Topic-Topic-ElasticityElasticity

TensorsTensors

Cartesian tensor of order 1Cartesian tensor of order 1

A cartesian tensor of order 1 is an A cartesian tensor of order 1 is an entity that may be represented by entity that may be represented by components acomponents aii

(i=1,2,3) w.r.t a R.C.C.S ox(i=1,2,3) w.r.t a R.C.C.S oxii and the and the same entity be represented by same entity be represented by aaii’(i=1,2,3) w.r.t R.C.C.S ox’(i=1,2,3) w.r.t R.C.C.S oxii’ which is ’ which is obtained by rotating oxobtained by rotating oxii about an about an angle. Then aangle. Then aii and a and aii’ obey the law’ obey the law

aaii’=l’=lijijaajj

This law is called tensorial law of order This law is called tensorial law of order 1.or it can be written as1.or it can be written as

aaii=l=lij ij aaj’j’

Examples of 1Examples of 1stst order tensorsorder tensors

Every vector is a tensor of order 1.Every vector is a tensor of order 1.

Pf:-AsPf:-As a a=a=a11ee 1 1+a+a22ee 2 2+a+a3 3 ee33

Also Also aa=a=a11’’ee11’+a’+a22’’ee22’+a’+a33’’ee33’’

Therefore Therefore

aeaeii’’=a=a11ee11eeii’+a’+a22ee22eeii’+a’+a33ee33eeii’’

=l=li1i1aa11+l+li2i2aa22+l+l1313aa33

=l=lijijaaj j (j=1,2,3)(j=1,2,3)

But But aeaeii’=a’=aii’’

aaeeii’=a’=a11ee11’e’eii’ +a’ +a22ee22’e’eii’+a’+a33ee33’e’eii’’

=a=aii’’

So we have So we have aaeeii’=a’=aii’=l’=lijijaajj

This shows that vector is a tensor of This shows that vector is a tensor of order 1.order 1.

Velocity is a tensor of order 1.Velocity is a tensor of order 1. Gradient of a scalar pt. function is a Gradient of a scalar pt. function is a

tensor of order 1.tensor of order 1.

Tensor of order 2Tensor of order 2

Let a physical entity be represented by Let a physical entity be represented by aaij ij w.r.t R.C.C.S oxw.r.t R.C.C.S oxi i and same entity and same entity be represented by abe represented by apqpq’ w.r.t R.C.C.S ’ w.r.t R.C.C.S oxoxii’ and if they are related by the law’ and if they are related by the law

aaijij’=l’=lipiplljqjqaapqpq

aaijij=l=lpipillqjqjaapqpq’’

(I,j,p,q=1,2,3)(I,j,p,q=1,2,3)

Sum of Two Sum of Two TensorsTensors

If aIf aijij and b and bijij are two tensors of order are two tensors of order two.then their sum is again a tensor two.then their sum is again a tensor of order two,and it is defined asof order two,and it is defined as

ccijij=a=aijij+b+bijij

(I,j=1,2,3)(I,j=1,2,3)

Difference of Difference of two Tensorstwo Tensors

If aIf aijij and b and bijij are two tensors of order are two tensors of order two then their difference is again two then their difference is again atensor of order two.It is defined as:atensor of order two.It is defined as:

ccijij=a=aijij-b-bijij

(i,j=1,2,3)(i,j=1,2,3)

Scalar InvariantScalar InvariantIf an entity be represented by a w.r.t If an entity be represented by a w.r.t

R.C.C.S oxR.C.C.S oxi i and by a’ w.r.t another and by a’ w.r.t another R.C.C.S oxR.C.C.S oxii’ and a’=a’ and a’=a

then that entity is called scalar then that entity is called scalar invariant and simply scalar.invariant and simply scalar.

Zero TensorZero Tensor Tensor of order n is called zero Tensor of order n is called zero

tensor if all the components of the tensor if all the components of the tensor w.r.t co-ordinate system are tensor w.r.t co-ordinate system are zero.zero.

If all the components of a tensor are If all the components of a tensor are zero w.r.t some co-ordinate system zero w.r.t some co-ordinate system oxoxii then it has zero components w.r.t then it has zero components w.r.t every co-ordinate system oxevery co-ordinate system oxii’.’.

Equality of TensorsEquality of Tensors

Two Tensors A and B are equal if the Two Tensors A and B are equal if the corresponding components of A and B corresponding components of A and B are equal.are equal.

