Vibration and buckling analysis of bio-inspired helicoidal composite plates with crack using XIGA

Abstract

Bio-inspired helicoidal composite laminates offer enhanced damage tolerance and superior mechanical performance, however, their vibration and buckling behavior in the presence of cracks remains insufficiently understood. Therefore, this study investigates the free vibration and buckling responses of cracked bio-inspired laminated composite plates using an extended isogeometric analysis (XIGA) framework. Reddy’s higher-order shear deformation theory is employed to accurately capture interlaminar stresses and kinematic behavior. The XIGA framework enables efficient modeling of displacement discontinuities associated with cracks while preserving precise geometric representation and field approximation, without mesh dependency. Consequently, the novelty of this work lies in the integration of bio-inspired helicoidal laminate configurations with XIGA-based crack modeling to perform an analysis of vibration and buckling behavior, which has not been previously reported. Three helicoidal configurations, including helicoidal exponential (HE), helicoidal recursive (HR), and helicoidal semicircular (HS), are examined for different plate geometries and crack lengths. The results indicate that increasing crack length leads to a reduction in both natural frequencies and critical buckling loads, with varying sensitivity across configurations. The HS configuration exhibits the most significant reduction in natural frequencies, whereas the HR configuration demonstrates comparatively better resistance. In terms of buckling behavior, the HR configuration shows the highest reduction for circular plates but the lowest for square plates. These findings provide new insights into the damage-sensitive response of bio-inspired composites and offer valuable guidance for the design of resilient structures in aerospace, marine, and automobile engineering applications.

Keywords

Bioinspired composites XIGA HSDT Through-thickness crack Natural frequency Critical buckling loads

Introduction

Bio-inspired structures are gaining attention for their unique designs,^1–3 optimized by nature over millions of years through evolutionary processes in various animals and plant organs. One of the successful examples of this type of mimicry in engineering is the design of the Japanese Shinkansen bullet train, where mimicking the kingfisher’s beak has resulted in minimized shockwave and minimal aerodynamic drag.⁴ Examples like this one have motivated researchers to mimic other natural structures such as plants, bone, nacre, and shrimp. Detailed studies have since uncovered mechanisms that yield many interesting combinations of mechanical properties found in various biological structures.^5–9 One such natural structure is a type of helicoidal structure known as the Bouligand structure found in the exoskeletons of crustaceans such as Odontodactylus scyllarus.^10,11 The unique type of helicoidal structure is being adapted to optimize the structural performance of carbon fiber reinforced plastics (CFRP) laminates. Many of these components and structures are subjected to dynamic loading, which requires careful determination of natural frequencies corresponding to different mode shapes to avoid resonance, which can cause excessive deformations, stresses, and catastrophic failures in the structure. Studying natural frequencies allows engineers to avoid or dampen the potential resonant conditions. However, under repeated dynamic loading, crack-like defects may appear due to fatigue. These defects alter the natural frequencies and buckling loads of the structures, potentially compromising their integrity and performance.

Extensive research has been conducted on the free vibration and buckling behavior of cracked structures by analytical techniques to assess their mechanical performance. One of the earliest studies, by Stahl et al.,¹² examined the vibration and buckling behavior of cracked plates using dual series equations. Ali et al.¹³ introduced a simplified approach based on Rayleigh’s principle to analyze the vibration behavior of plates with rectangular cutouts. Wu et al.¹⁴ conducted a study to investigate the dynamic instability of edge-cracked plates by using the incremental harmonic balance method. Rjoub et al.¹⁵ predicted the free vibration behavior of functionally graded porous plates with cracks through analytical modeling and artificial neural networks (ANNs). Song et al.¹⁶ presented an analytical solution to study the buckling behavior of cracked plates. However, these analytical methods are limited in their ability to analyze complex geometries, loading conditions, and boundary conditions and in considering complex environmental factors. Owing to the limitations faced by analytical techniques, researchers have utilized numerical methods to analyze the mechanical behavior of structures. These numerical methods include boundary element method,¹⁷ meshless local Petrov-Galerkin method,¹⁸ phantom-node method,¹⁹ Galerkin method,²⁰ differential quadrature method,²¹ strip-yield method,²² Ritz method,²³ extended cell-based smoothed discrete shear gap method,²⁴ dynamic stiffness method,²⁵ mesh-free method,²⁶ finite element method (FEM),^27–30 phase field method,³¹ extended finite element method (XFEM).³² Among these, FEM has been extensively used for analyzing advanced composite and functionally graded structures due to its versatility and robustness, as demonstrated in recent studies.^33–37 Despite their advantages, these numerical techniques have their own challenges, including discretization errors, difficulties in geometric representation, the need for higher-order continuity, and high computational costs. These limitations reduce their efficiency in accurately analyzing cracked bioinspired composite structures, highlighting the need for more advanced and adaptive modeling approaches.

With benefits like better geometry representation, smoother basis functions, and reduced meshing complexity, isogeometric analysis (IGA) was developed by Hughes et al.³⁸ is increasingly being adopted in recent studies examining the mechanical behavior of plate structures.^39,40 Later, Ghorashi et al.,⁴¹ enhanced IGA by introducing the partition of unity concept, leading to the development of XIGA. XIGA outperforms other discrete techniques, such as XFEM, meshfree method, boundary element method, etc., in analyzing cracked structures. XIGA utilizes NURBS basis functions for more accurate and efficient analysis. Like XFEM, XIGA eliminates the need for re-meshing as cracks propagate. However, it provides greater reliability by using CAD-based NURBS functions for both geometry representation and analysis, unlike XFEM, which relies on Lagrangian basis functions. In recent years, XIGA has been successfully applied to analyze the buckling and vibration behavior of cracked structures. Yin et al.⁴² studied the buckling and vibration behavior of cracked Reissner-Mindlin FG plates using NURBS based XIGA, demonstrating its high accuracy in modeling cracked plates. Singh et al.⁴³ employed higher-order shear deformation theory (HSDT) within the XIGA framework to analyze cracked FG plates. Shoheib et al.⁴⁴ employed XIGA to examine the effect of welding residual stress and cyclic internal pressure on crack propagation and fatigue life. Zhang et al.⁴⁵ employed locally refined NURBS based XIGA to study the behavior of cracked FG composite plates. Their findings reveal that incorporating adaptive local refinement offers high precision at a low computational cost with a fast convergence rate. Zhong et al.⁴⁶ presented a three-dimensional XIGA model to analyze the vibration behavior of cracked blades. Kumar et al.⁴⁷ studied the fracture behavior of edge-cracked FG structures using XIGA. Parihar et al.⁴⁸ conducted vibration analysis of cracked bi-directional FG plates using XIGA. Based on the reviewed literature, XIGA provides greater flexibility in selecting higher-order basis functions, which is not possible with the Lagrangian-based shape functions used in XFEM. Additionally, using higher-order shape functions for both geometry representation and solution approximation helps prevent shear locking in thin plates.

Fiber-reinforced composites are increasingly replacing conventional aluminum alloys in the aerospace industry due to their superior strength-to-weight ratio and customizable mechanical properties. Driven by constrained resources such as depleting fossil fuels, the need for further improvement has led researchers to explore solutions in natural structures such as the Bouligand structure, found in the exoskeleton of crustacean-type creatures. The Bouligand structure is a layered architecture composed of unidirectional laminae arranged in a helicoidal pattern. Mimicking this architecture in fiber-reinforced composites has shown an increase in the out-of-plane strength of the structure.⁴⁹ Apichattrabrut et al.⁵⁰ found that the helicoidal stacking sequence exhibited greater debonding resistance and enhanced damage tolerance. Grunenfelder et al.⁵¹ have demonstrated that bio-inspired helicoidal structure reduces through-thickness damage propagation during impact, which leads to higher toughness. Although out-of-plane damage tolerance and impact strength of Bouligand architecture-inspired fiber-reinforced architecture are numerous, the studies on free vibration and buckling are limited in the literature. Lu et al.¹⁰ found that bio-inspired carbon fiber-reinforced polymer composite improves the buckling load and the buckling temperature along with the thermal and compressive post-buckling strength.

