### Damage criteria methods

#### Failure criteria method

Tow most popular approaches to estimate damages in composites are stress-based failure criteria and damage mechanics methods. The maximum stress/strain criterion (Eqs. 2.1, 2.2, 2.3, 2.4) is the easiest to numerically perform and straightforward to understand for each individual damage mode, but unable to account for mode interaction.

$$ f=\mathit{\max}\ \left(\left|\frac{\sigma_{11}}{X}\right|,\left|\frac{\sigma_{22}}{Y}\right|,\left|\frac{\sigma_{12}}{S_{12}}\right|\right) $$

(2.1)

where

$$ \begin{array}{l}{\sigma}_{11}\ge 0\ \Rightarrow\ X={X}^T;\kern1.5em {\sigma}_{11}<0\ \Rightarrow\ X={X}^C\\ {}\ {\sigma}_{22}\ge 0\ \Rightarrow\ Y={Y}^T;\kern1.5em {\sigma}_{22}<0\ \Rightarrow\ Y={Y}^C\end{array} $$

(2.2)

or

$$ f=\mathit{\max}\ \left(\left|\frac{\varepsilon_{11}}{\varepsilon_{11}^0}\right|,\left|\frac{\varepsilon_{22}}{\varepsilon_{22}^0}\right|,\left|\frac{\gamma_{12}}{\gamma_{12}^0}\right|\right) $$

(2.3)

where

$$ \begin{array}{l}{\varepsilon}_{11}\ge 0\Rightarrow {\varepsilon}_{11}={\varepsilon}_{11}^{0\mathrm{T}};\kern2em {\varepsilon}_{11}<0\Rightarrow {\varepsilon}_{11}={\varepsilon}_{11}^{0\mathrm{C}}\\ {}{\varepsilon}_{22}\ge 0\Rightarrow {\varepsilon}_{22}={\varepsilon}_{22}^{0\mathrm{T}};\kern1.3em {\varepsilon}_{22}<0\Rightarrow {\varepsilon}_{22}={\varepsilon}_{22}^{0\mathrm{C}}\end{array} $$

(2.4)

where, *σ*
_{11}, *σ*
_{22}, *ε*
_{11} and *ε*
_{22} are the tensile or compressive stress and strain in the axial (1) and transverse (2) directions, respectively. *σ*
_{12} and *γ*
_{12} are the in-plane shear stress and strain. *X*
^{T}, *X*
^{C}, *Y*
^{T}, and *Y*
^{C} represent tensile and compressive strength for failure prediction in their respective directions. *S*
_{
12
} and \( {\gamma}_{12}^0 \) denote the in-plane shear strength and failure strain, respectively.

To predict the interactive effect on various damage modes, the stress/strain based failure envelop was developed by Tsai-Hill criterion (Tsai 1965) (Eqs. 2.5–2.6) and Tsai-Wu (Tsai and Wu 1971) criterion (Eqs. 2.7–2.8).

$$ \begin{array}{l}f={F}_{11}{\sigma}_{11}^2+{F}_{22}{\sigma}_{22}^2+{F}_{66}{\sigma}_{12}^2+2{F}_{12}{\sigma}_{11}{\sigma}_{22}+{F}_1{\sigma}_{11}+{F}_2{\sigma}_{22}+{F}_6{\sigma}_{12}\\ {}{F}_{11}=\frac{1}{X^2}, {F}_{22}=\frac{1}{Y^2}, {F}_{12}=-\frac{1}{2{X}^2}, \kern0.5em {F}_{66}=\frac{1}{S_{12}^2}\end{array} $$

(2.5)

$$ {F}_1=0,\kern1.25em {F}_2=0,\kern0.75em {F}_6=0 $$

(2.6)

where *X, Y* and *S*
_{
12
} in Eq. (2.6) have the same definition as in Eq. (2.1).

$$ {F}_{11}=\frac{1}{X^T{X}^C},\kern2.5em {F}_{22}=\frac{1}{Y^T{Y}^C},\kern2em {F}_{66}=\frac{1}{S_{12}^2} $$

$$ {F}_1=\frac{1}{X^T}-\frac{1}{X^C},\kern1.25em {F}_2=\frac{1}{Y^T}-\frac{1}{Y^C},\kern0.75em {F}_6=0 $$

(2.7)

with the coefficient F_{12} defined as:

$$ {F}_{12}=\frac{F_{12}^{\ast }}{\sqrt{X^T{X}^C{Y}^T{Y}^C}} $$

(2.8)

