1 Introduction

Hyperelastic structures often undergo large strains when subjected to external forces. The stress–strain relation in such structures is highly complicated, making the linear stress–strain relationship and linear elastic models invalid for simulating their mechanical behaviour. The hyperelastic behaviour can be seen in different soft structures such as rubbers, foams and human body organs. Along with understanding the characteristics of such structures, having accurate modelling of hyperelastic structures could also provide us with further potential applications in different fields.

1.1 Necessity for this review

By analysing the available database in Scopus on hyperelasticity, the significance of the dialogue between scientists and researchers on this topic was obtained. Figure 1a demonstrates the number of published works on hyperelasticity from 1990 to 2020. It can be seen that, during this period, the number of published papers on this subject has increased noticeably, reaching more than 1100 research studies published in 2020 alone.

Fig. 1
figure 1

Tables for the number of documents from 1990 to 2020 on (a) hyperelasticity and (b) hyperelastic beams, plates and shells

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Moreover, analysing the mechanical behaviours (bending, buckling and vibration) of hyperelastic structures (e.g. beams, plates, shells and membranes) shows the same incremental trend indicated in Fig. 1b; from which it can be seen that many studies on hyperelasticity are focused on the mechanics of such structures. The phenomenal growth of studies on this subject clarifies the importance of having a systematic literature review to summarise the achievements to date on this topic.

1.2 Applications of hyperelastic structures

In general, soft structures present hyperelastic behaviours while confronting different conditions. One of the main applications of hyperelastic structures is soft robotics [1,2,3]; since soft structures can provide higher-order degrees of freedom, movement in robotic parts could potentially become more smooth than when using rigid and/or firm structures.

Robotic rehabilitation systems for stroke patients can be significantly improved by using soft robotic as they can provide a smooth motion with a safer operation [4]. For instance, soft robotic gloves are capable of helping patients with muscular dystrophy, amyotrophic lateral sclerosis or post-stroke hand function assistance [5,6,7,8]. In the work presented by Polygerinos et al. [4], a soft robotic glove is presented to study the hand and fingers’ joint motion. In their model, the different mechanical behaviours of bending, twisting and extensions of soft beam-shaped structures were obtained.

Developing soft structures to explore unknown environments is another important application of hyperelastic structures as it can tolerate different types of loadings and impacts. In a study done by Antol et al. [9], it is shown that expensive wheel rovers can be replaced with tumbleweed rovers for Mars exploration. In another study, Trivedi et al. [10] used hyperelastic tubes for soft robotic modelling of Oct-Arm.

Besides, soft robots made of hyperelastic materials have been used for sensing and monitoring environments. For example, a dragonfly-inspired soft robot (DraBot) has been fabricated for measuring the contaminants (such as the presence of oil), pH, and temperature of water surfaces [11,12,13].

The application of hyperelastic structures in soft robotics has reached a turning point with the capability of 3D printing and the utilisation of soft actuators [14,15,16]. Hyperelastic structures also have other types of applications, of which some of the main ones are wearable devices [17], stretchable electronics [17], biomedical engineering [17], and energy harvesters [18, 19].

Figure 2 presents a simple robot that is capable of crawling using twisted and coiled actuators [20]. This simple model has been fabricated using hyperelastic beams, and the smooth motion of the robot was obtained by bending. Figure 3 also presents some useful examples of soft structures as actuators in soft robotics, which can be used for grabbing, twisting, motion, lifting and other purposes [15].

Fig. 2
figure 2

Schematic view of a crawling robot using soft structures [20]. (Permission obtained from Elsevier)

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Fig. 3
figure 3

Actuators made of soft structures with different purposes: a contractor; b bender; c grabber; d twister [15]. (This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).)

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Another application of soft structures can be found in belt operating systems. Belt conveyor systems are mainly used for power transmission from the driving pulley to the driver one in different engineering fields [21,22,23].

Layered hyperelastic structures have been used for packaging, especially food industry, as a soft safe layer is required for inside and a stiffer layer for outside. The proper design and material usage are of high importance as the packaging is around 15% of the total variable costs [24, 25]. Waste management and environmentally friendly (biodegradable) packaging [26,27,28] are also important topics making the discussion of using proper hyperelastic materials for packaging an ongoing novel research topic.

Since human body organs show nonlinear elastic behaviour, researchers have worked on fabricating prosthetics with similar hyperelastic behaviour. Using hyperelastic structures for firstly modelling the human body organs and secondly accurately designing prosthetics have been a novel topic for researchers to invest [29,30,31,32].

1.3 Contribution of this paper to the field

The importance of modelling hyperelastic structures accurately has been discussed in the previous subsections. It is shown that there has been a considerable number of researches on hyperelasticity with promising, growing trends over the past few years. The demonstration of the high application potential of these structures in the early years of their development and their future in engineering design [14, 15, 20, 33, 34] emphasises the necessity of having a systematic literature review through the need for categorisation, discussion and explanation of the achievements to date. Accordingly, this review intends to clarify the achievements and goals of this research field by analysing diverse critical hyperelastic studies in the framework of nonlinear dynamics.

1.4 Structure of this review paper

To present a comprehensive investigation on hyperelastic structures, this review is structured in the following order: as shown in the flowchart of Fig. 4, in Sect. 1, a brief introduction to hyperelastic structures is given, indicating the importance of understanding the hyperelastic mechanical behaviour, emphasizing the application’s potential and the future of hyperelastic structures, and lastly, demonstrating the contribution of this review to this field. In Sect. 2, some well-known constitutive hyperelastic models for isotropic soft materials are discussed in detail by presenting the fundamental continuum mechanics formulation and definitions related to hyperelastic behaviour. Some of the well-known techniques and models in continuum mechanics are then provided, followed by the sensitivity of the model in tracking hyperelastic behaviour. In Sect. 3, the application of the given and other hyperelastic continuum models on obtaining the nonlinear dynamics of hyperelastic beam structures is discussed. Section 4 concentrates on analysing hyperelastic plate and shell structures in the framework of nonlinear dynamics. Different plate and shell theories together with hyperelastic constitutive models are discussed for accurately modelling the nonlinear dynamics of soft plate and shell structures. Section 5 presents a detailed explanation on hyperelastic models and nonlinear dynamics of soft membranes and balloons using different continuum mechanics models. Lastly, in Sect. 6, a comprehensive summary of the analysis performed through this paper is provided, and the achievements and possible potential for improving the modelling of such structures are presented.

