Linear Viscoelastic Kelvin Voigt Model - GCMLab/GCMLab-FEM GitHub Wiki

The stress field is related to the strains and the strain rate through the constitutive law,

in which is the 4^th order linear elasticity tensor and is the damping tensor.

The fully discretized weak form is the following,

where, is the damping matrix, is the stiffness matrix, is external force vector, is the time-step and is the time-step parameter.

C is defined as follows, where is the damping coefficient of the element.

• One-dimensional problems:

  

• Two-dimensional problems:

  Plane strain

  

  in which is Poisson's ratio.

  Plane stress

  

T3 elements used over temporal domain of t = [0,10s].

Parameter	Symbol	Value
Young's Modulus	$E$	1 GPa
Poisson's Ratio	$\nu$	0.25
Damping Coefficient	$\eta$	$100\text{ }MN\cdot s/m$
Start Time	$t_{0}$	0s
End Time	$t_{end}$	10s
Time-Step	$\Delta t$	0.02s
Integration Parameter	$\alpha$	0.5
Thickness	$t$	1m

B.C $\rightarrow u_{x}(x=0,y,t) = 0$, $u_{y}(x=0,y,t) = 0$
I.C $\rightarrow u_{x}(x,y,t=0) = 0$, $u_{y}(x,y,t=0) = 0$

Displacement over the time interval of 0 to 10s. Displacement over the time interval of 0 to 10s at x = 0.5m, y = 0.9m.