The study of large deformations in hyperelastic beams has attracted considerable attention in recent years because of its relevance to a wide range of engineering applications, including biomechanics, soft robotics, aerospace structures, and flexible electronic devices [1–5]. Unlike classical linear elastic materials, hyperelastic materials exhibit highly nonlinear stress–strain behavior, which makes their mechanical response more complex and difficult to model accurately [6].

The dynamic response of hyperelastic materials has been investigated using both analytical and numerical approaches. Analytical solutions provide valuable insight into the fundamental behavior of these materials [7]. However, because large-deformation problems are inherently complex, numerical techniques particularly the finite element method (FEM) have become essential tools for their analysis [8,9]. Advances in computational mechanics have made it possible to simulate large-deformation responses with increasing accuracy, enabling the investigation of a broad range of loading scenarios and boundary conditions [10,11]. In addition, experimental studies have confirmed the occurrence of nonlinear wave phenomena in soft materials, further highlighting the need for robust and reliable modeling approaches [12,13].

Among various hyperelastic material models, the Gent model [14] is a widely accepted framework for capturing the finite deformation behavior of such materials, especially those with limited chain extensibility. This model effectively addresses the stiffening observed in polymeric and biological materials, rendering it a crucial tool for researchers in material science and structural mechanics [15,16]. A key challenge in modeling hyperelastic beams lies in selecting appropriate constitutive models. The Gent model, specifically, is frequently employed due to its capacity to incorporate the finite extensibility of polymer chains, thus offering a more realistic representation for biological and soft materials [15,17]. When contrasted with other hyperelastic models, such as the Mooney–Rivlin and Ogden models, the Gent model provides a more accurate approximation for materials subjected to extreme deformations [18].

Understanding the dynamic response of hyperelastic beams undergoing large deformations is essential for accurately predicting their performance in real-world applications. Classical beam theories are inadequate for capturing the nonlinear and time-dependent behavior inherent in hyperelastic structures [19,20]. Therefore, advanced formulations that incorporate finite elasticity and nonlinear kinematics are necessary [21]. Such approaches offer a more comprehensive understanding of the intricate interaction between material nonlinearity, large deformations, and dynamic effects; an understanding that is critical for designing and analyzing flexible medical devices and morphing aerospace structures [22–25].

Numerous studies have addressed the dynamic analysis of beam structures for various applications [26–29]. Destrade et al. [23] investigated the influence of the nonlinear characteristics of hyperelastic materials and showed that such behavior can introduce dispersion and attenuation effects in wave propagation. Using nonlocal continuum theory in conjunction with Euler–Bernoulli beam theory, Civalek and Demir [30] examined the buckling behavior of a cantilever carbon nanotube (CNT) beam. Hameury et al. [31] studied large-amplitude vibrations through an experimental setup developed for the active control of a sandwich beam.

Arefi et al. [32] analyzed the nonlinear vibration of functionally graded (FG) Timoshenko nanobeams, emphasizing the effects of flexoelectricity, surface energy, and residual stress. Alibakhshi et al. [33] investigated the nonlinear vibration of a microcantilever using the Euler–Bernoulli beam model. The dynamics of a multilayered microbeam with an axially functionally graded core and an intermediate elastic support were examined in [34], where the modal decomposition method was employed to handle the coupled equations of motion.

The dynamic instability of a sandwich beam based on the third-order shear deformation theory was comprehensively analyzed in [35]. Adela Mejia-Nava et al. [36] also investigated the instability of geometrically exact beams under both static and dynamic conditions. The deformation behavior of an Euler–Bernoulli FG nanobeam containing a crack was examined in [37]. In-plane buckling of small- and large-curvature FG microbeams was studied in [38] using the isogeometric collocation method. The coupled dynamic response of a double-beam system reinforced with functionally graded CNTs was investigated in [39].

