Shear response of ultra-lightweight CFRP cores

Composite Structures 238 (2020) 111879

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Shear response of ultra-lightweight CFRP cores a,b

Pablo Vitale

a

b

, Gaston Francucci , Helmut Rapp , Ariel Stocchi

a,⁎

T

a

Research Institute for Materials Science and Technology (INTEMA), Universidad Nacional de Mar del Plata, CONICET, Av. Juan B. Justo 4302, B7608FDQ Mar del Plata, Argentina Institute of Lightweight Structures, Universität der Bundeswehr München, Werner-Heisenberg-Weg 39, 85577 Neubiberg, Germany

b

ARTICLE INFO

ABSTRACT

Keywords: Sandwich panels Honeycomb cores Carbon fiber composites Lightweight structures

In this work the shear performance of ultra-lightweight (ULW) carbon fiber reinforced polymer (CFRP) cores with 3-dimensional (3D) geometries is analyzed. The cores are made from a machined CFRP laminate. Using the interlocking method, CFRP sheets are assembled in order to obtain a square pattern. The density of all manufactured cores is achieved below than 48 kgm−3. Through simple analytical mechanical models, it is possible to predict the failure behavior of the panels. Finite element analyses (FEA) are carried in order to contrast and validate the theoretical models. Good agreement among theoretical, finite element and experimental results is found. The experimental results show that the controlling failure mechanisms are shear failure of the base material or debonding, also predicted successfully by the numerical and analytical approaches. Finally, the cores are compared favorably as well with other competing known materials.

1. Introduction Sandwich structures have been implemented as basic materials for weight sensitive structures, as beams or panels, mainly due to its great versatility and mechanical efficiency [1–3]. The main reason is the relative low-cost/high-performance during the service life of the part, when implemented, for example, in the aircraft or automotive companies [4,5]. Both industries demand parts with high stiffness and strength to-weight ratios in a way to use the minimum raw materials as possible, as a manner of decreasing weight and enhancing loading capabilities or fuel savings, while still reaching the loading requirements. However, it is remarkable that the core must have sufficient mechanical properties (that may include, out-of-plane and/or in-plane properties) to enable high cost benefits [6]. Thus, there have been substantial efforts to develop and study low density cores in a way to undergo high mechanical solicitations and other properties like blast resistance, impact behavior or thermal response. Recent proposal for lightweight core constructions includes traditional foams (metallic [7–9] or polymeric [10–12]), 2D honeycomb [13–15], 3D honeycombs [16,17], 3D lattices [18–20], prismatic cores [21,22] or hybrids [23–25] as well. Sandwich panels are frequently subjected to shear loading and the shear stresses are almost completely resisted by the core. Many authors worked on paths to better understand the core response during this

loading mode. Some of them studied the response of cores made from metallic parent material. Coté et al. [26] studied the effects of different parameters such as relative density , shear stresses and unit cell heightto-cell size ratio (h/L) of lab-made 304 stainless steel square 2D honeycomb cores, indicating the higher specific shear performance of the stainless steel when comparing with aluminum honeycomb. Furthermore, they determined that the specific mechanical properties of square honeycombs are relatively unaffected to the cell aspect ratio for aspect ratios in the range 0.5 < h/L < 4. On the other hand, sandwich panels made from CFRP have attracted the attention of designers and researchers, because of its high specific performance in contrast to metallic ones. For instance, having relative densities (with = c / s ; where c is the density of the core and s is the density of the base material, exhibited as a general expression) in the range of 0.021–0.1, square honeycomb cores with CFRP as pattern material have 10 times the observed core peak strength cpk than those made from 304 stainless steel [15]. George [27] studied the mechanical behavior of CFRP sandwich panels with pyramidal truss cores. The research brought insights about the capabilities of lattices structures when they are obtained from CFRP laminates. When compared with known CFRP honeycombs or Titanium lattices on shear loading cases, the pyramidal CFRP structures demonstrated to have better shear performance (strength and modulus) in the range 0.01 ≤ ≤ 0.1. In a similar manner, Dong [28] manufactured sandwich panels with lattice octet-

Corresponding author. E-mail addresses: [email protected] (P. Vitale), [email protected] (G. Francucci), [email protected] (H. Rapp), [email protected] (A. Stocchi). ⁎

https://doi.org/10.1016/j.compstruct.2020.111879 Received 17 October 2019; Received in revised form 20 December 2019; Accepted 3 January 2020 Available online 07 January 2020 0263-8223/ © 2020 Elsevier Ltd. All rights reserved.

