Coupled multi‐scale cohesive modeling of failure in heterogeneous adhesives

A multi‐scale cohesive numerical framework is proposed to simulate the failure of heterogeneous adhesively bonded systems. This multi‐scale scheme is based on Hill's variational principle of energy equivalence between the higher and lower level scales. It provides an easy way to obtain accurate homogenized macroscopic properties while capturing the physics of failure processes at the micro‐scale in sufficient detail. We use an isotropic rate‐dependent damage model to mimic the failure response of the constituents of heterogeneous adhesives. The finite element method is used to solve the equilibrium equation at each scale. A nested iterative scheme inspired by the return mapping algorithm used in computational inelasticity is implemented. We propose a computationally attractive technique to couple the macro‐ and micro‐scales for rate‐dependent constitutive laws. We introduce an adhesive patch test to study the numerical performance, including spatial and temporal convergence of the multi‐scale scheme. We compare the solution of the multi‐scale cohesive scheme with a direct numerical simulation. Finally, we solve mode I and mode II fracture problems to demonstrate failure at the macro‐scale. Copyright © 2010 John Wiley & Sons, Ltd.     

Derivation of 3D masonry properties using numerical homogenization technique

Lots of research work has been conducted on homogenization technique, which derives global homogenized properties of masonry from the behaviour of the constitutive materials (brick and mortar). Such a technique mainly focused on two‐dimensional media in the previous studies with the out‐of‐plane properties of masonry material neglected. In this paper, homogenization technique and damage mechanics theory are used to model a three‐dimensional masonry basic cell to numerically derive the equivalent elastic properties, strength envelope, and failure characteristics of masonry material. The basic cell is modelled with distinctive consideration of non‐linear material properties of mortar and brick. Various displacement boundaries are applied on the basic cell surfaces in the numerical simulation. The detailed material properties of mortar and brick are modelled in a finite element program in the numerical analysis. The stress–strain relations of masonry material under various conditions are obtained from the simulation. The homogenized elastic properties and failure characteristics of masonry material are derived from the simulation results. The homogenized 3D model is then utilized to analyse the response of a masonry panel to airblast loads. The same panel is also analysed with distinctive material modelling. The efficiency and accuracy of the homogenized model are demonstrated. The homogenized material properties and failure model can be used to model large‐scale masonry structure response. Copyright © 2006 John Wiley & Sons, Ltd.     

Two‐scale diffusion–deformation coupling model for material deterioration involving micro‐crack propagation

In this paper, we introduce a two‐scale diffusion–deformation coupled model that represents the aging material deterioration of two‐phase materials involving micro‐crack propagations. The mathematical homogenization method is applied to relate the micro‐ and macroscopic field variables, and a weak coupling solution method is employed to solve the two‐way coupling phenomena between the diffusion of scalar fields and the deformation of quasi‐brittle solids. The macroscopic mechanical behavior represented by the derived two‐scale two‐way coupled model reveals material nonlinearity due to micro‐scale cracking induced by the scalar‐field‐induced deformation, which can be simulated by the finite cover method. After verifying the fundamental validity of the proposed model and the analysis method, we perform a simple numerical example to demonstrate their ability to predict aging material deterioration. Copyright © 2010 John Wiley & Sons, Ltd.     

Numerical modeling of the in-plane behavior of historical brick masonry walls

This paper addresses numerical modeling of historical brick masonry subjected to in-plane loads. Based on experimentally determined material parameters for the masonry constituents brick and mortar numerical simulations of masonry specimens under various loading conditions are performed. These simulations aim at calibrating and validating parameter sets for historical brick masonry on the meso-scale. For computations on a larger scale, a macro-model in the framework of multi-surface plasticity theory is implemented in a finite element program, which captures the fundamental failure modes of in-plane loaded masonry. Material parameters for the macro-model are derived in a numerical homogenization procedure yielding the transition from the meso-scale to the macro-scale. Results from comparative simulations on discrete and homogenized models show both efficiency and limitations of the proposed methodology.     

Thermal contact conductance characterization via computational contact homogenization: A finite deformation theory framework

In order to predict the macroscopic thermal response of contact interfaces between rough surface topographies, a computational contact homogenization technique is developed at the finite deformation regime. The overall homogenization framework transfers macroscopic contact variables, such as surfacial stretch, pressure and heat flux, as boundary conditions on a test sample within a micromechanical interface testing procedure. An analysis of the thermal dissipation within the test sample reveals a thermodynamically consistent identification for the macroscopic thermal contact conductance parameter that enables the solution of a homogenized thermomechanical contact boundary value problem based on standard computational approaches. The homogenized contact response effectively predicts a temperature jump across the macroscale contact interface. The strong dependence of this homogenized response on macroscale solution variables of interest is demonstrated via representative three‐dimensional numerical investigations. The proposed contact homogenization framework is suitable for the analysis of similar energy transport phenomena across heterogeneous contact interfaces where the investigation of the sources for energy dissipation is of concern. Copyright © 2010 John Wiley & Sons, Ltd.     

