A Micromorphic Beam Theory for Beams with Elongated Microstructures

A novel micromorphic beam theory that considers the exact shape and size of the beam’s microstructure is developed. The new theory complements the beam theories that are based on the classical mechanics by modeling the shape and size of the beam’s microstructure. This theory models the beam with a microstructure that has shape and size and exhibits microstrains that are independent of the beam’s macroscopic strains. This theory postulates six independent degrees of freedom to describe the axial and transverse displacements and the axial and shear microstrains of the beam. The detailed variational formulation of the beam theory is provided based on the reduced micromorphic model. For the first time, the displacement and microstrain fields of beams with elongated microstructures are developed. In addition, six material constants are defined to fully describe the beam’s microscopic and macroscopic stiffnesses, and two length scale parameters are used to capture the beam size effect. A case study of clamped-clamped beams is analytically solved to show the influence of the beam’s microstructural stiffness and size on its mechanical deformation. The developed micromorphic beam theory would find many important applications including the mechanics of advanced beams such as meta-, phononic, and photonic beams.


Micromorphic Beam theory
In the context of the classical mechanics, the material is a composition of an infinite number of material particles, each of which is a mass point that can only move with no rotation or deformation. Whereas the particle represents the building-block of the material's microstructure, the classical mechanics does not account for its shape or geometry. In contrast, in micromorphic mechanics, the material particle has a shape, and it does not only move but also rotates and/or deforms. The micromorphic mechanics instills the idea of microstructures with independent degrees of freedom. Given the aforementioned merits of the micromorphic mechanics over the classical mechanics, a novel micromorphic beam theory is developed here.
Reduced micromorphic model: review. The original micromorphic theory, developed by Eringen 20,21 , represents a micromorphic continuum with 12 degrees of freedom. Thus, the equilibrium of the continuum under surface tractions is governed by 12 equations, and the constitutive equations depend on 18 material coefficients for homogeneous-isotropic linear elastic materials. By modeling the microstructure of a micromorphic material, Shaat 18 revealed some redundant degrees of freedom in the context of the original micromorphic theory. It was demonstrated that all the microstructural deformation patterns can be captured by less number of degrees of freedom. The elimination of the redundant degrees of freedom from the original micromorphic theory yielded the reduced micromorphic model 18 . Only 8 material coefficients are required to fully describe the microstructural deformation of a micromorphic continuum according to the reduced micromorphic model. The reduced micromorphic model defines the kinematics of materials using the following kinematical variables 18 : where ε ij is the strain tensor, which is the symmetric dyadic of the displacement gradient tensor u i,j . Because the microstrain field is independent, γ ij is a dyadic that accounts for the difference between microstrain (s ij ) and macro-strain (ε ij ) fields. χ ijk is a triadic tensor that represents the gradient of the microstrain dyadic (s ij ). For a linear elastic-isotropic material, the strain energy density function is defined according to the reduced micromorphic model, as follows 18  and the constitutive equations that involve six material coefficients (λ, μ, λ m , μ m , λ c , and μ c ) and two length scale parameters ( 1 and  2 ) are expressed as follows 18  where t ij and τ ij are two symmetric stress dyadics, which measure the residual stresses in the elastic micromorphic material; m ijk is a triadic that is conjugate to the microstrain gradient tensor (χ ijk ); λ and μ are the conventional Lame moduli; λ m and μ m are microstructural Lame moduli that describe the stiffness of the material's microstructure; and λ c and μ c are coupling moduli used to capture the coupling between microstrains and macroscopic strains.
The dynamic equilibrium is governed by the following nine equations of motion 18 : ijk i jk jk jk m jk , with the following boundary conditions: where ρ is the material mass density; ρ m and J denote the particle's mass density and micro-inertia, respectively; f i and H jk are body forces and body higher-order-moments, respectively;  i and  jk are surface forces and moments; and n i is the unit normal. The reduced micromorphic model is simple but is an effective approach to investigate the mechanics of micromorphic media. The model depends on eight material coefficients, which can be related to the material's micro/ macro-stiffnesses and microstructural topology. Recently, the material coefficients of the reduced micromorphic model were determined depending on the microstructure topology of different multiscale photonic materials and composite metamaterials 17,18 . In this study, the reduced micromorphic model is employed to develop a novel micromorphic beam theory.