Garg et al.⁵² carried out free vibration analysis of cross and double-helicoidal laminated composite plates using higher-order zigzag theory and found that the bioinspired cross-helicoidal plates exhibit behavior like quasi-isotropic plates. Paruthi and coworkers⁵³ studied the buckling and free vibration of similar bio-inspired composites under thermal conditions. Kiran et al.⁴⁰ investigated the buckling and vibration behavior of helicoidal bioinspired composite plates using inverse hyperbolic shear deformation theory (IHSDT) within a NURBS-based IGA framework. Saurabh et al.⁵⁴ conducted a comprehensive study on the nonlinear vibration and bending behavior of bioinspired helicoidal laminated plates using IHSDT. In a subsequent study, they investigated the effect of hygrothermal environment on buckling and vibration performance of carbon nanotube-reinforced bioinspired composite plates using HSDT within a NURBS-based IGA framework.⁵⁵ Further, they have employed an IHSDT-based data-driven machine learning (ML) approach to analyze the nonlinear vibration behavior of bioinspired helicoidal laminated plates.⁵⁶ Gourdache et al.⁵⁷ studied free vibration behaviour of bioinspired helicoidal laminate plates in hygrothermal environments using simplified integral HSDT. Padhy et al.⁵⁸ developed a surrogate ML-based automated regression workflow for the bending analysis of bioinspired laminated composite plates by employing Reddy’s HSDT.

HSDTs are widely utilized for the analyses of laminated composite plates, as classical laminated plate theory (CLPT) fails to capture transverse shear strain effects. To solve this inadequacy of CLPT, first-order shear deformation theory (FSDT) was proposed, assuming a constant shear deformation.⁵⁹ However, to satisfy the traction free boundary conditions at the free surfaces of the plates, a shear correction factor is used. The value of the shear correction factor is quite dispersed, and its value depends upon the boundary conditions and other parameters. The limitations of FSDT have been addressed with the advent of the HSDT plate theory, proposed by Reddy⁶⁰ and widely known as Reddy’s HSDT plate theory.^60,61 This theory offers improvements to both CLPT and FSDT plate theories. Recently, several refined and advanced HSDT formulations have been proposed to further enhance accuracy in the analysis of complex composite and functionally graded structures. For example, Tounsi et al.⁶² investigated wave dispersion in functionally graded cylindrical shells, while Amar et al.⁶³ and Rebai et al.⁶⁴ analyzed the dynamic behavior of porous functionally graded spherical shells using refined HSDT models. Abualnour et al.⁶⁵ studied dynamic behaviour of advanced sandwich structures using quasi-3D HSDT. Garg et al.⁶⁶ conducted free vibration analysis of annular and flat plates using quasi-3D shear deformation theory. Additionally, quasi-3D and nonlocal refined theories have been developed for nanostructures and sandwich plates, as reported by Boussalem et al.⁶⁷ and Kaddari et al.⁶⁸ Despite these advancements, Reddy’s HSDT remains a reliable and efficient choice due to its balance between computational simplicity and accuracy. Therefore, in the present work, Reddy’s HSDT is employed in conjunction with the XIGA framework to accurately capture transverse shear deformation and interlaminar stress effects in cracked bio-inspired composite plates.

The above review highlights that while bioinspired composite structures have been extensively studied, their behavior in the presence of cracks remains unexplored. Composite structures used in advanced engineering applications, particularly in aerospace, marine, and energy systems, are frequently subjected to defects such as cracks arising from manufacturing imperfections, impact damage, or fatigue loading. These defects can significantly alter the dynamic characteristics and stability of the structure, potentially leading to resonance-induced failures or premature buckling. While bio-inspired helicoidal composite laminates have demonstrated superior damage tolerance and resistance to crack propagation, their structural response in the presence of existing cracks remains inadequately understood, especially under vibration and buckling conditions. From a scientific perspective, accurately capturing the influence of cracks on structural behavior requires advanced numerical techniques capable of modeling discontinuities without loss of accuracy. Although XFEM has been widely used, its integration with geometrically exact representations is limited. In this context, XIGA offers a powerful alternative by combining the geometric accuracy of isogeometric analysis with the capability to model cracks efficiently.

To the author’s best knowledge, no published research examines the influence of the crack on the natural frequency and buckling behavior in bio-inspired composites using XIGA. Therefore, the present study is motivated by the need to develop a robust computational framework to investigate the vibration and buckling behavior of cracked bio-inspired composite plates and to provide deeper insight into their damage-sensitive response for improved structural design and reliability. Hence, this study aims to investigate the behavior of a bio-inspired composite plate with a pre-existing crack on free vibration and buckling load of the structure. This work focuses particularly on the alteration of natural frequencies and buckling load in the presence of a crack, which may be of interest to the service and maintenance engineers seeking to safeguard the structure from resonance and buckling. To facilitate crack modeling within the finite element domain, this study employs a customized approach of IGA, known as XIGA. Despite the significance of understanding the changes in natural frequencies and buckling load in biomimetic composite plates with cracks, the literature, to the best of the authors’ knowledge, lacks investigation on this topic. Based on these gaps and the significance of the topic, the following objectives have been decided in this work:

(1) Development of an XIGA-based framework for analyzing cracked bio-inspired helicoidal composite plates.

(2) Incorporation of Reddy’s higher-order shear deformation theory for improved interlaminar stress prediction.

(3) First comprehensive investigation of both vibration and buckling responses of cracked helicoidal laminates.

(4) Comparative assessment of different helicoidal configurations (HE, HR, HS) under varying crack lengths and geometries.

The outcomes of the present study have important implications for the design of advanced composite structures where damage tolerance and structural stability are critical. Bio-inspired helicoidal laminates, due to their inherent resistance to crack propagation, are particularly suitable for aerospace structures such as wings, fuselage panels, and rotor components, where vibration characteristics and buckling resistance play a crucial role in performance and safety. Furthermore, these materials can be effectively utilized in marine structures, including ship hulls and offshore components, where resistance to fatigue and impact-induced damage is essential. The insights gained from the present vibration and buckling analyses of cracked helicoidal plates can also support the development of lightweight automotive components and wind turbine blades, where improved durability and stability under complex loading conditions are required.

The remainder of the paper is structured as follows: Section 2 introduces the formulation based on Reddy’s higher-order shear deformation theory. Section 3 presents the detailed XIGA formulation. Section 4 derives the governing equations for the free vibration and buckling analysis of cracked bioinspired laminated composite plates. Section 5 conducts a convergence and validation study of the proposed model. Section 6 provides several numerical results and discussions. Finally, Section 7 provides the conclusions of the study.

Reddy’s higher-order shear deformation theory

Let us consider a composite plate of length a, width b, thickness h, and crack length c as depicted in Figure 1, having mid-plane (i.e. $z = 0$ ) displacement, $u_{0}, v_{0}$ and $w_{0}$ on point $(x, y)$ and $β_{x}$ , and $β_{y}$ are the rotations of the normal to the mid-plane about the $y$ and $x$ axes respectively.

Figure 1.

Schematic of the typical composite plate with a through crack.

Reddy’s higher-order formulation begins with the following displacement field⁶⁹

u (x, y, z) = u_{0} (x, y) + z β_{x} (x, y) - c_{1} z^{3} (β_{x} + \frac{\partial w_{0}}{\partial x})

(1)

v (x, y, z) = v_{0} (x, y) + z β_{y} (x, y) - c_{1} z^{3} (β_{y} + \frac{\partial w_{0}}{\partial y})

(2)

w (x, y) = w_{0} (x, y)

(3)

where the constant

c_{1} = (\frac{4}{3 h^{2}})

h

indicating plate thickness.

The linear strains are expressed by

ε = ε^{(0)} + z ε^{(1)} + z^{3} ε^{(3)}

(4)

and

γ = γ^{(0)} + z^{2} γ^{(2)}

(5)

where,

\begin{array}{l} ε = {[\begin{array}{l} ε_{x x} & ε_{y y} & γ_{x y} \end{array}]}^{T}, γ = {[\begin{array}{l} γ_{x z} & γ_{y z} \end{array}]}^{T}, ε^{(0)} = {[\begin{array}{l} ε_{x x}^{(0)} & ε_{y y}^{(0)} & γ_{x y}^{(0)} \end{array}]}^{T}, ε^{(1)} = {[\begin{array}{l} ε_{x x}^{(1)} & ε_{y y}^{(1)} & γ_{x y}^{(1)} \end{array}]}^{T}, \\ ε^{(3)} = {[\begin{array}{l} ε_{x x}^{(3)} & ε_{y y}^{(3)} & γ_{x y}^{(3)} \end{array}]}^{T}, \end{array}

γ^{(0)} = {[\begin{array}{l} γ_{x z}^{(0)} & γ_{y z}^{(0)} \end{array}]}^{T}, a n d

γ^{(2)} = {[\begin{array}{l} γ_{x z}^{(2)} & γ_{y z}^{(2)} \end{array}]}^{T}, w i t h

