# A hysteretic multiscale formulation for nonlinear dynamic analysis of composite materials

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## Abstract

A new multiscale finite element formulation is presented for nonlinear dynamic analysis of heterogeneous structures. The proposed multiscale approach utilizes the hysteretic finite element method to model the micro-structure. Using the proposed computational scheme, the micro-basis functions, that are used to map the micro-displacement components to the coarse mesh, are only evaluated once and remain constant throughout the analysis procedure. This is accomplished by treating inelasticity at the micro-elemental level through properly defined hysteretic evolution equations. Two types of imposed boundary conditions are considered for the derivation of the multiscale basis functions, namely the linear and periodic boundary conditions. The validity of the proposed formulation as well as its computational efficiency are verified through illustrative numerical experiments.

### Keywords

Heterogeneous materials Multiscale finite elements Hysteresis Nonliner dynamics## 1 Introduction

Composite materials have long been utilized in construction and manufacturing in various forms. Nowadays, their scope of applicability spans a large area including, though not limited to the aerospace, automobile and sports industries [28]. Their appeal lies in the fact that composites exhibit some enhanced mechanical properties, such as high strength to weight ratio, high stiffness to weight ratio, high damping, negative Poisson’s ratio and high toughness. In the field of Civil Engineering, composite materials are used either in the form of fiber reinforcing or more recently as textile composites in various applications such as retrofitting and strengthening of damaged structures [11], or supporting cables for cable stayed bridges and high strength bridge decks [26] amongst many others. This vast and multidisciplinary implementation of composites results in the need for better understanding of their mechanical behaviour. Research efforts are oriented towards further improving the mechanical properties of composites while at the same time alleviating some of their disadvantages such as high production/ implementation costs and damage susceptibility [52].

Composites are mixtures of two or more mechanically separable solid materials. As such, they exhibit a heterogeneous micro-structure whose specific morphology affects the mechanical behaviour of the final product [34]. Within this framework, composites are intrinsically multiscale materials since the scale of the constituents is of lower order than the scale of the resulting material. Furthermore, the resulting structure, that is an assemblage of composites, can be of an even larger scale than the scale of the constituents (e.g. a textile strengthened masonry structure [24], a bio-sensor consisting of several nano-wires [44]). Thus, the required modelling approach has to account for such a level of detail that spreads through scales of significantly different magnitude. Throughout this paper, the term macroscopic (or coarse) scale corresponds to the structural level whereas the term microscopic (or fine) scale corresponds to the composite micro-structure properties such as the sizes, morphologies and distributions of heterogeneities that the material consists of.

The derivation of reliable numerical models for the simulation of mechanical processes occurring across multiple scales can aid both the design and/or optimization of new composite systems. Using appropriate modelling assumptions accounting for plasticity and damage [38], estimates on the damage susceptibility of composites can be readily derived and parametric models can be established where micro-material properties are identified based on experimentally measured quantities.

Modelling of structures that consist of composites could be accomplished using the standard finite element method [65]. However, a finite element model mesh accounting for each micro-structural heterogeneity would require significant computational resources (both in CPU power and storage memory). In general, the computational complexity of a finite-element solution procedure is of the order of \(O\left( n_{z} ^{3/2} \right) \) where \(n_{z} \) is the number of degrees of freedom of the underlying finite element mesh [37]. Therefore, the finite element scheme is usually restricted to small scale numerical experiments of a representative volume element (RVE) [1, 53].

To properly capture the micro-structural effects in the large scale more refined methods have been developed. Instead of implementing the standard finite element method, upscaled or multiscale methods have been proposed to account for such types of problems, therefore significantly reducing the required computational resources [36, 59, 67]. Upscaling techniques rely on the derivation of analytical forms to describe a coarser (i.e. large scale) model based on smaller scale properties [40]. Usually this is accomplished by analytically defining a homogenized constitutive law from the individual constitutive relations of the constituents. Thus, a continuous mathematical model that is problem dependent replaces the fine scale information. On the other hand, multiscale methods use the fine scale information to formulate a numerically equivalent problem that can be solved in a coarser scale, usually through the finite element method [2, 55]. An extensive review on the subject can be found in [33].

