Hyperelastic Body Research Articles

Recoverability degradation of adhesion between soft matters under uniaxial cyclic bonding–debonding: Modified cohesive interface model and numerical implementation

The main objective of this work is to propose a partial recoverable cohesive interface model, coupled with a bi-potential contact algorithm, to simulate the phenomenon of adhesion recoverability degradation under cyclic bonding–debonding between hyperelastic bodies. For this end, the proposed adhesion recoverability degradation model is constructed by defining the recovery of interface damage during the rebonding when two bodies come into contact, and a degradation factor related to the number of bonding–debonding cycles is introduced into the fully recoverable adhesion model to reduce energy dissipation after multiple cycles. Recoverability degradation includes adhesive stiffness and strength degradation, which is physically described as parallel, series and mixed arrays of adhesive bonds. Then, a finite element framework coupling the adhesion recoverability degradation model and the bi-potential contact algorithm is proposed, with Mooney–Rivlin hyperelastic material used to describe the soft matters. This framework is implemented in an in-house finite element code, with numerical examples demonstrating the model’s reliability. The proposed approach could be applied to investigate interfacial adhesion effects in fields such as flexible electronics and intelligent robotics.

Calculation of 3D Shape of a Hyperelastic Body for Nonlinear Elasticity Models Using the Newton Method

This article discusses methods for calculating the deformations of objects with hyperelastic materials within the framework of the nonlinear elasticity theory. This topic is relevant due to the use of new technological materials in industry, and as a result, the emerging task of preliminary numerical calculations for operational reliability, research and modeling of deformation behavior. The first part provides a brief overview of the main provisions and formulas of the nonlinear elasticity theory. Then the spatial discretization of 3-dimensional objects and the calculation of the deformation gradient using the finite element method are considered. The article provides an algorithm for solving this problem, such as minimizing the functional of stored energy, and also considers the class of permissible deformations. Afterwards, a detailed description is given, along with pseudocode, of the method of implementing and calculating the minimization by Newton’s method. The last part demonstrates an example of the calculations performed, based on the developed software that allows for numerical experiments and computer modeling of deformations of hyperelastic bodies, and one of the capabilities of which is to perform these calculations using Newton’s method.

Shape-Programming in Hyperelasticity Through Differential Growth

This paper is concerned with the growth-driven shape-programming problem, which involves determining a growth tensor that can produce a deformation on a hyperelastic body reaching a given target shape. We consider the two cases of globally compatible growth, where the growth tensor is a deformation gradient over the undeformed domain, and the incompatible one, which discards such hypothesis. We formulate the problem within the framework of optimal control theory in hyperelasticity. The Hausdorff distance is used to quantify dissimilarities between shapes; the complexity of the actuation is incorporated in the cost functional as well. Boundary conditions and external loads are allowed in the state law, thus extending previous works where the stress-free hypothesis turns out to be essential. A rigorous mathematical analysis is then carried out to prove the well-posedness of the problem. The numerical approximation is performed using gradient-based optimisation algorithms. Our main goal in this part is to show the possibility to apply inverse techniques for the numerical approximation of this problem, which allows us to address more generic situations than those covered by analytical approaches. Several numerical experiments for beam-like and shell-type geometries illustrate the performance of the proposed numerical scheme.

Reconstruction of Cam-clay model based on multiplicative decomposition of the deformation gradient

This study formulated the Cam-clay model, an elastoplastic constitutive model for fully remolded and normally consolidated clay, using the multiplicative decomposition of the deformation gradient and a hyperelastic body to correspond to the finite deformation theory. Finite deformation elastoplastic theory in this framework has been developed for metallic materials that do not yield plastic volume change. If this theory is directly applied to a model that yields plastic volume change, the Cauchy stress cannot be determined only from the current elastic deformation. Therefore, the existing multiplicative decomposition Cam-clay models cannot accurately reproduce the experimental facts underlying the core of critical-state soil mechanics. Through thermodynamic considerations that consider the volume changes occurring in the intermediate configuration, this study determined a common stress suitable for describing the elastic and plastic components, in addition to forms of hyperelastic body and plastic flow rule. Consequently, the proposed model can draw the state boundary surface in the Cauchy stress-specific volume space, which previous models could not achieve. In addition, an implicit stress-update algorithm for the proposed model was constructed, and a tangent modulus consistent with the algorithm was derived. The proposed model demonstrated the same numerical calculation convergence as that of existing models.

