Structural analysis: matrix question

[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?

 

[Nino MMP]

In this case of plane strain, σ33 is equal to zero. This is because, in a plane strain condition, there are no displacements or displacement gradients in the x3 direction.

 

[Blueshift5]

I understand your concerns about not fully understanding the problem due to language barriers and missing information in the course notes. I suggest seeking clarification from your instructor or colleagues who may have a better understanding of the material. In the meantime, I can provide some guidance on the structural analysis and derivation of the "rheology" matrix [D].

Firstly, the constitutive equation provided is a representation of Hooke's law for an isotropic material, which relates stress (σ) to strain (ε). The terms R and S represent temperature and pressure dependent material properties, respectively. In order to derive the [D] matrix, we need to express the constitutive equation in terms of the stress and strain matrices as follows:

{
σ11
σ22
σ12
}
=[D]
{
ε11
ε22
ε12
}

In this case, we are dealing with plane strain, which means there are no displacements or displacement gradients in the x3 direction. This can be represented in the stress and strain matrices as follows:

{
σ11
σ22
σ12
σ33
}
=[D]
{
ε11
ε22
ε12
0
}

Note that σ33 has been added as a zero value, as there is no strain in the x3 direction. Now, we can substitute the constitutive equation into the stress and strain matrices to obtain the [D] matrix:

{
σ11
σ22
σ12
0
}
=[D]
{
Rθδ11 – Sε11
Rθδ22 – Sε22
Rθδ12 – Sε12
0
}

From here, we can simplify the [D] matrix by rearranging terms and substituting the definition of cubical dilatation (θ = ε11 + ε22 + ε33) to obtain:

{
σ11
σ22
σ12
0
}
=[D]
{
R(ε11 + ε22 + ε33) – Sε11
R(ε11 + ε22 + ε33) – Sε22
R(ε11 + ε22 + ε33) – Sε12
0
}

This simplified [D] matrix can then be used in your finite element programming code for structural analysis of the newly discovered rheology. I hope this helps you better understand the problem and how to approach it. As for the value of σ33 in this case of plane