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truss cores from CFRP laminates previously cut by a water-jet cutting machine. The snap-fit/adhesive bonding technique was employed to successfully assemble the cores, leading to relative core densities in the range 0.017 ≤ ≤ 0.159 and peak shear strengths in the range of 0.38–5.03 MPa, respectively. The research identified three possible shear failure mechanisms: elastic buckling, delamination and plastic micro-buckling, governed mostly by delamination mode. When employing no fit-in nodes, the probability of failure due to skin-core debonding is high and this failure mode must be taken into account. Vitale [13] studied also debonding of sandwich materials made from natural fibers honeycomb cores. It is well known that the glued region between the skins and the core members must be large enough to resist shear stresses within the discontinuity. Xiong et al. [29] proposed two alternative paths for studying bonding failure. On the first place, considering debonding phenomena within a unit-cell, when shear stresses in the bonded area exceed the bond strength. On the other hand, since the glued area in a unit-cell is not easily measurable, because during the manufacturing process may it change due to local deformations, the analysis includes the total region with the number of involved unitcells, giving as a result an average value. Hence, the interest in ultra-lightweight materials (ULW) for improving structural efficiency has grown over the last years [30]. The ULW are defined as those materials that meet the core density condition below 48 kgm−3 [31]. The aim of this project is to develop and study the shear behavior of three different core geometries for ultra-lightweight applications, proposed as an alternative for traditional honeycomb cores [16]. The samples are obtained from a CFRP laminate by means of the vacuum infusion (VI), water-jet cutting (WJC) and slotting/interlocking method, resulting in square-honeycomb cores with large internal holes. In this way, the cores may be treated as hybrid materials, not only because they are made from raw material and empty spaces [32], but they resemble both honeycomb cores and lattice materials. The obtained cores reached the ultra-low weight statement, with potential multifunctional applications [30], given also by the openedcell characteristics (e.g. heat transfer, foam filling capabilities and its benefits such as isolation or impact response, damping properties, shape morphing, among others [33–36]). The in-plane shear response of the cores is studied and the predicted failure modes are compared with the obtained analytical and numerical results. At the end of this research, the final peak shear properties are compared along with similar known cores found on the literature.

Chemicals Inc.) was employed as hardener. It is interesting to remark that the susceptibility to delamination caused by machining is reduced when woven laminates are employed, instead of unidirectional fibers [37]. The final base material elastic properties based on classical laminate theory [38] are shown in Table 2. 2.2. Fabrication method The steps for manufacturing the cores can be described as a combination of several known processing stages, such as the vacuum infusion technique (Fig. 1a) with central-radial infusion, the water-jet cutting phase (WJC) (Fig. 1b) and the slotting/interlocking method (Fig. 1c). For further details about the manufacturing process please see [16]. Square honeycomb cores were manufactured from previous machined CFRP composite laminates of a mean thickness of t = 0.65 ± 0.05 mm and an average density of about 1300–1350 kgm−3, with an average fiber volume content of 50.8%. The CFRP panel was cut into rectangular sheets with the desired geometries patterns employing the WJC method (Fig. 1a). The machined slots had a final width of 0.65 mm. The clearance between sheet thickness and slot width was less than ± 0.05 mm, which gives a suitable snap-fit while implementing the slotting/interlocking method (Fig. 1b). A machined plate was used to assure the correct position of the CFRP sheets (assembly-in-position) [29] and the cross-slots were bonded together using the same base resin employed, satisfying the desired square pattern (Fig. 1c). Rectangular samples were cut from the cores to obtain shear test samples with a size of 200 mm in length, 100 mm in width and 25.4 mm in height (Fig. 1d). The honeycomb core cells are defined by the width “W”, the length “L”, the height “H” and the wall thickness “t” (Fig. 2a). In this work, the unit-cells dimensions are kept constant with a size of 20 by 20 mm for the cores made with machined composite sheets and 50 by 50 mm for the cores made with full walled cells that means both sizes “W” and “L” are the same for this particular work. The main reason for this is to fulfill the objective function as the low density, followed by a symmetric number of members sheets along the core in both x-y directions. The relative density of the cores was previously calculated by the area occupied by material within a CFRP sheet and multiplying it by its average thickness. Then this is related and scaled to an elementary unitcell volume. The resulting of each core is then less than 0.035, which means a core density ρc of less than 48 kgm−3, verified by lab measurements. Catenary-like shapes have been widely used in civil engineering or architecture, when high compressive strength is required to fulfill the mechanical demands [16]. The core cell patterns analyzed in this work are sketched on Fig. 2 with the dimensions specified on Table 3. To obtain stabilized samples, a 2-component epoxy adhesive (UHU plus Endfest 300®) was used for bonding CFRP skins to the cores. Photographs of the shear samples obtained are provided in Fig. 3.

2. Materials and methods 2.1. Base material The preform consisted in four layers of plain woven 0/90° Torayca T300-3k carbon fiber fabric. Table 1 shows the properties of this reinforcement and also of the epoxy resin used as the polymeric matrix. The polymeric matrix was a DGEBA (DER 383, Dow Chemicals). Glycidyl aliphatic ether (Novarchem S.A.) was used as an epoxy reactive diluent, in order to decrease the resin viscosity and facilitate the vacuum infusion process. A cycloaliphatic amine (Air Products and Table 1 Carbon fiber fabric and epoxy/amine resin main properties. T300 – Torayca Density (gcm−3)

Filament diameter (μm)

Tow size number

1.75 7 3k Epoxy (DER383) - glycidyl aliphatic ether - cycloaliphatic amine Epoxide equivalent (geq−1) Epoxide percentage Density mixture at (%) 298 K (gcm−3) 176–183 23.5–24.4 1.14

Textile weight (gm−2)

Tensile strength (GPa)

Tensile modulus (GPa)

Fracture strain (%)

198

3.53

230

1.5

Viscosity mixture at 298 K (Pa s) 0.57–0.6

Tensile strength (cured) (MPa) 65–80

Tensile modulus (cured) (GPa) 3–3.5

Fracture strain (cured) (%) 2–2.5

2

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Table 2 Theoretical elastic properties of the composite material made from plain T300/epoxy. Composite elastic modulus E0°s = E90°s (GPa)

Composite shear modulus G90°/0°s (GPa)

Composite Poisson’s Modulus ν90°/0°s = ν0°/90°s (–)

Fiber volume fraction vf (%)

Composite tensile strength σstf (MPa)

Composite shear strength τs (MPa)

60.5

3.45

0.0366

50.8

1360

110

Note: suffix s represents the base material.

