String-based cohesive zone model: implicit integration scheme and calibration method

Abstract

The objective of this paper is to develop the implicit integration scheme and calibration method for a recently developed cohesive zone model (CZM), for improved predictive capabilities in interlaminar delamination. The concept of string-based CZM is briefly introduced. An associated implicit integration scheme, which can handle complex separation paths and mixed-mode fracture, is developed, and the implicit CZM is implemented in an implicit solver via a user subroutine, for structural analysis. The constant interface parameters are identified from a thermodynamic perspective, and a systemic method for model calibration is subsequently developed. The present CZM is validated by calibrating its constant parameters via a series of flexural tests. The integration scheme is found to produce improved convergence and accuracy. The calibration method is found to provide a reliable guide to model calibration. The present CZM can be modified to accommodate other types of debonding or fracture.

Appendices

A Path dependence function

In this appendix, the construction of the path dependence function, $J\left( {\hat{\varvec{\gamma }}} \right)$, will be briefly introduced [see Zhang et al. (2017) for more details]. Borg et al. (2002) expressed the total energy release rate as

$$\begin{aligned} G = \int _0^t {{\varvec{\tau }} \cdot \dot{\varvec{\gamma }}{\mathrm{d}}t} - {\varPsi _e} = \int _0^t {{\varvec{\tau }} \cdot \dot{\varvec{\gamma }}{\mathrm{d}}t} - \frac{1}{2}{\varvec{\tau }} \cdot {\varvec{\gamma }}, \end{aligned}$$

(45)

where t denotes the current time. Clearly, G is a path-dependent line integral whose integration path is the separation path along which a cohesive element is deformed. Let

$$\begin{aligned} {{\varvec{G}}} = {\left\lfloor {\begin{array}{ccc} {{G_1}}&{{G_2}}&{{G_3}} \end{array}} \right\rfloor ^T} = {\left\lfloor {\begin{array}{ccc} {{G_{\mathrm{{I}}}}}&{{G_{{\mathrm{II}}}}}&{{G_{{\mathrm{III}}}}} \end{array}} \right\rfloor ^T} \end{aligned}$$

(46)

denote the energy release rate vector such that

$$\begin{aligned} G = {G_{\mathrm{{I}}}} + {G_{{\mathrm{II}}}} + {G_{{\mathrm{III}}}} = {G_1} + {G_2} + {G_3}. \end{aligned}$$

(47)

Combining Eqs. (46) and (47) gives

$$\begin{aligned} {{\varvec{G}}} = \int _0^t {{\varvec{\tau }} \cdot {\varvec{{\mathcal {I}}}} \cdot \dot{\varvec{\gamma }}{\mathrm{d}}t} - \frac{1}{2}{\varvec{\tau }} \cdot {\varvec{{\mathcal {I}}}} \cdot {\varvec{\gamma }}, \end{aligned}$$

(48)

where

$$\begin{aligned} {\varvec{{\mathcal {I}}}} = \sum \limits _{i = 1}^3 {{{{\varvec{e}}}_i} \otimes {{{\varvec{e}}}_i} \otimes {{{\varvec{e}}}_i}} . \end{aligned}$$

(49)

with ${{{\varvec{e}}}_i}$ denoting the ${i^{{\mathrm{th}}}}$ local unit vector. Let

$$\begin{aligned} {\varvec{\beta }} = \frac{{{\varvec{G}}}}{G} \end{aligned}$$

(50)

denote the mode mixture vector. The following assumption is invoked in Sect. 4: during each test, each element experiences the same proportional separation path. This assumption is the equivalent of setting $\dot{\hat{\varvec{\gamma }}} = {{\varvec{0}}}$ everywhere. Setting $\dot{\hat{\varvec{\gamma }}} = {{\varvec{0}}}$ in Eqs. (45) and (48) gives

$$\begin{aligned} G = \int _0^t {K\gamma {{\dot{\gamma }}} {\mathrm{d}}t} - \frac{1}{2}K{\gamma ^2}, \end{aligned}$$

(51)

$$\begin{aligned} {{\varvec{G}}}&= \left( {\int _0^t {K\gamma {{\dot{\gamma }}} {\mathrm{d}}t} - \frac{1}{2}K{\gamma ^2}} \right) \hat{\varvec{\gamma }} \cdot {\varvec{{\mathcal {I}}}} \cdot \hat{\varvec{\gamma }} \\&= G\left( {\sum \limits _{i = 1}^3 {{\hat{\gamma }} _i^2{{{\varvec{e}}}_i}} } \right) . \end{aligned}$$

