Monolithic FE2 approach for the thermomechanical modeling of beam structures

In the present contribution, the FE2 scheme for beam elements is extended to thermomechanically coupled problems. Beam elements have the advantage of drastically reducing the number of degrees of freedom compared to solid elements. However, the major challenge in modeling structures with beam elements lies in developing sophisticated non‐linear beam material models. This drawback resides in the fact that these elements require effective cross‐sectional properties involving material and geometric properties. The FE2 method, combined with a homogenization scheme based on the Hill‐Mandel condition, solves this problem. Within this scheme, homogenization of a representative volume element (RVE) on the mesoscopic scale provides effective cross‐sectional properties for the macroscopic scale. This homogenization procedure allows the consideration of non‐linear material formulations and cross‐sectional deformation within the analysis of a beam structure. The applicability of such a FE2 scheme for purely mechanical problems was already shown. In the present contribution, an extension to thermomechanically coupled problems is provided. In the proposed setting, the macroscopic scale is represented by beam elements with displacement, rotation, and temperature degrees of freedom. Solid elements with displacements and temperature degrees of freedom describe the behavior of the RVE. Hence, the proposed extension solves both scales in a monolithic approach. The assumption of a steady state problem at both scales allows a focus on a consistent scale transition and a discussion about the choice of suitable boundary conditions under the assumption of beam kinematics.


INTRODUCTION
In recent decades, an increasing interest in the computational multiscale modeling of functional composite materials has been shown [1,2].Especially FE 2 methods have proven to be a versatile and powerful tool in the modeling of complex materials across scales and have substantially improved the understanding of these systems.These computational multiscale frameworks make constructing phenomenological constitutive models at the macroscale level superfluous since the macroscopic overall stresses and tangent moduli are directly derived from an underlying microscopic boundary value problem.However, the developed frameworks are mainly restricted to continuum-based models at both scales, which makes the solution of boundary problems involving large three-dimensional structures at the macroscale level computationally not feasible.With the prominent usage of structural finite element models in civil engineering, and a distinct need for more sophisticated simulation models accounting for multiscale effects, there is a trend to develop FE 2 frameworks, that incorporate structural finite elements at the macroscale level.In the case of purely mechanical behavior, such computational multiscale methods have recently been proposed for beams [3], plates [4], and shells [5].In the present work, an extension of the model of Klarmann et al. [3] is given, that covers the variational-based modeling of thermoelastic beam structures across scales.

THEORETICAL ASPECTS
Steady-state thermoelasticity is a coupled two-field problem, that -in the absence of source terms -is governed by the homogeneous balance of linear momentum and the homogeneous balance of energy where  denotes the Cauchy stress tensor and  the heat flux.Mechanical boundary conditions can be prescribed in terms of surface tractions t and displacements ū as where  denotes a unit normal vector pointing outwards from the domain.The corresponding thermal boundary conditions are given by  ⋅  = q on   and  = T on   , in terms of a prescribed normal heat flux q and a prescribed temperature  = T.In linear isotropic thermoelasticity the constitutive equations for the Cauchy stress tensor and the heat flux are given by Hooke's law and Fourier's law where  ∶= sym[∇] denotes the infinitesimal strain tensor.Furthermore,  and  are the Lamé parameters,  the heat expansion coefficient,  0 the reference temperature and  the thermal conductivity.The weak formulation of the problem reads

Beam kinematics
In line with the Timoshenko beam theory the displacement field  and the temperature field  of a beam are given by (, , ) =   () − (, ) × (), and (, , ) =   () −    () +    () with respect to a Cartesian coordinate system, where the coordinate  coincides with the beam axis while the coordinates  and  are the cross-section coordinates.Both fields are composed of constant contributions   = [      ]  ,   and linear contributions  = [      ]  ,   ,   with respect to the cross-section.In consequence the non-vanishing components of the infinitesimal strain tensor due to the kinematical assumptions read while the components of the temperature gradient due to the field assumptions are given by Rearranging the components of the Cauchy stress tensor  and heat flux  in a generalized vector   , the components of the material tangent in a matrix , and the components of the infinitesimal strain tensor and temperature gradient in a generalized vector   , allows us to reformulate part  of Equation ( 5) to Assuming that the center of gravity of the cross-section coincides with the beam axis, matrix  evaluates to The matrix  serves as a reference solution for the homogenization process.Herein, the terms are:  the tension,   ,   the shear,   the torsional and   ,   the bending stiffness;  results in the normal force due to constant heating of the beam, and    and    represent bending moments due an increased temperature difference between top and bottom of the beam; the terms   ,   ,   ,   , and   represent Fourier's Law in terms of the beam kinematics.All the geometric values, namely  and , result from the integration over the cross-section.The different indices indicate the possible necessity of considering correction values, for example, shear correction factors.

