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International audience; Cellular materials, in particular metal or polymer foams and beam lattices, are characterized by a complex architecture where the material is concentrated in slender struts. This study is concerned with the construction of numerically efficient samplespecific digital twin for the analysis and characterization of the mechanical behavior of such meta-materials. A method to build an explicit, light, and analysis-suitable isogeometric beam model directly from digital images is proposed. More precisely, as a main contribution, the Virtual Image Correlation method is extended to the case of multiple interconnected beams with varying cross-sections. Special care is also given to the initialization for better efficiency and robustness. Synthetic and real examples of increasing complexity, i.e. curved beams, lattices, and foams, are presented to account for the performance of the developed workflow from digital images to an isogeometric twin for mechanical modelling and simulation.

In the search for more efficient and less environmentally harmful cooling technologies, the field of magnetocalorics is considered a promising alternative. To generate cooling spans, rotating permanent magnet assemblies are used to cyclically magnetize and demagnetize magnetocaloric materials, which change their temperature under the application of a magnetic field. In this work, an axial rotary permanent magnet assembly, aimed for commercialization, is computationally designed using topology and shape optimization. This is efficiently facilitated in an isogeometric analysis framework, where harmonic mortaring is applied to couple the rotating rotor-stator system of the multipatch model. Inner, outer and co-rotating assemblies are compared and optimized designs for different magnet masses are determined. These simulations are used to homogenize the magnetic flux density in the magnetocaloric material. The resulting torque is analyzed for different geometric parameters. Additionally, the influence of anisotropy in the active magnetic regenerators is studied in order to guide the magnetic flux. Different examples are analyzed and classified to find an optimal magnet assembly for magnetocaloric cooling.

A sequential nonlinear multiscale method for the simulation of elastic metamaterials subject to large deformations and instabilities is proposed. For the finite strain homogenization of cubic beam lattice unit cells, a stochastic perturbation approach is applied to induce buckling. Then, three variants of anisotropic effective constitutive models built upon artificial neural networks are trained on the homogenization data and investigated: one is hyperelastic and fulfills the material symmetry conditions by construction, while the other two are hyperelastic and elastic, respectively, and approximate the material symmetry through data augmentation based on strain energy densities and stresses. Finally, macroscopic nonlinear finite element simulations are conducted and compared to fully resolved simulations of a lattice structure. The good agreement between both approaches in tension and compression scenarios shows that the sequential multiscale approach based on anisotropic constitutive models can accurately reproduce the highly nonlinear behavior of buckling-driven 3D metamaterials at lesser computational effort.

In the present work, a machine learning based constitutive model for electro-mechanically coupled material behavior at finite deformations is proposed. Using different sets of invariants as inputs, an internal energy density is formulated as a convex neural network. In this way, the model fulfills the polyconvexity condition which ensures material stability, as well as thermodynamic consistency, objectivity, material symmetry, and growth conditions. Depending on the considered invariants, this physics-augmented machine learning model can either be applied for compressible or nearly incompressible material behavior, as well as for arbitrary material symmetry classes. The applicability and versatility of the approach is demonstrated by calibrating it on transversely isotropic data generated with an analytical potential, as well as for the effective constitutive modeling of an analytically homogenized, transversely isotropic rank-one laminate composite and a numerically homogenized cubic metamaterial. These examinations show the excellent generalization properties that physics-augmented neural networks offer also for multi-physical material modeling such as nonlinear electro-elasticity.

A computational method for optimizing the shape of the centerline curve and the spatial variation of geometric and material sizing parameters of the cross-sections of elastic, 3-dimensional beams and beam structures subject to large deformations is presented in this work. The approach is based on the concept of isogeometric analysis, i.e., the representation of geometry and the discretization of the numerical solution using spline functions. Here, mixed isogeometric collocation methods are used to discretize the geometrically exact 3D beam model. These spline representations are extended to the parameterization of the design variables, which are the initial centerline curves of the beams, as well as cross-sectional sizing properties, which may be varying along the beam axis and can be functionally graded through the cross-sections. To tailor the mechanical deformation behavior of a beam or beam structure, a nonlinear optimization problem is formulated and solved using gradient-based methods. For this purpose, all required gradients and sensitivities are derived analytically. The potential of this holistic design optimization approach is demonstrated in application to tailoring of elastic metamaterials and beam lattice structures, as well as 4D printing of multi-material laminate beams.