Microstructural degrees of freedom. Here, the elastic beam is represented with a microstructure, as shown in Fig. 1. Two independent elastic domains are considered, Ω x y z ( , , ) and Ω ′ ′ ′ x y z ( , , ) m , for the elastic beam and its microstructure, respectively. Beams are usually produced by rolling where grains are elongated in the rolling direction. Therefore, elongated-particles (beam-like particles) are considered to represent the beam microstructure ( Fig. 1). An elongated-particle moves with u x and u z along x and z-directions, respectively (Fig. 1). The beam can be modeled such that the normal to its mid-plane remains straight after deformation, according to Timoshenko beam assumptions. Therefore, the displacements, u x and u z , can be expressed as follows: x y z 0 0 and w x t ( , ) are, respectively, the axial and transverse displacements of an elongated-particle located on the beam axis (i.e., = X x ( , 0, 0)), and ϕ x t ( , ) is the angle of rotation of the beam's cross-section about y-axis with respect to z-axis.
In addition to the macroscopic displacements, u x and u z , the elongated particle deforms exhibiting microstrains. These microstrains are produced due to a micro-displacement field, ′ ′ ′ u x z x z t ( , , , , ) i , of a point belongs to the particle's microscopic domain, Ω m . The micro-displacement, ′ u i , is a slowly varying field over the microscopic domain, Ω m . When it is observed from the macroscopic domain Ω, the micro-displacement is a fast varying field. Therefore, it can be decomposed as follows: . ji ψ is a fast-varying field within the macroscopic domain Ω, which is introduced as a micro-deformation tensor 19 .
As previously mentioned, the beam microstructure is a composition of elongated-particles, each of which can be modeled as a micro-beam. Therefore, the micro-deformation field can be defined as follows: xx , p x t ( , ), and θ x t ( , ) are fast-varying functions that map microscopic fields, e.g., κ ′ x x ( , ) which is the micro-rotation of the cross-section of the micro-beam, to the macroscopic domain, Ω.
According to the reduced micromorphic model, all microstructural deformations can be detected utilizing a micro-strain tensor, which is the symmetric part of the micro-deformation tensor ψ ij 18 . According to Eq. (14), the non-zero components of the microstrain tensor can be derived as follows: It follows from the preceding discussion that a micromorphic beam exhibits six independent degrees of freedom (i.e., u 0 , w, ϕ, s 0 , p, θ). These degrees are generated due to four independent degrees of freedom of the beam microstructure (i.e., u x , u z , s xx , s xz ). Thus, a particle located at x y z ( , , ) exhibits displacements and microstrains defined by u x , u z , s xx , and s xz in Eqs. (12) and (15). . The micro-beam (i.e., elongated-particle) deforms such that its cross section rotates with an angle, κ ′ x x ( , ). the reduced micromorphic model. The substitution of Eqs. (12) and (14) into Eqs. (1)(2)(3) gives the non-zero components of the kinematical variables, ε ij , γ ij , and χ ijk , of the considered micromorphic beam with the form: where τ ij are the typical Cauchy-type stresses; t ij are the components of the microstresses of the elongated microstructure of the beam; and m ijk are the components of the double-stresses.
Micromorphic beam material parameters. It follows from Eqs. (19)(20)(21) that the constitutive equations of the developed micromorphic beam theory depend only on 6 material coefficients and two length scales. These material coefficients can be defined for a specific beam with an elongated microstructure. λ and μ are the classical Lame moduli of the beam. These moduli can be determined based on the typical flexural bending of beams test. λ m and μ m are the Lame moduli of the beam's microstructure. A typical example of a beam with an elongated microstructure is a polycrystalline material beam made by rolling. The stiffness of the grains of a polycrystalline material is usually considered the same as the stiffness of the entire material. In this case, the Lame moduli, λ m and μ m , would be considered the same as those of the classical Lame moduli of the beam, i.e., λ λ = m and μ μ = m . However, for the case that the beam microstructure is of a different stiffness, the Lame moduli of the microstructure would be independently defined. The Lame moduli of the microstructure can be determined experimentally by the direct testing a single crystal of the beam material, or by testing the shifts in the natural frequencies of the beam, as we decrease its size 44 .