In general, multiscale methods can be separated in two groups, namely multiscale homogenization methods [45] and multiscale finite element methods (MsFEMs) [20]. Within the framework of the averaging theory for ordinary and partial differential equations, multiscale homogenization methods are based on the evaluation of an averaged strain and corresponding stress tensor over a predefined space domain (i.e. the RVE) [5]. Amongst the various homogenization methods proposed [25], the asymptotic homogenization method has been proven efficient in terms of accuracy and required computational cost [61].

However, these methods rely on two basic assumptions, namely the full separation of the individual scales and the local periodicity of the RVEs. In practice, the heterogeneities within a composite are not periodic as in the case of fiber-reinforced matrices . In order to adapt to general heterogeneous materials, the size of RVE must be sufficiently large to contain enough microscopic heterogeneous information [3, 54], thus increasing the corresponding computational cost. Furthermore, in an elasto-plastic problem, periodicity on the RVEs also dictates periodicity on the damage induced which could result in erroneous results.

The MsFEM is a computational approach that relies on the numerical evaluation of a set of micro-scale basis functions. These are used to map the micro-structure information onto the larger scale. These basis functions depend both on the micro-structural geometry and constituent material properties. Therefore, the heterogeneity can be accounted for through proper manipulation of the underlying finite element meshes defined at different scales. MsFEM was first introduced in [31] although a variant of the method was earlier introduced in [7] for one-dimensional problems and later for the multi-dimensional case [6]. Along the same lines, domain-decomposition [66] and sub-structuring [68] approaches have also been introduced for the solution of elastic micro-mechanical assemblies.

Although MsFEMs have been extensively used in linear and nonlinear flow simulation analysis [19, 27] the method has not been implemented in structural mechanics problems. This is attributed to the inherent inability of the method to treat the bulk expansion/ contraction phenomena (i.e. Poisson’s effect). To overcome this problem, the enhanced multiscale finite element method (EMsFEM) has been proposed for the analysis of heterogeneous structures [62]. EMsFEM introduces additional coupling terms into the fine-scale interpolation functions to consider the coupling effect among different directions in multi-dimensional vector problems. The method has been also extended to the nonlinear static analysis of heterogeneous structures [63]. Recently, the geometric multiscale finite element method was introduced [14] along with a novel approach for the numerical derivation of displacement based shape functions for the case of linear elastic problems.

However, a limiting factor in a nonlinear analysis procedure, is the fact that the numerical basis functions need to be evaluated at every incremental step due to the progressive failure of the constituents. In [63] the initial stiffness approach is implemented for the solution of the incremental governing equations, thus avoiding the re-evaluation of the basis functions. Nevertheless, this method is known to face serious convergence problems and usually requires a large number of iterations to achieve convergence [46]. The computational cost increases even further for the case of a nonlinear dynamic analysis, where a time integration scheme is also required on top of the iterative procedure [30].

In this work, a modified multiscale finite element analysis procedure is presented for the nonlinear static and dynamic analysis of heterogeneous structures. In this, the evaluation of the micro-scale basis functions is accomplished within the hysteretic finite element framework [56]. In the hysteretic finite element scheme, inelasticity is treated at the element level through properly defined evolution equations that control the evolution of the plastic part of the deformation component. Using the principle of virtual work, the tangent stiffness matrix of the element is replaced by an elastic and a hysteretic stiffness matrix both of which remain constant throughout the analysis.

