Supports - Preventive Biomechanics - page 165

In Figs. 4.16 and 4.17 the displacement-time data (creep curves) and the cor-
responding simulation data based on plate and indenter creep test of foam mate-
rials HR and VE are depicted (cross markers). Due to the particular choice of
indenter masses in conjunction with the sample heights of the HR-foam material
the displacement-time outputs in Fig. 4.16 of both tests were similar.

4.2.1.5 Material Identification II: Viscoelasticity (Creep)

Constitutive Equation: Based on the long-term parameter optimization results of
the foam materials HR and VE of Sect. 4.2.1.3 as well as the results of creep
testing in the previous section, appropriate material parameters regarding foam
viscoelastic behaviour are characterized. Under the assumption of linear visco-
elasticity at finite strain, the continuum mechanical description is performed using
the material model proposed by (Simo 1987) and implemented in A BAQUS FE-
solver. The resulting K IRCHHOFF stress tensor s based on ( 3.334 ) reads (cf. outlines
in Sect. 3.2 , 3.2.6.5 )

ðÞ Z

t

X

N K

t 0
s K

k i
s i

s ðÞ¼ s H ðÞþ s D ðÞ¼ s 0 ðÞþ s 0

i s 0

t t 0 Þ dt 0

e ð

i ¼ 1

t 0 ¼ 0

2
4

3
5

Z

t

X

N G

t 0
s G

g i
s i

ð 4 Þ

i F 1
t

t t 0 Þ s 0 t t 0 Þ F T

t

t t 0 Þ dt 0

p e ð ð ð

i ¼ 1

t 0 ¼ 0

where the relative moduli k i and g i as well as the relaxation times s i and s i

are to

be identified.

Identification of the Prony-Series Parameters: Due to the fact that spatial
stress and strain states are to be expected during experimental testing, and similar
to the experiments regarding material equilibrium elasticity (cf. outlines in Sect.
4.2.1.3 ), characterization of Prony series parameters k i , g i , s i and s i in ( 3.334 )is
performed employing the algorithm based on iFEM, as described in Sect. 3.4 . The
model function f i M (h i , p) of the quality functional ( 3.364 ) corresponds to the
specialized displacement-time relation, which results from the inverted material
model given in ( 3.334 ) (this is automatically done by means of the FE-solver when
solving the boundary value problem). The f i E are the measured displacements, the
h i is the measured time and the coordinates of the vector p relate to the material
parameters k i , g i , s i and s i .
The creep test scenarios regarding Prony-series material parameter optimization
of k i , g i , s i and s i were FE-modeled and iteratively simulated during the
optimization process. Corresponding FE-models of the plate creep test and the
indenter creep test are depicted in Figs. 4.15 a, b and c. In both deformation modes,
foam material was modelled using fully integrated 8-node first-order incompatible
mode continuum elements with an average element side length of 8 mm. All three