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Constitutive Relations in SolidsElasticity

H. Garmestani, ProfessorSchool of Materials Science and Engineering

Georgia Institute of Technology

Outline: Materials Behavior

Tensile behavior…

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The Elastic Solid and Elastic Boundary Value Problems

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



(cont.)

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

I

lM

eke

k

EE


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

Examples of Materials Behavior


Constitutive Equations: Elastic

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


Constitutive Equation

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

Cwhere

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

Linear elastic CNonlinear-elastic C(


Equations of Infinitesimal Theory of Elasticity

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(2

1., 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

Equations of the Infinitesimal Theory of Elasticity (Cont'd)

In general, We know that

For small displacement

Thus

ij

x j

Bi ai

Bi is the body force/mass

Bi is the body force/volume X i

ai is the acceleration

ii Xx

j

ij

x

iii x

uv

t

u

Dt

Dxv

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

EdVdV

t

u

t

va

t

uv

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 Ekkij 2E ij

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

ji

i

i

kkio

io xx

u

x

EB

t

u

2

2

2



For static Equilibrium Then

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

02

2

t

ui

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

1t

uBudivE ookk


Plane Elasticity

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.


Plane Strain &Plane Stress

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

xx

E

x

E

x

E

EEE


Plane Strain &Plane Stress

(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

v

E

vE

GGE

vE

vE

E

vE

E

Y

Y

Y

Y

12121212

1122222

2211211

121

1

1

Gv

EE

v

E

vEEv

E

vEEv

E

YY

Y

Y

12

22

11

2

12

22

11

100

01

01

1E

E

E

v

v

v

v

EY

2,1,01212 ,,

jibuv

Eu

v

Eijii

Yjji

Y admission.edhole.com

Plane Strain (b) (Cont'd)

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

GE

vE

vvE

vE

vvE

vE

Y

Y

Y

22

12

11

11

121212

112222

221111

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

2,1,0211212 ,,

jibu

vv

Eu

v

Eijij

Yjji

Y



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

l

l0

engineering strain

E


Linear Elastic Solid If is Cauchy tensor and is small strain tensor, then in general,

ij ijE

ij Cijkl Ekl

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


Linear Elastic Solid

Show that if for a linearly elastic solid, then

Solution:Since for linearly elastic solid , therefore

Thus from , we have

Now, since

Therefore,

ij U

E ijklijijkl CC

ij Cijkl Ekl

ij

E rs

Cijrs

ij U

E ij ijrsijrs EE

UC

2

rsijijrs EE

U

EE

U

22

klijijkl CC


Linear Elastic Solid (cont.)

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


Linear Elastic Solid (cont.)

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


Linear Isotropic Solid

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


Linear Isotropic Solid

Thus for isotropic material

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

ij Cijkl Ekl

(ijkl ik jl il jk )Ekl

ijkl Ekl ik jl Ekl il jk Ekl

ij Ekk 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

Relationship between Youngs Modulus EY, Poisson's Ratio Shear modulus =G and Bulk Modulus k

We know that

So we have

Also, we have

ij eij 2E ij

kk 3 2 e or e 1

3 2 kk

E ij 1

2 ij

3 2

kkij


Relationship between EY, =G and k (Cont'd)

vE

vvv

vkEvEE

kE

E

v

v

v

Ek

v

vk

v

Ev

kv

E

E

v

v

vv

vEvkEvvE

Y

YYY

Y

YY

Y

Y

YY

YY

122

2131223

33213

12

2133

212

213

12

1

3

3

2

21

2

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, admission.edhole.com

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