Along these lines, a multi-axial smooth hysteretic model is implemented to control the evolution of the plastic strains that is derived on the basis of the Bouc–Wen model of hysteresis [10]. The smooth model used in this work accounts for any kind of yield criterion and hardening law within the framework of classical plasticity [38]. Smooth hysteretic modelling has proven very efficient with respect to classical incremental plasticity in computationally intense problems such as nonlinear structural identification [12, 35, 43], hybrid testing [13] and stochastic dynamics [58]. Furthermore, the proposed hysteretic scheme can be extended to account for cyclic damage induced phenomena such as stiffness degradation and strength deterioration [4, 22]. The thermodynamic admissibility of smooth hysteretic models with stiffness degradation has proven on the basis of an equivalence principle to the endochronic theory of plasticity [21]. However, such concepts are beyond the scope of this work.

The present paper is organized as follows. The smooth hysteretic model together with the hysteretic finite element scheme that form the basis of the proposed method are described in Sect. 2. In Sect. 3, the enhanced multiscale finite element method (EMsFEM) is briefly described. In Sect. 4, the proposed hysteretic multiscale finite element method is presented. The method used for the solution of the governing equations at the coarse mesh is described in Sect. 5. The latter is based on the simulation of the governing equations of motion in time using the Newmark direct-integration method [17]. In Sect. 6 a set of benchmark problems is presented to verify both the accuracy and the efficiency of the proposed multiscale formulation.

## 2 Hysteretic modelling

### 2.1 Multiaxial modelling of hysteresis

Classical associative plasticity is based on a set of four governing equations, namely the additive decomposition of strain rates, the flow rule, the hardening rule and the consistency condition [38, 49].

### 2.2 Test case

In Fig. 2c, the time history of the smoothed Heaviside function \(H_{1} \) is presented. The graph displays subsequent regions of elastic loading, yielding and elastic unloading corresponding to the stress–strain hysteresis loop presented in Fig. 1b. In Fig. 2d \(H_{1} \) is multiplied by the sign of the corresponding normal stress and plotted with respect to the strain. Small values of imposed strain correspond to small values of \(H_{1} \) and the elastic response is retrieved in Fig. 2b. Finally, in Fig. 2e and f the evolution of function \(H_{2} \) is presented with respect to time and strain respectively. As predicted by the model, in elastic loading it holds that \(H_{1} =1\) in both directions of strain. However, during unloading the value of \(H_{1} \) turns into \(H_{1} =\beta -\gamma =-0.8\). As long as the value \(H_{1} \) is not sufficiently small, the stiffness retrieved during unloading is different than that of the elastic loading.

The smooth hysteretic model implemented in this work is based on the Karray–Bouc model of hysteresis [16]. However, instead of relying on the assumptions of von-Mises yield and linear kinematic hardening, the constitutive formulation proposed herein accounts for any type of yield function and kinematic hardening, within the framework of classical rate-independent plasticity. The advantages of a Bouc–Wen type model accounting for deformation dependent hardening were recently highlighted in [47, 60] where the linear kinematic hardening coefficient of the Bouc–Wen model is substituted by a continuous function derived from calibration of experimental data.

### 2.3 The hysteretic finite element scheme

The exact form of the interpolation matrix \(\left[ N_{\sigma } \right] \) depends on the element formulation and is also relevant to the stress recovery procedure implemented within the finite element formulation [56]. In this work the collocation points are chosen to coincide with the Gauss quadrature points where stresses are evaluated in standard FEM [65]. Furthermore, smooth evolution equations of the form of relation (26) are implemented. The classical formulation of classical plasticity however can be also used by considering the flow rule defined in relation (3).

## 3 The enhanced multiscale finite element method

### 3.1 Overview

In the MsFEM the structure consists of two layers, namely a fine-meshed layer up to the scale of the heterogeneities and a coarse mesh of the macro-scale where the solution of the discrete problem is performed. In Fig. 3, the fine element mesh consists of 54 quadrilateral micro-elements and 70 micro-nodes while the coarse mesh consists of 6 quadrilateral macro-elements and 12 macro-nodes. Furthermore, two displacement fields are established corresponding to each level of discretization.

