Nonlinear Large Deformation Research Articles

Isogeometric analysis of adhesion between visco-hyperelastic material based on modified exponential cohesive zone model

This work suggests a reliable contact algorithm for simulating adhesion between soft bodies based on the isogeometric analysis. To accurately characterize adhesion-contact regional effects, a modified exponential cohesive zone model is proposed by incorporating a real contact area influence factor into the original exponential cohesive zone model. Considering the nonlinear large-deformation effect on soft bodies, numerical implementations of isogeometric analysis and bi-potential contact algorithm are proceeded, which provide a continuous normal field of contact interface and precise determination of contact forces. This comprehensive framework is implemented on our in-house isogeometric computing platform, and depicting the ability to recover from large deformations as well as time-dependent mechanical properties of soft bodies through the visco-hyperelastic model. Numerical examples encompassing debonding scenarios involving edges, centers, as well as smooth and rough surfaces, demonstrate the validity on present algorithm with adhesive contact mechanism. The adhesive effects on soft bodies are investigated detailedly via the influences on material viscosity and surface roughness. It has a prospect on providing a valuable reference for future studies especially on analyzing the interfacial adhesion behaviors of biomimetic structures.

Nonlinear Isogeometric Analyses of Instabilities in Thin Elastic Shells Exhibiting Combined Bending and Stretching

Abstract Thin structures like shells are highly susceptible to mechanical instabilities. They can undergo large nonlinear elastic deformations while exhibiting combined bending and stretching modes of deformation. Here, we propose a new displacement-based finite element (FE) formulation based on the isogeometric discretization of Koiter’s nonlinear shell theory and use of the method of dynamic relaxation (DR) to solve equilibrium configurations of problems involving fine-scale and hierarchical wrinkling and buckling. No imperfection seeding is required to trigger instabilities. The use of the NURBS basis provides a rotation-free, conforming, higher-order spatial continuity such that curvatures and membrane strains can be computed directly from interpolating the position vectors of the control points using spatial finite differences. The pseudo-dissipative FE dynamical system is updated through explicit time integration and made scalable for parallel computing using a message passing interface (MPI). The proposed FE method is successfully benchmarked against several numerical and experimental results.

Magnetomechanical Behaviors of Hard-Magnetic Elastomer Membranes Placed in Uniform Magnetic Field.

This paper aims to develop a theoretical model for a viscoelastic hard-magnetic elastomer membrane (HMEM) actuated by pressure and uniform magnetic field. The HMEM is initially a flat, circular film with a fixed boundary. The HMEM undergoes nonlinear large deformations in the transverse direction. The viscoelastic behaviors are characterized by using a rheological model composed of a spring in parallel with a Maxwell unit. The governing equations for magneto-visco-hyperelastic membrane under the axisymmetric large deformation are constructed. The Zeeman energy, which is related to the magnetization of the HMEM and the magnetic flux density, is employed. The governing equations are solved by the shooting method and the improved Euler method. Several numerical examples are implemented by varying the magnitude of the pre-stretch, pressure, and applied magnetic field. Under different magnetic fields, field variables such as latitudinal stress exhibit distinct curves in the radial direction. It is observed that these varying curves intersect at a point. The position of the intersection point is independent of the applied magnetic field and only controlled by pressure and pre-stretch. On the left side of the intersection point, the field variables increase as magnetic field strength increases. However, on the other side, this trend is reversed. During viscoelastic evolution, one can find that the magnetic field can be used to modulate the instability behaviors of the HMEM. These findings may provide valuable insights into the design of the hard-magnetic elastomer membrane structures and actuators.