Mechanical metamaterials such as open- and closed-cell lattice structures, foams, composites, and so forth can often be parametrized in terms of their microstructural properties, for example, relative densities, aspect ratios, material, shape, or topological parameters. To model the effective constitutive behavior and facilitate efficient multiscale simulation, design, and optimization of such parametric metamaterials in the finite deformation regime, a machine learning-based constitutive model is presented in this work. The approach is demonstrated in application to elastic beam lattices with cubic anisotropy, which exhibit highly nonlinear effective behaviors due to microstructural instabilities and topology variations. Based on microstructure simulations, the relevant material and topology parameters of selected cubic lattice cells are determined and training data with homogenized stress-deformation responses is generated for varying parameters. Then, a parametric, hyperelastic, anisotropic constitutive model is formulated as an artificial neural network, extending a recent work of the author extending a recent work of the author, Comput Mech., 2021;67(2):653-677. The machine learning model is calibrated with the simulation data of the parametric unit cell. The authors offer public access to the simulation data through the GitHub repository https://github.com/CPShub/sim-data. For the calibration of the model, a dedicated sample weighting strategy is developed to equally consider compliant and stiff cells and deformation scenarios in the objective function. It is demonstrated that this machine learning model is able to represent and predict the effective constitutive behavior of parametric lattices well across several orders of magnitude. Furthermore, the usability of the approach is showcased by two examples for material and topology optimization of the parametric lattice cell.

We present a novel isogeometric collocation method for nonlinear dynamic analysis of three-dimensional, slender, elastic rods. The approach is based on the geometrically exact Cosserat model for rod dynamics. We formulate the governing nonlinear partial differential equations as a first-order problem in time and develop an isogeometric semi-discretization of position, orientation, velocity and angular velocity of the rod centerline as NURBS curves. Collocation then leads to a nonlinear system of first-order ordinary differential equations, which can be solved using standard time integration methods. Furthermore, our model includes viscoelastic damping and a frictional contact formulation. The computational method is validated and its practical applicability shown using several numerical applications of nonlinear rod dynamics.

Advancement of additive manufacturing is driving a need for design tools that exploit the increasing fabrication freedom. Multimaterial, three-dimensional (3D) printing allows for the fabrication of components from multiple materials with different thermal, mechanical, and “active” behavior that can be spatially arranged in 3D with a resolution on the order of tens of microns. This can be exploited to incorporate shape changing features into additively manufactured structures. 3D printing with a downstream shape change in response to an external stimulus such as temperature, humidity, or light is referred to as four-dimensional (4D) printing. In this paper, a design methodology to determine the material layout of 4D printed materials with internal, programmable strains is introduced to create active structures that undergo large deformation and assume a desired target displacement upon heat activation. A level set (LS) approach together with the extended finite element method (XFEM) is combined with density-based topology optimization to describe the evolving multimaterial design problem in the optimization process. A finite deformation hyperelastic thermomechanical model is used together with an higher-order XFEM scheme to accurately predict the behavior of nonlinear slender structures during the design evolution. Examples are presented to demonstrate the unique capabilities of the proposed framework. Numerical predictions of optimized shape-changing structures are compared to 4D printed physical specimen and good agreement is achieved. Overall, a systematic design approach for creating 4D printed active structures with geometrically nonlinear behavior is presented which yields nonintuitive material layouts and geometries to achieve target deformations of various complexities.

