Gregory Washington, Ph.D.Farzad Ahmadkhanlou, Ph.D., P.E.

GLOBEX 20191

Smart Materials and Intelligent SystemsModule 2 – Mathematical Preliminaries

2

Preliminaries: Tensors and Vectors• Much of the mathematical theory utilized in the study of smart

materials involves physical quantities that are independent of any

particular coordinate system that may be used to describe them

• These quantities however are most conveniently described by referring

to a particular set of coordinates. In order to mathematically represent

this entity, tensors are used.

• A Tensor is defined as an operator that maps (or transforms) vectors

into other vectors.

• Tensors are independent of any coordinate system, yet they may be

specified in a particular coordinate system by a certain set of quantities

known as components.

3

Preliminaries: Matrices• A matrix is an m × n array of numbers, (m is related to the number

of rows and n is related to the number of columns) e.g.

• A transpose of a matrix (A) is given by AT or A’

• It is obtained by swapping the rows and columns, e.g.

2 03 −14 2

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

3x2! "# $#

1000

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

4 x1!"#

−1 3 1 2 00 2 0 1 00 x1 0 0 1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

A =2 03 −14 2

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥,A ' = 2 3 4

0 −1 2⎡

⎣⎢

⎤

⎦⎥

4

Preliminaries: Matrices• To multiply matrices A · B, we require the number of columns in the

left matrix to be the number of rows in the right matrix. Suppose

we had a row vector A and a column

• A transpose of a matrix (A) is given by AT or A’

• It is obtained by swapping the rows and columns, e.g.

A =2 03 −14 2

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥,A ' = 2 3 4

0 −1 2⎡

⎣⎢

⎤

⎦⎥

a = a11 a12 a13 a14⎡⎣

⎤⎦ B =

b11b12b13b14

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

5

Preliminaries: Tensors and Vectors• Cartesian coordinates: Coordinate patterned after the two and three

dimensional orthogonal coordinate systems of Analytic geometry

• Curvilinear Coordinates: coordinate systems other than Cartesian

• Tensors that give transformations from one homogeneous coordinate

system to another are called Cartesian Tensors or Tensors (we will focus

on these).

• Tensors that describe transformations between arbitrary curvilinear

coordinate systems are called General Tensors.

• Tensors are defined by rank or order.

• In 3-D Euclidean space the number of components is described based on

(3N) where N is the order of the Tensor.

• A zero order tensor is called a scalar and has only one coordinate.

6

Tensor/Vector Types

• Tensors/Vectors should not be confused with matrices. Tensors have components with respect to a defined basis (or coordinate system).

ui =u1u2u3

ì

í ï

î ï

ü

ý ï

þ ï , i =1,2,3

s ij =s11 s12 s13

s22 s23s33

é

ë

ê ê ê

ù

û

ú ú ú , i, j = 1,2,3

Cijkl

1st Order Tensor: Vector

2nd Order Tensor:

4th Order Tensor:

31 = 3 components

32 = 9 components

34 = 81 components

7

Vector Basics: Einstein Convention• Tensors and vectors are described using indicial notation.

• In indicial notation there are range indices and summation indices. Range indices

appear only once on each side of the equation. Summation indices appear twice (but

can be repeated no more than one time) on one side of the equation.

• Whenever an index is repeated once it is a dummy index representing a summation

with the index running through the integers 1,2…n.

Si = cijTj

aij x j ¹ akj x j Unless (i = k)

Example: For n = 3 expand

Si = cijTj

S1 = c11T1 +c12T2 +c13T3S2 = c21T1 +c22T2 +c23T3S3 = c31T1 +c32T2 +c33T3

8

Vector Basics: Einstein Convention• Expressions that have an index that is repeated more

than once are not defined within the Einstein

convention.

• In these cases a summation must be used. Thus the

following expression must retain its summation sign

• Expressions that have more than one summation index

are called multiple sums

aibixii=1

n

å

crsxrxs( ) - Double Sum

9

Vector Basics: Einstein Convention• Example: For n=3, solve the expression for T’

12

¢ T ij = aikajlTkl

T12 = a1ka2lTkl= a11a2lT1l + a12a2lT2l + a13a2lT3l= a11a21T11+ a12a21T21+ a13a21T31+ a11a22T12 + a12a22T22 + a13a22T32+ a11a23T13 + a12a23T23 + a13a23T33

10

yi = aij x j

Vector Basics: Einstein Convention

• Proper method: First identify any dummy indices in the expression to

be substituted that coincide with indices in the main expression. Then

change these indices to non duplicated characters

• Step 1. (dummy index j is duplicated)

• Step 2. (change dummy index from j to r

•Substitutions:–We have to change dummy variables to permit proper substitutions.

