# How to solve stress/strain constitutive equation for system of springs and dampers with a Coulomb friction element added in?

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: 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 $$E_3$$:

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?

Thanks.