Proceedings e report - Firenze University Press

NUMERICAL SIMULATION OF THE STRENGTH OF WOODEN STRUCTURES: APPLICATION TO HISTORIC PIANOFORTE
3.2. Ductile Behaviour
Primarily at compression loading, wood is characterised by ductile behaviour and failure with
irreversible deformations. To simulate this material behaviour, a three-dimensional multi-surface
plasticity model is developed, which allows a coupled consideration of the elastic-plastic failure
behaviour of wood as a result of compression, tension or shear loading. In addition, material direction
dependent post-failure behaviour is taken into account by defining different softening and hardening
relationships. The multi-surface-plasticity is validated using experimental data from [1].
(a) (b)
Fig. 5. Yield surfaces in r-, t- and l-direction for compression and tension (a) and
damage function q and stress-strain-relationship (b)
The limit between the elastic and plastic zone is defined by the flow rule including yield conditions for
each surface in form of:
= : σ + σ : : + − 1 ≤ 0
(3)
f a
m
b σ q
m
m m
where a m and b m are strength tensors and q m is a function of softening and hardening. Seven possible
failure modes need to be considered. Therefore, seven yield conditions m are defined. Fig. 5a shows
the yield surfaces of Norway Spruce due to tension and compression in every material direction.
The damage function:
( 1−
κ ) ⋅
⎛ −η
⋅α
⎞ ( α − α )
2
m u m
m m d
q = ⎜ − e ⎟
,
1
, − ζ
m,
u m,
u ⎜
⎟ m,
u α − α
⎝
⎠ m,
max m
allows the simulation of softening in the post-failure behaviour. In case of compression loading, an
additional term defines hardening. Both are shown exemplary in Fig. 5b. A stress-strain function for
compression is given in Fig. 5b, too. Further information about the multi-surface plasticity model with
regard to material parameters, implementation and verification as well as applications is given in [2]
and [3]. Fig. 6 shows deformations of the pin block at compression loading, which can be simulated
using the multi-surface plasticity model.
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(4)