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Size-dependent modelling of elastic sandwich beams with prismatic cores
Accepted Manuscript
Size-dependent modelling of elastic sandwich beams with prismatic cores Bruno Reinaldo Goncalves , Jani Romanoff PII: DOI: Reference:
S0020-7683(17)30534-6 10.1016/j.ijsolstr.2017.12.001 SAS 9819
To appear in:
International Journal of Solids and Structures
Received date: Revised date: Accepted date:
17 August 2017 10 November 2017 1 December 2017
Please cite this article as: Bruno Reinaldo Goncalves , Jani Romanoff , Size-dependent modelling of elastic sandwich beams with prismatic cores, International Journal of Solids and Structures (2017), doi: 10.1016/j.ijsolstr.2017.12.001
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Size-dependent modelling of elastic sandwich beams with prismatic cores Bruno Reinaldo Goncalves* and Jani Romanoff Aalto University, Department of Mechanical Engineering, FI-00076 Aalto, Finland *Corresponding author: [email protected]
ABSTRACT Sandwich panel strips with prismatic cores are modelled using the modified couple stress
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theory and their elastic size-dependent bending behavior investigated. Compatibility between the discrete sandwich and continuum beam kinematics is first discussed. A micromechanicsbased framework to estimate effective mechanical properties is provided and unit cell models constructed with elementary beam elements to determine the stiffness parameters of various prismatic cores. Numerical studies show that the modified couple stress Timoshenko beam
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enhances the static deflection predictions of the classical Timoshenko model. A sensitivity measure based on structural ratios is proposed to estimate the influence of size effects in the global beam-level response. The parameters governing size effects in elastic sandwich beams are identified: face-to-core thickness ratio, core density and topology, vertical corrugation
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order and set of load and boundary conditions. Size effects are shown more pronounced in low-density cores that rely on corrugation bending as shear-carrying mechanism. Based on
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the external load, boundary conditions and sensitivity factor, one can assess whether size effects are non-negligible in a given engineering structure.
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Keywords: Size effect, sandwich panels, couple stress theory, prismatic cores
1. Introduction
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All-metal prismatic sandwich panels are a lightweight alternative to conventional plates used in civil, mechanical, vehicle engineering and other applications. The structure is composed of two skins separated by a low-density, visibly discrete core, which governs the mechanical properties and can be tailored based on the application requirements. In the marine industry, sandwich panels with various cores have been studied to substitute the conventional plate with ribs arrangement (Kujala and Klanac, 2005). Prismatics such as the X- and Y-cores have been used for the double hull design due to their relative simple production and impact performance (Naar et al., 2002; Ehlers et al., 2012). Advances in manufacturing encouraged the development of micro-architectured prismatics, whose multifunctional capacity includes
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superior heat and energy absorption performances (Valdevit et al., 2004; Gu et al., 2001; Evans et al., 2001). Overall, the discrete character of prismatic cores leads to computational modelling challenges due to the presence of consecutive length scales, as representing their actual geometry is usually inefficient. Effective continuum models are therefore sought to facilitate the analysis and optimization of such structures. In a broad sense, continuum models of sandwich structures are divided into ply-level
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layerwise and equivalent single layer (ESL) categories; see Carrera and Brischetto (2009) for details. In the context of low-complexity single-layer models, the sandwich theory (e.g. Allen, 1969; Zenkert, 1995) is the classical reference, where deflections are decomposed into bending and shear terms and compatibility between faces and core is invoked. More recently, the homogenization theory based on asymptotic expansion of cell quantities has been used to
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describe the response of periodic structures such as sandwich panels (Hassani and Hinton, 1998; Buannic et al., 2003) based on classical and first-order shear deformation theories. Prismatic sandwich panels compose an application case where relatively flexible cores result in non-negligible local effects and size dependency. An enhanced model is therefore needed to include the internal cell stiffness to the continuum, improving the shear stress distribution
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description near discontinuities and capturing size effects.
In recent years, microstructure-dependent theories re-emerged as an alternative to account
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for the influence of different scales in the global structural response. Liu and Su (2009) have proposed using a two-dimensional couple stress model and theoretically derived length scale
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parameters to capture size effects in cellular solids. The modified couple stress theory of Yang et al. (2002) and generalizations to beams (e.g. Park and Gao, 2006; Ma et al., 2008;
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Reddy, 2011) and plates have received attention due to its relative simplicity in parameter characterization and interpretation of higher-order terms, unlike earlier works on the topic. It
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has been shown in Romanoff and Reddy (2014) and Romanoff et al. (2015) that the modified couple stress beam theory as given in Reddy (2011) satisfactorily predicts the response of web-core sandwich beams with semi-rigid joints. The concepts were extended by Reinaldo Goncalves et al. (2017) to study buckling and free vibrations of transversely flexible sandwich beams with web- and homogeneous cores. The results obtained in these studies indicate that the couple stress theory has potential to model size effects, while retaining simplicity and requiring a single additional parameter that can be determined with sound physical meaning. Yet, the parameters governing the size effect magnitude and relevance in practical modelling of sandwich structures are still unknown.
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The objective of this work is to define an enhanced continuum model for sandwich beams with prismatic cores and study their size-dependent elastic bending response. The main assumptions are followed by a description of the global scale beam, which is founded on the modified couple stress theory. A micromechanics-based framework to determine effective stiffness properties is then shown. Unit cell models constructed with elementary beam elements and analyzed using the direct stiffness method are used to estimate stiffness properties of various prismatic cores. The couple stress beam is then used to investigate size
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effects, which are also validated against corresponding discrete finite element models. In particular, structural ratios that govern the size-dependent behavior are identified, and the effect of cell density, core topology, vertical corrugation order, face-to-core thickness ratio
2. Size-dependent sandwich continuum 2.1. Modelling assumptions
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and set of load and boundary conditions studied.
Consider sandwich beams composed of n periodic cells of length s to a total length L. Continuum modelling requires the identification of two interacting scales, global and local, ) and (
) respectively,
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with coordinates (
(
⁄ )
, where i = 0,1..n is
the unit cell index (Fig. 1). The cells have one plane of topological symmetry whose normal
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is aligned with the x-axis. An equivalent continuum based on the Timoshenko beam theory assumes that the cell average shear deformation can be described in the global scale through a
reference point,
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constant shear angle with horizontal and vertical components, (
⁄ )
(
, located at the
), see Fig. 2a. This implies that the cell internal
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shear fluctuations of order higher than one must have the form of an odd function, which in deflections corresponds to (Fig. 2a)
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( )
(
)
(1)
If Eq. (1) is fulfilled, the derivatives are alternating even and odd functions of the
coordinate x, such that the shear-induced curvature and bending stresses also have zero average by definition; the equivalent continuum is then energetically consistent. Sandwich panels with prismatic cores have the condition (1) violated near local shear discontinuities such as concentrated loads or slope boundary conditions. The shear response becomes a combination of periodic frame-like and local bending modes, being a function of beam and
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internal cell parameters. Once this effect is sufficiently large, excluding the localized effects
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from the global response might not be a satisfactory assumption.
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Fig. 1. Global and local scales of the continuum model with adopted conventions indicated.
To account for the non-periodic shear stress distribution near discontinuities, it is necessary to introduce the unit cell stiffness in the continuum model. A local rotation measure based on the second displacement gradient is used here (see Eq. 3c), which can be thought as the shear angle variation between consecutive cell boundaries
and
as
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additional kinematical compatibility between discrete and continuum (Fig. 2b). That way, the unit cell is no longer represented by a point only, but also by the variation between two
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consecutive points. Assuming a constant cell rotation implies homogeneous cell internal stiffness, rendering piecewise constant local stresses decaying asymptotically away from the
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discontinuity point.
Fig. 2. (a) Constant shear angle definition in the classical first-order continuum and internal cell fluctuations (b) local rotation measure as the variation of shear angle between cell boundaries and .