\( {F}_{12}^{\ast } \) in the above equation ranges from −1 to 1, and can be obtained by fitting the equibiaxial experimental data. Hoffman (Hoffman 1967) derived the same coefficients as in Eq. (2.7) but defined *F*
_{
12
} as

$$ {F}_{12}=-\frac{1}{2{X}^T{X}^C} $$

(2.9)

Farooq et al. (Farooq and Myler 2016) attempted to use the Tsai-Hill and Tsai-Wu criteria to predict the ply level failure in their research and compared to the experimental results. However, the Tsai-Hill criterion underestimated the maximum loads compared to other failure theories. This might be due to its transverse compressive strength that is far larger than transverse tensile strength and therefore the Tsai-Hill criterion failed to predict ply based level failure. The Tsai-Wu criterion showed similar scenario that ply-by-ply failure index quantities never exceeded one, which indicated no failure occurred. The most likely reason might be due to these polynomial based criteria could not use through-thickness stresses to predict ply failure. Therefore, it can be seen the polynomial failure criteria are not ideal to model composite failure and a ply-by-ply progressive damage method is thus more popular in developing the predictive damage model.

#### Ply based progressive damage method

### Damage initiation

Ply based individual damage mode can be generally modelled for two stages: initiation and evolution. Hashin criteria have been widely used in academic research and industry (Eqs. (2.10)–(2.13)).

Fibre tension (*σ*
_{11} ≥ 0):

$$ {F}_{ft}={\left(\frac{\sigma_{11}}{X^T}\right)}^2+\kappa {\left(\frac{\sigma_{12}}{S_{12}}\right)}^2=1 $$

(2.10)

Fibre compression (*σ*
_{11} < 0):

$$ {F}_{fc}={\left(\frac{\sigma_{11}}{X^C}\right)}^2=1 $$

(2.11)

Matrix tension (*σ*
_{22} ≥ 0):

$$ {F}_{mt}={\left(\frac{\sigma_{22}}{Y^T}\right)}^2+{\left(\frac{\sigma_{12}}{S_{12}}\right)}^2=1 $$

(2.12)

Matrix compression (*σ*
_{22} < 0):

$$ {F}_{mc}={\left(\frac{\sigma_{22}}{2{S}_{23}}\right)}^2+\left[{\left(\frac{Y^C}{2{S}_{23}}\right)}^2-1\right]\frac{\sigma_{22}}{Y^C}+{\left(\frac{\sigma_{12}}{2{S}_{12}}\right)}^2=1 $$

(2.13)

In the above equations, *σ*
_{
ij
}
*(i, j = 1, 2, 3)* are the stress components defined in the material coordinate system. *X*
^{T} and *X*
^{C} denote the fibre tensile and compressive strengths, *Y*
^{T} and *Y*
^{C} are the transverse tensile and compressive strength, *S*
_{
i, j
}
*(i, j = 1, 2, 3)* denote the longitudinal and transverse shear strengths of the composite, respectively. The coefficient *κ* in Eq. (2.10) accounts for the contribution of shear stress to fibre tensile failure and generally ranges between 0 and 1. Hashin criteria comprehensively considers the various damage modes such as fibre ruptures in tension, matrix cracking due to transverse tension and shear, fibre compressive damage (buckling or kinking) and matrix crushing under transverse compression and shear effect (Hoffman 1967). Hashin criteria have been proved that it is effective method to predict ply based damage initiation (Shi et al. 2012; Feng and Aymerich 2014; Farooq and Myler 2016; Johnson et al. 2006; Liu et al. 2016; Li et al. 2014b), however, based on the experimental observation, the matrix compressive crack is always formed with an angle through the ply thickness whereas Hashin criteria are less able to simulate this process. Puck and Schurmann (1998) developed a damage model for transverse compression. They proposed to use the failure criteria of Mohr (Salencon 2001) instead of the yield criterion of von Mises which is normally applied. Puck’s damage criterion for compression damage mode can be expressed as Eq. (2.14).

$$ {F}_{mc}={\left(\frac{\sigma_{NT}}{S_{23}^A+{\mu}_{NT}{\sigma}_{NN}}\right)}^2+{\left(\frac{\sigma_{NL}}{S_{12}+{\mu}_{NL}{\sigma}_{NN}}\right)}^2=1 $$

(2.14)

