A nonlinear couple stress-based sandwich beam theory
AA nonlinear couple stress-based sandwich beam theory
Bruno Reinaldo Goncalves a, ∗ , Anssi T. Karttunen a,b , Jani Romanoff a a Aalto University, Department of Mechanical Engineering, FI-00076 Aalto, Finland b Texas A&M University, Department of Mechanical Engineering, College Station, TX 77843-3123, USA
Abstract
A geometrically nonlinear sandwich beam model founded on the modified couple stress Timoshenkobeam theory with von K´arm´an kinematics is derived and employed in the analysis of periodic sand-wich structures. The constitutive model is based on the mechanical behavior of sandwich beams,with the bending response split into membrane-induced and local bending modes. A micromechan-ical approach based on the structural analysis of a unit cell is derived and utilized to obtain thestiffness properties of selected prismatic cores. The model is shown to be equivalent to the classicalthick-face sandwich theory for the same basic assumptions. A two-node finite element interpolatedwith linear and cubic shape functions is proposed and its stiffness and geometric stiffness matricesare derived. Three examples illustrate the model capabilities in predicting deflections, stresses andcritical buckling loads of elastic sandwich beams including elastic size effects. Good agreement isobtained throughout in comparisons with more involved finite element models.
Keywords:
Couple stress, Sandwich structures, Timoshenko beam, Sandwich theory, Size effect
1. Introduction
Sandwich panels are lightweight structures composed of two faces separated by a low-densitycore. Over the past decades, these structures have found numerous applications where highstiffness-to-weight/strength ratios are important, such as in vehicle engineering. The two facesare relatively thin and stiff, whereas the core is comparatively thick and soft. The face and corematerials as well as the core topology can be tailored based on the desired mechanical proper-ties of the assembly [1, 2]. In recent years, technological advances such as aluminium brazing,laser-welding and additive manufacturing have allowed the production of sandwich structures withhighly optimized periodic cores, see Fig. 1 and Refs. [3–5] for examples. Continuum modelling is anefficient way to predict their response without discretely modelling all structural details involved.Continuum models for sandwich structures can be broadly divided into single-layer and layer-wise categories according to the dependency of layer-level variables [6]. Oftentimes, refinements ofclassical theories are referred to as higher-order theories, such as the works in [7, 8]. Low-complexitysingle-layer models are used in applications where their through-thickness behavior assumptionshold approximately. In particular, first-order shear deformation theories have been used exten-sively to analyze sandwich beams and plates coupled with standard mechanics or homogenization (cid:73)
This is only an example ∗ Corresponding author. Tel.: +358453308058
Email address: [email protected] (Bruno Reinaldo Goncalves)
Preprint submitted to Composite Structures March 20, 2019 a r X i v : . [ phy s i c s . c l a ss - ph ] M a r pproaches [3, 9]. Conventional first-order theories are, however, inaccurate near discontinuitiessuch as point forces or restrictive boundary conditions. They assume that cell and structure scalesare clearly separated and hence that local effects are negligible. Still, periodic sandwich structurescan display scale interactions in the presence of relatively large, flexible cells. To include such inter-actions, enhanced single-layer sandwich theories have been proposed under different assumptions.In particular, the thick-face sandwich theory [4, 10–12] and other enhanced models that includethe effect of thick faces [13, 14] have been used to analyze periodic sandwich structures. In recentyears, non-classical continuum theories have gained footing in the study of sandwich structures.For instance, couple stress [15–18] and micropolar theories are suitable for the analysis of periodicsandwich beams [19–22] and other lattice structures [23–25] involving different levels of complexityand accuracy.In this work, we develop a couple stress-based model for the analysis of sandwich beams. Themodel is founded on the modified couple stress Timoshenko beam theory [15, 16] and utilizes aconstitutive matrix tailored based on the mechanical behavior of sandwich structures. A microme-chanical approach based on the structural analysis of a unit cell is defined and utilized to determinethe stiffness properties of selected periodic sandwich beams. The model is shown to match the thick-face sandwich beam theory [10–12] in the linear case for the same basic assumptions. A two-nodefinite element is formulated and interpolated using linear Lagrangian and cubic Hermitian shapefunctions, and the underlying stiffness and geometric stiffness matrices are derived. Three exam-ples validate the theory against more involved finite element models and compare the results withthe conventional Timoshenko beam theory and thick-face sandwich theory. The examples concernlinear and geometrically nonlinear bending and linear buckling of periodic sandwich beams withdifferent core shear flexibility levels. Web-core Corrugated core X-core Y-frame core
Figure 1: Unit cells of periodic sandwich beams with applications in civil, mechanical and vehicle engineering.
