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In the Maxwell Spring-Dashpot model : Maxwell Material - Wikipedia, It is assumed that, $$\sigma_{\text{total}} = \sigma_{\text{dashpot}} = \sigma_{\text{spring}} \text{ ...(1)}$$ $$\epsilon_{\text{total}} = \epsilon_{\text{dashpot}} + \epsilon_{\text{spring}} \text{ ...(2)}$$ Where $\sigma$ stand for stress and $\epsilon$ stand for strain.

While I can arrive at equation (1) understandably, my own calculations do not agree with equation (2). Here's what I mean by that,

I have assumed that: $$\epsilon_{\text{dashpot}} = \frac{\Delta L_{\text{dashpot}}}{L_{\text{dashpot}}}, \text{ where }\Delta L_{\text{dashpot}} \text{ is the change in dashpot length and }L_{\text{dashpot}}\text{ is the original dashpot length}$$ $$\epsilon_{\text{spring}} = \frac{\Delta L_{\text{spring}}}{L_{\text{spring}}}, \text{ where }\Delta L_{\text{spring}} \text{ is the change in spring length and }L_{\text{spring}}\text{ is the original spring length}$$ and, $$\epsilon_{\text{total}} = \frac{\Delta L_{\text{dashpot}} + \Delta L_{\text{spring}}}{L_{\text{dashpot}} + L_{\text{spring}}}$$ $$\text{ where }(\Delta L_{\text{total}} = \Delta L_{\text{dashpot}} + \Delta L_{\text{spring}}) \text{ is the change in dashpot length and }$$ $$( L_{\text{total}} = L_{\text{dashpot}} + L_{\text{spring}} ) \text{ is the original dashpot length}$$

These relations then give an entirely different expression : $$\epsilon_{\text{total}} = \left(\frac{L_{\text{dashpot}}}{L_{\text{total}}}\epsilon_{\text{dashpot}}\right) + \left(\frac{L_{\text{spring}}}{L_{\text{total}}}\epsilon_{\text{spring}}\right)$$

This is not same as equation (2), which I do not understand how it is arrived at then for in that case the arguments $$L_{\text{total}} = L_{\text{dashpot}} + L_{\text{spring}}\text{ and }\Delta L_{\text{total}} = \Delta L_{\text{dashpot}} + \Delta L_{\text{spring}} $$ does not make sense, which is what I based my calculations on.

What am I assuming wrong here?

This question does not demand any check-my work, and rather it is a question about the validity of the physical basis of arguments used to construct the concerning equation(s). As described by people in the comments, this is really concerned with the CORRECT usage of Lump Modelling of the system with spring and dash-pot. The problem of the Maxwell model then, is just a motivating example to understand how to do that. Therefore I request you to re-open this question.

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  • $\begingroup$ The dashpot and the spring are in series so their stresses are equal at equilibrium and the total strain is the sum of the two individual strains (by definition of "in series", see page 9 here: web.mit.edu/course/3/3.11/www/modules/visco.pdf "Roylance, David (2001). Engineering Viscoelasticity" ) $\endgroup$
    – Quillo
    May 2, 2023 at 12:37
  • $\begingroup$ I thnk that the problem in your derivation is due to the fact that you are using a "finite" strain $\Delta L/L$ rather than the differential definition, see physics.stackexchange.com/q/430714/226902 physics.stackexchange.com/q/712152/226902 $\endgroup$
    – Quillo
    May 2, 2023 at 12:52
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    $\begingroup$ Does this answer your question? Limitations of spring and dashpot models in terms of strain meaningfulness $\endgroup$ May 2, 2023 at 15:02
  • $\begingroup$ @Quillo , I had already checked the MIT paper, it does not really give justification for the strain additivity argument, as you pointed out, takes it as the definition which is not my problem rather I am trying to reason why such a defintion, even as an assumption can be said to be valid. $\endgroup$
    – Aditya
    May 2, 2023 at 17:55
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    $\begingroup$ Hello Aditya, Try to have a look at the two questions I linked (one is exactly the one also linked by @Chemomechanics ). The point is that $\epsilon$ is defined via a derivative. $\endgroup$
    – Quillo
    May 2, 2023 at 18:03

1 Answer 1

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You've identified a failure mode (or edge case, or limitation) of applying certain lumped-component models: It's not generally meaningful to talk about their associated length.

Relatedly, we never worry about the dashpot “bottoming out” or “topping out” if we contract or extend it too far, or the spring leaving its elastic range, or the viscoelastic assembly taking up too much room in our system, or exhibiting lateral (Poisson) strain.

For convenience of applying the stress equation you give, considering that the tension in a series connection must be the same everywhere, we assume that all springs and dashpots have equal effective cross-sectional area—or rather, we don’t worry about it, simply taking the force as a surrogate for stress and vice versa when it suits us for ease of modeling constitutive behavior.

For convenience of applying the strain equation you give, since we’d like two springs placed in series to have double the compliance, we assume that all components and assemblies of these components have equal length—or rather, we don’t worry about it, simply taking the strain as a surrogate for length and vice versa when it suits us for ease of modeling constitutive behavior.

All this is to say that can't treat these idealized lumped models as real objects with finite geometry—at least without breaking some other rules of how real objects behave and how we define mechanical terms.

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