Q = bij yix j

Q ¹ bijaij x j x j Improper method

11

yi = aij x j

Vector Basics: Einstein Convention

• Example

• Step 1. (dummy index j is duplicated)

• Step 2. (change dummy index from j to r)

• Step 3. Make substitution and rearrange

Q = bij yix j

Q= bij airxr( )x j = bijairxrx j

yi = air x r

12

Vector Basics: Kronecker Delta• This symbol has the effect of annihilating the “off

diagonal” terms in a double summation

δ rj xr( )x j

dij =1 i = j0 i ¹ jì í î

dij =d ji

Example: Solve the following expression

δ rj xr( )x j = x1x1 + 0x1x2 + 0x1x3 + 0x2x1 + x2x2+0x2x3 + 0x3x1 + 0x3x2 + x3x3

δ rj xr( )x j = x1x1 + x2x2 + x3x3

13

Vector Basics: Indentities• Be careful of the manner in which the distributive, associative, and

commutative laws of addition and multiplication are used.

aij xi + yj( ) ≠ aij xi + aij yjaij xiyj ≠ aij yix jaij + aji( )xiyj ≠ 2aij xiyj

aij x j + yj( ) = aij x j + aij yjaij xiyj = aij yj xiaij + aji( )xix j = 2aij xix j

aij − aji( )xix j = 0

14

Mathematical Preliminaries• Mathematical notation and symbols

�

∈ Belongs to

�

∀ For all

�

⇒ Implies

�

⇔ If and only if

�

Rn N – dimensional real vector space

�

∃ There exists

�

! Maps an element of a vector space to element of another

�

→ Maps the whole vector space

�

∍ Such that

$ -v( ) ' v+ -v( )= 0Ex.

There exists a negative vector (-v) such that (v+ (-v)) = 0

15

Mathematical Definitions (vectors)• Physical quantities which possess both magnitude and direction (such as force

and velocity) may be represented in three dimensional space by directed line

segments that obey the parallelogram law of addition.

• Such directed line segments are the geometrical representations of first order

tensors and are called vectors.

a b c de

Arbitrary

VectorUnit Vector Equivalent

Vectors

16

Mathematical Definitions (Vector Space)• We will define a vector space (X) to be a set or collection of objects (vectors or

functions) for which the following statements hold

For a vector space X,

"u,v,wÎ X

"a,b Î scalar field

u + v Î X, auÎ Xu + v = v + uu + v + w( )= u + v( )+ w$0 ' u + 0 = u$ -u( )' u + -u( )= 0

a bu( )= ab( )u1u = ua + b( )x = ax + bx

a x + y( )= ax + ay

Vector addition is commutative

Vector addition is associative

Vector multiplication is associative

Additive Identity (existence of origin)

Additive Inverse

Multiplicative Identity

Closure

Multiplication by scalars is distributive

17

Mathematical Definitions• The real vector space (Rn

) consists of real vectors

and has dimension n. Where n is an arbitrary

positive integer.

• For example we can have

• R2= 2-dimensional vector space (plane)

• R3= 3-dimensional vector space (3-D space)

• When n > 3 the space is often called hyper-

dimensional or a hyper-plane.

18

Mathematical Definitions• A list is a finite length, ordered collection of n objects (vectors in

this class) that is separated by commas and surrounded by

parentheses. The list of length n looks like

• (v1, v2, …, vn)

• A linear combination of a list (v1,….,vm) of vectors in V is a vector of

the form

• V = a1 v1+…+am vm, where a1,…, am R.

• The set of all linear combinations of (v1,….,vm) is called the span of

(v1,….,vm), denoted span(v1,….,vm). In other words,

• span(v1,….,vm)={ a1 v1+…+am vm : a1,…, am Î R}. The set of vectors that

span a space don’t have to be independent.

