On the equivalence between the $s$-method, the XFEM and the ply-by-ply discretization for delamination analyses of laminated composites

Abstract

Two hierarchical approaches, the $s$-method and the extended finite element method (XFEM), are compared to the classical ply-by-ply discretization approach in terms of their effectiveness in modeling delamination in laminated composites. In the two hierarchical approaches, a smooth approximation field based on the mesh made of through-the-thickness solid laminated elements is first introduced to resolve a delamination-free response of the composite structure. The initiation and propagation of delamination is modeled by either superposition of element patches (s-method) or by enrichment functions (XFEM). A cohesive zone model is employed to model decohesion at the inter-ply interfaces. In terms of representing strong discontinuities, the two hierarchical methods have been shown to represent an identical approximation space as the classical ply-by-ply discretization approach, even though the s-method gives rise to sparser matrix structure. In terms of representing weak discontinuities, the s-method has been shown to be equivalent to the ply-by-ply discretization approach and provides a seamless transition from weak to strong discontinuity.

Appendices

Appendix 1: 20-node Isoparametric serendipity laminated element

A solid element employed in this study is a 20-node isoparametric serendipity laminated element, which contains multiple plies in a single element. Numerical integration of the weak form is carried out over each ply to account for material heterogeneity within an element. All the plies are stacked along the same parametric coordinate axis and all the interfaces are planes perpendicular to the normal axis in the parametric space. Figure 15 shows an example of the solid laminated element in both its parametric and physical spaces. Without loss of generality, the stack direction of the plies is assumed to be along the $\zeta$ axis in the parametric space of the element.

Fig. 15

A 20-node isoparametric serendipity laminate element in two different spaces: a the parametric space; b the physical space

In the present work, the weak form is integrated in the element domain using $3\times 3$ Gauss quadrature in $\xi \,\eta$ plane and Simpson’s rule in the $\zeta$ direction in each ply in the parametric space of the element. The integration scheme results in 27 integration points for each element ply.

Appendix 2: Cohesive zone model for inter-ply interfacial behavior

The cohesive zone model (CZM) describes decohesion by a traction-separation relation at inter-ply interfaces. The CZM considered in the present manuscript is a combination of a bilinear cohesive zone model proposed by Song et al. (2006) and friction-sliding model of Tvergaard (1990) in post failure regime.

Prior to introducing the traction-separation relation, we will define nine model parameters:$\sigma _{\mathrm{cn}}$, $\sigma _{\mathrm{cs}}$, $\sigma _{\mathrm{ct}}$, $G_\mathrm{I}$, $G_{\mathrm{II}}$, $G_{\mathrm{III}}$, $\mu _\mathrm{s}$, $\mu _\mathrm{t}$ and $\lambda _{\mathrm{cr}}$. $\sigma _{\mathrm{cn}}$, $\sigma _{\mathrm{cs}}$, and $\sigma _{\mathrm{ct}}$ are mode I, II, and III interface strengths, respectively; $G_\mathrm{I}$, $G_{\mathrm{II}}$, and $G_{\mathrm{III}}$ mode I, II, and III fracture energies, respectively; $\mu _\mathrm{s}$ and $\mu _\mathrm{t}$ mode II and III friction coefficients, respectively; $\lambda _{\mathrm{cr}}$ the non-dimensional critical value of effective interface separation to be subsequently defined.

A non-dimensional scalar effective separation is defined as

$$\begin{aligned} \lambda _\mathrm{e} =\left\{ {\begin{array}{ll} \sqrt{\left( {\frac{\delta _\mathrm{n} }{\delta _{\mathrm{cn}} }} \right) ^{2}+\left( {\frac{\delta _\mathrm{s} }{\delta _{\mathrm{cs}} }} \right) ^{2}+\left( {\frac{\delta _\mathrm{t} }{\delta _{\mathrm{ct}} }} \right) ^{2}}, &{}\quad \delta _\mathrm{n} \ge 0 \\ \sqrt{\left( {\frac{\delta _\mathrm{s} }{\delta _{\mathrm{cs}} }} \right) ^{2}+\left( {\frac{\delta _\mathrm{t} }{\delta _{\mathrm{ct}} }} \right) ^{2}}, &{}\quad \delta _\mathrm{n} <0 \\ \end{array}} \right. \end{aligned}$$

