Modeling the unusual mechanical properties of metamaterials is a challenging topic for the mechanics community and enriched continuum theories are promising computational tools for such materials. The so-called relaxed micromorphic model has shown many advantages in this field. In this contribution, we present significant aspects related to the relaxed micromorphic model realization with the finite element method (FEM). The variational problem is derived and different FEM-formulations for the two-dimensional case are presented. These are a nodal standard formulation \(H^1({{\mathcal {B}}}) \times H^1({{\mathcal {B}}})\) and a nodal-edge formulation \(H^1({{\mathcal {B}}}) \times H({\text {curl}}, {{\mathcal {B}}})\), where the latter employs the Nédélec space. In this framework, the implementation of higher-order Nédélec elements is not trivial and requires some technicalities which are demonstrated. We discuss the computational convergence behavior of Lagrange-type and tangential-conforming finite element discretizations. Moreover, we analyze the characteristic length effect on the different components of the model and reveal how the size-effect property is captured via this characteristic length parameter.
Hinweise
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1 Introduction
Metamaterials are receiving tremendous attention in both academia and industry due to their unconventional mechanical properties. These are not solely governed by the bulk mechanical properties but also by the geometry of the unit cells which can be designed to attain the desired functionality, see [14, 19, 53, 55, 56]. Moreover, recent advances of additive manufacturing (AM, or 3D printing) techniques are empowering the fabrication process of three-dimensional architected metamaterials, c.f. [16, 20, 32, 42]. To simplify the design process of novel metamaterials, suitable computational tools are needed to capture their unprecedented effective mechanical properties. The classical Cauchy-Boltzmann theory and first-order homogenization procedures often fail to describe the mechanical macroscopic behavior of mechanical metamaterials since they exhibit the size-effect phenomenon, i.e. small specimens are stiffer than big specimens, and therefore other generalized theories are needed such as the classical Mindlin-Eringen micromorphic theory [11, 12, 21, 22, 29, 52], the Cosserat theory [7, 35, 36], gradient elasticity [2, 13, 30] or others.
The relaxed micromorphic model, which we adopt in this work, has been introduced recently in [15, 37, 38]. It keeps the full kinematics of the micromorphic theory but employs the matrix Curl operator of a non-symmetric second-order micro-distortion field for the curvature measurement. The relaxed micromorphic model reduces the complexity of the classical micromorphic theory by decreasing the number of material parameters and has shown many advantages such as the separation of material parameters into scale-dependent and scale-independent ones, see for example [9]. Furthermore, it has already been used to obtain the main mechanical characteristics (stiffness, anisotropy, dispersion) of targeted metamaterials for many well-posed dynamical and statical problems, e.g. [1, 3, 4, 23‐28, 39]. Recently, the scale-independent short range elastic parameters in the relaxed micromorphic model were determined for artificial periodic microstructures in [40] which are used to capture band-gaps as a dynamical property of mechanical metamaterial in [9]. Analytical solutions of the relaxed micromorphic compared to solutions of other generalized continua for some essential boundary value problems, i.e. pure shear, bending, torsion and uniaxial tension, are discussed in [44‐47], emphasizing the validity of the relaxed micromorphic model for small sizes (bounded stiffness), where most of the other generalized continua exhibit unphysical stiffness properties.
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As a result of employing the matrix Curl operator of the micro-distortion field for the curvature measurement, the relaxed micromorphic model seeks the solution of the micro-distortion in \(H({\text {curl}},{{\mathcal {B}}})\), while the displacement solution is still in \(H^1({{\mathcal {B}}})\). The appropriate finite elements of such case must be conforming in \(H({\text {curl}},{{\mathcal {B}}})\) (tangentially conforming). The first formulation of edge elements was presented in [43]. In fact, the name “edge” elements was used because the degrees of freedom (dofs) are associated only with edges for a first-order approximation. \(H({\text {curl}},{{\mathcal {B}}})\)-conforming finite elements of first kind were introduced in [33] and second kind in [34], which are comparable with \(H({\text {div}},{{\mathcal {B}}})\)-conforming elements of first kind in [43] and second kind in [6]. An extension to elements with curved edges, based on covariant projections, was developed by [8]. A general implementation of Nédélec elements of first kind is presented in [41] and a detailed review about \(H({\text {div}},{{\mathcal {B}}})\)- and \(H({\text {curl}},{{\mathcal {B}}})\)-conforming finite elements is available in [17] and [48]. Furthermore, hierarchical \(H({\text {curl}})\)-conforming finite elements are used to solve Maxwell boundary and eigenvalue problems in [49]. A \(H^1({{\mathcal {B}}}) \times H({\text {curl}},{{\mathcal {B}}})\) finite element formulation for a simplified anti-plane shear case of the relaxed micromorphic model utilizing a scalar displacement field and a vectorial micro-distortion field is available in [50].
In our work, we demonstrate the main technologies related to the finite element realization of the theoretically-sound relaxed micromorphic model. The proper finite element approximation of the micro-distortion field is the Nédélec space which utilizes tangential-conforming vectorial shape functions. We provide a comprehensive description of the construction of \(H^1({{\mathcal {B}}}) \times H({\text {curl}}, {{\mathcal {B}}})\) elements with Nédélec formulation of first kind on triangular and quadrilateral meshes. Six finite elements are built, which differ in the approximation space of the micro-distortion: two triangular elements with first- and second-order Nédélec formulation, two quadrilateral elements with first- and second-order Nédélec formulation, and two nodal triangular elements with standard first- and second-order Lagrangian formulation. The paper is organized as follows. In Sect. 2, we introduce the relaxed micromorphic model and derive the variational problem with the resulting strong forms and associated boundary conditions which are modulated in a physical point of view by the so-called consistent coupling condition. We cover in Sect. 3 the main components of the implementation of standard nodal and nodal-edge elements. Two numerical examples are introduced in Sect. 4. The first numerical example is designed to check the convergence behavior of the different finite elements when the solution is discontinuous in the micro-distortion field. We investigate the influence of the characteristic length in a second example which covers the size-effect property. We conclude the paper in Sect. 5.
