Analysis of planes within reduced micromorphic model

A plane within reduced micromorphic model subjected to external static load is studied using the finite element method. The reduced micromorphic model is a generalized continuum theory which can be used to capture the interaction of the microstructure. In this approach, the microstructure is homogenized and replaced by a reduced micromorphic material model. Then, avoiding the complexity of the microstructure, the reduced micromorphic model is analyzed to reveal the interaction of the microstructure and the external loading. In this study, the three-dimensional formulation of the reduced micromorphic model is dimensionally reduced to address a plane under in-plane external load. The governing system of partial differential equations with corresponding consistent boundary conditions are discretized and solved using the finite element method. The classical and nonclassical deformation measures are then demonstrated and discussed for the first time for a material employing the reduced micromorphic model.

An expected feature of the generalized continua is their numerous material constants which are to be determined for any specific microstructure via real or numerical experiments, e.g. see 13 . Although this task is commonly referred to as a "challenge" for the application of the generalized continua, one should not forget that capturing the nonclassical features are only possible due to the presence of such nonclassical material constants. Nevertheless, the generalization of the continuum theory should be performed with care to employ the most efficient model with the minimum number of material constants for the microstructure and the objective physical phenomenon to be captured. For this purpose, recently, computational homogenization has been successfully used for identifying and realizing a suitable continuum theory [14][15][16][17][18][19][20]38 . To this end, a comprehensive methodology capable of identifying the most efficient continuum theory for a specific heterogeneous material is not yet identified.
In addition to more classical versions of generalized continua, recent developments have been reported in the literature such as relaxed micromorphic model 21 and reduced micromorphic model 22 . These models aim for filling the gaps in the realm of the generalized continua to provide an efficient selection of a generalized model without the need to include unnecessary extensions. In an attempt to reduce the number of material parameters and degrees of freedom, Neff et al. 21 proposed a relaxed micromorphic model which uses the curl of the micro-deformation tensor as a micro-dislocation measure. Furthermore, the coupling between the micro-strain and the macro-strain is eliminated in this model. The relaxed micromorphic model has been used to model the wave propagation in metamaterials [24][25][26][27] .
Another form of the micromorphic model is proposed by Shaat 22 which is referred to as the reduced micromorphic model (referred to, in this text, as RMM). In this model the author considers the coupling between the micro-strain and the macro-strain to measure the concept of so-called residual strain and consequently the residual stress. The model has been employed to investigate the wave propagation in metamaterials and composite materials. An application for the model is also introduced in 28 to define the equivalent shear modulus of composite metamaterials. The reduced micromorphic model is also formulated in orthogonal curvilinear coordinates and an application to a metamaterial hemisphere has been reported using spherical coordinates 29 . Later, Shaat et al. introduced a micromorphic beam theory 30 based on the reduced micromorphic model.
In the absence or due the complexity of an analytical solution 31 , the finite element method is an efficient numerical treatment of continuum theories. In this paper, the reduced micromorphic model is used together with the finite element method for the analysis of a plane. The framework is dimensionally reduced for planar analysis. Such dimension reduction is motivated as it can efficiently be used for the analysis of 2D structures 32 . The corresponding measures of deformation are elaborated to demonstrate the response of a plane within RMM subjected to static in-plane loading.
The paper is organized as follows. The reduced micromorphic model is reviewed in Section "The RMM in Cartesian coordinates". In Section "The RMM model in 2-Dimension", the proposed domain in two dimensions and the planar formulation are presented. After introducing the governing field equations and the variationally consistent boundary conditions, two case studies are defined. In Section "Numerical results". the finite element analysis and numerical simulation of the two case studies are reported. Finally, the conclusion is given in Section "Conclusion".

