Artificial Soft Elastic Media with Periodic Hard Inclusions for Tailoring Strain‐Sensitive Thin‐Film Responses

Engineering the coupling behavior between a functional thin film and a soft substrate provides an attractive pathway for controlling various properties of thin‐film materials. However, existing studies mostly rely on uniform deformation of the substrate, and the effect of well‐regulated and nonuniform strain distributions on strain‐sensitive thin‐film responses still remains elusive. Herein, artificially strain‐regulated elastic media are presented as a novel platform for tailoring strain‐sensitive thin‐film responses. The proposed artificial soft elastic media are composed of embedded arrays of inkjet‐printed polymeric strain modulators that exhibit a high modulus contrast with respect to that of the soft matrix. This strain‐modulating lattice induces spatially regulated strain distributions based on localized strain‐coupling. Controlling the structural parameters and lattice configurations of the media leads to spatial modulation of the microscopically localized as well as macroscopically accumulated strain profiles. Uniform thin films coupled to these media undergo artificially tailored deformation through lattice‐like strain‐coupled pathways. The resulting phenomena yield programmable strain‐sensitive responses such as spatial arrangement of ternary‐state surface wrinkles and stepwise tuning of piezoresistive responses. This work will open a new avenue for addressing the issue of controlling strain‐sensitive thin‐film properties through structural engineering of artificial soft elastic media.     

饱和多孔介质中动力渗流耦合分析的Biot-Cosserat连续体模型与应变局部化有限元模拟

提出了适用于饱和多孔介质中应变局部化分析及动力渗流耦合分析的Biot-Cosserat连续体模型。基于饱和多孔介质动力渗流耦合分析的Biot理论,将固体骨架看作Cosserat连续体,并考虑旋转惯性,建立了饱和多孔介质动力渗流耦合分析的Biot-Cosserat连续体模型。基于Galerkin加权余量法,对所发展的模型推导了以固体骨架广义位移(包含旋转)及孔隙水压力为基本未知量的有限元公式。利用所发展的数值模型,对包含压力相关弹塑性固体骨架材料的饱和多孔介质进行了动力渗流耦合分析与应变局部化有限元模拟,结果表明,所发展的两相饱和多孔介质动力渗流耦合分析的Biot-Cosserat连续体模型能保持饱和两相介质应变局部化问题的适定性及模拟饱和多孔介质中由应变软化引起的应变局部化现象的有效性。     

A variational framework for fiber‐reinforced viscoelastic soft tissues

The mechanical properties of soft biological tissues vary depending on how the internal structure is organized. Classical examples of tissues are ligaments, tendons, skin, arteries, and annulus fibrous. The main element of such tissues is the fibers which are responsible for the tissue resistance and the main mechanical characteristic is their viscoelastic anisotropic behavior. The objective of this paper is to extend an existing model for isotropic viscoelastic materials in order to include anisotropy provided by fiber reinforcement. The incorporation of the fiber allows the mechanical behavior of these tissues to be simulated. The model is based on a variational framework in which its mechanical behavior is described by a free energy incremental potential whose local minimization provides the constraints for the internal variable updates for each load increment. The main advantage of this variational approach is the ability to represent different material models depending on the choice of suitable potential functions. Finally, the model is implemented in a finite‐element code in order to perform numerical tests to show the ability of the proposed model to represent fiber‐reinforced materials. The material parameters used in the tests were obtained through parameter identification using experimental data available in the literature. Copyright © 2011 John Wiley & Sons, Ltd.     

Combined continuum damage‐embedded discontinuity model for explicit dynamic fracture analyses of quasi‐brittle materials

In this paper, a novel constitutive model combining continuum damage with embedded discontinuity is developed for explicit dynamic analyses of quasi‐brittle failure phenomena. The model is capable of describing the rate‐dependent behavior in dynamics and the three phases in failure of quasi‐brittle materials. The first phase is always linear elastic, followed by the second phase corresponding to fracture‐process zone creation, represented with rate‐dependent continuum damage with isotropic hardening formulated by utilizing consistency approach. The third and final phase, involving nonlinear softening, is formulated by using an embedded displacement discontinuity model with constant displacement jumps both in normal and tangential directions. The proposed model is capable of describing the rate‐dependent ductile to brittle transition typical of cohesive materials (e.g., rocks and ice). The model is implemented in the finite element setting by using the CST elements. The displacement jump vector is solved for implicitly at the local (finite element) level along with a viscoplastic return mapping algorithm, whereas the global equations of motion are solved with explicit time‐stepping scheme. The model performance is illustrated by several numerical simulations, including both material point and structural tests. The final validation example concerns the dynamic Brazilian disc test on rock material under plane stress assumption. Copyright © 2014 John Wiley & Sons, Ltd.     