In Eq. (2.14) *σ*
_{
ij
} (*i, j = L, T, N*) are the stresses *σ*
_{
ij
} (*i, j = 1, 2, 3*) rotated to the fracture plane, by reference to the axes shown in Fig. 1 (Shi 2014):

$$ {\sigma}_{NN}={\sigma}_2{m}^2+{\sigma}_3\left(1-{m}^2\right)+2{\tau}_{23}mn $$

(2.15)

$$ {\sigma}_{NT}=-{\sigma}_2mn+{\sigma}_3mn+{\tau}_{23}\left(2{m}^2-1\right) $$

(2.16)

$$ {\sigma}_{TT}={\sigma}_2\left(1-{m}^2\right)+{\sigma}_3{m}^2-2{\tau}_{23}mn $$

(2.17)

$$ {\sigma}_{NL}={\tau}_{12}m+{\tau}_{13}n $$

(2.18)

$$ {\sigma}_{LT}=-{\tau}_{12}n+{\tau}_{13}m $$

(2.19)

$$ {S}_{23}^A=\frac{Y_C}{2}\left(\frac{1- \sin \varphi }{cos\varphi}\right) $$

(2.20)

$$ \varphi =2\alpha -{90}^{\mathrm{o}} $$

(2.21)

where *m = cos(α)* and *n = sin(α)* in Eqs. (2.15)–(2.19). \( {S}_{23}^A \) is the transverse shear strength along the fracture plane, which can be determined by the transverse compression strength *Y*
_{
C
} and the angle of the fracture plane as shown in Eqs. (2.20) and (2.21).

The friction coefficients *μ*
_{
NT
} and *μ*
_{
NL
} in Eq. (2.14) can be defined based on the material friction angle (see Eq. (2.21)), *φ*, and material properties by reference to the Mohr failure criteria.

$$ {\mu}_{NT}= tan\varphi = \tan \left(2\alpha -{90}^{\mathrm{o}}\right) $$

(2.23)

$$ {\mu}_{NL}=\frac{\mu_{NT}}{S_{23}^A}{S}_{12} $$

(2.24)

In general, the fracture plane is oriented at *α* = 53° through the thickness direction under a uniaxial compressive load (Puck and Schurmann 1998). However, impact loading usually leads to various values of the fracture angle which can be numerically determined by executing the damage initiation index in a swept angle range. This method was performed by Shi et al. (2012), Faggiani and Falzon (2010) and Feng and Aymerich (2014) where the fracture plane and increase of shear strength due to normal compressive stress acted on fracture plane have been successfully simulated.

Besides, Sun et al. (1996) also appropriately modified Hashin’s criteria to improve the accuracy of modelling matrix compressive damage mode:

$$ {F}_{mc}={\left(\frac{\sigma_{22}}{Y^C}\right)}^2+{\left(\frac{\sigma_{12}}{S_{12}-{\varsigma \sigma}_{22}}\right)}^2=1 $$

(2.25)

where *ς* is a constant determined experimentally and generally regarded as an internal material friction parameter.

Camanho et al. (2005) developed a failure criterion called LaRC03 based on continuum damage mechanics (CDM) method. In order to demonstrate their damage model, the failure envelope plotted between the transverse stress *σ*
_{22} and in-plane shear stress *τ*
_{12} was generated and compared with other criteria. All predicted results have been validated by World Wide Failure Exercise (WWFE) test results. For the tensile damage mode, it rarely find big difference for simulation accuracy by proposed criteria, except the maximum stress criterion because the maximum stress criterion only defines the single failure without interaction modelled. Therefore, it cannot give a satisfactory prediction of the failure of composites, especially when damage is matrix dominated.

For matrix compression damage mode, it was illustrated as similar conclusion from experimental observation. Due to the fracture plane existed, Puck’s envelope offered the most accurate prediction while both Sun’s criterion (Sun et al. 1996) and LaRC03 (Davila et al. 2005) also demonstrate an appropriately predictive capacity. Hashin’s criteria showed an underestimation due to fracture plane was ignored. Topac et al. (2017) employed the damage criteria LaRC04 developed by Pinho et al. (2005) to further improve the accuracy of modelling matrix cracking using such continuum damage mechanics based failure criteria where the transverse cracks through the thickness have been accurately simulated compared to experimental measurement under the relatively fine meshing strategy.

