Computational Methods for Debonding in Composites

8 A Numerical Material Model for Predicting the High Velocity Impact Behaviour 173
In which ε f mat is defined as:
ε f mat = 2Γ
σ 0 matLmat
The driving stress at the onset of matrix failure, σ 0 mat, is defined as [10]:
σ 0 �
�σ �
mat = n 2 � �
b ′ + τn 2 � �
b ′ c ′ + τn 2
ab ′
(8.30)
(8.31)
Equation 8.30 indirectly depends on strain rate through the “dynamic matrix
fracture toughness” Γ . This is a function of strain rate defined by assuming that
a relationship of direct proportionality exists between shear traction strengths and
fracture toughness: the same scaling function that was used for predicting the
enhancement of shear strength with strain rate is used for scaling fracture toughness
with strain rate:
Γ = Γb
� σ 0 b ′
σ 0 mat
� 2
+ z( ˙γ b ′ c ′)Γ b ′ c ′
� τ 0 b ′ c ′
σ 0 mat
� 2
+ z( ˙γ ab ′)Γ ab ′
� τ 0 ab ′
σ 0 mat
� 2
(8.32)
This is a simplification and dynamic fracture energies should be experimentally
characterised. However, the formulation proposed here can be used with an arbitrary
scaling function.
The stress vector is then updated on the fracture plane as follows:
{�σ} n+1
ab ′ c ′ = {σa σ b ′ σ c ′ �τ ab ′ �τ b ′ c ′ �τ ac ′} n+1 = {�σ} n
ab ′ c ′ + {∆ �σ} ab ′ c ′
And the relevant stresses are degraded on this plane:
� �dmat n+1
n+1
σa = σa �
1 − d inst
� n+1
σb mat
′
��
� n+1
σb ′
�dmat =
� n+1
σc ′
�dmat n+1
= σc ′
� �τ n+1
ab ′
� �τ n+1
b ′ c ′
�σ n+1
b ′
�dmat � � inst
= 1 − dmat �τ n+1
ab ′
�dmat � � inst
= 1 − dmat �τ n+1
b ′ c ′
�
�τ n+1
ac ′
�dmat
= �τ n+1
ac ′
σ n+1
b ′
(8.33)
(8.34)