(32)

where the subscripts $\hbox {e}$ and $\hbox {c}$ denote effective and critical values, respectively; $\delta _\mathrm{n}$, $\delta _\mathrm{s}$, and $\delta _\mathrm{t}$ are the separations along the normal and two tangential directions, respectively; $\delta _{\mathrm{cn}}$, $\delta _{\mathrm{cs}}$, and $\delta _{\mathrm{ct}}$ are the critical separations of totally delaminated interface along the normal and two tangential directions, respectively. Assuming bilinear traction-separation relations, $\delta _{\mathrm{cn}}$,$\delta _{\mathrm{cs}}$, and $\delta _{\mathrm{ct}}$ can be computed from the corresponding fracture energies and strengths by

$$\begin{aligned} \delta _{\mathrm{cn}} =\frac{2G_\mathrm{I} }{\sigma _{\mathrm{cn}} }; \delta _{\mathrm{cs}} =\frac{2G_{\mathrm{II}} }{\sigma _{\mathrm{cs}} }; \delta _{\mathrm{ct}} =\frac{2G_{\mathrm{III}} }{\sigma _{\mathrm{ct}} } \end{aligned}$$

(33)

Let $\lambda _{\mathrm{max}}$ be the maximum value of the effective separation $\lambda _\mathrm{e}$ throughout the deformation history. Assuming that damage is irreversible, $\lambda _{\mathrm{max}}$ serves as a state variable indicating the local state of damage. When $\lambda _{\mathrm{max}} \ge 1$, the interface is totally damaged forming a cohesive traction-free interface crack.

The onset of interface damage is defined by $\lambda _{\mathrm{cr}}$. When $0\le \lambda _{\mathrm{max}} <\lambda _{\mathrm{cr}}$, the interface is intact and the corresponding traction-separation response is linear elastic

$$\begin{aligned} t_\mathrm{n} =\frac{\sigma _{\mathrm{cn}} }{\lambda _{\mathrm{cr}} }\left( {\frac{\delta _\mathrm{n} }{\delta _{\mathrm{cn}} }} \right) \hbox {; }t_s =\frac{\sigma _{\mathrm{cs}} }{\lambda _{\mathrm{cr}} }\left( {\frac{\delta _\mathrm{s} }{\delta _{\mathrm{cs}} }} \right) \hbox {; }t_t =\frac{\sigma _{\mathrm{ct}} }{\lambda _{\mathrm{cr}} }\left( {\frac{\delta _\mathrm{t} }{\delta _{\mathrm{ct}} }} \right) \nonumber \\ \end{aligned}$$

(34)

where $t_\mathrm{n}$, $t_\mathrm{s}$, and $t_\mathrm{t}$ are mode I, II, and III tractions. When $\lambda _{\mathrm{cr}} \le \lambda _\mathrm{e} \le \lambda _{\mathrm{max}}$, the interface is damaged. The corresponding traction-separation relation is given by

$$\begin{aligned} t_\mathrm{n}&= \left\{ {\begin{array}{ll} \sigma _{\mathrm{cn}} \frac{1-\lambda _{\mathrm{max}} }{1-\lambda _{\mathrm{cr}} }\frac{1}{\lambda _{\mathrm{max}} }\left( {\frac{\delta _\mathrm{n} }{\delta _{\mathrm{cn}} }} \right) , &{}\delta _\mathrm{n} \ge 0 \\ \frac{\sigma _{\mathrm{cn}} }{\lambda _{\mathrm{cr}} }\left( {\frac{\delta _\mathrm{n} }{\delta _{\mathrm{cn}} }} \right) , &{}\delta _\mathrm{n} <0 \\ \end{array}} \right. \end{aligned}$$

(35)

$$\begin{aligned} t_\mathrm{s}&= \sigma _{\mathrm{cs}} \frac{1-\lambda _{\mathrm{max}} }{1-\lambda _{\mathrm{cr}} }\frac{1}{\lambda _{\mathrm{max}} }\left( {\frac{\delta _\mathrm{s} }{\delta _{\mathrm{cs}} }} \right) ;\nonumber \\ t_\mathrm{t}&= \sigma _{\mathrm{ct}} \frac{1-\lambda _{\mathrm{max}} }{1-\lambda _{\mathrm{cr}} }\frac{1}{\lambda _{\mathrm{max}} }\left( {\frac{\delta _\mathrm{t} }{\delta _{\mathrm{ct}} }} \right) \end{aligned}$$

(36)

Note that when $\delta _\mathrm{n} <0$, $t_\mathrm{n}$ exhibits linear elastic response even though the interface may have been damaged already. The linear elastic response is employed herein to prevent the penetration between the contiguous plies.