Straight beams, rods and trusses are common elements in structural and mechanical engineering, but recent advances in additive manufacturing now also enable efficient freeform fabrication of curved, deformable beams and beam structures, such as microstructures, metamaterials and conformal lattices. To exploit this new design freedom for applications with nonlinear mechanical behavior, we introduce an isogeometric method for shape optimization of curved 3D beams and beam structures. The geometrically exact Cosserat rod theory is used to model nonlinear 3D beams subject to large deformations and rotations. The initial and current geometry are parameterized in terms of NURBS curves describing the beam centerline and an isogeometric collocation approach is used to discretize the strong form of the balance equations. Then, a nonlinear optimization problem is formulated in order to optimize the positions of the control points of the NURBS curve that describes the beam centerline, i.e., the geometry or shape of the beam. To solve the design problem using gradient-based algorithms, we introduce semi-analytical, inconsistent analytical and fully analytical approaches for calculation of design sensitivities. The methods are numerically validated and their performance is investigated, before the applicability and versatility of our 3D beam shape optimization method is illustrated in various numerical applications, including optimization of an auxetic 3D metamaterial.

A geometrically nonlinear, shear-deformable 3D beam formulation with inelastic material behavior and its numerical discretization by a mixed isogeometric collocation method are presented. In particular, the constitutive model captures elasto-visco-plasticity with damage/softening from Mullin's effect, which applies to the modeling of metallic and polymeric materials, e.g., in additive manufacturing applications and meta-materials. The inelastic material behavior is formulated in terms of thermodynamically consistent internal variables for viscoelastic and plastic strains and isotropic and kinematic hardening variables, as well as accompanying evolution equations. A mixed isogeometric collocation method is applied for the discretization of the strong form of the quasi-static nonlinear differential equations. Thus, the displacements of the centerline curve, the cross-section orientations, and the stress resultants (forces and moments) are discretized as B-spline or NURBS curves. The internal variables are defined only locally at the collocation points, and an implicit return-mapping algorithm is employed for their time discretization. The method is verified in comparison to 1D examples as well as reference results for 3D beams. Furthermore, its applicability to the simulation of beam lattice structures subject to large deformations and instabilities is demonstrated.

In this work, we introduce KnitKit , a flexible and customizable system for the computational design and production of functional, multi-material, and three-dimensional knitted textiles. Our system greatly simplifies the knitting of 3D objects with complex, varying patterns that use multiple yarns and stitch patterns by separating the high-level design specification in terms of geometry, stitch patterns, materials or colors from the low-level, machine-specific knitting instruction generation. Starting from a triangular 3D mesh and a 2D texture that specifies knitting patterns on top of the geometry, our system generates the required machine instructions in three major steps. First, the input is processed and the KnitNet data structure is generated. This graph structure serves as an abstract interface between the high-level geometric and knitting configuration and the low-level, machine-specific knitting instructions. Second, a graph rewriting procedure is applied on the KnitNet that produces a sequence of abstract machine actions. Finally, the low-level machine instructions are generated by adapting those abstract actions to a specific machine context. We showcase the potential of this computational approach by designing and fabricating a variety of objects with complex geometries, multiple yarns, and multiple stitch patterns.

This work investigates the capabilities of anisotropic theory-based, purely data-driven and hybrid approaches to model the homogenized constitutive behavior of cubic lattice metamaterials exhibiting large deformations and buckling phenomena. The effective material behavior is assumed as hyperelastic, anisotropic and finite deformations are considered. A highly flexible analytical approach proposed by Itskov (Int J Numer Methods Eng 50(8): 1777–1799, 2001) is taken into account, which ensures material objectivity and fulfillment of the material symmetry group conditions. Then, two non-intrusive data-driven approaches are proposed, which are built upon artificial neural networks and formulated such that they also fulfill the objectivity and material symmetry conditions. Finally, a hybrid approach combing the approach of Itskov (Int J Numer Methods Eng 50(8): 1777–1799, 2001) with artificial neural networks is formulated. Here, all four models are calibrated with simulation data of the homogenization of two cubic lattice metamaterials at finite deformations. The data-driven models are able to reproduce the calibration data very well and reproduce the manifestation of lattice instabilities. Furthermore, they achieve superior accuracy over the analytical model also in additional test scenarios. The introduced hyperelastic models are formulated as general as possible, such that they can not only be used for lattice structures, but for any anisotropic hyperelastic material. Further, access to the complete simulation data is provided through the public repository https://github.com/CPShub/sim-data.