Fig. 4
figure 4

Flowchart of the structure of this review

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2 Some constitutive hyperelastic models for isotropic materials

2.1 Fundamental continuum definitions

In order to define the mechanical characteristics of hyperelastic structures, there are key continuum mechanics definitions that must be presented. In general, the deformation gradient (F) is defined as [35]:

$$ \left\{ \begin{array}{l} F_{ij} = \delta_{ij} + \frac{{\partial u_{i} }}{{\partial x_{j} }}\,, \hfill \\ J = \det \left( F \right), \hfill \\ \end{array} \right. $$
(1)

with J is the determinant of the deformation gradient, δ is the Kronecker delta and ui is the displacement field, which could be rewritten in the principal directions of the structures as [36]:

$$ F = \left[ {\begin{array}{*{20}c} {\lambda_{1} } & 0 & 0 \\ 0 & {\lambda_{2} } & 0 \\ 0 & 0 & {\lambda_{3} } \\ \end{array} } \right]\,\,\,\,\,\, \to \,\,\,\,F_{ij} = \delta_{ij} \lambda_{i} , $$
(2)

where \(\lambda_{i}\) indicates the principal stretch through the principal direction i, defined as

$$ \lambda_{i} = \frac{{L_{i}^{\,} }}{{L_{i}^{0} }}, $$
(3)

with Li and Li.0 are the deformed and undeformed lengths of the structure through the i direction, respectively. Another important definition in describing hyperelastic structures is the left Cauchy–Green strain tensor (B) which is [36]

$$ B = F \cdot F^{T} \,\,\,\, \to \,\,\,\,\,\,B_{ij} = F_{ik} F_{jk} $$
(4)

For conventional definitions, the left Cauchy–Green strain tensor’s invariants are defined as [37]:

$$ \begin{aligned} I_{1} &= \frac{Tr\left( B \right)}{{J^{\frac{2}{3}} }} = \frac{{B_{ii} }}{{J^{\frac{2}{3}} }}, \hfill \\ I_{2} &= \frac{1}{2}\left( {I_{1}^{2} - \frac{{B_{ij} B_{ji} }}{{J^{\frac{4}{3}} }}} \right), \hfill \\ I_{3} = J^{2} , \hfill \\ \end{aligned} $$
(5)

where I1, I2 and I3 are the first, second and third strain invariants for compressible structures, respectively; for incompressible analysis, the first and second invariants will be simplified by having J = 1 and the third invariant will be equal to 1 (interested readers are referred to Refs. [38,39,40] for more information regarding compressible and incompressible materials). The Green–Lagrange strain–displacement can be written regarding the deformation gradient as:

$$ E = \frac{1}{2}\left( {F^{T}F - I} \right) \to \,\,E_{ij} = \frac{1}{2}\left( {F_{pi} F_{pj} - \delta_{ij} } \right). $$
(6)

In the case of having principal stretches, invariants of the left Cauchy–Green strain tensor are:

$$ \begin{aligned} I_{1} &= \frac{{\lambda_{1}^{2} + \lambda_{2}^{2} + \lambda_{3}^{2} }}{{\left( {\lambda_{1}^{\,} \lambda_{2}^{\,} \lambda_{3}^{\,} } \right)^{\frac{2}{3}} }}, \hfill \\ I_{2} &= \frac{{\lambda_{1}^{2} \lambda_{2}^{2} + \lambda_{1}^{2} \lambda_{3}^{2} + \lambda_{2}^{2} \lambda_{3}^{2} }}{{\left( {\lambda_{1}^{\,} \lambda_{2}^{\,} \lambda_{3}^{\,} } \right)^{\frac{4}{3}} }}, \hfill \\ I_{3} &= \left( {\lambda_{1}^{\,} \lambda_{2}^{\,} \lambda_{3}^{\,} } \right)^{2} . \hfill \\ \end{aligned} $$
(7)

As for the hyperelasticity definition, a structure is hyperelastic if specific strain energy exists which is differentiable from the deformation gradient. In other words, to have a fully elastic behaviour, it is assumed that the strain energy density is directly dependent on the deformation gradient tensor. In another definition, it has been shown that a hyperelastic structure is isotropic if and only if the strain energy term can be rewritten via the three invariants of the left Cauchy–Green strain tensor [41].

According to Richter theorem [41, 42], the constitutive equation of an isotropic solid hyperelastic can be written as:

$$ T = \beta_{0} I + \beta_{1} B + \beta_{2} B^{2} , $$
(8)

if and only if the coefficients are defined following a specific relationship defined in the literature. It has been shown that [42], by having an isothermic process, changes in –T0dS (in which T0 is the absolute temperature and S is the entropy) to become equal to the variation of the Helmholtz free energy. In case of isothermic process, coefficients in Eq. (8) are expressed by Eq. (9) as

$$ \begin{aligned} \beta_{0} &= \frac{\partial W}{{\partial I_{1} }} + I_{1} \frac{\partial W}{{\partial I_{2} }} + I_{2} \frac{\partial W}{{\partial I_{3} }}, \hfill \\ \beta_{1} &= - \frac{\partial W}{{\partial I_{2} }} - I_{1} \frac{\partial W}{{\partial I_{3} }}, \hfill \\ \beta_{2} &= \frac{\partial W}{{\partial I_{3} }}. \hfill \\ \end{aligned} $$
(9)

For a simple definition, by having the principal direction stretches, the normal stresses (σi) can be defined as [43]:

$$ \sigma_{i} = \frac{\partial W}{{\partial \lambda_{i} }},\,\,\,\,\,\,\,i = 1,2,3 $$
(10)

after which, due to the definition of hyperelastic strain energy (W) being a function of its invariants, Eq. (10) can be rewritten as:

$$ \frac{\partial W}{{\partial \lambda_{i} }} = \frac{\partial W}{{\partial I_{3} }}\frac{{\partial I_{3} }}{{\partial \lambda_{i} }} + \frac{\partial W}{{\partial I_{2} }}\frac{{\partial I_{2} }}{{\partial \lambda_{i} }} + \frac{\partial W}{{\partial I_{1} }}\frac{{\partial I_{1} }}{{\partial \lambda_{i} }},\,\,\,\,\,\,\,i = 1,2,3 $$
(11)

which for incompressible structures, the third term (derivation with respect to I3) will be neglected. Different types of formulation and modelling for hyperelastic strain energy density have been presented in order to predict the nonlinear behaviour accurately. In further subsections, these isotropic models are presented and the formulation procedure for reaching the stress–strain equation is given. These models are used by many researchers to study the nonlinear dynamics of hyperelastic structures which is discussed in further sections.