When $\lambda _{\mathrm{max}} \ge 1$, the interface is totally delaminated and all cohesive tractions vanish. If the interface is under tensile loading $\left( {\delta _\mathrm{n} >0} \right)$, then $t_\mathrm{n} =t_\mathrm{s} =t_\mathrm{t} =0$. If the interface is under compressive loading $(\delta _\mathrm{n} \le 0)$, the traction-separation response is governed by a static friction law (Tvergaard 1990) as

$$\begin{aligned} t_\mathrm{n}&= \frac{\sigma _{\mathrm{cn}} }{\lambda _{\mathrm{cr}} }\left( {\frac{\delta _\mathrm{n} }{\delta _{\mathrm{cn}} }} \right) ; \dot{t}_\mathrm{s} = \frac{100\mu _\mathrm{s} \left| {t_n } \right| }{\delta _{\mathrm{cs}} }\dot{\delta }_\mathrm{s} ;\nonumber \\ \dot{t}_\mathrm{t}&= \frac{100\mu _\mathrm{t} \left| {t_n } \right| }{\delta _{\mathrm{ct}} }\dot{\delta }_\mathrm{t} \left( {\delta _\mathrm{n} \le 0} \right) \end{aligned}$$

(37)

Appendix 3: Proof of exact reproduction of shape functions of the 20-node isoparametric serendipity elements by local elements of the same type

In this section, we show that any global shape function of the 20-node isoparametric serendipity element can be exactly reproduced by local elements of the same type if the nodal coordinates of the local elements in the parametric space of the global element satisfy

$$\begin{aligned} \left\{ {\begin{array}{l} \xi _i^\mathrm{L} =a_1 \xi _i^\mathrm{G} +a_0 \\ \eta _i^\mathrm{L} =b_1 \eta _i^\mathrm{G} +b_0 \\ \zeta _i^\mathrm{L} =c_1 \zeta _i^\mathrm{G} +c_0 \\ \end{array}} \right. \left( {i=1,2,\ldots ,20} \right) \end{aligned}$$

(38)

where $\left( {\xi _i^\mathrm{G}, \eta _i^\mathrm{G}, \zeta _i^\mathrm{G} } \right)$ and $\left( {\xi _i^\mathrm{L}, \eta _i^\mathrm{L}, \zeta _i^\mathrm{L}}\right)$ are the coordinates of the $i\hbox {th}$ nodes of the global and local elements, respectively, in the parametric space of the global element; $a_0, a_1, b_0, b_1, c_0, c_1$ are constants and $a_1, b_1, c_1$ are not zero.

In order to prove the above statement, the shape functions of the local element are investigated in the parametric space of the global element. At first, one can verify that the interpolation based on the shape functions of the 20-node isoparametric serendipity element results in a polynomial of the form:

$$\begin{aligned}&P\left( {r,s,t} \right) =C_0 +C_1 r+C_2 s+C_3 t \nonumber \\&\quad +\,C_4 r^{2}+C_5 s^{2}+C_6 t^{2}+C_7 st+C_8 tr+C_9 rs\nonumber \\&\quad +\,C_{10} st^{2}+C_{11} s^{2}t+C_{12} tr^{2}+C_{13} t^{2}r\nonumber \\&\quad +\,C_{14} rs^{2}+C_{15} r^{2}s+C_{16} rst \nonumber \\&\quad +\,C_{17} r^{2}st+C_{18} rs^{2}t+C_{19} rst^{2} \end{aligned}$$

(39)

where $r$, $s$, and $t$ are the coordinates in the parametric space of the element; $C_i\,\left( {i=0,1,2,\ldots ,19} \right)$ are constant coefficients.

It is evident that both the global and local elements use polynomials in the form described in Eq. (39) in their respective parametric spaces. Next, it will be shown that the form of the polynomial obtained through interpolation based on the local element can be described by Eq. (39) in the parametric space of the global element. As an isoparametric element, the local element has the following coordinate transformation from its own parametric space to the parametric space of the global element:

$$\begin{aligned} \left\{ {\begin{array}{l} \xi =\sum _{i=1}^{20} {N_i \left( {{\xi }',{\eta }',{\zeta }'} \right) \xi _i^\mathrm{L} } \\ \eta =\sum _{i=1}^{20} {N_i \left( {{\xi }',{\eta }',{\zeta }'} \right) \eta _i^\mathrm{L} } \\ \zeta =\sum _{i=1}^{20} {N_i \left( {{\xi }',{\eta }',{\zeta }'} \right) \zeta _i^\mathrm{L} } \\ \end{array}} \right. \end{aligned}$$