ε^{(0)} = {[\begin{array}{l} \frac{\partial u_{0}}{\partial x} & \frac{\partial v_{0}}{\partial y} & \frac{\partial u_{0}}{\partial y} + \frac{\partial v_{0}}{\partial x} \end{array}]}^{T}, ε^{(1)} = {[\begin{array}{l} \frac{\partial β_{x}}{\partial x} & \frac{\partial β_{y}}{\partial y} & \frac{\partial β_{x}}{\partial y} + \frac{\partial β_{y}}{\partial x} \end{array}]}^{T}

(6)

ε^{(3)} = - c_{1} {[\begin{array}{l} \frac{\partial β_{x}}{\partial x} + \frac{\partial^{2} w_{0}}{\partial x^{2}} & \frac{\partial β_{y}}{\partial y} + \frac{\partial^{2} w_{0}}{\partial y^{2}} & \frac{\partial β_{x}}{\partial y} + \frac{\partial β_{y}}{\partial x} + 2 \frac{\partial^{2} w_{0}}{\partial x \partial y} \end{array}]}^{T}

(7)

γ^{(0)} = {[\begin{array}{l} β_{x} + \frac{\partial w_{0}}{\partial x} & β_{y} + \frac{\partial w_{0}}{\partial y} \end{array}]}^{T}, γ^{(2)} = - 3 c_{1} {[\begin{array}{l} β_{x} + \frac{\partial w_{0}}{\partial x} & β_{y} + \frac{\partial w_{0}}{\partial y} \end{array}]}^{T}

(8)

The constitutive relation of a layer with respect to the local or material coordinate system for plane stress is represented by,

{\begin{cases} σ_{x x} \\ σ_{y y} \\ σ_{x y} \\ σ_{x z} \\ σ_{y z} \end{cases}} = [\begin{array}{l} Q_{11} & Q_{12} & Q_{16} & 0 & 0 \\ Q_{21} & Q_{22} & Q_{26} & 0 & 0 \\ Q_{61} & Q_{62} & Q_{66} & 0 & 0 \\ 0 & 0 & 0 & Q_{55} & Q_{54} \\ 0 & 0 & 0 & Q_{45} & Q_{44} \end{array}] {\begin{cases} ε_{x x} \\ ε_{y y} \\ γ_{x y} \\ γ_{x z} \\ γ_{y z} \end{cases}}

(9)

where the material constants are given by,

Q_{11} = \frac{E_{11}}{1 - ν_{12} ν_{21}}, Q_{22} = \frac{E_{22}}{1 - ν_{12} ν_{21}}, Q_{12} = \frac{ν_{21} E_{11}}{1 - ν_{12} ν_{21}}, Q_{66} = G_{12}, Q_{44} = G_{23}, Q_{45} = G_{12}, Q_{55} = G_{13}

Here

E_{11}

and

E_{22}

are effective Young’s modulus of the composite plate in the principal material directions, and

G_{12}, G_{13}, G_{23}

are the effective shear moduli and

ν_{12}, ν_{21}

are the effective Poisson’s ratios.

As a laminate consists of many layers, the constitutive relation needs to be transformed into the laminate coordinate system. The transformed constitutive relation of $k$ ^th layer, in a laminate coordinate system, can be expressed as⁷⁰

{\begin{cases} σ_{x x} \\ σ_{y y} \\ σ_{x y} \\ σ_{x z} \\ σ_{y z} \end{cases}}^{(k)} = [\begin{array}{l} {\underset{\sim}{Q}}_{11} & {\underset{\sim}{Q}}_{12} & {\underset{\sim}{Q}}_{16} & 0 & 0 \\ {\underset{\sim}{Q}}_{21} & {\underset{\sim}{Q}}_{22} & {\underset{\sim}{Q}}_{26} & 0 & 0 \\ {\underset{\sim}{Q}}_{61} & {\underset{\sim}{Q}}_{62} & {\underset{\sim}{Q}}_{66} & 0 & 0 \\ 0 & 0 & 0 & {\underset{\sim}{Q}}_{55} & {\underset{\sim}{Q}}_{54} \\ 0 & 0 & 0 & {\underset{\sim}{Q}}_{45} & {\underset{\sim}{Q}}_{44} \end{array}] {\begin{cases} ε_{x x} \\ ε_{y y} \\ γ_{x y} \\ γ_{x z} \\ γ_{y z} \end{cases}}^{(k)}

(10)

where,

{\underset{\sim}{Q}}_{i j}

are the transformed material constant, which are

\begin{array}{l} {\underset{\sim}{Q}}_{11} = Q_{11} \cos^{4} (θ) + (Q_{12} + 2 Q_{66}) \sin^{2} (2 θ) + Q_{22} \sin^{4} (θ), \\ {\underset{\sim}{Q}}_{12} = \frac{1}{2} (Q_{11} + Q_{22} - 4 Q_{66}) \sin^{2} (2 θ) + Q_{12} (\sin^{4} (θ) + \cos^{4} (θ)), \\ {\underset{\sim}{Q}}_{22} = Q_{11} \sin^{4} (θ) + (Q_{12} + 2 Q_{66}) \sin^{2} (2 θ) + Q_{22} \cos^{4} (θ), \\ {\underset{\sim}{Q}}_{16} = \frac{1}{2} [(Q_{11} - Q_{12} - 2 Q_{66}) \cos^{2} (θ) + (Q_{12} - Q_{22} + 2 Q_{66}) \sin^{2} (θ)] \sin (2 θ), \\ {\underset{\sim}{Q}}_{26} = \frac{1}{2} [(Q_{11} - Q_{12} - 2 Q_{66}) \sin^{2} (θ) + (Q_{12} - Q_{22} + 2 Q_{66}) \cos^{2} (θ)] \sin (2 θ), \\ {\underset{\sim}{Q}}_{66} = \frac{1}{2} (Q_{11} + Q_{22} - 2 Q_{12} - 2 Q_{66}) \sin^{2} (2 θ) + Q_{66} (\sin^{4} (θ) + \cos^{4} (θ)), \\ {\underset{\sim}{Q}}_{44} = Q_{44} \cos^{2} (θ) + Q_{55} \sin^{2} (θ), \\ {\underset{\sim}{Q}}_{45} = \frac{1}{2} (Q_{55} - Q_{44}) \sin (2 θ), \\ {\underset{\sim}{Q}}_{55} = Q_{44} \sin^{2} (θ) + Q_{55} \cos^{2} (θ) \end{array}}

(11)

XIGA formulation of cracked plate

NURBS basis functions

The foundations of NURBS are B-splines. It is composed of $n$ piecewise polynomial of basis functions of degree $p$ . Knot vectors are used to define these basis functions. A knot vector is a non-decreasing sequence of parameter values that determines the intervals and continuity of B-splines, which are $C^{\infty}$ continuous in the interior of the knot span. At each knot, the basis functions are $C^{p - 1}$ continuous and the continuity for a knot of multiplicity $k$ is $C^{p - k}$ . The basis functions of the B-spline are defined as follows starting with $p = 0$ ⁶⁹,

N_{i, 0} = {\begin{cases} 1 ζ_{i} \leq ζ \leq ζ_{i + 1} \\ 0 otherwise \end{cases}

(12)

For

p > 0

, the basis functions are defined as follows,

N_{i, p} (ζ) = \frac{ζ - ζ_{i}}{ζ_{i + p} - ζ_{i}} N_{i, p - 1} (ζ) + \frac{ζ_{i + p + 1} - ζ}{ζ_{i + p + 1} - ζ_{i + 1}} N_{i + 1, p - 1} (ζ)

(13)

A B-spline curve with $n$ control points and coordinate of the control points $P_{i}$ , is given by

C (ζ) = \sum_{i = 1}^{n} N_{i, p} (ζ) P_{i}

(14)

B-spline surfaces are generated by extending the idea of B-spline curves into two dimensions. This is implemented by using two sets of basis functions $N_{i, p} (ζ)$ and $M_{j, q} (η)$ with knot vectors $Z = {\begin{cases} ζ_{1} & ζ_{2} & ζ_{3} & . . . & ζ_{n + p + 1} \end{cases}}$ and $Ψ = {\begin{cases} ψ_{1} & ψ_{2} & ψ_{3} & . . . & ψ_{n + p + 1} \end{cases}}$ as

S (ζ, η) = \sum_{i = 1}^{n} \sum_{j = 1}^{m} N_{i, p} (ζ) M_{j, q} (η) P_{i, j}

(15)

where

P_{i, j}

consists of a

n \times m

set of the control set.

