Hyperelastic Plate Research Articles

Realization of planar and surface conformal mappings through stress-free growth of hyperelastic plates: Analytical formulas and numerical calculations

Conformal mapping is a well-known concept and has a long research history in mathematics. Accompanying the developments of computational science and 3D digital scanning technology, conformal mapping has also found wide applications in geometric modeling, computer graphics, medical imaging and other fields. In the virtual image or animation world, conformal mapping offers a convenient approach to achieve various shape changes of planar regions and 3D surfaces. In this work, we propose a promising approach, i.e., through the stress-free growth of hyperelastic plates, to realize arbitrary planar and surface conformal mappings in the physical world. The growth field in a hyperelastic plate can be represented by a symmetric tensor field. For any given planar or surface conformal mappings with explicit analytical expressions, the formulas of growth functions in the growth tensor are derived. In the case that the target surfaces of conformal mappings have no explicit analytical expressions, we further propose a numerical scheme to determine the growth data in the hyperelastic plates. The efficiency of the approach proposed in the current work is validated through several typical examples, where the 3D finite element simulations are conducted. The results demonstrate that, by incorporating the obtained growth functions or growth data, the target surfaces can be generated accurately through growth deformations of hyperelastic plates. As an application of the current work, we further propose an approach of shape-control for double-layer grid structures with plate forms. According to any given target shapes, the values of elongation or contraction of the bar elements are determined, then the desired deformations of the grid structures can be realized.

Predicting fluid–structure interaction with graph neural networks

We present a rotation equivariant, quasi-monolithic graph neural network framework for the reduced-order modeling (ROM) of fluid–structure interaction systems. With the aid of an arbitrary Lagrangian–Eulerian (ALE) formulation, the system states are evolved temporally with two sub-networks. The movement of the mesh is reduced to the evolution of several coefficients via complex-valued proper orthogonal decomposition (POD), and the prediction of these coefficients over time is handled by a single multi-layer perceptron (MLP). A finite element-inspired hypergraph neural network is employed to predict the evolution of the fluid state based on the state of the whole system. The structural state is implicitly modeled by the movement of the mesh on the solid–fluid interface; hence, it makes the proposed framework quasi-monolithic. The effectiveness of the proposed framework is assessed on two prototypical fluid–structure systems, namely, the flow around an elastically mounted cylinder and the flow around a hyperelastic plate attached to a fixed cylinder. The proposed framework tracks the interface description and provides stable and accurate system state predictions during roll-out for at least 2000 time steps and even demonstrates some capability in self-correcting erroneous predictions. The proposed framework also enables direct calculation of the lift and drag forces using the predicted fluid and mesh states, in contrast to existing convolution-based architectures. The proposed reduced-order model via the graph neural network has implications for the development of physics-based digital twins concerning moving boundaries and fluid–structure interactions.

Asymptotically correct 3D displacement of the Mooney–Rivlin model using VAM

The current work focuses on the nonlinear analytical analysis (dimensional reduction and recovery relations) of a hyperelastic plate governed by the compressible Mooney–Rivlin model using the Variational Asymptotic Method (VAM). The geometric nonlinearity is accommodated through finite deformations, and generalized 3D warping functions, while material nonlinearity through the hyperelastic material model. VAM mathematically splits the 3D nonlinear elastic problem into the 1D through the thickness analysis and 2D nonlinear plate analysis, using the inherent small parameters. These are the geometric small parameter (ratio of thickness to the characteristic dimension), and the physical small parameter (moderate strains). This work results include the derivation of closed-form analytical expressions of 3D warping functions, 2D nonlinear constitutive relation, and recovery relation to express the 3D displacement field for a plate. The 2D nonlinear constitutive relation is given as an input to the in-house developed 2D nonlinear finite element analysis of the reference surface to determine the 2D displacements and 2D strains. In order to validate the current theory, standard test cases are solved and compared with 3D nonlinear Finite Element Analysis (FEA).

Monitoring prestress in plates by sideband peak count-index (SPC-I) and nonlinear higher harmonics techniques

Propagating guided waves in a homogeneous, isotropic, prestressed, hyperelastic plate show nonlinear characteristics that depend on the state of initial prestress. These nonlinear phenomena include higher harmonic generation, occurring when Lamb wave modes of different frequencies (omega _a and omega _b) are allowed to mix within the material generating secondary waves at frequencies 2omega _a,2omega _b, text {and},omega _apm omega _b. Further, if prescribed internal-resonance conditions are satisfied, the amplitude of secondary waves increases in space, providing a response quantity which is dependent on prestress and easy to be observed. Using the finite element method, in this paper we investigate the time and space evolution of higher harmonics arising in one-way wave mixing. The influence of prestress on the response is elucidated, observing the nonlinear parameter beta . It is further shown that the nonlinear ultrasonic technique called sideband peak count-index (SPC-I) can provide an effective monitoring tool for prestress.