### Damage evolution

To predict the damage evolution within composites, a material degradation strategy should be generally defined for each individual damage mode. The most common way is to apply a degradation parameter corresponding to the damage mode for simulating the softening effect. The parameter for degradation of material might be from experimental measurement. Tita et al. (2008) reduced the stiffness by using appropriate factors with respect to the various failure modes observed experimentally. For instance, the transverse Young’s modulus E_{22} and the in-plane Poisson’s ratio ν_{12} were reduced to zero directly to represent the complete damage in their work. Farooq et al. also defined the degraded modulus for composite IM7/8552 with complete failure of each single damage mode such as the tensile moduli of E_{1}, E_{2} and E_{3} as well as in-plane and out-of-plane shear moduli to simulate the impact induced damage within the laminate (Farooq and Myler, 2015, 2016). In order to find the accurate degradation parameters for tensile and compressive moduli of the composite, Louca et al. performed cyclic tests and eventually concluded the average values for stiffness loss once the composite had failed (Johnson et al. 2006). It can be seen that this method is straightforward and always accurate for predicting the composite moduli degradation, however, experimental measurements have to be necessary for different damage modes and different kinds of fibre and resin systems. Moreover, this method cannot numerically demonstrate a progressive damage evolution, which could be a constraint to composite engineers, since it relies heavily on experiments.

A progressive damage evolution based on fracture mechanics was reported by Chang and Chang (1987), Chang et al. (1991). Considering the matrix failure, they proposed a degradation law to reduce the moduli *E*
_{
11
} and *G*
_{
12
} based on an exponential decaying, but other moduli were reduced abruptly to zero once the damage initiation happened.

$$ \begin{array}{l}\ {E}_{11}^d={E}_{11}\ \mathit{\exp}\left[-\left(\frac{A}{A_o}\right)H\right]\\ {}{G}_{12}^d={G}_{12}\ \mathit{\exp}\left[-\left(\frac{A}{A_o}\right)H\right]\end{array} $$

(2.26)

where *A* wass the area of the damage zone; *A*
_{
o
} was the area of the interaction zone of the fibre failure from Chang and Chang (1987), Chang et al. (1991). *H* was a factor to control the degradation of the material stiffness. Following this work, Matzenmiller et al. (1995) developed a damage model called ‘MLT model’ for the non-linear analysis of the composite laminates. They constructed the model using damage variables with respect to the individual failure modes in the material principal directions. The model assumes that each unidirectional lamina in the composite acted as a continuum irrespective of the damage state. The damage growth was controlled based on a Weibull distribution. The post-damage softening behaviour of the composite can be predicted by an exponential function:

$$ d=1-\mathit{\exp}\left(-\frac{1}{me}{\left(\frac{E^0\varepsilon }{X}\right)}^m\right) $$

(2.27)

where *E*
^{0} was the individually fibre or transverse modulus, *ε* was the strain related to the progressive damage at different time steps, “*e”* was a mathematical constant. *X* was the tensile or compressive strength with regard to the different damage modes in the different loading directions. “*m*” was the strain softening parameter during damage progression, which is a key factor to determine the accuracy of prediction eventually. In general, a high value of “*m*” could result in brittle failure of the material while a low value of “*m*” indicated a ductile failure response that could lead to a high absorbed energy due to damage.

It is an effective approach to model the damage growth for a composite laminate using strain softening parameter “*m*” in MLT model (Williams and Vaziri 2001; Gama and Gillespie 2011; Tabiei and Aminjikarai 2009; Jung et al. 2017). The appropriate value of “*m*” was usually related to the mesh size and load conditions. The value of “*m*” for various damage modes can be determined using uniaxial tensile or compressive tests. A set of values of “*m*” can be applied for the strain softening to model the complex failure process of composites for different damage modes. Obviously, the “*m*” value has a strong effect on determining the accuracy of prediction of damage progression. An inappropriate value of “*m*” could give rise to numerical difficulties to simulate the process of damage growth and eventually lead to the unrealistic results (Williams and Vaziri 2001). For example, if a relatively small value of “*m”* was applied in the damage model for composite laminate, it will exhibit a ductile behaviour which is obviously contradictory to its brittle characteristics. Therefore, in order to avoid experiment dependent “*m”,* the damage variable was proposed by an exponential function that only referred to characteristic length and composite properties (Dassault Systemes Simulia Corp 2008).

