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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

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
hi
Loga
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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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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