A VFIFE-DKMT Formulation for Nonlinear Motion Analysis of Laminated Composite Thick Shells

Abstract

This study presents a new formulation for laminated composite thick shells by incorporating the discrete Kirchhoff–Mindlin triangular (DKMT) element into the vector form intrinsic finite element (VFIFE) method. This integration enables the accurate modeling of transverse shear effects, which are difficult to capture using conventional VFIFEs. In this framework, the shell is discretized into particles whose motions are analyzed over discrete time intervals, referred to as path elements. Euler’s law of motion governs particle dynamics, while triangular elements connect the particles and describe local deformation and internal forces. Quaternions represent rigid body rotations within the convected material frame, and internal forces are obtained from the shape functions of the VFIFE–DKMT element. The formulation is validated through numerical examples involving geometrically nonlinear displacements, dynamic responses, and large deformations in isotropic and composite shells. The results demonstrate the accuracy and robustness of the proposed method in predicting the nonlinear motion of thick shell structures.

1. Introduction

Composite materials are engineered by combining two or more constituents with distinct properties to produce a new material with enhanced mechanical, thermal, or chemical performance. Among the most widely used composites are polymer matrices reinforced with fibers, such as carbon, glass, or aramid. These composites provide several advantages, including high specific strength, low weight, and resistance to corrosion and fatigue. Furthermore, they offer design flexibility as they can be fabricated into complex geometries with tailored dimensions. Composite shell structures are extensively applied in the aerospace, aviation, automotive, renewable energy, sporting goods, and civil engineering industries because of their favorable mechanical performance and cost-effectiveness.

Shell structures carry in-plane loads primarily through membrane action, even when undergoing significant out-of-plane (lateral) deformation. This coupling of in-plane stiffness and out-of-plane flexibility is a key factor in their high structural efficiency and low weight. The analysis of shells remains challenging because of their complex and nonlinear behavior after deformation, often resulting from geometric instabilities. Issues such as large nonlinear displacements and dynamic responses are of particular concern. Consequently, numerous theoretical formulations and computational models have been proposed to address these complexities.

In finite element analysis of geometrically nonlinear problems, widely adopted approaches include the total Lagrangian and updated Lagrangian formulations [1], as well as the co-rotational scheme [2,3,4,5]. Unlike the first two, the co-rotational method [6] assumes that structures undergo large translations and rotations accompanied by relatively small strains. The formulation distinguishes between the original configuration and a co-rotated configuration, with the latter obtained by filtering out infinitesimal rigid body rotations from the initial state. Although this procedure renders the governing equations linear at each increment, it may accumulate inaccuracies in evaluating rigid body motion during long-term simulations.

To overcome these limitations, Ting et al. [7,8] introduced the vector form intrinsic finite element (VFIFE) method, which adopts a particle-based representation governed by Euler’s law of motion. Structures are discretized into groups of particles, and their motions under internal and external forces represent the overall structural behavior. The VFIFE method simplifies analysis by avoiding iterative solutions of the partial differential equations required in conventional FEM, thereby providing a unified framework for different structures [9]. The three main concepts of the VFIFE method are the point value description (PVD), path element, and convected material frame (CMF), the last of which is based on an incremental co-rotational approach [10]. The use of CMF and path elements enhances the accuracy of rigid body rotation estimation within each time increment.

Ting et al. [8] introduced the triangular membrane element within the VFIFE method to enable the analysis of shell-type structural behaviors. Wu and Ting [11] extended this concept to quadrilateral membrane elements. Later developments incorporated shell elements composed of membranes and discrete Kirchhoff components to handle nonlinear dynamics [12]. Other studies have employed VFIFE in the analysis of roof and suspension structures [13,14], SAR membrane structure vibrations [15], and buried pipeline simulations [16]. However, most of these works remain limited to thin shell formulations and isotropic materials. Recently, Chou et al. [17] proposed the composite VFIFE-DKT shell formulation, which did not consider the transverse shear effect.

The major challenges in shell analysis involve not only the coupling of membrane and bending effects but also the accurate representation of post-deformation geometry and the inherently complex bending behavior. In addition, avoiding shear locking is a critical concern. To address these issues, numerous numerical models have been proposed for analyzing composite plates and shells. For example, Pham et al. [18], and Nguyen et al. [19] employed an edge-based smoothed finite element method with the MITC3 element based on FSDT to study the vibrations of functionally graded (FG) shells; Chau-Dinh et al. [20] applied the ES-MITC3+ element with higher-order shear deformation theory (HSDT) to study the vibration and buckling of laminated composite plates; Yang et al. [21] adopted isogeometric analysis (IGA) with FSDT to examine the static and dynamic responses of piezoelectric functionally graded plates (PFGPs); Kim et al. [22] proposed IGA with Bézier extraction and the assumed natural strain (ANS) method to alleviate locking in highly continuous shell models; Gao et al. [23] proposed a layerwise third-order shear deformation theory for piezo-laminated plates; and Shi [24] applied IGA based on a 3D elasticity model to analyze the dynamic responses of FG-CNTRC plates.

Classical theories, such as Kirchhoff–Love, assume that the normal to the mid-surface remains straight and normal after deformation, thereby neglecting transverse shear effects. While this assumption is an acceptable simplification for thin shells, it is inaccurate for thick configurations. Reissner–Mindlin and first-order shear deformation theories (FSDT), incorporating transverse shear deformation for improved accuracy in thick shell analysis, are one of the efficient formulations to analyze shells.

However, the Reissner–Mindlin formulation suffers from shear-locking as shell thickness approaches zero [25,26]. To overcome this drawback, the discrete Kirchhoff assumption was adopted in constructing the DKT element, which was designed for thin plate analysis [27,28]. Later refinements led to elements such as DST-BL [29] and DST-BK [30], which offered varying degrees of success for the analysis of thick plate behavior. Ultimately, Katili [31] proposed the discrete Kirchhoff–Mindlin triangle (DKMT) element, combining Reissner–Mindlin theory, discrete Kirchhoff constraints, and assumed shear strain fields to provide a robust solution across thin-to-thick shell applications. Maknun et al. [32] extended the DKMT element to the analysis of composite plate structures. Subsequent work confirmed the convergence and accuracy of the DKMT element for thin-to-thick plate structures [33,34,35].

Despite its proven robustness in FEM, the DKMT element has not yet been incorporated into the VFIFE framework. In this study, constant strain triangular (CST) and DKMT elements are integrated into the VFIFE method to analyze the motion of laminated composite shell structures for the first time. The primary advantage of this modification is the ability to accurately capture transverse shear effects in thick shells, which had remained a limitation of previous VFIFE models. Given that the rotations in a path element are finite (Argyris [36]), quaternion-based formulations are adopted to address the complexities of 3D rigid body rotations [37,38]. The equivalent single-layer (ESL) model is utilized to represent composite shell sections. A series of numerical examples—including nonlinear displacements, large deformations, buckling, and dynamic responses—are conducted using the proposed VFIFE formulation for thin-to-thick composite shell structures. The results are compared against established analytical solutions and published numerical data from various finite element methods to demonstrate the accuracy and reliability of the approach.

2. Fundamentals of the VFIFE Method

The VFIFE method models structural behavior by discretizing the continuum into finite particles connected by suitable elements [39]. Particles define the shape of a structure in space, whereas internal forces are computed through the interaction of particles within connecting elements. In contrast to conventional FEM, which relies on continuum assumptions and partial differential equations, the VFIFE method adopts a particle-based framework governed directly by Euler’s laws of motion.

In the VFIFE method, each structural component is represented by a discrete set of particles. The governing equations for the translational and rotational motions of each particle $i$ are expressed as

$${\overline{\mathbf{m}}}_i{\ddot{\mathbf{u}}}_i={\mathbf{F}}_i-{\mathbf{f}}_i,$$

(1)

$${\overline{\mathbf{I}}}_i{\ddot{\mathsf{\phi }}}_i={\mathbf{M}}_i-{\mathbf{m}}_i,$$

(2)

where ${\mathbf{F}}_i$ and ${\mathbf{f}}_i$ are the external and internal force vectors, respectively; ${\mathbf{M}}_i$ and ${\overline{\mathbf{m}}}_i$ are the external and internal moment vectors; ${\mathbf{u}}_i$ and ${\mathsf{\phi }}_i$ denote the translational and rotational displacement vectors of the particle; ${\overline{\mathbf{m}}}_i$ is the mass, and ${\overline{\mathbf{I}}}_i$ is the mass moment of inertia of the element. The mass and mass moment of inertia for each particle are obtained by summing the contributions from the surrounding elements. This process ensures that the total inertial properties of the discretized system are correctly represented. The lumped-mass approach is used to construct the mass matrix. The calculation of the mass moment of inertia of shell particles in the VFIFE method is detailed in Appendix A.