Figure 2: Couple stress sandwich beam model with conventions indicated. . Couple stress-based sandwich theory Let us consider the modified couple stress theory [15, 16, 18] to construct an equivalent sandwichTimoshenko beam model that includes the local bending effects arising near discontinuities such aspoint loads. Figure 2 shows the general conventions of a couple stress beam of length l and height h . The displacement field of the beam can be written as U x ( x, z ) = u ( x ) + zφ ( x ) , U z ( x, z ) = w ( x ) (1)where w and φ are the transverse displacement and the cross-sectional rotation about the y -axis,respectively. The nonzero strains including the von K´arm´an nonlinearity are [15] (cid:15) x = ∂u∂x + 12 (cid:16) ∂w∂x (cid:17) + z ∂φ∂x = (cid:15) + zκ φ ,γ xz = φ − θ,χ xy = 14 (cid:16) ∂φ∂x − ∂ w∂x (cid:17) = 14 ( κ φ + κ ) (2)where θ = − ∂w∂x , and χ xy is the curvature related to the macrorotation. The beam model at handcan transmit couple stress m xy , as well as the usual normal stress σ x and the transverse shearstress τ xz . The axial N , periodic shear Q and two independent bending stress resultants M and M l are defined (Fig 2) N = (cid:90) A σ x dz, Q = K s (cid:90) A σ xz dz, M = (cid:90) A (cid:16) σ x z + 12 m xy (cid:17) dz, M l = 12 (cid:90) A m xy dz (3)Using the conventions here proposed and following the general steps in [15], the equilibrium equa-tions are obtained for the static case in absence of applied body couples ∂N∂x + f = 0 , ∂Q∂x + q = 0 , Q − ∂M ∂x = 0 , (4)where f and q are, respectively, distributed axial and transverse loads, and Q = Q + Q l + N ∂w∂x , Q l = ∂M l ∂x (5)The boundary conditions, defined at x = ± l/
2, are N or u, Q or w, M or φ, M l or θ (6) Consider in Fig. 3a a unit-width sandwich cell composed of two continuous faces and arbitraryperiodic core, with all members assumed to behave as conventional Euler-Bernoulli beam elements.The vertical cell limits are defined at the centerline of the faces, whose distance characterizes thedepth d . The unit cell boundaries at x = ± s/ ba Figure 3: (a) General arrangement of a sandwich beam unit cell (b) Membrane stretch and local bending of sandwichmembers. at the corner nodes to . A constitutive model is defined with coupling between normal stressand local curvature, and couple stress and axial strain σ x = Q (cid:15) x + Q χ xy , σ xz = Q γ xz , m xy = Q (cid:15) x + Q χ xy (7)where Q = Q results from coordinate system invariance of the constitutive equations andsymmetry of the stress tensor. Two independent bending modes are, in general, identified. Thefirst, related to the area term of the parallel axis theorem, is the sandwich effect, while the secondis the local bending of the prismatic members to the total curvature (Fig 3b). According to theelementary beam theory, the corresponding member-level stresses can be written σ (0) i = E i ∂u l ∂x = E i ( z − z i ( z )) ∂φ∂x , σ (1) i = − E i z i ( z ) ∂ w l ∂x , z i = z − z i , w l = w (8)where z i is the vertical coordinate of the local coordinate system with origin at z i . We thendefine the bending-inducing stresses of the couple stress beam based on the discrete member-levelresponse σ (0) i z ≡ σ x z + 12 m xy , σ (1) i z ≡ m xy (9)Let us expand force- and bending-inducing stresses of the couple stress beam σ x = Q (cid:15) + (cid:16) Q z + Q (cid:17) κ φ + Q κσ x z + 12 m xy = (cid:16) Q z + Q (cid:17) (cid:15) + (cid:16) Q z + Q z + Q (cid:17) κ φ + (cid:16) Q z + Q (cid:17) κ m xy = Q (cid:15) + (cid:16) Q z Q (cid:17) κ φ + Q κ (10)Substituting Eq. (9) into Eq. (10) and knowing that the modes are uncoupled, the following rela-tionships are obtained Q
