Hydrogen-Diffusion Coupled Cohesive-Zone Modelling of

© 2019, 4RealSim B.V. & 4RealSim Services B.V.

Hydrogen-Diffusion Coupled Cohesive-Zone

Modelling of Crack Propagation in Steel

Structures

Vincent Bouwman

Nils Götzen

Gaëtan van den Berg

4RealSim BV

© 2019, 4RealSim B.V. & 4RealSim Services B.V.2

Motivation

General

hydrogen induced embrittlement is well recognized threat for steel structures

hydrogen embrittlement leads to loss of ductility, strength, toughness

investigated intensely over past decades - various competing models / theories developed

numerical methods can help investigating the theories and compare obtained results with experimental data

Objective

customer from oil/gas industry: implement hydrogen-enhanced localized plasticity (HELP) model into FEM

together with ability to simulate crack growth by means of cohesive zone modelling

future use as ‘lab-tool’ for virtual testing & for analysis or industrial structures

Stress Corrosion of Ultra High Strength Steel – University of Nevada, Reno, 2017

mechanism of sulfide stress corrosion cracking


Coupled Stress Hydrogen Diffusion – Theoretical Background

total concentration of hydrogen is given by

𝐶𝑇 = 𝐶𝐿 + 𝐶𝑋 where

𝐶𝐿 is the concentration at the lattice sites and

𝐶𝑋 is the concentration at the trap sites (dislocation)

implemented in ABAQUS but w/o trap-sites

mass conversation

𝜕

𝜕𝑡නΩ

ҧ𝐶𝐿 + ҧ𝐶𝑋 𝑑Ω +න𝜕Ω

𝐉 ⋅ 𝐧 𝑑𝑆 = 0

divergence theorem

𝜕 ҧ𝐶𝐿𝜕𝑡

+𝜕 ҧ𝐶𝑋𝜕𝑡

+ ∇ ∙𝐷𝐿𝑉𝐻 ҧ𝐶𝐿𝑅𝑇

∇𝜎𝐻 − ∇ ∙ 𝐷𝐿∇ ҧ𝐶𝐿 = 0

introducing simplifications and equilibrium assumptions

𝜕 ҧ𝐶𝐿𝜕𝑡

1 +𝜕 ҧ𝐶𝑋

𝜕 ҧ𝐶𝐿− ∇ ∙ 𝐷𝐿∇ ҧ𝐶𝐿 + ∇ ∙

𝐷𝐿𝑉𝐻 ҧ𝐶𝐿𝑅𝑇

∇𝜎𝐻 +𝜕 ҧ𝐶𝑋

𝜕 ഥ𝑁𝑋

𝑑 ഥ𝑁𝑋𝑑𝜀𝑝

𝑑𝜀𝑝

𝑑𝑡= 0

Koyama et al. (2017) Recent progress in microstructural hydrogen mapping in steels: quantification, kinetic analysis, and multi-scale characterisation

work is based on

Krom, et al. 1999

Oh and Kim 2009

Barrera, et al. 2016


Coupled Stress Hydrogen Diffusion – Theoretical Background

Heat equation Mass diffusion equation

ρcp𝜕T

𝜕t− ∇ ∙ Jq + rq = 0

𝜕CT𝜕t

− ∇ ∙ 𝐉m + rm = 0

derivative of thermal energy per unit mass:

ሶUq = cp𝜕T

𝜕t

derivative of total hydrogen concentration:

𝜕തCT𝜕t

=𝜕 തCL + തCX

𝜕t

degree of freedom: temperature

T

degree of freedom: lattice concentration

തCL

heat flux:

ҧ𝐉q

hydrogen flux:

ҧ𝐉𝐦 =DLVHതCLRT

∇σH − DL∇തCL

heat sources:

rq = 0

hydrogen source:

rm = 0

density:

ρ

density:

1


Coupled Stress Hydrogen Diffusion – Implementation

analysis type: fully coupled temperature–stress

analysis

stress & BC influence H2

H2 concentration influences material behavior

diffusion equation is solved with the UMATHT

data to be provided in subroutine

ҧ𝐶𝑇 = see below

𝜕 ҧ𝐶𝑇

𝜕 ҧ𝐶𝐿

𝜕 ҧ𝐶𝑇

𝜕𝛻 ҧ𝐶𝐿= 0

ҧ𝐽𝑚 =𝐷𝐿𝑉𝐻 ҧ𝐶𝐿

𝑅𝑇𝛻𝜎𝐻 − 𝐷𝐿𝛻 ҧ𝐶𝐿

𝜕 ҧ𝑱𝑚

𝜕 ҧ𝐶𝑳

𝜕 ҧ𝑱𝑚

𝜕𝛻 ҧ𝐶𝑳

ҧ𝐶𝑇 𝑡 + ∆𝑡 = ҧ𝐶𝐿 𝑡 +𝜕 ҧ𝐶𝑇

𝜕 ҧ𝐶𝐿𝑑 ҧ𝐶𝐿 +

𝜕 ҧ𝐶𝑋

𝜕 ഥ𝑁𝑋

𝑑 ഥ𝑁𝑋𝑑𝜀𝑝

𝑑𝜀𝑝

source term

𝑁𝑇 𝜀𝑝 = 10 𝐹𝐴−𝐹𝐵∙𝑒−𝐹𝐶∙𝜀𝑝

+ 𝐹0

challenge: hydrostatic stresses gradients

hydrostatic stresses obtained via UVARM at

integration points – current data available within each

iteration – USDFLD not used as it only provides data

from previous increment

gradient is computed in UMATHT based on shape-

function derivatives

shape function derivatives are computed once at the

beginning of the analysis within the UEXTERNALDB

shape function derivatives shared with UMATHT via

COMMON BLOCK



Mechanical Behavior

yield stress as a function of the plastic strain, which can be extended to account for the impact of the hydrogen

concentration

𝜎𝑦 = 𝜎0𝐻 1 +

𝜀𝑝

𝜀0

1𝑛

𝜎0𝐻 = Ψ ҧ𝐶𝐿 𝜎0

scaling function

𝛹 ҧ𝐶𝐿 = 𝐹𝑠 − 1 − 𝜉ҧ𝐶𝐿 − ҧ𝐶𝑚𝑖𝑛

ҧ𝐶𝑚𝑎𝑥 − ҧ𝐶𝑚𝑖𝑛

mechanical behavior is implemented with the UHARD instead of full UMAT – only the yield stress is controlled

by the hydrogen concentration

UHARD provides the current plastic strain value and plastic strain increment

is needed in UMATHT and

shared across the subroutines via SDVs



Subroutine Overview

UEXTERNALDB

evaluate model size / element type / initiate shared arrays

compute shape function derivatives acc. element type

UVARM

compute hydrostatic stress during iterations

UHARD

compute yield stress as function of hydrogen concentration

obtain equivalent plasticity and store in SDV

UMATHT

compute hydrostatic stress gradients using

shape function derivatives (UEXTERNALDB) &

hydrostatic stress (UVARM)

compute needed diffusion parameters

compute diffusion equation using

hydrostatic stress gradients &

equivalent plasticity (UHARD)


Coupled Stress Hydrogen Diffusion – Examples

Hydrogen transport near a blunting crack tip

far field solution for mode-I cracks based on linear elastic fracture mechanics and a stress intensity factor of

89.2 MPa√m

influence of loading rate 1.3 sec – 1.3E+6 sec

exterior surface is insulated – no H2 flux

initial H2 concentration in entire component = 3.454 mol/m3



CL [mol/m3] distribution for loading rate of 1.3 sec CX [mol/m3] distribution for loading rate of 1.3 sec



CL [mol/m3] distribution for loading rate of 1.3E+06 sec CX [mol/m3] distribution for loading rate of 1.3E+06 sec



0,0

0,5

1,0

1,5

2,0

2,5

0,0 1,0 2,0 3,0 4,0 5,0 6,0 7,0 8,0 9,0 10,0

CL/CL0

R/b

1.3 sec

13 sec

26 sec

130 sec

1300 sec

1.3E+06 sec

CL distribution ahead of crack tip as a function of loading rate (complete insulation at boundary)



