corrugated core sandwich composite structures reinforced by horizontal stiffeners

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Flexural performances of fiber face-sheets/corrugated core sandwich composite structures reinforced by horizontal stiffeners Hongyong Jiang , Yiru Ren , Qiduo Jin , Guohua Zhu , Zhihui Liu PII: DOI: Reference:

S0020-7403(19)32545-7 https://doi.org/10.1016/j.ijmecsci.2019.105307 MS 105307

To appear in:

International Journal of Mechanical Sciences

Received date: Revised date: Accepted date:

13 July 2019 24 October 2019 4 November 2019

Please cite this article as: Hongyong Jiang , Yiru Ren , Qiduo Jin , Guohua Zhu , Zhihui Liu , Flexural performances of fiber face-sheets/corrugated core sandwich composite structures reinforced by horizontal stiffeners, International Journal of Mechanical Sciences (2019), doi: https://doi.org/10.1016/j.ijmecsci.2019.105307

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Highlights 

Design a horizontal stiffener to reinforce performances of corrugated core SCS.



Obtain accurate predicted results including load, damage and deformation.



Effects of stiffener thickness,

position and number

on reinforced

performances. 

Reveal complex reinforcement and coupling deformation and EA mechanisms.



Greatly improve SEA and EA of corrugated core SCS by a horizontal stiffener.

Flexural performances of fiber face-sheets/corrugated core sandwich composite structures reinforced by horizontal stiffeners Hongyong Jiang1,2, Yiru Ren1,2,*, Qiduo Jin1,2, Guohua Zhu3, Zhihui Liu,4 Abstract The light-weight and high-strength design concept enables sandwich composite structures (SCS) to maximize specific strength and specific energy-absorption (SEA). To this end, a horizontal stiffener is designed to reinforce the flexural performances of corrugated core SCS for the first time. To explore preferable reinforcement parameters, several stiffeners with different thicknesses, positions and numbers are designed and studied. The flexural responses of SCS are predicted by finite element simulation. The predicted load responses, damage and deformation of face-sheets and core are consistent with experimental observations. From predicted results, the SCS undergoes several stages including elastic deformation, fracture of face-sheet, initial and stable deformation of core. The core with a stiffener obviously exhibits higher SEA than that without a stiffener. Complex reinforcement mechanisms and coupling deformation between pure core and stiffener are revealed. It is found that the stiffener improves not only the EA of pure core but also the EA of face-sheets, which explains the coupling EA. Structural parameters including thickness, position and number have great effects on flexural performances of SCS. Notably, the lightest SCS with 0.25 mm thickness improves the SEA about 23.83%. The SCS with three stiffeners presents an increase in EA of 92.33% but followed by a great increase in mass. Design recommendations are finally concluded through comprehensive evaluations.

Keywords Sandwich; composite; stiffener; reinforcement; energy-absorption; deformation 1. State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, Hunan 410082, China 2. College of Mechanical and Vehicle Engineering, Hunan University, Changsha, Hunan 410082, China 3. School of Automobile, Chang’an University, Xi'an 710064, China 4. School of Aeronautic Science and Engineering, Beijing University 100191, Beijing, China Corresponding author: Yiru Ren, College of Mechanical and Vehicle Engineering, Hunan University, Changsha, Hunan 410082, China * E-mail address: [email protected]

1. Introduction

Sandwich composite structures (SCS) have been increasingly applied in aerospace, automotive and marine fields, because they provide more excellent structural performances with limited weight [1-9]. Typically, the SCS is composed of two thin but high-strength face sheets and a light-weight core [10]. Both face-sheets fabricated by the carbon fiber/epoxy resin materials are used to bear main structural load. The low-density core, bonded to both face-sheets, is responsible for higher energy-absorption (EA), post-carrying capacity and more other functions. These extensive functions can be determined by specific core materials and configurations such as honeycomb, foam, lattice, grid, X-, Y-, V-frame, etc. [11-18]. Although the SCSs have exceptional superiority in many realms, the most major restrictions to their applications are that their mechanical performances are vulnerable to the structural deformation and damage caused by the bending load, low-velocity impact and out-of-plane compression. The deformation and damage may even cause a dramatic degradation in structural carrying and EA capacity, possibly leading to a catastrophic damage for the whole structure. Hence, providing an insight into the damage and deformation mechanisms of SCS is of critical significance to lower structural damage and more reliable applications of such structures. Much research efforts have been extensively devoted to typical damage modes of sandwich composite structures with different cores [19-20]. As was reported, the face-sheets of SCS with any core had similar and slightly different damage modes. The top face-sheet in SCS mainly exhibited several following failure modes. The bending deformation was initiated slightly and then the interlaminar delamination occurred simultaneously due to different deformations of adjacent layers [19-22]. In severe cases, the accumulation of micro fiber breakage, matrix cracking and their debonding even led to fracture or penetration at the macro-scale [17]. Once the fracture or penetration was appeared, the reaction force reached its limit and subsequently presented a drop. It was also found that when applying three-point bending the top face-sheet presented the compressive damage while the bottom face-sheet showed the tensile damage [23]. For the core, different damage modes were recorded for different core configurations. For example, it was revealed that the honeycomb core mainly presented crushing, shearing and buckling [19]. Further, the depth of damage far from the loading position was less than that at the loading position. For the foam core, several modes including foam densification, foam

shearing, foam crushing and foam crack were concluded by Taghizadeh et al. [20] and Tang et al. [24]. The metal plastic deformation characterized by plastic hinges dominated the damage modes of other arisen metal cores such as grid, X-, Y-frame, and corrugated core, etc. [17,25-29]. From these summaries, it was indicated that the core had a positive effect on the mechanical behaviors after the maximum force. Despite that, there was still an important problem about weak interface between face-sheets and core. For any core bonded to both fiber face-sheets, two interfaces respectively existed between the top face-sheet and core and between the bottom face-sheet and core. Due to discrepancy of different materials, it was found from studies of Xue et al. [19] and Shi et al. [29] that the weak interfaces were greatly sensitive to the debonding damage. The interface debonding harmed the structural integrity and carrying capacity of sandwich structure. To remedy this issue, Sun et al. [30-31] adopted a short aramid-fiber toughening method to reduce the interface damage of SCS, and further conducted a systematic study on the surface treatment (e.g. resin pre-coating) based interface toughening [32-33]. From there results, a dramatic improvement in interface performances was obtained. Based on above overview, damage modes of SCS have been revealed and the structural performances can be improved through the full understanding of damage mechanisms. Among above cores, it was found that the corrugated core sandwich panels exhibited higher flexural resistance capacity, shear performance and specific energy absorption (SEA) [17-18,34-35]. Hence, the corrugated core sandwich panels have received increasing attentions for the potential applications in the design of morphing wing, wind turbine blade and other main bearing components [34-35]. At present, numerous studies have been carried out on the mechanical behaviors of corrugated core SCS subjected to typical loading conditions [23,36-40,]. He et al. [37] studied the low-velocity impact behaviors of hybrid corrugated core sandwich structures, experimentally and numerically. The study revealed fiber damage, matrix damage and delamination of face-sheets and buckling of core under the impact loading. They also studied the effects of impact energy, core thickness, and impact site on impact resistance and energy-absorption. Liu et al. [35] conducted an investigation into the effect of loading conditions on the low-velocity impact behaviors of corrugated core SCS. They considered different impactor shapes (i.e., conical, hemispherical and flat), impact energies, impact locations. It was revealed that the damage of face-sheets and

