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Introduction to Constitutive Modeling

and Elastic Models

Amit Prashant

Indian Institute of Technology Gandhinagar

Short Course on

Finite Element Method, Constitutive Modeling

and Applications

28 Jan – 01 Feb, 2013

What is Constitutive Model?

Say, for mild steel the stress strain curves during tensile test is

Constitutive modeling of this relationship could be

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Now using this model various prototypes or real pieces of products may be designed using some numerical tool, such as FEM.

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Procedure of Predicting the Structural Behavior

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Laboratory soil tests

Simple closed form solutions (elastic and limit analysis, etc.)

Simple soil parameters (c, f, E, n, etc)

Constitutive model

Prediction of prototype

Prediction and comparison with model or full scale test

Model or full scale tests

FE or FD program

Prediction and comparison with model or full scale test

Simple procedure Advanced procedure

Essentials for Constitutive Model

A constitutive model is used to capture the entire stress-strain relation obtained from Laboratory and/or in-situ tests.

It is important to employ realistic constitutive model which can reproduce the important aspects of the material behavior under various loading conditions.

It is also important that the employed constitutive model requires least number of and relatively simple tests for calibration.

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Requirements in Modeling

Complexities in real problems and the material response often lead to a challenge of modeling.

Analysis requires knowledge about: Geometry of the problem, material variation and boundary

conditions

Stress-strain behavior constitutive model

Numerical tools (FEM) and computer programming

Still some judgment is required

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Why do we need a Realistic Constitutive Model?

Better understanding of soil stress-strain behavior

Extrapolate to conditions which can not be produced in laboratory testing equipment

Use in Numerical tools: Design of structures on rational basis

Limited amount of testing: Perform simplest and most convenient types of laboratory

tests for calibration

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Constitutive Equations

A mathematical model that can permit reproduction of the observed response of a continuous medium.

A simple law for linear elastic deformation of spring

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F F

F

k

.F k

Deformation and Strain

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F F

D L

x

Total Displacement = D

Displacement at distance x, x

u u xL

D

Strain = and x L

du

dx L

D ??

x

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Strain State

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11 12 13

21 21 23

31 32 33

ij

ij ji

ε

Stress State

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11 12 13

21 21 23

31 32 33

ij

ij ji

σ

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Constitutive Relation in Mechanics

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Mathematical form of Constitutive Equation

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o

ij ijkl kl ijC

Fourth order stiffness

tensor based on some

material constants

Initial stress

corresponding to

initial strain free state

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Some Definitions

Plasticity:

The behavior of solid bodies in which they deform permanently under the action of external loads.

Elasticity:

The behavior of solid bodies in which they return to their original shape when the external forces are removed.

Yielding:

The behavior of solid bodies in which the material begins to deform plastically.

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Common Models

Rigid Plastic Simple

Widely used

Simulate only Failure

Linear Elasticity: Simple

Most widely used

No failure

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y

E

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Common Models, contd…

Bi-linear Elasticity: Fairly easy to use

Approximates better to true relationship

Linear Elastic Perfectly Plastic: Rough Approximation

Combines the simplest possible way deformation and strength of material

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E1

E2

E1

Common Models, contd…

Non-linear Elastic, Elasto-plastic, Hardening/softening plasticity: Model true behavior

Relatively complex

More realistic solutions

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Primary

Loading

Load Cycles

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Principal Stresses

Eigen values of the stress tensor

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1

2

1

Major Principal StressIntermediate Principal StressMinor Principal Stress

1

ij 2

3

0 0

0 0

0 0

2

Z

Y

X

1

3

3

1

2

Stress Invariants

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1 1 2 3

2 1 2 2 3 3 1

3 1 2 3

I

I

I

2

2

2 2 2

1 2 2 3 3 1

Second invariant of deviatoric stress tensor,

Octahedral shear stress,

3

2

1

3

oct

oct

J

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Reduced form of Strain and Stress state

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11 12 13

21 21 23

31 32 33

ij

ij ji

σ

11 12 13

21 21 23

31 32 33

ij

ij ji

ε

Constitutive Relation in Reduced Form

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ij ijkl klC

i ij jD

11 12 13

21 21 23

31 32 33

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Elastic Model – Isotropic 3D

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i ij jD

3D2D: Plane Stress

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0zz xz yz

Z - axis 0xz yz

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3D2D: Plane Strain

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0zz xz yz

0xz yz

Elastic Model – Isotropic 2D

Plane Stress

Plane Strain

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Elastic Parameters

Two parameters

Young’s Modulus, E

Poisson’s Ratio, n

Shear Modulus

Bulk Modulus

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2 1

EG

n

3 1 2

EK

n

Tensile Test – Elastic parameters

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F

F

Cross-sectional Area = A

zz

F

A

zz

L

L

D

xx yy

R

R

D

.

.

zz

zz

F LE

A L

D

.

.

rr

zz

R L

R L

n

D

D

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General Stress Strain Curve

q

q

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2Gmax

2G

q

q

Elastic zone

Gmax = Maximum Shear Modulus

G = Secant Shear Modulus

Strain Dependent Shear Modulus in Soils

Linear Elastic Model

Nonlinear Elastic Model

%

Include Plasticity

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Shear Modulus

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StrainShear

StressShear

Strain,Shear

Stress,Shear oG

1

Initial Shear Modulus secG

1

Secant Shear Modulus

tanG1

Tangent Shear Modulus

Cross-anisotropic Elasticity

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ijD

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Cross-anisotropic Elasticity

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ijD

* vE E

* hhn n

2

h vE E

hhvh

nn

*

2 1 *

hhhv

G EG

n

1

*E

Hyperbolic Model Nonlinear Force-Displacement relationship

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Pu

F

F

1/K0= a

b = 1/Pu 1

Calibration of parameters

Fa b

a b

F

2

dF a

d a b

0

0

1dFK

d a

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Hyperbolic Model A non-linear Elastic Model

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u

1/E0= a

b = 1/u 1

Calibration of parameters

a b

a b

2

d a

d a b

0

0

1dE

d a

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Thank You