$$ {d}_{\alpha }=1-\frac{1}{r_{\alpha }}\mathit{\exp}\left[-{A}_{\alpha}\left({r}_{\alpha }-1\right)\right]\kern2em and\kern1.25em {d}_{\alpha}\ge 0 $$

where the coefficient A_{α} was defined as

$$ {A}_{\alpha }=\frac{2{g}_0^{\alpha }{L}_c}{G_{IC}^{\alpha }-{g}_0^{\alpha }{L}_c}\kern1.25em with\kern2em {g}_0^{\alpha }=\frac{X_{\alpha}^2}{2{E}_{\alpha }} $$

(2.29)

All parameters used in this exponential function correspond to material properties and therefore the experiment for evaluating “*m”* can be effectively avoided. Schwab et al. employed this method to successfully predict the impact induced damage with perforation for woven composites under intermediate velocity impact (Schwab et al., 2015, 2016; Schwab and Pettermann 2016). More specifically, using this model, they simulated the impact behaviour of a large composite fan containment of a jet engine impacted by deformable bodies (Schwab and Pettermann 2016).

In addition, the energy based damage mechanics approach was also extensively developed and used to model the progressive failure in composite laminates (Iannucci and Willows 2006, 2007; Iannucci and Ankersen 2006). The damage model used the strains at damage onset and at complete damage, effectively defined the damage variable for degradation and accurately captured the damage progression for composite laminates (Shi et al. 2012; Donadon et al. 2008; Faggiani and Falzon 2010; Feng and Aymerich 2014; Liu et al. 2016; Topac et al. 2017), which a typical damage variable defined for tensile damage mode is shown in Eq. (2.30).

$$ {d}_{1,2}^t=\frac{\varepsilon_{1,2}^{ft}}{\varepsilon_{1,2}^{ft}-{\varepsilon}_{1,2}^{0t}}\left(1-\frac{\varepsilon_{1,2}^{0t}}{\varepsilon_{1,2}}\right) $$

(2.30)

where the subscript *1* and *2* denoted the fibre and transverse directions, respectively; \( {\upvarepsilon}_{1,2}^{0\mathrm{t}} \) was the strain when the damage initiation condition was fulfilled. Due to the irreversibility of the damage variable, the strain calculated at each time step could be updated in comparison with the strain at damage initiation \( {\varepsilon}_{1,2}=\mathit{\max}\left({\varepsilon}_{1,2},{\varepsilon}_{1,2}^{0t}\right) \) in Eq. (2.30). In order to avoid a zero or even negative energy absorption, the complete failure strain could be also defined to be greater than the initial failure strain \( {\varepsilon}_{1,2}^{ft}>{\varepsilon}_{1,2}^{0t} \).

The failure initiation strain is given by the following equation:

$$ {\varepsilon}_{1,2}^{0t}=\frac{\sigma_{1,2}^T}{E_{1,2}} $$

(2.31)

For tensile failure in fibres, *σ*
^{T} was the tensile strength *X*
^{T} while *Y*
^{T} was used for the matrix tensile failure mode.

*ε*
^{ft} can be derived from the fracture energy \( {G}_{1,2}^T \) for the individual failure mode, the failure strength of the material and the characteristic length:

$$ {\varepsilon}_{1,2}^{ft}=\frac{2{G}_{1,2}^T}{\sigma^T{l}^{\ast }} $$

(2.32)

In Eq. (2.32), *l*
^{*} is the characteristic length which can maintain an energy release rate per unit area of rack constant and also keep the predicted results independent of the mesh size in a FE model. The definition of characteristic length can be found for more details in (Shi 2014; Bazǎnt and Oh 1983; Olivier 1989; Pinho 2005).

Shi et al. identified the most effective damage evolution laws for individual damage mode and developed the damage model including non-linear shear with damage for prediction of low velocity impact induced damage of reinforced carbon composite laminates (Shi et al. 2012). The damage model was implemented into commercial FE code ABAQUS/EXPLICIT by user defined subroutine VUMAT. The various impact energies were tested to explore the multi-damage modes of the composite, which was experimentally recorded by X-ray radiography. The damage of the laminate compared well with X-ray images taking at different impact loads (see Fig. 2). However, in Fig. 3 it can be seen that there is a difference between experimental measurement and numerical prediction, since splitting was not simulated but observed in the X-ray radiographs. The solution to this issue was proposed by Shi et al. (2014a, 2014b), Shi and Soutis (2015) using cohesive zone elements, which will be discussed in the following section.