buckling of corrugated core highly depended on these three types of parameters. To explore the effects of geometric configurations, Rong et al. [38] compared the impact responses and planar compression of SCS with different corrugated cores including arc-shaped, sinusoidal, rectangular, trapezoidal and triangular cores. It was found that the geometric configurations had slight effects on the EA under high energy impact, but had great effects on the load under the planar compression. To study the post-impact residual flexural strength, He et al. [23] modeled multiple analysis steps to simulate the low-velocity impact responses and the post-impact flexural behaviors of SCS with corrugated core. It was concluded that the residual flexural strength was susceptive to low energy impact but not susceptive to high energy impact. To improve the EA, Shu et al. [39] proposed two-layered corrugated core sandwich panels and found that two-layered corrugated core SCS presented higher EA. However, the SEA was not very well due to the increase of structural mass. In conclusion, high maximum structural-strength, damage resistance, EA were presented by the corrugated core SCSs under impact, flexural and planar-compression loading. These advantages are mainly attributed to light-weight, thin and 2D periodic structure and hybrid materials. However, the unsatisfactory post-mechanical behaviors such as low carrying capacity and EA were recorded for corrugated core SCSs. Due to high sensitivity to various parameters, the mechanical performances of SCS can be improved through design. However, it is disruptive for the structural SEA if using unreasonable design to cause an uncompromising mass increase. For instance, Sun et al. [28] and Shi et al. [29] designed an extra vertical grid structure to improve the mechanical properties of honeycomb sandwich composite structures. For this type of design, a reinforced grid structure was only embedded in the core, which effectively improved the total EA of SCS. However, the SEA was not increased, which might be due to selection of structural parameters such as thickness, position, number, etc. To solve this problem, the reinforcement design concept enabling SCS to maximize the SEA is very necessarily considered for reasonable structural parameters. Hence, inspired from the design method of Sun et al. [28] and Shi et al. [29], a light-weight and easy-processable plate as a horizontal stiffener is designed and optimal structural parameters are further considered to improve the SEA and post-carrying capacity of corrugated core SCS as much as possible.

This work mainly aims to design a new type of horizontal stiffener embedded in the corrugated core to improve the flexural performances and SEA of corrugated core SCS. To design better structural parameters of stiffener, effects of the stiffener thickness, position, number on flexural behaviors of SCS are predicted based on the finite element (FE) method. The FE model is validated with available experiment. According to predicted results, typical flexural response process is analyzed. The flexural behaviors of SCS with a stiffener and without a stiffener are compared to explain the reinforcement, coupling deformation, coupling EA mechanism and SEA. Finally, some design proposals are concluded based on the structural parameter analysis.

2. Assessment criteria To evaluate the mechanical performances of sandwich composite structures under the flexural loading, four assessment criteria including maximum flexural load (Fmax), average or stable flexural load (Fave), load efficiency (  ), total energy absorption (EA) and specific energy absorption (SEA) are adopted in the present work. ● Fmax is the maximum flexural load in the load-displacement curve which represents the structural flexural strength of SCS. ● Fave is the stable flexural load after the fracture of the top face-sheet and during the core deformation stage. This parameter is used to the carrying capacity of core which also affects the total energy absorption. ●  is the load efficiency which is used to evaluate the stability of load. The value is computed by the following expression.



Fave Fmax

(1)

● EA is the total energy dissipated by the failure of fiber face-sheet and plastic deformation of metal. The value of EA is determined by areas of load-displacement curve.



EA   F ( )d 0

(2)

where F is the flexural load and  is the deflection. ●

SEA in the present work is defined as the specific energy absorption which is used to evaluate structural energy absorption capacity in terms of the qualitative analysis. This parameter is the most crucial index to evaluate the structural light-weight and high EA performances. Its value is computed by a ratio of EA to total mass (equation (3)). SEA 

EA m

(3)

where m is the total mass of structure. To improve the SEA, increasing the total energy absorption and reducing the structural mass should be considered in the designing scheme as soon as possible.

3. Sandwich structures with corrugated cores reinforced by different horizontal stiffeners 3.1. Available test

Figure 1. (a) The experimental setup for three-point bending tests [23]; (b) The schematic of corrugated core without a stiffener in the experimental specimen. The quasi-static three-point bending tests for the sandwich composite structures without a stiffener are based on available physical tests [23], as depicted in Figure 1a. From the assembly diagram (Figure 1a), the sandwich composite structures with

3-unit cells are composed of three thin-walled components including the top face-sheet, the bottom face-sheet and the trapezoidal corrugated core. A cylindrical loading indenter with a diameter of 20 mm is located in the center of the top face-sheet. Two cylindrical supports located in the bottom face-sheet have the diameter of 20 mm and their span is 200 mm. The flexural tests are carried out at a constant speed of 1 mm/min and the total vertical displacement δ of indenter is 10 mm [23]. During the bending test, the variations of flexural load and vertical displacement can be recorded by the testing machine. The experimental results for SCS without a stiffener provide a baseline for the comparison of model.

3.2. Specimen without a stiffener Figure 1b shows the schematic of corrugated core without a stiffener in the experimental specimen. The corrugated core walls are fabricated from a 2A12-T4 aluminium sheet [23]. The geometrical configuration and dimensions of a unit cell of the core are detailed in Figure 2, where the unit is mm. The total length and width are respectively 300 mm and 96 mm. According to Ref. [23], the material parameters are measured by tests which are listed in Table 1. Both the top and bottom face-sheets are manufactured from the same unidirectional carbon fibre/epoxy (T700/3234) materials [23]. The face-sheets with the stacking sequence of [0º/90º/0º/90º]s have an average thickness of 1 mm, the length of 300 mm and the width of 96 mm. The corrugated core is closely bonded to the top and bottom face-sheets.