2.2. Couple stress continuum model
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Consider the modified couple stress Timoshenko beam model in Ma et al. (2008) and Reddy (2011). The displacement field can be expressed in the form ( where
)
( )
(
)
( )
(2)
( ) is the transverse rotation of the cross-section at the reference axis and
( ) the
transverse displacement of the reference axis (Fig. 3). As the bending response is the only object of study, the mid-plane axial displacement and consequently the axial stress resultant
( where
and
are the normal and shear strains, whereas
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are not included in the model. The strain field is given by (Reddy, 2011)
)
(3)
is the local cell curvature.
( Likewise,
,
and
)
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The bending, shear and local bending stress resultants are respectively (Reddy, 2011) (
)
(4)
represent the bending, transverse shear and local bending stiffness
terms, which are defined for prismatic sandwich panels through unit-cell analysis (see section
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2.3). The static equilibrium equations for the homogeneous beam under a transverse
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distributed load q is given by (Reddy, 2011)
(5)
(6)
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And the boundary conditions (Reddy, 2011)
The underlying static equilibrium equations may be solved analytically, for instance using the
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Navier procedure (Reddy, 2011), or numerically, with the finite element method as the most common choice (Arbind and Reddy, 2013; Reinaldo Goncalves et al., 2017). Alternatively, the exact solution can be employed; closed-form equations for common loading cases have been derived in Karttunen et al. (2016). The classical Timoshenko beam theory is recovered by constraining the local rotation; meanwhile, the modified couple stress Euler-Bernoulli model is obtained if
.
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Fig. 3. Kinematics of deformation of the modified couple stress Timoshenko beam.
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2.3. Unit cell models and micromechanics-based effective properties
Consider a two-dimensional unit cell i = n of an arbitrary prismatic sandwich beam of unit thickness in the local coordinate system (see Fig. 1). The cell body is denoted Ω, with origin located at mid-length and vertical neutral axis. The boundaries
defined from topological
periodicity are subjected to boundary tractions, while top and bottom boundaries
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traction-free. Three cell boundary value problems, where equivalent
,
,
are
(see Eq. 4)
are enforced one at a time while the others are set to zero, are used to compute effective properties. The average cell stresses and strains are determined using the principles of cell micromechanics (Sun and Vaidya, 1996; Lok and Cheng, 2000). The unit cell models used
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for the computations and their underlying boundary conditions are shown next. In addition to the boundary conditions discussed, the cells must be further restrained at the horizontal
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symmetry plane to prevent rigid body motion.
Fig. 4. Unit cell models for equivalent stiffness determination (a) bending stiffness (b) transverse shear stiffness (c) local bending stiffness.
a) Bending stiffness Fig. 4a shows a unit cell under horizontal forces in opposite directions applied at top and bottom faces, i.e. proposed
at
⁄ . The following boundary conditions are
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⁄
)
(
⁄
)
( ⁄
7
)
( ⁄
)
(7)
The average bending curvature is determined (8) ⁄
⁄ )
(
⁄
⁄ ). The bending stiffness reduces to
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(
where
b) Transverse shear stiffness
(9)
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Fig. 4b shows a unit cell under the transverse force Qx and horizontal forces N required for static equilibrium
,
and
are the transverse force at top, bottom faces and ith core member across
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where
(10)
⁄
)
(
⁄
)
( ⁄ ( ⁄
)
)
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(
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the unit cell boundary respectively. The displacement boundary conditions are proposed
( )
(11)
Based on Eqs. (10-11), it is implied that core boundary members can deform horizontally in
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periodic fashion (see Fig. 2a). The cell does not necessarily deform as a parallelogram, which would be an overly stiff requirement (Sun and Vaidya, 1996). The vertical boundary
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displacement
is constant to ensure that the cell remains incompressible after deformation.
The average cell shear strain is then obtained
where
(12) ( ⁄ )
(
⁄ ). The transverse shear stiffness reduces to (13)
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c) Local bending stiffness Unlike the previous quantities, the deformation mode induced by Pxy is not periodic in Ω. Yet, by assuming that the local stiffness varies slowly in Ω, that is, that the cell is relatively homogeneous in terms of its internal constituents, an estimate of the local stiffness can be obtained. Consider the cell of Fig. 4c under distributed edge bending moments. In this mode, the internal cell constituents are flexural only, such that Pxy induces purely internal rotations.
(
)
The following slope boundary condition is proposed )
(
⁄
)
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( ⁄
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The couple-stress resultant in Eq. (4c) is defined (14)
(15)
Assuming that the local rotation is much higher than the whole beam curvature as expected in size-effect sensitive structures, the local rotation vector reduces to (
)
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and the local bending stiffness is obtained
(16)
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(17)
2.4. Elastic constants for selected prismatic cores
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Stiffness parameters are derived for selected prismatic cores according to the micromechanical models shown in 2.3. The cells are constructed using Euler-Bernoulli beam
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elements and analyzed using the direct stiffness method. For the derivation of the local bending stiffness
, the axial degree of freedom is constrained, thus the elements are purely
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flexural. Face and core material properties are the same, defined by the elastic modulus E and Poisson’s ratio ν. Top and bottom faces have equal thickness
in all cases. It is assumed in
the derivations that the cell constituents are thin-walled, that is,
. The
corrugations are considered perfectly connected to the faces, composing rigid assemblies with combined properties. Based on Eq. (9), the global bending stiffness is approximated in all cases by (18)
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Equivalent shear and local bending stiffness properties for sandwich beams with single order
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of corrugation, nc = 1, (Fig. 5) are summarized in Table 1.
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Fig. 5. Unit cell models of prismatic sandwich beams with single order of corrugation (nc = 1), (a) web-core (b) triangular corrugated core (c) hexagonal honeycomb core (d) Y-core.
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Fig. 6. Unit cell models of (a) diamond core (b) hexagonal honeycomb core sandwich beams with various orders of corrugation.
Web-core
DQ
(
) (
)
(Fig. 5a)
Triangular
Sxy
(
DQ
(
corrugated core (Fig. 5b)
Sxy
) (
)
(
) (
Hexagonal honeycomb core,
)
DQ
(
)
(
) )
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nc = 1 (Fig. 5c)
Sxy
Y-core
DQ
(
(
)
( )
)
) (
(
)
)
(
(
)
(
(Fig. 5d)
Sxy
10
)
)
(
(
)
)
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Table 1. Stiffness parameters for sandwich panels with selected prismatic cores and nc = 1.
Fig. 6 shows the unit cells of multi-layered sandwich panels with diamond and hexagonal honeycomb cores (e.g. Valdevit et al., 2004; Gu et al., 2001). The stiffness parameters for diamond core beams (Fig. 6a) is given in the compact form
)
(
)
(
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(
)
(19) (20)
Stiffness parameters for hexagonal honeycomb core (Fig. 6b) are provided in Appendix A for
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selected corrugation orders. 3. Size-dependent sandwich response
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Stiffness size dependency is observed in sandwich panels as result of the relative influence of non-periodic stress distribution near discontinuities in the global structural response. The
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effect of discontinuities, such as concentrated loads and slope boundary conditions, vanishes asymptotically away from their application point and may be relevant if this range is non-
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negligible in relation to the overall beam length. Classical continuum models with strain measure based on the first displacement gradient have no mechanism to include the cell
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structural length scale in the analysis, and hence predict size-independent response. Conversely, single-layer models that include higher-order displacement gradients can supplement the continuum with information from the unit cell scale. In the modified couple stress Timoshenko beam (MCSTBT), size effects arising from the local scale stiffness are estimated through the non-dimensional relation √
(21)
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If face and core material properties are equal, the magnitude of such size effects for a given set of loading and boundary conditions is uniquely dependent on the cell geometric parameters and length between supports L. For selected prismatic geometries in 2.4 (Figs. 5are given in Table 2.
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6) the resulting
Fig. 7. Example of three-dimensional finite element model used for validation throughout this work (web-core panel strip in three-point bending and mid-length symmetry).