A formulation to model the nonlinear viscoelastic properties of the vascular tissue

Nearly all soft tissues, including the vascular tissue, present a certain degree of viscoelastic material response, which becomes apparent performing multiple relaxation tests over a wide range of strain levels and plotting the resulting stress relaxation curves, nonlinear viscoelasticity of the tissue. Changes in relaxation rate at each strain may occur at multiple strain levels. A constitutive formulation considering the particular features of the vascular tissue, such as anisotropy, together with these nonlinear viscoelastic phenomena is here presented and used to fit stress?Cstretch curves from experimental relaxation tests. This constitutive model was used to fit several data set of in vitro experimental stress relaxation tests performed on ovine and porcine aorta. The good fitting of the experimental data shows the capability of the model to reproduce the viscoelastic response of the vascular tissue.     

Studying nonlinear thermomechanical wave propagation in a viscoelastic layer based upon the Lord-Shulman theory

Abstract

Based on the Lord–Shulman (L-S) theory, the thermally nonlinear coupled thermo-viscoelasticity of a layer is analyzed using a numerical approach. Using the Kelvin-Voigt theory of viscoelasticity in conjunction with the L-S theory, the coupled governing equations are first derived in variational form. Then, the variational differential quadrature (VDQ) method is applied to solve the problem numerically. Parametric studies are presented in order to reveal the effects of viscoelastic parameter, intensity of heat flux and relaxation time on the propagation of displacement, temperature and stress waves. Furthermore, the influence of considering nonlinearity in the energy equation is analyzed.     

An embedded cohesive crack model for finite element analysis of mixed mode fracture of concrete*

An embedded cohesive crack model is proposed for the analysis of the mixed mode fracture of concrete in the framework of the Finite Element Method. Different models, based on the strong discontinuity approach, have been proposed in the last decade to simulate the fracture of concrete and other quasi‐brittle materials. This paper presents a simple embedded crack model based on the cohesive crack approach. The predominant local mode I crack growth of the cohesive materials is utilized and the cohesive softening curve (stress vs. crack opening) is implemented by means of a central force traction vector. The model only requires the elastic constants and the mode I softening curve. The need for a tracking algorithm is avoided using a consistent procedure for the selection of the separated nodes. Numerical simulations of well‐known experiments are presented to show the ability of the proposed model to simulate the mixed mode fracture of concrete.     

Formulation and numerical exploitation of mixed variational principles for coupled problems of Cahn–Hilliard‐type and standard diffusion in elastic solids

This work develops variational principles for the coupled problem of standard and extended Cahn–Hilliard‐type species diffusion in solids undergoing finite elastic deformations. It shows that the coupled problem of diffusion in deforming solids, accounting for phenomena like swelling, diffusion‐induced stress generation and possible phase segregation caused by the diffusing species, is related to an intrinsic mixed variational principle. It determines the rates of deformation and concentration along with the chemical potential, where the latter plays the role of a mixed variable. The principle characterizes a canonically compact model structure, where the three governing equations involved, that is, the mechanical equilibrium condition, the mass balance for the species content and a microforce balance that determines the chemical potential, appear as the Euler equation of a variational statement. The existence of the variational principle underlines an inherent symmetry of the coupled deformation–diffusion problem. This can be exploited in the numerical implementation by the construction of time‐discrete and space‐discrete incremental potentials, which fully determine the update problems of typical time stepping procedures. The mixed variational principles provide the most fundamental approach to the monolithic finite element solution of the coupled deformation–diffusion problem based on low‐order basis functions. They induce in a natural format the choice of symmetric solvers for Newton‐type iterative updates, providing a speedup and reduction of data storage when compared with non‐symmetric implementations. This is a strong argument for the use of the developed variational principles in the computational design of deformation–diffusion problems. Copyright © 2014 John Wiley & Sons, Ltd.     

Equivalent localization element for crack band approach to mesh‐sensitivity in microplane model

The paper presents in detail a novel method for finite element analysis of materials undergoing strain‐softening damage based on the crack band concept. The method allows applying complex material models, such as the microplane model for concrete or rock, in finite element calculations with variable finite element sizes not smaller than the localized crack band width (corresponding to the material characteristic length). The method uses special localization elements in which a zone of characteristic size, undergoing strain softening, is coupled with layers (called ‘springs’) which undergo elastic unloading and are normal to the principal stress directions. Because of the coupling of strain‐softening zone with elastic layers, the computations of the microplane model need to be iterated in each finite element in each load step, which increases the computer time. Insensitivity of the proposed method to mesh size is demonstrated by numerical examples. Simulation of various experimental results is shown to give good agreement. Copyright © 2004 John Wiley & Sons, Ltd.     