2.1. Path Element, PVD, and CMF

Figure 1 shows that the motion of the structure over the entire analysis period is described by tracking the positions of grouped particles. The dotted line connecting the initial positions at time $t_0$ to the final positions at time $t_f$ represents the particle’s trajectories. The position vector ${\mathbf{x}}_{\alpha }^t\left(t\right)$ of particle $\alpha$ is considered only at the discrete time intervals $\left(t_0,t_1\right)$, $\left(t_1,t_a\right)$, and $\left(t_a,t_b\right)$. Three basic types of particles based on the physical constraint conditions are used: motion, displacement control, and constrained particles [39,40]. The representative particles used to characterize the overall motion of the deformable body are referred to as the point value description (PVD). An independent span of time, for example: $t_a<t<t_b$, is defined as a path element. Within each path element, a particle satisfies Euler’s law of motion, and small deformation is assumed. Accordingly, material and geometric properties are considered constant within each interval, and the fundamentals of the mechanics of materials can be applied.

At the starting time $t_0$, the configuration is represented as $V_0$. Within each path element, three states are defined [39]: the earlier configuration $V_a$ at time $t_a$, the present configuration $V$ at time $t$, and a fictitious configuration $V_r$. The relative coordinate $\mathrm{d}\mathbf{x}$ in the current state $V$ is determined with reference to the corresponding coordinate $\mathrm{d}{\mathbf{x}}_a$ from the preceding configuration:

$$\mathrm{d}\mathbf{x}=\mathbf{F}\mathrm{d}{\mathbf{x}}_a=\mathbf{R}\mathbf{U}\mathrm{d}{\mathbf{x}}_a.$$

(3)

In Equation (3), $\mathbf{F}$, $\mathbf{R}$, and $\mathbf{U}$ are the deformation gradient, the rigid body rotation matrix, and the deformation matrix, respectively. This decomposition, often referred to as polar decomposition, separates the total motion into rigid body rotation and pure deformation. The position vector corresponding to the fictitious configuration $V_r$, expressed with respect to the present configuration $V$, can be written as

$$\mathrm{d}{\mathbf{x}}_{\mathit{r}}={\mathbf{F}}_r\mathrm{d}\mathbf{x}={\mathbf{R}}_r^{\mathrm{T}}\mathrm{d}\mathbf{x},$$

(4)

where ${\mathbf{F}}_r$ is the virtual deformation gradient, and ${\mathbf{R}}_r^{\mathrm{T}}$ is the reverse rigid body rotation matrix. ${\mathbf{F}}_r$ is equal to the matrix ${\mathbf{R}}_r^{\mathrm{T}}$ in a path element because of the assumption of small deformation. The following equation can be obtained by substituting Equation (3) into Equation (4):

$$\mathrm{d}{\mathbf{x}}_r={\mathbf{F}}_r\mathbf{F}\mathrm{d}{\mathbf{x}}_a=\left({\mathbf{R}}_r^{\mathrm{T}}\mathbf{R}\right)\mathbf{U}\mathrm{d}{\mathbf{x}}_a\cong \mathbf{U}\mathrm{d}{\mathbf{x}}_a.$$

(5)

Given that small deformation is considered within each path element, the reversed rotation matrix ${\mathbf{R}}_r^{\mathrm{T}}$ is close to $\mathbf{R}$. The rigid body rotation component included in the deformation gradient $\mathbf{F}$ is removed through ${\mathbf{R}}_r$. As a result, the fictitious configuration $V_r$ approximates the deformation condition of $V_a$ ($\mathbf{U}\cong \mathbf{I}$). This fictitious state ($V_r$) is then adopted for evaluating the internal forces acting on particle $i$. The configuration $V_a$ is regarded as the reference state, which is subsequently updated and treated as the new initial configuration. The continuous renewal of this reference state within each path element defines the convected material frame (CMF). As shown in Figure 1, the previous configuration ($V_a$) becomes the reference for the current time step ($V$). This allows for the separation of total motion into a large rigid body motion and a small deformation within each time increment, or “path element”. By resetting the reference configuration at each step, the CMF prevents the accumulation of errors that can occur when estimating rigid body motion over long durations.

2.2. Motion Analysis of the VFIFE Method

Figure 2a illustrates that the element $i-j-k$ with surface normal $\mathbf{n}$ denotes the current configuration $V$, while the element $i_a-j_a-k_a$ with normal ${\mathbf{n}}_a$ corresponds to the reference configuration $V_a$. The reverse rigid body motion from $V$ to $V_a$ can be separated into translation and rotation. The translational component is determined by shifting the shell from $V$ to $V^{\prime }$ (i.e., from element $i-j-k$ to $i^{\prime }-j^{\prime }-k^{\prime }$), with respect to a chosen reference point such as the centroids $C$ and $C_a$. The expression for the relative translation ${\mathbf{u}}_{\alpha }$ is given as

$${\mathbf{u}}_{\alpha }={\mathbf{x}}_{\alpha }^t-{\mathbf{x}}_{\alpha }^a, \alpha =i,j,k,$$

(6)

where ${\mathbf{x}}_{\alpha }^t$ and ${\mathbf{x}}_{\alpha }^a$ denote the position vectors of particle $\alpha$ in the current and reference configurations, respectively.

The reverse rigid body rotation can be divided into two parts: an out-of-plane rotation and an in-plane rotation. The out-of-plane component is evaluated by calculating the angle between the surface normals of configurations $V$ and $V_a$, expressed as

$${\theta }_1={\sin }^{-1}(\left|{\mathbf{n}}_a\times \mathbf{n}\right|).$$

(7)

The corresponding out-of-plane rotation vector ${\mathsf{\theta }}_1$ is

$${\mathsf{\theta }}_1={\theta }_1{\mathbf{n}}_1={\theta }_1\left(u_1^{\prime }\mathbf{i}+v_1^{\prime }\mathbf{j}+w_1^{\prime }\mathbf{k}\right),$$

(8)

where ${\mathbf{n}}_1$ is the unit vector of out-of-plane rotation and can be calculated as

$${\mathbf{n}}_1={\displaystyle \frac{{\mathbf{n}}_a\times \mathbf{n}}{\left|{\mathbf{n}}_a\times \mathbf{n}\right|}}.$$

(9)

Following the reverse out-of-plane rotation, the element is transferred from the configuration $i^{\prime }-j^{\prime }-k^{\prime }$ ($V^{\prime }$) to $i_p-j_p-k_p$ ($V_p$). The configuration $V_p$ is on the same plane, which is parallel to $V_a$. The reverse in-plane rotation ${\mathsf{\theta }}_2$ is then applied to rotate $V_p$ to the fictitious configuration $V_r$. ${\mathsf{\theta }}_2$ is calculated as the average angular difference between the corresponding vertices of configurations $V_p$ and $V_a$ (Figure 2b).

$${\mathsf{\theta }}_2={\theta }_2{\mathbf{n}}_a={\theta }_2\left(u_2^{\prime }\mathbf{i}+v_2^{\prime }\mathbf{j}+w_2^{\prime }\mathbf{k}\right),$$

(10)

where

$${\theta }_2={\displaystyle \frac{1}{3}}\sum _{\mathsf{\alpha }=\mathrm{i}}^{\mathrm{k}}{\theta }_{2\mathsf{\alpha }},$$

(11)

and

$${\theta }_{2\alpha }={\sin }^{-1}\left({\displaystyle \frac{∆{\mathbf{x}}_{c\alpha }^a}{\left|∆{\mathbf{x}}_{c\alpha }^a\right|}}\times {\displaystyle \frac{∆{\mathbf{x}}_{\mathrm{c}\mathsf{\alpha }}^p}{\left|∆{\mathbf{x}}_{\mathrm{c}\mathsf{\alpha }}^p\right|}}\right), \alpha =i,j,k.$$

(12)

2.3. Synthetic Rotation by Quaternion

Once the two finite rotations of the rigid body are identified, a quaternion formulation is applied to determine the combined rotation from $V^{\prime }$ to $V_r$ within each path element. In this formulation, the out-of-plane rotation ${\mathsf{\theta }}_1$ and the in-plane rotation ${\mathsf{\theta }}_2$ are represented by the quaternions $q_1$ and $q_2$, respectively:

$$\begin{gathered} q_1=w_1+x_1\mathbf{i}+y_1\mathbf{j}+z_1\mathbf{k}=\cos {\displaystyle \frac{{\theta }_1}{2}}+u_1^{\prime }\sin {\displaystyle \frac{{\theta }_1}{2}}\mathbf{i}+v_1^{\prime }\sin {\displaystyle \frac{{\theta }_1}{2}}\mathbf{j}+w_1^{\prime }\sin {\displaystyle \frac{{\theta }_1}{2}}\mathbf{k}, \\ {q}_2=w_2+x_2\mathbf{i}+y_2\mathbf{j}+z_2\mathbf{k}=\cos {\displaystyle \frac{{\theta }_2}{2}}+u_2^{\prime }\sin {\displaystyle \frac{{\theta }_2}{2}}\mathbf{i}+v_2^{\prime }\sin {\displaystyle \frac{{\theta }_2}{2}}\mathbf{j}+w_2^{\prime }\sin {\displaystyle \frac{{\theta }_2}{2}}\mathbf{k}. \end{gathered}$$