(51′)

Substituting Eqs. (51) into Eq. (50) gives

$$\begin{aligned} {\varvec{\beta }} = \sum \limits _{i = 1}^3 {{\hat{\gamma }} _i^2{{{\varvec{e}}}_i}} \quad {\mathrm{or}}\quad \dot{\varvec{\beta }} = {{\varvec{0}}}. \end{aligned}$$

(52)

Clearly, $\dot{\hat{\varvec{\gamma }}} = {{\varvec{0}}}$ is a sufficient condition of $\dot{\varvec{\beta }} = {{\varvec{0}}}$. Let ${t_c}$ denote the time when delamination occurs. ${G_c}$ and ${{{\varvec{G}}}_c}$ can then be obtained as

$$\begin{aligned} {G_c} = \int _0^{{t_c}} {{\varvec{\tau }} \cdot \dot{\varvec{\gamma }}{\mathrm{d}}t} - {\left( {{\varPsi _e}} \right) _c} = \int _0^{{t_c}} {{\varvec{\tau }} \cdot \dot{\varvec{\gamma }}{\mathrm{d}}t} \end{aligned}$$

(53)

$$\begin{aligned} {\mathrm{and}}\quad {{{\varvec{G}}}_c} = \int _0^{{t_c}} {{\varvec{\tau }} \cdot {\varvec{{\mathcal {I}}}} \cdot \dot{\varvec{\gamma }}{\mathrm{d}}t} , \end{aligned}$$

(53′)

respectively, where ${\left( {{\varPsi _e}} \right) _c} \rightarrow 0$ is assumed.

Given a set of n data points $\left( {{{\hat{\varvec{\gamma }}}_i},{J_i}} \right)$’s mentioned in Sect. 2, one can construct a multivariate Lagrange polynomial, $J\left( {\hat{\varvec{\gamma }}} \right)$, such that each ${J_i}$ makes ${G_c}$ take a designated value. Introduce the Hadamard power of a vector, ${\left( \cdot \right) ^{ \circ 2}}$, e.g.,

$$\begin{aligned} {{\hat{\varvec{\gamma }}}^{ \circ 2}} = \sum \limits _{i = 1}^3 {{\hat{\gamma }} _i^2{{{\varvec{e}}}_i}} . \end{aligned}$$

(54)

Also introduce Lagrange basis polynomials (Luo 2010)

$$\begin{aligned} \begin{aligned} {l_i}\left( {{{\hat{\varvec{\gamma }}}^{ \circ 2}}} \right) ={}&\prod \limits _{j = 1;j \ne i}^n {\frac{{\left( {{{\hat{\varvec{\gamma }}}^{ \circ 2}} - \hat{\varvec{\gamma }}_j^{ \circ 2}} \right) \cdot \left( {\hat{\varvec{\gamma }}_i^{ \circ 2} - \hat{\varvec{\gamma }}_j^{ \circ 2}} \right) }}{{\left( {\hat{\varvec{ \gamma }}_i^{ \circ 2} - \hat{\varvec{\gamma }}_j^{ \circ 2}} \right) \cdot \left( {\hat{\varvec{\gamma }}_i^{ \circ 2} - \hat{\varvec{\gamma }}_j^{ \circ 2}} \right) }}} {} \\ \equiv {}&\prod \limits _{j = 1;j \ne i}^n {\frac{{\left( {{{\hat{\varvec{\gamma }}}^{ \circ 2}} - \hat{\varvec{\gamma }}_j^{ \circ 2}} \right) \cdot \left( {\hat{\varvec{\gamma }}_i^{ \circ 2} - \hat{\varvec{\gamma }}_j^{ \circ 2}} \right) }}{{{{\left\| {\hat{\varvec{\gamma }}_i^{ \circ 2} - \hat{\varvec{\gamma }}_j^{ \circ 2}} \right\| }^2}}}} \end{aligned} \end{aligned}$$

(55)

and normalized Lagrange basis polynomials

$$\begin{aligned} {{{{\hat{l}}}}_i}\left( {{{\hat{\varvec{\gamma }}}^{ \circ 2}}} \right) = \frac{{{l_i}\left( {{{\hat{\varvec{ \gamma }}}^{ \circ 2}}} \right) }}{{\sum \limits _{j = 1}^n {{l_i}\left( {{{\hat{\varvec{\gamma }}}^{ \circ 2}}} \right) } }} \equiv \frac{{{l_i}\left( {{{\hat{\varvec{\gamma }}}^{ \circ 2}}} \right) }}{{l\left( {{{\hat{\varvec{\gamma }}}^{ \circ 2}}} \right) }}. \end{aligned}$$

