Continuum damage model for Prony-

series type viscoelastic solids

Juan G. Londono, Luc Berger-Vergiat and Haim Waisman

Department of Civil Engineering and Engineering Mechanics

Columbia University in the City of New York

August 6th 2014

“Mechanics for Sustainable and Resilient Systems”

.

Continuum damage model for Prony-series type viscoelastic

solids

Juan G. Londono, Luc Berger-Vergiat and Haim Waisman

Department of Civil Engineering and Engineering Mechanics,Columbia University in the City of New York

2014 Engineering Mechanics Institute (EMI) Conference,McMaster University,

August 6th, 2014

.

Outline

• Introduction and motivation

• Viscoelastic models

• Continuum Damage Mechanics

• Viscoelastic damage and implementation

• Applications and results

• Conclusions

• References

3Outline

Introduction

• Adequate computational models is required for

engineering applications (Expensive physical testing)

• Materials display time dependent deterioration

• Viscoelastic behavior: Elastic + viscous properties

• Damage growth: Continuum damage mechanics

• Viscoelastic behavior and damage growth effects

combined

4Introduction

• Model is phenomenological: no related to chemical

composition or molecular structure

• Material experience Creep, Stress relaxation,

Recovery

Viscoelastic model

5Viscoelasticity

Creep Relaxation

• Prony series:

• Material modulus expressed in the Prony Series form leads to a

generalization of the Maxwell model

6Viscoelasticity

Viscoelastic model

𝐸 𝑡 = 𝐸0 +

𝑖=1

𝑛

𝐸𝑖𝑒−𝑡𝜆𝑖 = 𝐸 𝜇0+

𝑖=1

𝑛

𝜇𝑖𝑒−𝑡𝜆𝑖

𝜏𝑖 = 𝑐ℎ𝑎𝑟𝑡𝑒𝑟𝑖𝑠𝑡𝑖𝑐 𝑡𝑖𝑚𝑒;

𝜎 = −∞

𝑡

𝐸 𝑡 − 𝜏 휀 𝜏 𝑑𝜏

𝑖=0

𝑛

𝜇𝑖 = 1

• Some materials display viscoelastic behavior on the shear

component only,

and

Prony series of 𝐺 𝑡 ,

Viscoelastic model

7Viscoelasticity

𝜎 = 𝜎𝑣𝑜𝑙 + 𝜎𝑑𝑒𝑣 = 3𝐾 𝑡𝑟 휀 + 2𝐺(𝑡)휀𝑑𝑒𝑣(𝑡)

2𝐺 𝑡 휀𝑑𝑒𝑣 𝑡 = 2 −∞

𝑡

𝐺(𝑡 − 𝜏) 휀𝑑𝑒𝑣(𝜏) 𝑑𝜏

𝐺 𝑡 = 𝐺 𝜇0+

𝑖=1

𝑛

𝜇𝑖𝑒−𝑡𝜆𝑖

2𝐺 𝑡 휀𝑑𝑒𝑣 𝑡 = 2𝐺 −∞

𝑡

𝜇0+

𝑖=1

𝑛

𝜇𝑖𝑒−(𝑡−𝜏)

𝜆𝑖 휀𝑑𝑒𝑣(𝜏) 𝑑𝜏

• Progressive deterioration of material preceding the failure due to accumulation of voids and micro-cracks

• No cracks present in the material

• Damage evolution fully phenomenological• Degree of damage is quantified into the parameter D (0 ≤ D ≤ 1)

• Damage might be anisotropic

Damage model

8

Fig.. Isotropic damage in uniaxial tension (concept of effective stress).

