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

• ASTM E 561-98: Standard

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• In the test, the specimen response is measured in terms of applied load vs displacement under the load line (V

or crack mouth opening displacement, CMOD)

• As the load for the crack propagation is determined, the K

(Conditional

toughness) is calculated

• K

has to be verified: to guarantee the

conditions for MFLE assumption

Estimation of the fracture toughness

• ASTM E399

𝑎 ≥ 2.5 𝐾

𝜎

2

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• Fracture toughness dependence on the

specimen thickness

Condition for K-dominance

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

• K

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

K I   p a a G E

 p

K I  EG

1 2 I

K EG

 



Irwin’s solution for real material

• 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

r



x