In addition to the aforementioned Lame moduli, λ c and μ c are introduced to account for the difference in the deformation and stress fields between the elongated microstructure and the entire beam. These special Lame moduli depend on the stiffness of the boundary of the elongated microstructure, e.g., stiffness of the grain boundary. Indeed, the microstress field of the grains depends on the grain boundary stiffness and how it would slide to allow for the grain deformation. According to the preceding equations, the transmissibility of the external stresses through the grain boundary to the grain can be measured by The developed beam model account not only for the effects of the beam microstructure but also the beam size effects. The size effects of the beam are captured via the two length scales,  1 and  2 . These length parameters mainly scale the dependence of the beam mechanics on its microstructural deformations. Thus, the length (2020) 10:7984 | https://doi.org/10.1038/s41598-020-64542-y www.nature.com/scientificreports www.nature.com/scientificreports/ parameters can be considered to scale the microstructural stiffness to the macro-stiffness of the beam, i.e., On the other hand, these length parameters would be defined depending on the microstructure size-to-the beam size ratio. Thus, for a macroscopic beam with infinitesimal microstructural grains,  → 0 1 and →  0 2 . These length scales can be experimentally determined by measuring the shifts in the frequencies or the transmission phase of the beam, as we decrease its size 44,45 . equations of motion. In light of the defined kinematical variables (Eqs. (16)(17)(18)) and the derived constitutive Eqs. (19)(20)(21) of the micromorphic beam, the first variation of the deformation energy of the beam can be written in the form: where L is the beam length, and A is the beam cross-sectional area. The substitution of Eqs. (16)(17)(18)(19)(20)(21) into Eq. (22) yields: ; ; The explicit form of these resultants in terms of the displacement and microstrain fields are expressible in the form: where I is the beam's area moment of inertia, i.e., Similar to the deformation energy, the first variation of the kinetic energy, δK, is determined as follows: In addition, the first variation of the work done by body forces and surface tractions can be defined, as follows: 10:7984 | https://doi.org/10.1038/s41598-020-64542-y www.nature.com/scientificreports www.nature.com/scientificreports/ The substitution of Eqs. (12) and (15) into Eq. (28) gives the work done in the form: ; The equations of motion are obtained by employing Hamilton's principle, ∫ δ , and then setting the coefficients of δu 0 , δw, δϕ, δs 0 , δp, and δθ to zero:̈̈̈ü   Similarly, the boundary conditions (Eq. (32)) can be written in terms of the displacement and microstrain fields, as follows: It follows from Eqs. (40)(41)(42)(43)(44)(45) that the axial displacement and microstrain of the mid-plane, u 0 and s 0 , are independent of the transverse ones. Therefore, explicit solutions for u 0 and s 0 can be obtained by simultaneously solving Eqs. (40) and (43). In this way, eliminating s 0 between Eqs. (40) and (43), we obtain: α α α α α α The general solution of Eq. (46) is obtained as:  Now, Eqs. (42) and (45) are integrated with respect to x to give: in which C 3 and C 4 are two unknown constants. Substituting Eqs. (49) and (50) into Eqs. (41) and (44), the following equations are derived: By eliminating θ xx , between Eqs. (51) and (52), a differential equation in term of the variable ϕ is derived as follows: The general solution of the differential Eq. (53) can be obtained as: By integrating Eq. (52) twice, we obtain: Substituting w ,x from Eq. (49) into Eq. (50), integrating the resultant equation, and after making some simplifications, we get: The general solution of Eq. (56) is given by: x 10 11 0 Finally, the transverse displacement of the beam is obtained by integrating Eq. (49), which is obtained in the form:    Here, we assume that the dimensionless axial and transverse distributed loads have the following forms:    In addition, C i (i = 1, 2, …, 12) are dimensionless constants. Applying the boundary conditions (62), we obtain Using the finite difference Newton method, Eq. (78) can be solved to determine the unknown constants C 3 , C 4 , C 5 and C 9 . After finding the constants, the problem is completely solved.

Microscopic Moduli
where h is the beam thickness. Table 1. Geometrical and microstructural material parameters as considered in the performed analyses.

Results and Discussion
In this section, the derived analytical solutions of clamped-clamped micromorphic beams are employed to investigate effects of the microstructural topology on the mechanics of beams. To this end, a beam subjected to unit-uniform axial and transverse loads, i.e., = n 0 and = = A B 1 n n , is considered with the microstructural parameters as given in Table 1. In Figs. 2-4, the variations of the deformation parameters (i.e., u 0 , w , ϕ, s 0 , p, θ) over the beam length are depicted for different microstructural parameters, μ m and μ c . In addition, the   www.nature.com/scientificreports www.nature.com/scientificreports/ the beam deflection as obtained with the developed micromorphic beam theory is comparison to the ones obtained by Timoshenko and Euler-Bernoulli beam theories in Fig. 7.
