Non-linear effective properties for web-core steel sandwich panels in tension

International Journal of Mechanical Sciences 115-116 (2016) 428–437

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International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci

Non-linear effective properties for web-core steel sandwich panels in tension Mihkel Kõrgesaar n, Bruno Reinaldo Goncalves, Jani Romanoff, Heikki Remes Aalto University, Department of Applied Mechanics, Advanced Structures, Rakentajanaukio 2D, 00076 Aalto, Finland

art ic l e i nf o

a b s t r a c t

Article history: Received 19 October 2015 Received in revised form 12 June 2016 Accepted 18 July 2016 Available online 21 July 2016

All-metal sandwich panels with a low weight-to-stiffness ratio provide the means to develop cost and energy efficient large-scale thin-walled structures such as ships. However, the modeling and computational effort for a three-dimensional (3-D) sandwich structures may be significant especially in the limit state analysis of ship structures. This can become a significant drawback especially in the conceptual design stage. To this end, we determine the load–displacement curves until tensile instability for orthotropic web-core sandwich panels under multi-axial tension loading. Load–displacement curves are determined by dividing sandwich panel into discrete members and analyzing each of the constituents separately. Combined response is obtained by rule of mixtures. Tensile instability of panel is assessed using analytical necking criteria. Validity of the analytically obtained curves is established by comparing them with those obtained by a 3-D Finite Element Modeling (FEM) of the unit cell. Analyses with the unit cells show that while stiffness of the orthotropic sandwich panels is strongly direction dependent, tensile instability of panels is direction independent and governed by the faceplates. The analytical curves are then implemented to ABAQUS UGENS subroutine to describe the non-linear mechanical shell section behavior in the framework of Equivalent Single Layer (ESL) theory. The subroutine was used to simulate the response of idealized accommodation deck, with and without of a cut-out, of a passenger ship. Good agreement in response was found between 3-D FEM and ESL. & 2016 Elsevier Ltd. All rights reserved.

Keywords: Tensile instability Sandwich panel Equivalent single layer theory Load–displacement

1. Introduction Sandwich panels are three-layered structures that consist of two face sheets separated by the core. Incentive to develop light and strong constructions have placed sandwich panels in an excellent position compared with stiffened or isotropic plates since they can offer significant weight savings at equal strength [1]. This has sparked an interest in marine, civil, automotive and aviation industries to exploit the benefits of sandwich panels in design of cost and weight efficient structures; see [2–9] and Noor et al. [10] for generic overview of sandwich panels. Steel sandwich panels have been also proposed for ship hulls to be used in decks and double-hulls as well as in side structures for impact mitigation [4,11–15]. However, FE modeling and analysis of such structures numerically becomes prohibitive considering the level of details (face and core) that must be represented with the sufficiently fine mesh. Fine mesh is needed to reliably simulate different failure mechanisms such as folding and crushing (see for example Paik [16]) and potential ductile failure in these constituents. n

Corresponding author. E-mail address: mihkel.korgesaar@aalto.fi (M. Kõrgesaar).

Therefore, the analysis of sandwich panels during the design process is often performed in terms of linear elastic effective mechanical properties rather than by means of direct computational model [10,17]. Essentially this means that the sandwich panel is replaced with the conventional plate using the effective in-plane, bending and coupling stiffness; in case of significant outof-plane shear deformation also the shear stiffness has central role in response prediction. Libove and Batdorf [18] and Libove and Hubka [19] were the first ones to propose Equivalent Single Layer (ESL) theory for corrugated core sandwich panels. Since then several investigations have been carried out on ESL approach representing steel sandwich panels as reviewed in [3,20]. For linear elastic response important issue is the homogenization-localization scheme that defines the first fiber yield, which is often used as a design criterion in structural analysis [20,21]. This criterion should be used in optimization. Optimization of the plates for lateral loads is performed in [3,6,22] using ESL. However, these investigations lack the plate buckling and ultimate strength criterion. Recent investigations have been developing ESL approach in this direction [23–26]. Nevertheless, as part of the large structures, such as ship hull-girders or steel bridges, the panels can be simultaneously exposed to large in-plane tension loads. Therefore, there is a clear need to investigate how to embed material non-

http://dx.doi.org/10.1016/j.ijmecsci.2016.07.021 0020-7403/& 2016 Elsevier Ltd. All rights reserved.