Edge-selective reconfiguration in polarized lattices with magnet-enabled bistability

The signature topological feature of Maxwell lattices is their polarization, which manifests as an unbalance in stiffness between opposite edges of a finite domain. The manifestation of this asymmetry is especially dramatic in the case of soft lattices undergoing large nonlinear deformation under concentrated loads, where the excess of softness at the soft edge can result in the activation of sharp indentations. This study explores how this mechanical dichotomy between edges can be tuned and possibly extremized by working with soft magneto-mechanical metamaterials. The magneto-mechanical coupling is obtained by endowing the lattice sites with permanent magnets, which activate a network of magnetic forces that can interact with – either augmenting or competing with – the elasticity of the material. Specifically, under sufficiently large deformation that macroscopically alters the equilibrium positions of the sites, the attractive forces between the magnets can trigger bistable reconfiguration mechanisms. The strength of such mechanisms depends on the landscapes of elastic reaction forces exhibited by the edges, which are different due to the polarization, and is therefore inherently edge-selective. We show that, on the soft edge, the addition of magnets simply enhances the softness of the edge. In contrast, on the stiff edge, the magnets activate snapping mechanisms that locally reconfigure the cells and produce a lattice response reminiscent of plasticity, characterized by residual deformation that persists upon unloading.

An adaptive lumped-mass dynamic model and its control application for continuum robots

Dynamic modeling for continuum robots remains challenging due to their large nonlinear deformation and the variation of dynamic parameters during movement. In this paper, a lumped-mass dynamic model (LMD) for a continuum robot is constructed including elastic and viscous parameters in the robotic joints. Then the appropriate dynamic parameters (e.g. spring and damping coefficients of the LMD) with respect to the motion status (e.g. position and velocity of the robot) are estimated using a Genetic Algorithm (GA). Based on the obtained data set, a Multi-Layer Perception (MLP) is trained to establish a direct mapping from the motion status to the dynamic parameters, so the LMD can tune its parameters in real-time when moving within the workspace, resulting an adaptive lumped-mass dynamic model (ALMD). Compared to the fixed-parameter LMD, the modeling error of the ALMD is reduced by up to 60.2 %. Finally, a feedforward controller is implemented to control a continuum robotic prototype using the presented ALMD, reducing the maximum tracking error by 67.5 %.

Topological derivatives for shell structure optimization considering crash loadcases with material nonlinearities and large deformations

The topological derivatives for shell structures in dynamic loadcases, which are often used in lightweight designs, are developed. These sensitivities consider the highly nonlinear material behavior and large deformations of the structure. The aim of the investigation is the support of an efficient topology optimization for crashloaded structures. The considered functionals can in general be described by displacements, velocities and accelerations. In particular, the semi-analytical sensitivity calculations for the internal deformation energy and for single point displacements in arbitrary directions in the shell structure are presented. These functionals cover the most important functions for the development of crash structures. Using the material derivative in combination with the adjoint method, a terminal value problem has to be solved. Depending on whether first differentiation and then discretization in the time domain is performed or first discretization and then differentiation, a separate solution scheme for the adjoint is derived. Both schemes are presented in a comparable description. For the numerical evaluation of the topological derivatives, the basic equations for large deformed shell elements are included as well as a phenomenological material interpolation, which represents the tangential material behavior resulting from plastic strain and isotropic hardening. All assumptions made for the derivation are described in detail and their influences are discussed. The elaborated equations and the procedure for the calculation of the topological derivatives are applied, discussed and checked for plausibility on two clearly defined loading conditions for the internal deformation energy and single point displacements on different locations.

A flexible design framework for lattice-based chiral mechanical metamaterials considering dynamic energy absorption

While chiral mechanical metamaterials (CMMs) are reported promising in energy absorption due to the unique chiral effect, the energy-absorbing CMMs lack effective and generalized design methodologies and corresponding structure-property relationship studies. To this end, a design framework for lattice-based CMMs was proposed, and the dynamic compressive behaviors of CMMs were systematically investigated. Firstly, based on a predefined design baseline that considered a support-free metal additive manufacturing process, a screw-theory-based assembly rule was presented, which enabled the scalable twist effects and the characterization of chiral features. Secondly, an aperiodic design process that sequentially defines joints, strut connections, and geometrical features was proposed. This framework via parameterization enables the rapid generation of geometric and finite element models that contain a large number of unit cells. It also enables the integration of joint enhancement design, bio-inspired helical design, and gradient design. Finally, by finite element analysis and experiments of uniaxial medium-strain-rate (50 s−1) compression, the effects of chirality on mechanical properties (compressive strength, yield plateau, energy absorption, etc.) during the nonlinear large-deformation responses were elucidated. Results show that a comprehensive and flexible method is presented by independently defining each rod component or joint of the lattice type metamaterials, which enables the design from chiral to achiral, from rectangular to helical, and from uniform to gradient. The bidirectional gradient CMMs design along the axial and radial directions achieves a 52.0 % specific energy absorption enhancement compared with achiral lattices, demonstrating the energy absorption advantage of CMMs, and laying the foundation for further optimization, inverse design, and engineering applications.