Variational principles in dissipative electro‐magneto‐mechanics: A framework for the macro‐modeling of functional materials

This paper presents a general framework for the macroscopic, continuum‐based formulation and numerical implementation of dissipative functional materials with electro‐magneto‐mechanical couplings based on incremental variational principles. We focus on quasi‐static problems, where mechanical inertia effects and time‐dependent electro‐magnetic couplings are a priori neglected and a time‐dependence enters the formulation only through a possible rate‐dependent dissipative material response. The underlying variational structure of non‐reversible coupled processes is related to a canonical constitutive modeling approach, often addressed to so‐called standard dissipative materials. It is shown to have enormous consequences with respect to all aspects of the continuum‐based modeling in macroscopic electro‐magneto‐mechanics. At first, the local constitutive modeling of the coupled dissipative response, i.e. stress, electric and magnetic fields versus strain, electric displacement and magnetic induction, is shown to be variational based, governed by incremental minimization and saddle‐point principles. Next, the implications on the formulation of boundary‐value problems are addressed, which appear in energy‐based formulations as minimization principles and in enthalpy‐based formulations in the form of saddle‐point principles. Furthermore, the material stability of dissipative electro‐magneto‐mechanics on the macroscopic level is defined based on the convexity/concavity of incremental potentials. We provide a comprehensive outline of alternative variational structures and discuss details of their computational implementation, such as formulation of constitutive update algorithms and finite element solvers. From the viewpoint of constitutive modeling, including the understanding of the stability in coupled electro‐magneto‐mechanics, an energy‐based formulation is shown to be the canonical setting. From the viewpoint of the computational convenience, an enthalpy‐based formulation is the most convenient setting. A numerical investigation of a multiferroic composite demonstrates perspectives of the proposed framework with regard to the future design of new functional materials. Copyright © 2011 John Wiley & Sons, Ltd.     

Relaxed incremental variational formulation for damage at large strains with application to fiber‐reinforced materials and materials with truss‐like microstructures

In this paper, an incremental variational formulation for damage at finite strains is presented. The classical continuum damage mechanics serves as a basis where a stress‐softening term depending on a scalar‐valued damage function is prepended an effective hyperelastic strain energy function, which describes the virtually undamaged material. Because loss of convexity is obtained at some critical deformations, a relaxed incremental stress potential is constructed, which convexifies the original nonconvex problem. The resulting model can be interpreted as the homogenization of a microheterogeneous material bifurcated into a strongly and weakly damaged phase at the microscale. A one‐dimensional relaxed formulation is derived, and a model for fiber‐reinforced materials based thereon is given. Finally, numerical examples illustrate the performance of the model by showing mesh independency of the model in an extended truss, analyzing a numerically homogenized microtruss material and investigating a fiber‐reinforced cantilever beam subject to bending and an overstretched arterial wall. Copyright © 2012 John Wiley & Sons, Ltd.     

Energy-based variational modeling of adiabatic shear bands structure evolution

A novel energy-based variational approach is proposed for modeling adiabatic shear band (ASB) structure evolution, including elasticity, work hardening, and heat conduction. Conservation laws are formulated as a mathematical optimization problem with respect to a limited set of scalars. Consequently, by means of canonical expressions of displacement and temperature, the bandwidth and the central temperature can be accurately computed as internal variables. Based on this thermo-mechanical coupled variational framework, we can verify the generality of the proposed analytical approach with respect to constitutive models, as illustrated through various thermal softening laws such as power laws or the popular Johnson–Cook model. In addition, accounting for work hardening and elasticity, we propose an effective (or macroscopic) thermo-elasto-viscoplastic model of the shear localization zone in transient regime. A new loading/unloading condition, stemming as a Kuhn–Tucker relation, is introduced for this variational model. The stress evolution and the capacity of the approach to handle cyclic loading are analyzed, presenting a very good correspondence with ASB simulations by finite element method.     

