Computational Methods for Debonding in Composites

8 A Numerical Material Model for Predicting the High Velocity Impact Behaviour 169
shear. The generic strain rate dependent shear law is modelled by using a scaling
function, T ( ˙γ n ), and incremental calculation as follows:
τ n+1 = τ n + T ( ˙γ n ) f (γ n eff )∆τ (8.19)
where f (γ n eff ) is defined in Sect. 8.2.4. ˙γ n is the current shear strain rate, which is
defined as:
˙γ n = ∆γ
(8.20)
∆t
In which ∆γ is the shear strain increment at the current time step, ∆t and T ( ˙γ) is
a scaling function of shear strain rate, which is characterised using pure longitudinal
static and dynamic stress-strain data, as shown in [12].
8.3.3 Modelling Strain Rate Effects in Matrix Dominated Modes
of Deformation
Strain rate effects are modelled on the plane of maximum shear stress (with friction)
interaction as:
�
�
{∆ �σ} ab ′ c ′ = ∆σa ∆σb ′ ∆σc ′ ∆ �τ ab ′ ∆ �τ b ′ c ′ ∆ �τ ac ′
∆ �τ b ′ c ′ = T � ˙γ n b ′ c ′
∆ �τ ab ′ = T � ˙γ n ab ′
∆ �τ ac ′ = T � ˙γ n ac ′
� �
f
� �
f
� f
γ n b ′ c ′ , eff
�
�
∆τ b ′ c ′
γ n ab ′ , eff ∆τab ′
�
γ n ac ′ �
, eff ∆τac ′
Upon unloading the constitutive law is assumed as follows:
�
{∆ �σ} ab ′ c ′ =
∆σa ∆σb ′ ∆σc ′ ∆ �τ ab ′ ∆ �τ b ′ c ′ ∆ �τ ac ′
�
∆τb ′ c ′
∆ �τ b ′ c ′ = � d0,bc + d1,bcγ n b ′ c ′, eff � T � ˙γ n b ′ c ′
∆ �τ ab ′ = � d0,ab + d1,abγ n ab ′, eff � T � ˙γ n ab ′
�
∆τab ′
∆ �τ ac ′ = � d0,ac + d1,acγ n ac ′, eff � T � ˙γ n ac ′
�
∆τac ′
�
(8.21)
(8.22)
The stress increment vector is then rotated back to material principal axes. The
3D rotation results in more components of the stress increment vector that are strain
rate dependent.
8.3.4 Validation of the 3D Strain-Rate Dependent Plasticity Model
Single solid element simulations are conducted at constant strain rate and the
numerical results are compared with the experimental data available in the open literature,
Hsiao et al. [5]. They investigated the effect of strain rate on the mechanical