• A basis of V is a list of vectors in V that is linearly independent and

spans V. For example,

• ((1, 0,…, 0), (0, 1, 0,…, 0),…,(0, …,0, 1)) is a basis of Rn, called the

standard basis of Rn. For n = 3 this is the Cartesian coordinate system

19

Vector Basics: Basis• As stated earlier vector quantities must have a specified basis (or coordinate system). This is

evidenced by examining the following three different basis.

Orthonormal Basis Orthogonal Basis Nonorthogonal Basis

Perpendicular

elements all of

length = 1

Perpendicular

elements with

at least one with

length ≠ 1

Not perpendicular

elements with at

least one with

length ≠ 1

20

Vector Inner (Dot) Product

x1 ⋅x2 = x1x2 cos θ( )

Vector Cross Product

x3 = x1 × x2 = −x2 × x1 = x1x2 sin(θ )e

e

Vector Examples

• Scalar triple product is a dot product of two vectors, one of which is a

cross product

• Vector triple product is a cross product of two vectors, one of which

is a cross product

a ⋅ b × c( ) = a × b( ) ⋅c = a ⋅b × c = λ

a × b × c( ) = a ⋅c( )b − a ⋅b( )c = w

21

V. Basics - 2nd Order Tensors• In mechanics we generally use first order tensors to represent column and row matrices and

second order tensors to represent square matrices.

• Not merely just for bookkeeping. Tensors greatly simplify rigid body kinematics and dynamics

and many mechanics problems where rotations between coordinate systems are used to

simplify the mathematics

• A Tensor is defined as an operator that maps vectors into other vectors. A linear map from one 3-dimensional vector space to another is a defined as a second order tensor T.

• Tensors obey addition and scalar multiplication as follows:

T v+ w( )=Tv +Tw

T av( )=aTv

For all

v,w in R3and scalars a

Linearity

22

V. Basics - 2nd Order Tensors• In this class Tensors obey the following attributes of a vector space

For a vector space X,

"u,v,wÎ X

"a,b Î scalar field

u + v Î X, auÎ Xu + v = v + uu + v + w( )= u + v( )+ w

$0 ' u + 0 = u$ -u( )' u + -u( )= 0

a bu( )= ab( )u1u = u

Vector addition is commutative

Vector addition is associative

Vector multiplication is associative

Additive Identity

Additive Inverse

Multiplicative Identity

23

• In order to understand 2nd order tensor quantities we will

introduce basic vector quantities in this example

• The coordinate system is orthogonal (axes are perpendicular) and

normalized (magnitude = 1). Thus they are orthonormal.

• The basis vectors are defined by ei (i = 1, 2, 3)

e i =1

r = e ixi = e1x1+ e2x2 + e3x3

ei × ej = dij

dij is the Kronecker delta

Vector Basics

24

V. Basics: Coordinate Rotations• Given the vector representation in the figure

r = ¢ e l ¢ x l = e ixi

¢ e m × r = ¢ e m × ¢ e l ¢ x l = dml ¢ x l = ¢ x m

ei × ej = dij

e'l ×e'm = dlm

(1)

(2)

Dotting both sides of (1) with

e'mLeft side

Right side

¢ e m × e ixi = amixi (3)

Equating (2) to (3) yields

x'm= amixi

x j= alj x'lSimilarly,

25

V. Basics: Coordinate Rotations

• The a’s listed on the previous page are called the direction cosines.

• The direction cosines are used for rotations of coordinates and are

defined in the following manner.

Transformation Order (Rank) Number of Transforms ′ u i = aiju j 1st order (1) ija ′ T ij = aika jlTkl 2nd order

(2)

�

aik

�

a jl Cijkl = aima jna koalpCmnop 4th order (4)