16 = E i z i z, Q z + Q z + Q
16 = E i ( z − zz i ) , Q z Q
16 = 0 , (11)Solving the system of equations, the constitutive terms reduce to Q = E ( z ) , Q = G ( z ) , Q = − E ( z ) z i ( z ) , Q = 16 E ( z ) zz i ( z ) (12)4here Q is obtained following the same reasoning as in the conventional Timoshenko beamtheory. We substitute the Q ij terms into Eq. (10) and further into Eq. (3), acknowledging that (cid:82) E ( z ) z i dz = 0, to obtain the relations between stress resultants and displacement gradients, whichdefine the beam constitutive matrix C NM Q M l = A B
B D D Q
00 0 0 D l (cid:15) κ φ γ xz κ (13)where D = D g − D l . Note that the local bending moment does not induce axial strain as resultof the thickness-averaging process, i.e. C = C = 0. The axial A , bending D g , D l , and axial-bending coupling B terms of a sandwich beam are defined A = (cid:90) z E ( z ) dz, B = (cid:90) z E ( z ) zdz, D g = (cid:90) z E ( z ) z dz,D l = k c (cid:90) z E ( z ) zz i ( z ) dz, D Q = k s (cid:90) z G ( z ) dz (14)where A, B, D g , D Q are equal to the terms in the conventional Timoshenko beam theory, while D l is the additional term due to the local cell bending stiffness. The coefficient k l describes thenon-uniform distribution of couple stresses over the depth. In the present work, we assume thatcouple stresses are approximately uniform, thus k c = 1 .
3. Micromechanical approach for periodic sandwich structures
Let us define an alternative, straightforward approach to determine stiffness for highly discretesandwich beams, for which direct through-thickness integration as in Eq. (14) can become complex.Consider a periodic cell following the conventions of Fig. 2 and Fig. 3a and modelled using con-ventional, nodally-exact Euler-Bernoulli beam elements. Boundary displacements are applied tothe corners inducing deformation modes ∆, which are utilized to determine the constitutive termsin Eq. (13) for a sandwich beam with arbitrary periodic core. In addition to the displacementboundary conditions discussed next, the cell should be constrained to prevent rigid-body motion. a b c d
Figure 4: Displacement boundary conditions used in the micromechanical approach for stiffness computation. .1. Axial, bending and coupling terms Figures 4a-b show idealized strain cases that involve axial deformation of the sandwich facesand core. The displacement boundary conditions that define ∆ and ∆ are∆ : u l = u (cid:16) s , z (cid:17) = − u (cid:16) − s , z (cid:17) , ∆ : u l = u (cid:16) s , d (cid:17) = − u (cid:16) s , − d (cid:17) , u (cid:16) s , z (cid:17) = − u (cid:16) − s , z (cid:17) (15)The equivalent non-zero strain components are∆ : (cid:15) = 2 u l s , ∆ : κ φ = 2 u l sd (16)Based on the conventions of Fig. (2) and Eq. (3), the stress resultants are obtained N = N t + N b , M = ( N t − N b ) d (17)The combinations that define the axial, bending and axial-bending stiffness terms become∆ : A = N(cid:15) = ( N t + N b ) s u l , B = M (cid:15) = ( N t − N b ) sd u l , ∆ : B = Nκ φ = ( N t + N b ) sd u l , D = M κ φ = ( N t − N b ) sd u l (18)where B obtained with either model can be shown numerically to be equal. Consider in Fig. 4c a sandwich cell under periodic transverse shear deformation. The boundaryconditions that define this mode are given by w l = w (cid:16) s , z (cid:17) = − w (cid:16) − s , z (cid:17) , u (cid:16) s , z (cid:17) = u (cid:16) − s , z (cid:17) ,N (cid:16) s , d (cid:17) = − N (cid:16) − s , d (cid:17) = − N (cid:16) s , − d (cid:17) = N (cid:16) − s , − d (cid:17) = 2( V t + V b ) sd (19)The equivalent