Aria et al. [40] employed the finite element method to study the thermal vibration of a cracked nanobeam. Peng et al. [41] modeled the mechanical behavior of 3D-printed sandwich cellular structures. The dynamic analysis of porous composite beams with geometric imperfections was carried out in [42]. Karami et al. [43] applied nonlocal strain-gradient theory to examine the dynamic behavior of nanobeams in a thermal environment. Firouzi et al. [44] investigated nonlinear free vibrations of geometrically exact beams under different boundary conditions. Finally, Bayat et al. [45] presented a comprehensive review of engineering structures, including beams reinforced with carbon nanotubes and graphene platelet nanofillers.

A review of the existing literature shows that most studies on the dynamic response of beam structures have focused primarily on geometrically exact beam formulations. The aim of this study is to develop a comprehensive framework for analyzing the transient response of hyperelastic beams subjected to large deformations. Specifically, the objective is to model beams made of rubber-like hyperelastic materials whose strain-stiffening behavior at large stretches is accurately captured by the Gent model. Because the response is dynamic, inertia effects are explicitly included.

The principal nonlinearities considered in this work include material nonlinearity, represented through the Gent hyperelastic constitutive law; geometric nonlinearity, arising from finite strains and large rotations; and dynamic nonlinearity, which appears through inertia effects and nonlinear internal forces. By incorporating advanced numerical techniques, this research seeks to bridge the gap between theoretical modeling and practical applications.

In this study, the Gent hyperelastic model is adopted, the time integration is performed using the implicit Newmark method, and a nonlinear finite element formulation for a five-parameter beam element is developed. The results contribute to a deeper understanding of hyperelastic materials and their behavior in modern engineering applications.

The remainder of this paper is organized as follows. Section 2 presents the derivation of the kinematic quantities associated with finite beam deformations. Section 3 develops the constitutive equations for the Gent hyperelastic beam model. Section 4 is dedicated to the finite element formulation for large-deformation dynamic analysis of beams. Several illustrative examples are provided and solved in Section 5. Finally, Section 6 summarizes the key findings and conclusions of the study.

In this section, the governing kinematic relations for the finite deformation of a straight beam are derived. Two Cartesian coordinate systems {X1,X2,X3} and {x1,x2,x3} are considered: one representing the reference configuration and the other the deformed state. The reference coordinate system is initially aligned with the beam’s undeformed centroidal axis. The beam has an initial length L, while its cross-section is characterized by a width b and a thickness h.

When subjected to various loading conditions, the beam undergoes deformation into its current (deformed) configuration, as illustrated in Fig. 1. This framework forms the basis for analyzing large deformations and for establishing subsequent formulations for hyperelastic beam behavior.

The deformation of the beam is assumed to be two-dimensional and confined to the X1X3-plane. Accordingly, the position of an arbitrary point within the beam can be described as (1)X=X1r→1+X3r→3,where r→1 and r→3 denote the unit basis vectors in theX1- andX3-directions, respectively. After deformation, the position of the same point can be expressed as follows [46]:(2)x=x0(X1,t)+(X3+X32ξ(X1,t))w(X1,t).

Here, ξ denotes the stretch parameter introduced to prevent the Poisson locking phenomenon, and t represents time. In addition, w is the director vector that characterizes the rotation of the beam cross-section. A new parameter, referred to as the difference vector, is defined as (3)y={y1,y3}T=w−r→3.

Furthermore, by introducing the displacement vector (4)u={u1,u3}T=x0−X1r→1,the deformation gradient tensor can then be obtained as follows:(5)F=Grad(u)+r→1⊗Grad(X1)+(y+r→3)⊗Grad(X3+ξX32)+(X3+ξX32)Grad(y).

The linearized form of the deformation gradient tensor can be expressed as (6)F=F0+X3F1+𝒪(X32),in which F0 and F1 are defined as (7)F0=(1+u′)r→1⊗r→1+y1r→1⊗r→3+u3′r→3⊗r→1+(1+y3)r→3⊗r→3, (8)F1=y1,X1r→1⊗r→1+2ξy1r→1⊗r→3+y3,X1r→3⊗r→1+2ξ(1+y3)r→3⊗r→3.