(13)

The rotation matrix associated with a quaternion $q_n$ is written as

$$R_r\left(q_n\right)=\left[\begin{array}{ccc}1-2\left(y_n^2+z_n^2\right)& 2\left(x_n{y}_n-w_n{z}_n\right)& 2\left(x_n{z}_n+w_n{y}_n\right)\\ 2\left(x_n{y}_n+w_n{z}_n\right)& 1-2\left(x_n^2+z_n^2\right)& 2\left(y_n{z}_n-w_n{x}_n\right)\\ 2\left(x_n{z}_n-w_n{y}_n\right)& 2\left(y_n{z}_n+w_n{x}_n\right)& 1-2\left(x_n^2+y_n^2\right)\end{array}\right], n=1,2$$

(14)

The successive action of the two rotations on a vector $\mathbf{x}$ can be condensed into a single quaternion operation:

$${\mathbf{x}}^{\prime }=R_r\left(q_2\right)R_r\left(q_1\right)\mathbf{x}=R_r\left(q\right)\mathbf{x},$$

(15)

where ${\mathbf{x}}^{\prime }$ denotes the transformed position vector after both out-of-plane and in-plane rotations. The combined quaternion $q$ is obtained from the multiplication of $q_1$ and $q_2$, yielding:

$$\begin{gathered} q=q_2{q}_1=\left(w_2{w}_1-x_2{x}_1-y_2{y}_1-z_2{z}_1\right)+\left(w_2{x}_1+x_2{w}_1+y_2{z}_1-z_2{y}_1\right)\mathbf{i}+\left(w_2{y}_1-x_2{z}_1+y_2{w}_1+ \\ {z}_2{x}_1\right)\mathbf{j}+\left(w_2{z}_1+x_2{y}_1-y_2{x}_1+z_2{w}_1\right)\mathbf{k}=w+\left(v_1\mathbf{i}+v_2\mathbf{j}+v_3\mathbf{k}\right). \end{gathered}$$

(16)

The reverse rigid body motion, including translational and rotational components, is a key part of the VFIFE method. This enables the decomposition of total motion into rigid body motion and pure deformation within each path element.

2.4. Deformation Coordinates

After the removal of the rigid body motion, the current configuration $V$ is mapped to a fictitious configuration $V_r$, in which only the pure deformation between $V_r$ and $V_a$ remains (Figure 1). The deformation coordinate system $\left(\widehat{x},\widehat{y},\widehat{z}\right)$ is established to facilitate the calculation of deformation, with its origin chosen arbitrarily within the element. The particle deformation displacements are represented by ${\mathsf{\eta }}_j^d$ and ${\mathsf{\eta }}_k^d$, while the plane normal is denoted by ${\mathbf{n}}_a$. As illustrated in Figure 3, the orthogonal basis vectors $\left({\widehat{\mathbf{e}}}_1,{\widehat{\mathbf{e}}}_2,{\widehat{\mathbf{e}}}_3\right)$ of the deformation coordinate system can be calculated as

$$\begin{gathered} {\widehat{\mathbf{e}}}_1={\displaystyle \frac{{\mathsf{\eta }}_j^d}{\left|{\mathsf{\eta }}_j^d\right|}}, \\ {\widehat{\mathbf{e}}}_3={\mathbf{n}}_a, \\ {\widehat{\mathbf{e}}}_2={\displaystyle \frac{{\widehat{\mathbf{e}}}_3\times {\widehat{\mathbf{e}}}_1}{\left|{\widehat{\mathbf{e}}}_3\times {\widehat{\mathbf{e}}}_1\right|}}. \end{gathered}$$

(17)

The coordinate transformation matrix $\mathbf{Q}$ from the global coordinate system to the deformation coordinate system is given by

$$\mathbf{Q}={\left\{\begin{array}{ccc}{\widehat{\mathbf{e}}}_1& {\widehat{\mathbf{e}}}_2& {\widehat{\mathbf{e}}}_3\end{array}\right\}}^{\mathrm{T}}.$$

(18)

The translational deformation displacement field in the local deformation coordinates can be written as

$${\widehat{\mathsf{\eta }}}_{\alpha }^d=\mathbf{Q}{\mathsf{\eta }}_{\alpha }^d, \alpha =i,j,k,$$

(19)

where ${\widehat{\mathsf{\eta }}}_{\alpha }^d={\left\{\begin{array}{cc}{\widehat{\eta }}_{\alpha x}^d& {\widehat{\eta }}_{\alpha y}^d\end{array}\right\}}^{\mathrm{T}}$ and ${\mathsf{\eta }}_{\alpha }^d={\left\{\begin{array}{cc}{\eta }_{\alpha x}^d& {\eta }_{\alpha y}^d\end{array}\right\}}^{\mathrm{T}}$ are the deformation displacement in the deformation and global coordinate system, respectively.

In the deformation coordinate, the displacement vector ${\widehat{\mathbf{u}}}_{\alpha }$ of particle $\alpha$ can be expressed as

$${\widehat{\mathbf{u}}}_{\alpha }=\left\{\begin{array}{c}{\widehat{u}}_{\alpha }\\ {\widehat{v}}_{\alpha }\end{array}\right\}=\left\{\begin{array}{c}{\widehat{\eta }}_{\alpha x}^d\\ {\widehat{\eta }}_{\alpha y}^d\end{array}\right\},\alpha =i,j,k.$$

(20)

If the reference point (origin of the deformation coordinate system) is set at particle $i$ (as shown in Figure 3), then ${\widehat{u}}_i={\widehat{v}}_i={\widehat{v}}_j=0$.

3. VFIFE-DKMT Element

In the VFIFE method, internal forces are calculated by using suitable elements that connect particles to simulate structural behavior. In this study on shell structures, the CST [8] and DKMT [31] elements are combined into a VFIFE–DKMT element to represent the membrane and bending effects, respectively.

3.1. Stress–Strain Relations and Shape Functions

For a typical laminated composite shell, the strain in layer $\mathrm{k}$, which includes membrane and bending effects in the deformation coordinate system, is represented as

$${\widehat{\mathsf{\epsilon }}}^{\left(\mathrm{k}\right)}={\widehat{\mathsf{\epsilon }}}_m+\widehat{z}{\widehat{\mathsf{\kappa }}}_b,$$

(21)

where ${\widehat{\mathsf{\epsilon }}}_m$ is the membrane strain, ${\widehat{\mathsf{\kappa }}}_b$ is the bending curvature of the shell, and $\widehat{z}$ is the distance from the mid-surface to layer $\mathrm{k}$.

The stress–strain relationship for layer $\mathrm{k}$ is given by

$${\widehat{\mathsf{\sigma }}}^{\left(\mathrm{k}\right)}={\overline{\mathbf{Q}}}^{\left(\mathrm{k}\right)}{\widehat{\mathsf{\epsilon }}}^{\left(\mathrm{k}\right)},$$

(22)

where ${\widehat{\mathsf{\sigma }}}^{\left(\mathrm{k}\right)}={\left\{\begin{array}{ccc}{\widehat{\sigma }}_x^{\left(\mathrm{k}\right)}& {\widehat{\sigma }}_y^{\left(\mathrm{k}\right)}& {\widehat{\tau }}_{xy}^{\left(\mathrm{k}\right)}\end{array}\right\}}^{\mathrm{T}}$ is the in-plane stress vector, and ${\overline{\mathbf{Q}}}^{\left(\mathrm{k}\right)}$ is the stiffness matrix of layer $\mathrm{k}$. ${\overline{\mathbf{Q}}}^{\left(\mathrm{k}\right)}$ is defined by

$${\overline{\mathbf{Q}}}^{\left(\mathrm{k}\right)}=\left[\begin{array}{ccc}{\overline{Q}}_{11}& {\overline{Q}}_{12}& {\overline{Q}}_{16}\\ {\overline{Q}}_{12}& {\overline{Q}}_{22}& {\overline{Q}}_{26}\\ {\overline{Q}}_{16}& {\overline{Q}}_{26}& {\overline{Q}}_{66}\end{array}\right],$$

