# 0.6 Tensors, lagrange methods, and hausdorff measure

Presentation Notes Harry

The Stress Tensor:

Informally, tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors (say, linear maps or dot products). A tensor can be represented by a multi-dimensional array of values and the number of indices needed to specify a component in this array is said to be the order of the tensor.

Multi-dimensional arrays:

1. Scalar [a] - this is a tensor of order 0 because we don't need any indices to specify "a" within a 1x1 array

2. Vector [ai ]i=1,...,n - This is a tensor of order 1 because we need 1 index to specify a component of the vector

3. mxn matrix - this is a tensor of order 2

4. Tensors of higher order can be thought of as multidimensional "boxes"

Notice that we can create a tensor of a new order through the multiplication of tensors. For example, the products of a nxmxp tensor with an px1 vector will yield an nxm matrix.

Definition: The stress force on an object is said to be the force per unit area

So, the stress tensor is an order 2 tensor (a matrix) which takes a unit length direction vector normal to the surface of an object as an input and gives the stress force vector (with respect to the given coordinate system) at this surface as an output.

Then, for our problem, which is in 3 dimensions, the stress tensor is a 3x3 matrix with entries Tij, where i,j=1,2,3.

Imagine a beam and say that we would like to calculate the stress vector at a point, p, in the beam due to a force at the endpoint. Then, given a coordinate system, consider an "epsilon-cube" around p. Given an applied force at the endpoint of the beam (which translates to the same force at p) Tij is the stress vector acting on the ith face in the -jth direction - note that opposite faces have the same index. For an explanation of these properties, see [1].

One can check that in the context of a Michell Truss, the stress tensor will be a rank-one matrix and that the system of forces is in equilibrium if the divergence of the stress tensor is the negative of the applied forces. In other words, admissible trusses satisfy

$\nabla \sigma =-F$

where F is the vector sum of the applied forces. See [5] for an explanation and derivation of this equation.

Definition: Strain is the amount of deformation an object experiences as a result of external forces when compared with its original size and shape.

The strain tensor is operates similarly to the stress tensor, but the entries Tij are replaced by partial derivates of the position of a point, p, with respect to movement in the coordinate directions as a result of external forces. In the context of a Michell Truss, the strain tensor will always be a rank-one matrix as well.

One can investigate the relationship between stress and strain via a stress/strain curve.

Euler-Lagrange Equations

How do we optimize functionals? Consider a functional

$I\left(u\right)={\int }_{a}^{b}f\left(x,u\left(x\right),{u}^{\text{'}}\left(x\right)\right)\phantom{\rule{0.166667em}{0ex}}dx.$

over a region containing a and b.

Let u(x) be a minimizer for I(u). Then u(x) should satisfy the Euler-Lagrange equations as follows:

$\frac{d}{dx}\left[\frac{df\left(x\right)}{d{\mathbf{u}}^{\text{'}}\left(\mathbf{x}\right)}\left(x,\mathbf{u}\left(\mathbf{x}\right),{\mathbf{u}}^{\text{'}}\left(\mathbf{x}\right)\right)\right]\phantom{\rule{4pt}{0ex}}=\frac{df\left(x\right)}{d\mathbf{u}\left(\mathbf{x}\right)}\left(x,\mathbf{u}\left(\mathbf{x}\right),{\mathbf{u}}^{\text{'}}\left(\mathbf{x}\right)\right)$

what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
how did you get the value of 2000N.What calculations are needed to arrive at it
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