2.2 The neo-Hookean model

This strain energy density expression is one of the straightforward models of a hyperelastic material in which we only consider the first and third invariant terms as

$$ W_{NH} = \sum\limits_{i = 0}^{n} {C_{i} \left( {\overline{{I_{1} }} - 3} \right)^{i} } + D_{1} \left( {J - 1} \right)^{2} , $$
(12)

where WNH is the strain energy density of this model, Ci are the coefficients of the first invariant parameter and D1 is the compressibility factor, both of which must be obtained experimentally. For n = 1 (one-term neo-Hookean model), the axial stress (σuni) is written as [43]:

$$ \begin{aligned} \sigma_{uni} &= C_{1} \frac{{4\left( {1 + \nu } \right)}}{3}\lambda_{1}^{{ - \left( {5 + 2\nu } \right)/3}} \left( {\lambda_{1}^{2 + 2\nu } - 1} \right) \hfill \\ & \,\,\,\, + 2D_{1} \left( {1 - 2\nu } \right)\lambda_{1}^{ - 2\nu } \left( {J - 1} \right), \hfill \\ \end{aligned} $$
(13)

where v is the Poisson’s ratio. By assuming an incompressible structure, Eq. (13) is simplified as:

$$ \sigma_{uni} = 2C_{1} \left( {\lambda - \lambda_{\,}^{ - 2} } \right)\,,\, $$
(14)

and for equibiaxial (σbi) stress and pure shear (σs) stress, by using the same definition given in Eq. (12), the stress resultants become

$$ \sigma_{bi} = 2C_{1} \left( {\lambda - \lambda_{\,}^{ - 5} } \right)\,,\,\, $$
(15)
$$ \sigma_{s} = 2C_{1} \left( {\lambda - \lambda_{\,}^{ - 3} } \right)\,, $$
(16)

which coincides with those used in ref [44].

2.3 The Mooney–Rivlin model

One of the popular formulations and models used for predicting the hyperelastic behaviour of structures is the Mooney–Rivlin model [45], which is an extended form of the neo-Hookean model, considering the second invariant term. In the basic form, Mooney defined the strain energy density as a two-parameter model defined as:

$$ W_{M} = C_{1} \left( {\overline{{I_{1} }} - 3} \right) + C_{2} \left( {\overline{{I_{2} }} - 3} \right) + D_{1} \left( {J - 1} \right)^{2} , $$
(17)

where WM is the strain energy density of the two-parameter Mooney model and Ci and D1 are the coefficients that must be found via the experimental observations (such as the work done by Falope et al. [46] where the coefficients were calibrated using genetic algorithm), which can vary from one soft structure to another. Rivlin [47, 48] extended this equation by writing it in a general form as a polynomial series of the first and second invariant terms:

$$ W_{MR} = \sum\limits_{i = 0}^{n} {\sum\limits_{j = 0}^{m} {C_{ij} \left( {\overline{{I_{1} }} - 3} \right)^{i} \left( {\overline{{I_{2} }} - 3} \right)}^{j} } + D_{1} \left( {J - 1} \right)^{2} , $$
(18)

where WMR is the strain energy density of the Mooney–Rivlin model, of which some of the well-known models and special cases of this polynomial series are the Biderman model [49], Klosner model [50] and Haines–Wilson model [51]. It has been mentioned already that this model has been widely used for analysing rubbers with less than 200% deformation [52].

For the two-parameter Mooney–Rivlin model under axial load, by having \(\lambda_{2}\) = \(\lambda_{3}\) = \(\lambda_{1}^{ - \nu }\), Brown et al. [43] obtained the axial stress as:

$$ \begin{aligned} \sigma_{uni} &= \frac{{4\left( {1 + \nu } \right)}}{3}\lambda_{1}^{{ - \left( {5 + 2\nu } \right)/3}} \left( {\lambda_{1}^{2 + 2\nu } - 1} \right)\left( {C_{1} + C_{2} \lambda_{1}^{{ - 2\left( {1 + \nu } \right)/3}} } \right) \hfill \\ & \,\,\,\, + 2D_{1} \left( {1 - 2\nu } \right)\lambda_{1}^{ - 2\nu } \left( {J - 1} \right), \hfill \\ \end{aligned} $$
(19)

from which, by assuming an incompressible structure, Eq. (19) becomes

$$ \sigma_{uni} = 2C_{1} \left( {\lambda - \lambda_{\,}^{ - 2} } \right) + 2C_{2} \left( {1 - \lambda_{\,}^{ - 3} } \right),\, $$
(20)

and for equibiaxial and pure shear stresses, by using the same definition given in Eq. (18), the stress resultants are:

$$ \sigma_{bi} = 2C_{1} \left( {\lambda - \lambda_{\,}^{ - 5} } \right) + 2C_{2} \left( {\lambda_{\,}^{3} - \lambda_{\,}^{ - 3} } \right)\,,\,\, $$
(21)
$$ \sigma_{s} = \left( {2C_{1} + 2C_{2} } \right)\left( {\lambda - \lambda_{\,}^{ - 3} } \right), $$
(22)

which coincides with those used in ref [44]. To have a better understanding of the Mooney–Rivlin model, the effect of the coefficients C1 and C2 on the uniaxial stress is presented in Fig. 5 for axial strain up to 100%. It can be seen that the stress–strain behaviour is completely nonlinear and the curve model is highly sensitive to the two-parameter Mooney–Rivlin coefficients. By having [37] C1 = 0.39 MPa and C2 = 0.015 MPa, in Fig. 5a, the first coefficient term is varied as [0.5C1–1.5C1], while in Fig. 5b the second coefficient is varied as [0.5C2–5.5C2]. Since the formulation of a neo-Hookean model is somewhat similar to the Mooney–Rivlin hyperelastic model (by neglecting the second invariant term), the influence of varying the hyperelastic coefficient C1 in one-term neo-Hookean models will be very similar to the one presented in Fig. 5a.

Fig. 5
figure 5

Uniaxial stress sensitivity to the two-parameter Mooney–Rivlin coefficients: a C1; b C2

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2.4 The Ogden model

Ogden [53, 54] proposed a series of models of strain energy density as a direct function of principal stretches

$$ W_{Og} = \sum\limits_{i = 1}^{n} {\frac{{2\mu_{i} }}{{\alpha_{i}^{2} }}\left( {\lambda_{1}^{{\alpha_{i} }} + \lambda_{2}^{{\alpha_{i} }} + \lambda_{3}^{{\alpha_{i} }} - 3} \right)} + \sum\limits_{i = 1}^{n} {D_{i} \left( {J - 1} \right)^{2i} } $$
(23)

where WOg is the strain energy density of the Ogden model and μi, Di and αi are the constant properties that must be found using experimental testing. This model is well capable of simulating the typical hardening of rubber materials, which is not included in both neo-Hookean and Mooney–Rivlin models. Brown et al. [43] have shown the uniaxial stress for this model to be:

$$ \sigma_{uni} = \sum\limits_{i = 1}^{n} {\frac{{2\mu_{i} }}{{\alpha_{i}^{\,} }}\left( {\lambda_{\,}^{{\alpha_{i} - 1}} - 2\nu \lambda_{\,}^{{ - \alpha_{i} \nu - 1}} } \right)} + 2\left( {1 - 2\nu \lambda_{\,}^{ - 2\nu } } \right)\sum\limits_{i = 1}^{n} {iD_{i} \left( {J - 1} \right)^{2i - 1} } $$
(24)

which can be rewritten for incompressible structures as

$$ \sigma_{uni} = \sum\limits_{i = 1}^{n} {\frac{{2\mu_{i} }}{{\alpha_{i}^{2} }}\left( {\lambda_{\,}^{{\alpha_{i} - 1}} - \lambda_{\,}^{{ - \frac{{\alpha_{i} }}{2} - 1}} } \right)} \,. $$
(25)