2 The relaxed micromorphic model
The relaxed micromorphic model is a continuum model which describes the kinematics of a material point using a displacement vector \({{\varvec{u}}}:{{\mathcal {B}}}\subseteq {\mathbb {R}}^3\rightarrow {\mathbb {R}}^3\) and a non-symmetric micro-distortion field \({{\varvec{P}}}:{{\mathcal {B}}}\subseteq {\mathbb {R}}^3\rightarrow {\mathbb {R}}^{3\times 3}\). Both are defined for the static case by minimizing the potential
with \(({{\varvec{u}}},{{\varvec{P}}})\in H^1({{\mathcal {B}}})\times H({\text {curl}},{{\mathcal {B}}})\). The vector \({{\overline{{{\varvec{f}}}}}}\) and tensor \({\overline{{{\varvec{M}}}}}\) describe, respectively, the given body force and body moment, while \({{\overline{{{\varvec{t}}}}}}\) is the traction vector acting on the boundary \(\partial {{\mathcal {B}}}_t \subset \partial {{\mathcal {B}}}\). The elastic energy density W reads
Here, \({\mathbb {C}}_{{\mathrm {micro}}},{\mathbb {C}}_{{\mathrm {e}}}\) are fourth-order positive definite standard elasticity tensors, \({\mathbb {C}}_{{\mathrm {c}}}\) is a fourth-order positive semi-definite rotational coupling tensor, \({\mathbb {L}}\) is a fourth-order tensor, \(L_{{\mathrm {c}}}\) is a non-negative parameter describing the characteristic length scale and \(\mu \) is a typical shear modulus. The characteristic length parameter plays a significant role in the relaxed micromorphic model. This parameter is related to the size of the microstructure and determines its influence on the macroscopic mechanical behavior. The characteristic length allows to scale the number of considered unit cells keeping all remaining parameters of the model scale-independent. It accounts for non-localities in the considered metamaterial where the deformation of each unit cell is influenced by deformations and motions of the neighbouring cells. A relation of the relaxed micromorphic model to the classical Cauchy model was shown in [40] for limiting values of \(L_{{\mathrm {c}}}\), which we can also observe in our numerical examples.
The variation of the potential with respect to the displacement field, i.e. \(\delta _{{{\varvec{u}}}} \Pi = 0\), with
satisfying \(\partial {{\mathcal {B}}}_u \cap \partial {{\mathcal {B}}}_t = \emptyset \) and \(\partial {{\mathcal {B}}}_u \cup \partial {{\mathcal {B}}}_t = \partial {{\mathcal {B}}}\) and \({{\varvec{n}}}\) is the outward normal on the boundary. In a similar way, the variation of the potential with respect to the micro-distortion field, i.e. \(\delta _{{{\varvec{P}}}} \Pi = 0\), with
where \({\varvec{\sigma }}_{\text {micro}} \) and \({{\varvec{m}}}\) are the micro- and moment stresses, respectively, and \({{\varvec{m}}}^i\) and \(\delta {{\varvec{P}}}^i\) denote row vectors of the associated second-order tensors. Using the identity of the scalar triple vector product
with related boundary conditions, formulated in terms of the row vectors \({{\varvec{P}}}^i\) and \({{\varvec{m}}}^i\) of the associated second-order tensors,
where \(\partial {{\mathcal {B}}}_P\cap \partial {{\mathcal {B}}}_m = \emptyset \) and \(\partial {{\mathcal {B}}}_P\cup \partial {{\mathcal {B}}}_m= \partial {{\mathcal {B}}}\). A dependency between the displacement field and the micro-distortion field on the boundary was proposed by [40] and subsequently considered in [10, 45, 46, 50]. This so-called consistent coupling condition is defined by
where \({\varvec{\tau }}\) is a tangential vector on the Dirichlet boundary and \({\nabla {{\varvec{u}}}}^i\) are the row vectors of \(\nabla {{\varvec{u}}}\). This condition relates the projection of the displacement gradient on the tangential plane of the boundary to the respective parts of the micro-distortion.