(40)

where $\left( {\xi , \eta , \zeta } \right)$ and $\left( {{\xi }',{\eta }',{\zeta }'} \right)$ are coordinates in the parametric spaces of the global and local elements, respectively; $N_i$ denotes the standard shape function associated with the $i\hbox {th}$ node of the 20-node isoparametric serendipity element. After substituting Eq. (38) into Eq. (40), the following expressions are obtained:

$$\begin{aligned} \left\{ {\begin{array}{l} \xi =a_1 \sum _{i=1}^{20} {N_i \left( {{\xi }',{\eta }',{\zeta }'} \right) \xi _i^\mathrm{G} +a_0 \sum _{i=1}^{20} {N_i \left( {{\xi }',{\eta }',{\zeta }'} \right) } } \\ \eta =b_1 \sum _{i=1}^{20} {N_i \left( {{\xi }',{\eta }',{\zeta }'} \right) \eta _i^\mathrm{G} +b_0 \sum _{i=1}^{20} {N_i \left( {{\xi }',{\eta }',{\zeta }'} \right) } } \\ \zeta =c_1 \sum _{i=1}^{20} {N_i \left( {{\xi }',{\eta }',{\zeta }'} \right) \zeta _i^\mathrm{G} +c_0 \sum _{i=1}^{20} {N_i \left( {{\xi }',{\eta }',{\zeta }'} \right) } } \\ \end{array}} \right. \end{aligned}$$

(41)

Since the shape functions $N_i\,\left( {i=1,2,\ldots ,20} \right)$ are partitions of unity, $\sum \limits _{i=1}^{20} {N_i \left( {{\xi }',{\eta }',{\zeta }'} \right) }$ in the equation above is equal to one. In addition, it is evident that the global and local elements have the same nodal coordinates in their respective parametric spaces. $\xi _i^\mathrm{G}$, therefore, can be replaced by ${\xi }'_i$, which denotes the $\xi$ coordinate of the $i\hbox {th}$ node of the local element in its parametric space in Eq. (41). ${\xi }'_i$ can then be interpreted as the value of the function $f\left( {{\xi }',{\eta }',{\zeta }'} \right) ={\xi }'$ at the point $\left( {{\xi }'_i, {\eta }'_i, {\zeta }'_i}\right)$. The expression $\sum \limits _{i=1}^{20} {N_i \left( {{\xi }',{\eta }',{\zeta }'} \right) \xi _i^\mathrm{G} }$ in Eq. (41) can in turn be interpreted as the interpolation approximation of the function $f\left( {{\xi }',{\eta }',{\zeta }'} \right) ={\xi }'$ through the local element. According to Eq. (39), the polynomial obtained via interpolation based on the shape functions of the 20-node isoparametric serendipity element is complete up to order 2 in the parametric space. The linear function $f\left( {{\xi }',{\eta }',{\zeta }'} \right) ={\xi }'$ can thus be exactly reproduced by the interpolation, i.e. $\sum \limits _{i=1}^{20} {N_i \left( {{\xi }',{\eta }',{\zeta }'} \right) \xi _i^\mathrm{G} } ={\xi }'$. Analogously, the two expressions, $\sum \limits _{i=1}^{20} {N_i \left( {{\xi }',{\eta }',{\zeta }'} \right) \eta _i^\mathrm{G} }$ and $\sum \limits _{i=1}^{20} {N_i \left( {{\xi }',{\eta }',{\zeta }'} \right) \zeta _i^\mathrm{G} }$, in Eq. (41) can be written as ${\eta }'$ and ${\zeta }'$, respectively. Equation (41) can eventually be written in the following simplified form:

$$\begin{aligned} \left\{ {\begin{array}{l} {\xi }'=\frac{1}{a_1 }\left( {\xi -a_0 } \right) \\ {\eta }'=\frac{1}{b_1 }\left( {\eta -b_0 } \right) \\ {\zeta }'=\frac{1}{c_1 }\left( {\zeta -c_0 } \right) \\ \end{array}} \right. \end{aligned}$$

(42)

Since Eq. (39) describes the general form of the polynomial resulting from interpolation based on the shape functions of the local element in its parametric space, Eq. (42) can be substituted into Eq. (39) to obtain the general form of the polynomial in the parametric space of the global element. Herein let $r={\xi }'$, $s={\eta }'$, and $t={\zeta }'$. Noticing that the coordinate transformation shown in Eq. (42) is linear, the substitution makes each term in Eq. (39) become a polynomial of $\xi$, $\eta$, and $\zeta$ with the same order as the original term. The resulting overall polynomial, therefore, has the same order as the original polynomial $P\left( {r,s,t} \right)$. In order to prove that the resulting polynomial has the same form as the original one, one has to show that each term in the resulting polynomial has the same form as one of the terms in the original polynomial. Since the original polynomial $P\left( {r,s,t} \right)$ is complete up to order 2, all the terms with orders less than 3 in the resulting polynomial must have their respective corresponding terms in $P\left( {r,s,t} \right)$. Consequently, one only needs to pay attention to the terms with orders larger than or equal to 3 in the resulting polynomial. In fact, these terms can only result from those terms with orders 3 and 4 in the original polynomial considering the linearity of the coordinate transformation. Moreover, due to the similarity among the three coordinates in Eqs. (39) and (42), it is reasonable to expand and check only the three representative terms, $st^{2}$, $rst$, and, $r^{2}st$, in Eq. (39). By substituting Eq. (42) into the three terms, yields