Instead of implementing a one-step approach, i.e. solving the fine meshed FEM model, a two-step solution procedure is performed. In the first step, a mapping is numerically evaluated that maps the fine mesh within each coarse-element to the corresponding macro-nodes. Next, the solution procedure is performed in the coarse mesh. Finally, the fine-mesh stress and strain history is retrieved by implementing the inverse micro-mapping procedure onto the results obtained on the coarse mesh.

### 3.2 Numerical evaluation of micro-scale basis functions

The numerical mapping is established by considering each type of coarse element and its corresponding fine mesh as a sub-structure. Considering groups of coarse-elements that bare the same geometrical and mechanical properties these coarse element types can be grouped into sets of representative volume elements (RVE). In this work the term RVE will be used to denote the coarse element together with its underlying fine mesh structure as in [62]. For each RVE a homogeneous equilibrium equation is established considering specific boundary conditions. The solution of this equilibrium problem forms a vector of basis functions that maps the displacement components of the fine mesh within the element to the macro-nodes of the RVE.

In Fig. 4, the RVE finite element mesh of the periodic composite structure (Fig. 3) is presented. This mesh is assigned a local nodal numbering since it is solved as an independent structure.

Matrix \(\left[ N\right] _{m}\) in Eq. (45) is a \(32\times 8\) matrix containing the components of the micro-basis shape functions evaluated at the nodal points \(\left( x_{j} ,y_{j} \right) , j=1, \ldots , 16\) of the micro-mesh. According to the property introduced in Eq. (40), each column of \(\left[ N\right] _{m} \) corresponds to a deformed configuration of the RVE where the corresponding macro-degree of freedom is equal to unity and all of the remaining macro-degrees of freedom are equal to zero.

The RVE stiffness matrix \(\left[ K\right] _{RVE}\) is formulated using the standard finite element method [8]. Thus, \(\left[ K\right] _{RVE}\) is assembled by evaluating the contribution of the individual stiffness of each micro-element in the stiffness of the RVE, the latter being considered as a stand-alone structure. In this work, the direct stiffness method [65] is implemented for that purpose. In the example case presented in Fig. 4, the RVE consists of \(16\) nodes and \(9\) quadrilateral plane stress elements. Therefore, the corresponding \(\left[ K\right] _{RVE}\) is a \(32 \times 32\) matrix.

Each column of the shape function matrix \(\left[ N\right] _{m} \) in Eq. (45) corresponds to a displacement pattern derived from the solution of the linear system introduced in Eq. (47) for a specific set of boundary conditions. Thus, for the example case presented in Fig. 4, eight (8) different prescribed displacement vectors \(\left\{ \bar{d}\right\} \) need to be defined and the corresponding solutions need to be performed. In this work, the solution of the boundary value problem established in Eq. (47) is performed using the Penalty method [9, 23].

The type of the boundary conditions implemented for the evaluation of the micro-basis shape functions significantly affects the accuracy of EMsFEM. Four different types of boundary conditions are established in the literature namely linear boundary conditions, periodic boundary conditions, oscillatory boundary conditions with oversampling and periodic boundary conditions with oversampling. In the first case, the displacements along the boundaries of the coarse element are considered to vary linearly. Periodic boundary conditions are established by considering that the displacement components of periodic nodes lying on the boundary of the coarse element differ by a fixed quantity that varies linearly along the boundary of the coarse element. The oscillatory boundary condition method with oversampling considers a super-element of the coarse element whose basis functions are evaluated using the linear boundary condition approach. Finally, the periodic boundary conditions with oversampling combine the oversampling technique with the periodic boundary condition method, thus allowing for the implementation of the latter in non-periodic RVE meshes [39, 63].

In this work, the cases of linear and periodic boundary conditions are considered. An example on the application of the periodic boundary conditions is described in the Appendix, however further details on the procedure implemented for the derivation of the micro-basis functions can be found in [20, 63].