(56)

Let $J\left( {\hat{\varvec{\gamma }}} \right)$ take the form of

$$\begin{aligned} \left( {\hat{\varvec{\gamma }}} \right) = \sum \limits _{i = 1}^n {{{{{\hat{l}}}}_i}\left( {{{\hat{\varvec{\gamma }}}^{ \circ 2}}} \right) {J_i}} \end{aligned}$$

(57)

so that it is a subjective (and possibly injective) function of ${\varvec{\beta }}$ when $\dot{\hat{\varvec{\gamma }}} = {{\varvec{0}}}$. ${{\partial J} / {\partial \hat{\varvec{\gamma }}}}$ can then be obtained from Eq. (57) as

$$\begin{aligned} \frac{{\partial J}}{{\partial \hat{\varvec{\gamma }}}} = \sum \limits _{i = 1}^n {\frac{{\partial {{{{\hat{l}}}}_i}}}{{\partial {{\hat{\varvec{\gamma }}}^{ \circ 2}}}} \cdot \frac{{\partial {{\hat{\varvec{\gamma }}}^{ \circ 2}}}}{{\partial \hat{\varvec{\gamma }}}}{J_i}} , \end{aligned}$$

(58)

where

$$\begin{aligned} \frac{{\partial {{\hat{\varvec{\gamma }}}^{ \circ 2}}}}{{\partial \hat{\varvec{\gamma }}}} = \sum \limits _{i = 1}^3 {2{{\hat{\gamma }} _i}{{{\varvec{e}}}_i} \otimes } {{{\varvec{e}}}_i}, \end{aligned}$$

(59)

$$\begin{aligned} {\mathrm{and}}\quad \frac{{\partial {{{{\hat{l}}}}_i}}}{{\partial {{\hat{\varvec{\gamma }}}^{ \circ 2}}}} = \frac{1}{l}\frac{{\partial {l_i}}}{{\partial {{\hat{\varvec{\gamma }}}^{ \circ 2}}}} - \frac{{{l_i}}}{{{l^2}}}\frac{{\partial l}}{{\partial {{\hat{\varvec{\gamma }}}^{ \circ 2}}}} \end{aligned}$$

(60)

with

$$\begin{aligned}&{} \frac{{\partial {{{{\hat{l}}}}_i}}}{{\partial {{\hat{\varvec{\gamma }}}^{ \circ 2}}}} = {} \nonumber \\ {}&\sum \limits _{j = 1;j \ne i}^n \left[ {\prod \limits _{k = 1;k \ne i,j}^n {\frac{{\left( {{{\hat{\varvec{\gamma }}}^{ \circ 2}} - \hat{\varvec{\gamma }}_k^{ \circ 2}} \right) \cdot \left( {\hat{\varvec{\gamma }}_i^{ \circ 2} - \hat{\varvec{ \gamma }}_k^{ \circ 2}} \right) }}{{{{\left\| {\hat{\varvec{\gamma }}_i^{ \circ 2} - \hat{\varvec{\gamma }}_k^{ \circ 2}} \right\| }^2}}}} } \right] \nonumber \\&\times \frac{{\hat{\varvec{\gamma }}_i^{ \circ 2} - \hat{\varvec{\gamma }}_j^{ \circ 2}}}{{{{\left\| {\hat{\varvec{\gamma }}_i^{ \circ 2} - \hat{\varvec{\gamma }}_j^{ \circ 2}} \right\| }^2}}} \end{aligned}$$

(61)

$$\begin{aligned}&{\mathrm{and}}\quad \frac{{\partial l}}{{\partial {{\hat{\varvec{\gamma }}}^{ \circ 2}}}} = \sum \limits _{i = 1}^n {\frac{{\partial {l_i}}}{{\partial {{\hat{\varvec{\gamma }}}^{ \circ 2}}}}} . \end{aligned}$$

(62)

In Sect. 4, a cohesive element is assumed to have the same properties in all directions in the ${x_2}{x_3}$ plane. In this case, J degenerates to a univariate Lagrange polynomial of ${{\hat{\gamma }} _1}$. Introduce Lagrange basis polynomials

$$\begin{aligned} {l_i}\left( {{\hat{\gamma }} _1^2} \right) = \prod \limits _{j = 1;j \ne i}^n {\frac{{{\hat{\gamma }} _1^2 - \left( {{{\hat{\gamma }} _1}} \right) _j^2}}{{\left( {{{\hat{\gamma }} _1}} \right) _i^2 - \left( {{{\hat{\gamma }} _1}} \right) _j^2}}} . \end{aligned}$$