Nonlinear 2D-DQ volume-preservative global–local dynamic analysis of composite sandwich plates with soft hyperelastic cores and viscoelastic Winkler-Pasternak foundations

Nonlinear large amplitude vibration and dynamic deformations of sandwich plates with composite face sheets and hyperelastic cores that encounter large thickness changes during the vibration are investigated here for the first time. Moreover, a new global–local zigzag hyperelastic plate theory that considers the incompressibility and large transverse deformability of the core, and employs transverse normal strains whose order is higher than that of the in-plane strains is proposed. The plate is resting on a viscoelastic Winkler-Pasternak substrate, exposed to various spatial and time distributions of loads, and experiencing a diversity of edge conditions. The governing equations of motion are obtained by using Hamilton’s principle, von Kármán’s assumptions, and the series expansion of the volume-preserving strain energy density function. A 2D differential quadrature (2D-DQ) technique is employed for the spatial discretization of a structure with both constitutive and kinematic nonlinearities, for the first time. The time-dependent combination of the nonlinear coupled motion equations and boundary conditions is treated by an iterative/updating Runge-Kutta time-marching technique and the effects of the thickness and aspect ratio of the plate, viscoelastic and shear parameters of the foundation, types of the edge conditions, constitutive parameters of the composite materials, fiber orientation angle, and magnitude and distribution pattern of the loads on the resulting dynamic lateral deflections are studied. Results reveal that while the vibration frequencies and amplitudes of the face sheets are quite different, the damping is more remarkable when stiffer cores are utilized. Moreover, the effects of the higher vibration modes are more apparent in looser edge conditions, smaller fiber angles, and special excitation frequencies.

Experimental and numerical analysis of hyperelastic plates using Mooney-Rivlin strain energy function and meshless collocation method

In this research, the flexural behavior of a hyperelastic plate made of silicone under uniformly distributed loading was investigated. The displacements and strains were formulated using the first-order shear deformation plate theory (FSDPT). The right Cauchy-Green tensor and Mooney-Rivlin strain energy function were employed to formulate hyperelastic silicone plate governing equations. The strong-form of the governing equations of the silicone plate using Mooney-Rivlin strain energy function and Euler-Lagrange relations were derived for the first time. In the governing equations, the effects of geometrical and material nonlinearity effects were considered simultaneously. The meshless collocation method (MCM) and thin-plate spline radial basis function were utilized to discretize the governing equations. The arc-length continuation algorithm was also used for analyzing the system of non-linear equations. MCM results were compared to those of the experimental test to validate the results of the meshless method. For this purpose, a square silicone plate with simply supported boundary conditions under uniformly distributed loading was investigated via the digital image correlation (DIC) technique. According to the obtained results, the MCM based on radial basis function is an adequate method for non-linear analysis of hyperelastic plates with Mooney-Rivlin strain energy function under uniformly distributed loading.

Meshless method and experimental analysis of hyperelastic plates using Mooney-Rivlin strain energy function subjected to concentrated loading

The purpose of this research is to investigate a strong-form meshless method for experimentally and numerically analyzing hyperelastic plates subjected to concentrated loading. The silicone hyperelastic plate was formulated utilizing Mooney-Rivlin strain energy function and Lagrangian strains. The displacements and Lagrangian strains of the plate are also calculated using the first-order shear deformation plate theory (FSDPT). The hyperelastic plate’s governing equations are deduced in strong form and analyzed using the logarithmic thin-plate spline radial basis function (LTPS-RBF) and the meshless method for the first-time. The nonlinear system of governing equations is solved using the arc-length continuation algorithm. In the meshless method, the concentrated loading is modeled using the Kronecker delta function in the expression of external work. A silicone plate is employed experimentally to evaluate the accuracy of the meshless method’s results. Furthermore, the finite element method is utilized to compare the contours obtained from the meshless method. Concentrated loading is considered at various points to validate the meshless methods and results from the experimental analysis. The most significant difference between the meshless method and the experimental results is less than 15%. The results indicate that the meshless method is highly accurate when analyzing hyperelastic plates subjected to concentrated loading.