Continuum Damage

𝐷 =𝛿𝑆 − 𝛿 𝑆

𝛿𝑆

𝜎 = 𝜎𝛿𝑆

𝛿 𝑆=

𝜎

(1 − 𝐷)= 𝑀𝜎

Damaged State

Effective Undamaged State

• Kachanov-Rabotnov uniaxial creep damage

• Hayhurst's (1972) equivalent stress measure

where

• Murakami & Ohno (1981), Murakami (1983), Murakami (1988)

• Simplifying for isotropic damage,

9

(1) 3 II (1 )Idev

1

(1) I ii dev dev1II

2dev mn mn

1 (1) (1) (1) (1){ [(1 ) ( )]} [(1 ) ]r kD B Tr D 1 γ

anisotropic parameter (1) (1)eigenvector related to

Continuum Damage

𝐷 = 𝐵𝜒 𝑟

(1 − 𝐷)𝑘𝐵, 𝑘, 𝑟 = Material parameters

𝜒 = Hayhurst's equiv. stress

Damage model 𝑑 = 𝐵

𝜎 𝑟

(1 − 𝑑)𝑘𝜎

• Current stress, 𝜎𝑛+1:

where,

Viscoelastic damage 10

Viscoelastic damage implementation

𝜎𝑛+1𝑑𝑒𝑣 = (1 − 𝐷 𝑛

+1) 2𝐺 𝜇0휀𝑛+1

𝑑𝑒𝑣 +

𝑖=1

𝑛

𝜇𝑖 𝑒−𝑡𝜆𝑖 휀0

𝑑𝑒𝑣 + ℎ𝑛+1𝑖

𝐷𝑛+1 = 𝐷𝑛 + ∆𝑡 𝐵𝜒 𝑟

(1 − 𝐷𝑛)𝑘

𝜎𝑛+1 = 𝜎𝑛+1𝑣𝑜𝑙 + 𝜎𝑛+1

𝑑𝑒𝑣

𝜎𝑣𝑜𝑙 = 1 − 𝐷𝑛+1 3𝐾 𝑇𝑟 휀

ℎ𝑖 = 𝑒−Δ𝑡𝜆𝑖 ℎ𝑛

𝑖 + Δℎ𝑖

ℎ𝑛𝑖 = 𝑒

−𝑡𝑛𝜆𝑖

0

𝑡𝑛

𝑒𝜏𝜆𝑖 휀𝑑𝑒𝑣(𝜏) 𝑑𝜏 , Δℎ𝑖 = 𝜆𝑖 1 − 𝑒

−Δ𝑡𝜆𝑖

Δ휀𝑑𝑒𝑣

Δ𝑡

Viscoelastic damage implementation

11Viscoelastic damage

• Damage time integration by explicit forward Euler method

• Initial conditions:

• For the current time step, 𝑡𝑛+1 :

1. From previous time step:

2. Strain computation:

3. Effective stress:

4. Damage update:

5. Approxim. Stiffness, 𝐾𝑛+1:

𝜎𝑛+1𝑑𝑒𝑣 = 𝑓 ℎ𝑛

𝑖 , 𝛥ℎ𝑖 , 휀𝑛+1𝑑𝑒𝑣 , 휀𝑛

𝑑𝑒𝑣

𝜎𝑛+1 = 𝜎𝑛+1𝑣𝑜𝑙 + 𝜎𝑛+1

𝑑𝑒𝑣

휀𝑛+1 = 휀𝑛+1𝑣𝑜𝑙 + 휀𝑛+1

𝑑𝑒𝑣

𝑡 = 0, 𝐷 = 0

𝑡 = 𝑡𝑛, 휀 𝑡𝑛 = 휀𝑛,

𝑢𝑛+1

𝐷𝑛+1 = 𝑓( 𝜎𝑛+1) Δ𝐷𝑛+1 = Δ𝑡 𝐷𝑛+1

𝐷𝑛+1 = 𝐷𝑛 + Δ𝐷𝑛+1

𝐷 𝑡𝑛 = 𝐷𝑛

𝑡𝑛+1 = 𝑡𝑛 + ∆𝑡

𝑑𝜎 =𝜕𝜎

𝜕𝑢𝑑𝑢 +

𝜕𝜎

𝜕𝐷

𝜕𝐷

𝜕𝑢𝑑𝑢

Applications and Results• Parameters calibration:

o Constrained Optimization

o Damage Parameters

Where, 𝑘1, 𝑘2 and 𝐵1,𝐵2 are material parameters from linear fitting

and 𝜃 and Θ have components of 𝜎, 휀 or 휀

• Civil Engineering applications:o Polycrystalline Ice

o Asphalt concrete

12Applications and Results

∀𝑖 ∈ 0, 𝑛 , 𝜇𝑖 > 0

𝜇0+

𝑖=1

𝑛

𝜇𝑖 = 1

𝑘 = 𝑘1+𝑘2𝜃

𝐵 = 𝐵1+𝐵2Θ

Polycrystalline Ice• Finite Elements implementation: FEAP user element

• Values at central node, Plane Stress

• Tensile creep (Mahrenholtz, W. Z. 1992)