�

aim,a jn ,ako,alp

¢ x 1 = x1 cosq + x2 sinq

¢ x 2 = -x1 sinq + x2 cosq

x'3= x3

aij =cos q( ) sin q( ) 0- sin q( ) cos q( ) 00 0 1

é

ë

ê ê ê

ù

û

ú ú ú

2-D Rotation

26

V. Basics: Coordinate Rotations

• For 3-Dimensional rotations

r = ¢ e l ¢ x l = e ixi (1)

r = r × ei( )eiSince r is arbitrary we can set r equal to

e'l

(2)

e'l = e'l ×e i( )e i

e'1= e'1 ×e1( )e1 + e'1 ×e2( )e2 + e'1 ×e3( )e3e'2= e'2 ×e1( )e1 + e'2 ×e2( )e2 + e'2 ×e3( )e3e'3= e'3 ×e1( )e1 + e'3 ×e2( )e2 + e'3 ×e3( )e3

e'1e'2e'3

é

ë

ê ê ê

ù

û

ú ú ú =a11 a12 a13a21 a22 a23a31 a32 a33

é

ë

ê ê ê

ù

û

ú ú ú

e1e2e3

é

ë

ê ê ê

ù

û

ú ú ú

a11 = cos e1, ¢ e 1( ), a12 = cos e2, ¢ e 1( ), a13 = cos e3, ¢ e 1( ), etc

(3)

The a’s are called the direction cosines

27

Two broad Approaches used to Describe Actuator Behavior

1. Statistical mechanics (microscopic description)

• Good for understanding the physics

• Complex to use in real time engineering applications

2. Phenomenological method (macroscopic description)

• Thermodynamic phenomenological theories (Mueller, Devonshire, Smith)

• Semi-atomic phenomenological theories (Weiss)

3. General (input/output) phenomenological theories ( Preisach)

• Limitations (single actuator, congruency, Amplifier effects, complex loading)

• Too many stored values

28

Energy Based Actuator Modeling• One may ask how are the actuator relationships developed for

piezoceramic, electrostrictive, piezomagnetic and magnetostrictiveactuators are developed.

• We can develop scalable actuator relationships from Thermodynamic formulations

• If an energy based formulation is used from the onset greater synergy can be employed in subsequent models since energy is a generic quantity

• Features• Scalability• High Fidelity• Energy basis• Can easily incorporate novel metrics (e.g. volumetric energy density), Easy

Applicability to nonlinear systems

29

Actuator Modeling (Definitions)• We will the Thermodynamic based modeling approach in the class. An number

of definitions are needed:• Crystal: We define the crystal as the actuated or sensing element• Internal Energy(U): The total amount of energy unaccounted for by gravitational potential

energy and kinetic energy. When a spring is compressed energy is stored in the spring. When a battery is charged the energy stored in the battery increases. In each of these examples the system energy is lumped into the internal energy.

• Heat Transferred (Q) the heat (or energy other than work) given to the unit volume of the crystal.

• Work (W): Work is done by system on its surroundings if the sole effect on everything external to the system could have the effect of raising a weight. There are many forms of work (extension of bar, stretching of a liquid film, work done by magnetic and electric fields, etc.

30

Actuator Modeling (Definitions)• The condition of a material can be described totally in terms of intensive and

extensive parameters• Extensive parameters can vary with time and are related to internal conditions.

Mass, volume, entropy and polarization are all extensive. They are additive and their values are dependent on size or amount.

• Intensive parameters are not additive and their values are independent of size or amount. These parameters can vary with both position and time. Field, temperature, and stress are all intensive and these values lead to external conditions.

31

Actuator Modeling (Definitions)• Property: a macroscopic characteristic of a system such as mass, volume,

and energy to which a numerical value can be assigned at a given time without knowledge of the history of the system.

• State: The condition of a system as described by its properties.• Exact (Perfect) Differential: The change in a property between two states

depends in no way on the details of process linking the two states.• For example the change in volume between two states can be determined by

integrating the differential dV, without regard for the details of the process.

32

Energy Based Actuator Modeling

• The work terms and the second law are substituted into the first law• This gives a general framework that can be applied to any actuator system

dU = dQ + dW

dS ³ dQT

!dW = Tijdsij + EmdDm + HldBl + EM(terms) + FM(terms) +K

!!

dU =TdS +TijdSij +EmdDm +HldBl +EM(terms) +FM(terms) +"

First Law of Thermodynamics

Internal Energy

Work

Second Law of Thermodynamics

(1)

(2)

33

Energy Based Modeling - Con’t

• The energy functions determine what independent variables will be used and thus what constitutive equations will be developed

