Strain Energy Density Function Research Articles

Data-driven continuum damage mechanics with built-in physics

Soft materials such as rubbers and soft tissues often undergo large deformations and experience damage degradation that impairs their function. This energy dissipation mechanism can be described in a thermodynamically consistent framework known as continuum damage mechanics. Recently, data-driven methods have been developed to capture complex material behaviors with unmatched accuracy due to the high flexibility of deep learning architectures. Initial efforts focused on hyperelastic materials, and recent advances now offer the ability to satisfy physics constraints such as polyconvexity of the strain energy density function by default. However, modeling inelastic behavior with deep learning architectures and built-in physics has remained challenging. Here we show that neural ordinary differential equations (NODEs), which we used previously to model arbitrary hyperelastic materials with automatic polyconvexity, can be extended to model energy dissipation in a thermodynamically consistent way by introducing an inelastic potential: a monotonic yield function. We demonstrate the inherent flexibility of our network architecture in terms of different damage models proposed in the literature. Our results suggest that our NODEs re-discover the true damage function from synthetic stress-deformation history data. In addition, they can accurately characterize experimental skin and subcutaneous tissue data.

Visco-Hyperelastic Constitutive Modeling for High-Damping Rubber Materials During Combined Quasi-Static Compression–Cyclic Shear Deformation Process

High-damping rubber materials utilized in high-damping rubber isolation bearings are frequently subjected to multiple deformations during the occurrence of earthquakes. Typically, large combined compression–shear deformations of the material could potentially cause compressive shear damage to rubber bearings. During this process, visco-hyperelastic properties of rubber materials will greatly change, which would significantly impact the seismic performance of rubber bearings. Thus, to give out a deep insight into their variations, it is necessary and urgent to develop a high-performance numerical method to investigate this process. This paper proposed a visco-hyperelastic constitutive modeling approach for high-damping rubber materials based on the experimental assessment of combined quasi-static compression–cyclic shear deformation process. Within the thermodynamic framework, the Clausius–Duhem inequality associated with the intrinsic dissipation of the material was firstly derived in accordance with the Lagrangian formulism. Then, stress–strain relations were obtained upon considering the occurrence of entropy production due to viscous dissipation. In the model, Stumpf–Marczak strain energy density function, which satisfies the Baker–Ericksen (B–E) inequality, was harnessed to describe the hyperelasticity of the material. By introducing higher orders of strain and strain rates and taking their couplings into account, a generalized viscous dissipation potential was proposed to capture nonlinear strain and strain rate-sensitivity effects of the material. To identify constitutive parameters, the deformation gradient was particularized for the combined quasi-static compression–cyclic shear deformation process. And, an inverse identification procedure was carried out at different levels of compression stress. The prediction results revealed that the proposed model exhibits remarkable prediction ability and adaptivity for different rubber materials during this process. Several new insights were highlighted on the variations of visco-hyperelastic characteristics of high-damping rubber materials with respect to the compression stress. The accuracy of the model was further validated by design parameters including initial shear modulus, secant shear modulus and equivalent viscous damping factor. This work could provide a fundamental guideline for the optimization and reliability analysis of high-damping rubber isolation bearings used in the field of seismic engineering.

Computational homogenization of the elastic properties of polycrystalline fcc metals within Mindlin’s second strain-gradient theory

The lack of interest in Mindlin’s second strain-gradient theory, despite its success in characterizing various phenomena for which there is no explanation in the classical elasticity, is predominantly due to the numerous higher-order elastic constants involved therein. As an attempt to overcome this shortcoming, a computational homogenization method for determining such elastic constants of polycrystalline face-centered cubic (fcc) metals is proposed in the present study. In this homogenization method, using the Voigt-type averaging scheme, a polycrystalline material is modeled by an isotropic aggregate of randomly oriented single crystals without accounting for the grain size effect and the complexities due to grain boundaries and junctions. Subsequently, analytical expressions for the strain energy due to certain modes of loading are determined, and molecular simulations of a few fcc metals under such modes of loading are performed. Then, by fitting the corresponding analytical expressions to the results obtained from these simulations, the effective elastic constants and consequently the effective characteristic lengths of the fcc metals in their polycrystalline form are determined. Moreover, the free-surface-induced reconstruction in a thin layer of a solid is addressed both analytically and by molecular simulations, as a result of which the effective moduli of cohesion of the fcc metals in their polycrystalline form are calculated. In addition, the complete set of conditions for the positive-definiteness of the strain-energy-density function of isotropic materials within the adopted theory is derived, and subsequently a discussion of whether or not the numerical values obtained for the effective elastic moduli of the polycrystalline fcc metals satisfy such conditions is provided.

