DAMAGE AND CRACKING MODELLING

DAMAGE AND CRACKING MODELLING

INTRODUCTION TO COMPUTATIONAL MECHANICS OF MATERIALS Civil Engineering, 1st cycle studies, 7th semester

elective subject academic year 2014/2015

Institute L-5, Faculty of Civil Engineering, Cracow University of Technology

Jerzy Pamin Adam Wosatko

With thanks to: Andrzej Winnicki

Definition of damage

dS _{~ n} = dS _{~ n} ^d + dS _{n ~} ^u ω(~ n) = ^{dS ~ n} _dS ^{−dS u ~ n}

~

n = ^dS _dS ^{~ n d}

~

n

Damage vs. plasticity

σ

 (1 − ω)E

E

Damage

 σ

E E

Plasticity

Exemplary descriptions of damage

Anisotropic damage

Constitutive equation for fourth order damage tensor:

σ = (I − Ω) : E : 

Isotropic damage

Constitutive equation for two-parameter desription:

σ = (I − Ω) : E : ; Ω = ω ₁ 1 ⊗ 1 + ω ₂ I Constitutive equation for scalar descritpion:

σ = (1 − ω) E : 

Strain equivalence – effective stress

Postulate

The strain associated with the damaged state under the applied stress is equivalent to the strain associated with the undamaged state under effective stress.

Strain equivalence

 = ˆ 

Effective stress

σ = (1 − ω) ˆ σ ˆ

σ = E

Scalar damage

Loading function in strain space

f ^d = ˜ () − κ ^d = 0

˜

 – equivalent strain measure κ ^d – damage history parameter

Loading/unloading conditions

˙κ ^d ≥ 0 f ^d ≤ 0 ˙κ ^d f ^d = 0

Equivalent strain measure definition

Principal strain space

Normalized elastic energy release (EER) measure

˜

 = q 1

E  : E : 

ν = 0.0 ν = 0.2

Equivalent strain measure definition

Principal strain space

Mazars definition – positive principal strains

˜

 = q

P

I (H( I ) I ) ² I = 1, 2, 3

−0.04

−0.03

−0.02

−0.01 0

−0.04 −0.03 −0.02 −0.01 0

 1

plane stress, ν = 0.2 plane strain

 2

Equivalent strain measure definition

Principal strain space

Modified von Mises definition

˜

 = _2k(1−2ν) ^k−1 I  ₁ + _2k ¹ r

 k−1 1−2ν I  ₁  ²

+ _(1+ν) ^12k ₂ J ₂ k = ^f _f ^c

t

−0.04

−0.03

−0.02

−0.01 0

−0.04 −0.03 −0.02 −0.01 0

 1

plane stress, ν = 0.2 ν = 0.0 plane strain, ν = 0.2

 2

Exemplary damage growth functions

ω = ω(κ ^d )

I Linear softening

I Exponential softening: ω(κ ^d ) = 1 − ^κ _κ ^o _d 

1 − α + αe ^{−η(κ d −κ o )} 

G f – fracture energy is the amount of energy necessary to open a unit crack area.

Crack closure phenomenon

Crack closure E

 σ

Tension

E

Compression (1 − ω)E

(1 − ω)E

Cantilever beam under load reversals

Example

100 mm

250 mm P(w)

Which solution is correct?

What is the cause of mesh sensitivity of results?

ω – animation

¯

 – animation

Types of discontinuities

Continuous model

Weak discontinuity

Strong discontinuity

G. N. Wells. Discontinuous modelling of strain localisation and failure,

Delft University of Technology, 2001.

Fracture modes

Mode:

I. Opening – caused by tension or bending, tensile stress normal to the plane of the crack

II. In-plane shear (sliding) – caused by shearing, shear stress acting parallel to the plane of the crack and perpendicular to the crack front III. Out-of-plane shear (tearing) – caused by shearing, shear stress acting

parallel to the plane of the crack and parallel to the crack front

Stress at crack tip

In linearly elastic fracture mechanics some stress tensor components grow to infinity at the crack tip, therefore a notion of stress intensity factor (SIF) has been introduced.

However, the material load-carrying capacity is finite and cannot exceed

the value of limit stress (strength, yield stress). Therefore, in front of the

crack tip an inelastic zone is envisaged.

Discontinuous models

It is possible to use discontinuous description in which parts of the body are connected along planes/surfaces (e.g. composites structures) or discrete cracks occur. To model discontinuities either interface finite elements or XFEM approach are employed.

G. N. Wells. Discontinuous modelling of strain localisation and failure, Delft University of Technology, 2001.

Discrete interfaces Partition of unity method

along boundaries – crack propagates

of finite elements arbitrarily (XFEM)

Interface elements

Usually in interface elements nonlinear relations are introduced,

representing e.g. friction, adhesion, fracture.

