Theroretical derivations of constitutive equations - Constitutive Models
https://polymerfem.com/community/constitutive-models/theroretical-derivations-of-constitutive-equations/
PolymerFEM.com Discussion Boarden-USFri, 19 Apr 2024 05:34:18 +0000wpForo60
https://polymerfem.com/community/constitutive-models/theroretical-derivations-of-constitutive-equations/#post-6795
Wed, 01 Apr 2009 06:14:50 +0000Constitutive ModelsJorgenhttps://polymerfem.com/community/constitutive-models/theroretical-derivations-of-constitutive-equations/#post-67952 follow up questions
https://polymerfem.com/community/constitutive-models/theroretical-derivations-of-constitutive-equations/#post-6789
Tue, 31 Mar 2009 08:44:27 +0000I hope I am not bothering too much.. we all have to learn right. I have differentiated eq. 3.12 wrt gamma, and this leads to: dT12/dgamma= 3*Langevin inv(1/lambda lock)This is not yielding the result I want, since it is a factor three too much. I am regarding the differentiation as 3 chains:x=gammaR=everything not dependent on gammaK=(1+(x^2)/3)^0.5 = dependent on gammaG=lambda lockLang=Langevin invT12=R*(x*K*Lang(K/G)) this needs a lot of chain rulesdT12/d x =1/R * Because we need the derivative at x=0 a lot of terms can be cancelled, i think..(furthermore I can disregard a couple of further chains because of multiplication by x which is zero)Now I am left withdT12/d x = 1/R * [K(x=0)*Lang(1/G)+K(x=0)*Lang(1/G)+K(x=0)Lang(1/G)=1/R*3Lang(1/G)So this is my reasoning and my result is faulty, I am not able to track the fault down. I understand that reading (aka understanding what I am meaning :) ) this kind of equation is difficult, I am hoping it will be possible.2. Another question is the following:With eq 3.23 is it true that B*=?Because now I dont understand why P isnt simply 1/3 tr(B*). If you calculate this then P=1/3(lambda^2+2/Lambda)If I follow your hint (first set S22 to zero for P), I get P=1/lambda. Which is correct leading to eq 3.23I dont understand why the first method does not yield the correct answer. The hydrostatic pressure was always the trace divided by 3. Can you help me with these follow up questions?Kind regards Harm]]>Constitutive Modelsxbzihanhttps://polymerfem.com/community/constitutive-models/theroretical-derivations-of-constitutive-equations/#post-6789Thanks a lot
https://polymerfem.com/community/constitutive-models/theroretical-derivations-of-constitutive-equations/#post-6786
Tue, 31 Mar 2009 01:58:06 +0000I will check out Holtzapfels book. Thanks a lot for your quick and clear reply mr. Bergstrom.:)]]>Constitutive Modelsxbzihanhttps://polymerfem.com/community/constitutive-models/theroretical-derivations-of-constitutive-equations/#post-6786
https://polymerfem.com/community/constitutive-models/theroretical-derivations-of-constitutive-equations/#post-6779
Mon, 30 Mar 2009 22:32:29 +0000 3.16:
This derivation is provided in different text books, for example, Holtzapfelss book.
(2) Eq. 3.17 -> 3.18:
Yes, your equation for I1 is correct. When I apply the chain rule I get Eq 3.18 as written.
(3) Eq. 3.13:
No, beta is not neglected. If you perform the calculation, a lot of terms cancel out.
(4) The Eq. 3.23 was derived using the standard approach:
* Derive expression for S22,
* Set S22=0, and solve for pressure p
* insert expression for p in expression for S11
(5) I recommend that you checkout, for example, Holzapfels book for the details.
-Jorgen]]>Constitutive ModelsJorgenhttps://polymerfem.com/community/constitutive-models/theroretical-derivations-of-constitutive-equations/#post-6779Explanation
https://polymerfem.com/community/constitutive-models/theroretical-derivations-of-constitutive-equations/#post-6772
Mon, 30 Mar 2009 03:05:15 +0000This thread is regarding the PHD thesis of Jorgen Bergstrom.]]>Constitutive Modelsxbzihanhttps://polymerfem.com/community/constitutive-models/theroretical-derivations-of-constitutive-equations/#post-6772Theroretical derivations of constitutive equations
https://polymerfem.com/community/constitutive-models/theroretical-derivations-of-constitutive-equations/#post-27173
Mon, 30 Mar 2009 03:03:21 +0000Before I can do this, I have however a couple of questions that I am hoping you can answer. I apologize if they seem trivial or dumbOn page 91 of your Phd thesis equation 3.15 states the continuum mechanics expression for the Cauchy stress. Is it possible to elaborate the derivation between eq. 3.15 and 3.16? This would help a lot. Regarding eq. 3.17 and 18 is it true that 3*lambda^2=I1? Which I found in another paper on polymerfem.com. Because if this is true I cant rewrite 3.17 to become 3.18 since Im stuck with lambda^2. When differentiating eq.3.13 with respect to lambda the dependence of beta (inverse langevin) on lambda seem to be ignored.is this correct? How is expression 3.23 for incompressible uniaxial deformation obtained from eq 3.19, and why is this eq (3.23) different from the one stated in Modeling of the dynamic mechanical response of elastomers. (No division through a langevin term in the latter) How would one obtain the Cauchy stress from a strain energy density function like mooney rivlin? And how would uniaxial tension stress be described? I hope I havent bothered you too much, but Im really stuck.Kind regards Harm Kooiker]]>Constitutive Modelsxbzihanhttps://polymerfem.com/community/constitutive-models/theroretical-derivations-of-constitutive-equations/#post-27173