Similarly, the equibiaxial and pure shear stresses are:

$$ \sigma_{bi} = \sum\limits_{i = 1}^{n} {\frac{{2\mu_{i} }}{{\alpha_{i}^{2} }}\left( {\lambda_{\,}^{{\alpha_{i} - 1}} - \lambda_{\,}^{{ - 2\alpha_{i} - 1}} } \right)} ,\,\, $$
(26)
$$ \sigma_{s} = \sum\limits_{i = 1}^{n} {\frac{{2\mu_{i} }}{{\alpha_{i}^{2} }}\left( {\lambda_{\,}^{{\alpha_{i} - 1}} - \lambda_{\,}^{{ - \alpha_{i} - 1}} } \right)} \,. $$
(27)

In order to elaborate the impact of each coefficient term on the stress–strain behaviour of Ogden hyperelastic models, uniaxial loading of a three-parameter Ogden model is considered with coefficients and power terms as [37], where μ1 = 0.62 MPa, μ2 = 0.00118 MPa, μ3 = 0.00981 MPa, α1 = 1.3, α2 = 5 and α3 =  − 2. Figures 6a–f indicate the strong effect of varying three-parameter Ogden model coefficients and power terms by [0.5 μ1–1.5 μ1], [0.5α1–1.5α1], [0.5μ2–50.5μ2], [0.5 α2–15.5α2], [0.5μ3–5.5μ3] and [0.5α3–15.5α3], respectively. It can be seen that each parameter has its own effect on the axial stress magnitude through which, by properly accounting for these terms, the hyperelastic behaviour of rubbery structures can be obtained.

Fig. 6
figure 6

Uniaxial stress sensitivity to the three-parameter Ogden coefficients and power terms: a μ1; b α1; c μ2; d α2; e μ3; f α3

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2.5 The eight-chain (Arruda–Boyce) model

Arruda and Boyce [55] proposed the eight-chain model in which the strain energy is described as a function of a polynomial series of the first invariant as

$$ W_{AB} = \sum\limits_{i = 1}^{n} {C_{i} \left( {\overline{{I_{1}^{i} }} - 3^{i} } \right)} + \sum\limits_{i = 1}^{n} {D_{i} \left( {J - 1} \right)^{2i} } , $$
(28)

where WAB is the strain energy density of the Arruda–Boyce model, for which, under uniaxial loading, the stress-stretch equation will become [43]

$$ \begin{aligned} \sigma_{uni} &= \frac{{4\left( {1 + \nu } \right)}}{3}\sum\limits_{i = 1}^{n} {iC_{i} \left( {\overline{{I_{1}^{i - 1} }} } \right)} \lambda_{\,}^{{ - \left( {5 + 2\nu } \right)/3}} \left( {\lambda_{\,}^{2 + 2\nu } - 1} \right) \hfill \\ &\,\,\,\, + 2\left( {1 - 2\nu } \right)\sum\limits_{i = 1}^{n} {iD_{i} \left( {J - 1} \right)^{2i - 1} } \lambda_{\,}^{ - 2\nu } , \hfill \\ \end{aligned} $$
(29)

which for incompressible structures it is simplified as

$$ \sigma_{uni} = 2\sum\limits_{i = 1}^{n} {iC_{i} \left( {\overline{{I_{1}^{i - 1} }} } \right)} \left( {\lambda - \lambda_{\,}^{ - 2} } \right), $$
(30)

with equibiaxial and pure shear stresses as

$$ \sigma_{bi} = 2\sum\limits_{i = 1}^{n} {iC_{i} \left( {\overline{{I_{1}^{i - 1} }} } \right)} \left( {\lambda_{\,}^{\,} - \lambda_{\,}^{ - 5} } \right), $$
(31)
$$ \sigma_{s} = 2\sum\limits_{i = 1}^{n} {iC_{i} \left( {\overline{{I_{1}^{i - 1} }} } \right)} \left( {\lambda_{\,}^{\,} - \lambda_{\,}^{ - 3} } \right). $$
(32)

For analysing the coefficient sensitivity of this model, a five-parameter Arruda–Boyce model is considered by having [56] C1 = μ/2, C2 = μ/20, C3 = 11μ/1050, C4 = 19μ/7000, C5 = 519μ/673750 where μ is the rubbery shear modulus assumed to be 0.4 MPa. The influence of each coefficient on the uniaxial stress variation can be seen in Fig. 7a–f by varying [0.5μ–1.5μ], [0.5C1–1.5C1], [0.5C2–2C2], [0.5C3–2.5C3], [0.5C4–5C4], and [0.5C5–5C5], respectively. Here, all the coefficients have a considerable effect on the uniaxial stress term.

Fig. 7
figure 7

Uniaxial stress sensitivity to the five-parameter Arruda–Boyce (eight-chain) model coefficients: a μ; b C1; c C2; d C3; e C4; f C5