transverse shear strain and resultant Q are given by γ xz = 2 w l s + 1 d (cid:104) u (cid:16) s , d (cid:17) − u (cid:16) s , − d (cid:17)(cid:105) , Q = V t + V b , (20)whose quotient defines D Q D Q = Q γ xz = ( V t + V b ) sd w l d + (cid:104) u ( s , d ) − u ( s , − d ) (cid:105) s (21) Figure 4d shows an unit cell composed of flexural-only elements under constant curvature. Thedisplacement boundary conditions, strain and stress resultants that define ∆ are given by θ l = θ (cid:16) s , z (cid:17) = − θ (cid:16) − s , z (cid:17) , κ = 2 θ l s , M l = M t + M b (22)The local bending term D l is defined D l = ( M t + M b ) s θ l (23)which is equivalent to the length-average bending stiffness of the cell prismatic members.6 . Selected periodic sandwich cores Figure 5 shows the unit cells of simple periodic sandwich beams discretized with Euler-Bernoullibeam elements. The faces (1)-(4) have equal thickness t f , while the core members (5),(6) havethickness t c . The material properties are equal for all elements, described by the elastic constants E and ν . The stiffness terms of Eq. (13) are determined by enforcing the displacement conditionsof the micromechanical model to nodes - , evaluating the boundary resultants and computingthe relations of Eq. (18), (21) and (23). Node has all degrees of freedom constrained to preventrigid-body motion. In addition to the stiffness properties, discrete resultants at the sandwich facesare determined based on the homogeneous beam solution. (1) (5)(6) (3) (4)(2) (1) (2) (5) (6)(3) (4) a b Figure 5: (a) Conventions used in the analysis of web-core and triangular corrugated core cells (b) Periodic shear-induced moment distribution.
The axial, bending and axial-bending stiffness terms for the web-core cell in Fig. 5a are A = 2 Et f , B = 0 , D = 12 Et f d , D l = Et f k θ , is given by D Q = 2 Et f t c s ( k t f t c + 2 dt f + st c ) , k = Ek θ (25) The axial, bending and axial-bending stiffness terms for a triangular corrugated cell as in Fig. 5aare given by A = E (cid:16) t f + st c pd (cid:17) , B = − Est c pd , D = E (cid:16) t f d st c p (cid:17) , D l = E (cid:16) t f st c p (cid:17) (26)and the transverse shear stiffness D Q = 4 Esd t f t c p t f + s t c (27)7 .3. Discrete response at the sandwich faces Let us define a simplified discrete stress analysis scheme for the selected structures based onthe sandwich continuum quantities. We assume that t c (cid:28) d for the triangular corrugated core,and exclude the nonlinear term in Eq. (2)a. The interval between two consecutive hard points isdefined n = [ h, h + 1] (Fig. 5a). The membrane force and bending moment at top and bottomfaces, i = t, b , result for both sandwich beams N ni ≈ s (cid:90) x h +1 i x hi (cid:16) N ± M d (cid:17) dx, M ni ≈ D l,i D l s (cid:90) x h +1 i x hi M l dx + M nQ ( x ) (28)where x hi is the x -coordinate of the hard point h at the face i = t, b . The shear-induced bendingmoment is given by M nQ = ( k Q n + k l Q nl ) (cid:16) x ni − s (cid:17) , Q n = 1 s (cid:90) x h +1 i x hi Q dx, Q nl = 1 s (cid:90) x h +1 i x hi Q l dx (29)where x ni = x − x hi and k , k l define, respectively, the share of periodic and local shear forces towhich each face is subjected. The factor k is obtainable from the boundary conditions in Eq. (19).For the web-core cell of Fig. 5, k = k l = 1 /
2, while k l ≈ k is assumed for the corrugated cell.The discrete stress for the sandwich faces i = t, b within the interval n is given by σ ni ( x ni , z i ) = N ni t f + 12 M ni z i t f (30)where z i is the local vertical coordinate consistent with the definition of Fig. 3b.