Biomechanical evaluation of a sheep tracheal scaffold.

Tissue engineering is a set of techniques for producing or reconstructing tissue that primarily aims to restore or improve the function of tissues in the human body. The aim of the present study was to evaluate the mechanical and histological characteristics of decellularized tracheal scaffolds prepared in comparison with fresh trachea for use in tracheal repair. In order to prepare the scaffold, sheep's trachea was prepared and after cleaning the waste tissues, they were decellularized. Then decellularized scaffolds were evaluated histologically and laboratory and numerical study of the nonlinear mechanical behavior of tracheal tissue and scaffold and their comparison. Examining the results of histological evaluations showed that the decellularization of the scaffolds was completely done. These results were confirmed by hematoxylin-eosin staining. Also, the exact hyperelastic properties of tracheal tissue and scaffold were used in biomechanical models, and according to the presented results, the five-term Mooney-Rivlin strain energy density function became a suitable behavioral model for modeling the hyperelastic behavior of trachea and scaffold. In total, the results of this research showed that the scaffolds obtained from decellularization by preserving the main compositions of the desired tissue can be a suitable platform for investigating cell behaviors.

Experimental and numerical investigations on the acoustoelastic effect in hyperelastic waveguides

Guided ultrasonic wave based structural health monitoring has been of interest over decades. However, the influence of prestress states on the propagation of Lamb waves in thin-walled structures is not fully covered yet. So far experimental work presented in the literature only focuses on a few individual frequencies, which does not allow a comprehensive verification of the numerous numerical investigations. Furthermore, most work is based on the strain-energy density function by Murnaghan. To validate the common modeling approach and to investigate the suitability of other nonlinear strain-energy density functions, an extensive experimental and numerical investigation covering a large frequency range is presented here. The numerical simulation comprises the use of the Neo-Hooke as well as the Murnaghan material model. It is found that these two material models show qualitatively similar results. Furthermore, the comparison with the experimental results reveals that the Neo-Hooke material model reproduces the effect of prestress on the difference in the Lamb wave phase velocity very well in most cases. For the [Formula: see text] wave mode at higher frequencies, however, the sign of this difference is only correctly predicted by the Murnaghan model. In contrast to this, the Murnaghan material model fails to predict the sign change for the [Formula: see text] wave mode.

Micropolar theory of solid subjected to initial stress

The basic of the initial stress in an elastic continuum is revisited, then this investigation studies micropolar elastic solid subjected to initial stress using the classical method. The strain energy density function was developed to derive the constitutive relations and field equations when the influence of initial stress is taken into account. The case of linear-homogeneous isotropic and anisotropic condition of field and constitutive relations of the solid is obtained.

A novel approach for generalized Green-Naghdi-type electro-magneto-thermo-hyperelasticity wave propagation and reflection investigations in near-incompressible layers under shock loads

A novel nonlinear coupled finite-strain electro-magneto-thermo-hyperelasticity (EMTHE) model is developed for the first- and second-sound wave propagation and reflection investigations in media that are exposed to electromagnetic fields and mechanical shocks. In contrast to the previous studies which studied behaviors of the finite-strain elastic or incompressible materials with small deformations, the current research proposes a much more complicated but much more accurate novel practical model for wave propagation and reflection analyses in near-incompressible finite-strain materials. Furthermore, to evaluate the effects of the electro-magneto-thermomechanical coupling, the strain energy density function of the hyperelastic material is expanded in a new way. The governing equations are obtained according to a nonlinear version of the Helmholtz free energy. The energy equations comprise the first- and second-order time rates of the temperature to enable the modeling of the finite-speed heat transfer; i.e., the establishment of a second-sound model. A nonlinear iterative finite element solution algorithm is proposed and implemented for the resulting coupled time-dependent generalized electro-magneto-thermo-hyperelasticity equations. The results show significant differences between the predicted wave propagation and reflection characteristics and behaviors of the near-incompressible and incompressible finite-strain models.