(23)

where

$$\begin{gathered} {\overline{Q}}_{11}=Q_{11}{\cos }^4\alpha +2\left(Q_{12}+2Q_{66}\right){\sin }^2\alpha {\cos }^2\alpha +Q_{22}{\sin }^4\alpha , \\ {\overline{Q}}_{12}=Q_{12}\left({\sin }^4\alpha +{\cos }^4\alpha \right)+\left(Q_{11}+Q_{22}-4Q_{66}\right){\sin }^2\alpha {\cos }^2\alpha , \\ {\overline{Q}}_{22}=Q_{11}{\sin }^4\alpha +2\left(Q_{12}+2Q_{66}\right){\sin }^2\alpha {\cos }^2\alpha +Q_{22}{\cos }^4\alpha , \\ {\overline{Q}}_{16}=\left(Q_{11}-Q_{12}-2Q_{66}\right)\sin \alpha {\cos }^3\alpha +\left(Q_{12}-Q_{22}-2Q_{66}\right){\sin }^3\alpha \cos \alpha , \\ {\overline{Q}}_{26}=\left(Q_{11}-Q_{12}-2Q_{66}\right){\sin }^3\alpha \cos \alpha +\left(Q_{12}-Q_{22}-2Q_{66}\right)\sin \alpha {\cos }^3\alpha , \\ {\overline{Q}}_{66}=\left(Q_{11}+Q_{22}-2Q_{12}-2Q_{66}\right){\sin }^2\alpha {\cos }^2\alpha +Q_{66}\left({\sin }^4\alpha +{\cos }^4\alpha \right). \end{gathered}$$

(24)

The engineering constants $E_1$, $E_2$, $G_{12}$, ${\nu }_{12}$, and ${\nu }_{21}$ for each ply in the laminated composite section are used to compute $Q_{ij}$ in Equation (25):

$$Q_{11}={\displaystyle \frac{E_1}{1-{\nu }_{12}{\nu }_{21}}}, Q_{12}={\displaystyle \frac{{\nu }_{12}{E}_2}{1-{\nu }_{12}{\nu }_{21}}},Q_{22}={\displaystyle \frac{E_2}{1-{\nu }_{12}{\nu }_{21}}},Q_{66}=G_{12}.$$

(25)

In this study, transverse shear deformation is also considered. The transverse shear stress in the deformation coordinate is shown in the following equation:

$${\widehat{\mathsf{\tau }}}_s^{\left(\mathrm{k}\right)}={\overline{\mathbf{Q}}}_s^{\left(\mathrm{k}\right)}{\widehat{\mathsf{\gamma }}}^{\left(\mathrm{k}\right)},$$

(26)

where ${\widehat{\mathsf{\tau }}}_s^{\left(\mathrm{k}\right)}={\left\{\begin{array}{cc}{\widehat{\tau }}_{xz}^{\left(\mathrm{k}\right)}& {\widehat{\tau }}_{yz}^{\left(\mathrm{k}\right)}\end{array}\right\}}^{\mathrm{T}}$ and ${\widehat{\mathsf{\gamma }}}^{\left(\mathrm{k}\right)}$ are the transverse shear stress vector and shear strain, respectively; and ${\overline{\mathbf{Q}}}_s^{\left(\mathrm{k}\right)}$ is the shear stiffness matrix in layer $\mathrm{k}$:

$${\overline{\mathbf{Q}}}_s^{\left(\mathrm{k}\right)}=\left[\begin{array}{cc}{\overline{Q}}_{44}& {\overline{Q}}_{45}\\ {\overline{Q}}_{45}& {\overline{Q}}_{55}\end{array}\right].$$

(27)

The transformation of shear stiffness from the material principal direction to the deformation coordinates follows the equations:

$$\begin{gathered} {\overline{Q}}_{44}=G_{13}{\cos }^2\alpha +G_{23}{\sin }^2\alpha , \\ {\overline{Q}}_{45}=\left(G_{13}-G_{23}\right)\cos \alpha \sin \alpha , \\ {\overline{Q}}_{55}=G_{13}{\sin }^2\alpha +G_{23}{\cos }^2\alpha . \end{gathered}$$

(28)

Figure 4 illustrates that $\alpha$ denotes the angle formed between the material’s principal orientation and the deformation coordinate system, and this angle is recalculated at every path element.

For the membrane component, the CST element uses a linear interpolation function to represent the in-plane displacement field:

$${\widehat{\mathbf{u}}}_m=\sum _{\alpha =j}^k{N}_{\alpha }{\widehat{\mathbf{u}}}_{\alpha },$$

(29)

where $N_{\alpha }$ is the shape function of the three-node constant strain triangle [8]. ${\widehat{u}}_j$, ${\widehat{u}}_k$, and ${\widehat{v}}_k$ indicate the deformation displacements ${\widehat{\eta }}_{jx}^d$, ${\widehat{\eta }}_{kx}^d$, and ${\widehat{\eta }}_{ky}^d$, respectively. Assuming small deformation in each path element, the membrane strain is expressed as

$${\widehat{\mathsf{\epsilon }}}_m={\mathbf{B}}_m{\widehat{\mathbf{u}}}_m,$$

(30)

where ${\mathbf{B}}_m$ is the strain–displacement matrix for the CST element.

In this study, the DKMT element [31,32] is adopted in the calculation of the thick plate bending effect. The bending curvature ${\widehat{\mathsf{\kappa }}}_b$ in the DKMT element is shown as the following equation:

$${\widehat{\mathsf{\kappa }}}_{\mathit{b}}=\left({\mathbf{B}}_{b\beta }+{\mathbf{B}}_{b∆\beta }{\mathbf{A}}_{w∆\beta }\right){\widehat{\mathbf{u}}}_b={\mathbf{B}}_b{\widehat{\mathbf{u}}}_b,$$

(31)

where ${\widehat{\mathbf{u}}}_b$ contains ${\widehat{\theta }}_{\alpha x}$ and ${\widehat{\theta }}_{ay}$, and $\alpha =i, j, k$, are the rotational degrees of freedom associated with bending effects in the deformation coordinate system. Given that the rigid body displacements are removed from the current configuration $V$ to the reference configuration $V_a$, transversal displacements are excluded from the bending formulation under deformation coordinate.

The transverse shear strain $\widehat{\mathsf{\gamma }}$ is represented in the following equation:

$$\widehat{\mathsf{\gamma }}={\mathbf{B}}_s{\widehat{\mathbf{u}}}_b,$$

(32)

where ${\mathbf{B}}_s$ is the shear strain–displacement matrix:

$${\mathbf{B}}_s={\mathbf{B}}_{s∆\beta }{\mathbf{A}}_{w∆\beta }.$$

(33)

Detailed expressions for ${\mathbf{B}}_{b\beta }$, ${\mathbf{B}}_{b∆\beta }$, ${\mathbf{A}}_{w∆\beta }$, ${\mathbf{B}}_{s∆\beta }$, and related parameters in the deformation coordinate system are provided in Appendix B.

3.2. Internal Forces in Deformation Coordinates

The expression of virtual strain energy with respect to the reference configuration $V_a$ is given by

$$\delta U={\int }_{V_a}\delta {\left({\widehat{\mathsf{\epsilon }}}^{\left(\mathrm{k}\right)}\right)}^{\mathrm{T}}{\widehat{\mathsf{\sigma }}}^{\left(\mathrm{k}\right)}\mathrm{d}{V}_a.$$

(34)

From this formulation, the contribution of the membrane component to the incremental virtual strain energy can be written as

$$\delta U_m=\delta {\widehat{\mathbf{u}}}_m^{\mathrm{T}}\left[\left({\int }_{A_a}{\mathbf{B}}_m^{\mathrm{T}}\mathbf{A}{\mathbf{B}}_b\mathrm{d}{A}_a\right){\widehat{\mathbf{u}}}_m+\left({\int }_{A_a}{\mathbf{B}}_m^{\mathrm{T}}\mathbf{B}{\mathbf{B}}_b\mathrm{d}{A}_a\right){\widehat{\mathbf{u}}}_b\right].$$

(35)

By applying the principle of equivalence between internal virtual work and virtual strain energy, the membrane force vector is obtained as

$${\widehat{\mathbf{f}}}^{\ast }=\left({\int }_{A_a}{\mathbf{B}}_m^{\mathrm{T}}\mathbf{A}{\mathbf{B}}_b\mathrm{d}{A}_a\right){\widehat{\mathbf{u}}}_m+\left({\int }_{A_a}{\mathbf{B}}_m^{\mathrm{T}}\mathbf{B}{\mathbf{B}}_b{\widehat{\mathsf{\sigma }}}_m\mathrm{d}{A}_a\right){\widehat{\mathbf{u}}}_b={\left\{\begin{array}{ccc}{\widehat{f}}_{jx}& {\widehat{f}}_{kx}& {\widehat{f}}_{ky}\end{array}\right\}}^{\mathrm{T}}.$$