$$\begin{aligned}&st^{2}={\eta }'{\zeta }'^{2}=\frac{1}{b_1 }\left( {\eta -b_0 } \right) \left[ {\frac{1}{c_1 }\left( {\zeta -c_0 } \right) } \right] ^{2} \nonumber \\&= \frac{1}{b_1 c_1^2 }\left( \eta \zeta ^{2}-b_0 \zeta ^{2}-2c_0 \eta \zeta +c_0^2 \eta +2b_0 c_0 \zeta \right. \nonumber \\&\left. \quad -b_0 c_0^2 \right) \end{aligned}$$

(43)

$$\begin{aligned}&rst={\xi }'{\eta }'{\zeta }'=\frac{1}{a_1 b_1 c_1 }\left( {\xi -a_0 } \right) \left( {\eta -b_0 } \right) \left( {\zeta -c_0 } \right) \nonumber \\&= \frac{1}{a_1 b_1 c_1 }\left( {\begin{array}{lll} \xi \eta \zeta -a_0 \eta \zeta -b_0 \zeta \xi -c_0 \xi \eta + \\ b_0 c_0 \xi +c_0 a_0 \eta +a_0 b_0 \zeta -a_0 b_0 c_0 \\ \end{array}} \right) \end{aligned}$$

(44)

$$\begin{aligned} r^{2}st&= {\zeta }'^{2}{\eta }'{\zeta }'=\left[ {\frac{1}{a_1 }\left( {\xi -a_0 } \right) } \right] ^{2}\frac{1}{b_1 }\left( {\eta -b_0 } \right) \frac{1}{c_1 }\left( {\zeta -c_0 } \right) \nonumber \\&= \frac{1}{a_1^2 b_1 c_1 }\left( {\begin{array}{lll} \xi ^{2}\eta \zeta -b_0 \zeta \xi ^{2}-c_0 \xi ^{2}\eta -2a_0 \xi \eta \zeta + \\ b_0 c_0 \xi ^{2}+a_0^2 \eta \zeta +2a_0 b_0 \zeta \xi +2a_0 c_0 \xi \eta - \\ 2a_0 b_0 c_0 \xi -c_0 a_0^2 \eta -a_0^2 b_0 \zeta +a_0^2 b_0 c_0 \\ \end{array}} \right) \nonumber \\ \end{aligned}$$

(45)

In the three expressions above, the terms that have orders higher than or equal to 3 are $\eta \zeta ^{2}$, $\xi \eta \zeta$, $\xi ^{2}\eta \zeta$, $\zeta \xi ^{2}$, and $\xi ^{2}\eta$ all of which are included by a polynomial of $\xi$, $\eta$, and $\zeta$ in the same form as shown in Eq. (39). Consequently, the polynomial resulting from interpolation based on the shape functions of the local element has the form shown in Eq. (39) in both its own and the global element’s parametric spaces.

It is evident that an individual shape function associated with a single node of the local element can be written in the same form as $P\left( {r,s,t} \right)$ in the parametric space of the global element. The following equation can thus hold:

$$\begin{aligned} \varvec{\phi } ^{\mathrm{L}}=\mathbf{C}^{\mathrm{L}}\mathbf{t} \hbox { in } \Omega ^{\mathrm{L}} \end{aligned}$$

(46)

where $\Omega ^{\mathrm{L}}$ is the domain of the local element; $\varvec{\phi } ^{\mathrm{L}}$ is the column matrix of all the shape functions of the local element. It can be written as

$$\begin{aligned} \varvec{\phi } ^{\mathrm{L}}=\left[ {{\begin{array}{cccc} {\phi _1^\mathrm{L} }&{} {\phi _2^\mathrm{L} }&{} {\cdot \cdot \cdot }&{} {\phi _{20}^\mathrm{L} } \\ \end{array} }} \right] ^{\mathrm{T}} \end{aligned}$$

(47)

where $\phi _i^\mathrm{L}\,\left( {i=1,2,\ldots ,20} \right)$ is the shape function associated with the $i\hbox {th}$ node of the local element. $\mathbf{t}$ in Eq. (46) denotes the column matrix of all the terms appearing in the polynomial in Eq. (39) with the variables being the coordinates, $\xi$, $\eta$ and $\zeta$, in the parametric space of the global element; $\mathbf{C}^{\mathrm{L}}$ is the matrix of the specific coefficients of the terms in $\mathbf{t}$. Noticing that both $\varvec{\phi } ^{\mathrm{L}}$ and $\mathbf{t}$ have 20 entries, $\mathbf{C}^{\mathrm{L}}$ is a square matrix of the size of $20\times 20$. Furthermore, if the relation in Eq. (38) is given, the values of the entries of $\mathbf{C}^{\mathrm{L}}$ are determined.