Multi‐scale modeling of moving interface problems with flux and field jumps: Application to oxidative degradation of ceramic matrix composites

Problems involving reaction and species diffusion involve field and flux jumps at a moving reaction front. In multi‐scale problems such as carbon fiber composite oxidation, these effects need to be tracked at the microscopic scale of individual carbon fibers. A multi‐scale model is derived in this paper for predicting species distribution in such problems using a fully coupled multi‐scale homogenization approach. The homogenized fluxes from the micro‐scale are derived using Hill's macro‐homogeneity condition accounting for both flux jumps and species density field jumps at the reacting interface in the micro‐scale unit cell. At the macro‐scale, the competition between the transport of reacting species (oxygen) and the reaction product (carbon dioxide) is modeled using homogenized mass conservation equations. The moving reaction front in carbon fibers at the micro‐scale is tracked using level set method and an adaptive meshing strategy. The macroscopic weight loss of the composite when exposed to oxygen is simulated as a function of time using a coupled finite element methodology at various locations in a validated macroscopic model. Copyright © 2010 John Wiley & Sons, Ltd.     

Finite heterogeneous element method using sliced microstructures for linear elastic analysis

A new finite heterogeneous element consisting of sliced microstructures (FHES) is applied in a multi?scale technique. The FHES represents a heterogeneous material with microscopic constituents without homogenization or microscopic finite element analysis. A representative volume element extracted from a heterogeneous structure is thinly sliced. Each slice is modeled as a combined spring to calculate properties of the FHES. Each FHES has the same number of nodes as an ordinary finite element, and the macroscopic analysis cost is the same as that for ordinary finite element analysis. However, the FHES retains information about the microscopic material layout (i.e., the distribution of a material's property) in itself that is lost during homogenization. In the proposed approach, materials are not homogenized. The FHES does not have a constant (homogenized) material property and can ‘change stiffness’ depending on its deformation behavior. This reduces error due to coarse?graining and allows us to calculate the macroscopic deformation behavior with sufficient accuracy even if a large gradient of strain is generated in the macroscopic field. The novelty of the research is the development of rational heterogeneous finite elements. The paper presents the theory behind the FHES and its practical application to a linear elastic problem. Copyright © 2014 John Wiley & Sons, Ltd.     

A homogenization method for ductile‐brittle composite laminates at large deformations

This paper presents a high fidelity homogenization method for periodically layered composite structures that accounts for plasticity in the matrix material and quasi‐brittle damage in the reinforcing layers, combined with strong geometrical nonlinearities. A set of deliberately chosen internal kinematic variables results in a rigorous representation of the kinematics of the 2 constituents, which in turn allows for complex constitutive laws per constituent to be used directly in the formulation. The model accounts for hyper‐elastoplastic behavior in the matrix phase and hyper‐elastic behavior in the reinforcement as well as for the bending stiffness of the reinforcement layers. Additionally to previously proposed models, the present method includes Lemaitre‐type damage for the reinforcement, making it applicable to a wider range of engineering applications. The capability of the proposed method in representing the combined effect of plasticity, damage, and buckling at microlevel within a homogenized setting is demonstrated by means of direct comparisons to a reference discrete model.     

Multiscale modeling of impact on heterogeneous viscoelastic solids containing evolving microcracks

Multiscale computational techniques play a major role in solving problems related to viscoelastic composites due to the complexities inherent to these materials. In this paper, a numerical procedure for multiscale modeling of impact on heterogeneous viscoelastic solids containing evolving microcracks is proposed in which the (global scale) homogenized viscoelastic incremental constitutive equations have the same form as the local‐scale viscoelastic incremental constitutive equations, but the homogenized tangent constitutive tensor and the homogenized incremental history‐dependent stress tensor at the global scale depend on the amount of damage accumulated at the local scale. Furthermore, the developed technique allows the computation of the full anisotropic incremental constitutive tensor of viscoelastic solids containing evolving cracks (and other kinds of heterogeneities) by solving the micromechanical problem only once at each material point and each time step. The procedure is basically developed by relating the local‐scale displacement field to the global‐scale strain tensor and using first‐order homogenization techniques. The finite element formulation is developed and some example problems are presented in order to verify the approach and demonstrate the model capabilities. Copyright © 2009 John Wiley & Sons, Ltd.     