Energy Type Function Name Indep. Variables

Helmholtz Free Energy

T, sij, D, B

Gibbs Free Energy

T, Tij ,E, H

Elastic Gibbs T, Tij, D, B

Electric Gibbs T, sij, E, H

A = U - TS

G =U -TS -TijSij -EmDm -HmBm

G1 =U -TS -Tijsij

G2 =U -TS -EiDi -HiBi

34

Energy Based Modeling - Con’t• The most useful set of constitutive relationships

are those that have stress (Tij) and electric field (E) or stress (Tij) and magnetic field (H) as independent variables, thus we will use Gibbs free energy as a function.• In this formulation we will model quasi-static

behavior. Dynamic modeling will be incorporated in future work.• Substituting equation (2) into the Gibbs free

energy expression gives the following

35

Energy Based Modeling - Con’t

• Since the Gibbs energy is a perfect differential we have

dG =-SdT - sijdTij -DmdEm -BmdHm

ThermalComponent

ElasticComponen

t

PiezoelectricComponent

Piezomagnetic Component

(3)

(4)

36

Energy Based Modeling - Con’tComparing (3) and (4) yield:

S = - ¶G¶Tæ è ç

ö ø ÷ Tij ,E ,H

, Sij = - ¶G¶Tij

æ

è ç ç

ö

ø ÷ ÷ T ,E ,H

, Dn = - ¶G¶En

æ

è ç

ö

ø ÷ Tij ,T ,H

, Bl = - ¶G¶Hl

æ

è ç

ö

ø ÷ Tij ,T ,E

(5)

Defining (G) using a Taylor’s series expansion one gets

G = Go +∂G∂T

⎛⎝⎜

⎞⎠⎟Qo

θ + ∂G∂Tij

⎛

⎝⎜⎞

⎠⎟Qo

Tij +∂G∂Em

⎛⎝⎜

⎞⎠⎟Qo

Em +∂G∂Hk

⎛⎝⎜

⎞⎠⎟Qo

Hk

12

∂ 2G∂T 2

⎛⎝⎜

⎞⎠⎟Qo

θ 2 + ∂ 2G∂Tij ∂Tpq

⎛

⎝⎜⎞

⎠⎟Qo

TijTpq +∂ 2G

∂Em ∂En

⎛⎝⎜

⎞⎠⎟Qo

EmEn +∂ 2G

∂Hk ∂Hl

⎛⎝⎜

⎞⎠⎟Qo

HkHl

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥+…

+ ∂ 2G∂T ∂Tij

⎛

⎝⎜⎞

⎠⎟Qo

Tijθ + ∂ 2G∂Tij ∂Em

⎛

⎝⎜⎞

⎠⎟Qo

TijEm +∂ 2G

∂T ∂Em

⎛⎝⎜

⎞⎠⎟Qo

θEm +

∂ 2G∂T ∂Hk

⎛⎝⎜

⎞⎠⎟Qo

Hkθ + ∂ 2G∂Tij ∂Hk

⎛

⎝⎜⎞

⎠⎟Qo

HkTij +∂ 2G

∂Em ∂Hk

⎛⎝⎜

⎞⎠⎟Qo

HkEm

T -To( )=q

Eo =so =Ho = 0

Given:

(6)

Subscripts indicate the variables to be held constant

37

Energy Based Modeling - Con’t

• The final constitutive relationships can be determined by applying equations (5) to equation (6)

Sij = ¶ 2G¶Tij¶Tpq

æ

è ç ç

ö

ø ÷ ÷ Qo

Tpq + ¶ 2G¶T¶Tij

æ

è ç ç

ö

ø ÷ ÷ Qo

q + ¶ 2G¶Tij¶Em

æ

è ç ç

ö

ø ÷ ÷ Qo

Em + ¶ 2G¶Tij¶Hk

æ

è ç ç

ö

ø ÷ ÷ Qo

Hk + h.o.t

Dm = ¶ 2G¶Tij¶Em

æ

è ç ç

ö

ø ÷ ÷ Qo

Tij + ¶2G¶T¶Em

æ

è ç

ö

ø ÷ Qo

q + ¶ 2G¶Em¶En

æ

è ç

ö

ø ÷ Qo

Em + ¶ 2G¶En¶Hk

æ

è ç

ö

ø ÷ Qo

Hk + h.o.t

Bm = ¶ 2G¶Tij¶Hm

æ

è ç ç

ö

ø ÷ ÷ Qo

Tij + ¶ 2G¶T¶Hm

æ

è ç

ö

ø ÷ Qo

q + ¶ 2G¶En¶Hm

æ

è ç

ö

ø ÷ Qo

En + ¶ 2G¶Hk¶Hl

æ

è ç

ö

ø ÷ Qo

Hk + h.o.t

(7)

