How to get the elemental stiffness/damping matrix of a Maxwell spring-damper element?

A book maybe useful to you:

"Mathematical Analysis in Engineering" by Prof. Chiang C.Mei in MIT

in chapter 13, there are a few introduction to damped elastic spring with weak nonlinearity.

the perturbation method is applied to solve this model.

Regards

LG

Hi!

I might be completely wrong here, but it seems to me that what you are really after is some equation that will give you the displacements for a given force F? And you assume that these equations can be written in the form M\dot{u} + Ku = F, where M and K would be the stiffness and damping matrix? I do not think its really that easy. Here are some of my ideas (I have no experience with these matters, so I could be completely of here):

Assuming small strains (not necessary), the total elongation can be expressed in terms of
the displacements u as
d = b^{t}u (1)
where, in 2D, b = (-cos(a) -sin(a) cos(a) cos(a))^{t}, in which a gives the
orientation of the bar relative to the x-axis.

Now, the principle of virtual work states that the external work is equal to
the internal work, so
F^{t}u = fd = fb^{t}u
where the last equality follows from (1). This can be rewritten as
(F^{t} - fb^{t})u = 0
and since u is arbitrary, we get
F = bf (2)

which is equlibrium for internal and external forces. The problem here seems to be that the internal force must be
determined by solving the nonlinear ordinary differential equation that you obtained:

\dot{f}/k2 + (f/c)^{\beta} = \dot{d} (3)

My guess is that you end up with a numerical scheme where u (and f) is updated iteratively while making sure that (2) holds.

In the book 'Nonlinear Finite elements ... ' by Belytscko et al, a very brief section
on viscoelasticity is found on pages 274-276. Some references are also given. Also, perhaps more useful, is "Nonlinear Solid Mechanics" by Holzapfel, chapter 6.

Best regards

Bashir

Hi again,

sorry for the late answer. Perhaps you have already solved it? Anyway, I have some more comments. First, you say that

" In the nonlinear finite element analysis, we intend to
solve a equation of the form

K(u)u =f"

I would say that we wish to solve systems f_int(u) = f_ext, where f_int often have the form K(u)u. The stiffness matrix (or Jacobian)

is really not central to the analysis, but is required in certain numerical procedures, e.g. Newton's method, that we may use to solve

the residual equations. This is also noted in the book by Belytscko et. al.

Now, regarding your question, I do not think that you can find an "element matrix". The internal force can, I think, not be viewed as

a function of the displacement. Intuitively: for a fixed u, we can have several different values of f, which is not what we expect from

a function f(u). Instead, my guess is that you must solve the following DAE (differential algebraic equation) for f AND u:

F = bf

\dot{f}/k2 + (f/c)^{\beta} = b\dot{u},

with some suitable initial conditions (notation as in my previous post). Systems of this type can be solved using some of the Matlab ODE-solvers,

or maybe the SUNDIALS solvers.

If you come up with a solution, would you please post a description?

best regards

/Bashir