An effective nonlinear dynamic formulation to analyze grasping capability of soft pneumatic robotic gripper

ABSTRACT Soft pneumatic robotic grippers have found extensive applications across various engineering domains, which prompts active research due to their splendid compliance, high flexibility, and safe human-robot interaction over conventional stiff counterparts. Previously simplified rod-based models principally focused on clarifying overall large deformation and bending postures of soft grippers from static or quasi-static perspectives, whereas it is challenging to elaborate grasping characteristics of soft grippers without considering contact interaction and nonlinear large deformation behaviors. To address this, based on absolute nodal coordinate formulation (ANCF), comprehensively allowing for structural complexity, geometric, material and boundary nonlinearities, and incorporating Coulomb’ friction law with a multiple-point contact method, we put forward an effective nonlinear dynamic modeling approach for exploring grasping capability of soft gripper. Moreover, we solved the established dynamic equations using Generalized-α scheme, and conducted thorough numerical simulation analysis on a three-jaw soft pneumatic gripper (SPG) in terms of grasping configurations, displacements and contact forces. The proposed dynamic approach can accurately both describe complicated deformed configurations along with stress distribution and provide a feasible solution to simulate grasping targets, whose effectiveness and precision were analyzed theoretically and verified experimentally, which may shed new light on devising and optimizing other multifunctional SPGs.

MODELING THE DYNAMICS OF DEFORMABLE OBJECTS BASED ON VOLUMETRIC PATCHES OF FREE FORMS

A method for solving nonlinear implicit numerical integration problems based on volumetric patches of free forms for modeling deformable objects with high-speed dynamics is proposed. This method improves the accuracy, consistency and controllability of deformation modeling and animation. The method is characterized by speed, accuracy, stability and uses optimization with region decomposition. The method is well suited for modeling deformable bodies with a large time step, in a wide range of deformation dynamics. We propose decomposed optimization, an optimization method based on free-form patches with region decomposition to minimize incremental potentials at each time step. The method uses quadratic matrix decomposition to combine non-overlapping subregions. The Hessian is evaluated once at the beginning of the time step. The advantages of the method are as follows. Geometric primitives and their mathematical models are proposed, which allow the reasonable application of these primitives to solve problems of volume-oriented modeling. Such requirements are met by volumetric patches of free forms based on analytical perturbation functions relative to the base triangles. The decomposed Hessian is constructed for each subregion and calculated using a set of vertices taken from a complete non-decomposed grid. The weights add the missing second-order Hessian data to the vertices of the subregions from the neighbors along the decomposition boundaries. Thanks to this, the descent is performed along the grid coordinates. There is no need to add gradients. During the descent, the gradient is determined. The Hessians under the region are calculated and factorized in parallel once per time step. They are used as an initializer at each iteration. Then the results are mixed together. This ensures stable and continuous high-quality modeling. An automated and reliable optimization method adapted for modeling nonlinear materials, high-speed dynamics and large deformations is proposed.

A parallel multi-resolution Smoothed Particle Hydrodynamics model with local time stepping

Smoothed Particle Hydrodynamics (SPH) model has exhibited remarkable efficacy in addressing strongly nonlinear large deformation problems. However, the expensive computational cost could limit its application. To address this, we propose a novel parallel multi-resolution SPH model with a local time step strategy. This model integrates the multi-resolution model with a Message Passing Interface-based parallelization framework, wherein subdomains discretize the computational domain at varying resolutions. Meanwhile, a local fourth-order Runge-Kutta time integration scheme is introduced, enabling different subdomains to use different time steps. Subdomains of varying resolutions are devoid of overlapping regions, and changes in particle resolution at interfaces are achieved through a dynamic strategy involving particle merging and splitting. Besides that, we conducted a comparative analysis of different SPH discretization formats concerning their impact on the accuracy of multi-resolution particle discretization. Finally, several numerical tests were conducted to validate the accuracy and efficiency of the present multi-resolution SPH model.