Interface elements

t = D ∆u

L =

 −1 1 0 0

0 0 −1 1



∆u = LN a

E.g. for shear.

Matrices are computed using Lobatto integration rule

Different applications of interface elements

Composite structures

P. Wronka. Interface finite elements in two-dimensional simulations of cracking and of

deformation of composite structures, Cracow University of Technology, 2005, master thesis.

Phenomena around crack tip

Degradation process in quasi-brittle materials

Scope of damage and fracture mechanics

DAMAGE PROGRESS FRACTURE PROGRESS

Initial Micro- Macro- Cracks Overload Failure

state damage damage

Microcracks Microdefects Voids Heterogeneities

Coales- cence

−−−−−−−−→

Growth

−−−−−−−−→

Propaga- tion

Macroc- racks Material degradation

Propaga-

−−−−−−−−→ tion Growth

−−−−−−−−→

Coales- cence

Stress

concentration

Secondary

cracks

Scope of damage and fracture mechanics

Discrete or smeared cracks?

Fracture energy G f (dissipated when a unit crack area opens)

Idea of smeared cracking

Stress vector:

σ =



 σ nn

σ tt

τ _nt





Strain vector:

 =





 _nn

 _tt γ _nt





Fixed cracks

I Strain decomposition:

 =  ^e +  ^cr

I Elastic part:

 ^e = C ^e σ C ^e = (D ^e ) ⁻¹

I Cracked part:

 ^cr = C ^cr σ

I Compliance relationship:

 = (C ^e + C ^cr ) σ = C σ C = C ^e + C ^cr

I Stiffness relationship:

σ = D

D = (C ^e + C ^cr ) ⁻¹

Constitutive relations for cracking

Strain in cracked part:

 ^cr =





 ^cr _nn 0 γ _nt ^cr



 Compliance operator:

C = _E ¹





1 −ν 0

−ν 1 0

0 0 2(1 + ν)



 +







1−µ

µE 0 0

0 0 0

0 0 ^1−β _βG





 Stiffness operator:

D = C ⁻¹ =







µE 1−µν ²

µνE 1−µν ² 0

µνE 1−µν ²

E 1−µν ² 0

0 0 βG





 Constitutive equation:

σ = D

µ : 1 → 0, β : 1 → 0

Stress-strain relationship in global coordinates

Relationship in local coordinates:

σ = D

Transformation formulae:

 = T  gl

σ gl = T ^T σ

Transformation matrix:

T =





n ² _x n _y ² n x n y

t _x ² t _y ² t x t y

n x t x n y t y n x t y + n y t x



 Relationship in global coordinates:

σ gl = T ^T DT  gl

Normal stress

σ _nn = µE  ^cr _nn

Shear stress

τ nt = βG γ nt

β = const lub β = β( nn )

Experimental results for concrete (Kupfer)

Plane stress case

Yield surfaces for concrete

Plane stress case

Plasticity versus smeared crack approach

Brazilian test – splitting effect

Load action → splitting Wedge formation Primary and secondary cracks lin/lin , ¯ 

quad/lin , ¯ 

Averaged strain measure ¯ 

Simulation of cracking in brick walls using DIANA package

P.B. Lourenc ¸o. Computational strategies for masonry structures. Doctoral dissertation, Delft University of Technology, 1996.

Animation of cracking under shear

Simulation of cracking in RC wall using ATENA

Continuous-discontinuous modeling

A. Simone, G.N.Wells and L.J. Sluys. From continuous to discontinuous failure in a gradient-enhanced continuum damage model, Comput. Methods Appl. Mech. Engrg., 192 (2003) 4581–4607.

Beam with a notch under nonsymmetric shear

Nonlocal continuum Discontinuous model

References

M.G.D. Geers. Continuum Damage Mechanics. Fundamentals, Higher-order Theories and Computational Aspects. Lecture notes, Eindhoven University of Technology, 1998.

R. de Borst and L.J. Sluys. Computational Methods in Nonlinear Solid Mechanics.

Lecture notes, Delft University of Technology, 1999.

B. Karihaloo. Application of Fracture Mechanics in Design. Lecture notes of Summer Course on Mechanics of Concrete, Cracow, 1996.

G. Rakowski, Z. Kacprzyk. Metoda elementow skończonych w mechanice kostrukcji. Oficyna Wyd. PW, Warszawa, 2005.

M. Kwasek Advanced static analysis and design of reinforced concrete deep beams.

Diploma work, Politechnika Krakowska, 2004.

P.B. Lourenc ¸o. Computational strategies for masonry structures. Doctoral dissertation, Delft University of Technology, 1996.

M. Jir´ asek. Plasticity, Damage and Fracture. Modeling of Localized Inelastic Deformation. Technical University of Catalonia (UPC), Barcelona, 2002.

DIANA Finite Element Analysis - User’s manual, release 9.4.2. TNO Building and

Construction Research, Delft, 2010.