(36)

The other three internal force components are derived from equilibrium relations as

$$\begin{gathered} {\widehat{f}}_{ix}=-\left({\widehat{f}}_{jx}+{\widehat{f}}_{kx}\right), \\ {\widehat{f}}_{jy}=-\left({\widehat{f}}_{ky}{\widehat{x}}_k^a-{\widehat{f}}_{jx}{\widehat{y}}_j^a-{\widehat{f}}_{kx}{\widehat{y}}_k^a\right)/{\widehat{x}}_j^a, \\ {\widehat{f}}_{iy}=-\left({\widehat{f}}_{jy}+{\widehat{f}}_{ky}\right). \end{gathered}$$

(37)

The complete membrane forces are presented as

$${\widehat{\mathbf{f}}}_m={\left\{\begin{array}{cccccc}{\widehat{f}}_{ix}& {\widehat{f}}_{iy}& {\widehat{f}}_{jx}& {\widehat{f}}_{jy}& {\widehat{f}}_{kx}& {\widehat{f}}_{ky}\end{array}\right\}}^{\mathrm{T}}.$$

(38)

The incremental virtual strain energy due to bending deformation is

$$\delta U_b=\delta {\widehat{\mathbf{u}}}_b^{\mathrm{T}}\left[\left({\int }_{A_a}{\mathbf{B}}_b^{\mathrm{T}}\mathbf{B}{\mathbf{B}}_m\mathrm{d}{A}_a\right){\widehat{\mathbf{u}}}_m+\left({\int }_{A_a}{\mathbf{B}}_b^{\mathrm{T}}\mathbf{D}{\mathbf{B}}_b\mathrm{d}{A}_a\right){\widehat{\mathbf{u}}}_b\right].$$

(39)

In Equations (35) and (39), $\mathbf{A}$, $\mathbf{B}$, and $\mathbf{D}$ are the extensional, coupling, and bending stiffnesses, respectively. They can be calculated by using the following equation:

$$\left(A_{ij},B_{ij},D_{ij}\right)=\sum _{\mathrm{k}=1}^n{\int }_{{\widehat{z}}_{\mathrm{k}-1}}^{{\widehat{z}}_{\mathrm{k}}}{\overline{Q}}_{ij}^{\left(\mathrm{k}\right)}\left(1,\widehat{z},{\widehat{z}}^2\right)\mathrm{d}\widehat{z},$$

(40)

where $n$ is the number of layers of the laminated composite section.

The equivalent internal bending forces can be obtained in accordance with the equivalence of internal virtual work and virtual strain energy as follows:

$$\begin{gathered} {\widehat{\mathbf{f}}}_{bb}^{\ast }=\left({\int }_{A_a}{\mathbf{B}}_b^{\mathrm{T}}\mathbf{B}{\mathbf{B}}_m\mathrm{d}{A}_a\right){\widehat{\mathbf{u}}}_m+\left({\int }_{A_a}{\mathbf{B}}_b^{\mathrm{T}}\mathbf{D}{\mathbf{B}}_b\mathrm{d}{A}_a\right){\widehat{\mathbf{u}}}_b \\ ={\left\{\begin{array}{cccccc}{\widehat{m}}_{ix}^b& {\widehat{m}}_{iy}^b& {\widehat{m}}_{jx}^b& {\widehat{m}}_{jy}^b& {\widehat{m}}_{kx}^b& {\widehat{m}}_{ky}^b\end{array}\right\}}^{\mathrm{T}}. \end{gathered}$$

(41)

The incremental virtual strain energy due to transverse shear in a path element is given by

$$\delta U_s={\int }_{V_a}{\delta \widehat{\mathsf{\gamma }}}^{\mathrm{T}}\widehat{\mathsf{\tau }}\mathrm{d}{V}_a=\delta {\widehat{\mathbf{u}}}_b^{\mathrm{T}}\left({\int }_{A_a}{\mathbf{B}}_s^{\mathrm{T}}{\mathbf{H}}_s{\mathbf{B}}_{\mathrm{s}}\mathrm{d}{A}_a\right){\widehat{\mathbf{u}}}_b,$$

(42)

where ${\mathbf{H}}_s$ is the corrected shear stiffness [32].

The transverse shear correction factors $k_{11}$, $k_{12}$, and $k_{22}$ for composite materials are applied to fit the 3D distribution of the transverse shear stress. The corrected shear stiffness is calculated by using the following equation:

$${\mathbf{H}}_s=\left[\begin{array}{cc}{k}_{11}{\overline{H}}_{s_{11}}& k_{12}{\overline{H}}_{s_{12}}\\ k_{12}{\overline{H}}_{s_{12}}& k_{22}{\overline{H}}_{s_{22}}\end{array}\right],$$

(43)

where the shear stiffness ${\overline{H}}_{s_{ij}}$ of the entire section is determined from the following:

$${\overline{H}}_{s_{ij}}=\sum _{\mathrm{k}=1}^n{\int }_{{\widehat{z}}_{\mathrm{k}-1}}^{{\widehat{z}}_{\mathrm{k}}}{\overline{Q}}_{s_{ij}}^{\left(\mathrm{k}\right)}\mathrm{d}\widehat{z}.$$

(44)

Similarly, the equivalent internal bending force caused by transverse shear can be obtained as

$${\widehat{\mathbf{f}}}_{sb}^{\ast }=\left({\int }_{A_a}{\mathbf{B}}_s^{\mathrm{T}}{\mathbf{H}}_s{\mathbf{B}}_s\mathrm{d}{A}_a\right){\widehat{\mathbf{u}}}_b={\left\{\begin{array}{cccccc}{\widehat{m}}_{ix}^s& {\widehat{m}}_{iy}^s& {\widehat{m}}_{jx}^s& {\widehat{m}}_{jy}^s& {\widehat{m}}_{kx}^s& {\widehat{m}}_{ky}^s\end{array}\right\}}^{\mathrm{T}}.$$

(45)

The total internal bending moment is the sum of contributions from bending and shear effects:

$${\widehat{\mathbf{f}}}_b^{\ast }={\widehat{\mathbf{f}}}_{bb}^{\ast }+{\widehat{\mathbf{f}}}_{sb}^{\ast }={\left\{\begin{array}{cccccc}{\widehat{m}}_{ix}& {\widehat{m}}_{iy}& {\widehat{m}}_{jx}& {\widehat{m}}_{jy}& {\widehat{m}}_{kx}& {\widehat{m}}_{ky}\end{array}\right\}}^{\mathrm{T}}.$$

(46)

The internal shear force caused by bending and transverse shear effects can be calculated by using the equilibrium equations:

$$\begin{gathered} \sum {\widehat{F}}_z=0, {\widehat{f}}_{iz}+{\widehat{f}}_{jz}+{\widehat{f}}_{kz}=0, \\ \sum {\widehat{M}}_x=0, {\widehat{m}}_{ix}+{\widehat{m}}_{jx}+{\widehat{m}}_{kx}=-\left({\widehat{f}}_{jz}{\widehat{y}}_j+{\widehat{f}}_{kz}{\widehat{y}}_k\right), \\ \sum {\widehat{M}}_y=0, {\widehat{m}}_{iy}+{\widehat{m}}_{jy}+{\widehat{m}}_{ky}=-\left({\widehat{f}}_{jz}{\widehat{x}}_j+{\widehat{f}}_{kz}{\widehat{x}}_k\right), \end{gathered}$$

(47)

where ${\widehat{f}}_{\alpha z}$, $\alpha =i, j, k$, is the internal shear force acting on the three particles of the triangular shell element.

The total internal force in the three particles of the VFIFE–DKMT element in the deformation coordinate system, as illustrated in Figure 5, is

$$\begin{gathered} \widehat{\mathbf{f}}={\widehat{\mathbf{f}}}_m+{\widehat{\mathbf{f}}}_s={\left\{\begin{array}{ccccccccc}{\widehat{f}}_{ix}& {\widehat{f}}_{iy}& {\widehat{f}}_{iz}& {\widehat{f}}_{jx}& {\widehat{f}}_{jy}& {\widehat{f}}_{jz}& {\widehat{f}}_{kx}& {\widehat{f}}_{ky}& {\widehat{f}}_{kz}\end{array}\right\}}^{\mathrm{T}}, \\ \widehat{\mathbf{m}}={\left\{\begin{array}{ccccccccc}{\widehat{m}}_{ix}& {\widehat{m}}_{iy}& 0& {\widehat{m}}_{jx}& {\widehat{m}}_{jy}& 0& {\widehat{m}}_{kx}& {\widehat{m}}_{ky}& 0\end{array}\right\}}^T, \end{gathered}$$

(48)

where ${\widehat{m}}_{\alpha z}=0$, $\alpha =i, j, k$, is due to the removal of plane rotation and small deformation from the current configuration $V$ to the reference configuration $V_a$.

