John R Yates

The Modelling and Simulation Centre

The University of Manchester

Some thoughts on the non-

linearity of cracks in structural

materials

Where, and what is, the crack tip?

2.0 mm

2.0 mm

7.0 mm

15.0 mm

Back to basics

• Real cracks are blunt, plastically deformed and have a failure process zone

• Crack growth is the permanent displacement of atoms from crack tip

Sources of non-linearity

• Plastic deformation

– Local to crack tip or globally

• Contact and closure of crack faces

– Residual stresses, roughness, oxide debris

• Changes in crack path

– Microstructure or macrostructure

• Multiple and competing cracking and damage

mechanisms

Competing damage mechanisms

• Ductile void nucleation, growth and coalescence– Inclusion cracking, debonding

• Transgranular and intergranular cleavage

• Fatigue growth– Crystallographic shear in Stage I

– Stage II blunting

• Creep cavitation nucleation, growth and coalescence

• Corrosion– Pitting

– Intergranular cracking

– Transgranular cracking

• Radiation

Crack tip events

Increasing load generates a sequence of local events from blunting to crack extension

Wei Zhang, Yongming Liu. Int.J.Fat., In Press, 2011

Key question

• How can a single global parameter, K, d or J,

correlate to such a complex set of

mechanisms?

Global parameters vs local events

Function of Kmax

Function of Kmax, Kmin

Function of dmax, dmin Function of J-Da

response

Tearing crack

Crack tip during tearing 2

.5 m

m

Strategy

If we can separate the solid mechanics problem

from the material failure problem,

then we can use FEM to deal with the

continuum mechanics

and solve material state problem separately, at

the appropriate length scale

11

Simplify the question

Given the stresses and strains in this region of

the structure,

has the material locally failed by cleavage,

ductile tearing, creep, fatigue, or corrosion?

How might we deal with the material

state question?

Consider a damage model for a given mechanism

• Rousselier’s model for void growth

– Depends on stress state, deformation response and microstructure

– Has a characteristic length scale, L

• Calculate the state of the material over the volume L3,

– using the local stresses and strains

• If the material is damaged, update the FE stiffness matrix

Multiple damage mechanisms

• Do each calculation independently and in

parallel for each mechanism,

– at its own length scale

• But

– need to understand inter-relations and

hierarchies to return loss of stiffness into FE

model

Idea

Combine Code_Aster for non-linear FEA

with a site-bond method for the material

states

managed by Salome-Meca

Site-bond methods

• Smoothed particle hydrodynamics

• Cellular automata

• Pore models

• Lattice Boltzmann methods

• Peridynamics

Key features– coordination number of sites

– physical attributes of links: stiffness, strength, transport or any other property

– statistical variation of these parameters

Coordination number

• Cube, coordination = 6

• Truncated octahedron, coordination = 14

Intergranular crack propagation

Intergranular SCC with ductile

bridging ligamentsSimulated 3D crack propagation with a

lattice model

Truncated octahedra with variable grain

boundary properties

Failed boundaries shown; bridges appear as

holes in crack surface

Jivkov

General approach to fracture

• Ductile Cellular Automata arrays

– 2 Rousselier models for different inclusion and

precipitate distributions

• Shear array

– Low constraint shear model

• Brittle CA array

– divided into different cleavage nucleation

micromechanisms

Da

ma

ge

Architecture

Finite element mesh

Cleavage fracture lattice of sites and bonds

Da

ma

ge

Ductile damage lattice

New ideas

• Develop site-bond methods for

– Cleavage

– Fatigue

– Stress corrosion cracking

• Include nucleation, triaxiality, anisotropy and

inhomogeneity

Site-bond approach to ductile fracture

• Sites = void formation region around

inclusions

– Distribution of orientations, sizes, interfacial

strengths, yield and hardening properties

• Bonds = matrix between voids

– Distribution of yield and hardening properties,

and coalescence parameters

– Incorporate texture

Based on microstructure:• inclusion size and spatial distributions

• void nucleation, growth and coalescence

• anisotropic hardening

• grain size and crystallographic

orientation

• …

Rousselier damage model

Modification to Rousselier model

By integration of Rousselier model using the modified damage

variable the effects nucleation could be introduced

Needleman

model of

nucleation

Cleavage modelling

Weakest link model with grain size, orientation angle,

and mis-orientation threshold assigned to each cell

�Fracture stress of each cell

�Propagation through grains and propagation from

one cell to another

Cleavage nucleation micromechanisms

Inclusions in grains Pearlite-ferrite boundary

Pearlite microstructure Cracked carbides

1μm 1μm

1μm

1μm

Microcracks at failure

(+25oC) (-196oC)

30μm2

Maybe not a weakest link problem?

Site-bond approach to cleavage

• Sites = grains

– Distribution of orientations, sizes, lattice

resistances, inclusions

• Bonds = grain boundaries

– Distribution of strengths, inclusions

Fatigue crack initiation

• Sites = grains

– Distribution of orientations, sizes, lattice

resistances, inclusions

• Bonds = grain boundaries

– Distribution of strengths

Two processes in early fatigue

Crack incubation

• Tanaka-Mura dislocation model. Incubation life related to macroscopic

plastic strain range:

Microstructurally small crack growth

• Navarro –De Los Rios model of blocked dislocations

• Or Hobson-Brown

• Or Fatemi-Socie

Comments

• Non-linear crack behaviour arises from the

many competing physical processes, each

operating at different length and time scales,

• Site-bond methods offer the opportunity for

better modelling and simulation

www.mace.manchester.ac.uk/research/research/centres/masc