Fig. 1. The manufacturing route for obtaining square-honeycomb cores.

3. Shear response

analyzing the displacements due to the shear load. In all cases, the cell walls were modeled as thin orthotropic plates clamped at the bottom and the top, neglecting the transverse cell walls (α = π). Fig. 5 shows the approximation model used for calculations. Due to the symmetry of the model, it is possible to work only with the half of the plotted sheet material in order to simplify the calculation. However, this means that only the half of the main displacement will be attained. The model is then approximated to an equivalent Timoshenko-like beam that represents the same displacement as the unit-cell wall assembly.

In this section, by analyzing a single unit-cell (Fig. 4), the shear stiffness and the shear strength of the proposed square honeycomb cores are estimated. Fig. 4b represents a view from above of the loading case, with a generic shear load P applied following the orientation angle α and distributed along the core cells. The present work studies the particular case, when the shear load vectors are coincident with x-axis (α = 0). In this way, different analyses are carried out, in order to investigate the in-plane shear stress τ31 vs. γ31 core response.

3.1.1. In-plane shear modulus The general expression of the shear modulus can be defined by simple mechanical analyses of the behavior of the cores subjected to shear stresses, while considering a uniform straining of the cell walls. First of all, it is necessary to define the relative density of a squarehoneycomb core. Thus, the average area occupied by the base material

3.1. Analytical models This section describes three approaches from different theoretical models. These models provided the necessary insights for estimating the failure behavior of the cores. The shear modulus was estimated 3

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Fig. 2. (a) Unit repetitive cell render. (b), (c) and (d) Core cell patterns. Table 3 Unit-cells mean dimensions and core densities employed for analyses: designs 1, 2 and 3.

Design 1 Design 2 Design 3

H (mm)

L (mm)

t (mm)

a (mm)

b (mm)

ρc (kgm−3)

25.40 25.40 25.40

50 20 20

0.65 0.65 0.65

– 21.40 21.40

– 13.09 4.10

37.80 47.40 47.77

Arel =

Acell = LW =

L2

(3)

Then, considering the general expression of the relative density of the base material as the mass of the base material ms within a unit cell over the base material volume as s

=

ms ms = As H 2tLH

(4)

The density of the unit-cell core, ergo the density of the full core sample, is defined by Eq. (5), as the ratio of the unit-cell core mass mcell and the unit-cell area Acell.

As within a square unit-cell as W = L (Fig. 4b), is defined by Eq. (1). The unit-cell area Acell is shown as Eq. (2) and considering a relative area Arel ≤ 1 as the ratio As/Acell, Eq. 3 is attained.

As = tL + tW = 2tL

As 2t = A cell L

c

(1)

=

mcell m = 2cell Acell LH

(5)

The relative density of a square unit-cell honeycomb is obtained by Eq. (6), where combining both Eq. (4) and Eq. (5) the ratio c / s is obtained ( c / s ≤ 1). The unit-cell mass mcell and the base material mass

(2)

Fig. 3. Stabilized core shear samples obtained by proposed manufacturing route. 4

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Fig. 4. (a) A schematic loading case render. (b) A top-view render with a schematic unit-cell used for calculations.

Fig. 5. Sketches of the calculation models. (a) Unit-cell wall sheet, disregarding transversal cell walls. (b) Unit-cell wall loaded with a shear load before straining. (c) Unit-cell wall deformed. (d) Model of an equivalent cantilever beam with a shear load. (e) Sketch of a deformed Timoshenko-like beam.

within a unit-cell ms are equal in magnitude.

=

c s

2t = L

(6)

In this way, the relative density of a square-honeycomb core can be defined as well by the relative area. When a shear load is applied throughout the director angle α, the shear vector P (Fig. 4) is set by its components by Eq. (7).

c

=

P = Gc Acell

1

=

P1 = G 90°/0°s As

1

(10a)

2

=

P2 = G 90°/0°s As

2

(10b)

(9)

c

Similarly, each component expressions are shown in Eq. 10, where G90°/0°s the shear modulus of the base material, 1 and 2 are the unit-cell strains according 1-direction and 2-direction also given by Eq. 11.

P = P1 + P2

(7)

P1 = |P| cos

(8a)

1

= | c| cos

(11a)

P2 = |P| sin

(8b)

2

= | c| sin

(11b)

The general expression of the shear stress τc applied on a unit-cell (Fig. 4b) is defined by Eq. (9), where c is the strain of the unit-cell and Gc represents the shear modulus of the unit-cell.

