To model progressive failure of braided textile composites, numerous studies combined two damage-evolution theories for inter- and intra-laminar damages, respectively. The first theory was a cohesive-zone model (CZM) widely used to capture inter-laminar delamination [62]. The CZM combines strength-based criteria used to predict damage initiation with fracture energy criteria to simulate damage propagation, yielding acceptable results with fewer limitations. Application of CZM requires a-priori knowledge of an intended crack path and a use of cohesive elements [62]. Another theory to evaluate intra-laminar failure was continuum damage mechanics (CDM) [94-96]. In CDM, damage is described by introducing internal state variables (𝐷𝑖𝑗) to an algorithm of continuum mechanics to represent micro-defects in a damage process in the material. Stiffness values of composites degraded with the growing damage variables (DVs) 𝐷𝑖𝑗 homogeneously when a material met its failure criteria. The CDM models are not able to capture the initiation and propagation of macroscopic cracks; however, it is not necessary to know exactly where damages occur when modelling failure with CDM.
(1) Cohesive Zone Models (CZM)
With CZM, the interface between fibre and epoxy is modelled by cohesive elements with a bilinear traction-separation law as shown in Figure 2.4. According to the traction- separation law, the area under the curve represents the fracture toughness (critical energy release rate) in specific fracture mode [96]. The crack is initiated when the Equation 2.15
below equals to 1: (𝑡𝑛 𝑁)2+ ( 𝑡𝑠 𝑆)2+ ( 𝑡𝑡 𝑆)2 = 1, (2.15)
where 𝑡𝑛, 𝑡𝑠, 𝑡𝑡 represent the interface stresses and 𝑁, 𝑆, 𝑇 are the interface strengths under mode I (opening), mode II (shear) and mode III (tearing), respectively. Damage evolution is defined based on fracture energy. Linear softening behaviour is utilised. The dependency of fracture energy on mixed fracture modes is expressed by the widely used Benzeggagh-Kenane formulation [97], which gives an analytical formula shown in
30 𝐺𝐶 = 𝐺 𝑛𝐶+ (𝐺𝑠𝐶− 𝐺𝑛𝐶) { 𝐺𝑠 𝐶+𝐺 𝑡𝐶 𝐺𝑛𝐶+𝐺𝑠𝐶+𝐺𝑡𝐶} 𝜂 (2.16) where, Gn, Gs and Gt are the work done by tractions and their conjugate relative displacements corresponding to mode I, mode II and mode III, respectively. The power, η, is a material parameter, may selected to 1.45 for a carbon fibre composite [98].
Figure 2. 4 Traction-separation behaviour bilinear mixed-mode [98].
However, there are still shortcomings using CZ elements to model interface damage [84]. For instance, the location of crack initiation should be known, although automatic insertion of cohesive zone elements is possible. In the braided structure, changes of fibre orientation result in an efficient-costly re-meshing. Moreover, CZM generally uses surface- and element-based approaches. In the former, the interface is regarded as interaction between two adjoining surfaces, and thickness of the interface is neglected. Long et al. [99] and Qiu et al. [100] successfully developed a cohesive interaction scheme for prediction of initiation and propagation of delamination during impact. Zhang et al. [101] reduced the computation time by using a quasi-static load with a surface- based cohesive contact model available in the ABAQUS FE software package. In the element-based method, COH3D8 cohesive elements (available in ABAQUS) were inserted at the interfaces between composite layers. Using this approach, Feng et al. [102]
investigated the influence of simulated intra-laminar damage modes on prediction of interface delamination. Kim et al. [103] studied the effect of delamination damage on performance of a whole structure. Although both approaches are acceptable, there is a lack of systematic studies to compare their advantages and shortcomings.
31 (2) Continuum Damage Mechanics (CDM)
The most direct way for damage modelling is a fracture-mechanics-based approach, in which cracks are directly introduced into the model. Still, introducing cracks inside complex yarns-matrix architecture and re-meshing are computationally intensive. Continuum damage mechanics (CDM), which can provide a tractable framework for modelling damage initiation and development, with strategy of stiffness degradation, is one of the important and effective methods to model progressive damage behaviour of fibre-reinforced composites supported by FE procedures. The main advantage of CDM is the straightforwardness of its implementation into FE analysis since the material is continuous throughout the damage process, it does not require re-meshing [104]. CDM provides not only the final failure load, but also information concerning the integrity of the material during the load history [104].
In CDM, damage is described by introducing internal state variables (𝐷) in the algorithm of continuum mechanics to represent micro-voids during damage process in the material. Damage modelling by variation in elastic modulus approach is one of the three fundamental methods [104]. For instance, in an isotropic bar under uniaxial loading, the damage variable (DV) is introduced as the ratio of damaged surface area (𝐴𝑑) to undamaged cross sectional area (𝐴) as 𝐷 =𝐴𝑑
𝐴 (see Figure 2.5). Damage variable (𝐷)
values of 1 indicates complete damage in the material, i.e. damaged surface area equals to the initial area of cross section at completely damaged state.
Figure 2.5 Uniaxial effective stress concept based on strain equivalence [40, 105].
32 Effective stress in the pseudo undamaged state:
𝜎̅ =𝐴−𝐴𝐹
𝑑=
𝜎
(1−𝐷)= 𝜀𝐸̅, (2.18)
From the hypothesis of strain equivalence as given by Lemtaire [106]: 𝜀 = 𝜀̅.
By combining Equation 2.17 and Equation 2.18,
𝐸̅
𝐸= 1 − 𝐷. (2.19)
It can be been seen from Equation 2.19 that damaged state Young’s modulus (𝐸̅) reduced
as the DV (𝐷) increases. The maximum value of 𝐷 can be ≈1 since the stiffness and compliance matrices should always be positive defined. For undamaged and elastic orthotropic composite materials, the stress-strain relationship can be written as:
{ 𝜎11 𝜎22 𝜎33 𝜏12 𝜏23 𝜏13} = [ 𝐶11 𝐶12 𝐶13 𝐶21 𝐶22 𝐶23 𝐶31 𝐶32 𝐶33 0 0 0 0 0 0 0 0 0 𝑠𝑦𝑚 𝐶44 0 0 𝐶55 0 𝐶66]{ 𝜀11 𝜀22 𝜀33 𝛾12 𝛾23 𝛾13} , (2.20)
where 𝜎𝑖𝑗 and 𝜏𝑖𝑗 are normal and shear stresses, 𝜀𝑖𝑗 and 𝛾𝑖𝑗 are normal and shear strains,
𝑪𝒊𝒋 are stiffness matrix. Therefore, post-peak behaviour of materials could be described
by a degraded stiffness matrix 𝑪(𝑫𝒊𝒋) or compliance matrix 𝑺(𝑫𝒊𝒋) , as shown in
Equation 2.21, 𝜎𝑖𝑗∗ = 𝑪(𝑫
𝒊𝒋)𝜀𝑖𝑗 or 𝜀𝑖𝑗 = 𝑺(𝑫𝒊𝒋)𝜎𝑖𝑗∗, 𝑖, 𝑗 = 1,2 and 3, respectively. (2.21)
Although many methods were developed, it is still an open question how to define DVs considering complicated failure modes of braided composites. In the following section, some stiffness-degradation approaches most broadly used in recent investigations are discussed.