Based on the deformation gradient tensor, the right Cauchy–Green deformation tensor is expressed as (9)C=FTF=C0+X3C1.

Here, C0 and C1 are defined as follows (10)C0=[(1+u1,X1)2+u3,X12](r→1⊗r→1)+[(1+u1,X1)y1+u3,X1(1+y3)]⋅(r→1⊗r→3+r→3⊗r→1)+[y12+(1+y3)2](r→3⊗r→3). (11)C1=[2y1,X1(1+u1,X1)+2y3,X1u3,X1](r→1⊗r→1)+[4ξy12+4ξ(1+y3)2]⋅(r→3⊗r→3)+[y1y1,X1+y3,X1(1+y3)+2ξy1(1+u1,X1)+ 2ξ(1+y3)u3,X1](r→1⊗r→3+r→3⊗r→1).

It should be noted that the Green–St. Venant strain tensor is calculated as (12)E=E0+X3E1

Here, E0 and E1 are given by:(13)E→0=12[(1+u1,X1)2+u3,X12−1]r→1⊗r→1+12[y12+(1+y3)2−1]r→3⊗r→3+12[(1+u1,X1)y1+u3,X1(1+y3)](r→1⊗r→3+r→3⊗r→1), (14)E→1=[y1,X1(1+u1,X1)+y3,X1u3,X1]r→1⊗r→1+[2ξ(y12+(1+y3)2)]r→3⊗r→3+12[y1,X1y1+y3,X1(1+y3)+ξ(1+u1,X1)+ ξ(1+y3)u3,X1](r→1⊗r→3+r→3⊗r→1).

Accordingly, the degrees of freedom for the beam element consist of the displacement parameters {u1,u3}, the components of the difference vector {y1,y3}, and the stretch parameter ξ. Therefore, five unknown variables need to be determined at each point along the beam.

In this work, the Gent strain energy density function is adopted to model the finite deformation behavior of hyperelastic beams, which is expressed as follows [13]:(15)W=−12μJmln⁡[1−(I1C−2)2]−μln⁡J+cβ−2(βln⁡J+J−β−1).

In this expression, I1C is the first invariant of the Cauchy–Green strain tensor. The quantity J denotes the Jacobian of the deformation gradient tensor, that is, J=det(F). In addition, μ, c and β are material parameters, where Jm represents the chain-extensibility parameter. It is also noted that as Jm→∞, the Gent model reduces to the classical neo-Hookean hyperelastic formulation.

Based on the strain energy density function, the corresponding stress measures can be derived. In large-deformation analysis, multiple stress definitions may be used depending on the formulation and requirements of the problem. In this study, the 2nd Piola–Kirchhoff stress tensor is employed, as it is energetically conjugate to the Green–Lagrange strain tensor within the Lagrangian framework. The 2nd Piola–Kirchhoff stress tensor is obtained as follows:(16)S=2W,C=−μJmI1C−Jm−2I−μC−1+cβ−1(1−2J−β)C−1,S=S0+X3S1

In beam structural analysis, it is customary to express quantities in a linearized form, where one component represents the in-plane deformation and the other corresponds to the curvature effects, as introduced in Eq. (1). To obtain the linear representation of the stress, several preliminary calculations are required [46]:(17)C−1=C0∗+X3C1∗,withC0∗=C0−1,C1∗=−C0−1C1C0−1, (18)J=J0+X3J1,withJ0=det(F0),J1=det(F1), (19)I1C=I10+X3I11,withI10=det(C0),I11=det(C1).

The linearized form of the stress is obtained as follows:(20)S0=−μJmI10−Jm−2I−μC0∗+cβ−1(1−2exp⁡(−βln⁡J0))C0∗, (21)S1=μJmI10−Jm−2I−μC1∗+cβ−1(1−2exp⁡(−βln⁡J0))C1∗+2cJ0−1J1exp⁡(−βln⁡J0)C0∗.