Next it will be shown that $\mathbf{C}^{\mathrm{L}}$ is invertible. Herein reduction to absurdity is again employed to prove this statement. Let us assume $\mathbf{C}^{\mathrm{L}}$ is not invertible. As a result, there exists a non-zero row matrix $\mathbf{v}^{\mathrm{L}}$ satisfying

$$\begin{aligned} \mathbf{v}^{\mathrm{L}}\mathbf{C}^{\mathrm{L}}=\mathbf{0} \end{aligned}$$

(48)

By pre-multiplying both sides of Eq. (46) by $\mathbf{v}^{\mathrm{L}}$, one obtains

$$\begin{aligned} \mathbf{v}^{\mathrm{L}}\varvec{\phi } ^{\mathrm{L}}=\mathbf{v}^{\mathrm{L}}\mathbf{C}^{\mathrm{L}}\mathbf{t}=0 \hbox { in } \Omega ^{\mathrm{L}} \end{aligned}$$

(49)

Equation (49) can be rewritten in the following form:

$$\begin{aligned} \sum _{i=1}^{20} {v_i^\mathrm{L} \phi _i^\mathrm{L} =0} \hbox { in } \Omega ^{\mathrm{L}} \end{aligned}$$

(50)

where $v_i^\mathrm{L}$ is the $i\hbox {th}$ entry of the row matrix $\mathbf{v}^{\mathrm{L}}$. Without loss of generality, it is assumed that $v_j^\mathrm{L}$ is one of the non-zero entries in $\mathbf{v}^{\mathrm{L}}$. According to the Kronecker delta property of the shape functions, the value of the left-hand side of Eq. (50) is $v_j^\mathrm{L}$ at the $j\hbox {th}$ node of the local element. Since $v_j^\mathrm{L}$ is assumed to be non-zero, Eq. (50) cannot hold at the $j\hbox {th}$ node of the local element and therefore $\mathbf{C}^{\mathrm{L}}$ must be invertible.

It is now straightforward to prove the statement made at the beginning of the “Appendix”. According to Eq. (39), the shape function associated with the $i\hbox {th}$ node of the global element can be written as

$$\begin{aligned} N_i^\mathrm{G} =\mathbf{c}^{\mathrm{G}}\mathbf{t} \hbox { in } \Omega ^{\mathrm{G}}\,\left( {i=1,2,\ldots ,20} \right) \end{aligned}$$

(51)

where $\Omega ^{\mathrm{G}}$ denotes the domain of the global element. $\mathbf{c}^{\mathrm{G}}$ is the row matrix of the coefficients of the terms in $\mathbf{t}$. Since $\mathbf{C}^{\mathrm{L}}$ has been proved to be invertible, Eq. (46) can be substituted into Eq. (51) yielding

$$\begin{aligned} N_i^\mathrm{G} \!=\!\mathbf{c}^{\mathrm{G}}\left( {\mathbf{C}^{\mathrm{L}}} \right) ^{-1}\varvec{\phi } ^{\mathrm{L}} \!=\!\mathbf{u}^{\mathrm{G}}\varvec{\phi } ^{\mathrm{L}} \hbox { in } \Omega ^{\mathrm{L}}\,\left( {i\!=1,2,\!\ldots ,20} \right) \nonumber \\ \end{aligned}$$

(52)

where $\mathbf{u}^{\mathrm{G}}$, which equals $\mathbf{c}^{\mathrm{G}}\left( {\mathbf{C}^{\mathrm{L}}} \right) ^{-1}$, is the row matrix of the coefficients of the shape functions of the local element. The Kronecker delta property of the shape functions requires that the $j\hbox {th}$ entry $u_j^\mathrm{G}$ of $\mathbf{u}^{\mathrm{G}}$ is equal to the value of the global shape function $N_i^\mathrm{G}$ at the $j\hbox {th}$ node of the local element. Equation (52) can therefore be written as

$$\begin{aligned}&N_i^\mathrm{G} \left( {\xi , \eta , \zeta } \right) =\sum _{j=1}^{20} {N_i^\mathrm{G} \left( {\xi _j^\mathrm{L}, \eta _j^\mathrm{L}, \zeta _j^\mathrm{L} } \right) \phi _j^\mathrm{L} \left( {\xi , \eta , \zeta } \right) } \nonumber \\&\quad \hbox { in } \Omega ^{\mathrm{L}}\,\left( {i=1,2,\ldots ,20} \right) \end{aligned}$$

(53)

This completes the proof.