Experimental and numerical evaluation of a rubber seal in a vacuum suction pad for an automatic mooring system

The utilization of automatic mooring systems is under extensive interest with the growing technological demands for autonomous ships and smart ports. A vacuum suction pad with a rubber seal, which endures external loads to the moored ship, such as mooring forces, is a critical element in an automatic mooring system. To develop a high-performance automatic mooring system, a vacuum suction pad and a rubber seal need to be thoroughly designed, manufactured, and evaluated. This work demonstrates a protocol for evaluating the performance of a vacuum suction pad through both simulation and experimental testing. Uniaxial tensile testing was conducted to understand the mechanical behavior of a rubber seal. Stabilized stress-strain curves were utilized to find an optimal strain energy density function model and to extract material parameters for 3D finite element method (FEM) simulations. The FEM simulations were conducted to calculate strain distributions, contact status and the maximum load capacity, and along with the FEM simulation results, experimental evaluations of the vacuum suction pad were designed and conducted against static and cyclic loads. Based on the simulation and experimental results, we can conclude that the vacuum suction pad can maintain the stable suction at least up to 25 kN suitable for the use in automatic mooring systems.

On the origin of Sanchez-Lacombe equation of state theory in hydrostatic strain energy model for rubber-like materials

To capture the volume deformation of rubber-like materials, the strain energy density (SED) function of a compressible hyperelastic model is formulated as the hydrostatic/liquid-like term from interchain interaction changes plus the compressible elastic term owing to elastic deformation of crosslinking network. Notably, the former term dominates volume responses. Although the physics of the hydrostatic term is relatively clear, to our knowledge, few physically-based models are available. However, in our previous work (Liu and Lu, 2023), we constructed a physically-based hydrostatic SED function inspired by the Flory-Orwoll-Vrij equation of state (EOS) theory for pure polymer fluids, and proposed the general strategy for developing hydrostatic functions according to the EOS theories. Sanchez-Lacombe EOS theory (note: a type of lattice-fluid EOS theory) inspired this work. We assume: (i) a rubber-like material consists of polymer segments occupying lattice sites and unoccupied/vacant lattice sites, and the compressibility of the material corresponds to changes in vacant sites fraction; (ii) the hydrostatic strain energy is associated with the interchain interaction energy change and the mixing entropy change between network and vacant sites. According to the Helmholtz free energy in Sanchez-Lacombe EOS theory and limiting our attention to the isothermal condition, another physically-based hydrostatic SED function is constructed; a specific compressible hyperelastic model is provided by further combining compressible 8-chain elastic SED function. The basic framework of this compressible model is inherently consistent with the Flory-Rehner framework for swollen elastomers. Our model provides good predictions for volume deformation data of ten different rubber-like materials in hydrostatic compression (HC), uniaxial tension (UT), and constrained uniaxial compression (CUC). The proposed model reveals that HC, CUC, or uniaxial compression correspond to the entropy-decreasing process and they all are controlled by entropy changes; while for UT, equibiaxial tension, or pure shear, the entropy first increases and then decreases, and the interchain interaction energy and entropy changes control the responses for initial small stretches and remaining large stretches, respectively. This study aims to provide a new physical insight and a valid physically-based hydrostatic SED function for the compressibility of rubber-like materials.

