Proceedings e report - Firenze University Press

WOOD SCIENCE FOR CONSERVATION OF CULTURAL HERITAGE
Fig. 6. Plastic deformations in pin block.
3.3. Brittle Behaviour
Wood loaded in shear and tension perpendicular to fibres fails markedly brittle. To simulate this
behaviour using FEM, a special element formulation as well as a suitable material model are needed.
In a 3-D structure, cracks will develop in a plane. This fracture surface can be suitably modelled using
so-called cohesive elements in a FE-model. These interface-elements are defined by the nodes of two
surfaces, which are congruent in the initial configuration (see Fig. 7a). Corresponding to the loading
and the interface-material law, the two planes of the element can move e.g. tangentially in the
directions 1 and 2. In this case, a shear loading yields a shear crack. Analogously, the planes shift apart
in direction 3 due to a loading normal to the surface.
(a) (b)
Fig. 7. Geometry of the interface element and definition of the local coordinate system (a) and
stress-displacement relationship for the interface-material model with status 1 to 4 (b)
To simulate brittle failure of wood and aligned to the interface-element formulation, a coupled
material-model is introduced. Basically, this model is characterised by a stress-deformation-relation.
Using the input parameters, the stress-deformation function (see Fig. 7b) is composed of a sinusoidalfunction
in the elastic range (status 1) and the function, which describes the softening behaviour in the
damage area (status 2). In case of unloading after softening or hardening, the path is defined by a
spherical surface (status 3) and linear function to the starting point (status 4). The material formulation
is continuously differentiable in the domain of definition due to the fact that all transitions are C1continuous.
The material formulation takes into consideration the anisotropy and, therefore, the not
coaxial orientation of the displacement and stress-vector. For further information about the material
law and its numerical implementation see [3] and [4].
As an example, Fig. 9 shows a beam with an eccentric hole. The centric loading of the beam causes
brittle failure. The FE-Model of Fig. 9 shows the expected crack propagation along the beam, starting
at the cutout. Using the interface-elements in combination with the material model, the correct type of
failure is predicted. Analogously to the example of Fig. 8, the introduced cohesive elements and
material formulation can be used to simulate cracks in pianofortes as shown in Fig. 8.
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