Formula of Cylindrical Spring Stiffness for Nonlinear Large Deformation and Its FEM Verification

Formula of Cylindrical Spring Stiffness for Nonlinear Large Deformation and Its FEM Verification

Effect of Resin Bleed Out on Compaction Behavior of the Fiber Tow Gap Region during Automated Fiber Placement Manufacturing.

Automated fiber placement is a state-of-the-art manufacturing method which allows for precise control over layup design. However, AFP results in irregular morphology due to fiber tow deposition induced features such as tow gaps and overlaps. Factors such as the squeeze flow and resin bleed out, combined with large non-linear deformation, lead to morphological variability. To understand these complex interacting phenomena, a coupled multiphysics finite element framework was developed to simulate the compaction behavior around fiber tow gap regions, which consists of coupled chemo-rheological and flow-compaction analysis. The compaction analysis incorporated a visco-hyperelastic constitutive model with anisotropic tensorial prepreg viscosity, which depends on the resin degree of cure and local fiber orientation and volume fraction. The proposed methodology was validated using the compaction of unidirectional tows and layup with a fiber tow gap. The proposed approach considered the effect of resin bleed out into the gap region, leading to the formation of a resin-rich pocket with a complex non-uniform morphology.

Effects of Mode Mixity and Loading Rate on Fracture Behavior of Cracked Thin-Walled 304L Stainless Steel Sheets with Large Non-Linear Plastic Deformation.

This study investigates the mixed-mode I/II fracture behavior of O-notched diagonally loaded square plate (DLSP) samples containing an edge crack within the O-notch. This investigation aims to explore the combined effects of loading rate and mode mixity on the fracture properties of steel 304L, utilizing DLSP samples. The DLSP samples, made from strain-hardening steel 304L, were tested at three different loading rates: 1, 50, and 400 mm/min, covering five mode mixities from pure mode I to pure mode II. Additionally, tensile tests were performed on dumbbell-shaped specimens at the same loading rates to examine their influence on the material's mechanical properties. The findings revealed that stress and strain diagrams derived from the dumbbell-shaped samples were largely independent of the tested loading rates (i.e., 1-400 mm/min). Furthermore, experimental results from DLSP samples showed no significant impact of the loading rates on the maximum load values, but did indicate an increase in the ultimate displacement. In contrast to the loading rate, mode mixity exhibited a notable effect on the fracture behavior of DLSP samples. Ultimately, it was observed that the loading rate had an insignificant effect on the fracture path or trajectory of the tested DLSP samples.

SEM2: Introducing mechanics in cell and tissue modeling using coarse-grained homogeneous particle dynamics.

Modeling multiscale mechanics in shape-shifting engineered tissues, such as organoids and organs-on-chip, is both important and challenging. In fact, it is difficult to model relevant tissue-level large non-linear deformations mediated by discrete cell-level behaviors, such as migration and proliferation. One approach to solve this problem is subcellular element modeling (SEM), where ensembles of coarse-grained particles interacting via empirically defined potentials are used to model individual cells while preserving cell rheology. However, an explicit treatment of multiscale mechanics in SEM was missing. Here, we incorporated analyses and visualizations of particle level stress and strain in the open-source software SEM++ to create a new framework that we call subcellular element modeling and mechanics or SEM2. To demonstrate SEM2, we provide a detailed mechanics treatment of classical SEM simulations including single-cell creep, migration, and proliferation. We also introduce an additional force to control nuclear positioning during migration and proliferation. Finally, we show how SEM2 can be used to model proliferation in engineered cell culture platforms such as organoids and organs-on-chip. For every scenario, we present the analysis of cell emergent behaviors as offered by SEM++ and examples of stress or strain distributions that are possible with SEM2. Throughout the study, we only used first-principles literature values or parametric studies, so we left to the Discussion a qualitative comparison of our insights with recently published results. The code for SEM2 is available on GitHub at https://github.com/Synthetic-Physiology-Lab/sem2.