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2.6 The polynomial model

The strain energy in the polynomial rubber model is defined as [57]

$$ W_{Po} = \sum\limits_{i + j = 1}^{n} {C_{ij} \left( {\overline{{I_{1}^{\,} }} - 3} \right)^{i} \left( {\overline{{I_{2}^{\,} }} - 3} \right)^{j} } + \sum\limits_{i = 1}^{n} {D_{i} \left( {J - 1} \right)^{2i} } , $$
(33)

where WPo is the strain energy density of polynomial model, which can also be written as [43]

$$ W_{Po} = \sum\limits_{i = 0}^{n} {\sum\limits_{j = 1 - i}^{n - i} {C_{ij} \left( {\overline{{I_{1}^{\,} }} - 3} \right)^{i} \left( {\overline{{I_{2}^{\,} }} - 3} \right)^{j} } } + \sum\limits_{i = 1}^{n} {D_{i} \left( {J - 1} \right)^{2i} } , $$
(34)

leading to an axial stress–stretch relationship as [43]

$$ \begin{aligned} \sigma_{uni} &= \frac{{4\left( {1 + \nu } \right)}}{3}\lambda_{\,}^{{ - \left( {5 + 2\nu } \right)/3}} \left( {\lambda_{\,}^{2 + 2\nu } - 1} \right)\left[ \begin{array}{l} \sum\limits_{i = 0}^{n} {\sum\limits_{j = 1 - i}^{n - i} {iC_{ij} \left( {\overline{{I_{1}^{\,} }} - 3} \right)^{i - 1} \left( {\overline{{I_{2}^{\,} }} - 3} \right)^{j} } } \hfill \\ + \lambda_{\,}^{{ - \left( {2 + 2\nu } \right)/3}} \sum\limits_{i = 0}^{n} {\sum\limits_{j = 1 - i}^{n - i} {jC_{ij} \left( {\overline{{I_{1}^{\,} }} - 3} \right)^{i} \left( {\overline{{I_{2}^{\,} }} - 3} \right)^{j - 1} } } \hfill \\ \end{array} \right] \hfill \\ &\,\,\,\, + 2\left( {1 - 2\nu } \right)\sum\limits_{i = 1}^{n} {iD_{i} \left( {J - 1} \right)^{2i - 1} } \lambda_{\,}^{ - 2\nu } , \hfill \\ \end{aligned} $$
(35)

which, for incompressible structures using Eq. (33), is rewritten as

$$ \begin{gathered} \sigma_{uni} = 2\sum\limits_{i = 0}^{n} {\sum\limits_{j = 1 - i}^{n - i} {iC_{ij} \left( {\overline{{I_{1}^{\,} }} - 3} \right)^{i - 1} \left( {\overline{{I_{2}^{\,} }} - 3} \right)^{j} } } \left( {\lambda_{\,}^{\,} - \lambda_{\,}^{ - 2} } \right) \hfill \\ \,\,\,\,\,\,\,\,\,\, + 2\sum\limits_{i = 0}^{n} {\sum\limits_{j = 1 - i}^{n - i} {jC_{ij} \left( {\overline{{I_{1}^{\,} }} - 3} \right)^{i} \left( {\overline{{I_{2}^{\,} }} - 3} \right)^{j - 1} } } \left( {1_{\,}^{\,} - \lambda_{\,}^{ - 3} } \right), \hfill \\ \end{gathered} $$
(36)

and for equibiaxial and pure shear stresses as

$$ \begin{gathered} \sigma_{bi} = 2\sum\limits_{i = 0}^{n} {\sum\limits_{j = 1 - i}^{n - i} {iC_{ij} \left( {\overline{I}_{1} - 3} \right)^{i - 1} \left( {\overline{I}_{2} - 3} \right)^{j} } } \left( {\lambda - \lambda^{5} } \right) \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\, + 2\sum\limits_{i = 0}^{n} {\sum\limits_{j = 1 - i}^{n - i} {jC_{ij} \left( {\overline{I}_{1} - 3} \right)^{i} \left( {\overline{I}_{2} - 3} \right)^{j - 1} } } \left( {\lambda^{3} - \lambda^{ - 3} } \right), \hfill \\ \end{gathered} $$
(37)
$$ \sigma_{s} = 2\left( {\lambda_{\,}^{\,} - \lambda_{\,}^{ - 3} } \right)\left[ \begin{gathered} \sum\limits_{i = 0}^{n} {\sum\limits_{j = 1 - i}^{n - i} {iC_{ij} \left( {\overline{{I_{1}^{\,} }} - 3} \right)^{i - 1} \left( {\overline{{I_{2}^{\,} }} - 3} \right)^{j} } } \hfill \\ + \sum\limits_{i = 0}^{n} {\sum\limits_{j = 1 - i}^{n - i} {jC_{ij} \left( {\overline{{I_{1}^{\,} }} - 3} \right)^{i} \left( {\overline{{I_{2}^{\,} }} - 3} \right)^{j - 1} } } \hfill \\ \end{gathered} \right]. $$
(38)

Similar to the previous subsections, by assuming the coefficients to be C10 = 1.44 MPa, C01 = 0.463 MPa, C11 =  − 0.029 MPa, C20 =  − 0.151 MPa, and C02 =  − 0.0042 MPa, Fig. 8a–f present the influence of varying the coefficients of the five-parameter polynomial model on the axial stress for strains up to 100%. Coefficients are assumed to vary as [0.5C10–1.5C10], [0.5C01–1.5C01], [0.5C11–1.5C11], [0.5C20–1.5C20] and [0.5C20–3C02], respectively.

Fig. 8
figure 8

Uniaxial stress sensitivity to the five-parameter polynomial model coefficients: a C10; b C01; c C11; d C20; e C02

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2.7 The Gent model

Gent [57, 58] proposed a simple model of hyperelasticity using the first invariant parameter:

$$ W_{G} = - \frac{\mu }{2}J_{m} \ln \left( {1 + \frac{{3 - I_{1} }}{{J_{m} }}} \right), $$
(39)

where WG is the strain energy density of the Gent model and Jm is the limiting stretch parameter [59], which for biological tissues is around (0.4–2.3) [60,61,62,63] and for plastic structures is in orders of 100 [57, 58]. By increasing the maximum permitted value Jm to infinity, the Gent model will be changed to the neo-Hookean incompressible model. The uniaxial, biaxial and pure shear stresses for this model will be

$$ \sigma_{uni} = \mu J_{m} \left( {\lambda - \frac{1}{{\lambda^{2} }}} \right)\left( {1 + \frac{{3 - I_{1} }}{{J_{m} }}} \right)^{ - 1} , $$
(40)
$$ \sigma_{bi} = 2\mu J_{m} \left( {\lambda - \frac{1}{{\lambda^{5} }}} \right)\left( {1 + \frac{{3 - I_{1} }}{{J_{m} }}} \right)^{ - 1} , $$
(41)
$$ \sigma_{s} = \mu J_{m} \left( {\lambda - \frac{1}{{\lambda^{3} }}} \right)\left( {1 + \frac{{3 - I_{1} }}{{J_{m} }}} \right)^{ - 1} , $$
(42)

of which Eq. (40) coincides with the uniaxial stress model presented by Ronald [64]. Figure 9a shows the sensitivity of the axial stress parameter to variations of μ for Jm = 5 and Fig. 9b shows the sensitivity of the axial stress parameter to variations of Jm for μ = 0.4 MPa. Although the main model presented by Gent is only dependent on the first invariant, modified versions of this model, designed to incorporate the compressibility and other invariant terms, have been extended by several researchers [65,66,67,68].