### 3.3 Macro equivalent micro-nodal forces

The evaluation of the “perturbed” micro-displacement vector is crucial for the efficiency of the multiscale scheme and will be further treated in Sect. 5.2 where the numerical aspects of the proposed method are presented. Equivalently, the actual stress field within the micro-element needs to be evaluated taking into account the contribution of both the micro-forces evaluated from the micro to macro-mapping and the “perturbed” forces.

## 4 The hysteretic multiscale analysis scheme

### 4.1 Equilibrium in the fine scale

Equation (52) is a multiscale equilibrium equation involving the displacement vector \(\left\{ d\right\} _{M}\) that accounts for the nodal displacements of the coarse-element nodes and the plastic part of the strain tensor \(\left\{ \varepsilon _{cq}^{pl} \right\} _{m\left( i\right) }\) that is evaluated at collocation points within the micro-scale element mesh. Using the micro-displacement to macro-displacement interpolation relation [Eq. (41)] the micro-element state matrices, namely the elastic stiffness matrix and the hysteretic matrix, defined in Eqs. (35) and (36) respectively are mapped onto their multiscale counterparts \(\left[ k^{el} \right] _{m\left( i\right) }^{M}\) and \(\left[ k^{h} \right] _{m\left( i\right) }^{M}\).

The derived multiscale elastic stiffness and hysteretic matrices are constant and need only be evaluated once during the analysis procedure. Therefore, the corresponding micro-basis functions introduced in relation (47) are also evaluated once, thus significantly reducing the required computational cost.

### 4.2 Micro to macro scale transition

Equations (66) and (67) are used to derive the equilibrium equation at the structural level as will be described in the next section. In analogy to the equilibrium equation of the micro-element (mapped onto the coarse element) defined in relation (56), the hysteretic force nodal load vector \(\left\{ f_{h} \right\} _{M}\) is the nonlinear correction to the external force vector \(\left\{ f\right\} _{M}\) at the coarse element level. However, the evolution of \(\left\{ f_{h} \right\} _{M}\) is manifested through the evolution of the plastic deformations at the micro-level and is therefore the link between the inelastic processes occurring at the fine scale and the macroscopically observed nonlinear structural behaviour.

The coarse element stiffness matrices are evaluated considering only their individual micro-mesh properties. Thus, they are independent and their evaluation can be performed in parallel.

## 5 Solution procedure

### 5.1 Governing equations in the macro-scale

The formulation of the mass matrix, defined at the coarse mesh, is established on the grounds of the micro-basis shape functions presented in Sect. 3. This leads to a multi-scale consistent mass matrix formulation where the derived mass matrix is non-diagonal. Well-known mass diagonalization techniques can then be performed to derive an equivalent lumped mass matrix [18]. However, the implications of such approaches are beyond the scope of this work. Similarly, the viscous damping can be of either the classical or non-classical type [17].

The global stiffness matrix of the structure, defined at the coarse mesh, is formulated through the direct stiffness method from the contributions of the coarse elements equivalent stiffness matrices \(\left[ K^{el}\right] _{CR\left( j\right) }^{M} \) [Eq. (64)]. Accordingly, the \(\left( ndof_{M} \times 1\right) \) vector \(\left\{ U\right\} _{M} \) consists of the nodal macro-displacements.

The external load vector \(\left\{ F\right\} _{M} \) and the hysteretic load vector \(\left\{ F_{h} \right\} _{M} \) are assembled considering the equilibrium of the corresponding elemental contributions \(\left\{ f\right\} _{M} \) and \(\left\{ f_{h} \right\} _{M} \), defined in Eqs. (58) and (67) respectively, at coarse nodal points.

Equations (69) are independent and thus can be solved in the micro-element level resulting in an implicitly parallel scheme. Both relations (69) and (72) depend on the current micro-stress state within each micro-element and consequently on the micro-strain and micro-displacement distribution. Thus, a procedure needs to be established that downscales the macro-displacements \(\left\{ U\right\} _{M} \) evaluated at the coarse mesh to the micro-displacements of the micro-nodes within the fine mesh.

