Virtual Fields Method

Institut Mines-Télé[email protected]

Recent advances in the Virtual Fields Method

for Nonlinear Elasticity

Dr. Yue Mei, Prof. Stéphane AVRIL

PARIS

AUVERGNE RHONE-ALPES

MINES SAINT-ETIENNE

First Grande Ecole

outside Paris

Founded in 1816

Institut Mines-Télé[email protected]@emse.fr

Non linear elasticity

Stéphane Avril - June 2021 - SEM annual conference 3

Soft biological tissues: many challenges

for continuum mechanics

Institut Mines-Télé[email protected]@emse.fr

Essais de Treloar (1944)

0 10 20 30 40 50 60 70

0 1 2 3 4 5 6 7

Lambda-1

C on tr a int e no m ina le

Traction simple Equibiaxiale Traction plane

Treolar’s tests on rubber (1944)

Nom inal str ess F /S0 (MPa )

Stretch λ=L/L ₀

Biaxial tension

Planar tension Uniaxial tension

5

1 2 3 4 5 6 7

Hyperelasticity

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0 10 20 30 40 50 60 70

0 1 2 3 4 5 6 7

Con tra int e no m ina le

Lambda-1

Nom inal str ess S =F/A 0 (MPa )

Stretch λ=L/L ₀

Stored energy per unit volume

න 𝑺: ሶ F𝑑𝑡 = න 𝝅: ሶ E𝑑𝑡 = Ψ(E)

1 2 3 4 5 6 7

Strain energy density function

Institut Mines-Télé[email protected]@emse.fr

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compressible hyperelastic behaviour

incompressible hyperelastic behaviour (𝐽=1)

 ?



Strain energy density:

𝛔 = 𝐽𝐅. 𝜕Ψ

𝜕𝐄 . 𝐅 ^𝑇

𝛔 = 𝐅. 𝜕Ψ

𝜕𝐄 . 𝐅 ^𝑇 + 𝑐𝐈

Stéphane Avril - June 2021 - SEM annual conference

Specific problem of the optical nerve head

Institut Mines-Télé[email protected]

9

Anisotropic strain energy density

(Holzapfel et al., 2000)

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9

The use of full-field

measurements in

nonlinear elasticity

Institut Mines-Télé[email protected]@emse.fr

DIC tend to be used for all kind of soft tissues

11

MEASUREMENT OF THE RESPONSE USING DIGITAL IMAGE CORRELATION

classical panoramic

Genovese. Optics Lasers Eng, 47, p 995-1008, 2009.

Badel et al. CMBBE, 15, p 37-48, 2012.

Institut Mines-Télé[email protected]@emse.fr

OCT-DVC applied to soft tissue mechanics

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Specific problem of the optical nerve head

Institut Mines-Télé[email protected]@emse.fr

Identification of

hyperelastic parameters:

state of the art

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Inverse approach – traditional FEMU approach

Set of materials properties

Resolution of forward problem

Deviation between predictions and measurements minimization

initialization

Identification of material properties

measurements

Institut Mines-Télé[email protected]@emse.fr Stéphane Avril - June 2021 - SEM annual conference 17

Set of materials properties

Resolution of forward problem

Deviation between predictions and measurements minimization

initialization

Identification of material properties

1. Use a gradient based method (steepest descent or BFGS)

2. Need to derive the gradient of J with respect to µ at each iteration. Many iteratons

3. Very unstable with hyperelastic models: many risks that the forward problems have a poor convergence

FEMU approach - limitations

The virtual fields method (VFM)

Set of materials properties

3D estimation of

stresses

Deviation to the principle

of virtual power minimization

initialization

Identification of material properties

Bersi et al., J Biomech Eng, 2016

Full-field strain

measurements

Institut Mines-Télé[email protected]@emse.fr

The principle of virtual work

19

Application to the identification of material parameters: the virtual fields method

Example in incompressible hyperelasticity

I c F

E . .

F ^t 





 





0 dS

u . T dV

: I c F

E . .

F ^*

V

* V

t     





 



 





 

  



Institut Mines-Télé[email protected]@emse.fr

Application to the identification of material parameters: the virtual fields method

Example for NeoHooke incompressible

I c F

. F C

2 ₁₀ ^t 





 ^{F . F c I}  ^{: dV T . u dS}

C

2 ^*

V

* V

t

10  









  ^{2  B c I  : dV}

dS u

. T dV

: I c F

. F 2

dS u

. T

C _*

V

* V

* V

t

* V

10   



 







 



21

Application in simple uniaxial tension

Institut Mines-Télé[email protected]@emse.fr

Application in simple uniaxial tension

FL dS

u . T ^*

V

 



 ^{B B FL B}  ^dV

C

V

33 22

11

10 ^  ^{ }

 

 



 

 









2 / z

2 / y

x u ^*

F

-F

L z

x

23

Set of materials properties

Resolution of forward problem

Deviation between predictions and measurements

Deviation to the principle

of virtual power

initialization

1. Method that can provide virtual fields for any 3D geometry

2. independent virtual fields are needed in order to separate the hydrostatic and deviatoric contributions of the stress for compressible hyperelastic materials

3. Lack of automatic approach

VFM approach - limitations

Customized

cost function

Institut Mines-Télé[email protected]@emse.fr

Successful applications for

reconstructing local hyperelastic

properties of arteries took 3 years…

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Bersi et al, BMMB 2018, Bersi et al, Scientific Reports 2020

How to apply the VFM on the optical nerve head problem without the

lengthy trial and error stage?

Institut Mines-Télé[email protected]@emse.fr

Identification of

hyperelastic parameters:

our new VFM approach

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Varying the material proerties result in variations of stresses

Variations of stresses:

measured

computed

Institut Mines-Télé[email protected]@emse.fr Stéphane Avril - June 2021 - SEM annual conference 29

Variations of stresses should satisfy the principle of virtual power

computed ?

computed

measured

Variations of stresses should satisfy the principle of virtual power

computed

measured

computed

Institut Mines-Télé[email protected]@emse.fr Stéphane Avril - June 2021 - SEM annual conference 31

Quadratic convergence

Institut Mines-Télé[email protected]@emse.fr

Application to the eye problem

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PLNT

ALC

PLC

Applications to other problems

Rubber membrane

Institut Mines-Télé[email protected]@emse.fr

Applications to other problems

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Elastography

Towards finite element software for identification

of material parameters in nonlinear elasticity

Institut Mines-Télé[email protected]@emse.fr

Acknowledgements

Stéphane Avril - June 2021 - SEM annual conference

Funding:

ERC-2014-CoG BIOLOCHANICS

 Vicky Nguyen

 Brandon Zimmerman

 Yue Mei

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