An inverse method for elastic constants identification of two-layer hyperelastic bodies with suction loading

Some soft tissues of the human body and biomaterials are considered as multi-layer structures with layer-specific material properties. Identification of the material parameters of these distinct layers is challenging and has attracted considerable attention in recent decades. In this research, an inverse finite element method, which uses the data obtained from simulated suction experiments, is developed. The method employs a limited number of surface-measured data to recover the layer-specific hyperelastic material parameters of a two-layer structure. The Gauss-Newton method is employed to minimize a cost function, which is defined in terms of the difference between measured and calculated displacements of the surface points. The sensitivity analysis during the optimization process is carried out by the finite difference method. The effects of pipette diameter, noise in experimental data, the initial guess and the hyperelastic material model, on the robustness and efficiency of the inverse method are investigated.

Modeling Elastomer Compression: Exploring Ten Constitutive Equations.

This paper presents the results of research aimed at assessing the effectiveness of ten selected constitutive equations for hyperelastic bodies in numerical modeling of the first compression load cycle of a polyurethane elastomer with a hardness of 90 Sh A depending on the methodology for determining the material constants in the constitutive equations. An analysis was carried out for four variants for determining the constants in the constitutive equations. In three variants, the material constants were determined on the basis of a single material test, i.e., the most popular and available in engineering practice, the uniaxial tensile test (variant I), the biaxial tensile test (variant II) and the tensile test in a plane strain (variant III). In variant IV, the constants in the constitutive equations were determined on the basis of all three above material tests. The accuracy of the obtained results was verified experimentally. It has been shown that, in the case of variant I, the modeling results depend to the greatest extent on the type of constitutive equation used. Therefore, in this case it is very important to choose the right equation. Taking into account all the investigated constitutive equations, the second variant for determining the material constants turned out to be the most advantageous.

Multi-material inverse design of soft deformable bodies via functional optimization

Controlling the deformation of a soft body has potential applications in fields requiring precise control over the shape of the body. Areas such as medical robotics can use the shape control of soft robots to repair aneurysms in humans, deliver medicines within the body, among other applications. However, given known external loading, it is usually not possible to deform a soft body into arbitrary shapes if it is fabricated using only a single material. In this work, we propose a new physics-based method for the computational design of soft hyperelastic bodies to address this problem. The method takes as input an undeformed shape of a body, a specified external load, and a user desired final shape. It then solves an inverse problem in design using nonlinear optimization subject to physics constraints. The nonlinear program is solved using a gradient-based interior-point method. Analytical gradients are computed for efficiency. The method outputs fields of material properties which can be used to fabricate a soft body. A body fabricated to match this material field is expected to deform into a user-desired shape, given the same external loading input. Two regularizers are used to ascribe a priori characteristics of smoothness and contrast, respectively, to the spatial distribution of material fields. The performance of the method is tested on three example cases in silico.

Finite-Deformation Cam-Clay Model Considering Elastic Dilatancy and Soil Skeleton Structure

Elastoplastic constitutive equations for soils often employ the nonlinear rate-type isotropic Hooke’s law, which is based on the additive decomposition of strain rate and considers the pressure dependency in the elastic part. Hooke’s law has independent isotropic and deviatoric components, and therefore, it cannot express elastic dilatancy as confirmed in experiments. In this model, although the elastic volume change is path-independent, the shear strain exhibits a significant residual when subjected to an effective stress cycle. This study proposes a rate-type elastic constitutive equation for soil applicable to elastoplastic constitutive equations based on finite deformation theory using an objective stress rate and additive decomposition of stretching; these equations cover these shortcomings of the rate-type Hooke’s law. The proposed rate-type elasticity constitutive equation is based on the hyperelastic body in the infinitesimal deformation theory proposed by Einav and Puzrin or Houlsby et al. and has confining pressure dependence, path independence of elastic volume change, and elastic dilatancy. Further, the residual strain is considerably smaller than that of the rate-type Hooke’s law. Moreover, this study improves the elasto-plastic constitutive equation, SYS Cam-Clay model, based on the skeleton structure concept by introducing the proposed elastic constitutive equation. The basic behavior of the elastoplastic constitutive equation for clay and sand is demonstrated to show that the newly acquired elastic dilatancy improves the mean effective stress reduction behavior during shear stress unloading under undrained conditions and the stiffness recovery behavior during liquefaction, which is essential for describing cyclic mobility.