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The first strong form in Equation (5) represents a generalized balance of linear momentum (force balance) while the second strong form in Equation (10) outlines a generalized balance of angular momentum (moment balance). The generalized moment balance invokes the Cosserat theory with the \({\text {Curl}}{\text {Curl}}\) operator rising from the matrix \({\text {Curl}}\) operator of the second-order moment stress \({{\varvec{m}}}\). In comparison to the classical micromorphic model, see [11, 37], the relaxed micromorphic model uses the same kinematical measures but employs a curvature measure from the Cosserat theory, see [36]. The micro-distortion field has the following general form for the three-dimensional case
where \({{\varvec{P}}}^i\) denotes the row vectors of \({{\varvec{P}}}\). We let the Curl operator act on the row vectors of the micro-distortion field \({{\varvec{P}}}\), i.e.,
Lowest-order (\(k=1\)) Nédélec elements: triangle \(\left[ \mathcal{ND}\mathcal{}^\triangle \right] ^{2}_1\) (right) and quadrilateral \(\left[ \mathcal{ND}\mathcal{}^\square \right] ^{2}_1\) (left). Definition of the individual edges \(e_i\). The red arrows indicate the orientation of tangential flux. (Color figure online)
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Fig. 2
Example of assembling of two neighboring elements which satisfy the orientation consistency via the orientation parameter \(\alpha _I\)
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Fig. 3
The implemented finite elements in the parameter space. Black dots represent the displacement nodes while red squares stand for micro-distortion field nodes associated with tensorial dofs. Red arrows and crosses indicate the edge and inner vectorial dofs, respectively, of the micro-distortion field used in Nédélec formulation. (Color figure online)
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3 Approximation spaces
3.1 Nodal elements \(({{\varvec{u}}},{{\varvec{P}}}) \in H^1({{\mathcal {B}}}) \times H^1({{\mathcal {B}}})\)
We introduce the formulation of a standard nodal element utilizing Lagrange-type shape functions for both displacement and micro-distortion field, see for example [54]. Let us assume that there are \(n^u\) nodes in each element for the discretization of the displacement field \({{\varvec{u}}}\) and \(n^P\) nodes for micro-distortion field \({{\varvec{P}}}\) in two dimensions. Geometry and displacement field are approximated employing related Lagrangian shape functions \(N^u_I\) defined in the parameter space with natural coordinates \({\varvec{\xi }}= \{ \xi , \eta \}\) by
where \({{\varvec{X}}}_I\) are the coordinates of displacement node I and \({{\varvec{d}}}^u_I\) are its displacement degrees of freedom. The deformation gradient is obtained in physical space by
where \({{\varvec{J}}}= \frac{\partial {{\varvec{X}}}}{\partial {\varvec{\xi }}}\) is the Jacobian, \(\nabla \) and \(\nabla _{\varvec{\xi }}\) denote gradient operators with respect to \({{\varvec{X}}}\) and \({\varvec{\xi }}\), respectively. The micro-distortion field \({{\varvec{P}}}\) for the 2D case is approximated using the relevant scalar shape functions \(N^P_I\)
where \({{\varvec{d}}}^{P^1}_I\) and \({{\varvec{d}}}^{P^2}_I\) are the micro-distortion row vectors degrees of freedom of node I. In order to calculate the Curl of \({{\varvec{P}}}\), the gradient of the row vectors in physical space can be calculated by
The error measurements \(||{{\varvec{u}}}-{{\overline{{{\varvec{u}}}}}}||_{L^2}\) and \(||{{\varvec{P}}}-{{\overline{{{\varvec{P}}}}}}||_{L^2}\) of the higher-order patch-test with quadratic displacement for element T2NT1 with varying number of equations in (d). Three different types of finite element meshing were chosen for the convergence study, see (a-c)
Fig. 6
The error measurements \(||{{\varvec{u}}}-{{\overline{{{\varvec{u}}}}}}||_{L^2}\) and \(||{{\varvec{P}}}-{{\overline{{{\varvec{P}}}}}}||_{L^2}\) of the higher-order patch-test with quadratic displacement for element Q2NQ1 with varying number of equations in (d). Three different types of finite element meshing were chosen for the convergence study, see (a-c)
The here presented formulation uses different spaces to describe the micro-distortion field. The geometry and displacement field are approximated in standard Lagrange space as in Equation (16). For the micro-distortion field, its solution is in \( H({\text {curl}},{{\mathcal {B}}})\) and the suitable finite element space is known as Nédélec space, see [33, 34]. In this work, we choose the Nédélec space of first kind. For more details the reader is referred to [5, 17, 31, 48]. Nédélec formulations use vectorial shape functions which satisfy tangential continuity at element interfaces. Lowest-order two-dimensional Nédélec elements are depicted in Fig. 1.
Triangular Nédélec elements of order k are based on the space
where \({\mathbb {P}}_{k-1}\) is the linear space of polynomials of degree \(k-1\) or less and \({\tilde{{\mathbb {P}}}}_{k}\) is the linear space of homogeneous polynomials of degree k. Equivalently, the space can be characterized by
with \( \dim \left( \left[ \mathcal{ND}\mathcal{}^\square \right] ^{2}_k \right) = 2k(k+1) \). The vectorial shape functions \({{\varvec{v}}}^k\) in parametric space are obtained by constructing a linear system of equations based on a set of inner and outer dofs. For the 2D case, the outer dofs of an edge \(e_i\) are defined by the integral
2D homogeneous rectangular bvp. Inspection line, used in Figs. 9 and 10, can be seen in red color. (Color figure online)
Fig. 8
Displacement and micro-distortion field components
Fig. 9
Illustration of \(P_{11}\) along the inspection line \(y=0.5\) using nodal elements with different mesh densities on regular structured meshes
Fig. 10
Illustration of \(P_{11}\) along the inspection line \(y=0.5\) using \(H^1({{\mathcal {B}}}) \times H({\text {curl}}, {{\mathcal {B}}})\) elements with different mesh densities on regular structured meshes
Fig. 11
Computational convergence study (b) of first mesh depicted in (a)
Fig. 12
Computational convergence study (b) of second mesh depicted in (a)
