Maxwell model of viscoelastic behaviour - Derivation of governing equation

[mark_d89]

Hi everyone,

i'm a 3rd year undergraduate chemical engineering student from Scotland, and i am wondering if anyone could help me with the following past paper question in my plastics engineering class. The question is based on the maxwell model of viscoelastic behaviour. Apologies if i have posted in the wrong section.

"In a tensile test on a plastic the material is subjected to a constant strain rate of 10^-5/s. The material may have it's behaviour modeled by a maxwell element with elastic constant 20GN/m^2 and viscous constant 1000 GNs/m^2.
(a) Starting from the governing equation for a maxwell element, derive the following expression for the stress in the material at any instant. stress=n2*strain*[1-exp(-G1*t/n2)]"
........

I have derived the governing equation down to the point, stress(t)=stress0*exp(-G1*t/n2) , but have not been able to get it to resemble the equation shown in the question. I'll attach a picture of the question for more clarity.

Any help is much appreciated, Thanks

 

[mark_d89]

Problem solved, spoke to lecturer today who gave me a few hints and was able to work through it.

I can post the solution if anyone is interested

 

[adnan jahan]

I need to know about the conductivity of visco-elastic material.

1) can it be considered as perfectly conductivity of electricity and heat?

2)what are the normal mode analysis method?
3) what information do we get from this method?
4) what are the cracks of mode I,II and III
how they change the stress and strain tensor?
any material or link related to that then please provide,,,,,

any comments and remarks will be informative,

 

[cammyt1]

Hey, I'm on the same course and really struggling with this one at the moment! I'd be really grateful if you could post your solution if you happen to read this, thanks.

 

[alphy]

.

Hi there,

I am familiar with the Maxwell model of viscoelastic behavior and its governing equation. I am happy to help you with your question.

Firstly, let's define the variables in the Maxwell model. The elastic constant, G1, represents the stiffness of the material and the viscous constant, n2, represents the rate at which the material deforms under an applied stress. The governing equation for the Maxwell model is:

stress(t) = G1 * strain(t) + n2 * integral of strain(t) dt

Using the given values for G1 and n2, we can rewrite the equation as:

stress(t) = 20GN/m^2 * strain(t) + 1000GNs/m^2 * integral of strain(t) dt

Now, since the material is subjected to a constant strain rate of 10^-5/s, we can express the strain as:

strain(t) = strain0 * t

Substituting this into the governing equation, we get:

stress(t) = 20GN/m^2 * strain0 * t + 1000GNs/m^2 * integral of strain0 * t dt

Integrating, we get:

stress(t) = 20GN/m^2 * strain0 * t + 1000GNs/m^2 * (strain0 * t^2)/2 + C

Where C is the constant of integration. Since we know that at t=0, the stress is 0, we can solve for C and get:

C = -1000GNs/m^2 * strain0 * t^2

Substituting this back into the equation, we get:

stress(t) = 20GN/m^2 * strain0 * t + 1000GNs/m^2 * (strain0 * t^2)/2 - 1000GNs/m^2 * strain0 * t^2

Simplifying, we get:

stress(t) = 20GN/m^2 * strain0 * t - 500GNs/m^2 * strain0 * t^2

Now, we can express the strain0 as a function of time using the given strain rate, 10^-5/s:

strain0 = 10^-5 * t

Substituting this into the equation, we finally get:

stress(t) = 20GN/m^2 * 10^-5