Fig. 9
figure 9

Uniaxial stress sensitivity to the Gent model coefficients: a μ; b Jm

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2.8 The Blatz–Ko model

For the proposed model by Blatz and Ko [69], which is used to characterise foam rubbery structures, the strain energy density is written as [43]:

$$ W_{BK} = \frac{\mu }{2}\left( {\frac{{I_{1}^{\,} }}{{I_{2}^{\,} }} + 2\sqrt {I_{3}^{\,} } } \right), $$
(43)

where WBK is the Blatz–Ko model of the strain energy density and for the incompressible structures, for axial loading, equibiaxial and pure shear stresses the stress–stretch relationship is, respectively, obtained as:

$$ \sigma_{uni} = \frac{\mu }{{2\lambda + \lambda^{ - 2} }}\left[ {\left( {\lambda - \lambda^{ - 2} } \right) + \frac{{\left( {\lambda^{2} + 2\lambda^{ - 1} } \right)}}{{\left( {2\lambda + \lambda^{ - 2} } \right)}}\left( {1 - \lambda^{ - 3} } \right)} \right], $$
(44)
$$ \sigma_{bi} = \frac{\mu }{{2\lambda + \lambda^{ - 2} }}\left[ {\left( {\lambda - \lambda^{ - 5} } \right) + \frac{{\left( {\lambda^{2} + 2\lambda^{ - 1} } \right)}}{{\left( {2\lambda + \lambda^{ - 2} } \right)}}\left( {\lambda^{3} - \lambda^{ - 3} } \right)} \right], $$
(45)
$$ \sigma_{s} = \frac{\mu }{{2\lambda + \lambda^{ - 2} }}\left( {\lambda - \lambda^{ - 3} } \right)\left[ {1 + \frac{{\left( {\lambda^{2} + 2\lambda^{ - 1} } \right)}}{{\left( {2\lambda + \lambda^{ - 2} } \right)}}} \right]. $$
(46)

Figure 10 presents the influence of varying the Blatz–Ko coefficient on the nonlinear stress–strain behaviour of uniaxial loaded structures. There are many other models presented by researchers to model the hyperelastic behaviour of such structures [41], such as the Rivlin–Saunders model [41], the Murnaghan model [70], the Ciarlet model [71], the Valanis–Landel model [72], the Hill model [73] and the Attard model [74].

Fig. 10
figure 10

Uniaxial stress sensitivity to the Blatz–Ko model coefficient

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The given models are mainly used for isotropic hyperelastic materials; however, some structures (especially biological tissues) show a more complicated behaviour which requires orthotropic and anisotropic modelling in their hyperelastic constitutive models. These models are developed mainly for a specific type of hyperelastic materials. A more detailed explanation on orthotropic and anisotropic modelling can be found in refs [75,76,77,78,79,80] and [81,82,83,84,85,86,87,88,89,90], respectively.

An important topic in hyperelasticity analysis is properly modelling the nonlinear dynamics of different soft structures. Since hyperelastic structures have been used lately in many different industrial needs such as soft robotics [91,92,93,94], understanding their nonlinear time-dependant behaviour in different mechanical conditions is of high importance. Studies in this field can be classified based on the structure type and the mechanical analysis. In this review, the nonlinear dynamics of hyperelastic structures is presented in three sections for beams, plates/shells and membranes/balloons.

3 Nonlinear dynamics of hyperelastic beams

One-dimensional structures, including beams, tubes and columns, as the critical part of many mechanical structures, undergo different types of dynamic loads. Since rubber-like beams undergo large strains and deformations, classical linear material models cannot define the nonlinear dynamics accurately; therefore, an accurate hyperelastic model of the structure should be derived and examined. In this section, the nonlinear dynamics of hyperelastic beams is reviewed using literature for different mechanical conditions. Since this review is focused on the nonlinear dynamics, interested readers on the statics of hyperelastic beam structures are referred to [17, 95,96,97,98,99,100].

For hyperelastic beams laying on a foundation, a mathematical formulation was presented by Forsat [101] to examine the nonlinear free vibration behaviour of silicone rubbers and natural rubbers. A higher-order shear deformation beam theory together with four different nonlinear elasticity models was utilised for modelling the soft beam. The structure was assumed to be sitting on a Pasternak–Winkler medium. Equations of motion were solved and obtained using Galerkin’s scheme and Hamilton’s principle (which have been used by researchers for studying elastic structures [102,103,104,105]), respectively; it was claimed that the shear strain effects were neglected in the Yeoh strain energy. In another study, Mirjavadi et al. [106] used the Euler–Bernoulli beam assumptions showing that for hyperelastic tube models, by increasing the amplitude of vibration, since the effect of nonlinear terms in the structure modelling increases and due to different stress invariant considerations, the difference between the Yeoh and two-parameter Mooney–Rivlin results increased. Figure 11 shows the nonlinear frequencies of hyperelastic tube models using Yeoh and Mooney–Rivlin strain energy density models. In both of these studies, the material was assumed to be incompressible; however, the incompressibility condition, which leads to strains in thickness directions, was neglected.

Fig. 11
figure 11

The nonlinear frequencies of hyperelastic tube models using Yeoh and Mooney–Rivlin strain energy density models for a pinned–pinned and b clamped–clamped boundary conditions [106]. (Permission obtained from Springer Nature)

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Since belt operating systems are one of the well-known applications of hyperelastic structures [107, 108], researchers focused on the nonlinear dynamics of axially moving hyperelastic beams to understand their mechanical behaviour in such conditions. Wang et al. [109] used a finite deformation leading-order model (presented in [110, 111]) to investigate the nonlinear oscillations of axially travelling soft beams. Using the Hamilton’s principle and neo-Hookean strain energy density model, the equations of motion were obtained. It was claimed that the natural frequencies of the Euler–Bernoulli beam model are lower than the ones obtained for this model. The variation of the natural frequencies with respect to the axial velocity parameter for this model is shown in Fig. 12. In another study by Wang et al. [112], they analysed accelerated longitudinal motion in soft beams using multiple scale perturbation methods and Galerkin’s scheme. The time traces for the axial and transverse vibrations of this model are shown in Fig. 13. Khaniki et al. [113] investigated the nonlinear forced oscillation of axially moving hyperelastic belts by employing the Yeoh’s strain energy. Different nonlinear elastic models were examined to find the best fit with the experimental testing of hyperelastic properties. The influence of the longitudinal speed on the natural frequencies, mode shapes and nonlinear frequency response was investigated showing a significant effect in changing the mechanical behaviour (Figs. 14 and 15).