Appendix 4: Relations between the nodal coordinates of the global, local and superimposed elements

Consider a 20-node isoparametric serendipity laminated element in its parametric space. The global element has $n$ interfaces all of which are planes perpendicular to the $\zeta$ axis in the parametric space. The interfaces are numbered from the bottom up with the $\zeta$ axis pointing upwards. The interfaces are located at $\zeta =\zeta _i^\mathrm{c}$.

If the ply-by-ply discretization approach is employed to model the displacement discontinuity across the interfaces, the global element is replaced by $\left( {n+1} \right)$ local elements one for each ply as illustrated in Fig. 5a. The nodal coordinates of the $i\hbox {th}$ local element in the parametric space of the global element can be written as

$$\begin{aligned}&\left\{ {\begin{array}{l} \xi _j^{\left( i \right) } =\xi _j \\ \eta _j^{\left( i \right) } =\eta _j \\ \zeta _j^{\left( i \right) } =\frac{\zeta _i^\mathrm{c} -\zeta _{i-1}^\mathrm{c} }{2}\zeta _j +\frac{\zeta _i^\mathrm{c} +\zeta _{i-1}^\mathrm{c} }{2} \\ \end{array}} \right. \nonumber \\&\quad \left( {i=1,2,\ldots ,n+1; j=1,2,\ldots ,20} \right) \end{aligned}$$

(54)

where $\left( {\xi _j^{\left( i \right) }, \eta _j^{\left( i \right) } ,\zeta _j^{\left( i \right) } } \right)$ and $\left( {\xi _j, \eta _j, \zeta _j} \right)$ denote the coordinates of the $j\hbox {th}$ nodes of the $i\hbox {th}$ local and global elements, respectively, in the parametric space of the global element. Note that the definition of $\zeta _i^\mathrm{c}$ is detailed in Eq. (54) by introducing $\zeta _0^\mathrm{c} =-1$ and $\zeta _{n+1}^\mathrm{c} =1$. Equations (54), which states the relation between the nodal coordinates of the local and global elements, have the form shown in Eq. (21).

If the displacement discontinuity across the delaminated interfaces is represented by the $s$-method, $n$ pairs of elements are superimposed onto the global element (see Fig. 5c). Each pair of superimposed elements compose a superimposed patch characterizing the crack in one of the interfaces. The superimposed patch has the same sequence number as its corresponding delaminated interface. The nodal coordinates of the lower superimposed element in the $i\hbox {th}$ superimposed patch can be written as

$$\begin{aligned}&\left\{ {\begin{array}{l} \xi _j^{\mathrm{S}\left( i \right) \mathrm{Low}} =\xi _j \\ \eta _j^{\mathrm{S}\left( i \right) \mathrm{Low}} =\eta _j \\ \zeta _j^{\mathrm{S}\left( i \right) \mathrm{Low}} =\frac{\zeta _i^\mathrm{c} +1}{2}\zeta _j +\frac{\zeta _i^\mathrm{c} -1}{2} \\ \end{array}} \right. \nonumber \\&\quad \left( {i=1,2,\ldots ,n; j=1,2,\ldots ,20} \right) \end{aligned}$$

(55)

where $\left( {\xi _j^{\mathrm{S}\left( i \right) \mathrm{Low}}, \eta _j^{\mathrm{S}\left( i \right) \mathrm{Low}}, \zeta _j^{\mathrm{S}\left( i \right) \mathrm{Low}} } \right)$ are the coordinates of the $j\hbox {th}$ node of the lower element in the $i\hbox {th}$ superimposed patch in the parametric space of the global element. Note that Eq. (55) is given in the form described by Eq. (21).

