# Structural analysis: matrix question

1. Mar 27, 2006

### hexa

I'm afraid I have another problem to be solved I don't quiet understand, thanks to my poor english and some information missing in the course notes.

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You are writing a 2-dimensional finite element programming code to be used in the structural analysis of a material with a newly discovered rheology. The constitutive equation, in index notation, for this rheology (in an isotropic medium) is:

σij = Rθδij – Sεij

where θ is cubical dilatation (take θ = ε11 + ε22 + ε33)
δij is Knonecker’s delta
R and S are temperature and pressure dependent material properties.

Derive the “rheology” matrix [D] that you will use in your algorithm, for plane strain (no displacements or displacement gradients in the x3 direction).

Define the 2-dimensional stress and strain matrices {σ} and
{ε} as they are in the course notes (they are not!), i.e.,

direction). Define the 2-dimensional stress and strain matrices {σ} and
{ε} as they are in the course notes (paragraph IV.2.8), i.e.,

{
σ11
σ22
σ12
}
=[D]
{
ε11
ε22
ε12
}

(written as {}=[D]{}

What is the value of σ33 in this case of plane strain?