Figure 2. The geometrical configuration and dimensions of a unit cell of the core (unit: mm).

3.3. Design of corrugated core reinforced by horizontal stiffeners To develop the light-weight and high-strength SCS, it is very valuable to increase

energy absorption while minimizing structural mass. It is a challenging task to achieve higher SEA. Hence, a light-weight and low-cost horizontal stiffener is proposed to reinforce the corrugated core SCS for the first time. The horizontal stiffener is embedded in the corrugated core to form a new type of combined core. During the loading process, the stiffener can profitably change the deformation of core, leading to an improvement of EA. Figure 3 shows the geometrical configuration and dimensions of the corrugated core with a horizontal stiffener, where the stiffener has the same thickness as core and is placed at intermediate height of the core.

Figure 3. The geometrical configuration and dimensions of the corrugated core with a horizontal stiffener (unit: mm). Because an additional structure is introduced in the corrugated core, the reinforced SCS exhibits more complicated deformation and failure. To reveal the reinforcement and deformation mechanisms of core, different horizontal stiffeners are designed including different thicknesses, different positions and different numbers. Five types of thickness including 0.25 mm, 0.5 mm, 1 mm, 2 mm and 3 mm are considered, as shown in Figure 4. Three positions including 2h/3, h/2 and h/3 are considered where h=12.85 mm, as shown in Figure 5. For different positions, one stiffener placed at the intermediate height of the core, two stiffeners with the adjacent distance of h/3 and three stiffeners with the adjacent distance of h/4 are considered, as shown in Figure 6. The flexural performances of all designed SCSs reinforced by stiffeners are studied and compared with the purpose of revealing optimal thickness, position and number.

Figure 4. Configurations for core reinforced by the stiffener with different thicknesses.

Figure 5. Configurations for core reinforced by the stiffener with different positions (unit: mm).

Figure 6. Configurations for core reinforced by the stiffener with different numbers.

4. Numerical modeling

4.1. Finite element model (FEM) The finite element model of the corrugated sandwich composite structure subjected to three-point bending is shown in Figure 7 according to the actual experiments. The FEM consists of the specimen, indenter and two supports. The indenter and supports are defined as rigid bodies due to little deformation. The rigid bodies are modeled with 4-node 3-D bilinear rigid quadrilateral elements (R3D4). A quasi-static vertical loading is assigned to the indenter and thereby the strain-rate sensitivity is neglected in the simulation due to a low and constant loading speed [23,41-42]. All degrees of freedom of supports are restricted and the indenter is constrained within 5 degrees of freedom except the loading direction. For deformed bodies, different element types are respectively modeled in the analysis including conventional shell (S4R) and continuum shell (SC8R) which can capture the delamination of face-sheets. S4R represents 4-node reduced integration thick shell with the hourglass control and SC8R represents 8-node reduced integration in-plane continuum shell with hourglass control. Here, three types of models are used for simulation: (i) model A (SC8R with interfaces), (ii) model B (S4R with interfaces) and (iii) model C (S4R without interface). For the model A (Figure 7a), the core is meshed with 3,168 SC8R elements. Each face-sheet is meshed with 8,640 SC8R elements and is stacked with 8-layer laminates. The interface between the core and face-sheets and the inter-ply delamination between adjacent plies are captured by surface-based cohesive behaviors [43-44]. Due to the fact that the present work is mainly focused on structural design, S4R is adopted in the simulation to reduce the computation cost and design period. Hence, in the model B and C (Figure 7b), the corrugated core without a stiffener is discretized with a total of 4,400 S4R elements. The horizontal stiffeners in the reinforced corrugated core are defined with a similar density of mesh. Both face-sheets with 8 layers are also meshed with a total of 2,500 S4R elements. For the model B, the interfaces between the core and face-sheets are considered. However, the interface in model C is not considered. Thus, a perfect bonding is defined with surface-based tie constrains due to the fact that little interface debonding occurs between the core and face-sheets [23,38]. In terms of contact, the general contact algorithm with the hard contact and friction coefficient of 0.2 that generates the required tangential and normal forces is defined in

the whole model. For the accurate and efficient computation, the explicit solver in ABAQUS is adopted to simulate the flexural response of SCS under the three-point bending.

Figure 7. The finite element model of the corrugated sandwich composite structure subjected to three-point bending.

4.2. Damage modeling The modeling of damage process of materials is a crucial factor to capture accurate flexural load responses of SCS. Thus, it is essential to establish reasonable damage criteria to predict the onset of any damage occurring in fiber composite and ductile metal materials. To better capture the material behaviors, the damage evolution is also modeled after the damage initiation criteria are satisfied [45]. In the present sandwich structure, fiber face-sheets and aluminium core have different material systems. The former is manifested as complicated brittle rupture while the latter is subjected to the plastic deformation. Both also have different energy dissipation modes. Hence, different damage initiation and evolution are adopted to simulate the deformation of materials. 4.2.1. Damage initiation and evolution of composite face sheets The most main damage modes of composite face-sheets are fiber tension fracture, fiber compression kinking, matrix tension cracking, matrix compression buckling. These basic damage modes considered are capable of reflecting typical mechanical features of composites. A common maximum stress damage criterion which is the simplest form has been widely applied in predictions, but the shear contribution to fiber and matrix is not considered. Hence, the Hashin damage criteria considering the effect of shear stress are adopted in the present prediction [46-47]. The initiation

criteria have the following general forms: Fiber rupture in tension:  ˆ   ˆ  F   11T     12L  X  S  2

2

t f

(4)

Fiber buckling and kinking in compression:  ˆ  F   11C  X 

2

c f

(5)

Matrix cracking under transverse tension and shearing:  ˆ   ˆ12  Fmt   22  L  T  Y  S  2

2

(6)

Matrix crushing under transverse compression and shearing: 2 2 2 C  ˆ  ˆ 22   Y   ˆ12  22 F   T    T   1 C   L   2S   2S  S   Y c m

(7)

In the above equations,  is a coefficient that determines the contribution of the shear stress to the fiber rupture in tension, and the value is set to one in this model. X T , X C , Y T , Y C , S L and S T respectively denote the longitudinal tensile

strength, longitudinal compressive strength, transverse tensile strength, transverse compressive strength, longitudinal shear strength and transverse shear strength. ˆ11 ,

ˆ 22 and ˆ12 are components of the effective stress tensor, where ˆ is used to evaluate the damage initiation criteria and is computed based on the true stress  and damage variables.