3.3. Internal Forces in Global Coordinates

Once the internal forces are evaluated in the deformation coordinate system, they must be transferred to the global system corresponding to the current configuration. This transformation of force and moment vectors is carried out according to

$$\begin{gathered} \begin{array}{cc}{\mathbf{f}}_{\alpha }={\mathbf{R}}^{\mathrm{T}}{\mathbf{Q}}^{\mathrm{T}}{\widehat{\mathbf{f}}}_{\alpha } \\ {\mathbf{m}}_{\alpha }={\mathbf{R}}^{\mathrm{T}}{\mathbf{Q}}^{\mathrm{T}}{\widehat{\mathbf{m}}}_{\alpha }& , \alpha =i,j,k,\end{array} \end{gathered}$$

(49)

where

$$\begin{gathered} {\mathbf{f}}_{\alpha }={\left\{\begin{array}{ccc}{f}_{\alpha x}& f_{\alpha y}& f_{\alpha z}\end{array}\right\}}^{\mathrm{T}}, \\ {\widehat{\mathbf{f}}}_{\alpha }={\left\{\begin{array}{ccc}{\widehat{f}}_{\alpha x}& {\widehat{f}}_{\alpha y}& {\widehat{f}}_{\alpha z}\end{array}\right\}}^{\mathrm{T}}, \\ {\mathbf{m}}_{\alpha }={\left\{\begin{array}{ccc}{m}_{\alpha x}& m_{\alpha y}& m_{\alpha z}\end{array}\right\}}^{\mathrm{T}}, \\ {\widehat{\mathbf{m}}}_{\alpha }={\left\{\begin{array}{ccc}{\widehat{m}}_{\alpha x}& {\widehat{m}}_{\alpha y}& {\widehat{m}}_{\alpha z}\end{array}\right\}}^{\mathrm{T}}. \end{gathered}$$

(50)

4. Numerical Integration of the Governing Equation

The VFIFE method essentially provides a framework for modeling the motion of solid bodies. In the present work, the central difference method is adopted due to its computational efficiency and inherent suitability for the particle-based dynamics of the VFIFE method [12]. The central difference method is conditionally stable. The time step $∆t$ can be chosen to satisfy the Courant–Friedrichs–Lewy (CFL) condition, ensuring numerical stability.

The acceleration and velocity of Equation (1) for particle $i$ at time $t$ within a path element can be calculated as follows:

$$\begin{gathered} {\ddot{\mathbf{u}}}_i\left(t\right)={\displaystyle \frac{{\mathbf{u}}_i\left(t+∆t\right)-2{\mathbf{u}}_i\left(t\right)+{\mathbf{u}}_i\left(t-∆t\right)}{{∆t}^2}}, \\ {\dot{\mathbf{u}}}_i\left(t\right)={\displaystyle \frac{{\mathbf{u}}_i\left(t+∆t\right)-{\mathbf{u}}_i\left(t-∆t\right)}{2∆t}}. \end{gathered}$$

(51)

For analyses requiring energy dissipation, such as achieving a stable static solution or modeling physical damping, a mass-proportional damping term is introduced into the governing equation:

$${\overline{\mathbf{m}}}_i{\ddot{\mathbf{u}}}_i={\mathbf{F}}_i-{\mathbf{f}}_i-{\mathbf{F}}_d,$$

(52)

where the damping force vector is defined as ${\mathbf{F}}_d={\alpha }_d\overline{\mathbf{m}}{\dot{\mathbf{u}}}_i$, with ${\alpha }_d$ being the mass damping coefficient. Combining Equations (51) and (52) yields the following displacement vector of particle $i$:

$${\mathbf{u}}_i\left(t+∆t\right)={\displaystyle \frac{1}{\frac{{\alpha }_d∆t}{2}+1}}\left[2{\mathbf{u}}_i\left(t\right)+{\displaystyle \frac{{\mathbf{u}}_i\left(t-∆t\right)}{\frac{{\alpha }_d∆t}{2}-1}}+{\overline{\mathbf{m}}}_i^{-1}∆t^2\left({\mathbf{F}}_i\left(t\right)-{\mathbf{f}}_i\left(t\right)\right)\right].$$

(53)

The rotational equation of motion is integrated through a similar central difference approach. Following the standard plate theory approach, the rotational governing equation for each particle is first expressed in the local coordinate system ($\tilde{x}$, $\tilde{y}$, $\tilde{z}$), after which the central difference scheme is applied:

$${\tilde{\mathsf{\phi }}}_i\left(t+∆t\right)={\displaystyle \frac{1}{\frac{{\alpha }_d∆t}{2}+1}}\left[2{\tilde{\mathsf{\phi }}}_i\left(t\right)+{\displaystyle \frac{{\tilde{\mathsf{\phi }}}_i\left(t-∆t\right)}{\frac{{\alpha }_d∆t}{2}-1}}+{\tilde{\mathbf{I}}}_i^{-1}∆t^2\left({\tilde{\mathbf{M}}}_i\left(t\right)-{\tilde{\mathbf{m}}}_i\left(t\right)\right)\right],$$

(54)

where ${\tilde{\mathsf{\phi }}}_i={\mathbf{Q}}_{loc}{\mathsf{\phi }}_i$, ${\tilde{\mathbf{I}}}_i={\mathbf{Q}}_{loc}{\mathbf{I}}_i{\mathbf{Q}}_{loc}^{\mathrm{T}}$, ${\tilde{\mathbf{m}}}_i={\mathbf{Q}}_{loc}{\mathbf{m}}_i$, and ${\tilde{\mathbf{M}}}_i={\mathbf{Q}}_{loc}{\mathbf{M}}_i$. The mapping from global to local coordinates is achieved through the transformation matrix ${\mathbf{Q}}_{loc}$, constructed from the orthogonal basis vectors ${\left\{\begin{array}{ccc}{\tilde{\mathbf{e}}}_1& {\tilde{\mathbf{e}}}_2& {\tilde{\mathbf{e}}}_3\end{array}\right\}}^{\mathrm{T}}$. As shown in Figure 6, the local normal vector ${\tilde{\mathbf{e}}}_3$ is determined by averaging the normals of the surrounding elements attached to particle $i$:

$${\tilde{\mathbf{e}}}_3={\mathbf{N}}_t^i={\displaystyle \frac{{\sum }_e{\mathbf{N}}_t^e}{\left|\sum _e{\mathbf{N}}_t^e\right|}}, e=l,m,n,\cdots .$$

(55)

To construct the remaining basis, a neighbor particle $l$ is chosen to define a vector ${\tilde{\mathbf{e}}}_{il}$. This vector is projected to be orthogonal to ${\mathbf{N}}_t^i$:

$${\tilde{\mathbf{e}}}_{il}={\mathbf{e}}_{il}-\left({\mathbf{e}}_{il}·{\tilde{\mathbf{e}}}_3\right){\tilde{\mathbf{e}}}_3,$$

(56)

and the base vectors ${\tilde{\mathbf{e}}}_1$ and ${\tilde{\mathbf{e}}}_2$ can be determined as

$$\begin{gathered} {\tilde{\mathbf{e}}}_1={\displaystyle \frac{{\tilde{\mathbf{e}}}_{il}}{\left|{\tilde{\mathbf{e}}}_{il}\right|}}, \\ {\tilde{\mathbf{e}}}_2={\displaystyle \frac{{\tilde{\mathbf{e}}}_3\times {\tilde{\mathbf{e}}}_1}{\left|{\tilde{\mathbf{e}}}_3\times {\tilde{\mathbf{e}}}_1\right|}}. \end{gathered}$$

(57)

5. Numerical Examples

Several benchmark problems are examined to verify the accuracy and effectiveness of the proposed VFIFE method for isotropic and laminated composite thick shell structures. These examples include comparisons with analytical solutions, published numerical results, and cases involving geometrically nonlinear behaviors, large deformations, buckling, and dynamic responses.