Permanent set and stress-softening constitutive equation applied to rubber-like materials and soft tissues

Many rubber-like materials present a phenomenon known as Mullins effect. It is characterized by a difference of behavior between the first and second loadings and by a permanent set after a first loading. Moreover, this phenomenon induces anisotropy in an initially isotropic material. A new constitutive equation is proposed in this paper. It relies on the decomposition of the macromolecular network into two parts: chains related together and chains related to fillers. The first part is modeled by a simple hyperelastic constitutive equation, whereas the second one is described by an evolution function introduced in the hyperelastic strain energy. It contributes to describe both the anisotropic stress softening and the permanent set. The model is finally extended to soft tissues’ mechanical behavior that present also stress softening but with an initially anisotropic behavior. The two models are successfully fitted and compared to experimental data.     

Buckling analysis of embedded nanotubes using gradient continuum theory

The buckling of nanotubes embedded in an elastic matrix is modeled within the framework of Timoshenko beams. Both a stress gradient and a strain gradient approach are considered. The energy variational approach is adopted to obtain the critical buckling loads. The dependences of the buckling load on the nonlocal parameter, the stiffness of the surrounding elastic matrix, and the transverse shear stiffness of the nanotubes are obtained. The results show a significant dependence of critical buckling load on the nonlocal parameter and the stiffness of the surround matrix. The Euler beam model, which neglects the shear stiffness of the nanotubes, over-predicts the critical buckling load. It is also found that the strain gradient model provides the lower bound and the stress gradient model provides the upper bound for the critical buckling load of nanotubes. In addition to mechanical buckling, thermally induced buckling of the nanotubes embedded in an elastic matrix is also studied. All results are expressed in closed-form and therefore are easy to use by materials scientists and engineers for the design of nanotubes and their composites.     

A finite‐strain gradient‐inelastic beam theory and a corresponding force‐based frame element formulation

This paper extends the gradient‐inelastic (GI) beam theory, introduced by the authors to simulate material softening phenomena, to further account for geometric nonlinearities and formulates a corresponding force‐based (FB) frame element computational formulation. Geometric nonlinearities are considered via a rigorously derived finite‐strain beam formulation, which is shown to coincide with Reissner's geometrically nonlinear beam formulation. The resulting finite‐strain GI beam theory: (i) accounts for large strains and rotations, unlike the majority of geometrically nonlinear beam formulations used in structural modeling that consider small strains and moderate rotations; (ii) ensures spatial continuity and boundedness of the finite section strain field during material softening via the gradient nonlocality relations, eliminating strain singularities in beams with softening materials; and (iii) decouples the gradient nonlocality relations from the constitutive relations, allowing use of any material model. On the basis of the proposed finite‐strain GI beam theory, an exact FB frame element formulation is derived, which is particularly novel in that it: (a) expresses the compatibility relations in terms of total strains/displacements, as opposed to strain/displacement rates that introduce accumulated computational error during their numerical time integration, and (b) directly integrates the strain‐displacement equations via a composite two‐point integration method derived from a cubic Hermite interpolating polynomial to calculate the displacement field over the element length and, thus, address the coupling between equilibrium and strain‐displacement equations. This approach achieves high accuracy and mesh convergence rate and avoids polynomial interpolations of individual section fields, which often lead to instabilities with mesh refinements. The FB formulation is then integrated into a corotational framework and is used to study the response of structures, simultaneously accounting for geometric nonlinearities and material softening. The FB formulation is further extended to capture member buckling triggered by minor perturbations/imperfections of the initial member geometry.     

Effect of magnetostriction on fracture of a soft ferromagnetic medium with a crack-like flaw

The magnetic‐induction field in the vicinity of an elliptical inclusion embedded in an infinite soft ferromagnetic medium is determined based on complex potential theory. By using a constitutive relation of magnetostriction for isotropic materials, the stress field in the vicinity of an elliptical flaw is obtained. Furthermore, the stress field at the tip of a slender elliptical crack is determined for the case in which only an external magnetic field perpendicular to the major axis of the ellipse is applied at infinity. The results indicate that the stress field in the neighbourhood of the tip is governed by the magnetostriction and permeability of the soft ferromagnetic material. The induction magnetostrictive modulus is a key parameter in determining which of the two mechanisms, i.e., magnetostriction and magnetic‐force‐induced deformation, is dominant in determining the stress field in the neighbourhood of the tip of a crack‐like flaw. With regard to the influence of the magnetic field on the apparent toughness of a soft ferromagnetic body with a crack‐like flaw, soft ferromagnetic materials can be roughly divided into two categories: one possesses a large induction magnetostrictive modulus and the other has a small modulus. An approximate criterion for categorizing the materials is presented. For the benefit of engineering design, the expressions of the stress‐intensity factor for these two categories of soft ferromagnetic materials are presented. The results show that the stress‐intensity factor is affected not only by the flaw geometry, but also by the permeability of the medium inside the flaw.