Controllable rogue waves in a compressible hyperelastic plate

In this paper, we investigate various rational solutions of a (2+1)-dimensional nonlinear equation with variable-coefficients which can describe the propagation of nonlinear waves in a compressible hyperelastic plate, and interestingly we find a new phenomenon in nonlinear hyperelastic mechanics—rogue waves. Based on the symmetry transformation and Hirota bilinear form, we obtain breather waves and rogue waves. And when we choose appropriate material parameters and transformation functions, we can control their directions of propagation. In addition, we present three new interaction solutions, i.e. the rogue waves keep moving from side to side between a pair of resonance stripe solitary waves. Finally, we analysis the dynamics characteristics and evolution behaviors of the obtained solutions by three-dimensional dynamic images. This paper exhibits the existence of rogue waves in nonlinear hyperelastic mechanics, and presents potential application value for control of rogue waves in elastic mechanics, fluid dynamics and other fields of nonlinear science.

A general multi-layered hyperelastic plate theory for growth-induced deformations in soft material samples

In this paper, we propose a general multi-layered hyperelastic plate theory to study the growth-induced deformations of soft material samples. First, by considering a multi-layered plate sample with an arbitrary geometrical shape and adopting a general hyperelastic constitutive model, the 3D governing system is established, which incorporates the growth effects of the different layers in the plate. 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. To show the efficiency of the plate theory, three typical examples regarding the growth-induced deformations and instabilities of multi-layered plate samples are studied, where the incompressible neo-Hookean constitutive model is adopted. Some analytical and numerical solutions to the plate equation system are obtained, which can provide accurate predictions on the growth behaviors of the plates. Furthermore, the problem of ‘shape-programming’ of multi-layered plates through differential growth is studied and the explicit formulas for some typical examples are derived. 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 applicable for the design of intelligent soft devices with multi-layered plate structures.

Theory and experiment for dynamics of hyperelastic plates with modal interactions

This paper presents a joint experimental and theoretical approach to the dynamics and mass sensitivity of hyperelastic plates including cases with modal interactions. For the theoretical approach, the plate structure is assumed to undergo large strains and deformations using the Mooney-Rivlin hyperelastic strain energy density model and the von-Kármán geometric nonlinearity, respectively. The plate is modelled using the continuum mechanics definitions and the Kirchhoff–Love plate theory. The coupled in-plane and out-of-plane equations of motion are obtained using Hamilton's equation and solved afterwards using a combination of Galerkin's procedure together with a dynamic equilibrium technique. For the experimental analysis, the properties of the material are first obtained by performing a set of stress-strain tests on the samples following the ASTM D-412 standard, and then from the same rubber material, a plate structure is fabricated, and an externally actuated vibration test is performed. The nonlinear frequency response of the structure both with and without a concentrated mass is investigated and the capability of the structure for mass sensing is discussed. By obtaining the amplitude-frequency response of the structure due to the applied external periodic load, using both theoretical and experimental approaches, it is shown that the given model has good accuracy in simulating the nonlinear dynamics of the hyperelastic plate structure under different conditions. Furthermore, a set of analyses on the nonlinear forced vibration behaviour of hyperelastic plates at different internal resonances, using concentrated mass and the length-to-width ratio is presented. The results of this study are useful in designing systems involving hyperelastic plates, such as soft robots and soft functional devices.

Non-linear analysis of orthotropic hyperelastic plate using VAM

An asymptotically accurate zeroth-order constitutive relation between 2D strains and stress tensor for an orthotropic hyperelastic material is derived using the Variational Asymptotic Method (VAM). Material nonlinearity is encompassed through the hyperelastic material model while geometric nonlinearity is by allowing finite displacements and rotations. Moderate strains of order 20% are considered. The small parameters such as the ratio of thickness to characteristic dimension, and maximum allowable strain (0.2) separate strain energy density into different orders. The analysis starts with the 3D strain energy density function and is divided into the thickness analysis (1D) and mid-surface analysis (2D). The 1D analysis leads to analytical expressions for warping functions, constitutive relations, and recovery relations. The 2D analysis is based on a non-linear finite element method that takes the constitutive relation as its input and obtains 2D displacements. The recovery relation converts them to 3D field variables i.e. displacements, strains, and stresses. The theory is applied to a square clamped plate subjected to uniformly distributed pressure. The results are accurate and obtained in a computationally efficient manner.

