A multi-scale enriched model for the analysis of masonry panels

A multi-scale model for the structural analysis of the in-plane response of masonry panels, characterized by periodic arrangement of bricks and mortar, is presented. The model is based on the use of two scales: at the macroscopic level the Cosserat micropolar continuum is adopted, while at the microscopic scale the classical Cauchy medium is employed. A nonlinear constitutive law is introduced at the microscopic level, which includes damage, friction, crushing and unilateral contact effects for the mortar joints. The nonlinear homogenization is performed employing the Transformation Field Analysis (TFA) technique, properly extended to the macroscopic Cosserat continuum. A numerical procedure is developed and implemented in a Finite Element (FE) code in order to analyze some interesting structural problems. In particular, four numerical applications are presented: the first one analyzes the response of the masonry Representative Volume Element (RVE) subjected to a cyclic loading history; in the other three applications, a comparison between the numerically evaluated response and the micromechanical or experimental one is performed for some masonry panels.     

A nonlinear homogenization procedure for periodic masonry

The present paper deals with the problem of the determination of the in-plane behavior of masonry material. The masonry is considered as a composite material composed by a regular distribution of blocks connected by horizontal and vertical mortar joints. The overall constitutive relationships of the regular masonry are derived by a rational micromechanical and homogenization procedure. Linear elastic constitutive relationship is considered for the blocks, while a new special nonlinear constitutive law is proposed for the mortar joints. In particular, a mortar constitutive law, which accounts for the coupling of the damage and friction phenomena occurring during the loading history, is proposed; the developed model is based on an original micromechanical analysis of the damage process of the mortar joint. Then, an effective nonlinear homogenization procedure, representing the main novelty of the paper, is proposed; it is based on the transformation field analysis, using the technique of the superposition of the effects and the finite element method. The presented methodology is implemented in a numerical code. Finally, numerical applications are performed in order to assess the performances of the proposed procedure in reproducing the mechanical behavior of masonry material.     

Homogenization of heterogeneous masonry beams

This study presents a two-scale model to describe the out-of-plane masonry response. One-dimensional (1D) structural elements, like masonry columns or strips of long wall characterized by the periodic repetition of bricks and mortar arranged in stack bond, are considered. A damage-friction plasticity law is adopted to model the mortar joint constitutive response, while the bricks are assumed as linear elastic. A 1D beam formulation is introduced at both the structural and micromechanical scale, linking the two levels by means of a kinematic map. This expresses the microscopic beam strains in the masonry unit cell (UC) as function of the macroscopic generalized strains. The kinematic field in the UC is completed by adding an unknown periodic fluctuation term. A nonlinear homogenization procedure is developed, proposing a semi-analytical solution for the micromechanical problem, based on the fiber discretization of the mortar joints. A force-based beam-column finite element procedure is adopted at the structural scale and the solution algorithm for the element state determination is illustrated in details. Some numerical applications, showing the UC constitutive response and the behavior of masonry structural elements, are finally presented.     

Non-linear micropolar and classical continua for anisotropic discontinuous materials

Non-linear Cosserat and Cauchy anisotropic continua equivalent to masonry-like materials, like brick/block masonry, jointed rocks, granular materials or matrix/particle composites, are presented.An integral procedure of equivalence in terms of mechanical power has been adopted to identify the effective elastic moduli of the two continuous models starting from a Lagrangian system of interacting rigid elements. Non-linear constitutive functions for the interactions in the Lagrangian system are defined in order to take into account both the low capability to carry tension and the friction at the interfaces between elements. The non-linear problem is solved through a finite element procedure based on the iterative adjustment of the continuum constitutive tensor due to the occurrence of some limit situation involving the contact actions of the discrete model.Differences between the classical and the micropolar model are investigated with the aid of numerical analyses carried out on masonry walls made of blocks of different size. The capability of the micropolar continuum to discern, unlike the classical continuum, the behaviour of systems made of elements of different size is pointed out. It is also shown that for anisotropic materials, even in the elastic case, the micropolar solution in general does not tend to the classical solution when the size of the elements vanishes.     

Multiscale micromechanical modeling of the constitutive response of carbon nanotube-reinforced structural adhesives

Appropriate formulations are developed to allow for the atomistic-based continuum modeling of nano-reinforced structural adhesives on the basis of a nanoscale representative volume element that accounts for the nonlinear behavior of its constituents; namely, the reinforcing carbon nanotube, the surrounding adhesive and their interface. The newly developed representative volume element is then used with analytical and computational micromechanical modeling techniques to investigate the homogeneous dispersion of the reinforcing element into the adhesive upon both the linear and nonlinear properties. Unlike our earlier work where the focus was on developing linear micromechanical models for the effective elastic properties of nanocomposites, the present approach extends these models by describing the development of a nonlinear hybrid Monte Carlo Finite Element model that allows for the prediction of the full constitutive response of the bulk composite under large deformations. The results indicate a substantial improvement in both the Young’s modulus and tensile strength of the nano-reinforced adhesives for the range of CNT concentrations considered.     

