Material Modelling

2.2.1 Wood and Steel

The wood is modelled as being a linear elastic orthotropic material. The inuence of taking into account the actual annual ring curvature of the timber, shown schematically in Figure 1, was investigated but it was found to have a negligible e ect on the results, and therefore the material directions were assumed to be constant in the timber. The numerical values of the elastic parameters that have been used are given in Table 1 (T = tangential, R= radial andL= longitudinal direction (MPa))

The steel rods are modelled as being linear elastic and isotropic. The Young's mod-ulus is set to 210 000 MPa and the Poisson's ratio = 0:3.

2.2.2 Bondline Model

The strain-softening bondline model used is a further development of a model by Wern-ersson 21], and applied by Serrano and Gustafsson 22]. The original model was two-dimensional and developed for thin bondlines which were assumed to fail along a line of failure, involving only one shear-stress component and the peel stress of the bondline.

An expansion of this model was made for the present study so that it now involves the two shear-stress components and the peel-stress component of an assumed plane of fail-ure. Thereby the model can be used for adhesive layers in three-dimensional structures.

For cases of axial pull-out with the wood bres aligned parallel to the rod, the state of deformation could be approximated using an axisymmetric approach. However, for other loading conditions and for cases were the loading is not applied parallel to the wood bres a full three-dimensional approach is necessary.

The behaviour of the bondline is dened by piecewise linear curves, describing the uniaxial behaviour for shear stress vs. shear slip (2 curves) and peel stress vs. normal displacement. Assuming a piecewise linear relation is the simplest way of obtaining a

τ1,1

Figure 2: Stress-slip curve for a bondline in uniaxial shear.

shapes. In the present study piecewise linear relations with three segments are used.

An example of such a piecewise linear curve is given in Figure 2.

The stress-strain relations in the bondline plane directions are assumed to be linear elastic. The three stress components in these directions (two normal and one shear stress) are not considered in the fracture model.

A general mixed-mode state of deformation of the bondline is given by two shear-slip deformations (s¹s²) and by the normal deformation across the bondline (n). The bondline response is assumed to retain its piecewise linear shape for radial deformation paths (constant ratio (_s¹:_s² :_n)), but vary smoothly with the degree of mixed mode, expressed by the mixed mode angles'ssand'sn:

'ss= arctans¹

s² (10)

'sn= arctans

n (11)

s=^q_s²¹+_s²² (12) The following criterion is used to determine whether the current state of deformation is elastic or not:

( s¹

⁰_s¹_^{1)m1+ (}s²

⁰_s²_^{1)m2+ (} n

_n^{01)p 1 (13)} In Eq. (13) s1 ands2 correspond to the two directions in a plane of failure. Subscript n refers to the peel response, subscript 1 refers to the rst breakpoint of the piecewise linear curve and nally, superscript 0 stands for the uniaxial properties.

If the current state is elastic, according to Eq. (13), the response is linear and un-coupled, according to the uniaxial responses. If not, the current state of mixed mode is calculated according to Eq. (10){(11). Following this, the new deformations (s¹i,

s²i andni) corresponding to the breakpoints on the piecewise linear curve are calcu-lated with an expression analogous to Eq. (13). The stresses corresponding to each such

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breakpoint, i, are then calculated according to:

¹i=¹⁰_i s¹¹

_s⁰¹_^{1 (14)}

²i=²⁰_i s²¹

_s⁰²_^{1 (15)}

i=_i⁰ n¹

_n^{01 (16)}

Knowing the stresses at the breakpoints, the stress for the current state of deforma-tion can be obtained by linear interpoladeforma-tion. In FE-analysis, it is necessary not only to dene the state of stress for the current state of deformation, but also to give the tan-gential sti ness of the material for this state, i.e. the derivative of the stress with respect to the strains. In the current implementation this is performed numerically, since for the present material model it is very di cult to obtain a general explicit equation for the derivative. The above model is concentrated on the severe state of simultaneously acting peel and shear stresses. For the case of a compressive stress perpendicular to the bondline, the shear stress-slip behaviour is assumed to coincide with the uniaxial response, and the normal stress-displacement behaviour is assumed to be linear elastic.

In the FE-implementation of the above model, a so-called crack band approach has been used. This means that the above stress-displacement relations are transformed into corresponding stress-strain relations by dividing the displacements by the thickness of the continuum nite element used to model the bond layer. This results in a material length being introduced as a material property apart from the uniaxial stress-strain relations and the powers m1, m2, p of Eq. (13).

In the present study it has been assumed that

m1 =m2 =p= 2: (17)

The tri-linear stress-displacement relation is dened through

¹⁰_²=¹⁰_¹=3 ¹⁰_³= 0: (18) Furthermore

⁰_s¹_²= 4_s⁰¹_¹ ⁰_s¹_³= 40⁰_s¹_¹: (19) The performance in thes2-direction was assumed to be the same as in thes1-direction.

In the n-direction it was assumed that

⁰² =¹⁰=4 ³⁰= 0: (20) It was further assumed that

n² = 30n¹ n³ = 180n¹: (21) The strengths in the three directions are given by

f =⁰_ =⁰_  f =⁰ (22)

which together with the relations

Gfs=^Z ¹⁰d⁰_s¹=^Z ²⁰d_s⁰² Gfn=^Z ⁰d⁰_n (23) dene the bond layer properties for the given numerical values of only four parameters:

f andf for the strengths in shear and peel, andGfs andGfn for the corresponding fracture energies.