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actually I'am working on a model to describe the creep-behaviour of a concrete specimen. This model is based on a 'generalized maxwell-chain'. I attached a picture, where you can see it:

The acting load is the tension [itex]\sigma[/itex], while [itex]E_0[/itex] and [itex]E_1[/itex] are the stiffnesses of the Hooke-springs and [itex]\eta_i[/itex] are the viscosities of the Newton-dashpots.

All these parameters are "hidden" in the parameters [itex]a_i, b_i[/itex] of the constitutive equation which describes the behaviour of this model by:

[itex]b_0\cdot\epsilon+b_1\cdot\dot\epsilon+b_2\cdot\ddot\epsilon+b_3\cdot\dddot\epsilon + b_4\cdot\ddddot\epsilon = a_0\cdot\sigma +a_1\cdot\dot\sigma + a_2\cdot\ddot\sigma + a_3\cdot\dddot\sigma + a_4\cdot\ddddot\sigma[/itex]

To simplify the whole problem, I assume that these parameters [itex]a_i, b_i[/itex] won't change within time.

Now I want to test the material by performing a "simple"

**creep-test.**This means that at a time [itex]t[/itex], a tension [itex]\sigma > 0[/itex] is applied and stays constant:

t = 0: σ = 0

t = 1: σ = 0

t = 2: σ = 0

t = 3: σ = 0

t = 3,001: σ = 1

t = 4: σ = 1

t = 5: σ = 1

t = 6: σ = 1

t = 7: σ = 1

(etc.)

If I want to know the resulting strain [itex]\epsilon[/itex] at the end of a time-step t-t0, i can solve the given equation which actually can be simplified to (because the [itex]\sigma[/itex] is constant in a time-step, all of it's derivatives are zero):

[itex]b_0\cdot\epsilon+b_1\cdot\dot\epsilon+b_2\cdot\ddot\epsilon+b_3\cdot\dddot\epsilon + b_4\cdot\ddddot\epsilon = a_0\cdot\sigma[/itex]

This is a fourth-order ODE with constant coefficients that can be solved analytically. In a time-step based calculation, I have to solve the equation for each time-step by the formulation of correct initial conditions. ... and there is the problem:

If I want to get the strain et the end of the time step [itex](t = 3,001) - (t_0 = 3)[/itex] I mean that the initial conditions for the [itex]\epsilon, \dot\epsilon, ...[/itex] are:

[itex]\epsilon(t = 3) = 0 + \frac{\Delta\sigma(t-t_0)}{E_0 + 4\cdot E_1}[/itex] (at the begin [itex]\epsilon[/itex] is zero, but there's always a kind of elastic deformation, when some stress is applied)

[itex]\dot\epsilon(t = 3) = \frac{\frac{\Delta\sigma}{t - t_0}}{E_0 + 4\cdot E_1}[/itex] (this is the velocity of the elastic strain)

[itex]\ddot\epsilon(t = 3) = 0[/itex] (I guess ... but I am not sure)

[itex]\dddot\epsilon(t = 3) = 0[/itex] (same here ...)

So far so good. If I come to the next time step [itex](t = 4) - (t_0 = 3,001)[/itex], I have to solve the equation again. Now with different initial conditions:

[itex]\epsilon(t = 4) = \epsilon(t = 3,001)+ \frac{\Delta\sigma(t-t_0) = 0}{E_0 + 4\cdot E_1}[/itex]

[itex]\dot\epsilon(t = 4) = ... [/itex] I don't know what... (I don't think that it is [itex]\dot\epsilon(t = 3,001)[/itex] and I also doubt that it is zero ...)

[itex]\ddot\epsilon(t = 4) = ...[/itex] (same here ...)

[itex]\dddot\epsilon(t = 4) = ... [/itex] (same here ...)

I hope I described the problem good enough that someone can help me ... this would be great!

Thank you in advance