Fig. 13
Computational convergence study (b) of third mesh depicted in (a)
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where \(r_{j}\) is a polynomial \({\mathbb {P}}_{k-1}\) along edge \(e_i\) and \({{\varvec{t}}}_i\) is the normalized tangential vector of edge \(e_i\). The inner dofs are introduced for triangular elements by
The scalar-valued and vectorial functions \(r_j\) and \({{\varvec{q}}}_i\) are linearly independent polynomials which are chosen as Lagrange polynomials in our work. For lowest-order element (\(k=1\)), only outer dofs occur. For higher-order elements (\(k \ge 2\)), the number of outer dofs is increased and additional inner dofs are introduced. E.g. for the \(\left[ \mathcal{ND}\mathcal{}^\triangle \right] ^{2}_2\) with a dimension 8, we have 6 outer dofs and 2 inner ones. The derivations of \(H({\text {curl}},{{\mathcal {B}}})\)-conforming vectorial shape functions are shown in Appendix B. Mapping vectorial shape functions \( {{\varvec{v}}}^k_I\) from the parametric space to \({\hat{{\varvec{\psi }}}}^k_I \) in the physical space must conserve the tangential continuity property. This is guaranteed by using the covariant Piola transformation, see for example [48], which reads
For our implementation of \(H({\text {curl}},{{\mathcal {B}}})\)-conforming elements, we modify the mapping to enforce the required orientation of the degrees of freedom at inter-element boundaries and to attach a direct physical interpretation to the Neumann-type boundary conditions. Hence, two additional parameters, \(\alpha \) and \(\beta \), appear for the vectorial shape functions associated with edge dofs
where \( \alpha _I = \pm 1\) is the orientation consistency function which ensures that on an edge, belonging to two neighboring finite elements, a positive tangential flux direction is defined. Therefore, this positive tangential direction is defined based on a positive x-coordinate. A tangential component pointing in negative x-direction is multiplied by a value \(\alpha _I = -1\) to obtain an overall positive tangential flux on each edge. If the tangential direction has no projection on the x-axis, the same procedure is employed in y-direction. Fig. 2 illustrates an example of calculating the orientation parameter values of two neighboring elements.
The normalization parameter \(\beta _I\) enforces that the sum of vectorial shape functions \({{\varvec{\psi }}}^k_I \) at a common edge scalar multiplied with the associated tangential vector has to be equal one in physical space. Furthermore, the sum of the shape functions belonging to one edge scalar multiplied with the tangential vector of the other edges must vanish. These conditions are reflected by
Here, \(\sum _J {{{\varvec{\psi }}}}_J^k \bigg |_{E_I} \) is the sum of shape vectors related to outer dofs of an edge \(E_J\) evaluated on edge \(E_I\) and \({\varvec{\tau }}_I\) is the normalized tangential vector of an edge \(E_I\) where E denotes the edges in the physical space. Based on Equations (28)\(_1\) and (29)\(_1\), we compute straightforward the parameters \(\beta _I\). In detail, we get for first- and second-order elements
respectively, where \(L_I\) denotes the length of edge \(E_I\) in physical space. For the 2D case, the rotation of vectorial shape functions only has one active component out of the plane which reads
The micro-distortion field \({{\varvec{P}}}\) is approximated by the vectorial dofs \({{\varvec{d}}}^P_I\) representing its tangential components at location \(I=1,\ldots ,n^P\). The micro-distortion field and its Curl are interpolated as
In this work, we present four nodal-edge elements based on the formulation in Sect. 3.2 and two standard nodal elements based on Sect. 3.1. All implemented finite elements employ scalar quadratic shape functions of Lagrange-type for the displacement field approximation with the notation T2 for triangles and Q2 for quadrilaterals. The micro-distortion field is approximated using different formulations introduced in Sects. 3.1 and 3.2. For the standard nodal elements, Lagrange-type ansatz functions are used resulting in element types T2T1 (linear ansatz for \({{\varvec{P}}}\)) and T2T2 (quadratic ansatz for \({{\varvec{P}}}\)). Different nodal-edge elements are built utilizing first- and second-order Nédélec formulations with tangential-conforming shape functions denoted as NT1 and NT2 for triangular elements and QT1 and QT2 for quadrilateral elements. The micro-distortion dofs in the standard nodal elements T2T1 and T2T2 are tensorial with \(2 \times 2\) entries while the nodal-edge elements use vectorial dofs for the micro-distortion field which represent the tangential components. The full micro-distortion tensor is restored based on Equation (32). The used finite elements are depicted in the parameter space in Fig. 3.
Fig. 14
The geometry of the second boundary value problem
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The expected convergence rates of \(H^1({{\mathcal {B}}}) \times H({\text {curl}},{{\mathcal {B}}})\) elements for the relaxed micromorphic model were discussed for anti-plane shear and 3D cases in [50, 51]. In a similar way for an element with a \(H^1({{\mathcal {B}}})\)-conforming formulation of order k for the displacement approximation and a first type Nédélec formulation of order k for the micro-distortion approximation, the solution converges with an optimal convergence rate k in \(H^1({{\mathcal {B}}}) \times H({\text {curl}},{{\mathcal {B}}})\) norm which is defined in Appendix A. Therefore, we expect that the elements T2NT2 and Q2NQ2 achieve an optimal convergence rate of two, see [50, 51].
3.4 Patch-test
The patch-test, introduced in the 1960s, is used for the following purposes: i) to check the performance of finite element formulations violating continuity requirements ii) as a necessary condition which all finite elements have to satisfy iii) to find out simple programming errors iv) as an established tool to check the convergence order for any element type, see [57]. For elements which satisfy required continuity conditions, a patch-test checks the correct programming.