A cell-based smoothed finite element method for multi-body contact analysis within the bi-potential formulation

This paper presents a cell-based smoothed finite element method (CS-FEM) for solving contact problems within the bi-potential formulation. The frictional contact and large deformation are modeled based on the CS-FEM. This method performs based on the generalized smoothed Galerkin weak-form which requires only value of shape functions. Because no isoparametric mapping is needed, the CS-FEM has an advantage when dealing with problems with large deformation which may cause mesh distortion. The contact forces and the relative velocity establish an implicit relationship within the bi-potential formulation in which no user-defined parameter is introduced. Numerical examples show the contact between elastic and hyperelastic bodies calculated by CS-FEM models with different smoothing domains. Compared with FEM, the CS-FEM produces robust and more accurate solutions.

Cushioning Performance of Hilbert Fractal Sandwich Packaging Structures under Quasi-Static Compressions

The sandwich structure of cushioning packaging has an important influence on the cushioning performance. Mathematical fractal theory is an important graphic expression. Based on Hilbert fractal theory, a new sandwich structure was designed. The generation mechanism and recurrence formula of the Hilbert fractal were expressed by Lin’s language, and the second-order Hilbert sandwich structure was constructed from thermoplastic polyurethane. The constitutive model of the hyperelastic body was established by using the finite element method. With the unit mass energy absorption as the optimization goal, the fractal sandwich structure was optimized, and the best result was obtained when the order was 2.5 and the unit layer thickness was 0.75 mm. The Hilbert sandwich structure was compared with the rice-shaped sandwich structure commonly used in industry, and the Hilbert fractal structure had better energy absorption. This has practical significance for the development and application of new cushioning packaging structures.

Unified Discrete Multisymplectic Lagrangian Formulation for Hyperelastic Solids and Barotropic Fluids

We present a geometric variational discretization of nonlinear elasticity in 2D and 3D in the Lagrangian description. A main step in our construction is the definition of discrete deformation gradients and discrete Cauchy-Green deformation tensors, which allows for the development of a general discrete geometric setting for frame indifferent isotropic hyperelastic models. The resulting discrete framework is in perfect adequacy with the multisymplectic discretization of fluids proposed earlier by the authors. Thanks to the unified discrete setting, a geometric variational discretization can be developed for the coupled dynamics of a fluid impacting and flowing on the surface of an hyperelastic body. The variational treatment allows for a natural inclusion of incompressibility and impenetrability constraints via appropriate penalty terms. We test the resulting integrators in 2D and 3D with the case of a barotropic fluid flowing on incompressible rubber-like nonlinear models.

Optimal design of planar shapes with active materials

Active materials have emerged as valuable candidates for shape morphing applications, where a body reconfiguration is achieved upon triggering its active response. Given a desired shape change, a natural question is to compare different morphing mechanisms to select the most effective one with respect to an optimality criterion. We introduce an optimal control problem to determine the active strains suitable to attain a target equilibrium shape while minimizing the complexity of the activation. Specifically, we discuss the planar morphing of active, hyperelastic bodies in the absence of external forces and exploit the notion of target metric to encompass a broad set of active materials in a unifying approach. For the case of affine shape changes, we derive explicit conditions on the body reference configuration for the optimality of homogeneous target metrics. More complex shape changes are analysed via numerical simulations to explore the impact on optimal solutions of different objective functionals inspired by features of existing materials. We show how stresses arising from incompatibilities contribute to reduce the complexity of the controls. We believe that our approach may be exploited for the optimal design of active systems and may contribute to gather insight into the morphing strategies of biological systems.

Non-isothermal viscoelastic flows with conservation laws and relaxation

We propose a system of conservation laws with relaxation source terms (i.e. balance laws) for non-isothermal viscoelastic flows of Maxwell fluids. The system is an extension of the polyconvex elastodynamics of hyperelastic bodies using additional structure variables. It is obtained by writing the Helmholtz free energy as the sum of a volumetric energy density (function of the determinant of the deformation gradient det F and the temperature [Formula: see text] like the standard perfect-gas law or Noble–Abel stiffened-gas law) plus a polyconvex strain energy density function of F, [Formula: see text] and of symmetric positive-definite structure tensors that relax at a characteristic time scale. One feature of our model is that it unifies various ideal materials ranging from hyperelastic solids to perfect fluids, encompassing fluids with memory like Maxwell fluids. We establish a strictly convex mathematical entropy to show that the system is symmetric-hyperbolic. Another feature of the proposed model is therefore the short-time existence and uniqueness of smooth solutions, which define genuinely causal viscoelastic flows with waves propagating at finite speed. In heat-conductors, we complement the system by a Maxwell–Cattaneo equation for an energy-flux variable. The system is still symmetric-hyperbolic, and smooth evolutions with finite-speed waves remain well-defined.