(63)

$J\left( {{{\hat{\gamma }} _1}} \right)$ is then given by

$$\begin{aligned} J\left( {{{\hat{\gamma }} _1}} \right) = \sum \limits _{i = 1}^n {{l_i}\left( {{\hat{\gamma }} _1^2} \right) {J_i}} , \end{aligned}$$

(64)

and

$$\begin{aligned} \frac{{\partial J}}{{\partial \hat{\varvec{\gamma }}}} = \frac{{\partial J}}{{\partial {{\hat{\gamma }} _1}}}{{{\varvec{e}}}_1} = \left( {\sum \limits _{i = 1}^n {2{{\hat{\gamma }} _1}\frac{{\partial {l_i}}}{{\partial {\hat{\gamma }} _1^2}}{J_i}} } \right) {{{\varvec{e}}}_1}, \end{aligned}$$

(65)

where

$$\begin{aligned}&\frac{{\partial {l_i}}}{{\partial {\hat{\gamma }} _1^2}} \nonumber \\&= \sum \limits _{j = 1;j \ne i}^n {\left[ {\prod \limits _{k = 1;k \ne i,j}^n {\frac{{{\hat{\gamma }} _1^2 - \left( {{{\hat{\gamma }} _1}} \right) _k^2}}{{\left( {{{\hat{\gamma }} _1}} \right) _i^2 - \left( {{{\hat{\gamma }} _1}} \right) _k^2}}} } \right] \frac{1}{{\left( {{{\hat{\gamma }} _1}} \right) _i^2 - \left( {{{\hat{\gamma }} _1}} \right) _j^2}}} .\nonumber \\ \end{aligned}$$

(66)

B UMAT algorithm

Figure 17 depicts a flowchart of the UMAT algorithm. The algorithm can be described as follows:

• 1. Read ${{\varvec{\tau }}_n}$ and $\varDelta {\varvec{\gamma }}$ passed in by Abaqus/Standard, while read ${{\varvec{\gamma }}_n}$, ${d_n}$, and ${\alpha _n}$ saved for the present element.

• 2. Check if damage initiates or evolves (see Eq. (10)).

• 3. If yes, (a) compute $\varDelta \alpha$, (b) update $\alpha$, d, and ${\varvec{\tau }}$, and (c) compute ${{{\varvec{L}}}^ * }$.

• 4. Otherwise, (a) update ${\varvec{\gamma }}$, (b) set $\alpha = {\alpha _n}$ and $d = {d_n}$, and (c) compute ${{{\varvec{L}}}^ * }$ (here ${{{\varvec{L}}}^ * } = K{{{{\varvec{I}}}^ + } + {{{{\tilde{K}}}}_c}\left( {{{\varvec{I}}} - {{{\varvec{I}}}^ + }} \right) }$) and ${\varvec{\tau }}$.

• 5. Save ${\varvec{\gamma }}$, d, and $\alpha$ for the present element, while return ${\varvec{\tau }}$ and ${{{\varvec{L}}}^ * }$ to Abaqus/Standard.

Once $d = {d_c}$, UMAT will set $K = 0$ and mark the present element as failed hereafter. More details on Abaqus/Standard and UMAT can be found in Simulia (2013).

Fig. 17

UMAT algorithm

C Method of nonlinear least squares

In this appendix, the method of NLLSQ, along with Monte Carlo experiments and Powell’s methods, will be briefly introduced. Unless otherwise specified, let the “local” variables defined in this appendix override the “global” variables defined in Sects. 2–5, having the same names.

Suppose that N data points $\left( {{x_i},{y_i}} \right)$’s are to be fitted to a nonlinear model depending on M adjustable parameters $a_j$’s ($N \ge M$). Let

$$\begin{aligned} {{\varvec{a}}} = {\left\lfloor {\begin{array}{ccc} {{a_1}}&\cdots&{{a_M}} \end{array}} \right\rfloor ^T}, \end{aligned}$$

(67)

and let the nonlinear model take the general form of $y = f\left( {x;{{\varvec{a}}}} \right)$. Further suppose that each $\left( {{x_i},{y_i}} \right)$ has its respective, known standard deviation ${\sigma _i}$, and introduce the so-called chi-square merit function,

$$\begin{aligned} {\chi ^2} \equiv {\sum \limits _{i = 1}^N {\left( {\frac{{{y_i} - f\left( {{x_i};{{\varvec{a}}}} \right) }}{{{\sigma _i}}}} \right) } ^2}, \end{aligned}$$

(68)

which is the sum of N squares of normalized, distributed residuals. The method of NLLSQ involves finding ${{\varvec{a}}}$ minimizing ${\chi ^2}$. An NLLSQ problem is therefore an optimization problem whose objective function is ${\chi ^2}\left( {{\varvec{a}}} \right)$.

