Constitutive Relations in SolidsElasticity

H. Garmestani, ProfessorSchool of Materials Science and Engineering

Georgia Institute of Technology

• Outline: • Materials Behavior

Tensile behavior…

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

The Elastic Solid and Elastic Boundary Value Problems (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 Dynamics

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

fixed

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

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 2

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

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

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

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 StressFor 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

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

The Elastic Solid and Elastic Boundary Value Problems

 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

Show that if for a linearly elastic solid, then

Solution:

Since for linearly elastic solid , therefore

Thus from , we have

Now, since

Therefore,

Linear Elastic Solid

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

• Thus for isotropic material

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

Linear Isotropic Solid

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 2E

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,