Please cite this article as: Kõrgesaar M, et al. Non-linear effective properties for web-core steel sandwich panels in tension. Int. J. Mech. Sci. (2016), http://dx.doi.org/10.1016/j.ijmecsci.2016.07.021i

M. Kõrgesaar et al. / International Journal of Mechanical Sciences 115-116 (2016) 428–437

Fig. 1. Fracture initiation strain as a function of stress triaxiality η = σm/σe (mean stress divided by equivalent stress) – plane stress fracture locus for Al2024-T351. Figure from [31].

linearity into ESL approach so that large plastic deformations and consequent plastic instability (including softening and fracture) under tension dominated loading could be simulated without the detailed FE mesh. Present investigation aims to fill this gap, as only then we can develop truly versatile ESL approach capable of handling both, compressive and tensions loads, respectively. Herein, we consider the behavior of panels until plastic instability that is a precursor to ductile fracture. Fracture ductility shows the material ability to go through significant plastic deformation before fracture. For isotropic material ductile fracture is a local phenomenon that depends on the stress state in the material, see Fig. 1. This relation needs to be properly upscaled in order to predict failure in realistic, large-scale structural models, where the shell elements are employed for their computational efficiency over detailed solid mesh. Herein, upscaling is defined as moving from one homogenous continuum scale to another, without consideration of heterogeneity of the microstructure. Different upscaling methods have been presented recently applicable to isotropic plates [27–30]. However, to the best of authors’ knowledge, there is no upscaling method available to predict plastic instability and consequent ductile failure in homogenized orthotropic sandwich panels, where orthotropy is induced by interaction of sandwich constituents while the sandwich panel itself is represented as a single ESL element. Therefore, the aim of the present paper is to analyze the behavior of orthotropic web-core steel sandwich panels under multi-

429

axial tension, and thus extend the versatility of the ESL approach by considering tensile loads. On the one hand, the present investigation contributes to the understanding of how stress state dependent plastic instability upscales from isotropic plate to orthotropic panel level, assuming that orthotropy is caused by the geometry of the panel while materials of the constituents are isotropic. On the other hand, the study provides a first step towards a computationally efficient method for accidental limit state prediction with FE method under multi-axial tensile loads. Therefore, the response of orthotropic web-core steel sandwich panel under tension is analyzed until tensile instability. One of the principal, strongly counterintuitive findings is that despite of the orthotropic nature of the web-core sandwich panel, the onset of tensile instability in the panel is independent of the loading direction while the load–displacement curve depends considerably on the loading direction. Tensile instability is direction independent because failure occurs in the faceplates of the sandwich, thus from the instability point of view orthotropic web core sandwich panel can be considered as isotropic panel. In other words, the direction independence arises due to the discrete and orthotropic nature of web-core sandwich panels, which is why the results cannot be generalized for other type of sandwich structures. Furthermore, for accurate analytical prediction of the point of instability we emphasize the selection of correct instability mode (localized vs. diffuse necking) under different global boundary constraints.

2. Panel geometry and boundary conditions The web-core orthotropic sandwich panel analyzed in this study is shown Fig. 2(A). The periodic nature of the web-core panel makes it possible to focus the analysis on a single repetitive unit cell shown in Fig. 2(B). Geometry of the unit cell is defined by the lengths in two principal directions L1 and L2, panel height of h, face sheets of equal thickness tF and web thickness of tW, whereas in the present investigation we assume that tF otW. The unit cell is deformed under two different type of global boundary constraints identified by the stress state they induce in the unit cell: PST – plane strain tension, and UAT – uniaxial tension; see Fig. 2(C). These explicitly defined constraints are termed global to distinguish them from local constraints (stress states) arising in the constituents: faces and web, of the unit cell. Because of the

Fig. 2. A) Web-core sandwich panel and B) unit cell of the analyzed web-core sandwich panel. C) loading modes considered in this study (PST – plane strain tension; UAT – uniaxial tension).

Please cite this article as: Kõrgesaar M, et al. Non-linear effective properties for web-core steel sandwich panels in tension. Int. J. Mech. Sci. (2016), http://dx.doi.org/10.1016/j.ijmecsci.2016.07.021i

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orthotropic nature of the web-core panel, deformations are applied in both in-plane, principal directions, denoted respectively as 1-direction and 2-direction.

3. Analytical predictions of the stiffness This section provides the analytical estimate for the response of the unit cell. The response is given in terms of generalized force (stress resultants per unit length) and engineering strain. This format allows straightforward implementation to commercial FEM packages through user subroutines, which will be used later to validate the present approach by comparing FE simulation results of ESL model and the results obtained by a full 3-D FEM. Once the response of the sandwich panel is known, we estimate the tensile instability using analytical forming limit diagram. Forming limit diagram provides the strain (or stress) levels of thin sheets in plane stress under multi-axial tension beyond which localized thinning and fracture takes place. Thus, the forming limit strains provide an excellent measure for engineering purposes of the maximum overall strains (or stresses) the sheet can sustain prior to fracture [32].

