I am attempting to solve the stress/strain constitutive equation for a system of springs, dampers, and a Coulomb friction element. Basic examples for such constitutive equations can be found in this article on Viscoelasticity. These expressions are made in terms of stress (σ) & strain (ε).

The goal is a final equation with all the stress terms on the left side and strain terms on the right side. For example, if you only had $E_1$, $E_2$, and $η_2$ in the following diagram (3 elements), it works out to:

$σ + \frac{η_2}{E_2}\dot{σ}= E_1ε + \frac{η_2(E_1+E_2)}{E_2}\dot{ε}$

Here is the system I am trying to solve: 3+2 Element System The first branch is a simple spring. The second branch is called a "Maxwell element" which is a spring and viscous damper in series. The third branch is called a "Jenkins element" which is a spring and Coloumb friction element in series. This is where I'm having trouble.

This article provides all the Jenkins arm equations I have to work from.

Maxwell only derivation (for comparison)

Here is an example of the correct solution for constitutive equation of the same system but with no Jenkins elements, with the branches as $A$ for $E_1$, $B$ for $E_2$, and $C$ for $E_3$:

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Stress ($σ$) relationships:

$σ = σ_A + σ_B + σ_C$

$σ_A = σ_{E1} = E_1ε_{E1}$

$σ_B = σ_{E2} = E_2ε_{E2} = σ_{η2} = η_2\dot{ε}_{η2}$

$σ_C = σ_{E3} = E_3ε_{E3} = σ_{η3} = η_3\dot{ε}_{η3}$

Strain ($ε$) relationships:

$ε = ε_A = ε_B = ε_C$

$ε_A = ε_{E1}$

$ε_B = ε_{η2} + ε_{E2}$

$ε_C = ε_{η3} + ε_{E3}$

Solve stress of each branch ($A/B/C$), eliminating individual element stresses and strains:

$\dot{ε}_A = \dot{ε}_{E1}$

$\dot{ε}_B = \dot{ε}_{η2} + \dot{ε}_{E2}$

$\dot{ε}_C = \dot{ε}_{η3} + \dot{ε}_{E3}$

$\dot{ε}_A = \frac{1}{E_1}\dot{σ}_A$

$\dot{ε}_B = \frac{1}{η_2}σ_B + \frac{1}{E_2}\dot{σ}_B$

$\dot{ε}_C = \frac{1}{η_3}σ_C + \frac{1}{E_3}\dot{σ}_C$

$\dot{ε} = \frac{1}{E_1}\dot{σ}_A$

$\dot{ε} = \frac{1}{η_2}σ_B + \frac{1}{E_2}\dot{σ}_B$

$\dot{ε} = \frac{1}{η_3}σ_C + \frac{1}{E_3}\dot{σ}_C$

$\dot{ε} = (\frac{1}{E_1}\frac{d}{dt})σ_A$

$\dot{ε} = (\frac{1}{η_2} + \frac{1}{E_2}\frac{d}{dt})σ_B$

$\dot{ε} = (\frac{1}{η_3} + \frac{1}{E_3}\frac{d}{dt})σ_C$

$σ_A = \dot{ε}\frac{1}{(\frac{1}{E_1}\frac{d}{dt})}$

$σ_B = \dot{ε}\frac{1}{(\frac{1}{η_2} + \frac{1}{E_2}\frac{d}{dt})}$

$σ_C = \dot{ε}\frac{1}{(\frac{1}{η_3} + \frac{1}{E_3}\frac{d}{dt})}$

Sum total stress:

$σ = σ_A + σ_B + σ_C$

$σ = \dot{ε}\frac{1}{(\frac{1}{E_1}\frac{d}{dt})} + \dot{ε}\frac{1}{(\frac{1}{η_2} + \frac{1}{E_2}\frac{d}{dt})} + \dot{ε}\frac{1}{(\frac{1}{η_3} + \frac{1}{E_3}\frac{d}{dt})}$

Final Solution:

$σ + (\frac{η_2}{E_2} + \frac{η_3}{E_3})\dot{σ} + \frac{η_2η_3}{E_2E_3}{σ_{tt}} = E_1ε + (\frac{E_1η_2}{E_2} + \frac{E_1η_3}{E_3} + η_2 + η_3)\dot{ε} + (\frac{η_2η_3}{E_2} + \frac{η_2η_3}{E_3} + \frac{E_1η_2η_3}{E_2E_3})ε_{tt}$

The Jenkins Problem

The Jenkins arm equation (given by (6) in the article) is, written in these terms:

$\dot{σ}_C = \frac{1}{2}E_3 \dot{ε}[1- \text{sign}(σ_C^2 - μ_3^2) - \text{sign}(\dot{ε}σ_C)(1+\text{sign}(σ_C^2 - μ_3^2))]$

Here, $μ_3$ is the "maximum Coulomb friction force" of the Jenkins element, and the $\text{sign}$ function is:

sign(y): -1 for y<0, 0 for y=0, 1 for y>0

My problem is this: I don't know how to rephrase this Jenkins arm equation in terms that allow me to sub it into $σ = σ_A + σ_B + σ_C$ like I did in the Maxwell solution.

I would need it as:

$σ_C =$ ... terms of $E_1, E_2, E_3, η_1, η_2, μ_3, ε, \frac{d}{dt}$ only ...

Or I can't substitute it into the overall stress equation and solve it.

I think the $\text{sign}$ functions make solving this the same way impossible. My thought is perhaps the only way is to develop multiple constitutive equations - one for each "condition" of the Jenkins arm.

But even this I don't think will solve it, because that would require me to know $σ_C$ to determine which equation I am using, and I won't know that. I need this in terms solely of $σ$, $ε$ and the constant coefficients, just like the Maxwell solution.

Is there any way this can be solved?



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