However, due to the inability of exactly representing the conic sections, B-splines are modified with weighted control points and a non-uniform knot vector. These modified curves are called non-uniform rational B-spline (NURBS) which can represent the conic sections exactly. A NURBS curve is mathematically represented as:

\tilde{C} (ζ) = \sum_{i = 1}^{n} R_{i, p} (ζ) P_{i}

(16)

where,

R_{i, p} (ζ) = \frac{N_{i, p} (ζ) w_{i}}{W (ζ)} = \frac{N_{i, p} (ζ) w_{i}}{\sum_{j = 1}^{n} N_{j, p} (ζ) w_{i}}

(17)

in which

R_{i} (ζ)

represents the NURBS basis function.

Similarly, a NURBS surface can be represented as,

\tilde{S} (ζ, η) = \sum_{i = 1}^{n} \sum_{j = 1}^{m} R_{i, p} (ζ, η) P_{i, j}

(18)

where,

R_{i, p} (ζ, η) = \frac{N_{i, p} (ζ) M_{j, q} (η) w_{i, j}}{\sum_{k = 1}^{n} \sum_{l = 1}^{m} N_{k, p} (ζ) M_{l, q} (η) w_{k, l}}

(19)

Figure 2 presents a set of two-dimensional B-spline basis functions of degree 2 for uniform knot vector $[0, 0, 0, 2.5, 0.5, 7.5, 1, 1, 1]$ .

Figure 2.

Two-dimensional B-spline basis functions for (a) without crack and (b) with crack.

Extended isogeometric analysis (XIGA)

IGA emerged as an accurate tool due to its use of NURBS basis functions, which can generate complex curves of machine components and structures. However, to customize this method to solving problems related to fracture mechanics, the strategies followed in the XFEM are used in isogeometric analysis, which is called extended isogeometric analysis. Thus, IGA enriched with the partition of unity enrichment enables the representation of discontinuity in the domain irrespective of the mesh, and it eliminates the necessity of remeshing in crack growth analysis.

Likewise, XFEM, the approximation of displacement is represented by

u^{h} (ζ) = \sum_{i = 1}^{n_{e n}} R_{i} (ζ) u_{i} + \sum_{j = 1}^{n_{c f}} R_{j} (ζ) [H (ζ) - H (ζ_{j}) a_{j}]

(20)

where,

u_{i} = {\begin{cases} u_{o i} & v_{o i} & w_{o i} & β_{x i} & β_{y i} \end{cases}}^{T}

represents the standard degree of freedom associated with the control point i, and

H (ζ)

represents the Heaviside function. The additional degrees of freedom to model the crack face has been denoted by

a_{j}

. The number of control points corresponding to Heaviside enrichments are

n_{c f}

. The Heaviside function takes the value of +1 or −1 based on the position with respect to the crack surface, as illustrated in Figure 2(b).

Governing equations

The weak form for the vibration⁷⁰ of the plate can be derived from the principle of virtual work:

\int_{Ω} δ ε^{T} D ε d Ω + {\int_{Ω} δ γ}^{T} D_{o} γ d Ω = \int_{Ω} δ u^{T} m \ddot{u} d Ω

(21)

where,

ε, γ

is the in-plane strain and out-of-plane shear strain vector, respectively, and

D = [\begin{array}{l} A & B & E \\ B & L & F \\ E & F & H \end{array}], D_{o} = [\begin{array}{l} A_{o} & B_{o} \\ B_{o} & G_{o} \end{array}]

(22)

(\begin{array}{l} A_{i j}, & B_{i j}, & L_{i j}, & E_{i j}, & F_{i j}, & H_{i j} \end{array}) = \int_{- \frac{h}{2}}^{\frac{h}{2}} (\begin{array}{l} 1, & z, & z^{2}, & z^{3}, & z^{4}, & z^{6} \end{array}) Q_{i j} d z

(23)

({\begin{array}{l} A_{i j}^{o} & B_{i j}^{o} & G \end{array}}_{i j}^{o}) = \int_{- \frac{h}{2}}^{\frac{h}{2}} (\begin{array}{l} 1, & z^{2}, & z^{4} \end{array}) Q_{i j} d z

The vector

u

indicates

u = {[\begin{array}{l} u_{1} & u_{2} & u_{3} \end{array}]}^{T}

where,

u_{1} = [\begin{array}{l} u_{0} \\ v_{0} \\ w_{0} \end{array}], u_{2} = [\begin{array}{l} β_{x} \\ β_{y} \\ 0 \end{array}], u_{3} = c [\begin{array}{l} β_{x} + \frac{\partial w}{\partial x} \\ β_{y} + \frac{\partial w}{\partial y} \\ 0 \end{array}]

m

in equation (21) represents mass matrix which is expressed as,

m = [\begin{array}{l} I_{1} & I_{2} & I_{4} \\ I_{2} & I_{3} & I_{5} \\ I_{4} & I_{5} & I_{7} \end{array}]

(24)

where

(\begin{array}{l} I_{1} & I_{2} & I_{3} & I_{4} & I_{5} & I_{7} \end{array}) = \int_{- \frac{h}{2}}^{\frac{h}{2}} ρ (\begin{array}{l} 1, & z, & z^{2}, & z^{3}, & z^{4}, & z^{6} \end{array}) d z

having

ρ = [\begin{array}{l} ρ & 0 & 0 \\ 0 & ρ & 0 \\ 0 & 0 & ρ \end{array}]

The weak form for buckling⁷⁰ is

{\int_{Ω} δ ε}^{T} D ε d Ω + {\int_{Ω} δ γ}^{T} D_{o} γ d Ω + h \int_{Ω} \nabla^{T} δ w {\tilde{σ}}_{o} \nabla w d Ω = 0

(25)

where

\nabla^{T} = [\begin{array}{l} \frac{\partial}{\partial x} & \frac{\partial}{\partial y} \end{array}]

and

{\tilde{σ}}_{o} = [\begin{array}{l} σ_{x}^{o} & τ_{x y}^{o} \\ τ_{x y}^{o} & σ_{y}^{o} \end{array}]

are the gradient operator and in-plane pre-buckling stresses, respectively.

XIGA discrete equations

Upon substituting equation (20) into equation (4), the in-plane strains and out-of-plane shear strains can be written in form as,

{[\begin{array}{l} ε & γ \end{array}]}^{T} =_{M = 1}^{n \times m} [\begin{array}{l} B_{M}^{m} & B_{M}^{b 1} & B_{M}^{b 2} & B_{M}^{s 0} & B_{M}^{s 1} \end{array}] \{q_{M}\} =_{M = 1}^{n \times m} [B_{M}] \{q_{M}\}

(26)

where

n \times m

indicates the number of basis functions,

q_{M}

denotes the degrees of freedom (both standard and enriched) associated with a control point

M

, respectively. Moreover, the expression of

B_{M}

is written as,

B_{M}^{m} = [\begin{array}{l} {\overset{⌢}{R}}_{M, x} & 0 & 0 & 0 & 0 \\ 0 & {\overset{⌢}{R}}_{M, y} & 0 & 0 & 0 \\ {\overset{⌢}{R}}_{M, y} & {\overset{⌢}{R}}_{M, x} & 0 & 0 & 0 \end{array}], B_{M}^{b 1} = [\begin{array}{l} 0 & 0 & 0 & {\overset{⌢}{R}}_{M, x} & 0 \\ 0 & 0 & 0 & 0 & {\overset{⌢}{R}}_{M, y} \\ 0 & 0 & 0 & {\overset{⌢}{R}}_{M, y} & {\overset{⌢}{R}}_{M, x} \end{array}]

(27)

B_{M}^{b 2} = [\begin{array}{l} 0 & 0 & {\overset{⌢}{R}}_{M, x x} & {\overset{⌢}{R}}_{M, x} & 0 \\ 0 & 0 & {\overset{⌢}{R}}_{M, y y} & 0 & {\overset{⌢}{R}}_{M, y} \\ 0 & 0 & 2 {\overset{⌢}{R}}_{M, x x} & {\overset{⌢}{R}}_{M, y} & {\overset{⌢}{R}}_{M, x} \end{array}]

(28)

B_{M}^{s 0} = [\begin{array}{l} 0 & 0 & {\overset{⌢}{R}}_{M, x} & {\overset{⌢}{R}}_{M} & 0 \\ 0 & 0 & {\overset{⌢}{R}}_{M, y} & 0 & {\overset{⌢}{R}}_{M} \end{array}], B_{M}^{s 1} = [\begin{array}{l} 0 & 0 & {\overset{⌢}{R}}_{M, x} & {\overset{⌢}{R}}_{M} & 0 \\ 0 & 0 & {\overset{⌢}{R}}_{M, y} & 0 & {\overset{⌢}{R}}_{M} \end{array}]

(29)

In the above expressions,

{\overset{⌢}{R}}_{M}

is different for standard elements (non-enriched) and enriched elements. For standard elements,

{\overset{⌢}{R}}_{M} = R_{M}

whereas

{\overset{⌢}{R}}_{M} = R_{M} {H (x) - H (x_{j})}

taken for split elements.