3.1. Equivalent single layer theory The macro-scale shell behavior follows a first-order shear deformation theory for plates, e.g. [8]; see Fig. 3(A). The assembly is reduced in the average sense to an equivalent single layer. The displacement field associated with the shell continuum is defined:

u=u0 +x3 ϕ1 v=v 0+x3 ϕ2 w =w 0

⎤ ⎡ 0 0 ⎞2 ⎛ ⎢ ∂u + 1 ⎜ ∂w ⎟ ⎥ ⎢ ∂x1 2 ⎝ ∂x1 ⎠ ⎥ ⎡ ∂ϕ1 ⎤ ⎡ ( 1) ⎤ ⎢ ⎥ ⎥ ⎢ ⎡ ε (0) ⎤ 2 κ11 ⎥ 0 0 ⎛ ⎞ ⎢ ∂x1 ⎥ ⎥ ⎢ ∂v 1 ∂w ⎢ ⎢ 11 ⎥ + ⎜ ⎟ ⎥ ⎢ ( 1) ⎥ ⎢ ⎡ ε11⎤ ⎢ (0) ⎥ ⎢ ∂ϕ2 ⎥ ⎢ κ22 ⎥ ⎢ ∂x2 2 ⎝ ∂x2 ⎠ ⎥ ⎢ ε22 ⎥ ⎢ ε22 ⎥ ⎥ ⎢ ⎢ ( 1) ⎥ ⎢ ∂x2 ⎥ ⎢ γ ⎥ ⎢ (0) ⎥ 0 ⎥+x ⎢ ∂w 3 ⎢ 23 ⎥=⎢ γ23 ⎥+x3 ⎢ γ23 ⎥=⎢ + ϕ2 ⎥ ⎥ ⎢ 0 ∂x2 ⎢ ( 1) ⎥ ⎢ ⎢ γ13 ⎥ ⎢ (0) ⎥ ⎥ ⎥ ⎢ ⎢⎣ γ12 ⎥⎦ γ13 0 ⎢ γ13 ⎥ ⎢ 0 ⎥ ⎥ ⎢ ∂w ⎥ ⎢ ⎥ ⎢ (0) + ϕ ⎥ ⎢ 1 ⎢ ∂ϕ1 ∂ϕ2 ⎥ ⎢⎣ γ12 ⎥⎦ ( 1) ∂x1 + ⎣⎢ γ12 ⎦⎥ ⎢ ⎥ ⎢⎣ ∂x2 ∂x1 ⎥⎦ ⎢ ∂u0 ∂v 0 ∂w 0 ∂w 0 ⎥ + + ⎥ ⎢ ⎣ ∂x2 ∂x1 ∂x1 ∂x2 ⎦

where subscript 12 refers to in-plane strains and curvatures and subscripts 13 and 23 for out-of-plane shear strains. The shell constitutive behavior is defined in terms of extensional ([A]), bending ([D]) and extensional-bending coupling ([B], [C]) nonlinear, deformation-dependent stiffness matrices. Coupling between shell section forces (Nαβ, Mαβ) and transverse shear forces (Qα) is neglected here for simplicity as it is assumed that stiffeners are parallel to main axes of the actual structure; in other case the terms 13 and 23 would be non-zero. Thus, in-plane shear (12-plane) and extension (in directions 1 and 2) are uncoupled. As the paper focuses on in-plane deformations, also the membranebending coupling and bending is ignored, i.e. B- and D- matrices are not needed. The assembled shell stiffness matrix in the general case is given as:

⎡ N11 ⎤ ⎡ A11 ⎥ ⎢ ⎢ ⎢ N22 ⎥ ⎢ A21 ⎢ N12 ⎥ ⎢ 0 ⎥=⎢ ⎢ ⎢ M11⎥ ⎢ B11 ⎢ M22 ⎥ ⎢ B21 ⎥ ⎢ ⎢ ⎣ M12 ⎦ ⎣ 0

A12 0 A22 0 0 A33 B12 0 B22 0 0 B33

C11 C21 0 D11 D21 0

(1)

where the superscript 0 refers to the shell mid-plane and ϕ is the total rotation, i.e. sum of slope –dw/dx of the deflection and shear angle γ, ϕ ¼ –dw/dx þ γ. The strain field, following the von Karman assumptions, is divided into extensional (0) and bending (1) components:

(2)

⎡ ε (0) ⎤ 11 ⎥ C12 0 ⎤ ⎢ ⎥ ⎢ ε (0) ⎥ C22 0 ⎥ ⎢ 22 ⎥ (0) 0 C33 ⎥ ⎢ γ12 ⎥ ⎥ ⎥⎢ D12 0 ⎥ ⎢ κ (1) ⎥ 11 D22 0 ⎥ ⎢ (1) ⎥ ⎥ ⎢ κ22 ⎥ 0 D33⎦ ⎢ (1) ⎥ ⎣ γ12 ⎦

⎡ Q 2 ⎤ ⎡ KA 44 0 ⎤ ⎡ γ23⎤ ⎢ ⎥=⎢ ⎥⎢ ⎥ ⎣ Q1 ⎦ ⎣ 0 KA55⎦ ⎣ γ13 ⎦

(3)

(4)

The transition between scales relies on defining the macroscale shell stiffness matrix. This is done in unit cell level as

Fig. 3. (A) Shell kinematics. (B) section forces and section moments acting on A plate section cut.