Wrinkling as a mechanical instability in growing annular hyperelastic plates

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. The governing differential equations for a two-dimensional plate system are derived using a variational principle with no apriori kinematic assumptions in the thickness direction. A linear bifurcation analysis is performed to investigate the stability behaviour of the growing hyperelastic annular plate by considering both axisymmetric and asymmetric perturbations. The resulting differential equations are then solved numerically using the compound matrix method to evaluate the critical growth factor that leads to wrinkling. 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. Our results are useful to model the physics of wrinkling phenomena in growing planar soft tissues, swelling hydrogels, and pattern transition in two-dimensional films growing on an elastic substrate.

A theoretical scheme for shape-programming of thin hyperelastic plates through differential growth

In this paper, a theoretical scheme is proposed for shape-programming of thin hyperelastic plates through differential growth. First, starting from the 3D governing system of a hyperelastic (neo-Hookean) plate, a consistent finite-strain plate equation system is formulated through a series expansion and truncation approach. Based on the plate equation system, the problem of shape-programming is studied under the stress-free assumption. By equating the stress components in the plate equations to be zero, the explicit relations between the growth functions and the geometrical quantities of target shape of the plate are derived. Then, a theoretical scheme of shape-programming is proposed, which can be used to identify the growth fields corresponding to arbitrary 3D shapes of the plate. To demonstrate the efficiency of the scheme, some typical examples are studied. The predicted growth functions in these examples are adopted in the numerical simulations, from which the target shapes of the plate can be recovered completely. The scheme of shape-programming proposed in the current work is applicable for manufacturing intelligent soft devices.

Dynamic behavior of heterogeneous neo-Hookean/Mooney–Rivlin plates reinforced nonuniformly by hyperelastic inclusions: Proposing the correct micromechanical model

Time-dependent nonlinear lateral vibrations of plates composed of a hyperelastic matrix and uniformly/nonuniformly distributed hyperelastic reinforcing inclusions are studied. Since the material constants of a hyperelastic must be extracted from the whole slope-varying stress-strain curve rather than a single slope, choosing power/exponential distributions for the material constants or using Voigt’s rule of mixtures is quite wrong. The neo-Hookean and Mooney–Rivlin constitutive models are adopted, and their results are compared. Another hint is incorporating the incompressibility condition. The governing equations of motion are derived by using Hamilton’s principle, a new energy-equivalence-based micromechanical model that can be employed for reinforcing phases with nonlinear constitutive laws, and von Kármán assumptions in the left Cauchy–Green deformation tensor, and solved by incorporation of an updating finite-element and Newmark’s techniques. Not only the displacement but typical stress results are also reported here. Results show that the neo-Hookean model overestimates the rigidity in comparison to the Mooney–Rivlin model, and unlike the elastic plates, the effect of the stiffer phase is more remarkable in the uniform distribution in comparison to the nonuniform distribution of the stiffening materials because the magnitudes of the tensile stresses of the hyperelastic plate are much larger than that of the bending stresses.

On a finite-strain plate theory for growth-induced plane-strain deformations and instabilities of multi-layered hyperelastic plates

In this paper, we propose a finite-strain plate theory to study the growth-induced deformations and instabilities of multi-layered hyperelastic plates. First, under the assumption of plane-strain deformation, we formulate the governing system for a multi-layered hyperelastic (neo-Hookean) plate. The layers in the plate have different geometrical, material and growth parameters. Starting from the governing system, we propose a scheme to establish the plate equation system, where one arbitrary layer is selected as the base layer and a series-expansion approach is adopted. Through a complicated derivation procedure, the iteration relations of the unknowns in the different layers are obtained, and then the plate equation system containing the unknowns in the base layer are derived, which can attain the accuracy of O(h2). To show the efficiency of the plate equations, we first study the growth-induced bending deformations of a multi-layered hyperelastic plate. By solving the plate equations, some analytical results are obtained, which provide accurate predictions on the deformations of the plate. The influences of the geometrical and material parameters on the response of the plate are also investigated. Besides that, the plate equation system is applied to study the growth-induced instabilities of a 3-layered hyperelastic plate. Through some conventional analyses, the critical values of the growth parameters corresponding to the different bifurcation modes are determined, which can fit the numerical results very well. In our opinion, the plate theory proposed in the current work is helpful for the design of intelligent soft devices with multi-layered plate forms.