To investigate size effects in the continuum modelling of sandwich beams with prismatic
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cores, selected topologies are studied in the linear elastic framework. The responses are compared to the ones obtained with the classical Timoshenko beam theory (TBT) throughout
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the study. To emphasize the size effect, deflections are presented relative to the maximum deflection obtained with the TBT. In all analyses, the material properties E = 206GPa and ν =
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0.3 are assumed (steel). Discrete three-dimensional finite element (3D FE) models are used for validation (Fig. 7). The validation models represent the actual panel geometry, finely
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discretized with Abaqus S4R shell elements (Dassault Systemes, 2013). Boundary conditions are applied as to ensure transverse incompressibility at the cross-sections under loading and supports. At the surfaces perpendicular to the beam main axis, plane strain conditions
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guarantee beam-like response. Size-effect sensitivity parameter ( ) (
Web-core
Triangular core
( ) ( )
(
) )(
)
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X-core (Diamond, n=2)
( ) ( )
Hexagonal core (n=1) Y-core (h = d/2 = s/2)
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( )
( )
(
(
)( (
√ )[ ( (
) )(
√ ) √ )
) (
√ )
(
]
√
√ )
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Table 2. Size-effect sensitivity parameter for sandwich panels with selected prismatic cores; face-to-core ⁄ . thickness ratio
3.1. Discrete response vs. classical and non-classical continuum descriptions
Consider prismatic sandwich panel strips with unit-cell properties shown in Fig. 8. The lengthwise deflection distribution is traced for limiting cases using classical and size-
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dependent continuum theories. In all cases, the beams are in three-point bending, with total lengths LA = 2.4m, LB = 0.64m and LC = 0.72m. The cases A-C are selected based on their shear-flexibility and sensitivity to size effects. In short, they can be summarized as follows, A) Shear stiff,
; low size-effect sensitivity
; moderate size-effect sensitivity
C) Shear flexible
; high size-effect sensitivity
; ;
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⁄
where
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B) Shear flexible
;
measures the shear flexibility of the beam (Zenkert, 1995). In
addition to the TBT and MCSTBT models, we also present the curves obtained with the
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Modified Couple Stress Euler-Bernoulli (MCSEBT) beam theory (Park and Gao, 2006),
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wherein the non-classical parameter is taken the same as in sections 2.3-2.4. The MCSTBT model is observed to predict the average static deflection distributions
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accurately in all cases. In case A (Fig. 8a), a bending-dominated problem, the three models render approximately the same deflection lines. Case B (Fig. 8b) shows a shear-sensitive problem; the MCSEBT has no mechanism to include shear deformations, thus predicts overly stiff response. The TBT and MCSTBT models behave similarly, and a kink is observed under the concentrated force as the unit cell has little internal bending stiffness. Local bending is observed only in the immediate vicinity of the concentrated force and its effect is nearly negligible. Case C (Fig. 8c) emphasizes the need of the MCSTBT for a size-dependent continuum; the unit cell stiffness is high, thus local bending is non-negligible in the entire cell near the concentrated force. This behavior cannot be described with TBT and MCSEBT
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theories due to the lack of length-scale parameter and shear deformation measure
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respectively.
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Fig. 8. Discrete cell geometry and deflection lines obtained with classical and non-classical continua and discrete finite elements.
3.2. Fundamental ratios governing the size effect: web-core case study
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Table 2 indicates that size effect in sandwich panels with prismatic cores is governed by certain structural ratios. To understand their influence in the response prediction with TBT and MCSTBT models, we select a simple geometry with weak coupling among geometric parameters. Consider a web-core sandwich panel strip of unit width in three-point bending. For this geometry, three ratios govern the size effect, which are studied here separately: ⁄ , ⁄ and
⁄
(see Fig. 5a). The sandwich panel depth is taken constant, d = 0.05m for all
analyses. In (a) and (b), tf = 0.004m. a) Relative unit cell size ⁄ (Fig. 9a)
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⁄
⁄ and
⁄
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be constant and the relative unit cell size ⁄
⁄ . As predicted in Table 2, the local effects become negligible as ⁄
⁄
, hence the TBT
model is incrementally accurate. The relative difference between beam models is a linear function of ⁄ , which accounts for the local effects magnitude. b) Unit cell aspect ratio ⁄ (Fig. 9b) ⁄
and
is here proportional to √ ⁄
⁄ are constant, while ⁄ . As ⁄
⁄
. The parameter
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In this analysis,
, the local effects are no longer dependent
on the cell aspect ratio, but still on the other ratios; the relative difference between beam models becomes therefore nearly constant. c) Face-to-core thickness ratio
⁄
(Fig. 9c)
⁄
⁄ and
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To study the role of member thicknesses, we select
⁄
⁄ , with ⁄
. It is shown that the magnitude of local effects does not depend substantially on
this ratio if
, in fact, converging to constant error as
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. Conversely, if
rapidly increasing as a function of ( ⁄ )
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incrementally increasing the ratio results in
⁄
, .
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Fig. 9. Maximum deflection predictions for web-core panel strip using beam and reference models as function of (a) relative unit cell size ⁄ (b) unit cell aspect ratio ⁄ (c) face-to-core thickness ratio ⁄ .
3.3. Role of core topology and relative density
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The position of corrugations and their relative density in relation to the overall unit cell area affect the structural ratios of Table 2 and thus the size-effect sensitivity. The core relative density is defined in the general case ∑
(24)
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where lc is the length of the ith to a total of j core members. We study common prismatic cores with single and multiple orders of corrugation to compare their behavior and validate
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the MCSTBT in terms of maximum deflection against three-dimensional finite element models. Face thicknesses and depths are scaled to obtain constant bending stiffness (Dxx =
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848.9kNm), while the thickness of the corrugations is adjusted as function of the core relative
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density.
Fig. 10. Core geometries and dimensions used for the core density analysis, nc = 1.
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Fig. 11. Core geometries and dimensions for panels with varying corrugation order nc.
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Fig. 12. Size-effect sensitivity parameter as function of the core geometry and relative density ratio (a) cores with order of corrugation nc = 1 (b) hexagonal and diamond cores with varying nc.
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a) Common prismatic cores with nc = 1
Consider prismatic sandwich panels with dimensions shown in Fig. 10. Fig. 12a shows the (Table 2) in terms of ρc for the geometries selected. An association of size-effect
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parameter
sensitivity with density and mechanism carrying transverse loads is observed. Cores whose
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main mechanism involves bending of the corrugations display dramatic increases in sensitivity as their slenderness is reduced. In contrast, cores that rely on membrane stretching
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have lower sensitivity overall, which is only marginally dependent on ρc. Fig. 13 shows the maximum deflection predicted by the models as function of the core
density,
, for a unit-width panel with L = 0.8m in three-point bending. In
size-effect sensitive cores, the TBT exhibits overly soft response as the density is reduced. In contrast, the MCSTBT is able to predict deflections very satisfactorily for all cases studied with maximum relative error under 2%. Differences between MCSTBT and 3D FE are attributed to internal cell fluctuations due to higher-order derivatives of displacement, which are averaged to a constant curvature as discussed in section 2.
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Fig. 13. Maximum deflection predictions for (a) web-core (b) triangular corrugated core (c) hexagonal honeycomb core (d) Y-core sandwich panels with varying core relative volume ratio using beam and validation models.
b) Multi-layered prismatic cores
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Consider unit-width panels with hexagonal honeycomb and diamond cores and varying core volume ratio as shown in Fig. 11. Fig. 12b shows the sensitivity parameter
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the corrugation order and core relative density,
as function of
. In hexagonal honeycomb
panels, the sensitivity increases with the corrugation order in low-density cores. The
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increments are increasingly smaller as nc grows, tending to a constant as the cross-section becomes homogeneous,
. In contrast, in diamond core panels, the corrugation order
plays a minimal role in the panel stiffness and size-effect sensitivity. The behavior is consistent with the shear-carrying mechanism dominant in each core: bending of cell walls in hexagonal honeycombs and corrugation stretching in diamond cores. Fig. 14 validates the MCSTBT against three-dimensional finite elements in terms of maximum deflections. The analyses are undertaken for a doubly-clamped unit-width 0.4m long panel strip under centered vertical force. It is shown that the maximum deflection
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increases with the corrugation order in hexagonal honeycomb cores due to the increased shear-flexibility as the wall thicknesses are reduced (Fig. 14a). The increase is considerably more subtle in diamond core panels. It is shown that the response of multi-layered sandwich panels can be accurately predicted with the MCSTBT regardless of their geometry, density or
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corrugation order.