5. Couple stress and thick-face sandwich theory equivalence
We revise the thick-face sandwich beam theory according to the conceptual framework of Allen[10] and Plantema [11]. In their works, the effect of thick faces is included by studying the sheardeformation compatibility between core and faces. The global sandwich beam response is defined q = − ∂Q ∂x , Q = ∂M ∂x , M = − D g ∂ w ∂x (31)Near discontinuities, the faces must bend to a finite curvature for faces and core to remain attached.Thus, they are locally subjected to a set of loads, shear forces and bending moments q = − ∂Q ∂x , Q = ∂M ∂x , M = − D f ∂ w ∂x (32)The corresponding total quantities are given by q = q + q , Q = Q + Q , M = M + M , w = w + w (33)Shear strain compatibility between face and core results in the following relations − Q = D g ∂ w ∂x = − D Q ∂w ∂x + D f ∂ w ∂x (34)8hich after some rearranging becomes ∂ Q ∂x − a Q = − a Q, a = D Q D g D f D (35)Eq. (35) can be adapted for linear buckling analysis of sandwich beams. Consider the presence of abending-inducing axial force P ; the total shear force becomes Q = P ( ∂w ∂x + ∂w ∂x ). Acknowledgingthat ∂w ∂x = Q a D f , the buckling equilibrium equation becomes ∂ w ∂x − (cid:16) a − PD f (cid:17) ∂ w ∂x − a PD g ∂w ∂x = 0 (36) Let us introduce some general assumptions to study the equivalence between the thick-facesandwich theory and the couple stress sandwich model • The axial degree of freedom u of the couple stress beam is removed; • The sandwich beam has an antiplane core. That is, the normal stress at the core is zero (i.e. E c = 0) and the cross-sectional shear stress distribution is constant over the core thickness; • Horizontal sliding, denoted γ in Allen [10], is excluded from the analyses;as a result of the first assumption, the equilibrium equations of the couple stress beam result ∂Q ∂x + ∂Q l ∂x − q = 0 , Q − ∂M ∂x = 0 (37)Based on the second assumption and considering the faces to be equal, we obtain the followingcommon stiffness terms for both, thick-face and couple stress sandwich models D = E f t f d , D l = D f = 2 E f I f , D g = D + D l (38)We shall now proceed with the equivalency derivations, which are developed for the static bendingcase. Substituting the relations in Eq. (13) into Eq. (37b), we obtain a relation between shearangle and cross-sectional rotation angle γ xz = D D Q ∂ φ∂x (39)Writing Eq. (37a) in terms of displacements, and substituting the shear angle definition of Eq. (2b) q = D g ∂ φ∂x − D f ∂ γ xz ∂x (40)Integrating Eq. (37a) and defining (cid:82) qdx = Q , we obtain Q = D g ∂ φ∂x − D f ∂ γ xz ∂x + C = Q g + Q γl (41)9here the constant of integration obtained is included in Q g . We now substitute Q g into Eq. (39)and differentiate twice ∂ γ xz ∂x = D D Q D g ∂ Q g ∂x (42)Isolating ∂ γ xz ∂x in Eq. (41) and substituting in Eq. (42), the following differential equation is ob-tained D f D D g D Q ∂ Q g ∂x = − Q + Q g → ∂ Q g ∂x − a Q g = − a Q, a = D Q D g D f D (43)Eq. (43) is equal to the governing equation of the thick-face sandwich theory, Eq. (35), acknowledg-ing that Q g = Q . Therefore, the two theories are shown to be equivalent for the basic assumptionsof [10].
6. Couple stress finite element model
Consider the finite element of length l e and height h e shown in Fig. 6. The element has twonodes and four degrees of freedom per node u = { u φ w θ u φ w θ } T (44)with positive directions following the conventions in Fig. 2. We approximate the primary variables u and φ using Lagrange linear polynomials ψ i , while w and θ are approximated using Hermitiancubic polynomials ϕ i u ( x ) = (cid:88) i =1 u i ψ i , φ ( x ) = (cid:88) i =1 φ i ψ i , w ( x ) = (cid:88) i =1 ∆ i ϕ i (45)where u = u ( − l e / , u = u ( l e / , φ = φ ( − l e / , φ = φ ( l e / = w ( − l e / , ∆ = θ ( − l e / , ∆ = w ( l e / , ∆ = θ ( l e / , (46)The stress resultants at the nodes are determined in relation to the positive directions of Fig. 1 N = − N ( − l e / , Q = − Q ( − l e / , M , = − M ( − l e / , M l, = − M l ( − l e / N = N ( l e / , Q = Q ( l e / , M , = M ( l e / , M l, = M l ( l e /
2) (47)We present next the finite element equations for linear, geometric nonlinear and eigenvalue bucklinganalyses.
