Chapter 4 Material Behavior – Linear Elastic Solids

Linear Elastic Constitutive Solid Model

Develop Force-Deformation Constitutive Equation in the Form of Stress-Strain Relations Under the Assumptions:

• Solid Recovers Original Configuration When Loads Are Removed• Linear Relation Between Stress and Strain• Neglect Rate and History Dependent Behavior• Include Only Mechanical Loadings• Thermal, Electrical, Pore-Pressure, and Other Loadings Can Also

Be Included As Special Cases

Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island

Typical One-Dimensional Stress-Strain Behavior

Tensile Sample

Steel

Cast Iron

Aluminum

Applicable Region for Linear Elastic Behavior

= E

Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island

Linear Elastic Material ModelGeneralized Hooke’s Law

zxyzxyzyxzx

zxyzxyzyxyz

zxyzxyzyxxy

zxyzxyzyxz

zxyzxyzyxy

zxyzxyzyxx

eCeCeCeCeCeC

eCeCeCeCeCeC

eCeCeCeCeCeC

eCeCeCeCeCeC

eCeCeCeCeCeC

eCeCeCeCeCeC

666564636261

565554535251

464544434241

363534333231

262524232221

161514131211

222

222

222

222

222

222

zx

yz

xy

z

y

x

zx

yz

xy

z

y

x

e

e

e

e

e

e

CC

C

CCC

2

2

2

6661

21

161211

klijklij eC

ijlkijkl

jiklijkl

CC

CC

or 36 Independent Elastic Constants

with

Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island

Anisotropy and NonhomogeneityAnisotropy - Differences in material properties under different directions. Materials like wood, crystalline minerals, fiber-reinforced composites have such behavior.

Nonhomogeneity - Spatial differences in material properties. Soil materials in the earth vary with depth, and new functionally graded materials (FGM’s) are now being developed with deliberate spatial variation in elastic properties to produce desirable behaviors.

(Body-Centered Crystal) (Fiber Reinforced Composite)(Hexagonal Crystal)

Gradation Direction

Typical Wood Structure

Note Particular Material Symmetries Indicated by the Arrows

Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island

Isotropic Materials

Although many materials exhibit non-homogeneous and anisotropic behavior, we will primarily restrict our study to isotropic solids. For this case, material response is independent of coordinate rotation

mnpqlqkpjnimijkl CQQQQC jkiljlikklijijklC

klijklij eC ijijkkij ee 2

zxzx

yzyz

xyxy

zzyxz

yzyxy

xzyxx

e

e

e

eeee

eeee

eeee

2

2

2

2)(

2)(

2)(

Generalized Hooke’s Law

- Lamé’s constant - shear modulus or modulus of rigidity

Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island

Isotropic MaterialsInverted Form - Strain in Terms of Stress

ijkkijij EEe

1

zxzxzx

yzyzyz

xyxyxy

yxzz

xzyy

zyxx

Ee

Ee

Ee

Ee

Ee

Ee

2

11

2

11

2

11

)(1

)(1

)(1

elasticity of modulusor modulus sYoung'...)23(

E

ratio sPoisson'...)(2

Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island

Physical Meaning of Elastic Moduli

000

000

00

ij

xeE /

000

00

00

ij

xy

xye

/

2/

ijij p

p

p

p

00

00

00

ModulusBulk ...)21(3

E

k

kkep kk

Simple Tension Hydrostatic Compression

Pure Shear

p

p

p

Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island

Relations Among Elastic Constants E k

E, E 213

E 12

E

211

E

E,k E k

Ek

6

3 k

Ek

kE

9

3

Ek

Ekk

9

33

E, E

2

2E E

E

33

E

E

3

2

E, E RE

2

6

3 RE

4

3 RE

,k 213k k

12

213k

1

3k

, 12

213

12

21

2

,

211

3

1

2

21

k,

k

k

6

9

26

23

k

k k

3

2k

k,

k

kk

3

9

k3 k )(

2

3k

,

23

)(2

3

23

EER 29 22

Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island

Typical Values of Elastic Moduli for Common Engineering Materials

  E (GPa) (GPa) (GPa) k(GPa) (10-6/oC)

Aluminum 68.9 0.34 25.7 54.6 71.8 25.5

Concrete 27.6 0.20 11.5 7.7 15.3 11

Cooper 89.6 0.34 33.4 71 93.3 18

Glass 68.9 0.25 27.6 27.6 45.9 8.8

Nylon 28.3 0.40 10.1 4.04 47.2 102

Rubber 0.0019 0.499 0.654x10-3 0.326 0.326 200

Steel 207 0.29 80.2 111 164 13.5

Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island

Hooke’s Law in Cylindrical Coordinates

zzrz

zr

rzrr

σ

zrzr

zz

rr

zzrz

zr

rzrr

e

e

e

eeee

eeee

eeee

2

2

2

2)(

2)(

2)(

x3

x1

x2r

z

dr

z

r

rrzz

d

Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island

Hooke’s Law in Spherical Coordinates

R

R

RRR

σ

RR

RR

R

R

RRR

e

e

e

eeee

eeee

eeee

2

2

2

2)(

2)(

2)(

R

x3

x1

x2

R

R

R

 

Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island