Please cite this article as: Kõrgesaar M, et al. Non-linear effective properties for web-core steel sandwich panels in tension. Int. J. Mech. Sci. (2016), http://dx.doi.org/10.1016/j.ijmecsci.2016.07.021i

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determined from the known form of equivalent von Mises stress and stress ratio β

L2

3 2

σ ̅ =σf =

1

⇒σ1= L1

Fig. 4. Axes of reference for the unit cell and parameter definitions.

described in Fig. 3. The stiffness parameters are obtained simply by integration of the stresses over panel thickness, i.e. using the relation

(9)

3.2.1. Load–displacement curves Load–displacement curves (generalized force vs engineering strains curves) are determined for both principal directions 1 and 2. Generalized normal force in direction 1 is defined using rule of mixtures

h /2

∫−h/2 [ σαβ ] dz,⎡⎣ Mαβ ⎤⎦= ∫−h/2 [ σαβ ] zdz, α, β =x, y

(5)

together with Eq. (3) and solving for stiffnesses, A, B, C and D. It should be noted here that the stress is a non-linear function of strain, thus the non-linear material behavior is needed before integration is carried out. 3.2. Stress determination

ΔNii̅ =Δ

σ 2 2α + 1 . = σ1 α +2

(6)

Note that this relation is valid for plastic strains only. Supplementing this with the constitutive law describing material hardn ening (we use power hardening rule σf =K ( ε ̅ p + ε0 ) ; where K is the strength coefficient, n is the hardening exponent, strain 1/ n allows the power law expression to intersect at ε0=( σ0/K ) p (ε ̅ =0,σ0 ), σf is the flow stress, ε ̅ p is the equivalent plastic strain, and σ0 is the yield stress) and assuming proportional loading, stress state in the material can be fully determined from the strain state. Hence, the approach presented is strain driven and generalized forces in the sandwich panel are evaluated based on the known value of applied displacement. Total strain ε and plastic strain ε p are given as

ε=ln ( 1 + εeng ), ε p=ε−σ 0/E

F faces+Fiiweb ∑ Fii =Δ ii Lj Lj

(10)

where ∑ Fii is the total normal force in direction i and Lj is the length in other principal direction, see the illustration in Fig. 4. In a displacement controlled approach forces for faces and web can be determined by converting principal stress to nominal stress and knowing the cross-sectional area of the constituents

Fii =

Since the faces and web are thin, through-thickness stress is assumed negligible and plane stress becomes a valid assumption. Furthermore, we assume that material follows standard J2 flow theory. Therefore, there is one-to-one mapping from the strain to the stress space; the principal strain rate ratio α = dε2/dε1 is related to principal stress ratio β = σ 2/σ1 through the expression (note that from this point onwards the symbols α and β take a different meaning than in previous section)

β=

1−β + β2

These principal stresses will be employed in the following section to determine the generalized normal forces in respective directions.

F11

h /2

1⎡ ( σ1 − σ2 )2 + ( σ2 − σ 3 )2 + ( σ1 − σ 3 )2⎤⎦ =σ1 1−β + β2 2⎣ σf

σ 2=σ1β

u1

⎡⎣ Nαβ ⎤⎦=

σi Ai 1 + εi, eng

(11)

where Ai is the cross-sectional area with normal in i-direction.

4. Tensile instability in orthotropic sandwich panel Under multi-axial tension condition, the constituents of the sandwich panel are assumed to follow different strain paths. In the principal strain space, strain and stress states associated with strain rate and stress ratio, respectively α and β, are illustrated in Fig. 5. Tensile instability is defined as the necking instability in one of the constituents (web or face) of the orthotropic sandwich panel. We distinguish between two types of necking instabilities: diffuse and localized, both of which are shown to prevail in webcore sandwich panels. Diffuse necking usually appears under uniaxial tension strain path over the width of the gauge or plate section. Diffuse necking is followed by localized necking, or severe thinning of the sheet metal, which is thus considered as a Uniaxial tension (UAT)

α = -1/2, β = 0

Plane strain tension (PST) α = 0, β = 0.5

0.5 Equi-biaxial tension (EBT) α = 1, β = 1

0.4

(7)

where E ¼206 GPa is the elastic modulus and εeng is the engineering strain. For an incompressible solid, the principal components of strain increments are related as dε1:dε2 :dε3=1 : α :−( 1 + α ), thus the equivalent plastic strain can be expressed as a function of strain rate ratio [33], from which the first principal plastic strain can be found: 2

2 2 ε ̅ p= ( dε12 + dε22 + dε32 = 3 3

)

1+α + α 2 ε1⇒ε1=

431

3

Swift 0.2

0.1

1+α + α 2 ε̅ p

BWH

0.3

(8)

The equivalent stress can be determined from the powerhardening rule presented above. Principal stresses are then

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Fig. 5. Localized necking according to BWH and diffuse necking according to swift.

Please cite this article as: Kõrgesaar M, et al. Non-linear effective properties for web-core steel sandwich panels in tension. Int. J. Mech. Sci. (2016), http://dx.doi.org/10.1016/j.ijmecsci.2016.07.021i

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precursor to ductile fracture. Therefore, in terms of ductile fracture localized necking criterion provides a better estimate than diffuse necking criterion. Consideration of diffuse necking (associated with geometric changes) is however relevant in present application, in which the obtained load–displacement curves describe the behavior of unit cell under assumed boundary conditions. These boundary constraints are violated when the geometry of the unit cell changes due to the diffuse necking. Therefore, diffuse necking condition provides, although conservatively, the first estimate for the tensile failure. Diffuse necking is described with the Swift [34] instability criterion formulated in the principal strain space with Eq. (12), for details see [33]. To describe localized necking we employ the stress based BWH (Bressan-Williams-Hill) instability criterion given with Eq. (13), see [35]. Both criteria are plotted in the principal strain space in Fig. 5.