The objective of this contribution is the numerical investigation of growth-induced instabilities of an elastic film on a microstructured soft substrate. A nonlinear multiscale simulation framework is developed based on the FE2 method, and numerical results are compared against simplified analytical approaches, which are also derived. Living tissues like brain, skin, and airways are often bilayered structures, consisting of a growing film on a substrate. Their modeling is of particular interest in understanding biological phenomena such as brain development and dysfunction. While in similar studies the substrate is assumed as a homogeneous material, this contribution considers the heterogeneity of the substrate and studies the effect of microstructure on the instabilities of a growing film. The computational approach is based on the mechanical modeling of finite deformation growth using a multiplicative decomposition of the deformation gradient into elastic and growth parts. Within the nonlinear, concurrent multiscale finite element framework, on the macroscale a nonlinear eigenvalue analysis is utilized to capture the occurrence of instabilities and corresponding folding patterns. The microstructure of the substrate is considered within the large deformation regime, and various unit cell topologies and parameters are studied to investigate the influence of the microstructure of the substrate on the macroscopic instabilities. Furthermore, an analytical approach is developed based on Airy’s stress function and Hashin–Shtrikman bounds. The wavelengths and critical growth factors from the analytical solution are compared with numerical results. In addition, the folding patterns are examined for two-phase microstructures and the influence of the parameters of the unit cell on the folding pattern is studied.

Lattice structures are frequently found in nature and engineering due to their myriad attractive properties, with applications ranging from molecular to architectural scales. Lattices have also become a key concept in additive manufacturing, which enables precise fabrication of complex lattices that would not be possible otherwise. While design and simulation tools for stiff lattices are common, here we present a digital design and nonlinear simulation approach for additive manufacturing of soft lattices structures subject to large deformations and instabilities, for which applications in soft robotics, healthcare, personal protection, energy absorption, fashion and design are rapidly emerging. Our framework enables design of soft lattices with curved members conforming to freeform geometries, and with variable, gradually changing member thickness and material, allowing the local control of stiffness. We model the lattice members as 3D curved rods and using a spline-based isogeometric method that allows the efficient simulation of nonlinear, large deformation behavior of these structures directly from the CAD geometries. Furthermore, we enhance the formulation with a new joint stiffening approach, which is based on parameters derived from the actual node geometries. Simulation results are verified against experiments with soft lattices realized by PolyJet multi-material polymer 3D printing, highlighting the potential for simulation-driven, digital design and application of non-uniform and curved soft lattice structures.

This article presents a numerical framework to predict the mechanical behavior of knitted fabrics from their discrete structure at the fabric yarn level, i.e., the mesostructure, utilizing the hierarchical multiscale method. Due to the regular distribution of yarn loops in a knitted structure, the homogenization theory for periodic materials can be employed. Thus, instead of considering the whole fabric sample under loading, a significantly less computationally demanding analysis can be done on a repeated unit cell (RUC). This RUC is created based on simple structural parameters of knitted yarn loops and its fabric yarns are assumed to behave transversely isotropic. Nonlinear finite element analyses are performed to determine the stress fields in the RUC under tensile and shear loading. During this analysis, contact friction among yarns is considered as well as the periodic boundary conditions are employed. The macroscopic stresses then can be derived from the stress fields in the RUC by means of the numerical homogenization scheme. The physical fidelity of the proposed framework is shown by the good agreement between the predicted mechanical properties of knitted fabrics and corresponding experimental data.