(8)

(9)

38

Energy Based Modeling - Con’t• Neglecting magnetic effects and higher order terms and assuming an isothermal

condition

Sij =∂ 2G

∂Tij ∂Tpq

⎛

⎝⎜⎞

⎠⎟Qo

Tpq +∂ 2G

∂Tij ∂Em

⎛

⎝⎜⎞

⎠⎟Qo

Em = sijpqE Tpq + dmijEm

4th order tensor representing compliances at constant field and temperature

3rd order tensor representing the coupling between field and strain (piezoelectric d constant)

Notice that (ij) is attached to the dependent variable (S), thus there must be an ij in each term on the right and the other terms must be dummy variables.

39

Energy Based Modeling - Con’t

• Neglecting magnetic effects and higher order terms and assuming an isothermal condition

• Neglecting electric effects and higher order terms and assuming an isothermal condition

Sij =∂ 2G

∂Tij ∂Tpq

⎛

⎝⎜⎞

⎠⎟Qo

Tpq +∂ 2G

∂Tij ∂Em

⎛

⎝⎜⎞

⎠⎟Qo

Em = sijpqE Tpq + dmijEm

Dm = ∂ 2G∂Tij ∂Em

⎛

⎝⎜⎞

⎠⎟Qo

Tij +∂ 2G

∂Em ∂En

⎛⎝⎜

⎞⎠⎟Qo

Em = dmijTij + εmnT En

Bm = ∂ 2G∂Tij ∂Hm

⎛

⎝⎜⎞

⎠⎟Qo

Tij +∂ 2G

∂Hm ∂Hn

⎛⎝⎜

⎞⎠⎟Qo

Hn = dmijE Tij + µmn

T ,EHn

Values agree with the IEEE standards

Dm = ∂ 2G∂Tij ∂Em

⎛

⎝⎜⎞

⎠⎟Qo

Tij +∂ 2G

∂Em ∂Hn

⎛⎝⎜

⎞⎠⎟Qo

Hm = dmijTij +mmnT Hn

(10)

(11)

40

Energy Based Modeling - Con’t• Nonlinear relationships (Anhysteretic). If one were to expand the previous

Taylor’s series to include higher order terms one can extrapolate many of the non-linear effects (Anomalous polarization, saturation, temperature effects, electrostrictives, etc.). • Once the Free Energy is expanded in a Taylor Series (higher order elasticity tensors

are assumed small). The result is the following:

G = − 12

∂ 2G∂Tij ∂Tpq

⎛

⎝⎜⎞

⎠⎟Qo

TijTpq +∂ 2G

∂Em ∂En

⎛⎝⎜

⎞⎠⎟Qo

EmEn

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥− ∂ 2G

∂Tij ∂Em

⎛

⎝⎜⎞

⎠⎟Qo

TijEm

− 13

∂ 3G∂Tij ∂Tpq ∂Trs

⎛

⎝⎜⎞

⎠⎟Qo

TijTpqTrs +∂ 3G

∂Em ∂En ∂Eo

⎛⎝⎜

⎞⎠⎟Qo

EmEnEo

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥−

14

∂ 4G∂Em ∂En ∂Eo ∂Ep

⎛

⎝⎜⎞

⎠⎟Qo

EmEnEoEp −∂ 3G

∂Em ∂En ∂Tij

⎛

⎝⎜⎞

⎠⎟Qo

EmEnTij −

∂ 3G∂Tij ∂Tpq ∂Em

⎛

⎝⎜⎞

⎠⎟Qo

TijTpqEm −∂ 4G

∂Em ∂En ∂Tij ∂Tpq

⎛

⎝⎜⎞

⎠⎟Qo

EmEnTijTpq −…h.o.t

41

Mechanics of Materials

• We will use the general constitutive relationship found in engineering mechanics. Towards that end stresses and strains will be considered. • Stresses: Man made quantities used in

failure theories. Defined mathematically as force per unit area.