### 5.2 Downscale computations

### 5.3 Newton iterative scheme

In this section, the nonlinear static analysis procedure implemented is presented for clarity, while the dynamic case is treated accordingly using the Newmark average acceleration method to integrate the equations of motion [17].

- 1.
Solve Eq. (49) for the fine-scale residual forces evaluated at the beginning of the step and retrieve the perturbed displacement vector \({}_{i}^{1} \left\{ \Delta \tilde{d}\right\} _{m(i)} \)

- 2.Evaluate the fine-scale incremental displacement components from Eq. (73)$$\begin{aligned} {}_{i}^{1} \left\{ \Delta d\right\} _{m(i)} =\left[ N\right] _{m(i)} {_{i}^{1} \left\{ d\right\} _{M}} \end{aligned}$$(86)
- 3.The total strains at the collocation points are then derived as$$\begin{aligned} {}_{i}^{1} \left\{ \varepsilon _{cq} \right\} _{m\left( i\right) }^{iq}&= \left[ B\left( \xi ,\eta \right) \right] \left( _{i-1} \left\{ d\right\} +_{i}^{1} \Delta \left\{ d\right\} _{m\left( i\right) }\right. \nonumber \\&\left. +{}_{i}^{1} \left\{ \Delta \tilde{d}\right\} _{m(i)} \right) \end{aligned}$$(87)

Relations (80)–(89) define an explicit Newton solution scheme, where the state matrices remain constant throughout the analysis procedure. The resulting iterative scheme relies on constant global matrices and does not require the re-evaluation and re-factorization of the global stiffness matrix. Inelasticity is introduced as an additional load vector that acts as a nonlinear correction to the externally applied load. This hysteretic load vector is evaluated by considering the evolution of the plastic strain at collocation points defined in the micro-scale.

Consequently, the re-evaluation of the micro to macro numerical mapping [relation (47)] is not required either. The numerical schema described herein can be extended for the case of nonlinear dynamic analysis by introducing a time-marching method on top of the iterative procedure. Both the static and dynamic analysis case has been treated and their corresponding results are discussed in the Sect. 6.

### 5.4 Comparison to the classical iterative solution procedure

The EMsFE method significantly reduces the size of the finite element mesh to be solved, since the solution procedure is applied in the coarse mesh. This is accomplished by the evaluation of a numerical mapping that interpolates the displacement components of the fine mesh onto the displacement components of the coarse mesh through relation (39).

However, in the proposed computational scheme that is schematically presented in Fig. 7 the need for re-evaluation of the micro to macro displacement mapping is alleviated. This is accomplished by treating inelasticity at the local micro-level through the introduction of the additional hysteretic components [Eq. (32)]. These, account for the plastic part of the strain tensor, measured at specific collocation points. In this work, these points are so chosen to coincide with the Gauss quadrature points of the micro-elements. The proposed procedure expands the vector of unknown quantities and introduces an additional set of nonlinear equations that need to be solved [Eq. (69)]. However, the solution of these equations is performed at the local micro-level. Each set of equations is independent and can be solved in parallel, thus significantly enhancing the computational efficiency of the proposed scheme.

## 6 Examples

In this section examples are presented for the verification of the proposed methodology. All analyses were performed on an Intel Xeon PC fitted with 16 GB of RAM. The Abaqus commercial code [29] is used for the validation of the derived multiscale numerical scheme. The implementation of the latter has been performed using the FORTRAN 2003 programming language.

### 6.1 Compression experiment of a cubic specimen

The model is considered fixed at its base, while a uniform pressure is applied at its top edge. The elastic parameters considered are \(E_{m} =10\) GPa and \(\nu =0.2\) for the Young’s modulus and the Poisson’s ration respectively. An associative linear Drucker–Prager plasticity model is used to model the nonlinear behaviour of the matrix. The following values are considered for the friction angle and the Drucker–Prager cohesion namely \(\phi =30^{\circ } \) and \(d=2000\) kPa respectively.