HOMOGENIZATION FRAMEWORK
In this section, a short survey on first-order homogenization in thermo-mechanics as well as its discussion in the context of meso-macro scale transition incorporating beam kinematics is given.

Definition of macro-variables
The macroscopic infinitesimal strain tensor ε and the macroscopic Cauchy stress tensor σ are related to their mesoscopic counterparts via averages over the RVE, as In analogy, the macroscopic temperature gradient ∇ T and the macroscopic heat flux q are defined as

Macrohomogeneity condition
Conceptually in line with the works of Özdemir et al. [1] and Temizer & Wriggers [2], the macrohomogeneity condition (Hill-Mandel lemma) in the thermomechanical context is given by Assuming a classical first-order homogenization scheme, the primary mesoscopic variables can be additively decomposed into linear macroscopic contributions and superimposed fine-scale fluctuation fields The Hill-Mandel lemma is satisfied by a suitable set of linear Dirichtlet-type (LD), uniform Neumann-type (UN) boundary conditions and periodic boundary conditions (PB), given by

Boundary conditions for beam kinematics
Under the kinematical assumptions of Timoshenko beam theory the components of the displacement gradient are given by  = [ 1  ], where the vector  1 represents the relation to the beam strains via and  denotes the three-dimensional zero vector.Periodic boundary conditions (PB) for the displacement are applied to opposite sides in length direction (-direction) of the representative volume element (RVE), see Figure 1.
Here ⟦(⋅)⟧ = (⋅) + − (⋅) − , denotes the jump of a quantity with respect to opposite boundaries of the RVE.On the remaining four sides of the RVE homogeneous Neumann boundary conditions (UN) are prescribed.Further constraints are required in order to remove the rigid body rotation of the RVE and the length dependency of the results due to shear deformation.For further details of the adapted approach we refer to Klarmann et al. [3].Furthermore, a suitable set of boundary condition for the thermal field has to be prescribed to ensure a energetically consistent scale transition between meso-and macroscale.In case of the assumed beam temperature field (6) and the resulting temperature gradient (8) the mesoscopic temperature field reads In this contribution either LD or PB are considered for the temperature field.In case of LD, the temperature fluctuation T vanishes on the boundaries.It is important to note that in comparison to the mechanical field, the temperatures are prescribed on all six sides of the RVE.Therefore, Equation (18) directly specifies the LD by setting T = 0 and evaluate it at the boundary coordinates of the RVE.In case of PB, it is assumed that T+ = T− is periodic on opposite faces.This leads to the following relations per direction: In As the PB in Equation ( 19) only define a relative temperature on opposite sides, the temperature field can still be shifted by a constant value, thus it is not unique.To avoid this problem, the temperature according to Equation (18) with T = 0 is prescribed on the 8 corner nodes of the RVE.This is a slightly different approach compared to Özdemir et al. [1], but with the proposed relation, the surface temperature of the macroscopic body can be specified with the assumed beam kinematics.Here, we have to keep in mind, that the faces on the sides  − ,  + ,  − , and  + of the RVE are the outer surfaces of the macroscopic body.
F I G U R E 2 Results -Beam heat expansion parameters.

NUMERICAL EXAMPLES
In this section, some numerical examples are investigated.The main focus of these studies lies on the benchmarking of the proposed FE 2 scheme.In the following for the sake of a more compact notation the subscripts  and  indicate macroscopic quantities, when LD and PB were used for the temperature in the computational homogenization framework, respectively.Furthermore, all the cross-section values in the diagrams are normalized by the analytical results based on Equation (10).