It follows from Fig. 2-4 that the macroscopic and microscopic parameters of the beam deformations, u 0 , w , ϕ, s 0 , p, θ, would decrease or increase depending on the coupling and microscopic moduli, μ c and μ m . When μ > 0 m , each of these parameters decreases as the coupling modulus, μ c , increases from μ c = −μ to μ c = μ. When μ < 0 m , the macroscopic parameters including u 0 , w, and ϕ decrease as the coupling modulus, μ c , increases from μ μ = − c to μ c = μ. When μ < 0 m , an increase in the microscopic parameters, s 0 , p, θ, is observed as the coupling modulus,   ). Results of Timoshenko beam theory (TM) and Euler-Bernoulli beam theory (EB) are also depicted. Note that the dotted curves represent the evolution of the depicted parameter, as it changes between the solid curves.
To elucidate microstructural topology effects on the static behavior of micromorphic beams, the variations of the nondimensional maximum deflection, = . w x ( 05), with the coupling modulus, μ c , and the microscopic modulus, μ m , are illustrated in Fig. 5. It is clear that the micromorphic beam model gives the same results as the classical beam model when μ = 1 c . When μ < 1 c , it is observed that the deflection decreases as μ m increases. However, when μ > 1 c , an opposite behavior is obtained. It can also be observed that the beam overall flexural stiffness increases as μ m increases where the maximum beam deflection decreases as μ m increases. Figure 5 shows cases in which the beam would deflect in a direction that is opposite to the direction of the applied load. Beams with such a behavior can be designed by tailoring the microstructure according to the results presented in Fig. 5. It should be mentioned that these designs should be carried out in parallel to analyses of the material stability 17,46-48 when elastic moduli with negative values would be used.
The effect of the material length scale parameter,  2 , on the maximum deflection of the beam is displayed in Fig. 6a,b for μ μ = .
0 5 m and μ μ = 2 m , respectively. It is seen that the maximum deflection decreases as the length scale parameter,  2 , increases. This can be attributed to the increase in the contribution of the microstrain gradient to the beam deformation as  2 , increases. It is also can be noticed that the influence of the microscopic and coupling moduli, μ m and μ c , to the beam deflection decreases as the length scale parameter,  2 , increases.
To demonstrate the superiority of the developed micromorphic beam theory over the Timoshenko beam theory (TM) and the Euler-Bernoulli beam theory (EB), the variation of the nondimensional maximum beam deflection ( = . w x ( 05)) as a function of the beam length-to-thickness ratio (L/h) is depicted in Fig. 7. The classical beam theories give the maximum beam deflection increases with an increase in the L/h ratio. However, according to the micromorphic beam theory, the beam deflection would increase or decrease as a function of the L/h ratio depending on the coupling modulus, μ c , (Fig. 7(a)) and the length scale,  2 . It is commonly known that the Timoshenko beam theory is preferred over the Euler-Bernoulli beam theory when L/h is lower than 20 where the two theories give the same results when L/h is higher than 20. However, it follows from Fig. 7 that the micromorphic beam theory is preferred over the classical beam theories in general. For advanced beams and for cases that require modeling the beam microstructure effects, the micromorphic beam theory is recommended over the classical theories. Thus, for accurate modeling of advanced beams, e.g., beams made of metamaterials, the developed micromorphic beam theory should be used.

conclusions
In this paper, we presented the first attempt to develop a micromorphic beam theory. Existing beam theories -including the Euler-Bernoulli and the Timoshenko beam theories -were developed based on the classical mechanics. These classical beam theories give no information about the beam microstructure, or the effect of its stiffness on the behavior of the entire beam. The developed micromorphic beam theory outweighs the classical beam theories in fully describing the beam deformation in relation to the size, shape, and deformation of its microstructure. In this study, we assumed that the beam was produced by rolling, and, therefore, the grains of its microstructure were elongated. Therefore, six independent degrees of freedom were considered to fully describe the displacements and microstrains of the beam. The implementation of the micromorphic beam theory for the elastostatic behavior of beams revealed that the mechanics of beams strongly depends on the stiffness of the beam microstructure. In addition, it was observed that the micromorphic beam theory can capture many of the exceptional properties of advanced beams. For example, we revealed that by tailoring the microstructure, beams with equivalent negative stiffnesses can be designed. In addition, beams that can deflect in a direction in that is opposite to the direction of the applied load. These results strongly agree with the recent development of metamaterial beams. Thus, the developed micromorphic beam theory can be applied to effectively model the mechanics of advanced beams, e.g., meta-, phononic, and photonic beams.

Data availability
The data that support the findings of this study are included in the article.