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Fig. 14. Maximum deflection predictions for doubly-clamped (a) hexagonal honeycomb core (b) diamond core sandwich panels with two orders of corrugation as function of the core relative density
3.4. Role of boundary conditions
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Besides the sensitivity parameter, the magnitude of size effects in prismatic sandwich structures depends on the loading scenario, as concentrated loads and restrictive boundary conditions induce the size-dependent behavior. In the context of concentrated forces, it is of
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particular interest how severe is the discontinuity in the shear force diagram. We compare three common loading cases in which size effect is observable: simply supported and doubly
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clamped beams under mid-span concentrated force and end-loaded cantilever. Consider panels with web-core, Y-core and hexagonal honeycomb core (nc = 1) and
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dimensions shown in Fig. 10. The relative difference in maximum deflections obtained with TBT and MCSTBT is shown in Fig. 15 as function of . Overall, the relative difference is a linear function of
when local effects are moderate, with slope governed by the discontinuity
type. It is largest for the doubly clamped beam, as multiple discontinuities are present, and lowest for the cantilever beam, where the only discontinuity is the clamped end. The results of previous sections can be readily scaled for other load conditions, with the size effect becoming more or less severe.
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Fig. 15. Relative difference between TBT and MCSTBT models for different sets of loading and boundary conditions as function of the parameter .
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4. Discussion and conclusions
In this study, the modified couple stress Timoshenko beam theory (Ma et al., 2008, Reddy, 2011) has been applied to study the size-dependent response of sandwich beams with prismatic cores. Unlike previous works (Dai and Zhang, 2008; Liu and Su, 2009), we
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consider solely local stiffening causing disturbances in the periodic shear field as size effect, as changes in flexural rigidity due to material positioning can be modelled using classical
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continua. Moreover, the size effects considered are only due to the length-scale interactions, and not to other local shear-field disturbances. It has been assumed that the structures are a repetition of transversely rigid complete cells. In reality, prismatic structures might be
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transversely flexible (Frostig and Baruch, 1990) and the edges may contain incomplete cells (Anderson and Lakes, 1994) promoting other types of size effects, whose influence in the
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response of lightweight sandwich structures is left for future work. In this study, we have chosen the modified couple stress Timoshenko beam theory
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(MCSTBT) given its simplicity; it includes a single non-classical stiffness parameter in its formulation, whose interpretation in the context of prismatic sandwich panels is tangible. A downside is that the local stiffness must be considered constant, while in reality the cells are highly discrete. Further consecutive scales could be introduced through higher-order gradients of displacement improving the cell stiffness description. The micromechanics-based stiffness determination framework generalizes the works of Romanoff and Reddy (2014) and Romanoff et al. (2015) in studying web-core panels with semi-rigid joints; in their work, equivalence with the classical sandwich theory (Allen, 1969) was invoked to determine the
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non-classical stiffness. It has been shown that the modified couple stress Euler-Bernoulli beam theory (Park and Gao, 2006) is not suitable to model the size-dependent behavior of these structures, as the size effects arise only when the shear flexibility is high. Size effects in prismatic sandwich beams are shown to be related to certain structural ratios involving cell and overall beam dimensions, being the exponent of each ratio a measure of its sensitivity. Such ratios have been studied in an example involving a simple geometry,
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for which the effects are easily isolated. The influence of core density and topology in the size-dependency has been then investigated. Overall, it has been shown that size effect is more pronounced in cores where corrugation bending is an important shear-carrying mechanisms. The core relative density governs the size effect magnitude, which is also influenced by the corrugation order in certain prismatic panels such as the hexagonal
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honeycomb type. Lastly, the influence of loading scenario in the size effect magnitude has been investigated; three simple cases have been selected, and the doubly clamped under midlength vertical load seen to be critical as it introduces the most severe set of discontinuities. While the Elastic Moduli of faces and core are taken for simplicity equal in this study, the
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size effect magnitude should otherwise also dependent on their ratio. Appendix A: Stiffness parameters for hexagonal honeycomb core panel
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The coefficients for the stiffness of hexagonal core sandwich panels (Fig. 6b) with selected
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orders of corrugation nc are presented in Table A.1. nc
(
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3
(
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2
5
(
(
)
)
(
( (
(
Sxy ) )
(
)
(
)
) )
( )
DQ
) )
(
)
Table A.1. Transverse shear stiffness DQ and local bending stiffness Sxy of hexagonal honeycomb core beams with selected orders of corrugation nc.
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Acknowledgements The authors gratefully acknowledge the financial support from the Graduate School of Aalto University School of Engineering and the partners of the Finland Distinguished Professor programme ―Non-linear response of large, complex thin-walled structures‖: Tekes, Napa, SSAB, Deltamarin, Koneteknologiakeskus Turku and Meyer Turku.
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Allen, H.G., 1969. Analysis and Design of Structural Sandwich Panels, Pergamon Press. Anderson, W.B. & Lakes, R.S., 1994. Size effects due to Cosserat elasticity and surface damage in closed-cell polymethacrylimide foam. Journal of Materials Science, 29(24), pp.6413–6419.
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Carrera, E. & Brischetto, S., 2009. A Survey With Numerical Assessment of Classical and Refined Theories for the Analysis of Sandwich Plates. Applied Mechanics Reviews, 62(1), p.10803.
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Dai, G.M. & Zhang, W.H., 2008. Size effects of basic cell in static analysis of sandwich beams. International Journal of Solids and Structures, 45(9), pp.2512–2533. Dassault Systèmes, 2013. ABAQUS documentation. (Dassault Systèmes Simulia Corp.)
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Ehlers, S., Tabri, K., Romanoff, J. & Varsta, P., ―Numerical and experimental investigation on the collision resistance of the X-core structure‖, Ships and Offshore Structures, Vol. 7, No. 1, 2012, pp. 21-29.
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Evans, A.G., Hutchinson, J.W., Fleck, N.A., Ashby, M.F. & Wadley, H.N.G., 2001. The topological design of multifunctional cellular metals. Progress in Materials Science, 46(3–4), pp.309–327. Frostig, Y. & Baruch, M., 1990. Bending of sandwich beams with transversely flexible core. AIAA Journal, 28(3), pp.523–531. Gu, S., Lu, T.J. & Evans, A.G., 2001. On the design of two-dimensional cellular metals for combined heat dissipation and structural load capacity. International Journal of Heat and Mass Transfer, 44(11), pp.2163–2175. Hassani, B. & Hinton, E., 1998. A review of homogenization and topology optimization I— homogenization theory for media with periodic structure. Computers & Structures,
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69(6), pp.707–717. Available at: http://linkinghub.elsevier.com/retrieve/pii/S004579499800131X. Karttunen, A.T., Romanoff, J. & Reddy, J.N., 2016. Exact microstructure-dependent Timoshenko beam element. International Journal of Mechanical Sciences, 111–112, pp.35–42. Kujala, P. & Klanac, A., 2005. Steel sandwich panels in marine applications. Brodogradnja, 56(4), pp.305–314.
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Liu, S. & Su, W., 2009. Effective couple-stress continuum model of cellular solids and size effects analysis. International Journal of Solids and Structures, 46(14–15), pp.2787– 2799. Lok, T.S. & Cheng, Q.H., 2000. Elastic stiffness properties and behavior of truss-core sandwich panel. Journal of Structural Engineering, 126(5), pp.552–559.