Three-dimensionally (3D) knitted technical textiles are spreading into industrial applications, since their geometric, structural and functional performance can be tailored and optimized on fibre-, yarn- and fabric levels by customizing yarn materials, knit patterns and geometric shapes. The ability to simulate their complex mechanical behaviour is thus an essential ingredient in the development of a digital workflow for optimal design and manufacture of 3D knitted textiles. Here, we present a multi-scale modelling and simulation framework for the prediction of the nonlinear orthotropic mechanical behaviour of single jersey knitted textiles and its experimental validation. On the meso-scale, representative volume elements (RVEs) of the fabric are modelled as single, interlocked yarn loops and their mechanical deformation behaviour is homogenized using periodic boundary conditions. Yarns are modelled as nonlinear 3D beam elements and numerically discretized using an isogeometric collocation method, where a frictional contact formulation is used to model inter-yarn interactions. On the macro-scale, fabrics are modelled as membrane elements with nonlinear orthotropic material behaviour, which is parameterized by a response surface constitutive model obtained from the meso-scale homogenization. The input parameters of the yarn-level simulation, i.e., mechanical properties of yarns and geometric dimensions of yarn loops in the fabrics, are determined experimentally and subsequent meso- and macro-scale simulation results are evaluated against reference results and mechanical tests of knitted fabric samples. Good agreement between computational predictions and experimental results is achieved for samples with varying stitch values, thus validating our novel computational approach combining efficient meso-scale simulation using 3D beam modelling of yarns with numerical homogenization and nonlinear orthotropic response surface constitutive modelling on the macro-scale.

Photopolymerization-based additive manufacturing methods like stereolithography and digital light processing only allow typically the monolithic fabrication of structures made from a single material. To overcome this limitation, grayscale digital light processing has been proposed for 3D printing of functionally graded materials. Here, this concept is extended to grayscale masked stereolithography (MSLA) printing using a LED light source and a LCD photomask to control the degree of photopolymerization of a UV-curable resin by varying grayscale pixels and thus light intensities. In this scale, tailorable hyperelastic material properties and functionally graded structures for finite deformations are realized. In this paper, the dependency of the resulting material properties on the parameters of the grayscale MSLA process is investigated and a grayscale-dependent hyperelastic material model is formulated. This parametric hyperelastic material model is fitted to the experiments and validated against experimental results for uniaxial tension and uniaxial compression tests. Then, functionally graded structures with tailored mechanical properties at finite deformations are designed and fabricated using grayscale MSLA printing. The hyperelastic material model is validated with experimental results for different geometries, showing good agreement between experimental tests and numerical calculations.

Generalized continuum mechanical theories such as second gradient elasticity can consider size and localization effects, which motivates their use for multiscale modeling of periodic lattice structures and metamaterials. For this purpose, a numerical homogenization method for computing the effective second gradient constitutive models of cubic lattice metamaterials in the infinitesimal deformation regime is introduced here. Based on the modeling of lattice unit cells as shear-deformable 3D beam structures, the relationship between effective macroscopic strain and stress measures and the microscopic boundary deformations and rotations is derived. From this Hill–Mandel condition, admissible kinematic boundary conditions for the homogenization are concluded. The method is numerically verified and applied to various lattice unit cell types, where the influence of cell type, cell size and aspect ratio on the effective constitutive parameters of the linear elastic second gradient model is investigated and discussed. To facilitate their use in multiscale simulations with second gradient linear elasticity, these effective constitutive coefficients are parameterized in terms of the aspect ratio of the lattices structures.

