Numerical-experimental assessment of Stress Intensity Factors in Ultrasonic Very-High-Cycle Fatigue (VHCF) Supervisors: ... state of the art. ... More...

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Seyed Husein Hasani Najafabadi Cycle XXX Student Started in 10.2.2015

Supervisors: Professor M. Rossetto Professor G. Chiandussi Professor D.S. Paolino

VHCF: Introduction

New needs:

232 Days

116 Days

14 Hour

50 Hz

100 Hz

20 kHz

New tests: ultrasonic VHCF

New trends: no fatigue limit?

Objects

Fracture toughness is a very important (critical) material parameter in VHCF. Nowadays many researchers evaluate the stress intensity factor in VHCF regime by using literature analytical equations validated in the static field. The validity of these equations in the dynamic field is an open question. Objective of present research is to search for a robust method to easily and effectively evaluate the Stress Intensity Factor(SIF) in the dynamic field (VHCF tests). Creating a strong and powerful tool(package) to calculate Stress Intensity Factor by different methods in Very High cycle Fatigue regime.

Outline

Analytic solutions: state of the art. FEM solutions: state of the art. Material & Experimental methods. FEM predictions in VHCF. Computational cost: Crisis. Hybrid Method: The present study. Solver: Harmonic Balance Method. Reduced order model (ROM) : Dynamic Sub-structuring (CB-CMS). SIF evaluation :Virtual Crack Closure Technique(VCCT). SIF evaluation :J-Integral. SIF evaluation :Equivalent Domain Integral. SIF evaluation :Interaction Integral (M-Integral). Results for a simple bar as a bench mark. Application of Hybrid Method on Hourglass and Gaussian Samples Conclusions .

Analytic solutions: state of the art Forman solution Shih & Cai solution

• One‐parameter K‐solution

• Two‐parameter K‐solution

FEM solutions: state of the art • • • •

Most of them in 2D geometries Poisson’s ratio neglected Crack in mid‐plane (perfectly symmetric case) Contact between crack surfaces neglected

4/1 2

Material & Experimental methods: present study

H13 steel quenched and tempered Two specimen geometries: Hourglass & Gaussian Constant stress amplitude with R=‐1 at 20 kHz Fractography through optical microscope Measurement with CAD software

These dimensions were used for FEM simulations Sample name

Diamete r [µm]

Dept h [µm]

Arc length [µm]

Crack radius [µm]

Displacement amplitude [µm]

σa [MPa]

Hourglass

5995

2524

2817

5820

44

640

Gaussian

10928

3172

3516

3624

52

510

FEM predictions in VHCF

Commercial software ABAQUS: 3D model with implicit solver (time step 10‐6s) Two geometries: Hourglass (4828 elements) and Gaussian (6192 elements) Seam configuration for crack modeling Crack tip: collapsed Barsoum elements (16 elements in hoop direction and 5 contours) Penalty method to model the contact between crack surfaces M‐integral for SIF computation

a): Gaussian

b): Hourglass

Computational cost: Crisis!!

Direct time integration method (Implicit) used in this study as reference solution(Time-domain analysis (TDA)) . The direct integration method used by ABAQUS is the Newmark Alpha method (Hilber-Hughes-Taylor scheme). The Newmark Alpha method is unconditionally stable. In order to obtain the fine high-frequency detail of the response it is desirable to use the fine step time.(10e-6~10e-7). The solution is in steady state regime. Solution time for 3D specimen is drastically long , computational cost is so high and this solution method is not efficient!!!! 4/1 1

s1 s2

Computational cost: Crisis!! 10KHz frequency Surfaced cracked bar with 192 element

10.2 9.7 9.2 8.7 8.2 7.7 7.2 6.7 9910

9920

9930

9940

9950

9960

9970

9980

9990

10/12

Hybrid Method: The present study

Suggestions to reduce the computational cost for this study: More efficient solver to solve the nonlinear problem fast , reliable and accurate : Harmonic Balance Method (HBM) Reduces solution time. Reduce the nonlinear core size of the problem by Dynamic Sub-structuring: Component Mode Synthesis- CraigBampton Method (CB-CMS) Drastically reduced the size of mass ,stiffness and damping matrices. Simplest method to evaluate Stress Intensify Factor : Virtual Crack Closure Techniques (VCCT), J-Integral, and M-Integral Ease of use and implement.