⎛ 2n (2 − β )(1 − β+β2) ⎞ ⎜ ⎟ 4 − 3β − 3β 2 + 4β 3 ε1 ⎜ ⎟ = ε2 ⎜ 2n (1 − 2β )(1 − β+β2) ⎟ ⎜− ⎟ 4 − 3β − 3β 2 + 4β 3 ⎠ ⎝

calculated and presented in Fig. 7(A). The point of instability in the panel members is evaluated analytically based on the stress state in the faces and web. Specifically, we use the weakest link approach meaning that panel is assumed to become unstable once its weakest member, face or web, has become unstable. For plane strain case (α ¼0) both instability criteria give the same critical value as shown in Fig. 6. According to BWH criterion equivalent plastic strain to instability is given as εPST ̅ =2/ 3 n − ε0=0.258 (for simpler power-law material expression would appear without ε0 ). This point is plotted in Fig. 7 (A) in terms of engineering strain. 4.2. Uniaxial tension Global constraints for the unit cell are now imposed so that the faces will be under uniaxial tension; at least this is the case when tension is applied in direction 1 (Fig. 6(B)). Because of the same reasoning as in PST1 loading case, the web is under uniaxial tension. When applying tension in direction 2, the web will constrain the movement of faces in direction 1, which leads to a stress state between uniaxial (η E 0.33) and plane strain (η E 0.57) tension. Simulations show that the stress state in the faces is close to η E 0.4. Generalized force – engineering strain curves are calculated based on this stress state and presented in Fig. 9(A). Under uniaxial tension path the faces are prone to diffuse necking, thus Swift diffuse necking criterion, Eq. (12), is used to predict the onset of instability for both loading directions. In direction 1 under uniaxial tension (η ¼1/3) εUAT ̅ =n and in direction 2 where stress state is more constrained (η E0.4) εUAT ̅ =0.225. The results are again converted to engineering strain format and presented in Fig. 9.

( )

(12)

⎧ 2K 1 + 0. 5α ⎛ 2 n ⎞n ⎪ 1+α + α 2 ⎟ if −1<α≤0 ⎜ ⎠ ⎪ 3 1+α + α 2 ⎝ 3 1+α ⎪ n 2 ⎨ σ1= n ⎪ 2K 3 , ifα>0 ⎪ 2 ⎪ 3 1− α 2+α ⎩

( )

( )

(13)

As we can link stress state with α and β, they can be determined by knowing the prevalent stress state in the constituents under global boundary constraints. Based on the engineering judgment inferred stress states are presented in Fig. 6 for plane strain and uniaxial tension loading. How each of the stress states arises in Fig. 6 requires an explanation.

5. Validation in the unit cell level The presented analytical equations are used to find the stiffness and tensile instability of the unit cell in Fig. 1(B). Unit cell dimensions are defined as follows: length L1 ¼360 mm, width L2 ¼120 mm, and height of h¼45 mm (including faces). Face (tF) and web thickness (tW) are varied to check the effect on the response and thus, test the validity of the approach. Material behavior is n characterized with the power hardening rule σf =K ( ε ̅ p + ε0 ) ; with typical values for steel of K ¼740 MPa, n ¼ 0.24, and ε0 ¼0.019. The unit cell is constrained as shown in Fig. 1(C). For each global constraint and loading type a corresponding stress state in the constituent was defined in Fig. 6, thus both α and β are known. Generalized force in the unit cell in the specific direction can be

4.1. Plane strain First, we note that no constraint is applied in direction 3 meaning that the unit cell is free to deform in direction 3 due to the Poisson effect. Consequently, in PST1 in Fig. 6(A) web is under uniaxial tension while the faces are under plane strain because of the applied global constraints. Both, the web and faces deform the same amount in the loading direction. In PST2 in Fig. 6(A), the web is under slight compression, but in contrast to faces, does not contribute to the stiffness in direction 2. Based on these assumed stress states the generalized force engineering strain curves are PST 1

A)

PST 2

B)

UAT 1

UAT 2

2

1

Faces: PST (α

= 0)

Web: UAT (α

Faces: PST (α

= −0.5)

= 0)

Web: Compression

Faces: UAT (α

= −0.5)

Web: UAT (α

Web: Compression

= −0.5) Face: Between UAT and PST, (α = −0.396)

Fig. 6. Stress states in the web and faces. unit cell constrained under (A) PST and (B) UAT.