Combining Eqs. 11 and 10, solving later Eq. (9) and Eq. 10 for each shear load into Eq. 7, the expression Eq. (12) resulted as normalized factors. 5

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so this approach cannot predict properly their mechanical properties because it does not consider the geometrical effects. To overcome this difficulty, proposed geometries were considered as well for the mechanical properties predictions as follows. When a beam is loaded by one concentrated load (Fig. 5), the deflection at the loading point can be determined by calculating the strain energy of the beam, matching it to the work of the conservative forces and solving for the displacement on the direction of the force. An equivalent method applied by the authors, is through Castiglianós second theorem (Eq. (14)) [39]. j

(12)

Since the director angle is set in this work as α = 0, only the component according to 1-direction will be taken into account. Then, solving Eq. (12) for the shear modulus of the unit-cell Gc (then extrapolated to the core) is

Gc = G 90°/0°s

t = G 90°/0°s L 2

Ui Pj

(14)

The partial derivative of the deformation energy Ui of a structure to any load Pj is equal to the displacement δj corresponding to that load. This form of derivative is applicable for short displacements and linear regimes of stress-strain. This statement is satisfied while calculating the shear modulus at the very beginning of the shear tests. As a first approach, due to the symmetry, the model (Fig. 5b and c) can be plotted as a cantilever beam with a punctual load P at the core mid-plane (Fig. 5d). The internal strain energy of the beam is assumed to be caused by bending and shear. In this way, the investigated model is treated as a Timoshenko-like beam that includes both bending and transverse shear effect. Since the transverse shear effect is neglected by Euler-Bernoulli classical beam theory [40,41], the successive planes that made the profile are not necessarily perpendicular to the bending line (Fig. 5e). The total displacement is defined by the sum of both effects, Uib the internal strain energy of a beam subjected just to a bending deformation and Uis the internal strain energy of a beam due to shear loading (Eq. (14a)). Eq. (14b) represents the expanded formula of Eq. (14a).

Fig. 6. Sketches of unit-cell wall sheet showing the glued area.

P = Gc c L2 = G 90°/0°s c cos tL + G 90°/0°s c sin tL

=

(13)

Finally, the elastic shear modulus of the square honeycomb core Gc can be related to the ratio c / c . The formula due to Eq. (13) is assumed to be valid for the full-walled cores (without machined geometries, Fig. 3a), for the particular case α = 0, where the normal cell walls undergo bending and carry an insignificant component of the load. In this work the densities of the cores were found to be almost equal,

Fig. 7. Render of the core design shear stress distribution with a generic line load of: (a) core design 1, (b) core design 2 and (c) core design 3. 6

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Fig. 8. Shear stresses vs. reference nodes extracted from simulations for: (a) core design 1, (b) core design 2 and (c) core design 3. j

= Uib + Uis

j

=

H /2 0

M

and fs is the form factor for shear [42]. The cross sectional area A varies along the profile for the present model, therefore a constant value was considered representing the behavior of the unit-cell wall. This equivalent cross sectional area will be approximated via numerical analyses. Also, this area is always represented as rectangle where A = tl, with t and l as width and length respectively of the equivalent

(14a)

M dz + P D

H /2 0

fs T

T dz P G 90°/0°s A

(14b)

where M and D are the internal bending moment and the bending stiffness respectively, coefficient T is the internal shear of the section 7

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beam cross section. Thus, in this particular case the factor fs is constant with a value of 6/5 [43]. Solving Eq. (14), the total displacement is then obtained by Eq. (15). j

fs PH PH 3 + 24E 0°s Iy 2G 90°/0°s A

=

(15)

The factor Iy in Eq. (15) represents the cross sectional second moment of inertia. Once the displacement is calculated, the shear modulus of the unit-cell is obtained by the ratio of the shear stress c to shear strain c , applying a generic shear force P (according α = 0 then P = P1, Figs. 4 and 5) and disregarding the contribution of the perpendicular sheets (when α = π) (Eq. (16)). c

Gc =

=

c

PH 2A cell

(16)

j

3.1.2. In-plane shear strength The analytical models for predicting the peak shear strength of the square honeycomb cores assuming uniform straining of the CFRP sheets, are thought as three governing failure modes of the unit-cell members, including: (i) maximum shear strength R , (ii) debonding between face sheet and cell member DB and (iii) elastic shear buckling B . The failure modes proposed below were determined imposing clamped constraints at the top and bottom of the CFRP cell walls of the honeycombs. Finally, the core peak shear strength cpk is given by the collapse mechanism that exhibits the minimum value Eq. (17). cpk

= min ( R,

DB , B )

(17)

In this work, each potential mechanism where the shear load is parallel to the 1-direction throughout the length of the sample (α = 0) is considered. 3.1.2.1. Maximum shear strength of the sheets. The core shear strength R depends upon the failure value of the base composite material. Considering the applied load P to the core (Figs. 4b and 5) and specifying it for the case α = 0, the failure load PR applied to the unit-cell walls, and thus to the core, is the same and it is then defined as Eq. (18)

PR =

90° /0°s As

=

R A cell

(18)

Solving for R Eq. (18), the core failure due to peak shear strength of the CFRP cell wall is obtained by Eq. (19). R

=

90° /0°s

2

(19)

Hereby it should be noted that Eq. (19) is valid for core case 1 (Fig. 3a) where the cores are made from full-walled cells. Other cases with machined geometries required similar analyses, but considering the local mid-plane cross-sectional area Amid of the sheet material (Fig. 5b). This area can be calculated as Amid = t(L − b). The maximum shear stress max on a beam with a rectangular crosssection is defined by Eq. (20) [39]. max

=

3 PR 2 Amid

(20)

The maximum load PR is applied to the core and consequently to the base material. The shear stress on the core is defined as Eq. (9) and referring it to the maximum load, Eq. (21) is obtained. R

Fig. 9. Load vs. displacement curves obtained from the simulations: (a) core design 1, (b) core design 2 and (c) core design 3.