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Ma, H.M., Gao, X.L. & Reddy, J.N., 2008. A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. Journal of the Mechanics and Physics of Solids, 56(12), pp.3379–3391. Naar, H., Kujala, P., Simonsen, B.C. & Ludolphy, H., 2002. Comparison of the crashworthiness of various bottom and side structures. Marine Structures, 15(4–5), pp.443–460.
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Noor, A.K., Burton, W.S. & Bert, C.W., 1996. Computational Models for Sandwich Panels and Shells. Applied Mechanics Reviews, 49(3), pp.155–199.
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Park, S.K. & Gao, X.-L., 2006. Bernoulli–Euler beam model based on a modified couple stress theory. Journal of Micromechanics and Microengineering, 16(11), pp.2355–2359. Available at: http://stacks.iop.org/09601317/16/i=11/a=015?key=crossref.5e13d8eb3f8bcf03dc33ffeb6bd15219.
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Reddy, J.N., 2011. Microstructure-dependent couple stress theories of functionally graded beams. Journal of the Mechanics and Physics of Solids, 59(11), pp.2382–2399. Available at: http://linkinghub.elsevier.com/retrieve/pii/S0022509611001566.
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Reinaldo Goncalves, B., Karttunen, A., Romanoff, J. & Reddy, J.N., 2017. Buckling and free vibration of shear-flexible sandwich beams using a couple-stress-based finite element. Composite Structures, 165, pp.233–241. Romanoff, J. & Reddy, J.N., 2014. Experimental validation of the modified couple stress Timoshenko beam theory for web-core sandwich panels. Composite Structures, 111, pp.130–137. Available at: http://linkinghub.elsevier.com/retrieve/pii/S026382231300620X. Romanoff, J., Reddy, J.N. & Jelovica, J., 2015. Using Non-Local Timoshenko Beam Theories for Prediction of Micro- and Macro-Structural Responses. Composite Structures. Available at: http://www.sciencedirect.com/science/article/pii/S0263822315005528.
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Sun, C.T. & Vaidya, R.S., 1996. Prediction of composite properties from a representative volume element. Composites Science and Technology, 56(2), pp.171–179. Valdevit, L., Hutchinson, J.W. & Evans, A.G., 2004. Structurally optimized sandwich panels with prismatic cores. International Journal of Solids and Structures, 41(18–19), pp.5105–5124.
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Yang, F., Chong, A.C.M., Lam, D.C.C. & Tong, P., 2002. Couple stress based strain gradient theory for elasticity. International Journal of Solids and Structures, 39(10), pp.2731– 2743.
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Zenkert, D., 1995. An introduction to sandwich construction, Sheffield, UK: Engineering Materials Advisory Service.
Size-dependent modelling of elastic sandwich beams with prismatic cores Bruno Reinaldo Goncalves , Jani Romanoff PII: DOI: Reference:
S0020-7683(17)30534-6 10.1016/j.ijsolstr.2017.12.001 SAS 9819
To appear in:
International Journal of Solids and Structures
Received date: Revised date: Accepted date:
17 August 2017 10 November 2017 1 December 2017
Please cite this article as: Bruno Reinaldo Goncalves , Jani Romanoff , Size-dependent modelling of elastic sandwich beams with prismatic cores, International Journal of Solids and Structures (2017), doi: 10.1016/j.ijsolstr.2017.12.001
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Size-dependent modelling of elastic sandwich beams with prismatic cores Bruno Reinaldo Goncalves* and Jani Romanoff Aalto University, Department of Mechanical Engineering, FI-00076 Aalto, Finland *Corresponding author: [email protected]
ABSTRACT Sandwich panel strips with prismatic cores are modelled using the modified couple stress
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theory and their elastic size-dependent bending behavior investigated. Compatibility between the discrete sandwich and continuum beam kinematics is first discussed. A micromechanicsbased framework to estimate effective mechanical properties is provided and unit cell models constructed with elementary beam elements to determine the stiffness parameters of various prismatic cores. Numerical studies show that the modified couple stress Timoshenko beam
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enhances the static deflection predictions of the classical Timoshenko model. A sensitivity measure based on structural ratios is proposed to estimate the influence of size effects in the global beam-level response. The parameters governing size effects in elastic sandwich beams are identified: face-to-core thickness ratio, core density and topology, vertical corrugation
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order and set of load and boundary conditions. Size effects are shown more pronounced in low-density cores that rely on corrugation bending as shear-carrying mechanism. Based on
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the external load, boundary conditions and sensitivity factor, one can assess whether size effects are non-negligible in a given engineering structure.
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Keywords: Size effect, sandwich panels, couple stress theory, prismatic cores
1. Introduction
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All-metal prismatic sandwich panels are a lightweight alternative to conventional plates used in civil, mechanical, vehicle engineering and other applications. The structure is composed of two skins separated by a low-density, visibly discrete core, which governs the mechanical properties and can be tailored based on the application requirements. In the marine industry, sandwich panels with various cores have been studied to substitute the conventional plate with ribs arrangement (Kujala and Klanac, 2005). Prismatics such as the X- and Y-cores have been used for the double hull design due to their relative simple production and impact performance (Naar et al., 2002; Ehlers et al., 2012). Advances in manufacturing encouraged the development of micro-architectured prismatics, whose multifunctional capacity includes
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superior heat and energy absorption performances (Valdevit et al., 2004; Gu et al., 2001; Evans et al., 2001). Overall, the discrete character of prismatic cores leads to computational modelling challenges due to the presence of consecutive length scales, as representing their actual geometry is usually inefficient. Effective continuum models are therefore sought to facilitate the analysis and optimization of such structures. In a broad sense, continuum models of sandwich structures are divided into ply-level
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layerwise and equivalent single layer (ESL) categories; see Carrera and Brischetto (2009) for details. In the context of low-complexity single-layer models, the sandwich theory (e.g. Allen, 1969; Zenkert, 1995) is the classical reference, where deflections are decomposed into bending and shear terms and compatibility between faces and core is invoked. More recently, the homogenization theory based on asymptotic expansion of cell quantities has been used to
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describe the response of periodic structures such as sandwich panels (Hassani and Hinton, 1998; Buannic et al., 2003) based on classical and first-order shear deformation theories. Prismatic sandwich panels compose an application case where relatively flexible cores result in non-negligible local effects and size dependency. An enhanced model is therefore needed to include the internal cell stiffness to the continuum, improving the shear stress distribution
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description near discontinuities and capturing size effects.
In recent years, microstructure-dependent theories re-emerged as an alternative to account
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for the influence of different scales in the global structural response. Liu and Su (2009) have proposed using a two-dimensional couple stress model and theoretically derived length scale
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parameters to capture size effects in cellular solids. The modified couple stress theory of Yang et al. (2002) and generalizations to beams (e.g. Park and Gao, 2006; Ma et al., 2008;
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Reddy, 2011) and plates have received attention due to its relative simplicity in parameter characterization and interpretation of higher-order terms, unlike earlier works on the topic. It
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has been shown in Romanoff and Reddy (2014) and Romanoff et al. (2015) that the modified couple stress beam theory as given in Reddy (2011) satisfactorily predicts the response of web-core sandwich beams with semi-rigid joints. The concepts were extended by Reinaldo Goncalves et al. (2017) to study buckling and free vibrations of transversely flexible sandwich beams with web- and homogeneous cores. The results obtained in these studies indicate that the couple stress theory has potential to model size effects, while retaining simplicity and requiring a single additional parameter that can be determined with sound physical meaning. Yet, the parameters governing the size effect magnitude and relevance in practical modelling of sandwich structures are still unknown.
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The objective of this work is to define an enhanced continuum model for sandwich beams with prismatic cores and study their size-dependent elastic bending response. The main assumptions are followed by a description of the global scale beam, which is founded on the modified couple stress theory. A micromechanics-based framework to determine effective stiffness properties is then shown. Unit cell models constructed with elementary beam elements and analyzed using the direct stiffness method are used to estimate stiffness properties of various prismatic cores. The couple stress beam is then used to investigate size
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effects, which are also validated against corresponding discrete finite element models. In particular, structural ratios that govern the size-dependent behavior are identified, and the effect of cell density, core topology, vertical corrugation order, face-to-core thickness ratio
2. Size-dependent sandwich continuum 2.1. Modelling assumptions
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and set of load and boundary conditions studied.