We design a patch-test for the relaxed micromorphic model. A domain \({{\mathcal {B}}}= [0,1] \times [0,1]\) is assumed. The isotropic case of the relaxed micromorphic model is considered with material parameters \( \lambda _{\text {micro}} = \mu _{\text {micro}} = \lambda _e = \mu _e = \mu = 1\), \(L_{{\mathrm {c}}}= 1\,\) and \(\mu _c=0\), c.f. [44, 47]. The solution is assumed to be known a priori (\( {{\overline{{{\varvec{u}}}}}} \) and \( {{\overline{{{\varvec{P}}}}}} = \nabla {{\overline{{{\varvec{u}}}}}}\)) and by solving the strong forms, the body moments and forces are defined. For the numerical setup, the boundary value problem is built by enforcing the derived body moments and forces with Dirichlet boundary conditions \({{\varvec{u}}}= {{\overline{{{\varvec{u}}}}}}\) and \({{\varvec{P}}}\cdot {\varvec{\tau }}= {{\overline{{{\varvec{P}}}}}} \cdot {\varvec{\tau }}\) on the whole boundary \(\partial {{\mathcal {B}}}\,\). Irregular meshes consisting of four quadrilaterals or triangles are used, see Fig. 4.
Simple patch-test: First, we analyze the case of a linear displacement and a constant micro-distortion field. The assumed solution with the related body-forces and body-moments reads
Non-vanishing micro-distortion components of the second boundary problem using 3040 Q2NQ2 elements for \(L_{{\mathrm {c}}}=5\)
Fig. 16
The non-vanishing components of \({{\varvec{P}}}\) along the radius for the \(H^1({{\mathcal {B}}}) \times H^1({{\mathcal {B}}})\) elements using three mesh densities and \(L_{{\mathrm {c}}}=5\)
Fig. 17
The non-vanishing components of \({{\varvec{P}}}\) along the radius for \(H^1({{\mathcal {B}}}) \times H({\text {curl}},{{\mathcal {B}}})\) elements using three mesh densities and \(L_{{\mathrm {c}}}=5\)
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Higher-order patch-test: Next, we analyze the case of a quadratic displacement and a linear micro-distortion field. The assumed solution with the related body-forces and body-moments reads
$$\begin{aligned}&{{\overline{{{\varvec{u}}}}}} = \left( \begin{array}{c} x^2 \\ y^2 \end{array} \right) , \quad {{\overline{{{\varvec{P}}}}}} = \left( \begin{array}{c c } 2 x &{} 0 \\ 0 &{} 2 y \\ \end{array} \right) , \nonumber \\&\quad {{\overline{{{\varvec{f}}}}}} = {\mathbf{0}}, \quad {{\overline{{{\varvec{M}}}}}} = \left( \begin{array}{c c } 6 x +2 y &{} 0 \\ 0 &{} 2 x+ 6 y \\ \end{array} \right) \,. \end{aligned}$$
(35)
Due to the polynomial order of the shape functions, the elements T2T1, T2T2, T2NT2 and Q2NQ2 are able to obtain the analytical solution (except for the typical computational inaccuracies). First-order Nédélec elements are unable to capture the analytical solution without numerical errors. Therefore, we use this test to study the convergence behavior of first-order Nédélec elements T2NT1 and Q2NQ1. The used structured and unstructured meshes as well as the convergence behavior are depicted in Figs. 5 and 6. The numerical solution converges to the analytical solution when refining the mesh.
4 Numerical examples
For our numerical examples, we assume isotropic material behavior which can be described by the set of material parameters \(\lambda _{\text {micro}},\mu _{\text {micro}},\lambda _e,\mu _e,\mu _c ,\mu \) and \(L_{{\mathrm {c}}}\), where \(\lambda _*\) and \(\mu _*\) denote the Lamé coefficients. Furthermore, we assume \({\mathbb {L}}\) as fourth-order identity tensor \({\mathbb {I}}\). Throughout all examples, we consider the Cosserat modulus \(\mu _c = 0\), cf. [35, 38], leading to symmetry of the force stress tensor. The simulations presented in this paper are performed within AceGen and AceFEM programs, which are developed and maintained by Jože Korelc (University of Ljubljana). The interested reader is referred to [18].
4.1 Discontinuous solution: convergence study
In this boundary value problem (bvp), we consider a homogeneous rectangular domain \({{\mathcal {B}}}\) with length \(l=2\) and height \(h=1\), see Fig. 7. We assume the following displacement and micro-distortion field
where the displacement field and tangential components of the micro-distortion \({{{\overline{{{\varvec{P}}}}}} \cdot {{\varvec{e}}}_2 = ({{\overline{P}}}_{12},{{\overline{P}}}_{22})^T}\) are continuous on an interface at \(x=1\) while the normal components \({{{\overline{{{\varvec{P}}}}}} \cdot {{\varvec{e}}}_1 = ({{\overline{P}}}_{11},{{\overline{P}}}_{21})^T}\) exhibit discontinuities. The isotropic case of the relaxed micromorphic model is considered with the material parameters \( \lambda _{\text {micro}} = \mu _{\text {micro}} = \lambda _e = \mu _e = \mu = 1\), \(L_{{\mathrm {c}}}= 1\,\) and \(\mu _c=0\). Solving the strong forms yields vanishing body forces while the body moments read
where the calculated body moments and different meshes are used. We show in Fig. 8 the displacement and micro-distortion field obtained by 882 Q2NQ2 elements on a structured rectangular mesh (see Fig. 11a). The tangential components of the micro-distortion are continuous on the interface while the normal components exhibit a jump.