Moreover, the elasticity tensor is obtained by differentiating the second Piola–Kirchhoff stress tensor with respect to the right Cauchy–Green tensor, leading to the following relation:(22)C=2μJm(I1−Jm−2)2I⊗I−2cJ−βC−1⊗C−1+[μ−cβ−1(1−2J−β)](C−1⊗¯C−1+C−1⊗_C−1).

Here, ⊗¯ and ⊗_ are the non-standard tensor product. Considering Q and H as two second-order tensors, the non-standard tensor products are defined as:(23)[(Q)⊗¯(H)]ijkl=QikHjl,[(Q)⊗_(H)]ijkl=QilHjk.

Representation of the fourth-order elasticity tensor in linearized form yields the following relations:(24)C0=2μJm(I10−Jm−2)2I⊗I+2cexp⁡(−βln⁡J0)C0∗⊗C0∗+[μ−cβ−1(1−2exp⁡(−βln⁡J0))](C0∗⊗¯C0∗+C0∗⊗_C0∗), (25)C1=−4μJmI11(I10−Jm−2)3I⊗I+2cexp⁡(−βln⁡J0)(C0∗⊗C1∗+C1∗⊗C0∗)−2cβJ0−1J1exp⁡(−βln⁡J0)C0∗⊗C0∗+2[μ−cβ−1(1−2exp⁡(−βln⁡J0))]⋅(C0∗⊗¯C1∗+C1∗⊗¯C0∗+C0∗⊗_C1∗+C1∗⊗_C0∗)−2cJ0−1J1exp⁡(−βln⁡J0)(C0∗⊗¯C0∗+C0∗⊗_C0∗).

In this section, a nonlinear finite element formulation for the large-amplitude dynamic response of hyperelastic beams is developed. For this purpose, Hamilton’s principle is expressed as follows [9]:(26)∫t1t2δT−δWint+δWext=0.

In this expression, δT, δWint, and δWext denote the virtual kinetic energy, the virtual internal energy, and the virtual work of external forces, respectively. Within each beam element, the generalized displacement field is approximated using the following discretization:(27)U={u1,u3,y1,y3,ξ}=∑I=1n~NI(η){u1I,u3I,y1I,y3I,ξI}.

Here, NI are the interpolation functions and n~ is the number of nodes per element. The virtual kinetic energy is derived as follows:(28)δT=∫VρU,t⋅δU,t,where ρ is the density. By taking integral by part, it is re-written as:(29)δT=−∫VρU,tt⋅δU.

Using Eq. (27), the virtual kinetic energy can be evaluated as follows:(30)δT=−∫LδUTmU¨dX1,where m is the element mass matrix, whose components are obtained from the following relation:(31)mIJ=∫AρNITNJdA,in which A represents the initial cross-sectional area of the beam. By assembling the element mass matrices over all elements, the final expression for the virtual kinetic energy is obtained (32)δT=−Ae=1NEδUTMU¨,M=∫LmdX1.

Here, Ae=1NE denotes the assembly operator. Next, the virtual work of the external loads can be written as follows:(33)δWext=∫Vf⋅δUdV+∫ΩT⋅δUdΩ,where f and T denote the body force per unit volume V and the external traction vector acting on a unit area, respectively. By neglecting the body force and assembling the external forces over all elements, the global expression for the external virtual work can be written as:(34)δWext=Ae=1NEδUTf ext=Ae=1NE∑I=1n~δUITfI ext,in which fI ext denotes the external force acting at the I’th node, given by:(35)fI ext=∫LNIT(X){T1,T2,T3,T4,T5}TdX1.

The virtual internal energy is expressed as:(36)δWint=∫VδW(C)dV=∫L∫AW,C:δCdAdX1=∫L∫AS:δEdAdX1.

It should be noted that the force and moment resultants are obtained as:(37)F→=∫AS→dA=∫A(S→0+X3S→1)dA=AS→0,P→=∫AS→X3dA=∫A(S→0+X3S→1)X3dA=I~S→1.