Consider sandwich beams composed of n periodic cells of length s to a total length L. Continuum modelling requires the identification of two interacting scales, global and local, ) and (
) respectively,
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with coordinates (
(
⁄ )
, where i = 0,1..n is
the unit cell index (Fig. 1). The cells have one plane of topological symmetry whose normal
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is aligned with the x-axis. An equivalent continuum based on the Timoshenko beam theory assumes that the cell average shear deformation can be described in the global scale through a
reference point,
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constant shear angle with horizontal and vertical components, (
⁄ )
(
, located at the
), see Fig. 2a. This implies that the cell internal
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shear fluctuations of order higher than one must have the form of an odd function, which in deflections corresponds to (Fig. 2a)
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( )
(
)
(1)
If Eq. (1) is fulfilled, the derivatives are alternating even and odd functions of the
coordinate x, such that the shear-induced curvature and bending stresses also have zero average by definition; the equivalent continuum is then energetically consistent. Sandwich panels with prismatic cores have the condition (1) violated near local shear discontinuities such as concentrated loads or slope boundary conditions. The shear response becomes a combination of periodic frame-like and local bending modes, being a function of beam and
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internal cell parameters. Once this effect is sufficiently large, excluding the localized effects
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from the global response might not be a satisfactory assumption.
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Fig. 1. Global and local scales of the continuum model with adopted conventions indicated.
To account for the non-periodic shear stress distribution near discontinuities, it is necessary to introduce the unit cell stiffness in the continuum model. A local rotation measure based on the second displacement gradient is used here (see Eq. 3c), which can be thought as the shear angle variation between consecutive cell boundaries
and
as
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additional kinematical compatibility between discrete and continuum (Fig. 2b). That way, the unit cell is no longer represented by a point only, but also by the variation between two
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consecutive points. Assuming a constant cell rotation implies homogeneous cell internal stiffness, rendering piecewise constant local stresses decaying asymptotically away from the
AC
CE
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discontinuity point.
Fig. 2. (a) Constant shear angle definition in the classical first-order continuum and internal cell fluctuations (b) local rotation measure as the variation of shear angle between cell boundaries and .
2.2. Couple stress continuum model
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Consider the modified couple stress Timoshenko beam model in Ma et al. (2008) and Reddy (2011). The displacement field can be expressed in the form ( where
)
( )
(
)
( )
(2)
( ) is the transverse rotation of the cross-section at the reference axis and
( ) the
transverse displacement of the reference axis (Fig. 3). As the bending response is the only object of study, the mid-plane axial displacement and consequently the axial stress resultant
( where
and
are the normal and shear strains, whereas
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are not included in the model. The strain field is given by (Reddy, 2011)
)
(3)
is the local cell curvature.
( Likewise,
,
and
)
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The bending, shear and local bending stress resultants are respectively (Reddy, 2011) (
)
(4)
represent the bending, transverse shear and local bending stiffness
terms, which are defined for prismatic sandwich panels through unit-cell analysis (see section
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2.3). The static equilibrium equations for the homogeneous beam under a transverse
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distributed load q is given by (Reddy, 2011)
(5)
(6)
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And the boundary conditions (Reddy, 2011)
The underlying static equilibrium equations may be solved analytically, for instance using the
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Navier procedure (Reddy, 2011), or numerically, with the finite element method as the most common choice (Arbind and Reddy, 2013; Reinaldo Goncalves et al., 2017). Alternatively, the exact solution can be employed; closed-form equations for common loading cases have been derived in Karttunen et al. (2016). The classical Timoshenko beam theory is recovered by constraining the local rotation; meanwhile, the modified couple stress Euler-Bernoulli model is obtained if
.
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Fig. 3. Kinematics of deformation of the modified couple stress Timoshenko beam.
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2.3. Unit cell models and micromechanics-based effective properties
Consider a two-dimensional unit cell i = n of an arbitrary prismatic sandwich beam of unit thickness in the local coordinate system (see Fig. 1). The cell body is denoted Ω, with origin located at mid-length and vertical neutral axis. The boundaries
defined from topological
periodicity are subjected to boundary tractions, while top and bottom boundaries
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traction-free. Three cell boundary value problems, where equivalent
,
,
are
(see Eq. 4)
are enforced one at a time while the others are set to zero, are used to compute effective properties. The average cell stresses and strains are determined using the principles of cell micromechanics (Sun and Vaidya, 1996; Lok and Cheng, 2000). The unit cell models used
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for the computations and their underlying boundary conditions are shown next. In addition to the boundary conditions discussed, the cells must be further restrained at the horizontal
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CE
PT
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symmetry plane to prevent rigid body motion.
Fig. 4. Unit cell models for equivalent stiffness determination (a) bending stiffness (b) transverse shear stiffness (c) local bending stiffness.
a) Bending stiffness Fig. 4a shows a unit cell under horizontal forces in opposite directions applied at top and bottom faces, i.e. proposed
at
⁄ . The following boundary conditions are
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⁄
)
(
⁄
)
( ⁄
7
)
( ⁄
)
(7)
The average bending curvature is determined (8) ⁄
⁄ )
(
⁄
⁄ ). The bending stiffness reduces to
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(
where
b) Transverse shear stiffness
(9)
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Fig. 4b shows a unit cell under the transverse force Qx and horizontal forces N required for static equilibrium
,
and
are the transverse force at top, bottom faces and ith core member across
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where
(10)
⁄
)
(
⁄
)
( ⁄ ( ⁄
)
)
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(
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the unit cell boundary respectively. The displacement boundary conditions are proposed
( )
(11)
Based on Eqs. (10-11), it is implied that core boundary members can deform horizontally in
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periodic fashion (see Fig. 2a). The cell does not necessarily deform as a parallelogram, which would be an overly stiff requirement (Sun and Vaidya, 1996). The vertical boundary
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displacement
is constant to ensure that the cell remains incompressible after deformation.
The average cell shear strain is then obtained
where
(12) ( ⁄ )
(
⁄ ). The transverse shear stiffness reduces to (13)
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c) Local bending stiffness Unlike the previous quantities, the deformation mode induced by Pxy is not periodic in Ω. Yet, by assuming that the local stiffness varies slowly in Ω, that is, that the cell is relatively homogeneous in terms of its internal constituents, an estimate of the local stiffness can be obtained. Consider the cell of Fig. 4c under distributed edge bending moments. In this mode, the internal cell constituents are flexural only, such that Pxy induces purely internal rotations.
(
)
The following slope boundary condition is proposed )
(
⁄
)
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( ⁄
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The couple-stress resultant in Eq. (4c) is defined (14)
(15)
Assuming that the local rotation is much higher than the whole beam curvature as expected in size-effect sensitive structures, the local rotation vector reduces to (
)
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and the local bending stiffness is obtained
(16)
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(17)
2.4. Elastic constants for selected prismatic cores
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Stiffness parameters are derived for selected prismatic cores according to the micromechanical models shown in 2.3. The cells are constructed using Euler-Bernoulli beam
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elements and analyzed using the direct stiffness method. For the derivation of the local bending stiffness
, the axial degree of freedom is constrained, thus the elements are purely
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flexural. Face and core material properties are the same, defined by the elastic modulus E and Poisson’s ratio ν. Top and bottom faces have equal thickness
in all cases. It is assumed in
the derivations that the cell constituents are thin-walled, that is,
. The
corrugations are considered perfectly connected to the faces, composing rigid assemblies with combined properties. Based on Eq. (9), the global bending stiffness is approximated in all cases by (18)
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Equivalent shear and local bending stiffness properties for sandwich beams with single order
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of corrugation, nc = 1, (Fig. 5) are summarized in Table 1.