=

PR A cell

(21)

Solving for PR and matching Eqs. (20) and (21), the core shear strength is now represented by Eq. (22), according to the nominal composite shear strength and leading to a shear fracture failure mode. The maximum shear stress is equal in this case to the shear strength of 8

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Fig. 10. Renders obtained from the simulations: (a) core design 1, (b) core design 2 and (c) core design 3.

the base material. R

=

2 3

90° /0°s

Amid Acell

FDB = A core (22)

=

DB

int

DB

= Abond

int N

2 fL N Acore

(23) (24)

Solving for DB , Eq. (24) represents the shear strength due to coreface debonding.

3.1.2.2. Debonding between face sheet and cell member. The bonding failure occurs when the shear stresses at the interface core/face sheet, exceed the adhesion strength or τint. The adhesion strength varies with processing temperature and time. In this work, it is assumed τint = 20 MPa. The mean bonded area Abond of the cells is calculated considering the mean area throughout the throat of the joint, while approximating the lateral meniscus to a uniform triangle (Fig. 6). This area depends upon the number N of CFRP sheets within a core, disregarding the transversal sheets at α = 0 (Fig. 4). The triangle is thought as an isosceles of flank f. Then the throat has 2 a width of 2 f and a length as the length of the unit cell L. The overall bonding strength depends upon the adhesion strength and the bonded surface ratio as the bonded area-to-core area or Abond/Acore. Due to the lack of homogeneity while gluing, the amount of glued surface is variable among the unit cells. Therefore, an average value of glued surface in the total sample surface was estimated. Eq. (23) is assumed to be valid, where the FDB represents the effective critical load for coreface bonding failure.

3.1.2.3. Buckling of CFRP sheets. The critical elastic shear buckling stress of the CFRP sheets Bs , is defined by Eq. 25 according to the theory developed by Seydel [44] that provided numerical approximations, considering the composite sheets as orthotropic materials. Bs

= ca

Bs

= cb

=

4

D1 D33 a2 D1 D3 a2

D1 D3 D31

if

>1

(25a)

otherwise,

(25b) (26)

where factors D1, D3 and D31 represent the mean bending stiffness for directions 1 and 3, and the mean bending distortion over the plain 3 – 1. It shall be noted that θ = 1 represents an isotropic plate. The general 9

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expressions of the stiffness are defined as follows.

D1 =

D3 =

E 0°s 1

°

0 /90° 90° /0°s

E 90°s 1

2D31 =

0° /90° 90° /0°s

Ix

Iz

E 0°s 1

0° /90° 90° /0°s

+ 4G 90°/0°s Ixz

(27) (28) 90° /0°s I x

+

E 90°s 1

0°/90° 90° /0°s

0° /90°s I z

(29)

where Ix and Iz are the second moment of inertia according x and zdirection respectively. The factor Ixz in Eq. (29), represents the polar moment of inertia. Coefficient a in Eq. 25 represents the half thickness of the core (or a = H/2); while ca and cb are tabulated values depending on border constraints and factor . Seydel [44] expressed the numerical results for two basic constraint cases: simple supported or fully clamped edges. The values of ca lie from 15.065 to 22.15; while cb varies from 18.59 to 22.15, both for fully clamped plate cases and 0 ≤ θ < ∞. In this work, the boundary conditions are assumed as an approximation of a plate clamped along the loaded edges and simply supported along the unloaded edges. Therefore, it is necessary to introduce further tabulated factors as [45,46].

Ks = Ks =

=

4ca 2

4cb 2

if

>1

(30a)

otherwise,

(30b)

L D1 4 H D3

(31)

In this way, Ks varies from ≈7 to 14 for values of 0 ≤ β ≤ 1, specified then by the obtained value of the factor θ. Finally, the critical elastic shear buckling stress of the core B is defined by Eq. (32), neglecting the contribution of the core walls perpendicular to the shear load (when α = π). B

=

Ks 2 4 D1 D33 ta2 2

(32)

3.2. Numerical models Numerical models were developed by using the software FEMAP™ 10.3 with NX™ Nastran® [47]. The simulations were carried out in order to have a further overview of the main structures behavior and to validate the theoretical analyses. Finite element analyses combined with analytical calculations were used for indirectly predicting the elastic shear modulus, validated later by experimental tests. The employed elastic properties of core walls are specified in Table 2 following the coordinate system showed in Fig. 4. The laminate was considered as an orthotropic material with an average thickness of 0.65 mm. 3.2.1. Static analyses CFRP sheet material are simulated by discretization of a regular mesh of shell elements (CQUAD4, 4-node reduced integration), representing the geometry of the studied cores (Fig. 2). The nodes on the plane z = −H/2 were fully clamped, while the nodes along the plane z = +H/2 are constrained not to move in z-axis and y-axis, considering short displacements. The rotation along the x-axis is also restrained, simulating the conditions imposed by the glued surface. A line load is applied over the top node-line, reproducing a shear load over the main set. An average width which represents the equivalent second moment of inertia of the entire section can be obtained studying the behavior of the struts while measuring the displacements.

Fig. 11. Load vs. node displacement curves obtained from the simulations: (a) core design 1, (b) core design 2 and (c) core design 3.

10

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Fig. 12. Scheme of the shear test set-up.

Fig. 13. Shear stress – strain response of proposed square – honeycomb cores.