Neural Metamaterial Networks for Nonlinear Material Design

Nonlinear metamaterials with tailored mechanical properties have applications in engineering, medicine, robotics, and beyond. While modeling their macromechanical behavior is challenging in itself, finding structure parameters that lead to ideal approximation of high-level performance goals is a challenging task. In this work, we propose Neural Metamaterial Networks (NMN)---smooth neural representations that encode the nonlinear mechanics of entire metamaterial families. Given structure parameters as input, NMN return continuously differentiable strain energy density functions, thus guaranteeing conservative forces by construction. Though trained on simulation data, NMN do not inherit the discontinuities resulting from topo-logical changes in finite element meshes. They instead provide a smooth map from parameter to performance space that is fully differentiable and thus well-suited for gradient-based optimization. On this basis, we formulate inverse material design as a nonlinear programming problem that leverages neural networks for both objective functions and constraints. We use this approach to automatically design materials with desired strain-stress curves, prescribed directional stiffness and Poisson ratio profiles. We furthermore conduct ablation studies on network nonlinearities and show the advantages of our approach compared to native-scale optimization.

Soft Magnetic Skin's Deformation Analysis for Tactile Perception

Soft magnetic skins have been widely adopted for tactile perception due to their high accuracy and simple wiring advantages. However, the perceptual properties of magnetic skins are limited by information mapping relationships with weak interpretation. To overcome existing limitations, dynamic Young's modulus (DYM) is proposed in this paper based on strain energy density function to precisely describe the compression stiffness of magnetic skins. Furthermore, a highly interpretable and broadly applicable method is derived using DYM to analyze a cylindrical magnetic skin's deformation process as the skin deformed under external mechanic load. Extensive experiments in simulated and real situations with different deformations are carried out to verify the proposed method. Experimental results demonstrate that 0.14% and 0.43% relative errors in simulated and real environments, respectively, can be reached. Moreover, the proposed method can achieve minimum errors in almost all situations than data-driven or state-of-the-art analysis approaches. And the generality of the proposed method is validated by experiments conducted on skins with two different shapes. These promising results indicate the potential of the proposed method in establishing practical information mapping relationships for magnetic skins, probably addressing the significant challenges for magnetic skin application in complex scenarios.

Study on deformation theory and resonance frequency of circular-plane PVC actuator

The polyvinyl chloride (PVC) gel actuator has the advantages of low mass density, large actuation strain, good compliance, and simple fabrication. It deforms when excited by a voltage, and the deformation amplitude of the actuator is higher when an alternating current voltage of a specific frequency causes the actuator to resonate. In this paper, a circular-plane actuator based on PVC gel film and flexible electrodes was developed. Then, a theoretical model is established based on the Yeoh strain energy density function model and the mechanical analysis of deformation region-boundary constraints, the key parameters affecting the deformation are obtained. Subsequently, finite element simulation tests were done on the basis of this model, and the values of uniaxial stretching were highly consistent with the previous actual experimental results, which verified the reasonableness of the model and simulation method. Then, we focus on the analysis of the resonance frequencies of PVC actuators with different plasticizer content, electrode diameter, and pre-stretching degree. Finally, we compare the differences in time-domain characteristics of PVC actuators in the resonant and non-resonant states.

Data-driven nonparametric identification of material behavior based on physics-informed neural network with full-field data

A Physics-Informed Neural Network (PINN) model is developed to extract material behavior from full-field displacement data. The PINN model consists of independent networks describing mechanical fields such as displacement, deformation gradient, stress, and strain energy density function. The displacement and deformation gradient networks learn from the deformation data of a specimen, while the stress and material networks predict the stress distribution within the specimen and learn the behavior of the target material, respectively. Each mechanical field should satisfy the physics laws of momentum balance, compatibility, constitutive relations, and boundary conditions. The constraints are wrapped up in the loss function, and the weights and biases of each hidden layer are determined by minimizing the loss function. This study introduces a two-stage learning strategy to deal with the minimization process, which is a multi-objective optimization problem. The accuracy of the proposed method is verified for linear elastic, power-law, and incompressible hyperelastic materials. The data-driven identification method successfully identifies the response of target materials in uniaxial tension, simple shear, and tension-shear-coupled loading cases. The robustness of the proposed method is verified on full-field data with noise and missing points. Data-driven identification can handle full-field data with missing points and maintain a reasonable accuracy level as the noise amplitude increases. Furthermore, data-driven identification results of an incompressible hyperelastic material are compared with conventional parametric identification results. The proposed method has advantages over the conventional method as it requires no predefined model. The results indicate that the data-driven identification method can be applied to new materials without a developed constitutive model or materials with manufacturing uncertainty.