0 0 1 (1  d f )    ˆ   0 1 (1  d m ) 0   0 0 1 (1  d s ) 

(8)

where d f , d m and d s are internal damage variables that characterize fiber, matrix, and shear damage. To accurately simulate the degradation of materials, the damage evolution laws corresponding to four damage modes are adopted to capture the mechanical behaviors of materials. Once the any damage criterion in Hashin criteria is satisfied, one value obtained from the corresponding damage evolution law is used to change or degrade the corresponding stiffness value Cd . The damage evolution law combines with the fracture energy method. Hence, the softening behavior of composites is defined by the fracture energy and stiffness degradation-based damage model [45,48]. In the present model, a linear damage evolution is employed which is theoretically described by Figure 8.

Figure 8. Theoretical equivalent stress-equivalent displacement curve. Four types of damage modes follow this equivalent stress-equivalent displacement relation [45,48], where they are respectively defined as follows: Fiber rupture in tension:

 eqft  Lc 11  2  122  eqft 

 11 11    1212  eqft Lc

(9)

(10)

Fiber buckling and kinking in compression:

 eqfc  Lc 11 

(11)

 eqfc 

 11 11   eqfc Lc

(12)

Matrix cracking under transverse tension and shearing:

 eqmt  Lc  22  2  122  eqmt 

 22  22   1212  eqmt Lc

(13)

(14)

Matrix crushing under transverse compression and shearing:

 eqmc  Lc  22  2  122  eqmc 

 22  22    1212  eqmc Lc

(15)

(16)

where Lc is the characteristic element length.  eqft ,  eqfc ,  eqmt and  eqmc are respectively the equivalent displacement of four damage modes.  eqft ,  eqfc ,  eqmt and

 eqmc are respectively the equivalent stress corresponding to the equivalent displacement for four damage modes. 11 ,  22 and 12 are components of the strain tensor.   represents the Macaulay bracket operator, which has the relation of

   (   ) 2 . Once any damage is initiated (i.e.,  eq   eq0 ), the damage variable d i for the corresponding damage mode is given by the following expression [45,48].

di 

 eqf ( eq   eq0 ) (i  ft , fc, mt , mc)  eq ( eqf   eq0 )

ds  1  (1  d ft )(1  d fc )(1  dmt )(1  dmc )

(17)

(18)

where d s is the shear damage variable.  eq0 and  eqf are respectively the equivalent displacement at which the damage is initiated and at which the material is

completely damaged. The values of  eqf for the various damage modes depend on the respective values of dissipation energy G C , which corresponds to the area of the triangle OAC. Thus,  eqf is defined as  eqf  2GC  eq0 . The material parameters used for the numerical simulation are listed in Table 1. Table 1. Numerical material parameters used to simulate the mechanical behaviors of carbon fiber face-sheets (T700/3234) [23,49-50]. Symbol E11

Property Longitudinal modulus

Value 123 GPa

E22

Transverse modulus

8.4 GPa

v12

Poisson’s ratio

0.32

G12 , G13

Shear modulus

4 Gpa

G23

Shear modulus

3 GPa

XT XC YT YC S L , ST 

G Cft

Longitudinal tensile strength Longitudinal compressive strength Transverse tensile strength Transverse compressive strength Shear strength Density Longitudinal tensile fracture energy

2100 MPa 800 MPa 25 MPa 120 MPa 40 MPa 1560 kg/m3 108 kJ/m2

G Cfc

Longitudinal compressive fracture energy

58.4 kJ/m2

C Gmt

Transverse tensile fracture energy

0.504 kJ/m2

C Gmc

Transverse compressive fracture energy

1.1 kJ/m2

4.2.2. Plastic deformation of aluminium core and stiffeners During the flexural loading, the large deformation mainly occurs in the aluminium core and stiffeners. Hence, the aluminium core can absorb amounts of energy through the plastic deformation of metal material. Due to the plastic behaviors of ductile metals, the ductile damage criterion is employed. The ductile criterion is a phenomenological model for predicting the onset of damage due to nucleation, growth, and coalescence of voids [48,51]. The corresponding expression is given as follow:

D  

d

pl

 Dpl ( ,  pl )

1

(19)

where D is a state variable that increases monotonically with the plastic deformation.  Dpl is the equivalent plastic strain at the onset of damage. 

pl

and

 pl are respectively the equivalent plastic strain and the equivalent plastic strain rate.

 is the stress triaxiality. After the damage initiation occurs at a given point, the evolution in the damage is specified in an exponential form as expressed by equation (20) [48].

d metal

 u pl  u pl  y   1  exp    0  Gf   

(20)

In equation (20), u pl  Lc pl . d metal is the damage variable of metal. u pl is the relative plastic displacement.  y is the equivalent yield stress. G f is the fracture energy dissipation. Lc is the characteristic element length. The material stiffness is degraded in terms of d metal . When the materials reach a critical failure value, the failed elements are deleted. The numerical material parameters of metals are listed in Table 2. Table 2. The numerical material parameters of aluminium alloy sheets (2A12-T4) [23,41]. Density

Young’s modulus 70 GPa

2700 kg/m3

Poisson’s ratio 0.3

Yield strength 460 MPa

Fracture energy 10.2 kJ/m2

4.2.3. Interface debonding In the model A, the debonding is considered in the interfaces between adjacent layers of carbon fiber face-sheets and between the core and face-sheets. In the model B, the interface debonding is only considered between the core and face-sheets. The interface debonding behavior is governed by the traction-separation model which is a built-in cohesive contact model [48]. The initial interface debonding is modeled by the quadratic nominal stress criterion having the following expression. 2

2

 tn    ts   tt   0   0   0   tn   ts   tt 

2

(21)

where tn , t s and tt are respectively are the normal traction stress and two shear traction stress. tn0 , t s0 and tt0 are respectively are the normal interface strength and two shear interface strength. After entering the degradation process of interface properties, the interface damage evolution is specified in a linear form as expressed by equation (22) [48,52].

D

 mf ( mmax   m0 )  mmax ( mf   m0 )

(22)

where D is the interface damage variable.  m0 and  mf are respectively the mixed-mode displacement at initial damage and complete damage.  mmax is the maximum value of the effective displacement. Further, the interface damage evolution is dependent on the mix-mode fracture energy which is defined by the Benzeggagh-Kenane (BK) criterion [53]. 

