| Soft materials composed of three-dimensional cross-linked networks and water molecules are elastic mixtures,which are common in nature and engineering,such as skin,leaves,polymer,gels and so on.This type of materials behaves the large and restorable volumetric deformation and complex deformation patterns under the geometric constraints when exposed to external physical and chemical stimuli.Meanwhile,they have biocompatibility,transparency and other good biological and physical properties.Therefore,soft materials present great potentials for the applications in bionic actuators,flexible optical devices,soft robots and other bionic structures.Moreover,they are one of smart materials investigated further in many core areas,such as new materials,smart device,biological health and other research fields.The swelling of soft materials under the external stimuli is a finite deformation problem.The external solvent molecules can flow into the soft materials,and diffuse in the cross-linked networks,which is a problem coupled multi physical fields.The mechanical problem during growth of biological tissues could help determine their functions and understand the pathogenic factors.Moreover,the mechanical problem during swelling of gels and other soft materials could help design the bionic structures and determine whether their special functions can be achieved.So far,the theoretical methods,numerical approaches and experimental methods are mainly adopted to investigate the mechanical problems in swelling of soft materials.However,the experimental methods are limited by the equipment and environment in experiment.The theoretical models are only effective for the soft structures with simple geometries and boundary conditions.The numerical methods,adopted to investigate the mechanical problem in swelling of soft materials widely,can circumvent the disadvantages of experimental and theoretical methods.But,the most numerical methods,based on the solid elements using the Lagrange polynomial as shape functions,are inefficient due to the large computational efforts and independent process of modeling and analysis for the soft structure with thin thickness and complex geometry.Based on the mentioned background,the effective numerical algorithms are proposed for the isotropic and anisotropic deformation of thin-walled structures and soft structures with complex geometries in the steady and transient swelling processes.The main contents of this dissertation are as follows.Firstly,a solid-shell based finite element method for the isotropic and anisotropic deformation of thin-walled soft structures in the steady swelling process is proposed.In the solid-shell framework,the deformation gradient,Green-Lagrange strain referring to the contravariant base vector and the second Piola-Kirchoff stress referring to the covariant base vectors are derived.For the isotropic swelling deformation,the multiplicative decomposition of deformation gradient is used to introduce swelling effects.However,the Flory-Rehner model,adding a term from the reinforced fibers,is applied to directly consider the influence of external environment for the anisotropic swelling.Using the enhanced assumed strain method,the virtual work equation under the two-field variational principle is derived based on the orthogonality condition.During linearization,the incremental forms of strain,stress and virtual work equations are obtained,and the consistent tangent modulus is derived.The terms related to the swelling tensor and chemical potential in the incremental stress drive the deformation of soft materials.The assumed natural strain method is adopted to modify the strain,and the interpolation scheme with 7 parameters is used to enhance strain.The stability and convergence of algorithms are promised.Moreover,the local degree of freedom in element is eliminated by the condensed technique to improve efficiency.In numerical examples,results obtained by the developed method,theoretical method and conventional solid element based method agree well.which indicate the correctness and efficiency of developed numerical method.In addition,the algorithm is adopted to model the wrinkle deformations of maize leaf and the anisotropic deformation of calla lily flower and lotus.Secondly,high-order NURBS solids elements and isogeometric analysis algorithm for the steady swelling of soft materials are proposed.The swelling effect is introduced based on the multiplicative decomposition of deformation gradient.The elastic deformation gradient is further decomposed into its volume-preserving and volumetric-dilatational parts multiplicatively.Meanwhile,the volumetric-dilatational part is modified by a linear projection operator.The element,in which the NURBS basis functions for the linear projection operator is calculated by ones for the displacement degree of freedom,is also designed to circumvent volumetric locking.In this framework,the virtual work equations and the variation of modified strain referring to the current configuration are derived.The modified elastic Green-Lagrange strain are also given.The consistent tangent modulus in the current configuration and the increments of variation of modified strain,modified Kirchoff stress and virtual work equations are derived in linearization.The terms from swelling and modification of elastic deformation gradient in the incremental virtual work equations promise the stability and convergence of algorithm.Base on the above theory,an iterative algorithm in framework of isogeometric analysis is proposed.For the swelling of one single layer and bilayer soft structures,the results based on the developed isogeometric analysis method using NURBS solid element based on the quadratic and cubic NURBS basis functions agree well with these obtained by theoretical method and conventional finite element method.Therefore,the correctness and efficiency of proposed algorithm are proved.Moreover,it is applied to simulate the swelling of bionic lotus and petunia and the wrinkle deformation of special-like fruits.At last,mixed NURBS solid elements and related isogemetric analysis approach for the transient swelling of soft materials are developed.For the transient swelling of soft materials coupling multi-physics problems,an auxiliary configuration for the multiplicative decomposition of deformation gradient is introduced.Then,the difficulty due to a negative infinite chemical potential for soft materials in a dry state is eliminated effectively.The virtual work equations are derived based on the balance equations for the mechanical and chemical fields.The stress for the mechanical field is calculated based on the Flory-Rehner theory.In the chemical field,the flux of solvent can be calculated by the relations between the gradient of chemical potential and mobility tensor on the basis of Darcy’s law.Adopting the subdivision properties of NURBS basis function,mixed isogeometric elements,providing a good balance between the number of displacement and chemical potential degree of freedom and circumventing the volumetric locking effectively,is designed.In linearization,the increments of Kirchoff stress,nominal flux,nominal solvent concentration are derived.Meanwhile,the consistent tangent moduli for displacement and coupling of displacement and chemical potential are obtained.The isogeometric formulations for the coupled multi-physics problems in transient swelling of soft materials are proposed.The swellings of a soft materials block and cylindrical soft materials pumped into water are simulated.Comparisons of results between the developed approach,theoretical method and conventional finite element method prove the correctness and efficiency of the isogeometric algorithm.In addition,the free swelling deformation of a cylindrical soft materials block,helical deformation of a bionic plant structure and the actuation of a flytrap-inspired robot are modeled. |
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