The geometric strain can be expressed as,

ε_{g} = \sum_{M = 1}^{n \times m} [B_{M}^{g}] {q_{M}}

(30)

where,

B_{M}^{g} = [\begin{array}{l} 0 & 0 & {\overset{⌢}{R}}_{M, x} & 0 & 0 \\ 0 & 0 & {\overset{⌢}{R}}_{M, y} & 0 & 0 \end{array}]

Upon substituting equation (20) and (26) into equation (21), the discrete finite element equation for free vibration analysis is formulated as,

(K - ω^{2} M) q = 0

(31)

where

ω

is the natural frequency and

K

is the global stiffness matrix, which can be expressed as

K = \int_{Ω} {{[\begin{array}{l} B^{m} \\ B^{b 1} \\ B^{b 2} \end{array}]}^{T} [\begin{array}{l} A & B & E \\ B & L & F \\ E & F & H \end{array}] [\begin{array}{l} B^{m} \\ B^{b 1} \\ B^{b 2} \end{array}] + {[\begin{array}{l} B^{s 0} \\ B^{s 1} \end{array}]}^{T} [\begin{array}{l} A_{o} & B_{o} \\ B_{o} & G_{o} \end{array}] [\begin{array}{l} B^{s 0} \\ B^{s 1} \end{array}]} d Ω

(32)

And mass matrix can be expressed as

M = \int_{Ω} {{[\begin{array}{l} N_{1} \\ N_{2} \\ N_{3} \end{array}]}^{T} [\begin{array}{l} I_{1} & I_{2} & I_{4} \\ I_{2} & I_{3} & I_{5} \\ I_{4} & I_{5} & I_{7} \end{array}] [\begin{array}{l} N_{1} \\ N_{2} \\ N_{3} \end{array}]} d Ω

(33)

where,

N_{1} = [\begin{array}{l} {\overset{⌢}{R}}_{M} & 0 & 0 & 0 & 0 \\ 0 & {\overset{⌢}{R}}_{M} & 0 & 0 & 0 \\ 0 & 0 & {\overset{⌢}{R}}_{M} & 0 & 0 \end{array}], N_{2} = [\begin{array}{l} 0 & 0 & 0 & {\overset{⌢}{R}}_{M} & 0 \\ 0 & 0 & 0 & 0 & {\overset{⌢}{R}}_{M} \\ 0 & 0 & 0 & 0 & 0 \end{array}]

And $N_{3} = [\begin{array}{l} 0 & 0 & {\overset{⌢}{R}}_{M, x} & {\overset{⌢}{R}}_{M} & 0 \\ 0 & 0 & {\overset{⌢}{R}}_{M, y} & 0 & {\overset{⌢}{R}}_{M} \\ 0 & 0 & 0 & 0 & 0 \end{array}]$

Upon substituting equation (20) and (26) into equation (25), the discrete finite element equation for buckling analysis is formulated as,

(K - λ_{c r} K_{g}) q = 0

(34)

where

K_{g}

is a geometrical stiffness matrix,

K_{g} = {\int_{Ω} (B^{g})}^{T} h {\tilde{σ}}_{0} B^{g} d Ω

and

λ_{c r}

indicates critical buckling load, and

B^{g}

represents a geometric strain-displacement matrix as expressed in equation (30).

Numerical integration

Standard Gauss quadrature-based numerical integration is performed for the evaluation of the stiffness matrix. Since this scheme cannot provide accurate results for enriched elements, the integration for split elements is done differently. In the current study, sub-triangulation and polar integration schemes are followed for the Heaviside-enriched elements. Since the approximation of the displacement variable is different for cut elements and the standard elements are different, different numbers of integration points are required for these elements. Different numbers of Gauss points are used for standard and split elements.⁴³

Convergence and validation study

A square plate (a/b = 1) with crack length-to-side ratio (c/a) of 0.5 and side-to-thickness ratio (a/h) of $0.01$ , is considered for the convergence and the validation study. Simply supported boundary conditions at all edges are considered. For determining the suitable mesh size, and the degree of the NURBS function, simulations at an increasing number of elements are performed for both degree 2 and 3 basis functions. The normalized natural frequency $(\bar{ω})$ corresponding to the first five modes is calculated. To consider all the first five modes, in the convergence study, the standard Euclidean norm is followed. In general, the first $n$ modes can be considered an element of $R^{n}$ . The Euclidean norm is a function $f : R^{n} \to R$ which takes a $n$ dimensional vector $x = (x_{1}, x_{2}, . . ., x_{n})$ and outputs a real number $f (x)$ such that $f (x) = \sqrt{x_{1}^{2} + x_{2}^{2} + . . . + x_{n}^{2}}$ . The norm of the first five modes is shown for both degree 3 and 2 in Figure 3. The absolute value of the slope of the best fit for degree 3 is slightly lesser than that of degree 2, indicating slightly lesser mesh sensitivity (which means better convergence) for degree 3. However, the difference in the slope is marginal, so degree 2 can be used for the simulation, providing easier computation than degree 3. The norm changed only 0.12% from element per edge 29 to 33 for degree 2 NURBS basis functions, so all the bio-inspired free vibration analysis would be done for this mesh size corresponding to 33 elements per edge. The $\bar{ω}$ corresponding to first five modes for degrees 2 and 3 are shown in Table 1 and Table 2 respectively. The results of the $\bar{ω}$ have been compared with the results of Emilija et al.²⁵ by HSDT. The obtained results of $\bar{ω}$ are found to conformal with Emilija et al.²⁵ thus validating the present model for further vibration analysis.

Figure 3.

Convergence study for vibration analysis.

Table 1.

The $\bar{ω}$ corresponding to the first five modes with the increasing number of elements for degree 2 NURBS basis function.

Modes	$\bar{ω}$
	Elements per edge					Emilija et al.²⁵
	17	21	25	29	33	Emilija et al.²⁵
1^st mode	18.56	18.53	18.51	18.5	18.49	18.45
2^nd mode	33.24	33.15	33.08	33.04	33	32.68
3^rd mode	60.53	59.9	59.58	59.36	59.2	57.99
4^th mode	67.82	67.72	67.75	67.76	67.76	67.74
5^th mode	73.74	73.55	73.4	73.29	73.19	71.79

Table 2.

The $\bar{ω}$ corresponding to the first five modes with the increasing number of elements for degree 3 NURBS basis function.

Modes	$\bar{ω}$
	Elements per edge					Emilija et al.²⁵
	17	21	25	29	33	Emilija et al.²⁵
1^st mode	18.58	18.56	18.54	18.53	18.52	18.45
2^nd mode	33.27	33.19	33.14	33.09	33.05	32.68
3^rd mode	60.96	60.44	60.07	59.79	59.58	57.99
4^th mode	67.63	67.7	67.73	67.75	67.76	67.74
5^th mode	73.86	73.69	73.55	73.43	73.33	71.79

For the convergence and validation study of buckling analysis, a square plate with a c/a of 0.4 is considered. The code is validated with the study of Milazzo et al.⁷¹ The boundary conditions were simply supported at all edges. Material properties taken, are $E_{1} = 113$ GPa, and $E_{2} = 9$ GPa, $ν_{12} = 0.302$ , $G_{12} = G_{13} = G_{23} = 3.82$ GPa where the fiber, longitudinal, transverse, and thickness directions are denoted by 1, 2, and 3, respectively. Symmetric layup with ${[0 / 45 / - 45 / 0]}_{S}$ is considered. The obtained critical buckling load was normalized by using the following formula and compared with the solution of Milazzo et al.⁷¹ by extended Ritz formulation.