G  G  (G  G )  S   G C (GS  Gs  Gt , GT  Gn  Gs  Gt )  GT  C n

C s

C n

(23)

where Gn , Gs , and Gt denote respectively the work done by the traction stresses and their conjugate separations in the normal, two second shear directions. GnC and GsC denote respectively the critical fracture energies in the normal and second shear

direction.  is a cohesive parameter. The interface parameters are listed in Table 3. Table 3. Numerical material parameters used to simulate the debonding of interfaces between core and face sheets [49]. Symbol Gn Gs , Gt tn0 t s0 , tt0 

Property Normal fracture energy Shear fracture energy Interface normal strength

Value 0.504 kJ/m2 1.566 kJ/m2 54 Mpa

Interface shear strength

70 Mpa

Cohesive parameter

1.45

5. Results and discussion

5.1. Model verification and flexural response analysis To verify the correctness of finite element model, available experimental results reported by He et al. [23] are used to compare with the present simulated results, including flexural load-displacement response features, flexural process and failure modes.

Figure 9. Comparison of three FEM and experimental [23] flexural load-displacement curves of SCS under three-point bending. Three FEMs and experimental flexural load-displacement responses of SCS under three-point bending are compared in Figure 9. Comparison of the flexural damage between simulation and experiment [23] for various stages is shown in Figure 10. From Figure 9, a reasonably good agreement with each other, except the stable load stage, is obtained in which the difference is explained in the following analysis. Typically, the flexural processes are divided into four steps including (i) elastic deflection stage A, (ii) matrix cracking and fiber fracture stage B, (iii) ductile damage and core deformation stage C and (iv) stable core deformation stage D. The overall flexural responses of corrugated core SCS are also similar to others reported by He et al. [23], Yu et al. [54], Jin et al. [55], etc. (i) At the stage of A, the experimental curve and simulated curves rapidly rise with little difference until the maximum load is reached. From Figure 10a, there is not damage recorded either by the experiment or the simulation. Thus, this stage is regarded as an elastic deformation process. During this process,

the SCS has relatively high structural stiffness, but the stiffness slightly decreases due to minor local damage. (ii) At the stage of B, the experimental and simulated curves reach the peak load. According to their values, there are only slight differences among experiment and simulation. According to Yu et al. [54], the delamination and interface debonding actually occur before the damage of the top face-sheet. However, the perfect interface is defined in the model C and thus, the delamination is not considered. Hence, the simulated maximum peak load is slightly higher than the experimental peak load. When reaching the peak load, the matrix cracking and fiber fracture begin to occur in the top face-sheet, which are similarly captured by both the experiment [23] and simulation. During this stage, the load cannot suddenly drop. This is due to the fact that 8-layer plies are not completely subjected to fracture and damaged face-sheets still have the residual strength. However, the top face-sheet continues to be damaged until all plies are damaged. (iii) At the stage of C, the flexural load drops down rapidly to a low value until the core deforms stably. This drop of load is mainly due to the loss of carrying capacity of the top face-sheet. At this moment, the core continues to bear the subsequent load and meanwhile, the damage of the top face-sheet is increasingly aggravated. Due to the ductile core material, the core presents the large plastic deformation and ductile damage which can dissipate large amounts of energy. From Figure 10c, it is found that both the experimental and simulated failure modes exhibit similar plastic deformation of core and a large area of fiber fracture damage. (iv) At the stage of D, the corrugated core gradually enters into a stable plastic deformation stage and energy absorption stage. Hence, the load in this stage almost remains unchanged but with slight fluctuations. By comparing simulated results with experimental results, it is observed that the values of average load are different. For the model A and C, their average loads are similar but both are higher than the experimental average load. The reason is because of some deviations existing in the prediction of interface debonding. It is explained from Figure 11a that the model A only captures little interface

debonding damage. For the model C, a tie constraint is modeled to simulate the perfect bonding of the core and face-sheets though the interface is not considered. Therefore, the strong bonding of the core and face-sheets contributes to overestimated average load. Additionally, it is found from Figure 9 that the model B exhibits underestimated average load as compared with the experimental observations. This is also due to the fact that a large area of debonding damage occurs in the center of the corrugated core, as shown in Figure 11b. Thus, the large interface debonding leads to the structural imperfection.

Figure 10. Comparison of the flexural damage between simulation and experiment [23] for various stages: (a) s=1.0 mm, (b) s=2.7 mm and (c) s=10.0 mm.

Figure 11. Final interface debonding damage between the core and face-sheets for (a) model A and (b) model B (the red areas represent the debonded interfaces). Typical damage morphologies of the top face-sheets for experiment [23] and simulation are shown in Figure 12. From this figure, it is evident that the top

face-sheet exhibits a central crack induced by compression which generates a crack line. The crack damage is similarly captured by both the experiment and simulation. Figure 13 shows the stress distributions of various components in SCS, where the max principal stress and Von Mises stress are given for carbon-fiber face-sheets and corrugated core respectively. It is indicated that for both the top and bottom face-sheet the stress concentrations take place only in the areas of large deflection. Differently, the bottom face-sheet suffers from the tensile stress while the top face-sheet suffers from the compressive stress. For the corrugated core, some local plastic hinges presenting high stress appear in the core, which is induced by the indentation. The phenomenon correlates well with experiment observations in Figure 10c, which correctly identifies complex failure mechanisms such as the compressive damage of face-sheets and plastic deformation of the corrugated core. Overall, the simulated results match very well with the experimental results including the initial structural stiffness, the peak load, average load, damage and deformation modes of the face-sheets and the corrugated core. Particularly, four stages of the flexural response clearly reflect the mechanical characteristic of fiber face-sheet/corrugated core SCS. Hence, the finite element model is validated to be reasonable and reliable for simulating the flexural behaviors of SCS. Further, the model can be effectively used to conduct a qualitative analysis to evaluate the flexural performances of reinforced corrugated core SCS.

Figure 12. Typical damage morphology of the top face-sheets for (a) experiment [23] and (b) model C.