Figure 6: Two-node beam element for the couple stress sandwich beam theory. .1. Linear finite element equations The linear finite element equations arise from removing the von K´arm´an term from the ax-ial strain description. The displacements are linearly proportional to the applied load, and theprinciple of superposition is valid. In this case, the elemental stiffness matrix becomes K = (cid:90) le − le B T CB dx (48)where B (Eq. (A.1)) is the linear strain-displacement matrix and C is the constitutive matrix inEq. (13). The standard finite element equations F = Ku are then solved after enforcing the loadsand boundary conditions. Let us now consider the von K´arm´an nonlinear problem. In an analogy with Eq. (48), theupdated finite element equations become [26, 27] K = (cid:90) le − le ( B + θ B σ ) T C ( B + θ B σ ) dx (49)where B σ is defined in Eq. (A.2). In short, the objective is to minimize the residual R = K ( u ) u − F (50)using a solution procedure for nonlinear differential equations such as the Newton-Raphson method.The tangent stiffness matrix derivations follow the standard steps as in [28]. Let us now consider the eigenvalue elastic buckling problem. Assume that the beam is subjectedto a constant axial force P that induces transverse displacements. Following the basic steps in [29],the geometric stiffness matrix becomes K σ = (cid:90) le − le P B σT B σ dx (51)The conventional eigenvalue buckling problem is then solved( K − λ K σ ) d σ = 0 (52)where the eigenvalues λ correspond to the buckling loads and the eigenvector d σ provides thebuckling modes. 11 . Numerical results In order to demonstrate the couple stress-based sandwich beam theory (CSS), we study thelinear and geometric nonlinear response of periodic sandwich beams. In the following analyses,the beams represent wide panels along the y -axis, whose response is two-dimensional. Plane strainconditions are then assumed, with elastic modulus set to E = E s / (1 − ν s ). The following materialproperties are taken: E s = 206 GPa and ν s = 0 .
3. The results are shown for an unit-width beam.The examples are validated using finite element models (3D FE) that represent the 3-D geometry(Fig. 7), constructed with Abaqus S4R shell elements. In the validation models, vertical point forcesare applied to the sandwich structural hard points at the relevant x -coordinate. Comparisons withthe thick-face sandwich theory (TFS) and the conventional Timoshenko beam (TBT) with effectiveproperties are shown. Pinned support Point forceSymmetry Symmetry
Structural hard point
Figure 7: Example of three-dimensional FE model with common load and boundary condition assumptions as usedfor validation.
Let us consider web-core and triangular-corrugated core sandwich beams of equal core density( ≈ . x -axis definethe structures, to a total of L = 0 . m . The beams are subjected to four-point bending within asingle linear, quasi-static step, to a total vertical displacement of − . m at x = 0 . m ( L/
3) andsymmetry conditions at mid-length. Figure 8 shows the vertical displacement distributions alongthe beam axes using the couple stress sandwich theory, thick-face sandwich theory [10–12] andvalidation models (bottom face). Good agreement is observed between couple stress and validationmodels, whereas couple stress and thick-face beams predict equal displacements.Figure 9a shows the bottom surface stress at the bottom face and top surface stress at the topface of the triangular corrugated core beam. Figure 9b shows the bottom surface stress at thebottom face of the web-core beam; the top face stress is qualitatively similar due to mid-depthsymmetry. Local bending stresses are also presented in Fig. 9. Stresses are localized from thehomogeneous solution as described in detail in Section 4.3. Overall, the couple stress sandwichbeam is able to predict the stress distributions with good accuracy against the validation model.Local bending stresses are comparatively higher in the web-core structure at hand due to its highershear flexibility. Yet, it is shown that local bending has a non-negligible contribution for an accuratestress analysis of both structures. 12 .06 0.12 0.18 0.24 0.3 0.3600-0.2-0.4-0.6-0.8-1.0-1.2 3D FECSSTFS 0.06 0.12 0.18 0.24 0.3 0.3600-0.2-0.4-0.6-0.8-1.0-1.2 D e f l e c t i on [ mm ] D e f l e c t i on [ mm ] x -coordinate [m] x -coordinate [m] sy mm e t r y sy mm e t r y
3D FECSSTFS
Figure 8: Deflections obtained with couple stress sandwich (CSS), thick-face sandwich (TFS) and 3D finite element(3D FE) models for (a) triangular corrugated and (b) web-core sandwich beams in four-point bending. Cell dimensionsare shown in [m].
