Orthotropic materials defined by 9 constants

[johnjaypl]

Could some one explain, or give me a pointer to a good explanation, of how the nine constants that are often used to define orthotropic materials are determined.

I understand what E is in each direction.

I understand poisson ratio.

I sort of understand G. (I understand it as the E equilivent for shear- is that right)

I don't understand D1111, D2222, etc.

I seam especially confused by G12, G13, and G23.

Thanks,

John

 

[Andy Resnick]

If I understand what you are asking, the number of independent material constants reflect the amount of symmetry in the material.

Most generally, the stress-strain relationship is tensor in nature:
$$\boldsymbol{\sigma} = \mathsf{c}:\boldsymbol{\varepsilon}$$

It's important to realize that the stress and strain terms themselves can be applied not only to mechanical behavior, but electrodynamic as well via, for example, the Maxwell stress tensor.

writing it out explicitly:

$$\sigma_{ij} = c_{ijk\ell}~ \varepsilon_{k\ell}$$

For mechanics, given a material with *no* symmetry,the 9 stress components are related to the 9 strain components via a 81-component tensor.

Symmetry reduces the number of independent components of the 4th rank tensor.

http://en.wikipedia.org/wiki/Orthotropic_material

About 1/2way down, there's some details.

 

[johnjaypl]

If I understand what you are asking, the number of independent material constants reflect the amount of symmetry in the material.

Most generally, the stress-strain relationship is tensor in nature:
$$\boldsymbol{\sigma} = \mathsf{c}:\boldsymbol{\varepsilon}$$

It's important to realize that the stress and strain terms themselves can be applied not only to mechanical behavior, but electrodynamic as well via, for example, the Maxwell stress tensor.

writing it out explicitly:

$$\sigma_{ij} = c_{ijk\ell}~ \varepsilon_{k\ell}$$

For mechanics, given a material with *no* symmetry,the 9 stress components are related to the 9 strain components via a 81-component tensor.

Symmetry reduces the number of independent components of the 4th rank tensor.

http://en.wikipedia.org/wiki/Orthotropic_material

About 1/2way down, there's some details.

Thanks. I'm so confused I can't even formulate a decent question. Let me try again.

What I'm trying to understand is described in the Orthotropic material section of this link

http://www.engin.brown.edu/courses/En222/Notes/Constitutive/Constitutive.htm

How do I go from understanding E (in 3 directions) v(in three directions) G(in three directions) to C11, c22, c33, c12, c13, c23, c44,c55, c66? What's the basic idea of what's going on here?

Thanks,

John

 

[Andy Resnick]

I didn't see a 'G' on that page, but did you follow the linear elasticity section down to the "Matrix form of the constitutive relations" section? That shows how these tensors can be written more compactly. The section "Elastic Symmetries" explains why some of 'c' components are zero- is that the part you are having trouble with?

 

[johnjaypl]

I didn't see a 'G' on that page,...

I believe they use u (mu) for shear modulus. So:

c44 = u12 = G12
c55 = u13 = G13
c66 = u23 = G23

Is that right?

Is there a formula that allows one to calculate Gij from other properties?

.. but did you follow the linear elasticity section down to the "Matrix form of the constitutive relations" section? That shows how these tensors can be written more compactly. The section "Elastic Symmetries" explains why some of 'c' components are zero- is that the part you are having trouble with?

Well I started with more conceptional missunderstanding than that but at this point I get the idea of what's going on and why you want 9 constants and how the linear equations solve the stress/strain in all directions. So that's progress.

At this point I'm thinking I better go back and make sure I understand all of the basic relationships for isotopic materials. I mostly do but the releationships betwen E G and v are a bit fuzzy. But with some thought I think I can clear that up.

Then maybe I need a book to cover more details of the orthotopic material constants if I need to get that far into it. That is, why is that matrix set up the way it is, etc.

Even though my questions were fuzzy this exchange somehow helped me get over the major problem that I was having.

Thanks,

John