We present a novel method for the mechanical simulation of slender, elastic, spatial rods and rod structures subject to large deformation and rotation. We develop an isogeometric collocation method for the geometrically exact, nonlinear Cosserat rod theory. The rod centerlines are represented as spatial NURBS curves and cross-section orientations are parameterized in terms of unit quaternions as 4-dimensional NURBS curves. Within the isogeometric framework, the strong forms of the equilibrium equations of forces and moments of the discretized Cosserat model are collocated, leading to an efficient method for higher-order discretizations. For rod structures consisting of multiple, connected rods we introduce a formulation with rigid, quasi- G 1 -coupling. It is based on the strong enforcement of continuity of displacement and change of cross-section orientation at interfaces. We also develop a mixed isogeometric formulation, which is based on an independent discretization of internal forces and moments and alleviates shear locking for thin rods. The novel rod simulation methods are verified by numerical convergence studies. Further computational examples include realistic applications with large deformations and rotations, as well as a large-scale rod structure with several hundreds of coupled rods and complex buckling behavior.

We present a design optimization and manufacturing approach for the creation of complex three-dimensional (3D) curved rod structures with spatially variable material distributions that exhibit active deformation behavior, enabled by the shape memory effect (SME) of 3D printed photopolymers���so-called four-dimensional (4D) printing. Our framework optimizes the cross-sectional properties of a rod structure, in particular the Young's modulus, such that under given loading conditions it can obtain one or more target shapes resulting from geometrically nonlinear deformation, from which the structure can then actively deform back to the original shape due to the SME. Our approach includes a novel algorithm to generate physical realizations from the computational design model, which allows their direct fabrication via printing of shape memory composites with voxel-level compositional control with a multi-material 3D printer. Our design and manufacture digital toolchain allows the continuous variation of multiple active materials as a route to optimize mechanical as well as active behavior of a structure, without changing the original shape of the 3D rod structure, which is not possible with a single material. We demonstrate the entire design-fabrication-test approach and illustrate its capabilities with examples including 3D characters, personalized medical applications, and complex structures that exhibit instabilities during their nonlinear deformation.

Summary Modal derivative is an approach to compute a reduced basis for model order reduction of large-scale nonlinear systems that typically stem from the discretization of partial differential equations. In this way, a complex nonlinear simulation model can be integrated into an optimization problem or the design of a controller, based on the resulting small-scale state-space model. We investigate the approximation properties of modal derivatives analytically and thus lay a theoretical foundation of their use in model order reduction, which has been missing so far. Concentrating on the application field of structural mechanics and structural dynamics, we show that the concept of modal derivatives can also be applied as nonlinear extension of the Craig–Bampton family of methods for substructuring. We furthermore generalize the approach from a pure projection scheme to a novel reduced-order modeling method that replaces all nonlinear terms by quadratic expressions in the reduced state variables. This complexity reduction leads to a frequency-preserving nonlinear quadratic state-space model. Numerical examples with carefully chosen nonlinear model problems and three-dimensional nonlinear elasticity confirm the analytical properties of the modal derivative reduction and show the potential of the proposed novel complexity reduction methods, along with the current limitations. Copyright © 2016 John Wiley & Sons, Ltd.

Soft lattice structures and beam-metamaterials made of hyperelastic, rubbery materials undergo large elastic deformations and exhibit structural instabilities in the form of micro-buckling of struts under both compression and tension. In this work, the large-deformation nonlinear elastic behaviour of beam-lattice metamaterials is investigated by micromechanical nonlinear buckling analysis. The micromechanical 3D beam finite element model uses a primary linear buckling analysis to incorporate the effect of geometric imperfections into a subsequent nonlinear post-buckling analysis. The micromechanical computational model is validated against tensile and compressive experiments on a 3D-printed sample lattice structure manufactured via multi-material jetting. For the development and calibration of macroscale continuum constitutive models for nonlinear elastic deformation of soft lattice structures at finite strains, virtual characterization tests are conducted to quantify the effective nonlinear response of representative unit cells under periodic boundary conditions. These standard tests, commonly used for hyperelastic material characterization, include uniaxial, biaxial, planar and volumetric tension and compression, as well as simple shear. It is observed that besides the well-known stretch- and bending-dominated behaviour of cellular structures, some lattice types are dominated by buckling and post-buckling response. For multiscale simulation based on nonlinear homogenization, the uniaxial standard test results are used to derive parametric hyperelastic constitutive relations for the effective constitutive behaviour of representative unit cells in terms of lattice aspect ratio. Finally, a comparative study for compressive deformation of a sample sandwich lattice structure simulated by both full-scale beam and continuum finite element models shows the feasibility and computational efficiency of the effective continuum model.