• Strains: are actual physical quantities related to displacement. Defined mathematically as the change in length over the original length.

T =σ = FA

S = ε = ΔLL

Strain

Stress

42

Mechanics of Materials

• The most general form of Hooke’s law is defined by the use of a 4th order tensor.• i, j, k, and l represent indices from 1 to 3. S is the

strain, s is the compliance and C is the elastic stiffness

Tij = CijklSkl Sij = sijklTkl

σ[ ] =σ 11 σ 12 σ 13

σ 21 σ 22 σ 23

σ 31 σ 32 σ 33

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥; ε[ ] =

ε11 ε12 ε13

ε21 ε22 ε23

ε31 ε32 ε33

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥;

Using mechanical engineering notation σ ij = Cijklε kl ε ij = sijklσ kl

43

Mechanics of Materials: Hooke’s Law

• Since the compliance and stiffness tensors are fourth order, there are at most 81 constants describing a material in a state of stress.

• Since we are assuming that there is no net moment acting on a stressed element we can reduce the number of constants to 54. When symmetry of the strains are considered the number of constants can be reduced to 36.

σ ij = Cijklε kl

σ[ ] =σ 11 σ 12 σ 13

σ 21 σ 22 σ 23

σ 31 σ 32 σ 33

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥; Where

σ 12 =σ 21

σ 13 =σ 31

σ 23 =σ 32

ε[ ] =ε11 ε12 ε13

ε22 ε23

ε33

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥;

44

Mechanics of Materials: Hooke’s Law

• In order to write the elastic tensors in the form of a matrix, a compressed notationis developed. This notation consist of replacing the (ij) or (kl) by p or q, where

σ 11

σ 22

σ 33

σ 23

σ 13

σ 12

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

=

C11 C12 C13 C14 C15 C16C21 C22 C23 C24 C25 C26

C31 C32 C33 C34 C35 C36

C41 C42 C43 C44 C45 C46

C51 C52 C53 C54 C55 C56

C61 C62 C63 C64 C65 C66

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

ε11ε22ε33ε23ε13ε12

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

σ 1

σ 2

σ 3

σ 4

σ 5

σ 6

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

=

C11 C12 C13 C14 C15 C16C21 C22 C23 C24 C25 C26

C31 C32 C33 C34 C35 C36

C41 C42 C43 C44 C45 C46

C51 C52 C53 C54 C55 C56

C61 C62 C63 C64 C65 C66

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

ε1ε2ε3ε4ε5ε6

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

i, j,k,l =1,2,3; p,q =1,2,3,4,5,6

�

ij or kl

�

p or q 11 22 33

23 or 32 31 or 13 12 or 21

1 2 3 4 5 6

45

Mechanics of Materials: Hooke’s Law

• Strain energy considerations reduce the amount from 36 to 21. We know that the strain energy (W) is an exact differential. This means that the following is true

dW = ∂W∂ε i

dε i +∂W∂ε j

dε j = Mdε i + Ndε j

∂∂ε j

∂W∂ε i

⎛⎝⎜

⎞⎠⎟= ∂∂ε i

∂W∂ε j

⎛

⎝⎜⎞

⎠⎟

W = 12Cijε iε j →σ i = Cijε j =

∂W∂ε i

and Cij =∂σ i

∂ε j

= ∂ 2W∂ε j ∂ε i

M and N have continuous first partial derivatives

Cij =∂ 2W∂ε j ∂ε i

= ∂ 2W∂ε i ∂ε j

= Cji

Applying the definition of work

Incorporating (a) into (b) gives the following:

(a)

(b)

46

Hooke’s Law: Anisotropic Materials• We now have a system with 21 constants. This type of material is defined

as anisotropic. Typical anisotropic materials are concrete and glass (on a microscopic scale). The question that now must be answered is what happens if there is symmetry in the atomic structure of a material? In general our 21 constant system looks like the following:

σ 1

σ 2

σ 3

σ 4

σ 5

σ 6

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

=

C11 C12 C13 C14 C15 C16C22 C23 C24 C25 C26

C33 C34 C35 C36

C44 C45 C46

C55 C56

C66

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

ε1ε2ε3ε4ε5ε6

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

47

Hooke’s Law: Monoclinic Materials• Let's assume that the 1-2 plane is the plane of symmetry. (This means

looking in the 3 direction is the same as looking in the -3 direction.)

