Application of the Open Hole Tensile Test to the Identification of the in-plane Characteristics of

Département

Application of the Open Hole Tensile Test to the Identification of the in-plane Characteristics of

Orthotropic Plates

J. Molimard, R. Le Riche, A. Vautrin, and J. R. Lee, -

GDR CNRS 2519, -

SMS/MeM, ENSM-SE, 158 Cours Fauriel, 42023 Saint Etienne, France

Département

Presentation summary

- Introduction

- Identification procedure - First results

- Interaction between anisotropy and geometrical modelling errors

- Conclusion

Département

Orthotropic laminate

E _xx , v _xy , E _yy , G _xy

Hypothesis : Uniform strain field

90 45

0

Load

Reference values

ε ε ε

Introduction: global overview

I- Introduction

Département

√√√√

FEM

Analytical Model

Hole on plate tensile test, T-shaped specimen, Modified Iosipescu shear test,

…

GI GS SS

Minimization algorithm

Mechanical properties

! " #

$

√√√√

√√√√ ^{= ( ε − ε ) ( + ε − ε ) ( + ε − ε )}

= %

& '(

: number of measures R : residue vector

Lekhnitskii solution for a hole on infinite orthotropic plate (plane stress conditions)

Modified Levenverg- Marquardt algorithm

Introduction: identification strategy

I- Introduction

Département

Experimental technique: digital phase-shifting grating interferometry

-1

1

Interference

resin

specimen grating

φ = − ⋅ = π

∆ ₋

)

* " !

=

−

= ₋

Phase difference

φ φ

φ = −

∆

spatial filter

10mW-laser lens1

Im a g in g syst e m

Illu m in atio

n s ys te m toward PC

Toward piezo system Tensile machine

Glass plate 3 mirrors

lens 2

lens 3

Rotating screen

CCD camera

x z

II- Identification procedure π ^∆ φ

=

+

Département

• Specimen: NC2 ^® reinforcement from Hexcel Composite + Epoxy resin processed in a RTM mold

Stacking sequence [{0/90} ₃ ] _s

Experimental technique: mechanical set-up and specimen

II- Identification procedure

• Mechanical set-up: Table-top tensile device Applied stress 10 MPa

Studied field

32.5mm

,&

256 mm

W = 26 mm

x

y

-

Département

Identification procedure: Unknown parameters

II- Identification procedure

Position of the hole center (x _c , y _c ) Hole geometry (a, b)

Covered field

αααα

Loading axis Material axis

. / . / ₀₀ . 1 ₀ . ν ₀

• Geometrical parameters:

• Material parameters:

2

Département

III- First results

First results: displacements and strains maps

ε ε ε

• Metrological aspects:

Field: 779 by 917 pixels Spatial resolution: 244 µm Resolution: 6 µstrains

Derivation: least square (7 pixels)

-208.5 -148.9 -89.4 -29.8 29.8 89.4 148.9 208.5 417nm/fringe +

x

y

3

Département

4(4+

( 2 15 2( 15

-( 15 Identified

values

4(4 ( 15

+ 15 +4 15

Reference values

+( 6 ( 6

(4 6 Difference ( 6

ν

1

/

/

Identified values Differences between experimental and

numerical strain maps

First identification results

III- First results

ε # ε (# ε # −ε #

ε

ε −

ε ε −

E[ ]

= -0.6957 µε

s.d.

[ ]

= 32.1672 µε statistics

ε # ε ^(# ε ^# −ε ^#

ε # ε ^(# ε ^# −ε ^#

ε

ε −

ε

ε −

ε

ε −

ε

ε −

E[ ]

= 0.4395 µε

s.d.

[ ]

= 32.3187 µε E[ ]

= 0.7691 µε

s.d.

[ ]

= 58.9253 µε

statistics

4

Département

0.0 7.5 15.0 22.5 30.0 37.5 45.0 52.5 60.0 67.5 0

5 10 15 20 25 30

21.21%

P e rc e n t e rr o r b e tw e e n F E M w it h f in it e w it d th a n d a n a ly ti c a l m o d e l w it h i n fi n it e w id th ( % )

Specimen width (W) in FEM / hole diameter (D)

v xy

5 7 & +

, &

R e la ti v e e rr o r b e tw e e n v a lu e s u s e d i n t h e F in it e E le m e n t s im u la ti o n a n d t h e i d e n ti fi e d v a lu e s u s in g t h e i n fi n it e a n a ly ti c a l m o d e (% )

Specimen width in FEM simulation (W) Hole diameter (D)

W > 240 mm, D = 4 mm W =26 mm, D < 0,43 mm

For Lekhnitskii Model

21.21%

0.96%

IV- geometrical errors

Study of geometrical modelling error

Département

• glass or carbon fibres

• 2 epoxy resins

• fibre volume fraction from 30 % to 60 %

• [0]

, [90]

or [0, 90]

stacking sequence

1 ^st - Simulation of various material cases using a micro/macro approach

2 ^nd - Calculation of the 3 required strain fields

3 ^rd - Identification of the 9 parameters

4 ^th - Results expressed as a ‘ratio’ vs. ‘anisotropy’

24 different cases

Anisotropy ratio from 0.20 to 0.74

• with the normalized test geometry

• using a FEM approach

• with the Lekhnitskii-based algorithm

Study of anisotropy vs geometrical modelling errors

IV- geometrical errors

Département

Study of anisotropy vs geometrical modelling errors

Poisson ratio νννν

Transverse Young modulus ΕΕΕΕ

Geometrical modelling errors grow with anisotropy

Use regression curves to correct identified material parameters

IV- geometrical errors

Département

An attempt to correct geometrical modelling errors

4(4 3 ( 15

+ ( 15 -(3 15

Identified values

4(4 ( 15

+ 15 +4 15

Reference values

3 (+ 6 8

-( 6 Mean error for all 8

studied cases

ν

1

/

/

Correction approach too simple and regression curves unable to follow data scattering induce low confidence in corrected values.

In particular, highest errors (up to 98%) for low Poisson’s ratio.

It is necessary to include the FEM within the identification procedure

IV- geometrical errors

Département

Final results using experimental maps and a FEM

(reference values from 3 classical tensile tests)

IV- geometrical errors

Département

Conclusions

Finally, identified mechanical parameters in agreement with classical tests (<6 %) Necessity of using a FEM within the identification procedure

These identification studies show that extrapolating the normalized open hole tensile test to other geometries is questionable because of the anisotropy / geometry interaction.

Perspectives

Experimental techniques

Use of a simpler OFFM Technique (Speckle shearography) Further results with other reference materials

V- Conclusion

Conclusions and Perspectives

+

Département σ

σ

σ

σ ∞

σ

σ ∞

A hole subjected to uniform remote load at infinity

Γ

Solution for in- plane deformation in the case of plane stress state

traction-free boundary condition along Γ

= 4

Γ

+

−

= _∞ ^∞ _∞ ^∞

σ σ σ

! σ

Under the condition of the uniform stress at infinity

+

= ^∞ _{∞ ∞} ^∞

∞

ε ε ε

ε

−

= _∞ ^∞ _∞ ^∞

∞

σ σ σ

σ

σ $

σ ^∞ =

= 4

= ^∞

∞ σ σ

Lekhnitskii’s solution in the case of the hole on a thin plate of which size are considerably larger than the hole diameter

[ ^{− −} ] ^{+ ∞}

= 9 # ς " !

[ ^{− −} ] ^{+ ∞}

= 9 " ς " ! Solution

Numerical simulation: Lekhnistskii’s analytical approach

II- Identification procedure