Let A=[aLet A=[aijij]]

B=[bB=[bijij] then ] then

A=BA=B

if aif aijij=b=bijij

(i,j=1,2,3)(i,j=1,2,3)

Product of Two Product of Two TensorsTensors

Let aLet ai i and band bj j & a& app’’ and band bqq’’ are the are the components of first order tensors components of first order tensors relative to two co-ordinate system oxrelative to two co-ordinate system ox i i

and oxand oxii’’

Let cLet cijij=a=aiibbjj

ccpqpq’=a’=app’b’bqq’’

As aAs ai i and band bj j are tensors of order 1 are tensors of order 1

So aSo app’=l’=lpipiaaii

bbqq’=l’=lqjqjbbjj

Multiply these equations:Multiply these equations:

aapp’b’bqq’=l’=lpipillqjqjaaiibbjj

ccpqpq’=l’=lpipillqjqjccijij

This is tensorial law of order two.This is tensorial law of order two.

ConclusionsConclusions If A is a tensor of order If A is a tensor of order αα and B is a and B is a

tensor of order tensor of order αα,then their sum and ,then their sum and difference is again a tensor of order difference is again a tensor of order αα..

If A is a tensor of order If A is a tensor of order αα and B is a and B is a tensor of order tensor of order ββ then their product then their product is a tensor of order is a tensor of order αα++ββ..

ContractionContraction

When we identify two suffixes in a When we identify two suffixes in a Tensor,then this operetion is called Tensor,then this operetion is called Contraction.Contraction.

We know that if a and b are two tensors We know that if a and b are two tensors of order 1 then their product is a tensor of order 1 then their product is a tensor of order 2.and it can be written asof order 2.and it can be written as

(a(ab)b)ijij=a=aiibbjj

(i,j=1,2,3)(i,j=1,2,3)

Then we put j=i then this process is Then we put j=i then this process is called contraction and expressed ascalled contraction and expressed as

((aabb))iiii=a=aiibbii

(i=1,2,3)(i=1,2,3)

=a=a11bb11+a+a22bb22+a+a33bb33

DefinitionDefinition Tensor Product:If A is a tensor of Tensor Product:If A is a tensor of

order 2 and C is a tensor of order order 2 and C is a tensor of order 1,then1,then

AAC=aC=aijijcckk

This is a tensor of order 3 and it is This is a tensor of order 3 and it is called called outer productouter product or or tensor tensor product.product.

Inner Product:If A is a tensor of order Inner Product:If A is a tensor of order 3 and C is a tensor of order 1.3 and C is a tensor of order 1.

thenthen

AC=aAC=aijijccjj

is a tensor of order 1.and is a tensor of order 1.and

It is called It is called Inner ProductInner Product i.e i.e I.PI.P

Scalar Product:If A and B are two Scalar Product:If A and B are two tensors of order 2.thentensors of order 2.then

A.B=aA.B=aijijbbijij

is a tensor of order 0.is a tensor of order 0.

It is called Scalar Product.It is called Scalar Product.

Isotropic TensorIsotropic Tensor If we change the frame of refrence or If we change the frame of refrence or

transformation and the values of transformation and the values of components of tensor do not change.components of tensor do not change.

Then this type of tensor is called Then this type of tensor is called Isotropic Tensor.Isotropic Tensor.

It is denoted by It is denoted by δδij.It is also called ij.It is also called ScalarScalar invariant invariant or or isotropic tensorisotropic tensor..

Alternate TensorAlternate Tensor Alternate Tensor Is defined asAlternate Tensor Is defined as

1 if i,j,k1 if i,j,k are in cyclic orderare in cyclic order

ijkijk = -1 if i,j,k are in anticyclic order. = -1 if i,j,k are in anticyclic order.

0 if any two suffixes are equal. 0 if any two suffixes are equal.

ijk ijk is a tensor of order 3.is a tensor of order 3.

123123= = 231231= = 312312=1=1

213213= = 321321= = 132132=-1=-1

222222=0…..=0…..

i.e. out of 27,6 values are not zero.i.e. out of 27,6 values are not zero.

TEST-1TEST-1

State and Prove Law of State and Prove Law of Transfomation in three Transfomation in three dimension.dimension.

Prove that Velocity is a tensor Prove that Velocity is a tensor of order 1.of order 1.

State and Prove Contraction State and Prove Contraction theorem.theorem.

Define Isotropic Tensor and Define Isotropic Tensor and Prove that it is Scalar Invariant.Prove that it is Scalar Invariant.

TEST-2TEST-2

Derive Strain quadric of Cauchy.Derive Strain quadric of Cauchy. Explain Cubical dilatation with diagram.Explain Cubical dilatation with diagram. Derive Saint-Venant’s Equation of Derive Saint-Venant’s Equation of

Compatibility.Discuss them in every Compatibility.Discuss them in every case.case.

Derive Stress-Strain relation for a Derive Stress-Strain relation for a Homogeneous Isotropic Elastic Solid Homogeneous Isotropic Elastic Solid Medium.Medium.

OROR Derive Stress Strain Relation for an Derive Stress Strain Relation for an

Orthotropic Medium.Orthotropic Medium.