Renders of the shear response of simulated CFRP sheets are shown in Fig. 7. A generic line-load of 10 N/mm was applied. Shear stresses in xz-plane are highlighted. Core design 1 denotes a shear parabolic distribution (Fig. 7a). Moreover, the highest shear stresses are located at the mid-plane (or neutral axis) and the middle sections. Core design 2 and core design 3 (Fig. 7b and c, respectively), show a stress concentration at the middle strut with a parabolic stress distribution. Remarkably for case 3, the struts at both sides of the center column do not carry significant share of the main load. Shear stresses vs. referenced node position curves (green horizontal mid-line) are extracted from the numerical calculations for the case of maximum shear strength of the base material (Fig. 8). The full-walled cell cores (Fig. 8a) present a combination of parabolic and constantlinear curves for the stress distribution at the mid-plane and a failure load of 65 N/mm is obtained. Core design 2 shows a parabolic-like

Fig. 14. Montage of photographs showing the shear deformation of core design 3.

curve for shear stress distribution at the mid-plane. In this case, a failure load of 16.8 N/mm is calculated. For the case 3, the node references were obtained from the mid-plane including the center and the two laterals struts. The highest shear stresses are observed at the center strut, with a failure load of 14.8 N/mm. In order to obtain the shear modulus of cores Gc, load-displacement curves were calculated for different element sizes (Fig. 9), with a minimum of three different element sizes. Convergence analyses were

Table 4 Resume of analytical, FEM and experimental results along with failure modes from in-plane shear tests. Core design

1 2 3

Analytical results

Numerical results

Experimental results

τcpk (MPa)

Gc (MPa)

τcpk (MPa)

Gc (MPa)

τcpk (MPa)

Gcpk (MPa)

Obs. fail. mode

1.59 0.82 0.51

50.08 61.72 28.64

1.37 0.84 0.74

51.22 59.53 27.66

0.453 ± 0.027 0.536 ± 0.008 0.334 ± 0.011

50.57 ± 1.14 52.52 ± 9.30 17.14 ± 2.14

DB SF SF

DB = debonding failure; SF = shear failure. 11

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carried-out in the limit as the element size tends to zero [48]. With the slope s calculated from the load-displacement curves, the core thickness c and applying Eq. (33) [49], the shear modulus of each core is calculated as

Gc =

sc Acell

(33)

The maximum element size was taken as 0.5 mm on core design 1 (Fig. 9a), due to the stiffness of the model, in order to get less discrepancy among the calculation with different displacements. 3.2.2. Non-linear analyses The CFRP cell-wall of the cores were simulated by shell elements (CQUAD4, 4-node reduced integration). A non-linear static model with modified Newton-Raphson method was employed for the iterations. The nodes at the bottom and the top were fully clamped. A linear load was applied at the top node-line to study the behavior of the struts while measuring the displacements, until the model stop due to instability. Fig. 10 represents schematically the material distortion due to the applied load. Core design 1 (Fig. 10a) shows a buckling behavior with two wrinkles. On the other hand, other cases exhibit one wavelength wrinkle. Fig. 11 presents the data extracted from the simulations for each core case, employing tracking points of nodes at both opposite sides of the material unit-cell sheet. The selected nodes, presents the major displacements in 2-direction (Fig. 4 coordinate system). Fig. 11a represents the selected node tracking of core design 1. Both curves begin with zero displacement until the instability load is reached, then they behave asymptotically to positive values of displacement, which indicates that the wrinkles exhibit two wave-length peaks to the same distortion direction. Through extrapolation of the curves, a load of approximately 6.1 kN are estimated as the start of the instability region; while a load of 8.2 kN represents the failure load. In a similar way the other cases were studied. Fig. 11b and c show the behavior of two opposite nodes selected, both exhibiting mirrored behavior accordingly to the zero positioning. These results suggest that only one-wave wrinkle is developed in both cases. The instability regime begins at load of almost 657.5 N and 422.5 N with an estimated failure load of 674.4 N and 443.12 N for core design 2 and core design 3, respectively.

Fig. 15. Photographs showing the debonding failure mode of core design 1.

3.3. Experimental results The test protocol was performed following the standard ASTM C273, for obtaining in-plane shear properties of the sandwich core materials [49]. Stabilized core samples were fixed to prismatic steel plates with a two-component epoxy glue. ASTM C273 defines a minimum length-to-thickness ratio of 12:1, aiming a desired shear failure of the samples. The employed length-to-thickness ratio in this work was 8:1, leading to an out-of-plane loading case higher than the standard recommendations. Nevertheless, it has been proven that even smaller ratios leaded to suitable results [28]. The test set-up is schematically shown in Fig. 12, where it can be observed that the load line was kept within the diagonal between opposite corners of the core, according to the standard recommendations. A Zwick/Roell Z150 screw-driven universal test machine, was employed to carried out three tests of each core sample at controlled room temperature. The cross-head speed was 0.5 mm/min. A displacement transducer HBM W2TK was employed. In-plane shear tests responses τ31 vs. γ31 of the three proposed cores are presented in Fig. 13 and the results are summarized in Table 4, including predicted shear stiffness and strength, and observed failure modes. Analyzing the curves in Fig. 13, a linear response is observed at the beginning of the curves in all the evaluated specimens, followed by a gradually change in the slope until rupture, to be expected during shear loading in particularly for this kind of fiber orientation [50]. Core design 1 exhibited catastrophic failure in all samples, dominated by

Fig. 16. Maximum shear modulus according to experimental tests, contrasted by FEM and analytical models.

Fig. 17. Comparative peak shear stresses among different theoretical approaches.

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Fig. 18. Photograph and FEM render: comparison for core design 2 deformation.