Please cite this article as: Kõrgesaar M, et al. Non-linear effective properties for web-core steel sandwich panels in tension. Int. J. Mech. Sci. (2016), http://dx.doi.org/10.1016/j.ijmecsci.2016.07.021i

M. Kõrgesaar et al. / International Journal of Mechanical Sciences 115-116 (2016) 428–437

433

Fig. 7. A)Generalized force – engineering strain curves for PST. B) point of instability in FE simulation. Color contours show the equivalent plastic strain. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

3500 total

3000 face

2500

F/L [kN/m]

defined based on these ratios as described in Section 3. The unit cell under prescribed boundary conditions (Fig. 1) was numerically analyzed with the commercial code ABAQUS/Explicit v6.13-2. All the simulations were displacement controlled and all the models were discretized with plane stress reduced integrated shell elements (S4R). Element length, Le, in the simulations was varied between 3 mm and 10 mm to check the convergence of results. Numerical results are presented together with analytical estimates in the next Section. No failure criterion was used in the simulations.

2000

Face nodes 2.5

1500

Web nodes

Analytical

4 2.5

1000

web

5.1. Case study: Faces tF ¼2.5 mm, web tW ¼4 mm The present approach is validated in the unit cell level by the excellent agreement between numerical simulation results and analytical estimates, see Fig. 7 and Fig. 9. As shown in Fig. 7 (A) under PST constraint localized necking criterion conservatively predicts the point of instability when compared with simulation results. Beyond instability, all the deformations localize into a narrow band (single row) of elements and force drops down. As expected and illustrated in Fig. 6(A), this happens first in the faceplates. Although FE simulations indicate some mesh size sensitivity, the effect is considered insignificant as the localization in simulations still occurs beyond the analytical estimate. Therefore, we consider that the converged solution of numerical simulations has been achieved. In Fig. 8, the response of the constituents is shown separately for web and faces. This plot is made only for direction 1, since in direction 2 only faces contribute to the response meaning that total response in direction 2 in Fig. 7 is due to face plates. The illustrative cross-section in Fig. 8 shows that nodes in the web-face intersection are shared due to the discretization. In extracting the force from simulation these nodes were considered as face nodes, while such split was not necessary in analytical calculations. Consequently, the face stiffness corresponding to FE results is slightly higher and stiffness of web is slightly lower than analytical estimate. Nevertheless, sum of curves produces the exact match between analytical and FE results as already shown in Fig. 7. In case of UAT, the negative strain rate ratio of α ¼  0.5 suggests that the second principal strain becomes active due to the Poisson effect. This renders diffuse necking a feasible deformation mode for the faceplates. Since diffuse necking takes place considerably earlier than localized necking under uniaxial tension, i.e. according to Swift εdiffuse =n versus according to BWH εlocalized =2n, ̅ ̅

500 0

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Engineering strain [-] Fig. 8. Evolution of forces in the faces and web: Comparison of the analytical and FE solution under PST loading in direction 1.

diffuse necking becomes more critical for this constraint scenario. The difference in diffuse necking strains in Fig. 9(A) is due to the small difference in the stress state in the faceplates as discussed in the previous section. Although diffuse necking condition provides a conservative estimate in terms of localization (see Fig. 9(A)), it provides an appropriate measure for tensile failure for engineering purposes in present application. It is noteworthy that stability of the panel is again defined by the faceplates, similarly as under PST loading. Furthermore, the engineering strain to tensile instability for both boundary conditions, PST and UAT, can be approximated to be equal for engineering purposes. Thereof, we conservatively advocate the use of Swift diffuse necking condition εdiffuse =n to ̅ determine instability in web-core steel sandwich panels. Similarly to PST case, the response of the constituents is shown separately for web and faces in Fig. 10. The differences between stiffnesses in the constituents are again due to the discretization as discussed already above. It is intriguing that despite the orthotropic nature of the webcore sandwich panel, it can be considered isotropic from the instability and thus, from the failure point of view. This neutrality to loading direction stems from the discrete nature of the web-core sandwich panel, which allows separate analysis of all the constituents. In combination with the weakest link analysis approach, the panel failure is governed by the failure of its weakest member – the faceplates. Essentially this means that the failure in web-core

Please cite this article as: Kõrgesaar M, et al. Non-linear effective properties for web-core steel sandwich panels in tension. Int. J. Mech. Sci. (2016), http://dx.doi.org/10.1016/j.ijmecsci.2016.07.021i

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Fig. 9. A) Generalized force – engineering strain curves for UAT. B) FE simulation results at specified steps. Color contours show the equivalent plastic strain. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.).

3000 total

2500 face

F/L [kN/m]

2000 Face nodes

1500

Periodicity with adjacent unit cells or supporting structure is lost

2.5

Web nodes

Analytical

4

1000

2.5

Diffuse type of necking web Fig. 11. Schematic illustration of diffuse necking taking place in simulations. Deformed shape of the unit cell would not conform with adjacent structure.