Modeling Dynamics of Laterally Impacted Piles in Gravel Using Erosion Method

Understanding the dynamic interaction between piles and the surrounding soil under vehicular impacts is essential for effectively designing and optimizing soil-embedded vehicle barrier systems. The complex behavior of pile–soil systems under impact loading, attributed to the soil’s nonlinear behavior and large deformation experienced by both components, presents significant simulation challenges. Popular computation techniques, such as the updated Lagrangian finite element method (UL-FEM), encounter difficulties in scenarios marked by large soil deformation, e.g., impacts involving rigid piles. While mesh-free particle and discrete element methods offer another option, their computational demands for field-scale pile–soil impact simulations are considerable. We introduce the erosion method to bridge this gap by integrating UL-FEM with an erosion algorithm for simulating large soil deformations during vehicular impacts. Validation against established physical impact tests confirmed the method’s effectiveness for flexible and rigid pile failure mechanisms. Additionally, this method was used to examine the effects of soil mesh density, soil domain sizes, and boundary conditions on the dynamic impact response of pile–soil systems. Our findings provide guidelines for optimal soil domain size, mesh density, and boundary conditions. This investigation sets the stage for improved, computationally efficient techniques for the pile–soil impact problem, leading to better pile designs for vehicular impacts.

A refined first-order shear deformation theory for large dynamic and static deflection investigations of abruptly/harmonically pressurized/blasted incompressible circular hyperelastic plates

In this paper, nonlinear large-deformation dynamic and static responses of the suddenly and harmonically pressurized or blasted compliant incompressible hyperelastic circular plates are investigated through the development of a novel enhanced circular plate theory that in contrast to the available plate theories, utilizes transverse normal strains whose order is much higher than those of the in-plane strains. For this reason, no shear correction factor is needed for the new plate theory. The deformation gradient and Green's strain tensors are first determined based on the first-order shear deformation plate theory as predictors and then, corrected by simultaneously including the transverse compliance and volume conservation concepts in the neo-Hookean framework of hyperelasticity. The equations of motion are obtained using Hamilton's principle. The resulting highly nonlinear equations that include both large deformation and constitutive nonlinearities are solved by a hybrid iterative/updating technique that utilizes a combination of the differential quadrature (DQ) spatial-discretization and Newmark's time-marching methods, after an exhaustive order analysis for the identification of the extremely small terms. The effects of the changes in geometric parameters, material properties, and loading rate and type are investigated on the dynamic lateral deflection of the plate and the approximate fundamental natural frequency. Results reveal that the contribution of the higher vibration modes in the responses is much more remarkable in the high-rate and high-frequency loadings. Furthermore, the deformed shape of the plate that indicates the superimposed effects of the higher vibration modes is plotted for successive time instants.

Analyzing the bending deformation of van der Waals-layered materials by a semi-discrete layer model

Van der Waals (vdW)-layered materials, such as graphite, exhibit unique mechanical properties owing to their structural and mechanical anisotropies. This study reports the development of a mechanical model that reproduces the characteristics of the nonlinear and reversible bending deformation of vdW-layered materials, while taking into account the microscopic mechanism of the discrete interlayer slips. The vdW-layered material was modeled as a stack of interacting discrete deformable layers (semi-discrete layer model), and the interlayer interaction was modeled using a cohesive zone model that reproduced the localized interlayer slip. Using the finite-element method, out-of-plane bending deformation analyses were performed on the cantilevers of the highly oriented pyrolytic graphite (HOPG) and MoTe2, and the validity of the model was verified by comparing it with the experimental results. The model accurately reproduced the loading and unloading behaviors in the experiments for the submicron HOPG cantilevers or the large nonlinear and reversible deformation with a hysteresis loop. Furthermore, the model reproduced well the characteristics of the bending experiments for the micro-MoTe2 cantilevers, or the intermittent decrease in stiffness during the loading process and deformation restoration during the unloading process. These results demonstrated that the designed semi-discrete layer model can be universally applied to reproduce the bending deformation characteristics of a variety of vdW-layered materials and can be employed to effectively elucidate the underlying deformation mechanisms.