Dynamic Sub-structuring: Craig-Bampton

In numerous cases of nonlinear systems having a large numbers of DOFs, the actual nonlinear components are spatially localized (like a crack). X

0

, ,

, ,

: Excitation force. : Non-linear forces. : : [M],[K],[C]: mass , stiffness and damping Matrices.

, ,

0

, 6/1 2

Dynamic Sub-structuring: Craig-Bampton

In the FEM of such cases, CB-CMS can be used to reduce the size of the system. Then, only the nonlinear nodes are retained in the nonlinear governing equations. ψ 0

T

0

ψ I

ψ

Craig-Bampton transformation matrix

Normal modes Constraint modes with fixed interface

, ,

0

,

The size of matrices drastically decreased. The computational cost decreases.

7/1 2

Nonlinear Solver: Harmonic Balance Method Time Domain to Frequency Domain

HBM is by far the most computationally efficient method for obtaining the steady-state solution to nonlinear dynamics problem. ̅

: number of harmonics.

̅ ,

̅

̅,

0 ̅

,

Harmonic Balance-Frequency Domain (New nonlinear Equation )

Solving of this new nonlinear equation is possible by iterative solution methods like Newton-Raphson. •

: Non-linear forces in our case is only the normal contact force.

AFT method applied to calculate contact forces in time domain and send back to frequency domain. Node-to-node frictionless contact element with normal contact stiffness (penalty). Compression force N = max(knv, 0), with v: normal relative displacement.

Stress Intensify Factor evaluation: VCCT Strain Energy Release Rate(SERR) : ∆

1 lim ∆ → 2 ·∆

,

·

∆

,

: the crack opening displacement function.

SERR (G) Equation based on 20-nodes non-singular elements is: 2

,

1 ∆

1 2

∗

∗

Finally Stress Intensity Factor can be obtained by : √

/ 1

]

∗

1 2

∗

Results for a simple bar: VCCT

shows the displacement of the one node during a cycle Geometry and mesh of the cracked bar

Computational time [s]

Implicit solver ABAQUS

HBM+VCCT

Reduction ratio

93600

158

~600

Comparison between computational costs 10/11

the computed SIFs during a cycle, according to different numbers of harmonics

Stress Intensify Factor evaluation: Contour Integral( J-Integral )

The energy release rate is described, J. Rice obtained a path independent contour integral, the J-integral, from his first-name Jim, which describe the energy release rate in LEFM, and the J-integral is equivalent to G: J=G The J-integral can be described by a path around a crack tip given by followed expression in 2D as path integral: ∮ᴦ

dy-

W=

The J-integral is effective for evaluating K in two-dimensional crack problems. For three-dimensional problems, however, it is difficult to distinguish K at different x3 locations, assuming the line integral is performed on the xl-x2 plane. Thus an alternative procedure needs to be developed to determine the distribution of K through the thickness.

Stress Intensify Factor evaluation: Contour Integral( Equivalent Domain Integral )

The Domain Integral is a numerical solution of the J-integral .

A closed contour containing an inner ᴦ and outer ᴦ contours is used where the inner is vanishingly small and the outer is finite .

Divergence theorem can be applied and instead of integrating around a path of the crack tip, integration over the area between the paths is used to evaluate the J-Integration.

∗

ᴦ ᴦ

Stress Intensify Factor evaluation: Contour Integral( 3D J-Integral )

With the EDI method, a point-wise value of J-integral along a three-dimensional crack front can be calculated, and therefore the value of K along the crack front can be obtained. In order to do J–integral: Chiarelli-Frediani formulation for 20 node brick elemet

Stress Intensify Factor evaluation: Contour Integral( 3D M- Integral )

A more convenient way to determine the mixed-mode SIF values compared to the domain integral is the interaction integral which gives a more robust and actual results. The interaction integral in a 3D domain is a contribution of the domain integral in called actual field, an auxiliary field and an interacting field between the actual and auxiliary field.

M(s)

J(s):actual field from the domain integral =auxiliary field in the vicinity of a crack, containing the auxiliary stress ,strain and displacement. M(s)=Interaction (M-Integral) , that interacting the auxiliary and actual field are given by, without the components of traction. In ABAQUS using M-Integral with collapsed elements create very good results with optimum number of meshes.