{\bar{λ}}_{c r} = \frac{12 λ_{c r} a^{2} (1 - ν_{12}^{2})}{π^{2} E_{1} h^{3}}

(35)

The result of

{\bar{λ}}_{c r}

can be seen from Table 3. The percentage difference between the smallest buckling load of degree 2 and that of Milazzo et al.⁷¹ is 0.74%. The result of

{\bar{λ}}_{c r}

is plotted concerning the increasing number of elements per edge. The values and their best-fitted curve are shown in Figure 4 Comparing the percentage difference of 0.73, between a number of elements per edge 29 and 33. It can be concluded that the results are acceptably converged, and the number of elements per edge 33 can be used for further buckling analysis.

Table 3.

The result of ${\bar{λ}}_{c r}$ with increasing value of the number of elements.

Degree	${\bar{λ}}_{c r}$
	Elements per edge					Milazzo et al.⁷¹
	17	21	25	29	33	Milazzo et al.⁷¹
2	1.41	1.39	1.37	1.36	1.35	1.34
3	1.42	1.4	1.38	1.37	1.36	-

Figure 4.

Convergence study for buckling analysis.

Results and discussions

This section presents the free vibration and the buckling analysis of four different laminate configurations, including quasi-isotropic and three bioinspired layup configurations, by employing NURBS based XIGA framework. The three bioinspired layup configurations are helicoidal exponential (HE), helicoidal semicircular (HS), and helicoidal recursive (HR). The parametric study presents the effect of crack length, crack locations, boundary conditions, geometrical shapes, and helicoidal layup configurations, of bioinspired composite plates. The stacking sequence and the configurations of these layups are presented in Table 4, and schematic representation of different bio-inspired and quasi-isotropic arrangements is shown in Figures 5 and 6, respectively.

Table 4.

Stacking sequence of various configurations.

Configurations	No of plies	General form	Description
HE	20	${[a_{1}, a_{2}, . . ., a_{i} = α^{i}]}_{2 S}$ [3, 9, 27, 81, 243]_2S	$α = 3; 1 \leq i \leq 5$
HS	20	${[a_{1}, a_{2}, . . ., a_{i} = \sqrt{φ_{\max}^{2} - (μ (i - 1) - φ_{\max}^{2})}]}_{S}$ [0, 82.5, 113.1, 134, 149.7, 161.2, 169.7, 175.5, 178.9, 180]_S	$\begin{array}{l} a_{1} = 0; 2 \leq i \leq 10 \\ μ = 20; φ_{\max} = 90 \end{array}$
HR	20	${[a_{1}, a_{2}, . . ., a_{i} = a_{i - 1} + β (i - 1)]}_{S}$ [0, 3, 9, 18, 30, 45, 63, 84, 108, 135]_S	$\begin{array}{l} a_{1} = 0; 2 \leq i \leq 10 \\ β = 3 \end{array}$
QI	20	${[a_{1}, a_{2}, . . ., a_{i}]}_{S}$ [ 0, 90, 45, −45, 0, 90, 45, −45, 0, 90]_S	$1 \leq i \leq 10$

Figure 5.

Schematic representation of the different bio-inspired laminate layup configurations, (a) helicoidal exponential, (b) helicoidal semicircular, and (c) helicoidal recursive.

Figure 6.

Schematic representation of the quasi-isotropic configuration.

Vibration analysis of bio-inspired composite plates with crack

In this section free vibration analysis of bio-inspired composite laminates is performed. Three different cases are considered which are square plate with center crack, square plate with edge crack, and circular plate of diameter, d with center crack. The meshed domains of these three cases are presented in Figure 7. These composite plates consist of 20 plies for each of the configurations shown in Table 4.

Figure 7.

XIGA domain for (a) square plate with center crack, (b) square plate with edge crack, and (c) circular plate with center crack.

The natural frequency $ω$ corresponding to the first five modes has been determined and normalized using the following formula, equation (36)²⁵ where $a$ is the side of the plates, $h$ is the thickness, $ρ$ is the density and $E_{2}$ is the transverse elastic modulus. The plate length is replaced by the radius of the circular plate in the normalizing formula of $ω$ for circular plates with a center crack.

N o r m a l i z e d n a t u r a l f r e q u e n c y, \bar{ω} = ω \frac{a^{2}}{h} \sqrt{\frac{ρ}{E_{2}}}

(36)

The study considers boundary conditions of simply supported and clamped edges with the details provided in the equations below.

simply supported (S) : {\begin{cases} u = w = β_{x} = 0 at x = 0, a \\ v = w = β_{y} = 0 at y = 0, b \end{cases}

(37)

clamped (C) : u = v = w = β_{x} = β_{y} = 0

(38)

Simply supported at all edges and clamped at all edges are denoted by ‘SSSS’ and ‘CCCC’, respectively.

For all cases, the modulus ratios (E₁/E₂) were set to 40, and the crack inclination angle was set to 0⁰. The a/h ratio for square plates and the diameter to thickness (d/h) for circular plates were maintained at 20. The crack length ratios, c/a of 0.1, 0.2, 0.3, 0.4 0.5, have been considered for HE, HS, HR, and QI configurations. The first five mode shapes for different geometries with a c/a of 0.5 are presented in Figure 8. The variation of $\bar{ω}$ for square plates with center cracks, square plates with edge cracks, and circular plates with center cracks for SSSS boundary conditions are presented in Tables 5–7, respectively. It is also observed that compared to the bioinspired laminates, the quasi-isotropic laminate shows a higher $\bar{ω}$ .

Figure 8.

Nature of mode shapes of vibration analysis for SSSS HE bioinspired composite plate with c/a or c/d = 0.5.

Table 5.

Effect of crack length on the $\bar{ω}$ for square SSSS bioinspired composite plate with center cracks.

c/a	Modes	$\bar{ω}$
c/a	Modes	HE	HS	HR	QI
0.1	1	18.15	19.27	18.34	20.14
	2	35.77	44.26	30.08	46.38
	3	52.13	48.84	49.92	49.38
	4	58.98	65.74	54.69	69.44
	5	68.43	84.03	63.04	83.5
0.2	1	18.02	19.04	18.27	19.93
	2	35.7	44.15	30.06	46.21
	3	52.1	48.75	49.39	49.37
	4	58.66	65.65	54.67	69.35
	5	67.65	82.68	62.9	82.19
0.3	1	17.62	18.36	18.06	19.29
	2	35.24	43.33	29.87	45.06
	3	51.88	48.21	47.83	49.29
	4	57.54	65.33	54.55	69.09
	5	65.79	78.87	62.56	78.55
0.4	1	17.13	17.55	17.8	18.53
	2	34	40.71	29.38	42.1
	3	51.27	47.33	46.03	49.08
	4	55.9	64.72	54.14	68.54
	5	64.31	75.24	62.2	74.97
0.5	1	16.88	17.15	17.65	18.15
	2	32.95	38.45	28.96	39.82
	3	50.76	46.91	45.16	48.91
	4	55.01	64.2	53.67	68.03
	5	63.74	73.74	62.03	73.47

Table 6.

Effect of crack length on the $\bar{ω}$ for square SSSS bioinspired composite plate with edge cracks.

c/a	Modes	$\bar{ω}$
c/a	Modes	HE	HS	HR	QI
0.1	1	18.19	19.35	18.35	20.22
	2	35.77	44.26	30.06	46.36
	3	52.07	48.81	50.07	49.35
	4	59.06	65.73	54.62	69.39
	5	68.74	84.5	63.06	83.97
0.2	1	18.15	19.3	18.32	20.18
	2	35.72	44.18	30.01	46.24
	3	51.94	48.68	49.91	49.25
	4	58.94	65.52	54.53	69.11
	5	68.56	84.16	63	83.74
0.3	1	18.03	19.14	18.26	20.04
	2	35.57	43.87	29.92	45.87
	3	51.71	48.34	49.47	48.98
	4	58.51	64.6	54.38	68.08
	5	67.82	82.8	62.76	82.72
0.4	1	17.72	18.65	18.08	19.59
	2	34.82	42.3	29.59	43.94
	3	51.22	47.27	48.1	48.43
	4	56.72	60.65	54.21	63.65
	5	65.33	79.08	61.4	79.59
0.5	1	17.34	18.05	17.87	19.03
	2	33.33	38.71	28.99	40.09
	3	50.41	46.29	46.52	48.16
	4	54.55	57.67	54.12	59.98
	5	62.26	76.1	57.75	76.71

Table 7.