Consider now in Fig. 10a a web-core unit cell with semi-rigid joints, whose rotational stiffnessis half of the average reported in [30]. A sandwich beam composed of 10 of such cells ( L = 1 . m )is subjected to bending. Two doubly-clamped settings are investigated, where the load is eitherconcentrated at the mid-length, or distributed over the top face plate. The Newton-Raphsonalgorithm is utilized in conjunction with the finite element method to obtain an approximatesolution to the nonlinear equilibrium equations. The convergence tolerance is set to 10 − and theanalysis is divided into fine load steps. Figure 8 shows the load vs. maximum deflection relations ineither case, as well as comparisons with the conventional Timoshenko beam and validation model.Overall, the couple stress sandwich theory satisfactorily predicts the nonlinear response in termsof displacements. In the linear range, a slight error (4-5%) related to the periodic shear descriptionof the couple stress model is observed. The example under consideration is an extreme case withrelatively few, highly shear-flexible cells. More involved models, such as the micropolar theory [22],can be used in such case for a more precise response prediction. The error is rapidly reduced asthe geometric nonlinearity increases and the membrane action becomes dominant. In both cases,the couple stress sandwich model is considerably more accurate than the conventional Timoshenkobeam theory with effective properties. Figure 10b shows the simplified unit cell of a Y-frame core sandwich beam with similar dimen-sions as in [5]. Y-frame structures used in ship design are often composed of relatively few cells,thus prone to size effects, which we investigate next. We analyze the changes in linear bucklingloads of axially compressed Y-frame beams as function of their relative unit-cell size s/L . Twoconfigurations are covered, namely an end-loaded cantilever and a pinned-pinned beam loaded atboth ends. Figure 11 shows the buckling load predicted with the validation models and the rel-ative errors obtained with classical Timoshenko and couple stress sandwich models. Overall, the13 t r e ss [ M P a ] S t r e ss [ M P a ] x -coordinate [m] x -coordinate [m] sy mm e t r y sy mm e t r y -100-60-2020 -2004080 3D FE CSS Local bend., CSS3D FE CSS Local bend., CSS top face, top surface bottom face, bottom surface bottom face, bottom surface Figure 9: Stress distributions of (a) triangular corrugated (top and bottom faces) and (b) web-core sandwich (bottomface) beams in four point bending as obtained from the couple stress sandwich beam model. a b
Figure 10: Unit cells of web-core and Y-frame core sandwich beams used in Sections 7.3 and 7.4. Dimensions in [m].
Maximum deflection [mm] P r e ss u r e [ k P a ] Maximum deflection [mm]
3D FE CSS TBT CSS (error) TBT (error) V e r t i c a l f o r c e [ k N ] R e l a t i v e e rr o r [ % ] R e l a t i v e e rr o r [ % ] Figure 11: Nonlinear bending analysis of clamped web-core sandwich beams under (a) centered point force (b)distributed load. B u ck li ng l oad [ k N ] s / L
3D FE (buckling load)CSS (error) TBT (error)0 0.1 0.2 0.3 s / L R e l a t i v e e rr o r [ % ] R e l a t i v e e rr o r [ % ] B u ck li ng l oad [ k N ] Figure 12: Elastic buckling load of Y-frame sandwich structures and relative error in the predictions using the couplestress sandwich and classical Timoshenko models. conventional Timoshenko model is progressively less accurate as the cell size approaches the totalstructural length. This supports the observations in [20, 21], where a conventional couple stressbeam was used to model the sandwich structures. The couple stress sandwich beam succeeds indescribing size effects in Y-frame core sandwich structures, displaying good accuracy against themore involved validation models.