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ED
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Fig. 5. Unit cell models of prismatic sandwich beams with single order of corrugation (nc = 1), (a) web-core (b) triangular corrugated core (c) hexagonal honeycomb core (d) Y-core.
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Fig. 6. Unit cell models of (a) diamond core (b) hexagonal honeycomb core sandwich beams with various orders of corrugation.
Web-core
DQ
(
) (
)
(Fig. 5a)
Triangular
Sxy
(
DQ
(
corrugated core (Fig. 5b)
Sxy
) (
)
(
) (
Hexagonal honeycomb core,
)
DQ
(
)
(
) )
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nc = 1 (Fig. 5c)
Sxy
Y-core
DQ
(
(
)
( )
)
) (
(
)
)
(
(
)
(
(Fig. 5d)
Sxy
10
)
)
(
(
)
)
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Table 1. Stiffness parameters for sandwich panels with selected prismatic cores and nc = 1.
Fig. 6 shows the unit cells of multi-layered sandwich panels with diamond and hexagonal honeycomb cores (e.g. Valdevit et al., 2004; Gu et al., 2001). The stiffness parameters for diamond core beams (Fig. 6a) is given in the compact form
)
(
)
(
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(
)
(19) (20)
Stiffness parameters for hexagonal honeycomb core (Fig. 6b) are provided in Appendix A for
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selected corrugation orders. 3. Size-dependent sandwich response
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Stiffness size dependency is observed in sandwich panels as result of the relative influence of non-periodic stress distribution near discontinuities in the global structural response. The
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effect of discontinuities, such as concentrated loads and slope boundary conditions, vanishes asymptotically away from their application point and may be relevant if this range is non-
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negligible in relation to the overall beam length. Classical continuum models with strain measure based on the first displacement gradient have no mechanism to include the cell
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structural length scale in the analysis, and hence predict size-independent response. Conversely, single-layer models that include higher-order displacement gradients can supplement the continuum with information from the unit cell scale. In the modified couple stress Timoshenko beam (MCSTBT), size effects arising from the local scale stiffness are estimated through the non-dimensional relation √
(21)
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If face and core material properties are equal, the magnitude of such size effects for a given set of loading and boundary conditions is uniquely dependent on the cell geometric parameters and length between supports L. For selected prismatic geometries in 2.4 (Figs. 5are given in Table 2.
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6) the resulting
Fig. 7. Example of three-dimensional finite element model used for validation throughout this work (web-core panel strip in three-point bending and mid-length symmetry).
To investigate size effects in the continuum modelling of sandwich beams with prismatic
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cores, selected topologies are studied in the linear elastic framework. The responses are compared to the ones obtained with the classical Timoshenko beam theory (TBT) throughout
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the study. To emphasize the size effect, deflections are presented relative to the maximum deflection obtained with the TBT. In all analyses, the material properties E = 206GPa and ν =
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0.3 are assumed (steel). Discrete three-dimensional finite element (3D FE) models are used for validation (Fig. 7). The validation models represent the actual panel geometry, finely
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discretized with Abaqus S4R shell elements (Dassault Systemes, 2013). Boundary conditions are applied as to ensure transverse incompressibility at the cross-sections under loading and supports. At the surfaces perpendicular to the beam main axis, plane strain conditions
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guarantee beam-like response. Size-effect sensitivity parameter ( ) (
Web-core
Triangular core
( ) ( )
(
) )(
)
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X-core (Diamond, n=2)
( ) ( )
Hexagonal core (n=1) Y-core (h = d/2 = s/2)
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( )
( )
(
(
)( (
√ )[ ( (
) )(
√ ) √ )
) (
√ )
(
]
√
√ )
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Table 2. Size-effect sensitivity parameter for sandwich panels with selected prismatic cores; face-to-core ⁄ . thickness ratio
3.1. Discrete response vs. classical and non-classical continuum descriptions
Consider prismatic sandwich panel strips with unit-cell properties shown in Fig. 8. The lengthwise deflection distribution is traced for limiting cases using classical and size-
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dependent continuum theories. In all cases, the beams are in three-point bending, with total lengths LA = 2.4m, LB = 0.64m and LC = 0.72m. The cases A-C are selected based on their shear-flexibility and sensitivity to size effects. In short, they can be summarized as follows, A) Shear stiff,
; low size-effect sensitivity
; moderate size-effect sensitivity
C) Shear flexible
; high size-effect sensitivity
; ;
ED
⁄
where
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B) Shear flexible
;
measures the shear flexibility of the beam (Zenkert, 1995). In
addition to the TBT and MCSTBT models, we also present the curves obtained with the
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Modified Couple Stress Euler-Bernoulli (MCSEBT) beam theory (Park and Gao, 2006),
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wherein the non-classical parameter is taken the same as in sections 2.3-2.4. The MCSTBT model is observed to predict the average static deflection distributions
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accurately in all cases. In case A (Fig. 8a), a bending-dominated problem, the three models render approximately the same deflection lines. Case B (Fig. 8b) shows a shear-sensitive problem; the MCSEBT has no mechanism to include shear deformations, thus predicts overly stiff response. The TBT and MCSTBT models behave similarly, and a kink is observed under the concentrated force as the unit cell has little internal bending stiffness. Local bending is observed only in the immediate vicinity of the concentrated force and its effect is nearly negligible. Case C (Fig. 8c) emphasizes the need of the MCSTBT for a size-dependent continuum; the unit cell stiffness is high, thus local bending is non-negligible in the entire cell near the concentrated force. This behavior cannot be described with TBT and MCSEBT
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theories due to the lack of length-scale parameter and shear deformation measure
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ED
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respectively.
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Fig. 8. Discrete cell geometry and deflection lines obtained with classical and non-classical continua and discrete finite elements.
3.2. Fundamental ratios governing the size effect: web-core case study
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Table 2 indicates that size effect in sandwich panels with prismatic cores is governed by certain structural ratios. To understand their influence in the response prediction with TBT and MCSTBT models, we select a simple geometry with weak coupling among geometric parameters. Consider a web-core sandwich panel strip of unit width in three-point bending. For this geometry, three ratios govern the size effect, which are studied here separately: ⁄ , ⁄ and
⁄
(see Fig. 5a). The sandwich panel depth is taken constant, d = 0.05m for all
analyses. In (a) and (b), tf = 0.004m. a) Relative unit cell size ⁄ (Fig. 9a)
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⁄
⁄ and
⁄
14
be constant and the relative unit cell size ⁄
⁄ . As predicted in Table 2, the local effects become negligible as ⁄
⁄
, hence the TBT
model is incrementally accurate. The relative difference between beam models is a linear function of ⁄ , which accounts for the local effects magnitude. b) Unit cell aspect ratio ⁄ (Fig. 9b) ⁄
and
is here proportional to √ ⁄
⁄ are constant, while ⁄ . As ⁄
⁄
. The parameter
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In this analysis,
, the local effects are no longer dependent
on the cell aspect ratio, but still on the other ratios; the relative difference between beam models becomes therefore nearly constant. c) Face-to-core thickness ratio
⁄
(Fig. 9c)
⁄
⁄ and
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To study the role of member thicknesses, we select
⁄
⁄ , with ⁄
. It is shown that the magnitude of local effects does not depend substantially on
this ratio if
, in fact, converging to constant error as
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. Conversely, if
rapidly increasing as a function of ( ⁄ )
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incrementally increasing the ratio results in
⁄
, .
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Fig. 9. Maximum deflection predictions for web-core panel strip using beam and reference models as function of (a) relative unit cell size ⁄ (b) unit cell aspect ratio ⁄ (c) face-to-core thickness ratio ⁄ .