Figure 13. The stress distributions of various components in SCS

5.2. Comparisons of flexural response between with and without a stiffener The qualitative analysis is conducted to verify the reinforcement function of stiffeners based on the reasonable model. To achieve better structural reinforcement design, the reinforcement, coupling deformation and coupling energy-absorption mechanisms and SEA are revealed to deepen the understanding for them. Further, typical parameter studies on the stiffener thickness, position and number are investigated to offer the design thought and experience. 5.2.1. Load-displacement curve Comparison of the load-displacement response between the core with a stiffener and the core without a stiffener is shown in Figure 14. From this figure, it is intuitively illustrated that the core reinforced by a stiffener exhibits much higher post carrying capacity than the core without a stiffener. As analyzed previously, the corrugated core SCS without a stiffener generally experiences four typical stages. However, the reinforced core SCS undergoes different flexural processes which are described as follows. At the initial stage, the core reinforced by a stiffener approximates an elastic rise of load being similar to the core without a stiffener. It is also observed from Figure 14 that both elastic regimes exhibit some oscillations while

the experimental linear-elastic regions are smooth as shown in Figure 9. This is mainly because of the use of the explicit solver to simulate the quasi-static three-point bending behaviors and acceptable different loading rate (or inertia). This difference between simulation and experiment can be also reported in many literatures [23]. After the elastic stage, both curves present large fluctuations due to the large damage of the top face-sheet. However, the curve for the reinforced SCS does not rapidly drop though the fiber fracture and matrix cracking occur at the top face-sheet. This is attributed to the fact that the stiffener plays a reinforced role in the structural load carrying. For the core without a stiffener, the large plastic deformation and damage of the core continue to occur after the damage of face-sheet. At the same time, there is only little metal damage in the core with a stiffener. With further loading of indenter, sudden drops are observed at both curves due to buckling of core. This leads to the formation of plastic hinges. The formation of the first plastic hinge represents a period of transition. After the period of transition, the variations of loads gradually become stable, and cores exhibit the progressive deformation. Differently, it is obvious that the stable load for the core with a stiffener is significantly high than that for the core without a stiffener. Similar to CFRP/metal honeycomb sandwich reinforced by grid [29], the post carrying capacity of SCS can be improved by a reinforced structure due to higher flexural resistance capacity of core. From most studies on non-reinforced corrugated core sandwich structures [54-56], it was found that after the peak load, the drops of stable loads were even more than 50%. However, it is shown from Figure 14 that the reinforced SCS only presented a reduction of about 30% in terms of stable load. This finding also explains the contributions of the present stiffener to the post carrying capacity of SCS.

Figure 14. Comparison of the load-displacement response between the core with a

stiffener and the core without a stiffener.

Figure 15. Deformed processes of SCS (a) without a stiffener and (b) reinforced by a stiffener including s=2.5 mm, s=5 mm, s=7.5 mm and s=10 mm. 5.2.2. Reinforcement and coupling deformation mechanisms According to Figure 14, a great improvement of structural load is disclosed when merely introducing a simple reinforced structure. However, the effect caused by a horizontal stiffener on the deformation process is fairly complicated. To clearly understand the reason for the improvement, the reinforcement mechanism is here studied by analyzing the coupling deformation. Figure 15 compares the deformed processes of SCS with a stiffener and without a stiffener. Figure 16 presents the theoretical analysis of the reinforcement and coupling deformation mechanisms. From Figure 15, it is indicated when s=2.5 mm both of SCS with and without a stiffener exhibit a slight elastic deformation. At this time, the horizontal stiffener does not have a large effect on the deformation of the core. When s=5 mm, the plastic hinges are firstly formed for both of SCS, but the deformation modes of the whole structure behave differently. For the non-stiffener (Figure 15a or 16a), the core presents an outer-convex deformation in the region A where both oblique planes are rotated by an angle of 1 . The outer-convex deformation observed from the present simulation is also consistent with the experimental observation [38]. However, for the stiffener (Figure 15b), the top end of core in the region A exhibits an inner-concave deformation resulting in a vertical crushing, and thereby improving the vertical load

efficiency. According to an experimental phenomenon observed by Rong et al.’s [38], it was shown that the corrugated core under the planar compression loading presented an asymmetric deformation due to micro defeats. This can affect the structural performances. In other word, the deformation of core without a stiffener is either outer-convex or inner-concave when subjected to a vertical loading. Hence, the horizontal stiffener here used in the corrugated core is able to change and unify the deformation of core. From the present simulation, it is observed that two oblique planes really present an inner-concave deformation once they are hindered by the horizontal force offered by the stiffener. In turn, with further loading, the pure core with the inner-concave deformation reacts upon the stiffener. As shown in Figure 16b, obvious bending deformation modes of stiffener occur in the region A and B of stiffener due to different mechanisms. The complex deformation mechanisms including ab  ab and bc  bc are theoretically analyzed in Figure 16b, which can be explained as follows:

Figure 16. Theoretical analysis of the reinforcement and coupling deformation mechanisms, taking s=7.5 mm as an example. (i)

For the region A ( ab  ab ). Under the deflection of  caused by indenter, it is shown that the intersection points of oblique planes and stiffener (i.e., a and

b ) respectively move to a ' and b ' . Both of them approximatively shift a small horizontal displacement of 1 2 . In this case, ab is subjected to the compressive load, thereby resulting in a bending mode as ab . (ii)

For the region B ( bc  bc ). Similarly, the intersection point b and c

respectively move to b ' and c ' by a small horizontal displacement of

2 2 due to the bending of stiffener in the region A. Thus, bc is in fact subjected to the tensile load. Then, it can be seen that the intersection angle between undeformed stiffener and undeformed oblique plane is  0 . After the deformation, the intersection angle between deformed stiffener and deformed oblique plane is supposed to remain unchanged if without any constraint. However, the intersection point b suffers from the tensile constraint of stiffener bc while b moves to b ' . Hence, during the deformation process, the intersection angle decreases by a certain angle  , and a bending mode as

bc occurs in the region B. During the stable deformation stage, there are some damaged areas and potential damaged areas appearing in the reinforced core, which is revealed by Figure 17. From the sectional view, it is illustrated that the damage mainly appears in red areas where the red areas represent the occurrence of ductile damage (DUCTCRT). The presented damage areas of metal core are also similar to the plastic hinge areas presented in Figure 10c [23]. Moreover, it is found from Figure 17 that the areas near six intersection points and other areas represented by the green and yellow present the potential damage.

Figure 17. Ductile damage of aluminium core reinforced by a stiffener during the stable deformation stage.