Fig. 12
figure 12

Variation of the a first and b second natural frequencies of neo-Hookean axially moving beams with respect to the velocity parameter for different geometrical parameters [109]. (Permission obtained from ELSEVIER)

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Fig. 13
figure 13

The time traces for the a axial and b transverse vibrations of axially accelerated neo-Hookean beams [112]. (Permission obtained from ELSEVIER)

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Fig. 14
figure 14

The natural frequency and mode shapes of axially moving hyperelastic beams versus the speed parameter a first and b second modes of vibration [113]. (Permission obtained from ELSEVIER)

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Fig. 15
figure 15

The transverse amplitude–frequency response of axially moving hyperelastic beams with different velocities a first, b second, and c third coordinates [113]. (Permission obtained from ELSEVIER)

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For the case of having both thermal and hyperelasticity effects, a wave propagation method was employed by Mirparizi and Fotuhi [114] to understand the nonlinear dynamics of thermo-hyperelastic one-dimensional structures. Hyperelasticity was modelled using a Mooney–Rivlin strain energy density model. It was elucidated that the maximum amplitude of oscillation in the structure is significantly higher than for elastic ones. Figure 16 shows the stress wave propagation with respect to time showing a large difference between the hyperelastic and linear elastic models response.

Fig. 16
figure 16

Stress wave propagation in hyperelastic and elastic structures [114]. (Permission obtained from ELSEVIER)

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Since hyperelastic structures are mostly made by moulding and 3D printing, the presence of voids and porosity is highly possible. To understand the effect of having porosities in the nonlinear dynamics of hyperelastic structures, Khaniki et al. [115] studied the characteristics of hyperelastic samples experimentally with different porosities (the infill rate). A general constitutive model for hyperelastic-porous was presented via the Mooney–Rivlin hyperelastic strain energy model, showing that the porosity has a nonlinear effect in varying the hyperelastic constitutive model (Fig. 17). For the derived model, they modelled the nonlinear vibrations of porous hyperelastic beams under externally time-dependent forces, showing that increasing the porosity in the structure has a significant effect in changing the stiffness softening behaviour of the structure to a combination of hardening and softening behaviour (Fig. 18).

Fig. 17
figure 17

Experimental results for stress–strain behaviours of porous hyperelastic structures with different infill rates (α) [115]. (Permission obtained from ELSEVIER)

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Fig. 18
figure 18

Influence of the infill rate (porosity) in varying the nonlinear frequency response of porous hyperelastic beams a first and b third dynamic coordinates. [115]. (Permission obtained from ELSEVIER)

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Layered hyperelastic structures have many applications in packaging industry (especially food packaging) [116,117,118,119], which makes it important to comprehend the behaviour of layered hyperelastic structures made of different materials. For this reason, Khaniki et al. [120] examined five different shear deformable beam theories together with the Mooney–Rivlin strain energy model for analysing the nonlinear dynamics of sandwich soft beams. It was shown that considering the shear effect, layering and material positioning can highly affect the nonlinear resonance behaviour of the thick sandwich soft beam. Figure 19 shows the effect of material ordering in changing the nonlinear dynamic behaviour of higher-order shear deformable three-layered beam structures.

Fig. 19
figure 19

The effect of hyperelastic material layering on the nonlinear dynamics Mooney–Rivlin shear deformable beams for the a axial, b rotational and c transverse motions [120]. (This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).)

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Longitudinal vibrations of neo-Hookean beams have been examined by Wang and Zhu [121, 122] in its subcritical buckling regime and under different axial loads. Using a linear bifurcation analysis, the critical buckling loads were obtained showing a high sensitivity of material and geometrical properties. Using the pseudo-arc-length method, the nonlinear frequency response of the system was calculated. Figure 20 shows the axial frequency response of the neo-Hookean beam model for different material and geometrical properties.

Fig. 20
figure 20

The axial nonlinear frequency response of the neo-Hookean beam model for different a material and b geometrical properties [121]. (Permission obtained from ELSEVIER)

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In recent years, hyperelasticity has been employed for modelling the static and dynamic responses of nano-/micromaterials [123,124,125,126], which have crucial importance in new technologies. For instance, micro-scale beam structures made by hyperelastic materials have been studied by Alibakhshi et al. [127] using the Euler–Bernoulli beam theory, modified couple stress theory (which have been used previously for studying small-scale elastic structures [128,129,130,131,132]), and Gent strain energy density model. It was shown that the force–amplitude response of the structure is highly sensitive to the stiffness parameter of Gent model (Fig. 21). Studies on the nonlinear dynamics of hyperelastic beams are summarised in Table 1.

Fig. 21
figure 21

The force–amplitude response of micro-hyperelastic beam for different Gent coefficients[127]. (Permission obtained from MDPI)

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Table 1 Studies on the nonlinear dynamics of hyperelastic beams
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4 Nonlinear dynamics of hyperelastic plates and shells

Both shells and plates are an important element in structural design, and the extensive usage of rubbery structures makes it necessary to comprehend the nonlinear dynamics of the hyperelastic plate and shell structures. For this reason, this section focuses on the investigations undertaken to comprehend the nonlinear dynamics of such structures.

For isotropic hyperelastic plate structures, a combination of the nonlinear von Kármán plate theory and the neo-Hookean strain energy density model was used by Breslavsky et al. [133] for examining the large amplitude vibrations of thin rectangular hyperelastic plates. The equations of motion were presented with quadratic and cubic nonlinear terms by considering both material and geometrical nonlinearities; it was shown that for small strains, the material nonlinearity terms have a weak effect, while this effect increases significantly by having larger strains. Figure 22 shows the amplitude response and backbone curves of the first mode of vibration with two peaks associated with the 2:1 in-plane resonance. Amabili et al. [134] investigated the vibration behaviour of hyperelastic plates. A two-parameter incompressible Mooney–Rivlin model was used to describe the nonlinearity of the structure using Novozhilov nonlinear shell theory to model the deformations. The governing equations were obtained using Hamilton’s principle; it was shown that the experimental results are in good agreement with the proposed dynamic model.

