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H. Garmestani, ProfessorSchool of Materials Science and Engineering

Georgia Institute of Technology Outline: Materials Behavior

Tensile behavior…

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Constitutive equation is the relation between kinetics (stress, stress-rate) quantities and kinematics (strain, strain-rate) quantities for a specific material. It is a mathematical description of the actual behavior of a material. The same material may exhibit different behavior at different temperatures, rates of loading and duration of loading time.). Though researchers always attempt to widen the range of temperature, strain rate and time, every model has a given range of applicability.

Constitutive equations distinguish between solids and liquids; and between different solids.

In solids, we have: Metals, polymers, wood, ceramics, composites, concrete, soils…

In fluids we have: Water, oil air, reactive and inert gases



a

d

d

a

AP

ll

Ratio sPoisson'

stress/strain diametral

strain axial/

0

Load-displacement response

axis)cylinder thealongmoment torsionala r to radius ofsection

corsscircular ofbar lcylindrica a(for modulusshear

material) elastican (for dilatation is modulus,bulk is

)elasticity of modulus(or modulus sYoung' is

p

t

Ya

Y

IlM

eke

k

EE



Uniaxial loading-unloading stress-strain curves for(a) linear elastic;(b) nonlinear elastic; and(c) inelastic behavior.

Examples of Materials Behavior



Elastic behavior is characterized by the following two conditions:

(1) where the stress in a material () is a unique function of the strain (),

(2) where the material has the property for complete recovery to a “natural” shape upon removal of the applied forces

Elastic behavior may be Linear or non-linear



The constitutive equation for elastic behavior in its most general form as

CwhereC is a symmetric tensor-valued function and is a strain tensor we introduced earlier.

Linear elastic CNonlinear-elastic C(


Boundary Value Problems we assume that the strain is small and there is no rigid body rotation. Further we assume that the material is governed by linear elastic isotropic material model. Field Equations

(1)

(2) Stress Strain Relations

(3)Cauchy Traction Conditions (Cauchy Formula)

(4)

)1(21

., ijjiij uuE

ij E kkij 2E ij (2)

ti jin j

ji, j X j 0

ji, j Bi 0 For Statics

ji, j Bi ai For Dynamicsadmission.edhole.com

In general, We know that

For small displacement

Thus

ij

x j

Bi ai

Bi is the body force/massBi is the body force/volume X i

ai is the acceleration

ii Xx

j

ij

x

iii x

uvt

uDtDxv

i

fixedadmission.edhole.com

Assume v << 1, then

For small displacement,

Thus for small displacement/rotation problem

okk

kkokk

kko

iii

x

ii

E

EE

EdVdVtu

tva

tuv

i

1

11

1

1 Since

01

2

2

fixed

o

ij

x j

Bi 2ui

t 2admission.edhole.com

Consider a Hookean elastic solid, then

Thus, equation of equilibrium becomes

ij E kkij 2E ij

uk,kij ui, j u j ,i ij , j uk,kjij ui,ij u j ,ij

ji

i

i

kkio

io xx

ux

EBtu

2

2

2


For static Equilibrium Then

The above equations are called Navier's equations of motion.In terms of displacement components

02

2

tui

0

0

0

3323

2

22

2

21

2

3

2223

2

22

2

21

2

2

1123

2

22

2

21

2

1

Buxxxx

E

Buxxxx

E

Buxxxx

E

okk

okk

okk

2

2

1tuBudivE ookk


In a number of engineering applications, the geometry of

the body and loading allow us to model the problem using

2-D approximation. Such a study is called ''Plane

elasticity''. There are two categories of plane elasticity,

plane stress and plane strain. After these, we will study

two special case: simple extension and torsion of a circular

cylinder.admission.edhole.com

For plane stress,

(a) Thus equilibrium equation reduces to

(b) Strain-displacement relations are

(c) With the compatibility conditions,

2,1,, 21 jixxijij

0

0

0

332313

22,221,21

12,121,11

b

b

1,22,1122,2221,111 2 uuEuEuE

21

122

12

222

22

112

12,1211,2222,11 2

xxE

xE

xE

EEE


(d) Constitutive law becomes, Inverting the left relations,

Thus the equations in the matrix form become:

(e) In terms of displacements (Navier's equation)