Effect of crack length on the $\bar{ω}$ for circular SSSS bioinspired composite plate with center cracks.

c/d	Modes	$\bar{ω}$
c/d	Modes	HE	HS	HR	QI
0.1	1	7.42	7.62	6.9	7.8
	2	12.72	14.19	11.08	15.06
	3	16.27	15.9	16.27	15.7
	4	19.17	21.75	16.53	23.02
	5	22.18	24.07	20.77	24.11
0.2	1	7.29	7.46	6.8	7.64
	2	12.68	14.11	11.05	14.95
	3	16.18	15.78	16.19	15.63
	4	18.79	21.59	16.21	22.95
	5	21.94	23.69	20.68	23.55
0.3	1	7.13	7.25	6.69	7.44
	2	12.5	13.82	10.96	14.52
	3	15.91	15.41	15.77	15.44
	4	18.29	21.29	15.97	22.55
	5	21.66	23.08	20.54	23.02
0.4	1	6.96	7.04	6.57	7.23
	2	12.09	13.07	10.75	13.57
	3	15.44	14.84	15.33	15.17
	4	17.78	20.84	15.59	21.82
	5	21.4	22.55	20.38	22.77
0.5	1	6.83	6.86	6.47	7.04
	2	11.28	11.65	10.32	12.04
	3	14.95	14.43	14.94	14.95
	4	17.37	20.33	15.17	21.23
	5	20.17	22.01	18.49	22.23

As the crack length increases, a decrease in $\bar{ω}$ is observed across all modes for both bio-inspired and quasi-isotropic composite laminates. The reduction of the $\bar{ω}$ with the increment of c/a can be attributed to the decrement of stiffness of the plates due to the increment of the crack length. As it is known that the presence of a crack effectively reduces the cross-sectional area for load bearing, which leads to a reduction of the stiffness of the plate. Nonetheless, the drop in $\bar{ω}$ impacted by crack length increment is not the same for all the bio-inspired composite laminates. The percentage reduction in the $\bar{ω}$ for crack length increment 0.1 to 0.5, is evaluated for all the laminates. The fundamental frequency is generally defined as the $\bar{ω}$ corresponding to the first mode. The percentage reduction of fundamental frequency is presented in Figure 9. It can be observed that the reduction in fundamental frequency is maximum for HS configuration irrespective of the geometric configuration and location of the plate. However, the quantitative value of the drop is dependent on the geometry of the plate and the location of the crack. In HS configuration, the square plate with a center crack shows the highest drop in fundamental frequency, followed by a circular plate with a center crack, and a square plate with an edge crack. The lowest reduction in fundamental frequency is observed in HR configurations. Among these, the circular plate with a center crack exhibits a greater reduction compared to the square plate with a center crack and a square plate with an edge crack. In the QI configuration, it is observed that the square plate with a center crack and the circular plate with a center crack demonstrate almost the same value of the reduction in fundamental frequency due to the increment of crack length from 0.1 to 0.5.

Figure 9.

Percentage drop of fundamental frequency in different bio-inspired and QI configurations for c/a or c/d changes from 0.1 to 0.5.

The influence of boundary conditions for the square plate with a center crack, square plate with an edge crack, and the circular plate with a center crack is shown in Tables 8–10 respectively. Two boundary conditions, SSSS and CCCC are considered with a c/a of 0.5. The a/h and E₁/E₂ for the square plates are considered as 20, and 40 respectively. For the circular plate, the diameter was used instead of width for defining the ratios, and the d/h = 20 are considered. The orientation of the crack was 0⁰ angle for both SSSS and CCCC boundary conditions. From the results, it is evident that the

\bar{ω}

exhibit higher values for the CCCC boundary condition compared to the SSSS boundary condition. The higher restriction of displacement and rotation for clamped boundary conditions leads to higher values of stiffness which results from higher natural frequency as the mass distribution does not change due to changes in the boundary conditions. It is observed that the increase in

\bar{ω}

for SSSS boundary conditions to the CCCC boundary condition is significant for square-shaped plates, whereas the increase is much smaller for circular plates. Likewise, the SSSS boundary conditions and the CCCC boundary conditions also demonstrate that the quasi-isotropic configuration exhibits higher normalized natural frequencies compared to the other bio-inspired configurations.

Table 8.

Effect of boundary conditions on $\bar{ω}$ for the square plate with a center crack, c/a = 0.5.

Boundary conditions	Modes	$\bar{ω}$
Boundary conditions	Modes	HE	HS	HR	QI
SSSS	1	16.88	17.15	17.65	18.15
	2	32.95	38.45	28.96	39.82
	3	50.76	46.91	45.16	48.91
	4	55.01	64.2	53.67	68.03
	5	63.74	73.74	62.03	73.47
CCCC	1	30.83	30.68	30.99	31.47
	2	45.22	48.99	41.23	50.06
	3	64.51	61.09	57.29	63.62
	4	68.29	79	66.4	82.23
	5	78.28	87.01	73.64	86.02

Table 9.

Effect of boundary conditions on $\bar{ω}$ for the square plate with an edge crack c/a = 0.5.

Boundary conditions	Modes	$\bar{ω}$
Boundary conditions	Modes	HE	HS	HR	QI
SSSS	1	17.34	18.05	17.87	19.03
	2	33.33	38.71	28.99	40.09
	3	50.41	46.29	46.52	48.16
	4	54.55	57.67	54.12	59.98
	5	62.26	76.1	57.75	76.71
CCCC	1	31.34	32.16	31.1	32.9
	2	47.67	53.49	42.38	54.89
	3	65.07	61.06	59.89	62.9
	4	68.97	72.46	67.31	74.49
	5	75.91	90.33	70.69	90.06

Table 10.

Effect of boundary conditions on $\bar{ω}$ for the circular plate with a center crack c/d = 0.5.

Boundary conditions	Modes	$\bar{ω}$
Boundary conditions	Modes	HE	HS	HR	QI
SSSS	1	6.83	6.86	6.47	7.04
	2	11.28	11.65	10.32	12.04
	3	14.95	14.43	14.94	14.95
	4	17.37	20.33	15.17	21.23
	5	20.17	22.01	18.49	22.23
CCCC	1	8.6	8.56	8.63	8.75
	2	12.42	12.93	11.74	13.25
	3	17.43	16.77	16.24	17.48
	4	19.06	22.87	17.71	23.72
	5	21.5	24.42	19.55	24.37

Buckling analysis of bio-inspired composite plates with crack

The buckling analysis of square-shaped bioinspired composite plates with a central and edge crack is performed considering a/h = 20 and E₁/E₂ = 40. The crack orientation was aligned at 0⁰ angle. Both uniaxial and biaxial loading conditions are considered for the analysis. The most critical buckling load of a structure, which is the smallest buckling load, is generally prioritized in engineering design to ensure the reliability and safety of the structure. Therefore, the current analysis evaluated only the smallest i.e. the critical buckling load $λ_{c r}$ for different cases. The evaluated $λ_{c r}$ has been normalized by using equation (39) for square-shaped plates. For normalizing the buckling load for the circular plate with crack, the length of the plate has been replaced by the radius of the circular plate.

N o r m a l i z e d c r i t i c a l b u c k l i n g l o a d, {\bar{λ}}_{c r} = λ_{c r} \frac{a^{2}}{E_{2} h^{3}}

(39)

The mode shapes corresponding to the ${\bar{λ}}_{c r}$ for the c/a, of 0.5 are presented in Figure 10. The influence of c/a on the ${\bar{λ}}_{c r}$ is presented in Tables 11, 12 and 13. The ${\bar{λ}}_{c r}$ of bio-inspired and quasi-isotropic laminates are found to decrease with the increase in c/a. This behavior is consistent for all the cases, including uniaxial and biaxial loading conditions. The decreased value of the ${\bar{λ}}_{c r}$ due to the increment of c/a is a consequence of the stiffness degradation of the structures caused by the increased amount of discrete damage in the form of crack. However, the impact of c/a on the composite laminates is not uniform across all laminate types.

Figure 10.

Mode shapes correspond to the ${\bar{λ}}_{c r}$ of square SSSS HE bioinspired composite plates, with c/a or c/d = 0.5.

Table 11.

Effect of crack length on ${\bar{λ}}_{c r}$ for square SSSS bioinspired laminate composite plates with center cracks.

Loading condition	c/a	${\bar{λ}}_{c r}$
Loading condition	c/a	HE	HR	HS	QI
Uniaxial	0.1	32.76	37.32	33.72	41.23
	0.2	32.26	36.41	33.47	40.34
	0.3	30.8	33.76	32.7	37.72
	0.4	29.1	30.81	31.73	34.76
	0.5	28.26	29.42	31.22	33.35
Biaxial	0.1	16.35	18.66	16.82	20.61
	0.2	16.07	18.17	16.66	20.14
	0.3	15.14	16.64	15.98	18.61
	0.4	13.89	14.83	14.71	16.74
	0.5	13.24	13.96	13.94	15.83

Table 12.