Fig. 19. Zoom-in into a photograph and FEM render: comparison for core design 3 deformation.

debonding. Core design 2 and core design 3 presented both a significant bending deformation, with a decrement on the bearing of stresses and an increment of the strain. The failure of design 2 and 3 samples was attained after reaching a maximum deformation, where the cores failed in a slower and softer manner while reaching the maximum shear strength of the CFRP base material. The highest shear modulus was exhibited by core design 2, with an average of 52.52 MPa, followed by core designs 1 and 3, with 50.57 MPa and 17.14 MPa respectively (Table 4). These differences can be attributed to the geometric imperfections and the number of struts bearing shear loads at the same time. Additionally, the predicted shear strengths were higher than the experimental results, presumably caused by base material defects, probably induced during the manufacturing process. A series of photographs of core design 3 sample at different deformation levels can be seen in Fig. 14. The core design 3 is presented as an example for showing the effect of bending induced by the shear loads. The snapshots show an initial elastic regime in which bending of the small columns takes place before the first signs of rupture are observed after reaching the maximum load. The failure does not occur at the same time on all the columns, but it happens gradually, suggesting a

misalignment in the support of the loads, possibly due to the non-automated manufacturing process. 4. Discussion Table 4 shows the predictions of the shear strength, both analytical and numerical models are reasonably in agreement, however it can be seen that in all cases they are moderately over estimated. The discrepancy among the experimental data and the predictions can be attributed to imperfections included while manufacturing the samples, during the test set-up or during tests (e.g. misalignments of shear plates or lack of parallelism between core sheets and shear plates). These issues may considerably affect the performance of the cores, especially of the core design 3 that presented the higher number of machined geometries and the smaller columns more susceptible to bending stresses. For core design 1, the predicted strength was based in failure due to fracture of the constituent base material (Table 4). However, when the predictions are expressed in terms of debonding of the face-core interface, the values are closer to experimental results and confirmed by the observed failure mode, with small cracking bands at the bottom of

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Table 5 Comparison between measured shear modulus and shear strength of similar materials of the literature. References

Topology/base material

ρc (kgm−3)

Russell [15]

Square-honeycomb/woven 0/90° CFRP

Xiong [29]

Pyramidal truss/laminated CFRP

Dong [28]

Octet-truss lattice/woven and laminated CFRP

Feng [53]

CSH/foam and woven CFRP HCSH/foam and woven CFRP

George [27]

Pyramidal truss/laminated CFRP

Wang [20]

X-type lattice truss/laminated CFRP

Present work

Machined square-honeycomb core/woven 0/90° CFRP

34.25 68.50 137 9.92 19.38 28.06 43.87 24.48 77.76 135.36 187.2 228.96 123 78 103 150 43 72 101 144 38.58 55.55 86.41 47.25

Notes: ρc = density of core,

(%) 2.5 5.0 10.0 0.64 1.25 1.81 2.83 1.7 5.4 9.4 13 15.9 N/A N/A N/A N/A 2.986 5 7.01 10 2.5 3.6 5.6 3.5

Gc (MPa)

Gc/ρc (103 m2 s−2)

τcpk (MPa)

τcpk/ρc (106 m2 s−2)

48 86 177 9.53 21.59 38.78 59.93 66 157 282 424 493 N/A N/A N/A N/A 174 252 365 500 20.95 21.09 45.51 61.82

1.401 1.255 1.292 0.961 1.114 1.382 1.366 2.696 2.01 2.0833 2.264 2.153 N/A N/A N/A N/A 4.04 3.5 3.61 3.47 0.54 0.38 0.53 1.31

0.65 1.52 3.9 0.075 0.421 0.519 0.548 0.46 2.21 3.47 4.45 5.91 1.31 1.63 2.03 2.62 2.8 4.9 6.3 7.6 0.59 0.72 1.55 0.54

18.97 22.19 28.46 7.56 21.729 18.49 12.49 18.79 28.42 25.64 23.77 25.81 10.65 20.89 19.71 17.46 65.11 68.11 62.37 52.77 15.29 12.96 17.93 11.34

= relative density.

the sample before the sudden rupture (Fig. 15). The smaller bonding surface influences the mechanical performance of the core, causing an abrupt premature failure (Fig. 13). In this case, the sample was de-attached from the support glued surface. The elastic shear moduli of the cores are in good accordance with predictions (Fig. 16). The analyses were addressed by introducing shear loads. The analytical models are in good accordance to the numerical predictions, and then were validated by the experimental tests. Core design 2 denotes a higher rigidity, less influenced by bending moments in comparison with core design 1, as a result of wider mean equivalent mid cross-area Amid. The discrepancy among the experimental data and models, is attributed to unexpected fabrications issues, less remarkable by core design 1 that presented fewer machining stages during the manufacturing process. A comparison of failure modes is presented in Fig. 17. The maximum shear stresses presented by the experimental tests (black bars) were predicted correctly by the proposed analytical models. Comparing simultaneously the theoretical failure modes, the modes with minimum peak stress values represents the appropriate failure mode, having a correspondence with the experimental data. It can be inferred, that the failure mode that need less deformation energy is the one that have more probability to occur, in accordance with the proposed methodology. The bonding failure was correctly predicted by the approximation used, where an average bonding surface is calculated among the obtained samples. Furthermore, both core design 2 and 3 showed a bending deformation (see also Figs. 18 and 19) followed by a shear failure, also predicted by analytical and numerical models, despite the discrepancy contrasted by the experimental results. The failure due to buckling of the plates was estimated as well, but none of the specimens showed such failure mode. A comparison between FE predictions of the deformation of core design 2, is shown in Fig. 19 in which a shear strain of γ = 0.045 was applied. The original position of the struts is represented by a shadow behind the FE sheet. In this case, the deformation results from a superposition of shear and bending loads. No wrinkling of the core cell walls was predicted. Nevertheless, good agreement is seen between the FE calculations and the core design 1 deformation at γ = 0.03 (Fig. 19). The bending load influence leads to a further distortion of the struts, also remarkable in Fig. 14. In both FE cases, a 2D-plate with fully