Hybrid unsupervised paradigm based deformable image fusion for 4D CT lung image modality

Deformable image registration plays a critical role in various clinical applications (e.g., image fusion, atlas creation, and tumors targeting). In radiation therapy, especially in the context of fast registration of computed tomography (CT) lung image modalities, it is used to determine the geometric transformation by relating the anatomic points in two images. The main challenge lies in effectively addressing the nonlinear large and small deformation between the inspiration and expiration phases. In this work, we propose an unsupervised hybrid paradigm-based registration network (HPRN) for the registration of 4D CT lung images without relying on ground truth data. The proposed HPRN exhibits effective learning of multi-scale and multi-resolution features, leading to the computation of a more accurate Deformation Vector Field (DVF). Furthermore, we incorporate the regularization, image similarity and Jacobian determinant loss functions, which results in improving capability in dealing with complex large and small deformations. We evaluate the effectiveness of the proposed model on the publicly accessible DIRLab 4DCT lung image dataset, which shows the effectiveness of the proposed framework by achieving better Target Registration Error (2.04±1.42mm) compared to other state-of-the-art unsupervised image registration algorithms.

Modelling of a general lumped-compliance beam for compliant mechanisms

Compliant mechanisms are crucial in a wide range of applications, and straight flexure beams with uniform thickness serve as their fundamental building blocks. While techniques and methods exist to estimate the sophisticated load-displacement behavior of these beams, dealing with the nonlinear large deformation and parametric design of complex compliant mechanisms using these beams remains a significant challenge. To address this challenge, an analytical model has been developed to integrate six independent geometric parameters into a general lumped-compliance beam, known as the general lumped-compliance beam model (GLBM). This approach enables the determinate synthesis of force-displacement characteristics in compliant mechanisms that feature any two flexure beams connected in series, such as the conventional lumped-compliance beam, distributed beam, inverted beam, and folded beam. The closed-form beam constraint model (BCM) is utilized to derive the GLBM that accurately captures geometric nonlinearity and load-dependent effects. To demonstrate the effectiveness of this modeling technique, we verified five specific configurations of the GLBM using nonlinear finite element analysis (FEA). In addition, we selected two representative compliant mechanisms, a revolute joint and a bistable mechanism, for nonlinear analysis and experimental validation, which further showcases the efficacy of this proposed GLBM.

Synergistic integration of deep neural networks and finite element method with applications of nonlinear large deformation biomechanics

Patient-specific finite element analysis (FEA) holds great promise in advancing the prognosis of cardiovascular diseases by providing detailed biomechanical insights such as high-fidelity stress and deformation on a patient-specific basis. Albeit feasible, FEA that incorporates three-dimensional, complex patient-specific geometry can be time-consuming and unsuitable for time-sensitive clinical applications. To mitigate this challenge, machine learning (ML) models, e.g., deep neural networks (DNNs), have been increasingly utilized as potential alternatives to finite element method (FEM) for biomechanical analysis. So far, efforts have been made in two main directions: (1) learning the input-to-output mapping of traditional FEM solvers and replacing FEM with data-driven ML surrogate models; (2) solving equilibrium equations using physics-informed loss functions of neural networks. While these two existing strategies have shown improved performance in terms of speed or scalability, ML models have not yet provided practical advantages over traditional FEM due to generalization issues. This has led us to the question: instead of abandoning or replacing the traditional FEM framework that can reliably solve biomechanical problems, can we integrate FEM and DNNs to enhance performance?In this study, we propose a synergistic integration of DNNs and FEM to overcome their individual limitations. Using biomechanical analysis of the human aorta as the test bed, we demonstrated two novel integrative strategies in forward and inverse problems. For the forward problem, we developed DNNs with state-of-the-art architectures to predict a nodal displacement field, and this initial DNN solution was then updated by a FEM-based refinement process, yielding a fast and accurate computing framework. For the inverse problem of heterogeneous material parameter identification, our method employs DNN as a regularizer of the spatial distribution of material parameters, aiding the optimizer in locating the optimal solution. In our demonstrative examples, despite that the DNN-only forward models yielded small displacement errors in most test cases; stress errors were considerably large, and for some test cases, the peak stress errors were greater than 50%. Our DNN-FEM integration eliminated these non-negligible errors in DNN-only models and was magnitudes faster than the FEM-only approach. Additionally, compared to FEM-only inverse method with errors greater than 50%, our DNN-FEM inverse approach significantly improved the parameter identification accuracy and reduced the errors to less than 1%.