Benchmark: Rectangular cross-section
This first example serves as a benchmark example for the proposed homogenization scheme.It considers a rectangular, homogeneous cross-section with a width and height of  = 1 and ℎ = 1, respectively.The material parameters are chosen as Young's modulus  = 100, Poisson ratio  = 0.3, thermal expansion coefficient  = 1, heat conductivity  = 1, and reference temperature  0 = 0. Figure 2 shows the results for the cross-section parameters connected to the thermal expansion, and Figure 3 shows the results for the thermal conductivity associated with the first-order gradients.In both cases, the results of the proposed homogenization scheme are independent of the length   of the RVE and are equivalent to the analytical results of Equation (10).In contrast, the cross-section parameters associated with the thermal curvatures depend on the length of the RVE, see Figure 4.For all cross-section parameters, the homogenization scheme leads to the same results independent of the chosen boundary conditions of the RVE.

Layered beam
In this example, the cross-section consists of three layers, see Figure 5.The cross-section has a total height of 1 and a width of  = 1, with a core height of ℎ  = 0. with Young's modulus   = 100, heat expansion parameter   = 0.001, thermal conductivity   = 1 for the outer layers, and   = 25,   = 0.00025, and   = 0.01 for the core material.Both materials have Poisson ratio of  = 0.3, and the reference temperature is taken  0 = 0. Homogenizing the RVE with the proposed scheme leads to the results for the thermal expansion in Figure 7. diagram shows a length dependency of the value     , using LD, the length of the RVE and a smaller number compared to the results with the PB.The term   is absent, as there is no difference between the results and no length dependency.Regarding the thermal conductivity, the results show a similar behavior as for the thermal expansion, see  the thermal conductivity associated with the heat flux in thickness direction is much smaller when using PB compared to LD.The results associated with the thermal curvatures depend on the length of the RVE, see Figure 9.If the length   → 0, the results converge against the analytical ones.
As there are differences in the thermal expansion parameters and conductivity values between LD and PB, the last example investigates the suitability of the homogenized values.Figure 6 shows the system with prescribed temperatures such that the resulting thermal gradient in the system is affected by the cross-section values, which differ when applying LD or PB.The lengths for the numerical investigation are  = 10 and   = 4.   = −1 and   = 1 are applied over the length   to trigger the wanted thermal state.The materials parameters are chosen as before.The RVE length is   = 0.1.Figures 10 and 11 show the results when using LD (Beam l.) and PB (Beam p.).In both cases, the results agree well with the reference solution of a solid model when using PB.In contrast, there is a big difference in the results when applying the LD to the RVE.Overall, using PB for the thermal field, the resulting effective cross-section properties are more suitable compared to the ones with LD in case of heterogeneous cross-sections.

CONCLUSION
The present contribution covers the variational-based modeling of thermoelastic beam structures across scales.We propose a monolithic FE 2 framework embedded in a suitable scale transition scheme between the macroscale -incorporating a structural beam model based on Timoshenkos assumptions -and the mesoscale incorporating a classical continuum model of the underlying RVE.For the sake of simplicity we restrict ourselves to a linearized setting and steady state conditions at both scales.Within a first order homogenization scheme we discuss a suitable set of boundary conditions that is compatible with Timoshenko's beam theory.Benchmark studies show that the proposed multiscale scheme is able to accurately capture the thermoelastic behavior of beam structures.Furthermore, a study including heterogenous cross-sections at the mesoscale reveals a superior performance of PB for the temperature field compared to LD boundary condition.Future work will be devoted to the extension of the framework to incorporate transient effects at the macroscale level and non-linearities at the mesoscale level.

A C K N O W L E D G M E N T S
Open access funding enabled and organized by Projekt DEAL.

F I G U R E 1
Schematic plot of a beam RVE.RVE, representative volume element.

5 ,F I G U R E 4 F I G U R E 5
and layer heights of ℎ  = 0.25.It is composed of two different materials F I G U R E 3 Results -Beam thermal conductivity for the first order gradients.Results -Beam thermal conductivity for thermal curvatures.Geometry of layer cross-section.F I G U R E 6 System with prescribed temperature in the center.
Figure 8.Here, F I G U R E 7 Results -Beam heat expansion parameters.

F I G U R E 8
Results -Beam thermal conductivity for the first-order gradients.F I G U R E 9Results -Beam thermal conductivity for thermal curvatures.

F I G U R E 1 0
Comparison of the deflection between Beam p. (PB), Beam l. (LD) and 3D solid.LD, linear Dirichtlet-type; PB, periodic boundary conditions.

F I G U R E 1 1
Comparison of the temperature at the top between Beam p. (PB), Beam l. (LD) and 3D solid.LD, linear Dirichtlet-type; PB, periodic boundary conditions.