Figure 18. Comparison of the total energy absorption between the core reinforced by a stiffener and the core without a stiffener. 5.2.3. Coupling energy-absorption mechanisms During the flexural process, the reinforced stiffener can not only absorb a partial of energy but also greatly affect and improve the energy absorption of other components. It is thus deduced that the pure core and face-sheets are interactively coupled with the stiffener in terms of EA. However, the underlying coupling energy absorption mechanisms are not very clear. Figure 18 shows comparison of the total energy absorption between the core reinforced by a stiffener and the core without a stiffener. It is indicated that before s=3 mm, both the SCS exhibit the same energy-absorbing behaviors. After s=3 mm, the difference of EA between the SCS and reinforced SCS is increasingly large. Finally, the total EA of reinforced SCS is 53.14 % higher than that of the SCS when s=10 mm. According to Shi et al. [29], they also found that the stiffener increased the total EA of sandwich by 499%. However, the vertical stiffener they used was not able to obviously improve the EA of other components due to similar deformation. From Figure 16, it is clearly indicated that the stiffener in the present work affects the deformation of other components, which indicates that the EA of other components may be changed. Hence, variations of energy-absorption of other components are analyzed as follows. The internal energy (IE) mainly denotes the energy dissipated by material damage, i.e. fiber fracture and matrix cracking for face-sheets, plastic deformation for metal core and stiffener. The IE is slightly lower than the total EA because other forms of energy dissipation are ignored. From Figure 19, it is shown that there are three

typical relations of energy which can explain the coupling EA. (i)

The relation I (A1
(ii)

The relation II (A2
(iii)

Relation III (A1+A2
Figure 19. Internal energy of various components in the SCS without a stiffener and reinforced by a stiffener (An and Bn represent the value of IE). 5.2.4. Specific energy absorption Total energy absorption, mass and SEA are listed in Table 4. Although the total energy absorption of the reinforced SCS increases, the total mass of structure also increases. To achieve structural light-weight and high EA, the total mass should be considered in the structural design to improve the SEA. To this end, the SEA is analyzed. It is indicated from Table 2 that the EA of reinforced SCS increases by 53.14% as compared with the SCS. However, the introduction of a stiffener leads to a mass increase of 0.039 kg. Despite of this, the reinforced SCS has higher SEA than the SCS, which presents an increase of 20.68% in SEA. This is due to a relatively low increase in mass (26.9%). As compared with Ref. [29], though the stiffener increased

much EA and flexural strength but it led to an increase of 157.8% in mass. Generally, higher mass is not beneficial for structural performances. Hence, the proposed stiffener may be better used for light-weight design and high-efficiency energy absorption design. Table 4. Total energy absorption, mass and SEA. Core

EA (J) △EA

Without stiffener With stiffener

40.16 61.50

+53.14

mtotal (kg) 0.145 0.184

mstiff (kg) 0.039

△mtotal (%) +26.90

SEA (J/g) 0.277 0.334

△SEA (%) +20.58

5.3. Structural parameter studies on the stiffener Different structure parameters of stiffener are speculated to have certain effects on the deformation of corrugated core so that the EA may be affected. Hence, to better achieve light-weight and high-efficiency EA design, structural parameters including the thickness of stiffener, position of stiffener and number of stiffeners are studied to explore the optimal design.

Figure 20. Effects of the thickness of stiffener on (a) the load response and (b) total energy absorption. 5.3.1. Effects of the thickness of stiffeners Figure 20 shows effects of the thickness of stiffener on the load response and total energy absorption. Here, five types of thickness are considered. It is indicated from Figure 20a that during the elastic stage the loads for all thicknesses are similar to each other. However, there are some differences among them during the deformation

of core. It can be seen from Figure 20a and 20b that the average load and EA increase with increasing the thickness to t=2 mm. However, both the values no longer increase when exceeding t=2 mm. For the positive correlation from t=0.25 mm to t=2 mm, it is explained from Figure 21a that the bending deformation of stiffener becomes smaller as the thickness increases. However, the pure cores similarly present an inner-concave plastic deformation mode. Thus, it is inferred that when the core exhibits the inner-concave bending, the stiffener with a smaller bending deformation plays a greater role in the reinforcement. When the thickness further increases such as t=3 mm, both the average load and EA no longer increase but slightly decrease. There are two reasons to explain this variation. The one is that there is almost no deformation occurring in both stiffeners for t=2 mm and t=3 mm. Both of them look like a rigid plate. Thus, it is confirmed that the reinforced capacity of stiffener for t=3 mm is not higher than that of stiffener for t=2 mm. Another one is that despite the similar deformation of stiffener for t=2 mm and t=3 mm, the pure core for t=3 mm exhibits an outer-convex plastic deformation which is opposite to that for t=2 mm. Because the outer-convex mode does not generate a vertical crushing, the core for t=3 mm has lower vertical load efficiency. Hence, the average load and EA for t=3 mm slightly decrease when the thickness exceeds t=2 mm. Figure 21b shows effects of the thickness of stiffener on the ductile damage of core where the red color denotes the ductile damage. It is indicated that the damaged areas and potential damaged areas occur at the intersection points and the pure core above the stiffener. With the increase of thickness, the potential damaged areas near the intersection points decrease gradually. When t≥2 mm, there is not any potential damage in the stiffener. In terms of EA of components, the variation trend is similar to the variation of total EA, which is presented in Figure 22. It is shown that before t=2 mm the IEs of whole core, face sheets and stiffener increase with the increase of stiffener thickness. When t=3 mm, the IEs of whole core and face sheets slightly decrease. However, the IEs of stiffener for t=2 mm and 3 mm are similar due to similar deformation of stiffener.

Figure 21. Effects of the thickness of stiffener on (a) the deformation modes and (b) ductile damage of core.

Figure 22. Effects of the thickness of stiffener on the internal energy of various components in the reinforced core.

Figure 23. Effects of the position of stiffener on (a) the load response and (b) total energy absorption. 5.3.2. Effects of the position of stiffener Figure 23 shows effects of the position of stiffener on the load response and total energy absorption. Here, three types of positions are considered. It is indicated that three positions of stiffener present similar load responses before the fracture of the top face-sheet. Differences among positions A, B and C only appear during the deformation of core. It is found that the similar average load and EA are recorded for the position A and B. However, the fluctuation of curve for the position A is relatively larger during the deformation of core. This results in the structural instability. From Figure 24b, the stiffener placed at the position A also leads to very large areas of potential damage including areas above the stiffener and areas near each intersection point. Due to these dangerous areas and the fluctuation of load, there is a potential threat for the structure. Hence, it is inappropriate to design too high position of stiffener. Oppositely, too low position of stiffener is also undesired such as the position C. It is observed from Figure 24b that the reinforced core for the position C presents larger deformation and damage, providing an insufficient support. Hence, the reinforced core for the position C presents the lowest average load and EA as shown in Figure 23. Effects of the position of stiffener on the internal energy of various components in the reinforced core are shown in Figure 25. It is indicated that when the loading deflection is 10 mm, the IEs of whole core and stiffener decrease with the decrease of position height. However, the IE of face-sheets for any position has only slight differences.

Figure 24. Effects of the position of stiffener on (a) the deformation modes and (b) ductile damage of core.

Figure 25. Effects of the position of stiffener on the internal energy of various components in the reinforced core.

Figure 26. Effects of the number of stiffeners on (a) the load response and (b) total energy absorption.