Analysis of transversely isotropic compressible and nearly-incompressible soft material structures by high order unified finite elements

This work proposes high-order beam (1D) and plate (2D) finite element models for the large strain analysis of compressible and incompressible transversely isotropic hyperelastic media, defined within the Carrera Unified Formulation (CUF) framework. The strain energy density function adopted in fiber-reinforced hyperelastic materials modeling is presented and expressed in terms of invariants and pseudo-invariants of the right Cauchy-Green strain tensor. The explicit expression of the tangent elasticity tensor is derived through the assumption of coupled formulation of strain energy functions. Refined fully nonlinear beam and plate models are defined in a total Lagrangian formulation, deriving the governing equations of the nonlinear static analysis through the Principle of Virtual Displacements in terms of fundamental nuclei, in resulting expressions of internal and external forces vectors, and tangent stiffness matrix independent of kinematic models and approximation theories adopted. The iterative Newton-Raphson linearization scheme coupled with the arc-length constraint is adopted to obtain actual numerical solutions. Different benchmark analyses in hyperelasticity are performed to assess the capabilities of our proposed model, analyzing the three-dimensional stress field for moderate to large strain states and comparing actual numerical results with exact closed-form solutions or results available in the literature, demonstrating the capabilities and reliability of CUF models in the analysis of fiber-reinforced soft materials and structures.

Controllable Deformations of Unconstrained Ideal Nematic Elastomers

We establish that, for ideal unconstrained uniaxial nematic elastomers described by a homogeneous isotropic strain-energy density function, the only smooth deformations that can be controlled by the application of surface tractions only and are universal in the sense that they are independent of the strain-energy density are those for which the deformation gradient is constant and the liquid crystal director is either aligned uniformly or oriented randomly in Cartesian coordinates. This result generalizes the classical Ericksen’s theorem for nonlinear homogeneous isotropic hyperelastic materials. While Ericksen’s theorem is directly applicable to liquid crystal elastomers in an isotropic phase where the director is oriented randomly, in a nematic phase, the constitutive strain-energy density must account also for the liquid crystal orientation which leads to significant differences in the analysis compared to the purely elastic counterpart.

Data-driven hyperelasticity, Part II: A canonical framework for anisotropic soft biological tissues

In this work, we present a novel anisotropic data-driven hyperelasticity framework for the constitutive modeling of soft biological tissues that allows direct incorporation of experimental data into the constitutive model, without requirement of a predetermined mathematical formula for the strain–energy density function. The data-driven framework is constructed through a dispersion-type anisotropic formulation based on a generalized structure tensor in the sense of Holzapfel et al. (2015) that take into account in- and out of plane dispersion. The partial derivatives of the strain energy density functions are replaced with appropriate B-spline interpolations where the control points are calibrated against experimental data obtained from uniaxial tension, triaxial shear, and (equi)biaxial tension deformations. The model calibration phase incorporates the normalization condition and the polyconvexity condition is enforced through the control points of the B-splines in order to ensure a stable constitutive response that allows unique solution in finite element analysis. The predictive capabilities of the proposed model are shown against linea alba, rectus sheath, aneurysmal abdominal aorta, and myocardium tissues. On the numerical side, the stress and moduli expressions of the model are derived and implemented into the finite element method. The performance of the model is demonstrated through representative boundary value problems.

Remarks on bifurcation of an inflated and extended swellable isotropic tube

We consider a tubular membrane whose mechanical behavior is defined by a strain energy density function that combines the neo-Hookean model with the Demiray model. Both, the neo-Hookean and the Demiray models are isotropic material models, which found their application in the modeling of the mechanical behavior of biological soft tissue. This tubular membrane is subjected to an inner pressure, an axial stretch, and a change in the material volume due to swelling or deswelling. The interplay between the cylinder geometry and the loading conditions and different instability modes, namely, bulging bifurcation, bending bifurcation, and prismatic bifurcation, are studied. It is shown that a change in the material volume has a strong effect on the occurrence of these bifurcation modes because a change in the material volume may stabilize the cylinder against a particular bifurcation mode and may trigger another bifurcation mode at the same time. Despite the restriction to isotropic material behavior, this article shows that the material response to pressure, axial stretch, and material volume change is quite complex.