Fig. 22
figure 22

Amplitude–frequency response and backbone curves of the first mode of vibration of a rectangular hyperelastic plate with two peaks associated with the 2:1 in-plane resonance [133]. (Permission obtained from ELSEVIER)

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The nonlinear dynamics of cylindrical shell structures have been examined lately by many researchers. Zhang et al. [135] modelled the nonlinear vibrations of thin-walled hyperelastic cylindrical shells using Donnell’s nonlinear shallow shell theory. Using the Lagrange equation together with the Mooney–Rivlin strain energy density model, the equations of motion were obtained. It was shown that radius-to-thickness ratio has a significant effect in changing the radial vibration behaviour. Figure 23 shows the Poincare section and bifurcation diagram of the hyperelastic cylindrical shell for different excitation forces. Xu et al. [136, 137] examined the nonlinear dynamics of thin and thick hyperelastic cylindrical shells subjected to a time-dependant thermal load. It was shown that the single-mode model of such structures gives inaccurate results and the diameter-to-length ratio has a significant influence in the internal resonance phenomena. Figure 24 shows the nonlinear response of the cylindrical shell with different temperature loads for the axisymmetric and asymmetric modes. Breslavsky et al. [44, 138] investigated the free and forced nonlinear responses of circular cylindrical hyperelastic shells and square hyperelastic plates in the framework of large deformations. Hyperelasticity was modelled using the neo-Hookean model in conjunction with the Fung model, while the shell was assumed to be made of arterial bio-tissues. It was shown that the single-mode response is weak; however, the resonant response considering both companion and driven modes, was found with nonlinear complicated dynamics.

Fig. 23
figure 23

a Bifurcation diagram and b Poincaré sections of a thin-walled Mooney–Rivlin cylindrical shell for different excitation forces [135]. (Permission obtained from Springer Nature)

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Fig. 24
figure 24

The nonlinear response of the cylindrical shell with different temperature loads for the (a) axisymmetric and (b) asymmetric modes [136]. (Permission obtained from WSPC)

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In the case of analysing electrostrictive–hyperelastic structures, Tripathi and Bajaj [139, 140] studied the internal resonances due to transverse vibration when designing hyperelastic and electrostrictive–hyperelastic plates. The Mooney–Rivlin and neo-Hookean hyperelastic strain energy models were investigated, while the plate was modelled using Kirchhoff plate theory. These showed that for nearly incompressible structures, the level of nonlinearity in the strain energy model of neo-Hookean is insufficient to lead to 1:2 internal resonances. A visco-hyperelastic model was presented by Zhao et al. [141] for modelling the chaotic motion of spherical shells. Other studies on the dynamic behaviour of hyperelastic plates and shells can be found in refs [142,143,144,145,146,147,148], emphasising the importance of considering hyperelasticity in studying the dynamic response of such structures. Studies on the nonlinear dynamics of hyperelastic plates and shells are summarised in Table 2.

Table 2 Studies on the nonlinear dynamics of hyperelastic plates/shells
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5 Nonlinear dynamics of hyperelastic membranes and balloons

Comprehending the nonlinear dynamics of hyperelastic membranes and balloons has been a challenging task for researchers in the past few years. For circular membranes, Goncalves et al. [149, 150] studied the nonlinear vibration behaviour of isotropic homogeneous circular hyperelastic membranes stretched radially. Hyperelasticity was modelled using the neo-Hookean strain energy density model, and the motion equations were derived via Hamilton’s principle. Natural frequencies were obtained analytically and compared with finite element modelling. It was revealed that all the hyperelastic models, namely the Arruda–Boyce model, Ogden model, Yeoh model and Mooney–Rivlin model, present similar nonlinear frequency–amplitude responses, qualitatively. The influence of the pre-stretch ratio on the nonlinear amplitude–frequency response of the circular hyperelastic membrane can be seen in Fig. 25.

Fig. 25
figure 25

Amplitude–frequency responses of hyperelastic circular membranes for different pre-stretch ratios [149]. (Permission obtained from ELSEVIER)

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The nonlinear breathing motion of hyperelastic spherical membranes has been examined by Soares et al. [151] using an incompressible isotropic Mooney–Rivlin strain energy density model. It was shown that the linear viscous damping term has a significant effect in changing the nonlinear resonance behaviour of the spherical membranes (Fig. 26).

Fig. 26
figure 26

The nonlinear resonance response of hyperelastic spherical membranes for different linear damping parameters [151]. (Permission obtained from Springer Nature)

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Since it was shown by Mangan and Destrade [152] that the hyperelastic Gent–Gent model shows a good accuracy in modelling the nonlinear elasticity of inflated balloons, Alibakhshi and Heidari [59] used this model for studying the chaotic motion of dielectric elastomer balloons; it was shown that the chaotic motion of the dielectric elastomer balloons could be suppressed by the presence of the second invariant term in the Gent–Gent model (Fig. 27). Ilssar and Gat [153] examined the fluid–structure interaction of liquid-filled balloons using the incompressible Mooney–Rivlin strain energy density model. By verifying the model with results obtained from the finite element scheme, it was shown that the simplified presented model was capable of studying the static and dynamic behaviour of such structures. Other studies on the dynamic behaviour of hyperelastic membranes and balloons can be found in refs [19, 154,155,156]. Studies on the nonlinear dynamics of hyperelastic balloons and membranes are summarised in Table 3.

Fig. 27
figure 27

Variation of the motion from a chaotic motion to b quasiperiodic vibration by increasing the second invariant parameter of the Gent–Gent model in the dielectric elastomer balloons [59]. (Permission obtained from ELSEVIER)

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Table 3 Studies on the nonlinear dynamics of hyperelastic membranes/balloons
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6 Summary and conclusions

Over the past few decades, accurately modelling the hyperelastic behaviour of polymeric structures has been a challenging task in terms of the complexity in both material and structural nonlinearities. Dozens of constitutive hyperelastic models have been presented by researchers for different materials, which some of the most practical and well-known isotropic models are presented and formulated, and the characteristics of each model are discussed. In terms of the case studies, each of these models has advantages and disadvantages in terms of accurate and inaccurate modelling for material nonlinearity and computational time cost.

Hyperelastic structures, such as rubbers and elastomers, have been analysed in different shapes and mechanical conditions. Most studies into these types of structures have been published over the past few years, since their applications are only now becoming understood. Recently, novel outcomes from rubbery structures in soft robotics have moved forward such study significantly, notably in terms of achieving smooth deformations and higher the degrees of freedom. The high potential for the application of soft-based structures helped researchers to realise the importance of modelling the dynamic behaviour of hyperelasticity accurately. To demonstrate the achievements and investigations made into hyperelastic structures, a comprehensive review was presented for different structures types (beams, plates, shells, membranes and balloons) in the frameworks of nonlinear dynamics. It was shown that there has been progress in simulating the hyperelastic dynamic response using various continuum mechanic techniques in conjunction with hyperelastic strain energy density models. Since the theoretical modelling of such structures could require highly complex and long theoretical models, many studies in this field are based on simplified models ignoring the higher terms of displacement and strain. Furthermore, since soft structures largely deform when subjected to external loads, the importance of using appropriate nonlinear large-deformation modelling is undeniable.

In summary, by analysing more than 150 research works related to this field from basic analysis to specified simulations, it can be seen that although there have been several studies on each subject of hyperelastic structure behaviour, the field is under-researched and requires further investigations to model and analyse hyperelastic structures.