5.1. Timoshenko Beam

To examine the effectiveness of the proposed VFIFE thick shell element, a cantilever beam with different thickness values under a concentrated end load is investigated. As shown in Figure 7, the model beam has a span length of $L=10$, a rectangular section with width $B=1$, material properties of Young’s modulus $E=1.2\times {10}^6$ and Poisson’s ratio $\nu =0$, and is subjected to a tip load of $P=0.4$ applied at the free end. Four thickness values, namely, $H=0.1$, $0.5$, $1$, and $2$, ranging from thin to moderately thick, are considered. The analytical solution is obtained from Timoshenko beam theory (TBT) [41,42]. The free end vertical deflection ${\mathsf{\Delta }}_z$ and the normal stress of the top (or bottom) surface ${\sigma }_x$ due to the fixed-end moment $M$, are given by

$$\begin{gathered} {\Delta }_z(x)={\displaystyle \frac{-PL}{kAG}}\left({\displaystyle \frac{x}{L}}\right)-{\displaystyle \frac{P{L}^3}{3EI}}\left[{\displaystyle \frac{-1}{2}}{\left({\displaystyle \frac{x}{L}}\right)}^3+{\displaystyle \frac{3}{2}}{\left({\displaystyle \frac{x}{L}}\right)}^2\right], \\ {\sigma }_x\left(z\right)={\displaystyle \frac{Mz}{I}},z={\displaystyle \frac{H}{2}}. \end{gathered}$$

(58)

where $k=10(1+\nu )/(12+11\nu )$ is the shear correction factor for the rectangular cross section [43], and $\nu$ is Poisson’s ratio.

As a result of symmetry, only half of the beam is modeled. The VFIFE model contains 63 particles and 80 elements. As presented in

Table 1. TBT and VFIFE calculation results.

$\raisebox{1ex}{$\mathit{L}$}\!\left/ \!\raisebox{-1ex}{$\mathit{H}$}\right.$	TBT		VFIFE
	$\frac{{\mathsf{\Delta }}_{\mathit{z}}(\mathit{x} = 10)}{\left(\mathit{P}{\mathit{L}}^3/\mathit{E}\mathit{I}\right)}$	${\mathit{\sigma }}_{\mathit{x}}(\mathit{x} = 0)$	$\frac{{\mathsf{\Delta }}_{\mathit{z}}(\mathit{x} = 10)}{\left(\mathit{P}{\mathit{L}}^3/\mathit{E}\mathit{I}\right)}$	${\mathit{\sigma }}_{\mathit{x}}(\mathit{x} = 0)$
$100$	$0.333$	$2.4\times {10}^3$	$0.327$	$2.36\times {10}^3$
$20$	$0.333$	$96$	$0.333$	$95.79$
$10$	$0.335$	$24$	$0.335$	$23.97$
$5$	$0.341$	$6$	$0.341$	$5.989$

, the tip deflection of VFIFE results exhibit good consistency with the TBT predictions across all thickness ratios. The stress obtained from VFIFE differs from the analytical solution by less than 1.6%. The results verify the accuracy of the VFIFE formulation.

5.2. Pendulum Motion of a Plate

A $10 \mathrm{cm}\times 10 \mathrm{cm}$ square plate with thickness $h=1 \mathrm{cm}$ and suspended at point D, as shown in Figure 8, is released from rest and allowed to oscillate in the $x-z$ plane. The VFIFE simulation model includes 121 particles and 200 elements. No damping is applied in the analysis.

The system is also modeled as a single-degree-of-freedom (SDOF) pendulum to provide a reference solution. The equation of motion for the simplified SDOF model is given by

$$\ddot{\theta }+{\displaystyle \frac{mgd}{I_p}}\sin \theta =0,$$

(59)

and the period of Equation (59) is expressed as [44]

$$\begin{gathered} T=2\pi \sqrt{{\displaystyle \frac{I_p}{mgd}}}\times {\displaystyle \frac{2}{\pi }}K\left(k\right); \\ K\left(k\right)={\int }_0^{\pi /2}{\displaystyle \frac{d\phi }{\sqrt{1-k^2{\sin }^2\phi }}},k=\sin {\displaystyle \frac{{\theta }_0}{2}}. \end{gathered}$$

(60)

In Equations (59) and (60), $m$ denotes the plate mass, $g$ the gravitational constant, $d$ the distance from the centroid G to the pivot point D, and $I_p$ the mass moment of inertia about D. The function $K\left(k\right)$ is the complete elliptical integral of the first kind. The initial conditions are ${\theta \left(0\right)=\theta }_0$ and $\dot{\theta }\left(0\right)=0$. Based on this formulation, the resulting periods of oscillation are $T=0.617 \mathrm{s}$ for ${\theta }_0=10°$, $T=0.641 \mathrm{s}$ for ${\theta }_0=45°$, and $T=0.941 \mathrm{s}$ for ${\theta }_0=135°$.

The angular displacements of the plate center calculated from VFIFE are plotted in Figure 9. The period of the oscillations in Figure 9 are $T=0.619 \mathrm{s}$ for ${\theta }_0=10°$, $T=0.642 \mathrm{s}$ for ${\theta }_0=45°$, and $T=0.944 \mathrm{s}$ for ${\theta }_0=135°$. In such cases, the oscillation period obtained from the VFIFE simulation closely matches that acquired with the SDOF model.

Despite the system’s nonlinearity, the VFIFE method successfully captures small- and large-amplitude swinging motions with high accuracy. This example confirms that the VFIFE method can simulate rigid-body-dominated dynamics in plate and shell structures undergoing large rotational displacements.

5.3. Double Pendulum Motion

The case study considers a double pendulum consisting of two rectangular plates, one hinged at point A and the other linked through an additional hinge at point C. As depicted in Figure 10, the assembly is initially placed in a horizontal position and released from rest. The two panels share identical material parameters: Young’s modulus $E=2\times {10}^9$, Poisson’s ratio $\nu =0$, and mass density $\rho =1$. Both are modeled with a constant thickness of $h=1$, and damping effects are not considered.

The VFIFE model contains 60 particles and 72 elements. The motion is simulated for a duration of $10 \mathrm{s}$. The resulting trajectories of points C and D are depicted in Figure 11. The motion of point C is shown to be a simple harmonic, whereas the motion of point D exhibits chaotic behavior. This example demonstrates the capability of the proposed VFIFE method to analyze complex and large nonlinear motions.

5.4. Post-Buckling Behavior of a Plate

A rectangular plate of dimensions $L_x\times L_y=30 \mathrm{cm}\times 10 \mathrm{cm}$ with thickness $h=0.1 \mathrm{cm}$ is subjected to uniform in-plane pressure $p$ along its longitudinal direction, as shown in Figure 12. All boundaries are simply supported. The considered plate is assigned material parameters of Young’s modulus $E=2.1\times {10}^6 \mathrm{N}/{\mathrm{cm}}^2$ and Poisson’s ratio $\nu =0.25$. To trigger buckling, a minor transverse point load of magnitude ${10}^{-3} \mathrm{N}$ is applied at its center.

Due to geometric symmetry, the analysis is restricted to one-quarter of the plate. The VFIFE discretization consists of 225 particles connected through 384 elements. Figure 13a presents the variation in the lateral ($z$-direction) displacement at the plate center (point A) with respect to the load factor $\lambda$. In contrast, Figure 13b shows the corresponding in-plane horizontal ($x$-direction) displacement at the loaded boundary (point B).

Comparison with the results reported by Izzuddin [45], Pacoste [46], and Lanzo et al. [47] shows that the VFIFE method accurately captures the critical buckling load and post-buckling path. Note that the results in [47] were obtained by using the quadratic perturbation algorithm and are stiffer than those reported in [45,46]. The post-buckling displacement from the VFIFE solution lies between the results in [45,47], but is close to the quadratic perturbation technique, and remains numerically stable in the extended post-buckling range.

5.5. Vibration of a Shallow Spherical Cap

The dynamic behavior of a shallow spherical cap subjected to a uniform pressure pulse is investigated in this example. As illustrated in Figure 14a, the cap geometry is defined by a radius of $R=0.577 \mathrm{m}$, a thickness of $h=0.01041 \mathrm{m}$, and a half-subtended angle of $\theta =26.67°$. The material parameters are specified as Young’s modulus $E=72.4 \mathrm{G}\mathrm{P}\mathrm{a}$, Poisson’s ratio $\nu =0.3$, and density $\rho =2618 \mathrm{k}\mathrm{g}/{\mathrm{m}}^3$. No damping is included in the analysis. The edge along the base is fully clamped, and due to geometric symmetry, only one-half of the cap is modeled. The VFIFE model consists of 400 particles and 720 elements. Figure 14b presents the time–history response of the vertical displacement at the cap apex (point C). The results agree well with the findings of Peng et al. [48] and Owen and Hinton [49], and demonstrate numerical stability over long simulation durations.