Here, I~ denotes the second moment of area (area moment of inertia). Substituting Eqs. (37) into (36) yields the following relation:(38)δWint=∫L(δE→0TF→+δE→1TP→)dX.

The Green-St Venant strain tensor can be approximated using the strain-displacement matrices as follows:(39){δEγ,ΔEγ}=∑I=1n~BγI{δU,ΔU},γ=0,1,where B0I and B1I denote the linear and nonlinear strain–displacement matrices, respectively. Consequently, the discretized form of the virtual internal energy is expressed as follows:(40)δWint=Ae=1NEδUTf int=Ae=1NE∑I=1n~δUITfI int,in which fI int is the internal force at I’th node and given by (41)fI int=b∫L(B0ITF→+B1ITP→)dX.

Finally, by substituting Eqs. (32), (34), and (40) into the governing formulation, the following expression is obtained:(42)Ae=1NEδUT(−MU¨+f ext−f int)=0.

As δU is not zero, it is concluded that:(43)MU¨+f int(U)=f ext.

Several approaches have been proposed in the literature for solving the nonlinear system of Eq. (43). In the present study, the Newmark method is adopted for time discretization. Assuming that the solution is known at time step tn, the objective is to determine the corresponding quantities at time step tn+1, where Δt=tn+1−tndenotes the time increment. Accordingly, the updated expressions for displacement and velocity can be written as follows [47]:(44)Un+1=Un+ΔtU˙n+Δt2(12−B )+B Δt2U¨n+1, (45)U˙n+1=U˙n+Δt(1−Λ)U¨n+ΛΔt2U¨n+1.

To ensure unconditional stability of the time integration scheme, the following parameter values are adopted:(46)B=0.25,Λ=0.5.

Eq. (43) can therefore be rewritten as follows:(47)MU¨n+1+fn+1 int=fn+1 ext.

Eq. (47) is linearized using the Newton–Raphson method. By substituting Eqs. (34) and (40), the residual vector can be obtained as follows:(48)R=f int−f ext.

By applying the Newton–Raphson method, the corresponding linearized form can be expressed as follows:(49)δWint=Ae=1NEδWeint=Ae=1NE∑I=1n~∑J=1n~δUITKIJΔUJ=δUTKΔUin which KIJ denotes the element stiffness matrix, defined as:(50)KIJ=KIJmat+KIJgeo.

This section is devoted to evaluating the performance and accuracy of the formulations developed in the previous sections. To this end, three representative examples involving complex loading and boundary conditions are investigated. In all examples, three-node quadratic beam elements with reduced integration are employed in order to prevent shear locking.

Although the Gent hyperelastic model is well established for describing the strain-stiffening response of soft materials, its use in beam-type structural members subjected to transient dynamic loading has been only marginally explored in the existing literature. Consequently, extending this model to finite transient deformations of beams represents a meaningful and original contribution.

In this study, the Gent hyperelastic model was employed to investigate the transient response of beams under various loading scenarios and boundary conditions. The main findings can be summarized as follows:

A formulation for the transient deformation of beam structures based on the Gent model was successfully developed.

Overall, the presented formulation provides a comprehensive basis for analyzing the dynamic behavior of hyperelastic beams undergoing large deformations.

Limitations and Future Developments

The present study extends the formulation of finite dynamic deformations of geometrically exact beams to the framework of the Gent hyperelastic model. Although the proposed model produces promising theoretical results, it currently remains primarily within a theoretical framework, as experimental investigations of this formulation for beam-type structures have not yet been conducted.

The Gent hyperelastic model has demonstrated strong capability in predicting the large-deformation behavior of rubber-like materials [14]. However, experimental studies focusing specifically on the dynamic response of rubber-like beam structures remain limited.

Future work may therefore focus on designing an experimental setup to investigate the dynamic behavior of rubbery beam-like structures. Such experiments would allow direct comparison between theoretical predictions and measured responses, thereby providing further validation and practical significance for the proposed model.