A convergence study of the component \(P_{11}\) along the inspection line \(y=0.5\) is plotted in Figs. 9 and 10. \(H^1({{\mathcal {B}}}) \times H^1({{\mathcal {B}}})\) elements, see Fig. 9, result in a continuous solution and cause a transition zone at the interface which needs to be resolved by increasing the mesh density tremendously in order to approximate the discontinuous solution at the interface. Meanwhile, the discontinuous solution of \(P_{11}\) can be captured by \(H^1({{\mathcal {B}}}) \times H({\text {curl}}, {{\mathcal {B}}})\) elements, see Fig. 10. The second-order Nédélec formulations in T2NT2 and Q2NQ2 achieve a numerical solution close to the analytical solution even with a coarse mesh while first-order Nédélec formulations in T2NT1 and Q2NQ1 require a denser mesh, because they only represent a constant micro-distortion field in each element.
We study the convergence rates of the implemented finite elements on three different meshes, see Figs. 11, 12 and 13. The mesh schemes are refined similarly to Figs. 5 and 6. Instead of plotting \(||\{{{\varvec{u}}},{{\varvec{P}}}\} - \{{{\overline{{{\varvec{u}}}}}},{{\overline{{{\varvec{P}}}}}}\} ||^2_{H^1({{\mathcal {B}}})\times H({\text {curl}},{{\mathcal {B}}})}\), see Appendix A, we show its individual components. These are the \(L^2\)-norms of the error of the displacement \({||{{\varvec{u}}}-{{\overline{{{\varvec{u}}}}}}||_{L^2}}\), its gradient \({||\nabla {{\varvec{u}}}- \nabla {{\overline{{{\varvec{u}}}}}}||_{L^2}}\), the micro-distortion \(||{{\varvec{P}}}-{{\overline{{{\varvec{P}}}}}}||_{L^2}\) and its Curl field \({||{\text {Curl}}{{\varvec{P}}}- {\text {Curl}}{{\overline{{{\varvec{P}}}}}}||_{L^2}}\). The analysis of the individual error norms is more targeted-oriented from an engineering point of view, since it treats the different energy terms separately. Second-order Nédélec elements T2NT2 and Q2NQ2 achieve convergence rates of three in the \(L^2\)-norm \(||{{\varvec{u}}}-{{\overline{{{\varvec{u}}}}}}||_{L^2}\), and a convergence rate of two in the remaining error norms. In the analyzed bvp the second-order Nédélec elements T2NT2 and Q2NQ2 lead to an optimal convergence rate of two in the space \(H^1({{\mathcal {B}}}) \times H({\text {curl}},{{\mathcal {B}}})\) norm, which meets the theoretical expectations. First-order Nédélec elements T2NT1 and Q2NQ1 show one order of convergence less compared to second-order ones.
4.2 Characteristic length analysis: pure shear problem
We introduce a second boundary value problem, see Fig. 14, consisting of a circular domain \({{\mathcal {B}}}\) with radius \(r_{\text {o}}= 25\) and a circular hole at its center with radius \(r_{\text {i}} = 2\). No body forces or moments are considered. We fix the displacement field \( {\overline{{{\varvec{u}}}}} = {\mathbf{0}}\) on the inner boundary \(\partial {{\mathcal {B}}}_{\text {i}}\) and we rotate the outer boundary \(\partial {{\mathcal {B}}}_{\text {o}}\) counter clockwise with \( {\bar{{{\varvec{u}}}}} = (-\frac{\Delta }{r_{\text {o}}} y, \frac{\Delta }{r_{\text {o}}} x)^T\) where \(\Delta = 0.01\). For the micro-distortion field, we apply the consistent coupling boundary condition (\({{\varvec{P}}}\cdot {\varvec{\tau }}= \nabla {\overline{{{\varvec{u}}}}} \cdot {\varvec{\tau }}\)) on all boundaries \(\partial {{\mathcal {B}}}= \partial {{\mathcal {B}}}_{\text {i}} \cup \partial {{\mathcal {B}}}_{\text {o}} \). Two different cases are discussed in the following. For case A, a single material is assumed whereas for case B two materials are considered. The second material is located as a ring with an outer radius \(r_{\text {m}} = 10\) and an inner radius \(r_{\text {i}} = 2\). The material parameters are shown in Table 1. For the analysis of the influence of the characteristic length \(L_{{\mathrm {c}}}\), the characteristic length is varied.
The problem results in a rotationally-symmetric solution where only the shear components (\(u_{r,\theta }, u_{\theta ,r}, P_{r \theta }, P_{\theta r} \ne 0\)) are non-vanishing. The convergence behavior of the different elements is investigated for case B and \(L_{{\mathrm {c}}}= 5\) using three different mesh densities (410, 3044 and 30620 triangular elements and 448, 3040 and 30256 quadrilateral elements). Since the micro-distortion field is in \(H({\text {curl}},{{\mathcal {B}}})\), the tangential shear component \(P_{r \theta }\) has to be continuous while the radial shear component \(P_{\theta r}\) exhibits a jump, see Fig. 15, where Q2NQ2 elements are used. Similar to Sect. 4.1, \(H^1({{\mathcal {B}}}) \times H^1({{\mathcal {B}}})\) elements are unable to capture this discontinuity in \(P_{\theta r}\), which is shown in Fig. 16. The discontinuous solution of the micro-distortion field is demonstrated in Fig. 17 using \(H^1({{\mathcal {B}}}) \times H({\text {curl}},{{\mathcal {B}}})\) elements. The higher-order Nédélec formulation in T2NT2 and Q2NQ2 elements exhibits very satisfactory results already with a coarse mesh.