A new 3-D finite element for nonlinear elasticity using the theory of a Cosserat point

The theory of a Cosserat point has been used to formulate a new 3-D finite element for the numerical analysis of dynamic problems in nonlinear elasticity. The kinematics of this element are consistent with the standard tri-linear approximation in an eight node brick-element. Specifically, the Cosserat point is characterized by eight director vectors which are determined by balance laws and constitutive equations. For hyperelastic response, the constitutive equations for the director couples are determined by derivatives of a strain energy function. Restrictions are imposed on the strain energy function which ensure that the element satisfies a nonlinear version of the patch test. It is shown that the Cosserat balance laws are in one-to-one correspondence with those obtained using a Bubnov–Galerkin formulation. Nevertheless, there is an essential difference between the two approaches in the procedure for obtaining the strain energy function. Specifically, the Cosserat approach determines the constitutive coefficients for inhomogeneous deformations by comparison with exact solutions or experimental data. In contrast, the Bubnov–Galerkin approach determines these constitutive coefficients by integrating the 3-D strain energy function using the kinematic approximation. It is shown that the resulting Cosserat equations eliminate unphysical locking, and hourglassing in large compression without the need for using assumed enhanced strains or special weighting functions.     

A MODEL CONSIDERING HYDROSTATIC STRESS OF POROUS NITI SHAPE MEMORY ALLOY

Based on the micromechanical method and thermodynamic theory,a constitutive model for the macroscopic mechanical behavior of porous NiTi shape memory alloy is presented.The hydrostatic stress is considered for porous NiTi according to the transformation function of dense NiTi.The present model takes account of the tensile-compressive asymmetry of NiTi,and can degenerate to model dense material.Numerical calculations,which only need material parameters of dense NiTi,are conducted to investigate the nonlinear and hysteretic strain of porous NiTi,and the predicted results are in good agreement with the corresponding experiments.     

Nonlinear ferro-electro-elastic beam theory

Motivated by the uniqueness and potential of the nonlinear range of piezoelectric and ferroelectric smart materials and structures, a static physically nonlinear ferro-electro-elastic beam theory which takes the effect of domain switching into account is developed. The kinematic assumptions adopt the geometrically linear Bernoulli–Euler form for the mechanical components and a first-order theory for the electrical potential, and lay the basis for further augmentation to higher order theories. The beam theory includes the field equations that correspond to the static case, the boundary conditions and the constitutive equations of ferro-electro-elasticity. The general 3-D constitutive equations are reduced to comply with the beam theory and formulated as ordinary differential equations by means of a set of generalized electro-mechanical stiffnesses. A micromechanical constitutive model that accounts for the loading history and for the domain switching phenomenon is adopted and an iterative solution procedure that incorporates the micromechanical approach is suggested. A numerical example that demonstrates the impact of the domain switching on the nonlinear electromechanical static response of a ferro-electro-elastic beam is presented and discussed. The quantitative assessment of this behavior takes a step towards new structural applications that cope with or even take advantage of the nonlinear ferro-electro-elastic range.     

各向同性超弹性本构模型数值计算及验证

现有多种形式的橡胶本构模型试图预测橡胶力学性质,其中部分模型已写入有限元软件中用于仿真计算,还存在较多拟合性较好的模型无法在有限元材料库中直接获得。本文详述了由不变量和主伸长率描写的各向同性超弹性本构模型的数值实现方法,并结合最新的本构模型开发了UHYPER和UMAT子程序。将UHYPER用于有限元实现对多孔橡胶板的拉伸仿真,对比仿真和试验结果,验证子程序的正确性以及评估本构模型预测复杂应变场的准确性;将UMAT用于单轴、等双轴和剪切拉伸的有限元仿真,对比仿真和本构模型理论结果,验证子程序的可靠性。结果表明,有限元仿真结果与理论结果拟合较好,子程序能够契合本构模型的力学描述,所述方法可以用于超弹性材料的数值计算。     

Compatible model for herringbone bond masonry: Linear elastic homogenization,failure surfaces and structural implementation