The Rastrigin function is frequently used for performance testing of optimization methods. Figure 18 shows its 3D surface and contour plots. As can be seen, the function has a global minimum at $\left( {0,0} \right)$ and numerous local minima. When handling such a function, an optimization method itself is not guaranteed to converge to the global minimum and often gets “lost” if started far from the solution. Fortunately, setting the guessed values close to the solution greatly improves the convergence. In this paper, such guessed values are obtained through Monte Carlo experiments consisting of the following steps:

• 1. Estimate the domains of $a_j$’s.

• 2. Create a lot of different combinations of $a_j$’s over these domains.

• 3. Compute the values of ${\chi ^2}$ for these combinations.

• 4. Find as many neighborhoods of local minima (i.e., where an optimization method converges to these local minima) as possible.

The global minimum can be found first by carrying out an optimization procedure at each of these neighborhoods and then by identifying the local minimum yielding the smallest ${\chi ^2}$. In this paper, each set of Monte Carlo experiments include less than $20 \cdot M$ (M the number of $a_j$’s) numerical tests of a single cohesive element, taking about 20 ms each.

Fig. 18

Rastrigin function

Optimization methods can be classified into (1) those only requiring evaluations of the objective functions (e.g., Powell’s method) and (2) those also requiring evaluations of the derivatives of the objective functions (e.g., Newton’s method). In this paper, Powell’s method is chosen due to the following reasons:

• Newton’s method often fails to converge if an $a_j$ is an exponent.

• In iterative optimization, it is difficult to compute the derivatives of ${\chi ^2}$ through finite element analysis.

• Given good initial guesses, Powell’s method produces good convergence and high efficiency.

Powell’s method involves successively minimizing the objective function along M mutually non-interfering directions (see Fig. 19 for the case of $M = 2$). These directions are defined so that Powell’s method converges quadratically (see Press et al. (1992) for more details). The procedure is repeated until the objective function effectively stops decreasing, or mathematically speaking, until

$$\begin{aligned} \left| {\chi _k^2 - \chi _{k - 1}^2} \right|< {\varepsilon _1}\quad {\mathrm{or}}\quad \left| {\frac{{\chi _k^2 - \chi _{k - 1}^2}}{{\chi _{k - 1}^2}}} \right| < {\varepsilon _2}, \end{aligned}$$

(69)

where $\chi _k^2$ and $\chi _{k - 1}^2$ are the values of ${\chi ^2}$ in the current and the previous iteration, respectively, and ${\varepsilon _1}$ and ${\varepsilon _2}$ are two prescribed tolerances.

Fig. 19

(adapted from Press et al. (1992))

Successive minimizations in a long, narrow “valley”

Table 6 Information required to generate ${\chi ^2}$’s (${\sigma _i} = {y_i}$)

In Sect. 4, the method of NLLSQ is used (1) when estimating Q and b, (2) when estimating $J_i$’s, and (3) in iterative optimization (see also Fig. 6). In the first two cases, both Monte Carlo experiments and Powell’s method are needed, while in the last case, only Powell’s method is needed thanks to the initial guess previously created (see Fig. 11, where the estimated curves are already close to the experimental ones). Still suppose that there were n flexural tests, and let subscript i denote the ${i{{\mathrm{th}}}}$ test, e.g., ${\left( {{G_c}} \right) _i}$ denotes the ${i{{\mathrm{th}}}}$ estimated value of $G_c$. Let ${P_{i\max }}$ and ${u_{i\max }}$ denote the ${i{{\mathrm{th}}}}$ peak load and its corresponding displacement, respectively, so that $\left( {{u_{i\max }},{P_{i\max }}} \right)$ is the peak point of the ${i{{\mathrm{th}}}}$ load–displacement curve. Table 6 lists the information required to generate ${\chi ^2}$ in each case. Here each nonlinear model is a black box only whose inputs and outputs are of interest. For simplicity’s sake, each $x_i$ in Table 6 is set to be a sequence number. Setting ${\sigma _i} = {y_i}$ (${y_i} \ne 0$) makes ${\chi ^2}$ the sum of N squares of percentage errors made in fitting, and $a_j$’s minimizing this measure of percentage errors are the best possible fitted parameters.