Numerical Analysis and Validation of Characterization of Polydimethylsiloxane Using Hyper-elastic Constitutive Models

The most researched elastomer in recent years is polydimethylsiloxane (PDMS), which has several uses in various engineering industries. One of the PDMS’s key characteristics is its hyper-elasticity nature, which enables the production of sensors, flexible electrical circuits, transducers, and antennas. This study used the hyper-elastic constitutive models to predict the mechanical behavior of incompressible, isotropic, and hyper-elastic material PDMS under uniaxial tension. These models are curve-fitting tools that consist of strain energy density and stress functions. To pursue the analysis, a new formulation of PDMS substrate was proposed, and a tensile test was performed to evaluate its stress-strain behavior. The experimental data was implemented on various hyper-elastic models using Abaqus, like Mooney-Rivlin, Yeoh, Ogden, and reduced polynomial models. The goodness of fit of every model was evaluated by calculating R2 values. Consequently, among these models, the reduced polynomial model with 6 material constants possessed the highest R2 value (0.9936) and was considered the best-fit model among the other models. Furthermore, the material constants of this model were applied to the 3D dumbbell-shaped model of PDMS in Abaqus for its validation. The boundary conditions were applied on the model similar to the experimental setup, as 33 mm displacement on one end and the other was fixed with all DOF. For mesh quality and mesh sensitivity of the material, various mesh sizes with the linear formulation (C3D8RH) were utilized, and the best mesh size was selected to evaluate very close results with the experimental.

New insight into large deformation analysis of stretch-based and invariant-based rubber-like hyperelastic elastomers

A nonlinear finite element formulation for large deformation analysis of hyperelastic membranes in Total Lagrangian framework is developed. In many cases, it is convenient to utilize the invariant-based strain energy density function to analyze the structures when employing the finite element method. However, in this contribution, both invariant-based and stretch-based formulations are developed. Formulations are completely general and not restricted to axisymmetric loading and geometry of membrane. Furthermore, initially flat and non-flat membrane can be investigated by this method. Incompressible as well as compressible materials can be modeled. A method for imposing the plane stress condition to the formulation for compressible materials is presented. Moreover, isotropic and anisotropic materials can be considered. Some problems are analyzed and results compared to the available ones in the literature. The comparison between solution based on invariant-based method and stretch-based method with exact fourth-order elasticity tensor obtained in this work demonstrates that the stretch-based method derived in this paper works very well. The presented examples demonstrate that the developed formulations have no restriction or simplification with respect to the modeled object geometry and inherent anisotropy.

An experimentally validated anisotropic visco-hyperelastic constitutive model for knitted fabric-reinforced elastomer composites

A new anisotropic visco-hyperelastic constitutive model has been developed to predict the time-dependent deformation response of flexible, knitted fabric-reinforced elastomer composites. These composites exhibit a time-dependent response due to the viscoelastic behavior of the elastomer, while the anisotropic reinforcement effect is attributed to the knitted fabrics. The proposed modeling approach features an invariant-based strain energy density function that incorporates the architecture of knitted fabrics and the hyper-viscoelastic behavior of elastomers. In this study, the elastomer is considered as a viscous neo-Hookean material, and the knitted fabrics are modeled as cords with negligible stiffness in bending. The proposed material model is implemented in commercial Finite Element Analysis (FEA) software, Abaqus (version 2017), through a user-defined material subroutine, UMAT. This approach enables the determination of the constitutive behavior of the composite based on the constituent elastomer, fiber properties, and fabric architectures. The proposed model is verified through a discrete 3D FEA model, in which the elastomer and fabric reinforcement are modeled explicitly. The approach is further validated through physical testing including uniaxial tension, torsion, and three-point bending. Since the proposed strain energy-based constitutive model can predict the deformation response of knitted fabric-reinforced elastomer composites based on the fabric architecture, it can be employed as an efficient modeling tool to aid the design of knitting pattern for target mechanical properties.