3.3. Role of core topology and relative density
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The position of corrugations and their relative density in relation to the overall unit cell area affect the structural ratios of Table 2 and thus the size-effect sensitivity. The core relative density is defined in the general case ∑
(24)
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where lc is the length of the ith to a total of j core members. We study common prismatic cores with single and multiple orders of corrugation to compare their behavior and validate
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the MCSTBT in terms of maximum deflection against three-dimensional finite element models. Face thicknesses and depths are scaled to obtain constant bending stiffness (Dxx =
PT
848.9kNm), while the thickness of the corrugations is adjusted as function of the core relative
AC
CE
density.
Fig. 10. Core geometries and dimensions used for the core density analysis, nc = 1.
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Fig. 11. Core geometries and dimensions for panels with varying corrugation order nc.
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Fig. 12. Size-effect sensitivity parameter as function of the core geometry and relative density ratio (a) cores with order of corrugation nc = 1 (b) hexagonal and diamond cores with varying nc.
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a) Common prismatic cores with nc = 1
Consider prismatic sandwich panels with dimensions shown in Fig. 10. Fig. 12a shows the (Table 2) in terms of ρc for the geometries selected. An association of size-effect
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parameter
sensitivity with density and mechanism carrying transverse loads is observed. Cores whose
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main mechanism involves bending of the corrugations display dramatic increases in sensitivity as their slenderness is reduced. In contrast, cores that rely on membrane stretching
AC
have lower sensitivity overall, which is only marginally dependent on ρc. Fig. 13 shows the maximum deflection predicted by the models as function of the core
density,
, for a unit-width panel with L = 0.8m in three-point bending. In
size-effect sensitive cores, the TBT exhibits overly soft response as the density is reduced. In contrast, the MCSTBT is able to predict deflections very satisfactorily for all cases studied with maximum relative error under 2%. Differences between MCSTBT and 3D FE are attributed to internal cell fluctuations due to higher-order derivatives of displacement, which are averaged to a constant curvature as discussed in section 2.
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Fig. 13. Maximum deflection predictions for (a) web-core (b) triangular corrugated core (c) hexagonal honeycomb core (d) Y-core sandwich panels with varying core relative volume ratio using beam and validation models.
b) Multi-layered prismatic cores
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Consider unit-width panels with hexagonal honeycomb and diamond cores and varying core volume ratio as shown in Fig. 11. Fig. 12b shows the sensitivity parameter
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the corrugation order and core relative density,
as function of
. In hexagonal honeycomb
panels, the sensitivity increases with the corrugation order in low-density cores. The
AC
increments are increasingly smaller as nc grows, tending to a constant as the cross-section becomes homogeneous,
. In contrast, in diamond core panels, the corrugation order
plays a minimal role in the panel stiffness and size-effect sensitivity. The behavior is consistent with the shear-carrying mechanism dominant in each core: bending of cell walls in hexagonal honeycombs and corrugation stretching in diamond cores. Fig. 14 validates the MCSTBT against three-dimensional finite elements in terms of maximum deflections. The analyses are undertaken for a doubly-clamped unit-width 0.4m long panel strip under centered vertical force. It is shown that the maximum deflection
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increases with the corrugation order in hexagonal honeycomb cores due to the increased shear-flexibility as the wall thicknesses are reduced (Fig. 14a). The increase is considerably more subtle in diamond core panels. It is shown that the response of multi-layered sandwich panels can be accurately predicted with the MCSTBT regardless of their geometry, density or
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corrugation order.
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Fig. 14. Maximum deflection predictions for doubly-clamped (a) hexagonal honeycomb core (b) diamond core sandwich panels with two orders of corrugation as function of the core relative density
3.4. Role of boundary conditions
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Besides the sensitivity parameter, the magnitude of size effects in prismatic sandwich structures depends on the loading scenario, as concentrated loads and restrictive boundary conditions induce the size-dependent behavior. In the context of concentrated forces, it is of
PT
particular interest how severe is the discontinuity in the shear force diagram. We compare three common loading cases in which size effect is observable: simply supported and doubly
CE
clamped beams under mid-span concentrated force and end-loaded cantilever. Consider panels with web-core, Y-core and hexagonal honeycomb core (nc = 1) and
AC
dimensions shown in Fig. 10. The relative difference in maximum deflections obtained with TBT and MCSTBT is shown in Fig. 15 as function of . Overall, the relative difference is a linear function of
when local effects are moderate, with slope governed by the discontinuity
type. It is largest for the doubly clamped beam, as multiple discontinuities are present, and lowest for the cantilever beam, where the only discontinuity is the clamped end. The results of previous sections can be readily scaled for other load conditions, with the size effect becoming more or less severe.
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Fig. 15. Relative difference between TBT and MCSTBT models for different sets of loading and boundary conditions as function of the parameter .
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4. Discussion and conclusions
In this study, the modified couple stress Timoshenko beam theory (Ma et al., 2008, Reddy, 2011) has been applied to study the size-dependent response of sandwich beams with prismatic cores. Unlike previous works (Dai and Zhang, 2008; Liu and Su, 2009), we
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consider solely local stiffening causing disturbances in the periodic shear field as size effect, as changes in flexural rigidity due to material positioning can be modelled using classical
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continua. Moreover, the size effects considered are only due to the length-scale interactions, and not to other local shear-field disturbances. It has been assumed that the structures are a repetition of transversely rigid complete cells. In reality, prismatic structures might be
PT
transversely flexible (Frostig and Baruch, 1990) and the edges may contain incomplete cells (Anderson and Lakes, 1994) promoting other types of size effects, whose influence in the
CE
response of lightweight sandwich structures is left for future work. In this study, we have chosen the modified couple stress Timoshenko beam theory
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(MCSTBT) given its simplicity; it includes a single non-classical stiffness parameter in its formulation, whose interpretation in the context of prismatic sandwich panels is tangible. A downside is that the local stiffness must be considered constant, while in reality the cells are highly discrete. Further consecutive scales could be introduced through higher-order gradients of displacement improving the cell stiffness description. The micromechanics-based stiffness determination framework generalizes the works of Romanoff and Reddy (2014) and Romanoff et al. (2015) in studying web-core panels with semi-rigid joints; in their work, equivalence with the classical sandwich theory (Allen, 1969) was invoked to determine the
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non-classical stiffness. It has been shown that the modified couple stress Euler-Bernoulli beam theory (Park and Gao, 2006) is not suitable to model the size-dependent behavior of these structures, as the size effects arise only when the shear flexibility is high. Size effects in prismatic sandwich beams are shown to be related to certain structural ratios involving cell and overall beam dimensions, being the exponent of each ratio a measure of its sensitivity. Such ratios have been studied in an example involving a simple geometry,
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for which the effects are easily isolated. The influence of core density and topology in the size-dependency has been then investigated. Overall, it has been shown that size effect is more pronounced in cores where corrugation bending is an important shear-carrying mechanisms. The core relative density governs the size effect magnitude, which is also influenced by the corrugation order in certain prismatic panels such as the hexagonal
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honeycomb type. Lastly, the influence of loading scenario in the size effect magnitude has been investigated; three simple cases have been selected, and the doubly clamped under midlength vertical load seen to be critical as it introduces the most severe set of discontinuities. While the Elastic Moduli of faces and core are taken for simplicity equal in this study, the
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size effect magnitude should otherwise also dependent on their ratio. Appendix A: Stiffness parameters for hexagonal honeycomb core panel
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The coefficients for the stiffness of hexagonal core sandwich panels (Fig. 6b) with selected
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orders of corrugation nc are presented in Table A.1. nc
(
AC
3
(
CE
2
5
(
(
)
)
(
( (
(
Sxy ) )
(
)
(
)
) )
( )
DQ
) )
(
)
Table A.1. Transverse shear stiffness DQ and local bending stiffness Sxy of hexagonal honeycomb core beams with selected orders of corrugation nc.
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Acknowledgements The authors gratefully acknowledge the financial support from the Graduate School of Aalto University School of Engineering and the partners of the Finland Distinguished Professor programme ―Non-linear response of large, complex thin-walled structures‖: Tekes, Napa, SSAB, Deltamarin, Koneteknologiakeskus Turku and Meyer Turku.
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