Effect of crack length on ${\bar{λ}}_{c r}$ for square SSSS bioinspired laminate composite plates with edge cracks.

Loading conditions	c/a	${\bar{λ}}_{c r}$
Loading conditions	c/a	HE	HR	HS	QI
Uniaxial	0.1	32.92	37.64	33.79	41.56
	0.2	32.77	37.45	33.69	41.42
	0.3	32.38	36.8	33.46	40.82
	0.4	31.23	34.72	32.81	38.82
	0.5	29.89	32.34	32.05	36.49
Biaxial	0.1	16.43	18.82	16.85	20.78
	0.2	16.34	18.72	16.78	20.7
	0.3	16.08	18.37	16.58	20.37
	0.4	15.26	17.18	15.89	19.21
	0.5	14.26	15.74	14.87	17.76

Table 13.

Effect of crack length on ${\bar{λ}}_{c r}$ for circular SSSS bioinspired laminate composite plates with a center crack.

Loading conditions	c/d	${\bar{λ}}_{c r}$
Loading conditions	c/d	HE	HS	HR	QI
Uniaxial	0.1	17.06	16.23	13.36	18.62
	0.2	16.33	15.46	12.93	17.8
	0.3	14.84	13.94	11.78	16.21
	0.4	13.14	12.45	10.36	14.5
	0.5	11.62	11.28	9.04	13.01
Biaxial	0.1	8.74	9.82	6.93	10.34
	0.2	8.24	9.33	6.59	9.85
	0.3	7.35	8.45	5.9	8.95
	0.4	6.43	7.51	5.14	7.97
	0.5	5.69	6.69	4.48	7.08

The percentage reduction in ${\bar{λ}}_{c r}$ with increasing crack length ratio (c/a or c/d = 0.1 to 0.5) is evaluated for both uniaxial and biaxial loading conditions, as shown in Figures 11 and 12, respectively. The results indicate that the reduction in ${\bar{λ}}_{c r}$ is strongly influenced by laminate configuration, plate geometry, and crack location. For circular plates with center cracks, the reduction is the highest across all configurations, reaching approximately 32% for HE, 31% for HS, about 33% for HR, and around 30% for QI under uniaxial loading, indicating a high sensitivity of circular geometries to crack-induced instability. In contrast, square plates with center cracks show comparatively lower reductions, typically in the range of 14–21% (uniaxial) and 19–25% (biaxial), with the HS configuration exhibiting the maximum reduction and HR showing relatively lower sensitivity. Square plates with edge cracks consistently exhibit the least reduction, generally limited to about 9–14% under uniaxial loading and 12–16% under biaxial loading, highlighting the less severe influence of edge cracks on global stability. Furthermore, biaxial loading results in consistently higher reductions in ${\bar{λ}}_{c r}$ compared to uniaxial loading across all configurations, due to the additional compressive stresses acting in multiple directions. Overall, these results demonstrate that circular plates with center cracks are the most vulnerable, while HR configurations tend to offer comparatively better resistance in square geometries, emphasizing the combined role of laminate architecture and geometry in governing buckling performance.

Figure 11.

Percentage drops in ${\bar{λ}}_{c r}$ for uniaxial loads with c/a or c/d changes from 0.1 to 0.5.

Figure 12.

Percentage drops in ${\bar{λ}}_{c r}$ for biaxial loads with c/a or c/d changes from 0.1 to 0.5.

The influence of different boundary conditions on cracked bio-inspired and quasi-isotropic plates is presented in Tables 14–16. Four different boundary conditions were considered, which are (i) all edges simply supported (SSSS), (ii) all edges clamped (CCCC), (ii) Simply supported and free boundary condition in alternation (SFSF), and (iii) Clamped and free boundary conditions in alternations (CFCF).

Table 14.

Effect of boundary conditions on ${\bar{λ}}_{c r}$ of square bioinspired laminate composite plates with center crack c/a = 0.5.

Loading conditions	Boundary conditions	${\bar{λ}}_{c r}$
Loading conditions	Boundary conditions	HE	HS	HR	QI
Uniaxial	SSSS	28.26	29.42	31.22	33.35
	CCCC	73.64	70.74	77.81	76.29
	SFSF	8.99	16.22	7.16	18.49
	CFCF	23.03	27.38	15.28	29.59
Biaxial	SSSS	13.24	13.96	13.94	15.83
	CCCC	26.72	31.24	21.22	33.51
	SFSF	4.47	7.91	2.16	8.03
	CFCF	12.2	20.9	7.69	22.89

Table 15.

Effect of boundary conditions on ${\bar{λ}}_{c r}$ of square bioinspired laminate composite plates with an edge crack c/a = 0.5.

Loading conditions	Boundary conditions	${\bar{λ}}_{c r}$
Loading conditions	Boundary conditions	HE	HS	HR	QI
Uniaxial	SSSS	29.89	32.34	32.05	36.49
	CCCC	78.34	77.8	80.31	84.1
	SFSF	5.92	10.71	6.05	13.82
	CFCF	14.7	22.26	12.21	29.83
Biaxial	SSSS	14.26	15.74	14.87	17.76
	CCCC	30.17	37.64	24.28	40.58
	SFSF	2.07	3.71	1.51	4.34
	CFCF	5.97	11.39	4.32	14.49

Table 16.

Effect of boundary conditions on ${\bar{λ}}_{c r}$ of circular bioinspired laminate composite plates with center crack c/d = 0.5.

Loading conditions	Boundary conditions	${\bar{λ}}_{c r}$
Loading conditions	Boundary conditions	HE	HS	HR	QI
Uniaxial	SSSS	11.62	11.28	9.04	13.01
Uniaxial	CCCC	13.95	13.12	10.87	15.31
Biaxial	SSSS	5.69	6.69	4.48	7.08
Biaxial	CCCC	6.95	8.17	5.51	8.64

From Tables 14–16, it is evident that the CCCC boundary conditions exhibit higher ${\bar{λ}}_{c r}$ compared to the SSSS, which is caused by the higher stiffness of the structures in CCCC boundary conditions as a result of the additional constraints present in the clamped boundary conditions. The SFSF boundary condition exhibits the lowest value of ${\bar{λ}}_{c r}$ in all boundary conditions for all laminate types. The less constraint boundary conditions imposed in the free edges are responsible for additional flexibility, which contributes to the low value of ${\bar{λ}}_{c r}$ . The boundary condition CFCF leads to a higher value of ${\bar{λ}}_{c r}$ compare to SFSF due to two clamped edges, however, the ${\bar{λ}}_{c r}$ in this boundary conditions are closer to the buckling load corresponding to the SSSS boundary condition in most of the cases.

Conclusions

This study investigated the influence of cracks on the free vibration and buckling behavior of bio-inspired laminated composite plates using a NURBS-based XIGA framework combined with Reddy’s higher-order shear deformation theory. The main findings are summarized as follows:

(a) The presence of cracks leads to a reduction in both natural frequencies and critical buckling loads, with the degree of reduction increasing as crack length increases.

(b) In free vibration analysis, the HS configuration exhibits the highest reduction in fundamental frequency, with the response strongly influenced by plate geometry and crack location.

(c) Among HS configurations, square plates with center cracks show greater frequency reduction compared to circular plates with center cracks and square plates with edge cracks.

(d) The HR configuration demonstrates the least reduction in natural frequency, indicating better resistance to stiffness degradation.

(e) In buckling analysis, the HR configuration shows the highest reduction in critical buckling load for circular plates with center cracks, whereas for square plates, the reduction is comparatively lower.

(f) Overall, circular plates with center cracks exhibit greater reductions in both natural frequencies and buckling loads compared to square plates, regardless of laminate configuration or loading condition.

Overall, the study emphasizes the importance of incorporating crack effects in the analysis and design of bio-inspired composite structures and demonstrates the robustness of XIGA for modeling discontinuities in complex laminates. The findings of this study highlight the potential of bio-inspired helicoidal composite plates for use in advanced engineering applications, particularly in aerospace, marine, and energy sectors, where enhanced damage tolerance, vibration control, and buckling resistance are essential.

Footnotes

ORCID iD

Sunil Kumar Singh

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Anusandhan National Research Foundation (formerly the Science and Engineering Research Board), New Delhi, under grant number SRG/2023/000818.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data Availability Statement

The data supporting the findings of this study are available from the corresponding author upon reasonable request.

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