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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

σ = - 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 ( u ) = a b f ( x , u ( x ) , u ' ( x ) ) d x .

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:

d d x [ d f ( x ) d u ' ( x ) ( x , u ( x ) , u ' ( x ) ) ] = d f ( x ) d u ( x ) ( x , u ( x ) , u ' ( x ) )

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
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Source:  OpenStax, Michell trusses study, rice u. nsf vigre group, summer 2013. OpenStax CNX. Sep 02, 2013 Download for free at http://cnx.org/content/col11567/1.2
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