Analytical study on growth-induced axisymmetric deformations and shape-control of circular hyperelastic plates

Growth of soft material plates is commonly observed in nature. However, the relations between growth fields and shape changes of the plate samples remain poorly understood. The current work aims to derive some analytical results for the growth-induced axisymmetric deformations of a circular hyperelastic plate. The problem of shape-control of the circular plate through differential growth will also be studied. First, starting from the 3D governing system, we formulate the 2D vector plate equation system through a series expansion and truncation approach. Then, the plate equations are solved by using the regular perturbation method. Some analytical (asymptotic) solutions are derived for both the out-of-plane bending and in-plane flat deformations of the plate, which can fit the numerical results quite well. By comparing the energy densities, the preferred deformation styles of the plate corresponding to any given growth functions can be determined, based on which the combined bending-flat deformation of the plate is also studied. Furthermore, by solving an inverse problem, we derived the explicit formula for shape-control of thin circular hyperelastic plates. With this formula, most of the 3D axisymmetric configurations can be generated by selecting proper growth functions in the plates. The results obtained in the current work can not only reveal the mechanisms of growth-induced deformations of soft material samples, but also have wide potential applications in the manufacture of intelligent soft devices.

Growth induced instabilities in a circular hyperelastic plate

In this work, we have explored growth-induced mechanical instability in an isotropic circular hyperelastic plate. Consistent two-dimensional governing equations for a plate under a general finite strain are derived using a variational approach. The derived plate equations are solved using the compound matrix method for two cases of axisymmetric growth conditions – purely radial, and combined radial and circumferential growth. The effect of growth on the buckling behaviour of the plate (in particular, the critical growth factor and the associated buckling mode shapes) is investigated for different thickness values. These results are applicable to model growth induced deformation in planar soft tissues such as skin.

Nonlinear Bending Analysis of Hyperelastic Plates Using FSDT and Meshless Collocation Method Based on Radial Basis Function

This paper investigates the nonlinear bending analysis of a hyperelastic plate via neo-Hookean strain energy function. The first-order shear deformation plate theory (FSDPT) is used for the formulation of the field variables. Also, the nonlinear Lagrangian strains are considered via the right Cauchy–Green tensor. The governing equations and nonlinear boundary conditions are derived using Euler–Lagrange relations. The meshless collocation method based on radial basis function is used to discretize the governing equations of the hyperelastic plate. Square and circular plates are studied to evaluate the accuracy of the meshless collocation method based on thin-plate spline (TPS) and multiquadric (MQ) and logarithmic thin-plate spline (LTPS) radial basis function. Also, the results of the meshless method are compared to those of the finite element method. In some cases, the meshless method is more efficient than the finite element method due to no meshing. The linear and nonlinear natural boundary conditions are directly imposed on the stiffness matrix and are compared to each other. The maximum differences between linear and nonlinear natural boundary conditions are 1.43%. The von-Mises stress using meshless collocation method based on TPS basis function is compared to those of the finite element method.

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

In this work, we study the growth-induced bending deformations of multi-layered hyperelastic plates. First, under the assumption of plane strain, we formulate the 3D governing system for a multi-layered hyperelastic (neo-Hookean) plate, which incorporates the growth parameters in the different layers. Based on the governing system and by adopting a series expansion–truncation approach, we derive a plate equation system. In the case of traction-free boundary conditions, we solve the plate equation system explicitly and obtain some simple analytical results, which can be used to describe the deformations of the multi-layered plates induced by growth. We verify the accuracy and efficiency of the analytical results through comparisons with some numerical and experimental results. We also provide a comparison between the analytical results obtained for single- and multi-layered hyperelastic plates. Furthermore, we analyze the properties of residual stresses in the multi-layered plates. The plate equation system and the analytical results obtained here have wide potential applications in the design of intelligent soft devices.

Stress-free bending of a neo-Hookean plate induced by growth: Exact solution and experiments

We study the growth-induced large deformations of a thin hyperelastic plate through both theoretical analyses and experimental tests. In particular, we investigate the stress-free states of the plate during the growth process systematically. First, for the biaxial growth case and with the assumption of plane-strain deformation, we formulate the governing PDE system of the plate model, where the traction-free boundary conditions are adopted. By considering the stress-free states of the plate, a constraint condition on the growth functions can be derived. Then, by choosing some specified growth functions, we obtain the exact solution to the governing system, which represents the stress-free bending deformation of the plate. With the obtained solution, the effects of the different growth parameters on the current configuration of the plate can be revealed. To show the prediction power of the exact solution, we conduct some experiments on the swellings of thin hydrogel samples in water. It is found that the deformation styles of the hydrogel samples are consistent with those described by the exact solution. The analytical results obtained in this paper is helpful for understanding the mechanical behaviours of hyperelastic plates in free growth, which would also be useful for the design of soft devices. Besides that, the problem considered in this paper can serve as a benchmark example for testing the correctness of approximate plate models and numerical schemes in growth theory.