The RMM in Cartesian coordinates
The kinematical variables. The classical theories of continuum mechanics do not have the ability to represent the nanoscale phenomena. Accordingly, the generalized continua such as the reduced micromorphic model (RMM) is introduced as an alternative for studying such phenomena at the micro-scale level, where the model introduces the micro-strain tensor as an additional measure besides the displacement field. Moreover, it introduces the coupling between the strain tensor and the micro-strain tensor as a coupling measure with the elimination of the repeated effects.
In addition to its ability to reducing the material parameters, the RMM generates additional field equations and reduces the order of the partial differential equations of the model. Such reduction sometimes facilitates obtaining the analytical solutions.
The deformation occurring for elastic materials is an accumulative movement for the material points at the nanoscale. Moreover, the tools used in classical theories to measure the displacement ignore the internal movement of such points and is only limited to the movement of the surface points. Consequently, introducing new measures to describe the internal movement of the material points and eliminating the repeated effects resultant from two different actions, the kinematics of RMM is given by where ε ij = ε ji denote the classical strain tensor, s ij = s ji is the micro-strain tensor, γ ij = γ ji is the coupling between the micro-strain s ij and the macro-strain ε ij and χ ijk = χ ikj is the gradient of the micro-strain tensor s ij . (1)

Equations of motion.
The variational method is a mathematical procedure that is used to obtain the field equations and the corresponding boundary conditions for the considered model. This procedure is used widely in continuum mechanics, fluid mechanics, optics, quantum mechanics, thermodynamics and electromagnetism. The total free energy function for the volume (volume of the body) bounded by the surface ∂�(surface of the body) is considered as a function with internal variables as The first variation for the total free energy reads The constitutive relations are defined as follows where t ij is the micro-stress tensor, τ ij is a Cauchy-like tensor or residual stress tensor, and m ijk is the higher order stress tensor, respectively. Substituting Eq. (4) into Eq. (3) results in Similar to the free energy, the kinetic energy is also generalized by considering the micro-inertia energy as where ρ is the macroscopic mass density of the metamaterial, ρ m is the mass density of the material particle, and J denotes its micro-inertia. Additional terms can be considered in the kinetic energy to describe complex phenomena in the metamaterials as in 33 .
The first variation of kinetic energy is Therefore, the total variation for internal energy reads The work done by the external forces, W , is defined as where f i is the body force, H jk is the body higher-order moment, t i is the Cauchy stress vector, m jk is the higher stress tensor or double force, and F ex i is the wedge forces at the corners of the domain 34 . Here, ∂∂� are the vertex points. The variation of external work reads Considering Eqs. (8) and (10), the Hamilton's principle results in δK = � ρü i δu i + ρ m Js jk δs jk dV .  (12) is satisfied for all volumes bounded by smooth and unsmooth boundaries if and only if the integrands are zero. Consequently, equations of motion and corresponding boundary conditions for smooth boundaries enclosed a bounded volume are and Internal energy. According to 22,[28][29][30] , the free energy in RMM is taken in the form where m and µ m are the elastic moduli of the microstructure, and µ are the elastic moduli of the matrix material between two particles, c and µ c are two elastic moduli accounting for the coupling between the micro-strain and the macro-strain, and ℓ 1 and ℓ 2 are length scale parameters. The free energy function in two dimensions can be written in a matrix form as 35 Therefore, the function W is positive definite if the following conditions are satisfied The constitutive relations. The constitutive relations are related to the free energy function by Substituting the Eq. (15) into Eq. (17) and considering the deformation measures (1), the stress measures can be expressed in terms of deformation measures as General field equations. Considering the Eqs. (13) and (18) and using the condition χ ijj=0 , the general field equations read  (19), (20) represent the governing field equations for the RMM model, which should be satisfied inside the domain of the solution. In the limiting case, by neglecting the micro-strain tensors and the second equation, the RRM model is reduced to the classical model of elasticity.

The RMM model in 2-Dimension
The field equations in 2D. We consider a rectangular ( a × b ) plane which behaves according to the RMM model. In order to study the interaction between the applied forces and the induced internal fields in the material, we consider planar displacement field and micro-strain field as follows: According to the reduced dimensions of the problem, the non-vanishing components of kinematic relations are while the following quantities are eliminated The components of the micro-stress tensor (Eq. (18) 1 ) take the form The components of the so-called residual stress tensor (Eq. (18) 2 ) read Finally. the components of higher order micro-stress tensor (Eq. (18)   The boundary conditions. To get a solution for the coupled system of partial differential Eqs. (26)(27)(28)(29)(30), we consider two boundary value problems where the domain of the solution is a rectangle with width a and height b . The first boundary value problem (BVP) and the second BVP are presented graphically in Figs. 2, 3.  2) Boundary conditions for the second boundary value problem:

Numerical results
The boundary value problem, i.e. Equations (26)(27)(28)(29)(30) with the corresponding boundary conditions (31) and (32), are numerically solved using the finite element method. For this purpose, the COMSOL Multiphysics 36,37 is used for the numerical treatment of the boundary value problem. It is noted that, similar to the other generalized continua, the RMM is not available in the finite element software such as COMSOL Multiphysics, and the boundary value problem should be implemented to use the software as a solver to the system of partial differential equations. In the absence of experimental data, the behavior of the solution and the results are interpreted qualitatively.
The material constants. The material constants given in the Table 1 are used for the analysis of the plane. A more realistic input values necessities the homogenization of the heterogeneous microstructure 38 . In order to implement the system of Eqs. (26)(27)(28)(29)(30), they are expressed in the following matrix form where (31) ∂s xy ∂y  The first boundary value problem. For numerical simulation in the first boundary value problem, we choose the following values for the applied displacement at the boundary.
We also consider vanishing coupling parameters µ c and c (i.e. µ c = c = 0 ) while the length scale parameters take the values ℓ 1 = ℓ 2 = 0.01, 0.004, 0.008, 0.001.  Reducing the length scale parameters ℓ 1 and ℓ 1 , as expected, eliminates the nonclassical measures i.e. microstrains. It is noted that setting the length scale parameters ℓ 1 and ℓ 1 equal to zero cancels the effect of the gradient of the micro-strain. In this case, with vanishing c and µ c , the RMM reduces to the classical model of elasticity for which the numerical treatment developed here is to be reformulated and replaced by the finite element for classical elasticity.
To elaborate the deformation pattern in the plane, the displacement and micro-strain components are demonstrated in Figs. 8, 9, 10, 11, 12 along the mid-line y = b/2 . It is observed that the length scale parameter has a greater effect on micro-strain rather than the displacement components. It is also observed that the displace- Considering these values together with µ c = c = 0, ℓ 1 = ℓ 2 = 0, 0.001, 0.004, 0.008, 0.01m , the internal fields of the RMM plane are obtained.    www.nature.com/scientificreports/ Figures 13,14,15,16,17 show the contour plot for the internal fields u x x, y , u y x, y , s xx x, y , s yy x, y , and s xy x, y , respectively. Similar to the first BVP, the micro-strain vanishes when reducing the length scale parameters l 1 and l 2 . Moreover, the s xx and s yy components of micro-strain only appears in the neighborhood of the external excitation while the s xy component of micro-strain is distributed over the entire domain. Figures 18,19,20,21,22 show the displacement and micro-strain fields of the homogenized domain along the midline y = b/2 with changing the parameter ℓ 1 and ℓ 2 while µ c = c = 0 . It is again noticed that the nonclassical quantities, i.e. micro-strains, are considerably affected with the variation of the length scales.

Conclusion
In this article, the reduced micromorphic model is used for the static analysis of a plane. The variational method is used to get the field equations and boundary conditions for the proposed quantities. The main difference between the present model and the classical model is the set of the deformation measures including classical strain tensor, micro-strain tensor, and residual strain tensor. One of the proposed quantities, micro-strain tensor, is used as a measure of the internal interaction of microelements due to the externally applied fields. The governing equations and the specified boundary conditions were introduced in Cartesian coordinates to describe a rectangular domain. The numerical solution based on the finite element method were derived using the COMSOL Multiphysics. In order to elaborate the material behavior in the reduced micromorphic model, two case studies were discussed. The first case was a rectangular plane possessing one displaced boundary and two supports while the second case was a rectangular plane with two displaced boundaries and one support. The results are illustrated graphically and discussed. The most admissible results are that the micro-strain tensor is concentrated around the displaced boundary and supports while the displacement is distributed uniformly over the entire domain of the solution.
In future studies, specific microstructures/microarchitectures will be considered and homogenized computationally towards reduced micromorphic model. The deformation patterns obtained in this study can be used for    www.nature.com/scientificreports/ the identification of the microarchitectures whose behavior can be captured by reduced micromorphic model. Furthermore, in the context of laboratory experiments and based on digital image correlation, the results of this study (in particular the contours of the field quantities) can also guide for identifying the heterogenous materials whose behavior can be well described with reduced micromorphic model.