sq
2005-10-11, 08:49
I recently had to do some modeling of a material using LVE. However, I had some difficulty in that the ususal tests (i.e., DMA, stress relaxation, creep) were unavailable at the strains I needed to work in (around 10%) and the timescales I was interested in (~0.01 to 1 ms). The best I could do was to use constant (or approximately constant) strain rate loading over a broad range of rates. So how do we use this?
For a Maxwell element (i) in a Prony series:
[TeX:d1ddc7029b]\sigma_{i} = E_{i}(\epsilon-\epsilon_{i})[/TeX:d1ddc7029b]
where the relaxation [TeX:d1ddc7029b]\epsilon_{i}[/TeX:d1ddc7029b] is governed by:
[TeX:d1ddc7029b]\dot{\epsilon_{i}} + \frac{\epsilon_{i}}{\tau_{i}} = \frac{\epsilon}{\tau_{i} [/TeX:d1ddc7029b]
Well, now, given a constant strain rate:
[TeX:d1ddc7029b] \epsilon=\dot{ \epsilon}t[/TeX:d1ddc7029b]
and performing a little calculus (assuming that the initial stress state of every element (i) is zero), we ultimately find that the overall stress is:
[TeX:d1ddc7029b]\sigma = \dot{\epsilon} \big[ E_{0}t + \sum_{i=1}^{n} E_{i} \tau_{i}(1-e^{-t/\tau_{i}} ) \big] [/TeX:d1ddc7029b]
So, now, the fun comes in when we examine stress data at a fixed strain [TeX:d1ddc7029b]\epsilon_{0}[/TeX:d1ddc7029b] for a number of strain rates. Then,
[TeX:d1ddc7029b]\sigma = E_{0}\epsilon_{0} + \dot{\epsilon} \sum_{i=1}^{n} E_{i}\tau_{i}(1-e^{\frac{-1}{\tau_{i}}\frac{\epsilon_{0}}{\dot{\epsilon}}}) [/TeX:d1ddc7029b]
This is a nice form for the solution, because now, if we plot [TeX:d1ddc7029b]\sigma/\epsilon_{0}[/TeX:d1ddc7029b] against log strain rate, we find a nice series of jumps associated with each relaxation (reminiscent of a creep plot). Each jump is located at:
[TeX:d1ddc7029b]\dot{\epsilon}=\frac{\epsilon_{0}}{\tau_{i}}[/TeX:d1ddc7029b]
With a strength of [TeX:d1ddc7029b]E_{i}[/TeX:d1ddc7029b] (naturally).
Beauty, eh? Now, it only remains to fix some number of time constants [TeX:d1ddc7029b]\tau_{i}[/TeX:d1ddc7029b], and fit the strengths to your data accordingly. Voila!
P.S.,
Maybe this has been done a million times before, I just haven't seen it in any of my books, and I thought maybe it would be useful.
For a Maxwell element (i) in a Prony series:
[TeX:d1ddc7029b]\sigma_{i} = E_{i}(\epsilon-\epsilon_{i})[/TeX:d1ddc7029b]
where the relaxation [TeX:d1ddc7029b]\epsilon_{i}[/TeX:d1ddc7029b] is governed by:
[TeX:d1ddc7029b]\dot{\epsilon_{i}} + \frac{\epsilon_{i}}{\tau_{i}} = \frac{\epsilon}{\tau_{i} [/TeX:d1ddc7029b]
Well, now, given a constant strain rate:
[TeX:d1ddc7029b] \epsilon=\dot{ \epsilon}t[/TeX:d1ddc7029b]
and performing a little calculus (assuming that the initial stress state of every element (i) is zero), we ultimately find that the overall stress is:
[TeX:d1ddc7029b]\sigma = \dot{\epsilon} \big[ E_{0}t + \sum_{i=1}^{n} E_{i} \tau_{i}(1-e^{-t/\tau_{i}} ) \big] [/TeX:d1ddc7029b]
So, now, the fun comes in when we examine stress data at a fixed strain [TeX:d1ddc7029b]\epsilon_{0}[/TeX:d1ddc7029b] for a number of strain rates. Then,
[TeX:d1ddc7029b]\sigma = E_{0}\epsilon_{0} + \dot{\epsilon} \sum_{i=1}^{n} E_{i}\tau_{i}(1-e^{\frac{-1}{\tau_{i}}\frac{\epsilon_{0}}{\dot{\epsilon}}}) [/TeX:d1ddc7029b]
This is a nice form for the solution, because now, if we plot [TeX:d1ddc7029b]\sigma/\epsilon_{0}[/TeX:d1ddc7029b] against log strain rate, we find a nice series of jumps associated with each relaxation (reminiscent of a creep plot). Each jump is located at:
[TeX:d1ddc7029b]\dot{\epsilon}=\frac{\epsilon_{0}}{\tau_{i}}[/TeX:d1ddc7029b]
With a strength of [TeX:d1ddc7029b]E_{i}[/TeX:d1ddc7029b] (naturally).
Beauty, eh? Now, it only remains to fix some number of time constants [TeX:d1ddc7029b]\tau_{i}[/TeX:d1ddc7029b], and fit the strengths to your data accordingly. Voila!
P.S.,
Maybe this has been done a million times before, I just haven't seen it in any of my books, and I thought maybe it would be useful.