Multi-scale computational method for elastic bodies with global and local heterogeneity

A multi-scale computational method using the homogenization theory and the finite element mesh superposition technique is presented for the stress analysis of composite materials and structures from both micro- and macroscopic standpoints. The proposed method is based on the continuum mechanics, and the micro–macro coupling effects are considered for a variety of composites with very complex microstructures. To bridge the gap of the length scale between the microscale and the macroscale, the homogenized material model is basically used. The classical homogenized model can be applied to the case that the microstructures are periodically arrayed in the structure and that the macroscopic strain field is uniform within the microscopic unit cell domain. When these two conditions are satisfied, the homogenization theory provides the most reliable homogenized properties rigorously to the continuum mechanics. This theory can also calculate the microscopic stresses as well as the macroscopic stresses, which is the most attractive advantage of this theory over other homogenizing techniques such as the rule of mixture. The most notable feature of this paper is to utilize the finite element mesh superposition technique along with the homogenization theory in order to analyze cases where non-periodic local heterogeneity exists and the macroscopic field is non-uniform. The accuracy of the analysis using the finite element mesh superposition technique is verified through a simple example. Then, two numerical examples of knitted fabric composite materials and particulate reinforced composite material are shown. In the latter example, a shell-solid connection is also adopted for the cost-effective multi-scale modeling and analysis. This revised version was published online in June 2006 with corrections to the Cover Date.     

An adaptive multi‐scale approach to the modelling of masonry structures

We present an adaptive multi‐scale approach for predicting the mechanical behaviour of masonry structures modelled as dynamic frictional multi‐body contact problems. In this approach, the iterative splitting of the contact problem into normal contact and frictional contact is combined with a semismooth Newton/primal‐dual active‐set procedure to calculate deformations and openings in the model structures. This algorithm is then coupled with a novel adaptive multi‐scale technique involving a macroscopic scale, which is the size of the masonry structure, and a mesoscopic scale, which is the size of the constituents (bricks, stone‐blocks), to predict appearance of dislocations and stress distribution in large‐scale masonry structures. Comparisons of the numerical results with data from experimental tests and from practical observations illustrate the predictive capability of the multi‐scale algorithm. Copyright © 2008 John Wiley & Sons, Ltd.     

The fracture process in quasi‐brittle materials simulated using a lattice dynamical model

The damage process in quasi‐brittle materials is characterized by the evolution of a micro‐crack field, followed by the joining of micro‐cracks, stress localization and crack instability. In network models, masses are lumped at nodal points which are interconnected by one‐dimensional elements with a bilinear constitutive relation, considering the energy consistency during the simulated process. In order to replicate the material imperfections, to render a realistic behaviour in damage localization, the model has not only random elastic and rupture properties, but also a geometric perturbation. In the present paper 2D plates with different levels of brittleness are simulated. The numerical results are presented in terms of global stress vs strain diagram, final network configuration, energy balance during the process and as geometric damage evolution. Therefore, the predictive potential of the lattice discrete element model to capture fracture processes in quasi‐brittle materials is demonstrated.     

Multiscale thermomechanical contact: Computational homogenization with isogeometric analysis

A computational homogenization framework is developed in the context of the thermomechanical contact of two boundary layers with microscopically rough surfaces. The major goal is to accurately capture the temperature jump across the macroscopic interface in the finite deformation regime with finite deviations from the equilibrium temperature. Motivated by the limit of scale separation, a two‐phase thermomechanically decoupled methodology is introduced, wherein a purely mechanical contact problem is followed by a purely thermal one. In order to correctly take into account finite size effects that are inherent to the problem, this algorithmically consistent two‐phase framework is cast within a self‐consistent iterative scheme that acts as a first‐order corrector. For a comparison with alternative coupled homogenization frameworks as well as for numerical validation, a mortar‐based thermomechanical contact algorithm is introduced. This algorithm is uniformly applicable to all orders of isogeometric discretizations through non‐uniform rational B‐spline basis functions. Overall, the two‐phase approach combined with the mortar contact algorithm delivers a computational framework of optimal efficiency that can accurately represent the geometry of smooth surface textures. Copyright © 2013 John Wiley & Sons, Ltd.     

Multiscale design of three-dimensional nonlinear composites using an interface-enriched generalized finite element method

A computational framework is developed to model and optimize the nonlinear multiscale response of three-dimensional particulate composites using an interface-enriched generalized finite element method. The material nonlinearities are associated with interfacial debonding of inclusions from a surrounding matrix which is modeled using C⁻¹ continuous enrichment functions and a cohesive failure model. Analytic material and shape sensitivities of the homogenized constitutive response are derived and used to drive a nonlinear inverse homogenization problem using gradient-based optimization methods. Spherical and ellipsoidal particulate microstructures are designed to match a component of the homogenized stress-strain response to a desired constructed macroscopic stress-strain behavior.     