The hysteretic multiscale finite element method is implemented considering 8 coarse elements. Each coarse element is meshed into 64 micro-elements so that the total number of fine elements remains equal to 512. The corresponding pressure-displacement path is presented in Fig. 9b. The obtained solution is compared to the derived solution from the standard FE analysis. The difference between the two formulations is less than 1.0 %. Furthermore, while the Abaqus analysis procedure concluded in 51 s, the multiscale analysis module concluded in 13 s resulting in a 70 % reduction of the computational time.

The derived pressure displacement path is presented in Fig. 11a, where the displacement is measured at node #6 (Fig. 10a). Although the multiscale solution with linear boundary conditions succeeds in capturing both the elastic stiffness of the body as well as the maximum attained pressure, the overall difference from the 512 finite element mesh solution is greater than 5 %. On the contrary, the multiscale solution obtained using the periodic boundary HMsFEM solution practically coincides with the FEM solution.

The linear boundary constraint imposed on the coarse element cannot compensate for the curvature variation along the edges of the solid as shown in Fig. 10b. Further increasing the number of coarse elements reduces the discrepancy at the cost of increasing the required computational time. In Fig. 17b, results obtained considering a multiscale model comprising of 64 coarse elements (each one including 8 fine-scale elements) are presented.

### 6.2 Cantilever with periodic micro-structure

Nodes in sector AB are considered fixed in both directions (Fig. 12a. A traction load \(T\) is applied at the free end of the cantilever.

Using the Abaqus commercial code [29] a detailed FEM model is formulated, to serve as a reference model for the validation of the proposed methodology. The derived model consists of 76380 nodes and 75686 quadrilateral plane stress elements.

Due to the periodicity of the structure, a periodic finite element mesh is derived accordingly. Thus, using the multiscale finite element method, a single fine mesh component needs to be evaluated comprising of 353 nodes and 320 quadrilateral plane stress elements. The corresponding coarse-element structure (Fig. 12a) consists of 217 nodes and 180 elements. Therefore, using the proposed methodology, the computational complexity of the initial finite element problem reduced from a magnitude of \(O\left( 76380^{2} \right) \) to that of \(O\left( 353^{2} \right) \).

These differences are observed during the inelastic regime of the cantilever response, with the HMsFEM-L solution being stiffer than the exact one and the HMsFEM-P solution being more flexible than the exact one. In this case, the error introduced by the linear boundary condition assumption are reduced, with respect to the case examined in Example 1. However in the case considered herein, the actual cantilever deformed configuration can be adequately reproduced with a piece-wise linear displacement distribution, provided that the number of coarse elements along the length of cantilever is sufficient enough.

A lumped mass matrix approach is implemented considering the following densities, namely \(\gamma _{m} =1\mathrm{{KN/ m^{3}} } \) and \(\gamma _{i} =0.1\mathrm{{KN/m^{3}}}\) for the matrix and the inclusion respectively. The time history of the tip vertical displacement for the two formulations is presented in Fig. 15a where in the multiscale case both linear (HMsFEM-L) and periodic boundary (HMsFEM-P) conditions are considered. Similar to the monotonic case, the solution derived with linear boundary conditions is stiffer. This is evident during the last cycle of the cantilever response where severe inelastic deformations occur.

However in this case, the relative error between the linear boundary condition case (HMsFEM-L) and the FEM solution assumes the maximum value of 2.75 % while the corresponding error for the HMsFEM-P solution is less than 1.5 %. The evolution of the relative error for the three different models is presented in Fig. 16. The relative error assumes its maximum value at the time instant \(t=8.20\) s where plastic deformation initiates and remains constant for the remaining of the analysis procedure. This error is attributed to the evaluation of the additional “perturbed” micro-displacements that are used to evaluate the total vector of micro-strains [Eqs. (73) and (74)]. As described in Sect. 3.3, the evaluation of the vector of “perturbed” micro-displacements \(\left\{ \tilde{d}\right\} _{m(i)}\) depends on the RVE boundary condition assumption.