500 0

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Engineering strain [-] Fig. 10. Evolution of forces in the faces and web: comparison of the analytical and FE solution under UAT loading in direction 1.

sandwich panels can be assessed based on the analysis of faceplates only, according to same principles applicable to isotropic plates, e.g. [28]. There are two main reasons why faceplates fail first under different boundary constraints: 1) When panel is constrained under plane strain the stress state in the faces is plane strain, but the web is under uniaxial tension. Moreover, the web is supported by the faceplates, which means that diffuse neck cannot develop and local necking becomes the design criterion for webs. However, local neck in plane strain (faces) develops much earlier than in uniaxial tension (web), see Fig. 5. 2) Under uniaxial tension all the constituents of the panel are under uniaxial tension. Diffuse neck develops in the faces while the localized neck develops in the web – again because of the supporting faceplates. Fig. 5 indicates that diffuse necking takes place earlier than localized necking. Hence, the faceplates become the critical constituents. One could argue that diffuse necking cannot take place at all in the faceplates because of the supporting structure, see the

illustration in Fig. 11. In other words, similarly as the faceplates support the web, the adjacent unit cells or primary structural members, e.g. girders in the actual structure, support the faceplates. This means that stress state in the faceplates would be that of plane strain and findings corresponding to plane strain state are applicable. However, for purpose of completeness and insight we believe that findings corresponding to uniaxial tension case are equally valuable. 5.2. Thickness effect: Thick face – thin web and thin face – thick web To check the effect of thickness variations on the analytical estimates, and thus further validate the approach, two assumptions are made regarding the thicknesses of the constituents. In the first case, faces are much thicker (tF ¼5 mm) than the web (tw ¼1 mm) and in the second case, faces are much thinner (tF ¼1 mm) than the web (tw ¼5 mm). Analyses are performed only under PST constraint. As the basic assumptions regarding stress states remain valid, the analytical approach still yields accurate stiffness for these extreme cases as shown in Fig. 12.

6. Application in large structures In order to demonstrate the applicability of the present approach, two case studies with an idealized accommodation deck of a passenger ship are considered. A deck with dimensions of 18  9 m is deformed under static tensile loading. In the first case we consider the whole deck and in the second case a deck with a

Please cite this article as: Kõrgesaar M, et al. Non-linear effective properties for web-core steel sandwich panels in tension. Int. J. Mech. Sci. (2016), http://dx.doi.org/10.1016/j.ijmecsci.2016.07.021i

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B) 6000

A) 2000

5000

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1600 F/L [kN/m]

435

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800

4000 3000 2000

Direction 1 Direction 2 Analytical

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Engineering strain [-]

0

0.05

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0.15 0.2 0.25 Engineering strain [-]

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Fig. 12. Effect of constituent thickness on the response under PST loading. A) thin face – thick web; B) thick face – thin web.

3.12  3 m cut-out in the middle of the panel. We emphasize that model with cut-out is studied to show the capabilities as well as limitations of the present approach. Two alternative simulations approaches are considered: ESL approach and full three-dimensional shell model of the structure. The main sandwich dimensions and material parameters remain the same as used in previous Section. The thickness of face and web correspond to case study 1, tF ¼2.5 mm and tw ¼4 mm. The equivalent single layer theory in Section 3.1 was implemented to ABAQUS/Standard v6.13-2 using a UGENS subroutine. The generalized force vs. engineering strain PST relations given in Fig. 7 (A) are used to determine the components A11 and A22 of the extensional shell stiffness matrix. The other extensional, bending and extensional-bending stiffnesses (Eq. (3)) are adjusted in the non-linear range based on the ratio between non-linear and linear elastic A11 and A22. For example, if the unit cell yields in the Direction 1, the non-linear stiffness A12 becomes

A12, nl =A12, lin

A11, nl A11, lin

(14)

As these curves were defined only for the plastic response, the elastic portion is modeled using the rule of mixtures, e.g. see [20], and by considering the strain state and resulting stress state up to yield point. Both FE models with respective mesh and boundary conditions for the structure with cut-out are shown in Fig. 13. It is clear that the assumption of plane strain that was made in deriving the analytical equations is not satisfied along the hole edges. The effect

of this will be discussed later. The modeling approach for the deck structure without the cut-out is the same in both cases, thus the models are not shown. The structure without the cut-out was analyzed first, whereas the ESL model was discretized with three different mesh densities – fine (0.25  0.25 m), medium (0.5  0.5 m) and coarse (1  1 m). The ESL model of structure with cut-out was analyzed using the medium mesh density of 0.5  0.5 m. Full 3D FEM models were discretized with S4R elements with dimensions shown in Fig. 13(B). Shell element reference surface in face plates was offset so that the total height of the panel would be 45 mm, see Fig. 13(B). All translational degrees of freedom were fixed in one end of the structure and constant displacement was applied in direction 1 to the nodes on the other end of the structure. A condition of zero normal and out-of-plane displacement was imposed along the longer edges of the panel. Essentially the structure is assumed to be supported by the rigid members, i.e. longitudinal girders, along these longer edges. 6.1. Numerical simulations and results Numerical geometric non-linear simulations using the Riks method were carried out with ABAQUS/Standard v6.13-2. In both simulations, constant displacement with increasing magnitude was applied on the plate edge. In the whole plate example, the maximum displacement is 3 m, while reduced to 0.5 m in the plate with cut-out. In the latter case, strains for 0.5 m end-displacement are already very large in elements near the corners. No failure criterion was used in the simulations.