2211221133

12121212

112222

221111

1 that Note

21

1

1

EEv

vEvE

GGEvE

vE

E

vE

E

Y

Y

Y

Y

12121212

1122222

2211211

121

1

1

Gv

EEv

E

vEEv

E

vEEv

E

YY

Y

Y

12

22

11

2

12

22

11

1000101

1EEE

vv

v

vEY

2,1,01212 ,,

jibuv

Euv

Eijii

Yjji

Y admission.edhole.com

(b) Inverting the relations, can be written as:

GE

vE

vvE

vE

vvE

vE

Y

Y

Y

2212

11

11

121212

112222

221111

(c) Navier's equation for displacement can be written as:

2,1,0211212 ,,

jibu

vvEu

vE

ijijY

jjiY


Relationship between kinetics (stress, stress rate) and kinematics (strain, strain-rate) determines constitutive properties of materials.Internal constitution describes the material's response to external thermo-mechanical conditions. This is what distinguishes between fluids and solids, and between solids wood from platinum and plastics from ceramics.

Elastic solid Uniaxial test: The test often used to get the mechanical properties

PA0

engineering stress

ll0

engineering strain

E


If is Cauchy tensor and is small strain tensor, then in general,

ij ijE

ij Cijkl E kl

where is a fourth order tensor, since T and E are second order tensors. is called elasticity tensor. The values of these components with respect to the primed basis ei’ and the unprimed basis ei are related by the transformation law

ijklC

mnrsslrknimiijkl CQQQQC

However, we know that and then lkkl EE

ij ji

We have symmetric matrix with 36 constants, If elasticity is a unique scalar function of stress and strain, strain energy is given by

iklkjiklijkl CCC 44C

dU ijdEkl or U ij E ij

Then ij UE ij

Cijkl Cklij

Number of independent constants 21

ijklC


Show that if for a linearly elastic solid, then

Solution:Since for linearly elastic solid , therefore

Thus from , we have

Now, since

Therefore,

ij UE ij

klijijkl CC

ij Cijkl E kl

ij

E rs

Cijrs

ij UE ij ijrs

ijrs EEUC

2

rsijijrs EEU

EEU

22

klijijkl CC


Now consider that there is one plane of symmetry (monoclinic) material, then One plane of symmetry => 13

If there are 3 planes of symmetry, it is called an ORTHOTROPIC material, then orthortropy => 3 planes of symmetry => 9

Where there is isotropy in a single plane, then Planar isotropy => 5

When the material is completely isotropic (no dependence on orientation) Isotropic => 2


Crystal structure Rotational symmetry

Number of independent

elastic constants

Triclinic Monoclinic Orthorhombic Tetragonal Hexagonal Cubic Isotropic

None 1 twofold rotation 2 perpendicular twofold rotations 1 fourfold rotation 1 six fold rotation 4 threefold rotations

21 13 9 6 5 3 2


A material is isotropic if its mechanical properties are independent of direction

Isotropy means

Note that the isotropy of a tensor is equivalent to the isotropy of a material defined by the tensor. Most general form of (Fourth order) is a function

ijklC

jkiljlikklij

ijklijklijklijkl HBAC

ij Cijkl E kl

ij C ijkl E kl

Cijkl C ijkl


Thus for isotropic material

and are called Lame's constants. is also the shear modulus of the material (sometimes designated as G).

ij Cijkl E kl

(ijkl ik jl il jk )Ekl

ijkl E kl ik jl E kl il jk E kl

ij E kk E ij E ji

ije ( )E ij

eij 2E ij

when i j ij 2E ij

when i j ij e 2E ij

eI 2Eadmission.edhole.com

We know that

So we have

Also, we have

ij eij 2E ij

kk 3 2 e or e 13 2

kk

E ij 1

2 ij

3 2

kkij


vEvvv

vkEvEE

kE

Evv

vEk

vvk

vE

vkv

EE

vv

vvvE

vkEvvE

Y

YYY

Y

YY

Y

Y

YY

YY

122

213122333213

122133

212

21312

13

32

212

211

,,,,,

Note: Lame’s constants, the Young’s modulus, the shear modulus, the Poisson’s ratio and the bulk modulus are all interrelated. Only two of them are independent for a linear, elastic isotropic materials,


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