restricted constraints at the top and bottom was assumed. 5. Comparison of the square honeycomb with competing sandwich cores The peak shear modulus and peak strength performance of the core design 2 are compared and briefly quoted in Table 5 in order to have a clear comparison with sandwich structures morphologies. The specific shear modulus presented in this work is comparable with other similar core morphologies like the full walled square-honeycomb cores. Remarkably, the cores presented in this work exhibited better shear moduli than some of the lattices made from CFRP. For example, they exhibited a specific shear modulus of 71% and 27% higher for the Xtype lattices and many of the pyramidal-truss cores respectively. This can be attributed to the higher rigidity shown by cores made from plates than the lattices structures made from bars or beams [51]. The fiber orientation plays a fundamental role when CFRP structures are used. It should be noted that the fiber orientation employed in this work is better implemented when the loading case is parallel to the fiber orientation. Based on that, an improvement for this kind of material performance would be by the coupling of two layer of ± 45° woven carbon fiber between two layers oriented at 0/90°, with a small lack of out-of-plane performance. Normally, the fibers are the component that withstand most of the load in a fiber reinforced composite material. Therefore, if the loading case is due to shear and the orientation of the fibers are at ± 45°, this type of fiber orientation would enhance the shear performance of the proposed cores, because the maximum shear stresses are at 45° [50]. The specific shear strength exhibited by core design 2 is still competitive with other materials, whether the full potential of this morphology is achieved. Despite not showing the same performance of commercial available cores [52], the cores developed in the present work are still comparable to those materials. Moreover, this new cores have potential multifunctional applications due to the hollow pattern. 6. Concluding remarks In this work, three different types of ultra-lightweight cores were designed, manufactured and tested. The core density was maintained 14

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below 48 kgm−3 in all cases. The results obtained from theoretical approaches to study the shear behavior due to the deformation of the cores, were in good agreement with experimental tests. The FE calculations also helped to understand the displacement performed by the CFRP sheets, validating the theory of a Timoshenko-kind beams. The hollow cores with interconnected holes allowed the material to be ultra-lightweight and also gives the cores great potential for multifunctional applications (e.g. heat transfer, embed foams, cables or electronics). The amount of sheets is an important point to review in structures that are made from non-monolithic parts. The glued surface ratio must be taken into account, not only by shear loading cases but also by flexural responses. The proposed cores in this paper do not achieve the full potential of the predicted shear strength. Whether the maximum theoretical shear load is reached by mitigating the defects introduced by the manufacturing method and the misalignment while loading during the test set-up, the competitiveness of these cores is relatively clear. Although the catenaries were thought firstly as an optimal shape of an arch for compressive loading, they are found in this work to still have comparably good shear performance. Furthermore, good specific properties in terms of shear modulus and strength are achieved. Further improvement for minimizing the manufacturing imperfections of the material is needed in order to overcome some lack in performance.

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CRediT authorship contribution statement Pablo Vitale: Investigation, Formal analysis, Writing - review & editing. Gaston Francucci: Investigation, Formal analysis, Writing review & editing, Methodology. Helmut Rapp: Writing - review & editing, Supervision. Ariel Stocchi: Conceptualization, Supervision, Investigation, Writing - review & editing, Project administration. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements This research was financial supported by the Bayerisches Hochschulzentrums für Lateinamerika (BAYLAT) and finished with the support of the Deutsche Akademische Austauschdienst (DAAD). The authors would like to acknowledge the Universidad Nacional de Mar del Plata, CONICET Argentina and the Universität der Bundeswehr München, Germany. Appendix A. Supplementary data Supplementary data to this article can be found online at https:// doi.org/10.1016/j.compstruct.2020.111879. References [1] Belingardi G, Martella P, Peroni L. Fatigue analysis of honeycomb-composite sandwich beams. Compos Part A Appl Sci Manuf 2007;38:1183–91. https://doi.org/ 10.1016/J.COMPOSITESA.2006.06.007. [2] Ashby MF, Cebon D. Materials selection in mechanical design, Le. J Phys IV 1993;3:C7-1-C7-9. [3] Hertel H. Struktur. Bewegung: Form; 1963. [4] Wang J, Shi C, Yang N, Sun H, Liu Y, Song B. Strength, stiffness, and panel peeling strength of carbon fiber-reinforced composite sandwich structures with aluminum honeycomb cores for vehicle body. Compos Struct 2018;184:1189–96. [5] Eschena H, Harnischa M, Schüppstuhla T. Flexible and automated production of sandwich panels for aircraft interior. Procedia Manuf 2018;18:35–42. [6] Thomsen OT, Bozhevolnaya E, Lyckegaard A. Sandwich Structures 7: Advancing with Sandwich Structures and Materials. Proceedings of the 7th International Conference on Sandwich Structures, Aalborg University, Aalborg, Denmark.

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