8. Conclusions
A sandwich beam model founded on the modified couple stress Timoshenko beam theory hasbeen defined and employed in the analysis of elastic periodic sandwich structures. A standardmicromechanical approach has been proposed as a generalization of previous works (e.g. [3, 21, 31,32]) for the couple stress sandwich model, and applied to determine effective stiffness propertiesof selected cores. The model was shown to be equivalent to the thick-face sandwich theory inthe linear case [10–12] for the same basic assumptions. Unlike the thick-face model, however,the couple stress sandwich beam relies on a single kinematical definition to describe the scalesinvolved. The kinematical variables are equal to the conventional Timoshenko beam theory exceptof a higher-order curvature term.The model herein derived improves the single-layer description of sandwich beams when com-pared to the conventional Timoshenko model. It is able to predict the structural behaviour neardiscontinuities such as point loads or clamped boundary conditions, and thus capture size effects.When compared to the modified couple stress Timoshenko model for layered structures [15, 16], theredefinition of resultants and introduction of coupling constitutive coefficients guarantee that themodel describes the deformation of a sandwich beam as in Refs. [10, 12]. The stress resultants havea direct correspondence to the sandwich beam member-level forces and moments, which facilitatesstress localization for any periodic core. Stress predictions were shown to be reasonably accuratefor selected bending- and stretch-dominated cores, which are respectively more and less prone tosize effects [21]. 15 cknowledgements
The authors gratefully acknowledge the financial support from the Graduate School of AaltoUniversity School of Engineering.
Appendix A. Finite element matrices
The elemental matrices needed for the finite element computations are given as follows. Thelinear strain-displacement matrix B that approximates the relations between nodal displacements(Eq. (44)) and linear strains of the couple stress sandwich model is given by B = ψ (cid:48) ψ (cid:48) ψ (cid:48) ψ (cid:48) ψ ϕ (cid:48) − ϕ (cid:48) ψ ϕ (cid:48) − ϕ (cid:48) − ϕ (cid:48)(cid:48) ϕ (cid:48)(cid:48) − ϕ (cid:48)(cid:48) ϕ (cid:48)(cid:48) (A.1)The geometric strain-displacement matrix, which interpolates the von K´arm´an nonlinear terms, isgiven by B σ = − ϕ (cid:48) ϕ (cid:48) − ϕ (cid:48) ϕ (cid:48) (A.2) References [1] C. A. Steeves, N. A. Fleck, Material selection in sandwich beam construction, Scripta Materialia 50 (2004)13351339.[2] N. Fleck, V. Deshpande, M. Ashby, Micro-architectured materials: past, present and future, Proc. R. Soc. A466 (2010) 2495–2516.[3] T.-S. Lok, Q.-H. Cheng, Elastic stiffness properties and behavior of truss-core sandwich panel, J. Struct. Eng.126 (5).[4] J. Romanoff, P. Varsta, Stress analysis of homogenized web-core sandwich beams, Compos. Struct. 79 (3) (2007)411–422.[5] L. St-Pierre, V. Deshpande, N. Fleck, The low velocity impact response of sandwich beams with a corrugatedcore or a y-frame core, Compos. Struct. 91 (2015) 71–80.[6] E. Carrera, S. Brischetto, A survey with numerical assessment of classical and refined theories for the analysisof sandwich plates, Appl. Mech. Rev 62 (1).[7] L. He, Y.-S. Cheng, J. Liu, Precise bending stress analysis of corrugated-core, honeycomb-core and x-coresandwich panels, Compos. Struct. 94 (5) (2012) 1656–1668.[8] Y. Frostig, M. Baruch, Bending of sandwich beams with transversely flexible cores, AIAA Journal 28 (3) (1190)523–531.[9] Buannic, P. N., Cartraud, T. Quesnel, Homogenization of corrugated core sandwich panels, Compos. Struct.59 (3) (2003) 299–312.[10] H. G. Allen, Analysis and Design of Structural Sandwich Panels, Pergamon Press, 1969.[11] F. Plantema, Sandwich construction: the bending and buckling of sandwich beams, plates and shells, Wiley,1966.[12] D. Zenkert, An introduction to sandwich construction, Emas, 1995.[13] J. Romanoff, P. Varsta, Bending response of web-core sandwich plates, Compos. Struct. 81 (2007) 292–302.[14] J. Jelovica, J. Romanoff, Buckling of sandwich panels with transversely flexible core: Correction of the equivalentsingle-layer model using thick-faces effect, J. Sand. Struct. Mat. 0 (0) (2018) 1–23.
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