Analytical study on growth-induced bending deformations of multi-layered hyperelastic plates

Du, Ping; Dai, Hui–Hui; Wang, Jiong; Wang, Qiongyu

doi:10.1016/j.ijnonlinmec.2019.103370

Cited by 22 publications

(12 citation statements)

References 29 publications

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“…), which formulate a closed equation system. To derive a 2D vector plate equation system from this closed system, we adopt an elaborate calculation scheme proposed in Du et al [42,43]. The procedure of this calculation scheme is introduced below (which is also shown in Fig.…”

Section: Derivation Of the 2d Vector Plate Equationmentioning

confidence: 99%

“…Ackermann et al [41] investigated the mechanics of growing epithelia based on a bilayer FvK plate theory. Following the approach of Wang et al [24], Du et al [42,43] proposed a finite-strain plate theory to study the growth-induced plane-strain deformations and instabilities of multi-layered hyperelastic plates. Some analytical solutions of the plate equations were obtained, which can provide accurate predictions on the response of the plate samples.…”

Section: Introductionmentioning

confidence: 99%

“…To establish the plate theory, we start from the 3D governing system and eliminate the thickness variables through a series expansion-truncation approach [20,24]. An elaborate calculation scheme is then applied to derive the iteration relations of the coefficient functions in the series expansions [42,43]. Through some further manipulations, a 2D vector plate equation system with the associated boundary conditions can be established.…”

Section: Introductionmentioning

confidence: 99%

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On a general multi-layered hyperelastic plate theory of growth

Du¹,

Li²,

Chen³

et al. 2022

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In this paper, we propose a multi-layered hyperelastic plate theory of growth within the framework of nonlinear elasticity. First, the 3D governing system for a general multi-layered hyperelastic plate is established, which incorporates the growth effect, and the material and geometrical parameters of the different layers. Then, a series expansion-truncation approach is adopted to eliminate the thickness variables in the 3D governing system. An elaborate calculation scheme is applied to derive the iteration relations of the coefficient functions in the series expansions. Through some further manipulations, a 2D vector plate equation system with the associated boundary conditions is established, which only contains the unknowns in the bottom layer of the plate. To show the efficiency of the current plate theory, three typical examples regarding the growth-induced deformations and instabilities of multi-layered plate samples are studied. Some analytical and numerical solutions to the plate equation are obtained, which can provide accurate predictions on the growth behaviors of the plate samples. Furthermore, the problem of 'shape-programming' of multi-layered hyperelastic plates through differential growth is studied. The explicit formulas of shape-programming for some typical multi-layered plates are derived, which involve the fundamental quantities of the 3D target shapes. By using these formulas, the shape evolutions of the plates during the growing processes can be controlled accurately. The results obtained in the current work are helpful for the design of intelligent soft devices with multi-layered plate structures.

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Section: Derivation Of the 2d Vector Plate Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On a general multi-layered hyperelastic plate theory of growth

Du¹,

Li²,

Chen³

et al. 2022

Preprint

Self Cite

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“…Wang et al (2018) derived a consistent finite-strain plate theory for growth-induced large deformation and investigated the buckling and post-buckling behaviour of a thin rectangular hyperelastic plate under axial growth. Recently, the finite strain asymptotic plate theory has been applied to study the plane strain problems of growth-induced deformation in single and multi-layered hyperelastic plates (Wang et al, 2019b;Du et al, 2020). Liu et al (2020a) discussed the large elastic deformation in nematic liquid crystal elastomer and the authors of the present paper (Mehta et al, 2021b) investigated the bifurcation behaviour of circular hyperelastic plate under the influence of growth using finite strain asymptotic plate theory.…”

Section: Introductionmentioning

confidence: 99%

On the stability of a growing annular hyperelastic plate

Mehta¹,

Raju²,

Saxena³

2022

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Growth-induced instabilities are ubiquitous in biological systems and lead to diverse morphologies in the form of wrinkling, folding, and creasing. The current work focusses on the mechanics behind growth-induced wrinkling instabilities in an incompressible annular hyperelastic plate subjected to two types of boundary constraints. Governing differential equations for a two-dimensional plate system are derived using a variational principle and series expansion along the thickness direction. The incremental differential equations for studying the bifurcation behaviour of the growing hyperelastic annular plate are derived by considering both axisymmetric and asymmetric perturbations. The resulting equations are then solved numerically using the compound matrix method to evaluate the critical growth stretch that causes bifurcation. The effect of boundary constraints, thickness, and radius ratio of the annular plate on the critical growth factor is studied. For most of the considered cases, an asymmetric bifurcation is the preferred mode of instability for an annular plate. The obtained results are relevant to modelling of wrinkling phenomena in planar soft tissues, pattern transition in two-dimensional films growing on an elastic substrate, and manufacturing of smart materials.

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“…Hyperelastic materials play a vital role in soft systems and structures, e.g., soft robotics [14], human organs [15,16], soft actuators [17,18], soft sensors [19,20], and soft energy harvesters [19][20][21][22]. Data from previous studies show that various mechanical structures such as beams, plates, membranes, and shells were made of hyperelastic materials [23][24][25][26][27][28][29][30][31][32][33][34][35]. It was reported that hyperelastic beams are an appropriate candidate to fabricate systems with high performance.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Free and Forced Vibrations of a Hyperelastic Micro/Nanobeam Considering Strain Stiffening Effect

et al. 2021

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In recent years, the static and dynamic response of micro/nanobeams made of hyperelasticity materials received great attention. In the majority of studies in this area, the strain-stiffing effect that plays a major role in many hyperelastic materials has not been investigated deeply. Moreover, the influence of the size effect and large rotation for such a beam that is important for the large deformation was not addressed. This paper attempts to explore the free and forced vibrations of a micro/nanobeam made of a hyperelastic material incorporating strain-stiffening, size effect, and moderate rotation. The beam is modelled based on the Euler–Bernoulli beam theory, and strains are obtained via an extended von Kármán theory. Boundary conditions and governing equations are derived by way of Hamilton’s principle. The multiple scales method is applied to obtain the frequency response equation, and Hamilton’s technique is utilized to obtain the free undamped nonlinear frequency. The influence of important system parameters such as the stiffening parameter, damping coefficient, length of the beam, length-scale parameter, and forcing amplitude on the frequency response, force response, and nonlinear frequency is analyzed. Results show that the hyperelastic microbeam shows a nonlinear hardening behavior, which this type of nonlinearity gets stronger by increasing the strain-stiffening effect. Conversely, as the strain-stiffening effect is decreased, the nonlinear frequency is decreased accordingly. The evidence from this study suggests that incorporating strain-stiffening in hyperelastic beams could improve their vibrational performance. The model proposed in this paper is mathematically simple and can be utilized for other kinds of micro/nanobeams with different boundary conditions.

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Analytical study on growth-induced bending deformations of multi-layered hyperelastic plates

Cited by 22 publications

References 29 publications

On a general multi-layered hyperelastic plate theory of growth

On a general multi-layered hyperelastic plate theory of growth

On the stability of a growing annular hyperelastic plate

Nonlinear Free and Forced Vibrations of a Hyperelastic Micro/Nanobeam Considering Strain Stiffening Effect

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