0,0

25,0

50,0

75,0

100,0

0,0 0,5 1,0 1,5 2,0 2,5

CX/C

L0

R/b

1.3 sec

13 sec

26 sec

130 sec

1300 sec

1.3E+06 sec

CX distribution ahead of crack tip as a function of loading rate (complete insulation at boundary)


Fracture Propagation – Theoretical Background

Crack Propagation

crack / future crack path is predefined by means of cohesive zone model: contact definitions between crack-

surfaces using cohesive behavior

cohesive behavior is defined at crack surfaces via General Contact

cohesive behavior is characterized by

cohesive stiffness (equivalent to contact stiffness)

cohesive strength

cohesive fracture energy

Simulation Approach

cohesive strength is defined with dependency on the hydrogen concentration at the surfaces

cohesive fracture energy is considered independent of the hydrogen concentration (but dependency could be

included easily)


Fracture Propagation – Theoretical Background

hydrogen-dependent cohesive stress σC(θH) is

computed as (strength-ratio)

𝜎𝐶 θ𝐻𝜎𝐶 0

= 1 − 1.0467 θ𝐻 + 0.1687 θ𝐻2

θH is the hydrogen coverage

Θ𝐻 =𝐶

𝐶 + 𝑒𝑥𝑝 −△ 𝐺𝑏0/𝑅𝑇

with C the bulk hydrogen concentration (unit mol

H/mol Fe) and

ΔGb0 the Gibbs energy difference between surface

and bulk material (surface being any microstructural

interface like a crystallographic plane, grain

boundary, etc.) Hydrogen coverage as a function of hydrogen concentration, for various

levels of Gibbs energy (kJ mol−1).


Fracture Propagation – Implementation

UMATHT

hydrogen-coverage (HC) and

strength-ratio (SR) are computed at the integration-point-level

URDFIL

SR is read in from the .fil-file as averaged nodal value at the end of each increment

UFIELD

SR is provided at nodes of cohesive surfaces as Field-Variable-1

cohesive strength is defined with a dependency on Field-Variable-1 FV1 (= strength-ratio SR)

cohesive fracture energy is defined as linear model with


Fracture Propagation – Examples

double-cantilever beam (DCB)

standard specimen for material testing (NACE

Standard Double-Cantilever-Beam Test)

2D-model

predefined crack at the mid-line surface

pre-cracked

Loading

1st step: mechanical loading: free ends are vertically

pulled – or metal wedge is pushed into the slit

2nd step|: hydrogen loading at the exterior boundary

surface

Investigations

mesh size

fracture strength

fracture energy

fracture viscous damping

hydrogen concentration / profiles

etc.






Fracture Propagation – ExamplesCL [mol/m3] distribution

cohesive strength ratio [/] distribution


Fracture Propagation – ExamplesCL [mol/m3] distribution

cohesive strength ratio [/] distribution


Fracture Propagation – Examples – Mesh Size

0,000

0,001

0,002

0,003

0,004

0,005

0,006

0,007

0,008

0,009

0 50000 100000 150000 200000 250000 300000 350000 400000

Cra

ck-L

engt

h [

m]

Time [sec]

MD1-FE1-DS1

MD2-FE1-DS1

MD3-FE1-DS3


Fracture Propagation – Examples – Fracture Energy

0,000

0,002

0,004

0,006

0,008

0,010

0,012

0 50000 100000 150000 200000 250000 300000 350000 400000

Cra

ck-L

engt

h [

m]

Time [sec]

MD2-FE1-DS1

MD2-FE1-DS2

MD2-FE1-DS3

MD2-FE2-DS1

MD2-FE2-DS2

MD2-FE3-DS1


Summary

stress driven hydrogen diffusion model implemented and verified

cohesive zone model implemented

crack growth / self arrest can be simulated

influencing factors such as material properties / fracture behavior / loading scenarios can be investigated

2D DCB model is a feasible tool

3D simulations (not shown) shows poor convergence

Documents

Hydrogen-Diffusion Coupled Cohesive-Zone Modelling of