5.6. Large Deformation of a Hemispherical Shell

A hemispherical shell is analyzed under a constant internal pressure load to examine large deformation behavior. A similar structure was also proposed by Macneal and Harder [50] to investigate the vibration. The model parameters follow those in the work of Zhang et al. [51] and are diameter $D=0.5 \mathrm{m}$, height $H=0.2 \mathrm{m}$, thickness $h=0.01 \mathrm{m}$, Young’s modulus $E=200 \mathrm{GPa}$, Poisson’s ratio $\nu =0.3$, and mass density $\rho =7800 \mathrm{kg}/{\mathrm{m}}^3$, as illustrated in Figure 15a. All edges, except for the clamped bottom boundary, are free. The inner pressure $q=1 \mathrm{GPa}$ is applied from the beginning, and perpendicular to the shell surface. Damping and contact force are not considered. As a result of symmetry, half of the shell is modeled by using 451 particles and 800 elements. The simulation is conducted to $t=0.4 \mathrm{ms}$, as illustrated in Figure 15b. The deformation snaps of the shell are also provided in Figure 16. The results of the VFIFE method align with those of the method reported by Zhang et al. [51], which was based on the Mindlin–Reissner plate theory, confirming the VFIFE method’s ability to capture large nonlinear responses without numerical instability.

5.7. Nonlinear Response of a Laminated Composite Cylindrical Shell

The dynamic response of a laminated cylindrical shell with fully clamped edges under internal surface pressure is investigated in this example. As summarized in Figure 17, the elastic constants of the shell are $E_1=20\times {10}^6\mathrm{psi}$, $E_2=1.43\times {10}^6\mathrm{psi}$, $G_{12}=G_{13}=G_{23}=0.76\times {10}^6\mathrm{psi}$, and ${\nu }_{12}=0.3$. The structure has a density of $\rho =0.146\times {10}^{-3} \mathrm{lb}·{\mathrm{s}}^2/{\mathrm{in}}^4$, and a total thickness of $h=0.05\mathrm{in}$. The principal direction of the material is along the $x$-direction. The lamination scheme is ${\left[0/-45/45/90\right]}_s$, with equal thickness for each ply. A distributed pressure $q=1 \mathrm{p}\mathrm{s}\mathrm{i}$ is applied on the inner face. Damping is not considered. The VFIFE model includes 961 particles and 1800 elements. The vertical response at the shell center is shown in Figure 18. This case has been studied by Reddy and Chandrashekhara [52], Wu et al. [53], To and Wang [54], Naidu and Sinha [55], and Kurtaran [56]. The VFIFE results agree well with the latest findings report by Kurtaran [56].

5.8. Laminated Composite Ring Plate Subjected to End Shear

A clamped ring plate subjected to a vertical distributed edge shear $q$ is studied. Figure 19 shows the geometry of the plate, with inner diameter $R_i=6$, outer diameter $R_o=10$, and thickness $h=0.03$.

In the first case, the ring plate is assumed to be isotropic with material properties $E=21\times {10}^6$ and $\nu =0$. For comparison, the benchmark results of Sze et al. [57], obtained using the ABAQUS version 5.8 S4R element, are adopted. The VFIFE model consists of 185 particles and 288 elements. The vertical displacements at points A and B ($W_a$ and $W_b$) are plotted in Figure 20, and the VFIFE predictions show good agreement with the results reported by Sze et al. [57] and Arciniega and Reddy [58].

In the second case, the ring plate is modeled as a laminated composite. The material properties are defined in the circumferential principal direction of the plate: $E_1=20\times {10}^6$, $E_2=E_3=6\times {10}^6$, $G_{12}=G_{13}=3\times {10}^6$, $G_{23}=2.4\times {10}^6$, ${\nu }_{12}={\nu }_{13}=0.3$, and ${\nu }_{23}=0.25$. Three lamination schemes, namely, $\left[0/90/0\right]$, $\left[-45/45/-45/45\right]$, and $\left[30/-60/-60/30\right]$, are investigated. The vertical displacement at point B under different values of $q$ is shown in Figure 21. The VFIFE results are consistent with the benchmark solutions of Arciniega and Reddy [58].

5.9. Composite Panel Subjected to Pulse Load

A laminated composite square plate with dimensions of $a\times a=2.54 \mathrm{m}\times 2.54 \mathrm{m}$ (Figure 22) subjected to three types of transverse pulse load (Figure 23) is investigated, as follows:

(1)

Blast load

$$P\left(t\right)=\left\{\begin{array}{c}{P}_0\left(1-{\displaystyle \frac{t}{t_1}}\right)e^{{\displaystyle \frac{-\alpha t}{t_1}}}, \mathrm{f}\mathrm{o}\mathrm{r} t\le t_1\\ 0 , \mathrm{f}\mathrm{o}\mathrm{r} t>t_1\end{array}\right.$$

(61)

(2)

Triangular load

$$P\left(t\right)=\left\{\begin{array}{c}{P}_0\left(1-{\displaystyle \frac{t}{t_1}}\right), \mathrm{f}\mathrm{o}\mathrm{r} t\le t_1\\ 0 , \mathrm{f}\mathrm{o}\mathrm{r} t>t_1\end{array}\right.$$

(62)

(3)

Step load

$$P\left(t\right)=P_0$$

(63)

$P_0=3447 \mathrm{kPa}$, $\alpha =2.0$, and $t_1=0.1 \mathrm{s}$ are used in the three loading conditions. The boundary conditions are: $u=v=w={\theta }_x=0 \mathrm{a}\mathrm{t} x=\pm 1.27 \mathrm{m}, u=v=w={\theta }_y=0 \mathrm{a}\mathrm{t} y=\pm 1.27 \mathrm{m}\left(\mathrm{S}\mathrm{S}\mathrm{S}\mathrm{S}\right)$. The material properties of the plate are $E_1=132.4 \mathrm{G}\mathrm{P}\mathrm{a}$, $E_2=10.8 \mathrm{G}\mathrm{P}\mathrm{a}$, $G_{12}=G_{13}=G_{23}=5.6 \mathrm{G}\mathrm{P}\mathrm{a}$, ${\nu }_{12}=0.24$, and $\rho =1443 \mathrm{k}\mathrm{g}/{\mathrm{m}}^3$. The plate has a thickness of $h=0.17 \mathrm{m}$, and the lay-up is $\left[0/90/0\right]$, with the center $90°$ ply twice as thick as each $0°$ ply. Damping is not considered. Only a quarter of the plate is used with 36 particles and 50 elements in the VFIFE modeling because of symmetry. The displacement at the center under each pause load is depicted in Figure 24, Figure 25 and Figure 26. Comparison with the methods of Kazancı and Mecitoğlu [59] and Upadhyay et al. [60] confirms the validity of the VFIFE model.

Two additional boundary conditions, namely, CCSS ($u=v=w={\theta }_x={\theta }_y=0$ at $x=\pm 1.27 \mathrm{m}$, $u=v=w={\theta }_y=0$ at $y=\pm 1.27 \mathrm{m}$) and CCCC ($u=v=w={\theta }_x={\theta }_y=0$ at four edges), are tested using blast load [60]. The resulting displacements are depicted in Figure 27, illustrating the influence of boundary stiffness. The VFIFE calculation results are in good agreement with the findings of Upadhyay et al. [60].

Two parametric studies are performed under the boundary condition SSSS with the blast load to examine span-to-thickness ratio effects. In the first study, $a=2.54 \mathrm{m}$ is retained whereas $h$ is varied; in the second study, $h=0.17 \mathrm{m}$ is retained whereas $a$ is varied. The corresponding $a/h=14.94$, $20$, $30$, $40$. The results in Figure 28 and Figure 29 show that displacement amplitudes increase with large span:thickness ratios.

6. Conclusions

This study extends the VFIFE method by incorporating DKMT elements to capture transverse shear effects in laminated composite thick shells. Compared with conventional FEM, the proposed VFIFE–DKMT formulation offers significant advantages in handling large rotations and dynamic motions within a particle-based framework. Nevertheless, certain limitations remain, including higher computational cost for large-scale models and sensitivity to time-step size, which may affect computational efficiency.

The effectiveness and versatility of the proposed approach have been demonstrated through a series of benchmark numerical examples. These validations confirm the method’s capability to capture static bending, post-buckling behavior, and dynamic motions involving large displacements and rigid body rotations. In particular, the results obtained for problems with different span-to-thickness ratios verify the robustness of the VFIFE–DKMT formulation across a broad range of nonlinear problems in both thin and thick composite structures.

In conclusion, the proposed VFIFE method proves to be efficient and accurate for analyzing the nonlinear deformation and dynamic response of laminated composite thin-to-thick shells and plates. This study successfully improves the VFIFE framework, emphasizing its potential as a valuable computational tool for advanced structural analysis involving thick composite shells.