Fig. 18
Elastic energy W along the radius
Fig. 19
The non-vanishing components of \({{\varvec{P}}}\) and \(\nabla {{\varvec{u}}}\) along the radius. \(\nabla {{\varvec{u}}}\) is not influenced by the value of the characteristic length \(L_{{\mathrm {c}}}\)
Fig. 20
Total potential varying the characteristic length
Fig. 21
Force stress shear component \(\sigma _{r \theta } = \sigma _{\theta r}\) plotted along the radius
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In the following, we will analyze the influence of a variation of the length-scale parameter \(L_{{\mathrm {c}}}\) on the response of the relaxed micromorphic model for case A. The relation of the relaxed micromorphic model to the classical Cauchy theory has been discussed in detail in [3, 40] for the limiting case \(L_{{\mathrm {c}}}\rightarrow 0\) and \(L_{{\mathrm {c}}}\rightarrow \infty \). \(L_{{\mathrm {c}}}\rightarrow 0\) relates to a macroscopic view on the material with microstructure, with the relaxed micromorphic model being equivalent to a linear elasticity model with stiffness \({\mathbb {C}}_{{\mathrm {macro}}}\) defined as the Reuss lower-bound of \({\mathbb {C}}_{{\mathrm {e}}}\) and \({\mathbb {C}}_{{\mathrm {micro}}}\), i.e. \({\mathbb {C}}_{{\mathrm {macro}}}:= ({\mathbb {C}}_{{\mathrm {e}}}^{-1} + {\mathbb {C}}_{{\mathrm {micro}}}^{-1})^{-1}\). The case \(L_{{\mathrm {c}}}\rightarrow \infty \) resembles an infinite zoom into the material, where an equivalence to linear elasticity with \({\mathbb {C}}_{{\mathrm {micro}}}\) can be derived, cf. [40]. In the latter case, it can be shown that \({{\varvec{P}}}= \nabla {{\varvec{u}}}\) holds. In our study, we approximate the limiting cases by \(L_{{\mathrm {c}}}=10^{-3}\) and \(L_{{\mathrm {c}}}=10^{3}\), respectively. Figure 18 shows the elastic energy W along the radius and Fig. 19 illustrates the non-vanishing components of \({{\varvec{P}}}\) together with the respective displacement gradient components using 30624 Q2NQ2 elements. In Fig. 20, we plot the total potential of the relaxed micromorphic model while varying the characteristic length parameter \(L_{{\mathrm {c}}}\). The figures clearly show the above described behavior. Bounding behavior of the relaxed micromorphic model for small sizes is an important advantage which most other generalized continua miss. Nevertheless, the previous results do not hold for different boundary conditions of the micro-distortion field. Using a different setting of Dirichlet boundary conditions (e.g. homogeneous Dirichlet boundary condition) will keep the role of the characteristic length (increasing \(L_{{\mathrm {c}}}\) makes the material stiffer) but the upper limit, when \(L_{{\mathrm {c}}}\rightarrow \infty \), will be reliant on the boundary value problem and the applied boundary conditions. Using the consistent coupling boundary condition allows the model to reproduce Cauchy linear elasticity with \({\mathbb {C}}_{{\mathrm {micro}}}\) and \({{\varvec{P}}}= \nabla {{\varvec{u}}}\) for \(L_{{\mathrm {c}}}\rightarrow \infty \) regardless of the boundary value problem, see [44‐47]. For the consistent coupling boundary condition, \({\mathbb {C}}_{{\mathrm {micro}}}\) can be related to the stiffest response of the material on the smallest reasonable scale such as one unit cell of a metamaterial. For the linear elasticity model, a standard T2 nodal element is implemented.
Fig. 22
Non-zero component of moment stress \(m_{r z}\) plotted along the radius
Fig. 23
Micro-stress shear component \((\sigma _{\text {micro}})_{r \theta } = (\sigma _{\text {micro}})_{\theta r}\) plotted along the radius
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Next, we investigate the behavior of the different stresses \({\varvec{\sigma }}\), \({\varvec{\sigma }}_{\text {micro}}\) and \({{\varvec{m}}}\) under a variation of \(L_{{\mathrm {c}}}\). The force stress tensor \({\varvec{\sigma }}\) shown in Fig. 21 vanishes for large value of the characteristic length, \(L_{{\mathrm {c}}}=1000\), while it is bounded from above by the classical linear elasticity stress with elasticity tensor \({\mathbb {C}}_{{\mathrm {macro}}}\) for \(L_{{\mathrm {c}}}=0.001\). The only non-vanishing component of the moment stress \(m_{r z}\) is shown in Fig. 22 (\(m_{\theta z} = 0\)), which behaves opposite to the force stress when varying \(L_{{\mathrm {c}}}\). It is nearly zero for \(L_{{\mathrm {c}}}=0.001\) and it increases for growing \(L_{{\mathrm {c}}}\). The micro-stress shown in Fig. 23 is confined between the linear elasticity stress with elasticity tensor \({\mathbb {C}}_{{\mathrm {micro}}}\) from above and \({\mathbb {C}}_{{\mathrm {macro}}}\) from below for large and small values of the characteristic length, respectively. Summarizing the previous findings shortly, increasing the characteristic length diminishes force stress and raises micro- and moment stresses, while both force and moment stresses vanish for large and small values of the characteristic length, respectively, micro-stress is always present.