### 6.3 Masonry wall under earthquake excitation

Stone and mortar material properties

Stone | Mortar | |
---|---|---|

Young’s modulus (MPa) | 20200 | 3494 |

Poisson’s ratio | 0.2 | 0.11 |

Plasticity | Von-Mises | Mohr–Coulomb |

Friction angle (\(^\circ \)) | – | 21.8 |

Cohesion (MPa) | – | 0.1 |

Yield stress (MPa) | 69.2 | – |

Textile composite material properties

Young’s modulus | \(E_{11} =54000\) | \(E_{22} =53200\) | \(E_{33} =53200\) |
---|---|---|---|

(MPa) | \(E_{12} =53200\) | \(E_{23} =54000\) | \(E_{12} =54000\) |

Poisson’s ratio | \(v_{12} =0.14\) | \(v_{23} =0.2\) | \(v_{13} =0.2\) |

Finally, the proposed formulation concludes in approximately 49 min while the standard FEM procedure requires 195 min, thus leading to a 75 % reduction of the required computational time.

## 7 Conclusions

In this work, a novel multi-scale finite element method is presented for the nonlinear analysis of heterogeneous structures. The proposed method is derived within the framework of the enhanced multiscale finite element method. However, the necessary re-evaluation of the the micro to macro basis functions is avoided by implementing the hysteretic finite element formulation at the micro-level. Consequently, inelasticity is treated at the micro-level through the introduction of local inelastic quantities. These are assembled at the macro-level in the form of an additional load vector that acts as a nonlinear correction to the externally applied loads. As a result, the state matrices of the multiscale problem need only to be evaluated once at the beginning of the analysis procedure.

The evolution of the additional inelastic quantities, e.g. the plastic part of the strain tensor, are bound to evolve according to a generic smooth hysteretic law. The hysteretic model implemented is a generalized form of the Bouc–Wen model of hysteresis, allowing for a more versatile approach on material modelling. In the application section, examples are presented that verify the computational efficiency of the proposed formulation as well as its accuracy.

## 8 Appendix

The stiffness matrix \(\left[ K\right] _{RVE}\) is used to evaluate the micro-basis shape functions that are readily derived as solutions of the boundary value problem defined in relation (47). The boundary conditions imposed are evaluated in such a way that the fundamental property of the micro-basis functions defined in relation (40) holds. A set of values satisfying relations (40) can be retrieved by means of the following reasoning; For the first set of equations (40) to hold it suffices that a micro-basis function mapping the micro-displacement components along \(x\) to a macro-displacement along the same direction \(x\) of a coarse-node is equal to unity at that specific coarse-node and zero to every other coarse-node. Moreover, the second set of equations (40) is satisfied if and only if a micro-basis function mapping the micro-displacement component along \(x\) to the macro-displacement component along the direction \(y\) is equal to zero in every coarse-node.

The periodic boundary conditions introduce a numerical perturbation on the displacement field of periodic boundary nodes. Thus, they can in principle be used in non-periodic media (i.e. RVEs with non-periodic material distribut0ions). However in this case the size of the RVE should be small enough for the considered perturbation to be valid, i.e. for the displacements of periodic boundary nodes to differ by a small variation of the displacement field. Furthermore, the applicability of the method is restricted on periodic micro-element meshes. To alleviate such problems, a procedure has been established for the generalization of the periodic boundary condition assumption allowing its application to non-structured, non-periodic meshes [42]. Also, refined boundary condition assumptions such as the oversampling technique [19] and the generalized periodic boundary condition method (combining periodic boundary conditions with oversampling) have been effectively used in [63] for non-periodic media. The effect of different boundary condition assumptions on the accuracy of the EMsFEM method is examined in [64].

## Notes

### Acknowledgments

This work has been carried out under the support of the Swiss National Science Foundation for Research Grant # 200021_146996: “Hysteretic Multi/Scale Modeling for the Reinforcing of Masonry Structures”.

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