Identification of muscle activity in tongue motion during swallowing through medical image data

This paper presents a formulation to identify muscle activities from the variation in shapes of organs during swallowing. We assume that each organ consists of a three-dimensional hyperelastic body, and the contraction movement of the muscle is caused by a contractive inelastic stress in the organ. A function distributed in the organ domain to control the magnitude of the inelastic stress is chosen as a design variable in the same manner as the density in the topology optimization problem of density variation type. The identification problem is formulated as a problem of determining the design variable that minimizes an objective cost function defined by the squared L2 norm of the reaction force in the normal direction on the boundary when an enforced displacement to fit the varied boundary of the organ and the inelastic stress modeling the muscle activity are applied. The finite deformation problem of the hyperelastic body is analyzed using the finite element method. The direction of the muscle fiber is assumed to be the direction of the minimum principal stress obtained as the solution to the finite deformation problem. The solution to the identification problem is presented based on a scheme using the H1 gradient method for the topology optimization problem of density variation type. A numerical example using a previously developed model of the tongue is introduced to demonstrate the effectiveness of the proposed approach.

Model order reduction using Karhunen-Loève expansion in shape optimization analysis of hyperelastic body

Model order reduction using Karhunen-Loève expansion in shape optimization analysis of hyperelastic body

Shape optimization of a fish-like hyperelastic body vibrating with a swimming mode

Shape optimization of a fish-like hyperelastic body vibrating with a swimming mode

A bi-potential contact formulation of orthotropic adhesion between soft bodies

An orthotropic adhesion model is proposed based on the bi-potential method to solve adhesive contact problems with orthotropic interface properties between hyperelastic bodies. The model proposes a straightforward description of interface adhesion with orthotropic adhesion stiffness, whose components are conveniently expressed according to the local coordinate system. Based on this description, a set of extended unilateral and tangential contact laws has been formulated. Furthermore, we use an element-wise scalar parameter \(\beta \) to characterize the strength of interface adhesive bonds, and the effects of damage. Therefore, complete cycles of bonding and de-bonding of adhesive links with the account for orthotropic interface effects can be modelled. The proposed model has been tested on cases involving both tangential and unilateral contact kinematics. The test cases allowed emergence of orthotropic interface effects between elastomer bodies involving hyperelasticity. Meanwhile, the model can be implemented with minimum effort, and provides inspiration for the modelling of adhesive interface effects in areas of applications such as biomechanics.

Elasticity theory in general relativity

The general relativistic theory of elasticity is reviewed from a Lagrangian, as opposed to Eulerian, perspective. The equations of motion and stress–energy–momentum tensor for a hyperelastic body are derived from the gauge–invariant action principle first considered by DeWitt. This action is a natural extension of the action for a single relativistic particle. The central object in the Lagrangian treatment is the Landau–Lifshitz radar metric, which is the relativistic version of the right Cauchy–Green deformation tensor. We also introduce relativistic definitions of the deformation gradient, Green strain, and first and second Piola–Kirchhoff stress tensors. A gauge-fixed description of relativistic hyperelasticity is also presented, and the nonrelativistic theory is derived in the limit as the speed of light becomes infinite.

Numerical simulation of the hyperelastic material using new strain measure

The three-dimensional model of the hyperelastic body under large strains is researched. A new strain measure, based on the QR-decomposition of the deformation gradient, is discussed. It is shown that this strain measure allows for more explicit analytical connections between different parameters of the stress–strain condition then the traditional approach. An example method of introducing this measure into numerical simulations for an arbitrary three-dimensional model with linear shape functions is presented.

Model order reduction by Karhunen-Lo`eve expansion in shape optimization analysis of hyperelastic body

Model order reduction by Karhunen-Lo`eve expansion in shape optimization analysis of hyperelastic body