¢ s ij = aikajlskl aij =1 0 00 1 00 0 −1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

′σ 11 =σ 11 = a1ka1lσ kl

= a11a1lσ 1l + a12a1lσ 2l + a13a1lσ 3l

= a11a11σ 11 + a12a11σ 21 + a13a11σ 31

+ a11a12σ 12 + a12a12σ 22 + a13a12σ 32

+ a11a13σ 13 + a12a13σ 23 + a13a13σ 33

Applying one plane of symmetry one can reduce the number of constants from 21 to 13. A material with 13 material constants is called monoclinic. There are no naturally occurring monoclinic materials, but composite materials can be formed synthetically

Applying Rotations

′σ 11 = a11a11σ 11, ′σ 12 = a11a22σ 12, ′σ 13 = a11a33σ 13

for i=j=1,

σ '11 =σ 11

σ '12 =σ 12

σ '13 = −σ 13

Similar of all other sij

48

Hooke’s Law: Monoclinic Materials• The procedure for reduction from 21 - 13 is as follows:

Expanding the tensor rotation relationships for i,j,k = 1,2,3 and applying the direction cosines for the rotation

′σ ij = aikajlσ kl aij =1 0 00 1 00 0 −1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

σ '11 =σ 11, σ '12 =σ 12, σ '13 = −σ 13, σ '21 =σ 21, σ '22 =σ 22, σ '23 = −σ 23, σ '31 = −σ 31, σ '32 = −σ 32, σ '33 =σ 33

Similar for strains

(1)

ε '11 = ε11, ε '12 = ε12, ε '13 = −ε13, ε '21 = ε21, ε '22 = ε22, ε '23 = −ε23, ε '31 = −ε31, ε '32 = −ε32, ε '33 = ε33

The original and rotated stresses are subject to Hooke’s law (in compressed notation)

σ 'i = Cijε ' j = C11ε '1+C12ε '2+C13ε '3+C14ε '4+C15ε '5+C16ε '6

σ i = Cijε j = C11ε1 +C12ε2 +C13ε3 +C14ε4 +C15ε5 +C16ε6 (3)

(4)

(2)

49

Hooke’s Law: Monoclinic Materials

• When this methodology is applied to all of the stresses and strains, 8 constants are equal to zero and we have a material defined by 13 constants

Applying (1) to (3) and (5) yields

σ '11 =σ 11 ⇒σ '1−σ 1 = 0

C14e4 +C15e5 = 0

σ 'i = Cijε ' j = C11ε1 +C12ε2 +C13ε3 −C14ε4 −C15ε5 +C16ε6

Which implies

The strains are independent so

Applying (2) to (4)

C14 =C15 = 0

50

Hooke’s Law: Orthotropic Materials• Now what if there are two planes of material symmetry? The first transformation is

the same as we have just seen.• This transformation reduces the constants from 13 to 9. A material with 9 constants

is defined as orthotropic.

aij =1 0 00 −1 00 0 1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

1st plane of symmetry13 constants

2nd plane of symmetry9 constants

aij =1 0 00 1 00 0 −1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

Monoclinic Orthotropic

51

Hooke’s Law: Transversely Isotropic• Most piezoelectric materials have three planes of symmetry with one

plane being isotropic. This is described by 5 constants. A system with 5 constants is called Transversely Isotropic. This means that an arbitrary rotation is permitted.

• Isotropic: 3 planes of material symmetry all planes are isotropic. Isotropy is different in the following manner. Isotropy means that if you stand in the center of a piece of isotropic material and look in any direction what you see will be exactly the same.

Classification # of const Type of material Anisotropic 21 Concrete, Glass (micro) Monoclinic 13 Synthetic composites Orthotropic 9 Wood, Barytes (BaSO4)

Triagonal Syngony 6 Calcite, Quartz (SiO2) Transversly Isotropic 5 Beryl, Piezoceramics

Isotropic 2 Most Metals

aij =cosθ sinθ 0−sinθ cosθ 00 0 1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

Thank You!

Obrigado!

¡Muchas Gracias!

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Merci Bien!

Asante Sana!

СПАСИВО!

Vielen Dank!

Grazie!

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Tak!