Fig. 13. Idealized web-core accommodation deck model with cut-out, 18  9 m. A) ESL representation. B) Full three-dimensional representation.

Please cite this article as: Kõrgesaar M, et al. Non-linear effective properties for web-core steel sandwich panels in tension. Int. J. Mech. Sci. (2016), http://dx.doi.org/10.1016/j.ijmecsci.2016.07.021i

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30

25 ESL

Force [MN]

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Fig. 14. Simulated force-displacement curves for idealized 18  9 m sandwich structure with and without cut-out.

The force-displacement curves for both case studies are shown in Fig. 14. The agreement between ESL and full three-dimensional response is very good, validating the approach also for larger structures. For the plate without cut-out the agreement is excellent throughout the analysis. When the cut-out is introduced the ESL model becomes slightly stiffer, producing a maximum external force difference of approximately 5% for same displacement. The difference is mainly explained by the violation of the plane-strain assumption near the cut-out edges. As they are allowed to move in the direction perpendicular to loading, planestrain assumption becomes overly restrictive. More accurate determination of the other ABD matrix components could also improve the method accuracy, which is left for future work. The power of the ESL method is further illustrated by comparing section force distributions with the equivalent 3D model. Fig. 15 shows the membrane force distribution in the Direction 1 for an axial displacement of 0.4 m according to both methods. The top face plate in the 3D model is expected to carry 39% of the total membrane force, and therefore the scaling in legend is different. The figure shows that the ESL is capable of describing the force distribution near the hole very consistently against the 3D despite the simplifications and coarse mesh.

7. Concluding remarks In this study, we determined the load–displacement curves, i.e. generalized force vs engineering strain, for web-core steel sandwich panels under tension loading and assessed the instability of those panels. Validity of the approach is established by comparing

analytically obtained curves with those obtained by a full threedimensional meshing of the web-core sandwich panel. The analytical curves were implemented to ABAQUS UGENS subroutine to describe the non-linear mechanical shell section behavior. The subroutine was used to simulate the response of idealized accommodation deck of a passenger ship. The paper shows that the rule of mixtures can be applied to predict analytically the stress resultant strain relation of web-core panels under multi-axial tensile loading until tensile instability, which is a precursor to ductile fracture. Thus, the paper extends the work of Libove and Hubka [19] and Romanoff and Varsta [20] to non-linear regime. It also complements the work by Reinaldo Goncalves et al. [26] who investigated non-linear behavior of steel sandwich panels in compression. The combination of compressive and tensile non-linear behavior produces so-called load-endshortening curves that are widely used in analysis of large structures; see for example Refs. [36–39]. On contrary to often-assumed uniaxial stress state on these curves, the present paper considered multi-axial tensile stresses. It was observed that the tensile instability in web-core sandwich panel under uniaxial or plane strain tension occurs in the faceplates. The point of tensile instability is given in terms of equivalent plastic strain and engineering strain. For engineering purposes, this point can be approximated to be independent of loading direction and boundary conditions. Therefore, despite the orthotropic nature of the webcore sandwich plate, the onset of tensile instability in the sandwich plate is independent of the loading direction. This knowledge can simplify practical FE simulations involving large displacements where orthotropic web-core sandwich panels are represented using ESL approach. One such application case was presented, though without considering failure. Furthermore, this work provides the first step in upscaling the ductile fracture criteria for isotropic plates used in large-scale structural analysis [27,28,40] towards sandwich plates with periodic microstructure. In a future work, the aim is to develop a truly versatile ESL approach that can handle both, tensile as well as compressive loads [23–26] without changing the meshing of large and complex structures. Such an approach would allow fast modification of material and geometrical parameters, and thus would open a way for tailoring structural topologies through optimization, e.g. see [6]. Furthermore, the presented approach should be extended to account for bending and asymmetry. Coupling could be basically done as given in Romanoff and Varsta [20] for stress and strain, but extension is definitely needed. Effect of bending coupled with ductile failure analysis were investigated by Woelke and Abboud [27] and Voyiadjis and Woelke [41] for isotropic plates. However, for isotropic plates the definition of localization is more straightforward than for web-core sandwich panels. In web-core panels, this depends on the ratio of out-of-plane-shear-induced secondary stresses to bending and membrane induced primary stresses. This causes some challenges and is the reason why we left this for future work.

Fig. 15. Comparison between response between A) full 3D and B) ESL analysis. Color contours show the x-directional membrane force. Deformations are scaled up by factor of 3. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.).

Please cite this article as: Kõrgesaar M, et al. Non-linear effective properties for web-core steel sandwich panels in tension. Int. J. Mech. Sci. (2016), http://dx.doi.org/10.1016/j.ijmecsci.2016.07.021i

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Please cite this article as: Kõrgesaar M, et al. Non-linear effective properties for web-core steel sandwich panels in tension. Int. J. Mech. Sci. (2016), http://dx.doi.org/10.1016/j.ijmecsci.2016.07.021i