Verification of Hybrid method with ABAQUS: Benchmark

Material properties: o Density:7850 Kg/M3 o Damping: Proportional (β=1e-7) o Elastic modulus : 210 GPa

Frequency: ~ 20 KHz Mesh: o Total number of nodes: 3350 o Total number of elements: 2856 o Elements of type C3D8R

Implicit solver in ABAQUS as a reference

Verification of HBM with ABAQUS: Benchmark 1000

ABAQUS

Stress Intensity Factor MPa÷mm

800

600

PRESENT STUDY 400

200

0

-200

-400

-600 0

SOLVER

10

20

30

40

50

Time Step

60

70

80

90

100

ABAQUS[IMPLICIT]

HBM+M-INTEGRAL

WALLCLOCK TIME (SEC) =3133 sec

156[HBM]+1374[M-INTEGRAL] =1530 sec

Results: static vs. dynamic

specimen core (bar) static simulation

specimen static simulation

specimen dynamic simulation (20 kHz)

Gaussian

Hourglass

analytic static solutions specimen type Gaussian Hourglass

Forman

Shih & Cai

48.3 74.4

49.0 71.5

bar static 46.7 72.4

FEM solutions specimen specimen static dynamic 46.5 39.5 76.3 61.6

SIF values [MPa √m]

error (percent) bar/specimen static/dynamic static 4.6 17.5 4.3 23.9

PDS3

Size eﬀect in VHCF → significant decrement of fatigue properties [Furuya, Murakami, Beretta].

Stress

Stres s

Discussion: size effects

Hourglass

Gaussian

Fatigue strength decrement

Critical Volume of material [mm3]

Experimental data from Furuya

8/1 2

Conclusions and Further research Conclusions An innovative procedure for the effective and efficient computation of relevant SIFs in a cracked body in resonance condition was proposed. The procedure is based on the use of Reduced Order Models(CB‐CMS), Harmonic Balance Method and Virtual Crack Closure Technique(VCCT),J‐Integral and M‐Integral. The applicability and the potentialities of the proposed hybrid method were successfully shown with a numerical example. It was found that a reduction factor of about 600 in terms of computational time can be achieved with the proposed procedure with no loss of accuracy. Gaussian vs. Hourglass specimens: significant different SIFs SIF smaller in Gaussian specimen: larger size induces a transition to plane strain condition SIF evaluated with Gaussian specimens closer to critical SIF (material fracture toughness)

Further research

Computation of threshold SIF at the boundaries of the Optically‐Dark‐Area.

Publications and Courses

Publications:

Experimental-Numerical Assessment of Critical SIF from VHCF Tests ,Key Engineering Materials, Advances in Fracture and Damage Mechanics XV, Volume 713, pp.62-65,September 2016.

Numerical computation of stress intensity factor in ultrasonic Very-HighCycle Fatigue tests, Key

Engineering Materials -754 September 2017

Courses: Analysis of structures subjected to impulsive loading-4 credits [20 hours], Politecnico di Torino The Boundary Element Method for Anisotropic Bodies and Multilayered - 3 credits [20 hours], Politecnico di Torino Nonlinear structural dynamics-2credits [10 hours], Politecnico di Torino Managing Ph.D. Thesis as a Project - 2 credits [16 hours ], Politecnico di Torino Applied probability and stochastic processes - 6 credits [30 hours], Politecnico di Torino Tools and applications of systems engineering - 6 credits [30 hours], Politecnico di Torino Topics In Internet & Society Interdisciplinary Studies - 4 credits [20 hours], Politecnico di Torino Programming in LabVIEW: Part 1 and Part 2- 8 credits [40 hours], Politecnico di Torino The redefinition of the International System of Units (SI)- 3 credits [15 hours], Politecnico di Torino Competitive funds for research: from the idea to the writing of the project- 2credits [10 hours], Politecnico di Torino Models and methods for the dynamics of mechanical components with contact interfaces-3 credits -[10 hours], Politecnico di Torino Probe scanning microscopy for physics and engineering-3 credits-[30 hours]-Politecnico di Torino TOTAL: 53 credits collected