A Fourier transformation‐based method for gradient‐enhanced modeling of fatigue

A key limitation of the most constitutive models that reproduce a degradation of quasi‐brittle materials is that they generally do not address issues related to fatigue. One reason is the huge computational costs to resolve each load cycle on the structural level. The goal of this paper is the development of a temporal integration scheme, which significantly increases the computational efficiency of the finite element method in comparison to conventional temporal integrations. The essential constituent of the fatigue model is an implicit gradient‐enhanced formulation of the damage rate. The evolution of the field variables is computed as a multiscale Fourier series in time. On a microchronological scale attributed to single cycles, the initial boundary value problem is approximated by linear BVPs with respect to the Fourier coefficients. Using the adaptive cycle jump concept, the obtained damage rates are transferred to a coarser macrochronological scale associated with the duration of material deterioration. The performance of the developed method is hence improved due to an efficient numerical treatment of the microchronological problem in combination with the cycle jump technique on the macrochronological scale. Validation examples demonstrate the convergence of the obtained solutions to the reference simulations while significantly reducing the computational costs.     

Topology optimization of multiscale elastoviscoplastic structures

This paper extends current concepts of topology optimization to the design of structures made of nonlinear microheterogeneous materials. The objective is to maximize the macroscopic structural stiffness for a prescribed material volume usage while accounting for the nonlinearity and the microstructure of the material. The resulting design problem considers two scales: the macroscopic scale at which the optimization is performed and the microscopic scale at which the material heterogeneities and the nonlinearities are observed. The topology optimization at the macroscopic scale is performed by means of the bi‐directional evolutionary structural optimization method. The solution of the macroscopic boundary value problem requires as inputs the effective constitutive response with full consideration of the microstructure. While computational homogenization methods such as the FE² method could be used to solve the nonlinear multiscale problem, the associated numerical expense (CPU time and memory) is highly unacceptable. In order to regain the computational feasibility of the computational scale transition, a recent model reduction technique of the authors is employed: the potential‐based reduced basis model order reduction with graphics processing unit acceleration. Numerical examples show the efficiency of the resulting nonlinear two‐scale designs. The impact of different load amplitudes on the design is examined. Copyright © 2015 John Wiley & Sons, Ltd.     

Many-scale finite strain computational homogenization via Concentric Interpolation

A method for efficient computational homogenization of hyperelastic materials under finite strains is proposed. Multiple spatial scales are homogenized in a recursive procedure: starting on the smallest scale, few high fidelity FE computations are performed. The resulting fields of deformation gradient fluctuations are processed by a snapshot POD resulting in a reduced basis (RB) model. By means of the computationally efficient RB model, a large set of samples of the homogenized material response is created. This data set serves as the support for the Concentric Interpolation (CI) scheme, interpolating the effective stress and stiffness. Then, the same procedure is invoked on the next larger scale with this CI surrogating the homogenized material law. A three-scale homogenization process is completed within few hours on a standard workstation. The resulting model is evaluated within minutes on a laptop computer in order to generate fourth-scale results. Open source code is provided.     

Nonlinear homogenization techniques to solve masonry structures problems

The behaviour of masonry material subjected to different in-plane loading combination is studied in this work. The masonry is considered as a periodic composite material composed by a regular distribution of brick and mortar and it is analyzed using a homogenization technique. The mechanical properties of the masonry, as an orthotropic homogeneous material, depend on the geometrical and mechanical properties of the components based on the study of the equilibrium and compatibility of a basic cell. The masonry is a frictional material and its behaviour depends on the loading direction, for these reasons, a unilateral damage model is chosen for the analysis. This model describes the behaviour of brittle materials subjected to tension-compression cyclic loads based on the introduction of two damage variables and it assumes that the damage is due to the beginning and growth of cracks only in the mortar joints. It is considered that the bricks have a linear elastic constitutive relationship. Numerical applications are performed with a nonlinear finite element code in order to test the proposed procedure by comparing the results with those available in the literature and also with experimental data.     

Micro-mechanical approach and algorithm for the study of damage appearance in elastomer composites

This paper aims with the development of a numerical non-incremental algorithm well suited for the homogenization of non-linearly elastic composites. On the one hand it allows to obtain the macroscopic behavior of such composites, without any numerical overcosts and on the other hand to estimate with a good accuracy the local forces sustained by the composite constituents. This latter property is very helpful in a further step when dealing with the simulation of damage evolution in structures of industrial relevance. In order to illustrate these homogenization and fast solution algorithm, they are both applied to the case of unidirectional composites. Thanks to the use of the finite element technique, the homogenized responses corresponding to some macroscopic loadings are first computed. Next, the analysis of three kinds of damage, which may be expected in such composites, is carried out. Finally the numerical performances of the proposed algorithm are presented and compared to these induced by an incremental algorithm.