5.3.3. Effects of the number of stiffeners Figure 26 shows effects of the number of stiffeners on the load response and total energy absorption, which considers one stiffener, two stiffeners and three stiffeners. According to Figure 26a, main differences are recorded for the displacement of 3 mm to the displacement of 10 mm. It can be observed that the average load and EA increase with increasing the number of stiffeners. This may be due to that several stiffeners have interactive effects on the deformation of pure core to reinforce the SCS. However, the post-flexural loads are obviously found to be unstable except one stiffener. The reason for the instability may be attributed to the change of stiffener position. For example, for two stiffeners and three stiffeners, the one is placed at the higher position and the other one is placed at the lower position. As analyzed in the last section, too high position of stiffener affects the stability of load. Thus, it is inferred that the stability of load is vulnerable to the position of the top stiffener. For the bottom stiffeners, they only have little effect on the stability of load due to the slight deformation as shown in Figure 27. Effects of the number of stiffeners on the internal energy of various components in the reinforced core are shown in Figure 28. It is concluded that increasing the number of stiffeners has increasing effects on the IEs of whole core and stiffener. For the face-sheets, the IE is basically unaffected by the number of stiffeners. Although more energy is dissipated by the SCS, the total mass is greatly increased, leading to a very low SEA.

Figure 27. Effects of the number of stiffeners on (a) the deformation modes and (b) ductile damage of core.

Figure 28. Effects of the number of stiffeners on the internal energy of various components in the reinforced core.

5.4. Comprehensive evaluations 5.4.1. SEA and EA The total energy absorption and specific energy absorption of SCS are analyzed to evaluate the energy absorption capacity. Table 5 lists total energy absorption, mass and specific energy absorption for all cases. Table 5. Total energy absorption, mass and specific energy absorption for all cases.

△EA (%) 40.16 Non-stiffener Stiffener with different thicknesses t=0.25 mm 56.31 +40.21 t=0.5 mm 61.50 +53.14 t=1 mm 70.11 +74.58 t=2 mm 75.27 +87.43 t=3 mm 70.95 +76.67 Stiffener with different positions Position A 62.60 +55.88 Position B 61.50 +53.14 Position C 52.38 +30.43 Stiffener with different numbers One stiffener 61.50 +53.14 Two stiffeners 66.49 +65.56 Three stiffeners 77.24 +92.33 Core

EA (J)

mtotal (kg) 0.145

mstiff (kg) 0

△mtotal (%) -

SEA (J/g) 0.277

△SEA (%) -

0.164 0.184 0.223 0.301 0.378

0.019 0.039 0.078 0.156 0.223

+13.10 +26.90 +53.79 +107.59 +153.79

0.343 0.334 0.314 0.250 0.188

+23.83 +20.58 +13.34 -9.74 -32.13

0.184 0.184 0.184

0.039 0.039 0.039

+26.90 +26.90 +26.90

0.340 0.334 0.285

+22.74 +20.58 +2.89

0.184 0.223 0.262

0.039 0.078 0.117

+26.90 +53.79 +80.69

0.334 0.298 0.295

+20.58 +7.58 +6.50

For the stiffener with different thicknesses, it is shown that the EA increases with

increasing the thickness of stiffener but the EA no longer increases for too thick stiffener. However, the SEA decreases with increase of thickness due to the increase of mtotal. Here, the SCS with the thickness of t=0.25 mm has the lightest mtotal, thereby presenting the highest SEA with an increase of 23.83% in all cases. For three positions of stiffener, the increased mtotal is unchanged. The EA and SEA increase when increasing the position height of stiffener. According to previous results, the position A presents larger fluctuation and lower load efficiency than the position B. However, the position B exhibits more stable and progressive load. Thus, the middle position is assumed to be relatively more appropriate. For the stiffener with different numbers, it is presented that increasing the number of stiffeners has an increasing effect on the EA but has a decreasing effect on the SEA. The SCS with three stiffeners absorbs the highest energy with an increase of 92.33%. However, two stiffeners and three stiffeners show the unstable post-flexural load which may cause a potential threat. 5.4.2. Design recommendation The proposed horizontal stiffeners in sandwich structure are similar to stiffened panels, reported by available literatures [57-63]. The stiffened panels have been widely used in the thin-walled structures to reinforce the mechanical performances. They have been applied in various fields such as aerospace (e.g., aircraft wing [57] and aircraft fuselage [58]), automotive (e.g., side wall [59] and roof wall [60]), ship [61], wind turbine blade [62] and tidal turbine blades [63], etc. The horizontal stiffeners in sandwich structure have the same role as all kinds of stiffened panels. Hence, the proposed SCS reinforced by horizontal stiffeners can be also extended to above fields. However, each application needs to satisfy their own specific safety requirements of structures. Hence, some structural design recommendations for this reinforced SCS are given as follows. ● Any horizontal stiffener embedded in the SCS can effectively improve the structural energy absorption. ● If requiring high EA but ignoring the total mass, increasing the number and thickness of stiffener can be firstly considered.

● If both the high EA and load stability need to be simultaneously satisfied, the stiffener can be placed at the middle position of the corrugated core to improve the load efficiency. ● If the structural light-weight and adequately high energy absorption are simultaneously considered in the design, the SEA can be improved by reducing the thickness of stiffener to decrease the total mass.

Conclusions A horizontal stiffener is proposed to efficiently reinforce the flexural performances of corrugated core SCS by FEM. The effectiveness of the adopted model is validated with available experiments for the SCS without a stiffener. Mechanical behaviors of any SCS are clearly reflected by four stages including elastic deflection, matrix cracking/fiber fracture, ductile damage/core deformation and stable deformation. Differently, the SCS with a stiffener exhibits higher flexural strength and EA than that without a stiffener due to different deformation mechanisms. Obviously, the stiffener interacts with the pure core during the deformation, which improves the carrying capacity. As a result, the EA of each component is further affected due to coupling EA mechanisms, and thus, the stiffener not only improve the EA of the pure core but also the EA of face sheets. It is shown that the total EA increases by 53.14%, while the SEA increases by 20.58 % despite of increasing the mtotal. To understand the stiffener-based reinforcement mechanisms, great effects of structural parameters of stiffener are revealed as: i) The SEA increases with decrease of thickness, position height and number; ii) The SCS with the stiffener placed at the middle position has the highest load efficiency and stability; iii) The SCS with 0.25 mm thick stiffener has the lightest mtotal and the highest SEA.

Acknowledgement The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: Authors would like to thank the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (No. 51621004) and the National Natural Science Foundation of

China (No. 11402011). It is also Supported by Hunan Provincial Innovation Foundation For Postgraduate (CX2018B204).

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Conflict of Interest Statement The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.