13Applications and Results

𝜎 = 0.93, 0.82, 0.64 [𝑀𝑃𝑎]𝑇 = −10°𝐶

Figure. Plane Stress

Polycrystalline Ice• Parameters selected

• Results

14Applications and Results

E [MPa] ν nvis µ1 τ1

9500 0.35 1.0 0.999 415.0

α β B1 B2 r k1 k2 γ εthreshold

0.2 0.63 5.232 × 10−7 0.0 0.43 -2.63 7.24 0.0d0 0.8 × 10−2

𝜃 = 𝜎𝑖𝑖 𝐷𝑐 = 0.45

Damage

Prony Series

Applications and Results 15

Asphalt Concrete• Unconfined compression:

• Experimental data (Katsuki D. & Gutierrez M., 2011)

Strain rates applied

휀 = 4.2 × 10−6, 휀 = 1.7 × 10−5, 휀 = 1.7 × 10−4, 휀 = 1.7 × 10−3.

Applications and Results 16

Asphalt Concrete• Results

• Parameters α β B1 B2 r k1 k2 γ εthreshold

1 0 1.14 × 10−4 −8.655 × 10-3 3.24 -10.986 3.31 0.0d0 0.0

ν n µ1 τ1

0.35 1.0 0.999 415.0

Damage

Prony Series

𝜃 = 휀𝐵𝐶 𝐷𝑐 = 0.0Θ = 휀𝐵𝐶

Conclusions• Material viscoelastic deterioration can be predicted with

simple implementation with proper material selection

• Prediction of material behavior under tension and

compression can be calibrated

• Semi-analytical time integration of the constitutive

equation and explicit Forward Euler for damage results in

fast and accurate prediction of material behavior

• Optimization of the material parameter required careful

selection of initial values. Global optimization method

could be beneficial

• In the finite elements implementation, the model

displayed mesh sensitivity

17Conclusions

References• Findley W. N., Lai J.S, Onaran K., “Creep and relaxation of nonlinear viscoelastic

materials - With an introduction to linear viscoelasticity” (1976)

• Krishnamachari S.I, “Applied Stress Analysis of Plastics-A mechanical engineering approach” (1993)

• Lakes, R.S., “Viscoelastic Solids” (1999)

• Murakami S., Kawai M., Rong H., “Finite Element Analysis of Creep Crack Growth By A Local Approach” (1988)

• Taylor, R.L, “FEAP-A Finite Analysis Program”, University of California at Berkley (2011)

• Waisman H., Duddu R., “A temperature dependent creep damage model for polycrystalline ice” (2011)

• Katsuki D., Gutierrez M., “Viscoelastic damage model for asphalt concrete”, ActaGeotechnica (2011

18Thank you

AcknowledgementThe authors are grateful to the funding support provided by the National Science

Foundation under Grant #PLR-1341472

Thank you!Questions ?

Thank you

Viscoelastic model• Springs and dashpots models:

• Elastic behavior (Hook’s law)

• Viscous behavior (Newton’s Law)

• Maxwell model:

• Represent relaxation very well but not creep or recovery

• Kelvin-Voigt model:

• Creep and recovery are well represented but not relaxation

20Thank you

𝜎 = 𝐸휀𝜎 = 𝜂 휀

𝜎 = 𝐸휀2 = 𝜂 휀1

𝜎 = 𝜎1 + 𝜎2 = 𝐸휀 + 𝜂 휀

𝜎 = 휀

1𝜂 +

1𝐸𝑑𝑑𝑡휀 = 휀1 + 휀2

휀 = 휀1 = 휀2𝜎 = 𝐸 + 𝜂

𝑑

𝑑𝑡휀