A simplified kinematic procedure at a cell level is proposed to obtain in-plane elastic moduli and macroscopic masonry strength domains in the case of herringbone masonry. The model is constituted by two central bricks interacting with their neighbors by means of either elastic or rigid-plastic interfaces with friction, representing mortar joints. The herringbone pattern is geometrically described and the internal law of composition of the periodic cell is defined.A sub-class of possible elementary deformations is a-priori chosen to describe joints cracking under in-plane loads. Suitable internal macroscopic actions are applied on the Representative Element of Volume (REV) and the power expended within the 3D bricks assemblage is equated to that expended in the macroscopic 2D Cauchy continuum. The elastic and limit analysis problem at a cell level are solved by means of a quadratic and linear programming approach, respectively.To assess elastic results, a standard FEM homogenization is also performed and a sensitivity analysis regarding two different orientations of the pattern, the thickness of the mortar joints and the ratio between block and mortar Young moduli is conducted. In this way, the reliability of the numerical model is critically evaluated under service loads.When dealing with the limit analysis approach, several computations are performed investigating the role played by (1) the direction of the load with respect to herringbone bond orientation, (2) masonry texture and (3) mechanical properties adopted for joints.At a structural level, a FE homogenized limit analysis is performed on a masonry dome built in herringbone bond. In order to assess limit analysis results, additional non-linear FE analyses are performed, including a full 3D numerical expensive heterogeneous approach and models where masonry is substituted with an equivalent macroscopic material with orthotropic behavior and possible softening. Reliable predictions of collapse loads and failure mechanisms are obtained, meaning that the approach proposed may be used by practitioners for a fast evaluation of the effectiveness of herringbone bond orientation.     

Finite element formulation of slender structures with shear deformation based on the Cosserat theory

This paper addresses the derivation of finite element modelling for nonlinear dynamics of Cosserat rods with general deformation of flexure, extension, torsion, and shear. A deformed configuration of the Cosserat rod is described by the displacement vector of the deformed centroid curve and an orthogonal moving frame, rigidly attached to the cross-section of the rod. The position of the moving frame relative to the inertial frame is specified by the rotation matrix, parameterised by a rotational vector. The shape functions with up to third order nonlinear terms of generic nodal displacements are obtained by solving the nonlinear partial differential equations of motion in a quasi-static sense. Based on the Lagrangian constructed by the Cosserat kinetic energy and strain energy expressions, the principle of virtual work is employed to derive the ordinary differential equations of motion with third order nonlinear generic nodal displacements. A cantilever is presented as a simple example to illustrate the use of the formulation developed here to obtain the lower order nonlinear ordinary differential equations of motion of a given structure. The corresponding nonlinear dynamical responses of the structures are presented through numerical simulations using the MATLAB software. In addition, a MicroElectroMechanical System (MEMS) device is presented. The developed equations of motion have furthermore been implemented in a VHDL-AMS beam model. Together with available models of the other components, a netlist of the device is formed and simulated within an electrical circuit simulator. Simulation results are verified against Finite Element Analysis (FEA) results for this device.     

Nonlinear damping in a micromechanical oscillator

Nonlinear elastic effects play an important role in the dynamics of microelectromechanical systems (MEMS). A Duffing oscillator is widely used as an archetypical model of mechanical resonators with nonlinear elastic behavior. In contrast, nonlinear dissipation effects in micromechanical oscillators are often overlooked. In this work, we consider a doubly clamped micromechanical beam oscillator, which exhibits nonlinearity in both elastic and dissipative properties. The dynamics of the oscillator is measured in both frequency and time domains and compared to theoretical predictions based on a Duffing-like model with nonlinear dissipation. We especially focus on the behavior of the system near bifurcation points. The results show that nonlinear dissipation can have a significant impact on the dynamics of micromechanical systems. To account for the results, we have developed a continuous model of a geometrically nonlinear beam-string with a linear Voigt–Kelvin viscoelastic constitutive law, which shows a relation between linear and nonlinear damping. However, the experimental results suggest that this model alone cannot fully account for all the experimentally observed nonlinear dissipation, and that additional nonlinear dissipative processes exist in our devices.     

Cosserat modeling of cellular solidsModélisation des solides cellulaires selon Cosserat

Cellular solids inherit their macroscopic mechanical properties directly from the cellular microstructure. However, the characteristic material length scale is often not small compared to macroscopic dimensions, which limits the applicability of classical continuum-type constitutive models. Cosserat theory, however, offers a continuum framework that naturally features a length scale related to rotation gradients. In this paper a homogenization procedure is proposed that enables the derivation of macroscopic Cosserat constitutive equations based on the underlying microstructural morphology and material behavior. To cite this article: P.R. Onck, C. R. Mecanique 330 (2002) 717–722.     