We present a nonlinear multiscale modeling and simulation framework for the mechanical design of machine-knitted textiles with functionally graded microstructures. The framework operates on the mesoscale (stitch level), where yarns intermesh into stitch patterns, and the macroscale (fabric level), where these repetitive stitch patterns are composed into a fabric. On the mesoscale, representative unit cells consisting of single interlocked yarn loops, modeled as geometrically exact, nonlinear elastic 3D beams, are homogenized to compute their effective mechanical properties. From this data, a B-Spline response surface model is generated to represent the nonlinear orthotropic constitutive behavior on the macroscale, where the fabric is modeled by a nonlinear Kirchhoff–Love shell formulation and discretized using isogeometric finite elements. These functionally graded textiles with locally varying properties can be designed and analyzed by parameterizing the stitch value, i.e., the loop length of a single jersey stitch, and the knitting direction as mesoscopic design variables of the macroscopic response surface constitutive model. To validate the multiscale simulation and design approach, numerical results are compared against physical experiments of different tensile loading cases for various grading scenarios. Furthermore, the versatility of the method for the design of functionally graded textiles is demonstrated.

A frictionless contact formulation for spatial rods is developed within the framework of isogeometric collocation. The structural mechanics is described by the Cosserat theory of geometrically nonlinear spatial rods. The numerical discretization is based on an isogeometric collocation method, where the geometry and solution fields are represented as NURBS curves and the strong forms of the equilibrium equations are collocated at Greville points. In this framework, a frictionless rod-to-rod contact formulation is proposed. Contact points are detected by a coarse-level and a refined search for close centerline points and reaction forces are computed by the actual penetration of rod surface points, so that the enforcement of the contact constraints is performed with the penalty method. An important aspect is the application of contact penalty forces as point loads within the collocation scheme, and methods for this purpose are proposed and evaluated. The overall contact algorithm is validated by and applied to several numerical examples. This is the authors' accepted and refereed manuscript to the article. Locked until 01 July 2019 due to copyright restrictions

In this paper we analyze the vibrations of nonlinear structures by means of the novel approach of isogeometric finite elements. The fundamental idea of isogeometric finite elements is to apply the same functions, namely B-Splines and NURBS (Non-Uniform Rational B-Splines), for describing the geometry and for representing the numerical solution. In case of linear vibrational analysis, this approach has already been shown to possess substantial advantages over classical finite elements, and we extend it here to a nonlinear framework based on the harmonic balance principle. As application, the straight nonlinear Euler–Bernoulli beam is used, and overall, it is demonstrated that isogeometric finite elements with B-Splines in combination with the harmonic balance method are a powerful means for the analysis of nonlinear structural vibrations. In particular, the smoother k-method provides higher accuracy than the p-method for isogeometric nonlinear vibration analysis.

Helical shapes are ubiquitous in both nature and engineering. However, the development of soft actuators and robots that mimic helical motions has been hindered primarily due to the lack of efficient modeling approaches that take into account the material anisotropy and the directional change of the external loading point. In this work, we present a theoretical framework for modeling controllable helical deformations of cable-driven, anisotropic, soft composite actuators. The framework is based on the minimum potential energy method, and its model predictions are validated by experiments, where the microarchitectures of the soft composite actuators can be precisely defined by 3D printing. We use the developed framework to investigate the effects of material and geometric parameters on helical deformations. The results show that material stiffness, volume fraction, layer thickness, and fiber orientation can be used to control the helical deformation of a soft actuator. In particular, we found that a critical fiber orientation angle exists at which the twist of the actuator changes the direction. Thus, this work can be of great importance for the design and fabrication of soft actuators with tailored deformation behavior.