5 Conclusions
The relaxed micromorphic model is a generalized continuum model which can suitably reproduce the macroscopic effective properties of mechanical metamaterials. First, we derived the variational problem with the relevant weak and strong forms and associated boundary conditions. We put together the main components of standard nodal and nodal-edge finite element formulations of the relaxed micromorphic model. The standard nodal elements \(H^1({{\mathcal {B}}}) \times H^1({{\mathcal {B}}})\) are incapable to achieve satisfactory results for a discontinuous solution. \(H^1({{\mathcal {B}}}) \times H({\text {curl}}, {{\mathcal {B}}})\) elements capture the jumps of the normal components of the micro-distortion field. In contrast to the standard nodal elements, we observe an efficient convergence in the sense of the error norm reduction with mesh refinement. We reveal the role of the characteristic length which governs the scale-dependency property of the relaxed micromorphic model. For \(L_{{\mathrm {c}}}\rightarrow 0\), the model is equivalent to the standard Cauchy linear elasticity model with \({\mathbb {C}}_{{\mathrm {macro}}}\) defined as the Reuss lower-limit of elasticity tensors \({\mathbb {C}}_{{\mathrm {e}}}\) and \({\mathbb {C}}_{{\mathrm {micro}}}\) while the model is corresponding to Cauchy linear elasticity model with \({\mathbb {C}}_{{\mathrm {micro}}}\) with \({{\varvec{P}}}= \nabla {{\varvec{u}}}\) for \(L_{{\mathrm {c}}}\rightarrow \infty \). Furthermore, we have shown the dependency of different stress measurements on the characteristic length. The force stress is at maximum for \(L_{{\mathrm {c}}}\rightarrow 0\) and it vanishes for \(L_{{\mathrm {c}}}\rightarrow \infty \) but the moment stress behaves in the opposite way. The micro-stress varies between Cauchy linear elasticity stresses with \({\mathbb {C}}_{{\mathrm {micro}}}\) and \({\mathbb {C}}_{{\mathrm {macro}}}\) for \(L_{{\mathrm {c}}}\rightarrow \infty \) and \(L_{{\mathrm {c}}}\rightarrow 0\), respectively.
Acknowledgements
Funded by the Deutsche Forschungsgemeinschaft (DFG, German research Foundation) - Project number 440935806 (SCHR 570/39-1, SCHE 2134/1-1, NE 902/10-1) within the DFG priority program 2256. The authors gratefully acknowledge Jože Korelc for the development and ongoing support when using AceGen and AceFEM.
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Construction of triangular Nédélec shape functions
The implemented finite elements in the parameter space are defined with natural coordinates \({{\varvec{\xi }}= (\xi ,\eta )^T}\). Triangular elements are defined on a domain \({{\mathcal {B}}}_e^\triangle = \{0 \le \xi \le 1, 0 \le \eta \le 1 - \xi \}\). The finite elements with the respective edge numbering are shown in Fig. 1.
First-order triangular element NT1
The Nédélec space of a first-order triangular element (NT1) reads
where \(a_i, i=1,2,3\) are coefficients yet to be defined based on the dofs. Starting from the definition in Equation (24), we set \(r_j =1\) for all edges. The tangential vectors of all edges, see Fig. 1 (right), are given by
We calculate the dofs following Equation (24) using \(\xi + \eta = 1\) on the first edge, \(\xi =0\) on the second edge and \(\eta =0\) on the third edge and obtain
In order to obtain the vectorial shape functions \( {{\varvec{v}}}^1_1, {{\varvec{v}}}^1_2\) and \( {{\varvec{v}}}^1_3\) from the general function in (B.2), we have to compute three associated combinations for \(a_1, a_2\) and \(a_3\). We derive the explicit expressions for the vectorial shape functions by enforcing
where \(a_i, i=1,\ldots ,8\) are coefficients yet to be defined based on the dofs. Explicit functions \({r_j, {j=1,2}}\) and \({{{\varvec{q}}}_i, { i=1,2}}\) in Equations (24) and (25) are assumed as
and the tangential vectors of the element edges are the same as in Equation B.3 .
Fig. 25
Tangential-conforming vectorial shape functions of NT2 element. Blue circles indicate the position where the dofs are defined. (Color figure online)
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The inner and outer dofs are calculated according to Equations (24) and (25) using \(\xi + \eta = 1\) on the first edge, \(\xi =0\) on the second edge and \(\eta =0\) on the third edge, such as
where \(a_i, i=1,..,4\) are coefficients yet to be defined based on the dofs. Starting from Equation (24), we set \(r_j = 1\) for all edges. The tangential vectors for the first and third edge are \({{\varvec{t}}}_1 = {{\varvec{t}}}_3 = (1,0)^T\) and for the second and fourth edge \({{\varvec{t}}}_2 = {{\varvec{t}}}_4 = (0,1)^T\), see Fig. 1 (left). We calculate edge dofs taking into consideration that \(\eta = -1\) on the first edge, \(\xi = 1\) on the second edge, \(\eta = 1\) on the third edge and \(\xi = -1\) on the fourth edge, leading to
where \(a_i, i=1,..,12\) are coefficients yet to be defined based on the dofs. Starting from Equations (24) and (26), explicit functions \(r_j, {j=1,2}\) and \({{\varvec{q}}}_i, {i=1,2,3,4}\) are set as
Tangential-conforming vectorial shape functions of NQ2 element. Blue circles indicate the position where the dofs are defined. (Color figure online)
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The edge and inner dofs are calculated according to Equations (24) and (26) considering that tangential vectors and coordinates coloration are identical to the NQ1 element, such that