Statics and kinematics of discrete Cosserat-type granular materials

A theoretical framework is presented for the statics and kinematics of discrete Cosserat-type granular materials. In analogy to the force and moment equilibrium equations for particles, compatibility equations for closed loops are formulated in the two-dimensional case for relative displacements and relative rotations at contacts. By taking moments of the equilibrium equations, micromechanical expressions are obtained for the static quantities average Cauchy stress tensor and average couple stress tensor. In analogy, by taking moments of the compatibility equations, micromechanical expressions are obtained for the (infinitesimal) kinematic quantities average rotation gradient tensor and average Cosserat strain tensor in the two-dimensional case. Alternatively, these expressions for the average Cauchy stress tensor and the average couple stress tensor are obtained from considerations of the equivalence of the continuum force and couple traction vectors acting on a plane and the resultant of the discrete forces and couples acting on this plane. In analogy, the expressions for the average rotation gradient tensor and the average Cosserat strain tensor are obtained from considerations of the change of length and change of rotation of a line element in the two-dimensional case. It is shown that the average particle stress tensor is always symmetrical, contrary to the average stress tensor of an equivalent homogenized continuum. Finally, discrete analogues of the virtual work and complementary virtual work principles from continuum mechanics are derived.     

A concurrent micromechanical model for predicting nonlinear viscoelastic responses of composites reinforced with solid spherical particles

A concurrent micromechanical model for predicting nonlinear viscoelastic responses of particle reinforced polymers is developed. Particles are in the form of solid spheres having micro-scale diameters. The composite microstructures are idealized by periodically distributed cubic particles in a matrix medium. Each particle is assumed to be fully surrounded by polymeric matrix such that contact between particles can be avoided. A representative volume element (RVE) is then defined by a single particle embedded in the cubic matrix. A spatial periodicity boundary condition is imposed to the RVE. One eighth unit-cell model with four particle and polymer subcells is generated due to the three-plane symmetry of the RVE. The solid spherical particle is modeled as a linear elastic material. The polymeric matrix follows nonlinear viscoelastic behaviors of thermorheologically simple materials. The homogenized micromechanical relation is developed in terms of the average strains and stresses in the subcells and traction continuity and displacement compatibility at the subcells’ interfaces are imposed. A stress–strain correction scheme is also formulated to satisfy the linearized micromechanical and the nonlinear constitutive relations. The micromechanical model provides three-dimensional (3D) effective properties of homogeneous composite responses, while recognizing microstructural geometries and in situ material properties of the heterogeneous medium. The micromechanical formulation is designed to be compatible with general displacement based finite element (FE) analyses. Experimental data and analytical micromechanical models available in the literature are used to verify the capability of the above micromechanical model for predicting the overall composite behaviors. The proposed micromodel is also examined in terms of computational efficiency and accuracy.     

Strength domain of non-periodic masonry by homogenization in generalized plane state

This paper deals with the evaluation of the strength domain for non-periodic masonry using a random media micromechanical approach. The generalized plane state formulation is used in order to more accurately describe the masonry behavior and failure criteria proposed in literature. Considering the masonry as a heterogeneous material with random microstructure, the elastic characteristics of the homogenized continuum and the strength domain are evaluated by using the hierarchy theory related to partitions with increasing dimensions through the application of natural and essential boundary conditions with proportionally growing values. An overall failure criterion based on the mean stress state of each phase is introduced. The proposed procedure is validated by comparison with the experimental results obtained with periodic masonry subjected to biaxial stress states recovering the main failure mechanisms. Then the approach is applied to an actual non-periodic masonry introducing peculiar algorithms in order to evaluate strength surfaces and to verify the convergence of the domains obtained through the application of natural and essential boundary conditions with the increasing size of portion dimensions at the mesoscale level.     

An endochronic plasticity formulation for filled rubber

The nature of elastomeric material demands the consideration of finite deformations, nonlinear elasticity including damage as well as rate-dependent and rate-independent dissipative properties. While many models accounting for these effects have been refined over time to do better justice to the real behavior of rubber-like materials, the realistic simulation of the elastoplastic characteristics for filled rubber remains challenging.The classical elastic-ideal-plastic formulation exhibits a distinct yield-surface, whereas the elastoplastic material behavior of filled rubber components shows a yield-surface free plasticity. In order to describe this elastoplastic deformation of a material point adequately, a physically based endochronic plasticity model was developed and implemented into a Finite Element code. The formulation of the ground state elastic characteristics is based on Arruda and Boyce (1993) eight-chain model. The evolution of the constitutive equations for the nonlinear endochronic elastoplastic response are derived in analogy to the Bergström–Boyce finite viscoelasticity model discussed by Dal and Kaliske (2009).