By combining Eqs. (54) and (55), one can obtain the relation between the nodal coordinates of the $i\hbox {th}$ local element and lower element in the $j\hbox {th}$ superimposed patch as follows:

$$\begin{aligned}&\left\{ {\begin{array}{l} \xi _k^{\left( i \right) } =\xi _k^{\mathrm{S}\left( j \right) \mathrm{Low}} \\ \eta _k^{\left( i \right) } =\eta _k^{\mathrm{S}\left( j \right) \mathrm{Low}} \\ \zeta _k^{\left( i \right) } =\frac{\zeta _i^\mathrm{c} -\zeta _{i-1}^\mathrm{c} }{\zeta _j^\mathrm{c} +1}\zeta _k^{\mathrm{S}\left( j \right) \mathrm{Low}} +\frac{\zeta _{i-1}^\mathrm{c} \zeta _j^\mathrm{c} +\zeta _i^\mathrm{c} }{\zeta _j^\mathrm{c} +1} \\ \end{array}} \right. \nonumber \\&\quad \left( {i=1,2,\ldots ,j; j\!=\!1,2,\ldots ,n; k\!=\!1,2,\ldots ,20} \right) \nonumber \\ \end{aligned}$$

(56)

Note that $i\le j$ in the equation above so that the $i\hbox {th}$ local element is within the domain of the lower element in the $j\hbox {th}$ superimposed patch.

Based on the proof of Eq. (42) in Appendix 3, the coordinate transformation from the parametric space of the lower superimposed element to that of the global element has the same form as Eq. (55), namely

$$\begin{aligned}&\left\{ {\begin{array}{l} \xi _={\xi }'^{\mathrm{S}\left( i \right) \mathrm{Low}} \\ \eta _={\eta }'^{\mathrm{S}\left( i \right) \mathrm{Low}} \\ \zeta _=\frac{\zeta _i^\mathrm{c} +1}{2}{\zeta }'^{\mathrm{S}\left( i \right) \mathrm{Low}} +\frac{\zeta _i^\mathrm{c} -1}{2} \\ \end{array}} \right. \left( {i=1,2,\ldots ,n} \right) \end{aligned}$$

(57)

where $\left( {{\xi }'^{\mathrm{S}\left( i \right) \mathrm{Low}}, {\eta }'^{\mathrm{S}\left( i \right) \mathrm{Low}}, {\zeta }'^{\mathrm{S}\left( i \right) \mathrm{Low}} } \right)$ and $\left( {\xi , \eta , \zeta } \right)$ are the coordinates in the parametric spaces of the lower element in the $i$th superimposed patch and the global element, respectively.

By applying the transformation in Eq. (57) to Eq. (56), the relation in Eq. (56) can be expressed in the parametric space of the lower element of the $j\hbox {th}$ superimposed patch as follows:

$$\begin{aligned}&\left\{ {\begin{array}{l} {\xi '}_{\!\!k}^{(i)} ={\xi '}_{\!\!k}^{\mathrm{S}\left( j \right) \mathrm{Low}} \\ {\eta '}_{\!\!k}^{\left( i \right) } \!=\!{ }^\mathrm{s}{\eta '}_{\!\!k}^{\mathrm{S}\left( j \right) \mathrm{Low}} \\ {\zeta '}_{\!\!k}^{\left( i \right) } =\frac{\zeta _i^\mathrm{c} -\zeta _{i-1}^\mathrm{c} }{\zeta _j^\mathrm{c} +1}{\zeta '}_{\!\!k}^{\mathrm{S}\left( j \right) \mathrm{Low}} \!+\!\frac{\zeta _i^\mathrm{c} \zeta _j^\mathrm{c} \!+\!\zeta _{i-1}^\mathrm{c} \zeta _j^\mathrm{c} \!-\!\left( {\zeta _j^\mathrm{c} } \right) ^{2} \!+\!\zeta _{i-1}^\mathrm{c} \!+\!\zeta _i^\mathrm{c} +1}{\left( {\zeta _j^\mathrm{c} +1} \right) ^{2}} \\ \end{array}} \right. \nonumber \\&\quad \left( {i=1,2,\ldots ,j; j=1,2,\ldots ,n; k=1,2,\ldots ,20} \right) \end{aligned}$$

(58)

where $\left( {{\xi '}_{\!\!k}^{\mathrm{S}\left( j \right) \mathrm{Low}}, {\eta '}_{\!\!k}^{\mathrm{S}\left( j \right) \mathrm{Low}}, {\zeta '}_{\!\!k}^{\mathrm{S}\left( j \right) \mathrm{Low}} } \right)$ are the coordinates of the $k\hbox {th}$ node of the lower element in the $j\hbox {th}$ superimposed patch in the parametric space of the lower element; $\left( { {\xi '}_{\!\!k}^{\left( i \right) }, {\eta '}_{\!\!k}^{\left( i \right) }, {\zeta '}_{\!\!k}^{\left( i \right) } } \right)$ are the coordinates of the $k\hbox {th}$ node of the $i\hbox {th}$ local element in the same parametric space. Equation (58) indicates that the lower superimposed element in the $s$-method and local element in the ply-by-ply discretization approach satisfy the conditions described in Eq. (21).

Following the same procedures, one can readily show that any upper superimposed and local element satisfy the conditions described above.