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Finite element inversion method for interfacial stress analysis of composite single-lap adhesively bonded joint based on full-field deformation
Author’s Accepted Manuscript Finite element inversion method for interfacial stress analysis of composite single-lap adhesively bonded joint based on full-field deformation Ruixiang Bai, Shuanghua Bao, Zhenkun Lei, Cheng Yan, Xiao Han www.elsevier.com/locate/ijadhadh
PII: DOI: Reference:
S0143-7496(17)30214-2 https://doi.org/10.1016/j.ijadhadh.2017.11.011 JAAD2086
To appear in: International Journal of Adhesion and Adhesives Received date: 10 June 2017 Accepted date: 4 November 2017 Cite this article as: Ruixiang Bai, Shuanghua Bao, Zhenkun Lei, Cheng Yan and Xiao Han, Finite element inversion method for interfacial stress analysis of composite single-lap adhesively bonded joint based on full-field deformation, International Journal of Adhesion and Adhesives, https://doi.org/10.1016/j.ijadhadh.2017.11.011 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Finite element inversion method for interfacial stress analysis of composite single-lap adhesively bonded joint based on full-field deformation Ruixiang Bai1, Shuanghua Bao1, Zhenkun Lei1,* , Cheng Yan1,2, Xiao Han1 1 State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, China 2 School of Chemistry, Physics and Mechanical Engineering, Science and Engineering Faculty, Queensland University of Technology, Brisbane, Queensland, Australia *Correspondent email: [email protected]
Abstract: A hybrid inversion method combining the finite element method and full-field displacement data was proposed to analyze the adhesive interface stress at a composite single-lap bonded joint. The displacement field distribution for the outer surface of the overlapped composite plate was measured through the two-dimensional digital image correlation method at different times by performing a tensile test. At the same time, the finite element analysis of the single-lap joint was carried out, which provided the data on structural deformation characteristics. Then, a polynomial interpolation method was used to establish the relation between the internal and external surface displacements. The displacements of the inner surface of the plate can be calculated from the measured displacement data of the outer surface of the plate by using the polynomial function. The displacements of the inner surface of the plate are regarded as the displacement load conditions for the outer and inner surfaces of the plate to establish the finite element model of the adhesive layer. The validity of the hybrid inversion method was proved by FEM analysis and experiments. Keywords: Composites; Finite element stress analysis; Lap-shear; Mechanical properties of adhesives
1. Introduction Fiber-reinforced composites are being widely used in aviation, aerospace, and other engineering fields because of their light weight, high performance, multifunctionality, and low cost. To reduce the weight of structures further, adhesive joint connections are widely adopted in composite structures. Compared to welding, riveting, and other traditional bonding methods, adhesive bonding ensures a more uniform stress distribution at the bonding interface so that the load transfer is continuous in plane. Therefore, it ensures the continuity of the reinforcing fiber and reduces plastic deformation caused by stress concentration at the edge of holes. Many experiments showed that the composite bonding interface is prone to manufacturing defects, and adhesive joints can be easily damaged owing to local stress concentration. Chai [1] says that these defects, along with damage propagation at the bonded interface, are likely to cause the early failure of adhesive joints. Therefore, the reliability of the bonding interface influences the overall mechanical properties of a composite structure, and it must be ensured that the bonding interface is not damaged first. Understanding stress distribution in adhesive joints, particularly near the adhesive interface, is very helpful to design composite adhesive joints for increased load capacity with reasonable accuracy. Many researches have been focus on the stress distribution of the adhesive joints in
theoretical analysis and FEM. Banea et al. [2] and Budhe et al. [3] have reviewed the typical mechanical problems of adhesively bonded joints in composite materials. The main factors affecting the performance of adhesive layer were described, the analytical and numerical methods for analyzing stress in joints were discussed, and several methods for predicting failure adhesive layer were presented. Adams et al. [4] have made great efforts at the aspect of the stress distribution of the adhesive joints by Finite element method (FEM) and experimental test. The results showed that the maximum shear stress in the adhesive layer appeared at the end of the overlapped plate. Many measures have been adopted to reduce the peak shear stress of the adhesive layer and increase the bearing capacity of adhesive joints. Sahoo et al. [5] have presented a non-linear analysis of adhesively bonded lap joints and drew a conclusion that the material non-linearity leads to beneficial effect by decreasing the stress concentration factors at the ends of the lap length. Li et al. [6] have performed a 2-D finite element analysis to determine the stress and strain distributions across the adhesive bond thickness of composite single-lap joints. The results showed that the peak shear and peel stresses increase with the bond thickness and elastic modulus. The oblique treatment is to cut the end of the lapping plate into an oblique angle. da Silva et al. [7] and Oterkus et al. [8] said that when the notch oblique angle changes, the stress singularity at the interface will correspondingly change. In particular, it is possible to design notch angles within a given range so that there is no singularity at the interface. Therefore, the optimization of end geometry can eliminate and reduce the stress concentration at the end of the interface. The experimental test method plays an important role in the establishment of the mechanical model. Nunes [9] investigated the shear deformation of single-lap joints using the digital image correlation (DIC) method, and the shear modulus was estimated. Tsai and Morton [10,11] demonstrated that there was a nonlinear response in aluminum single-lap joints by means of surface strain gauges and Moiré interferometry. Lee et al. [12] also used a surface strain gauge technique to measure the strain of double-lap joints and found that there was a linear relationship between surface strain and load. Cheng et al. [13] detected surface strain of the joint from the change of the electric signal from the piezoelectric material, and demonstrated that appropriate piezoelectric patches can be used to effectively reduce the concentration of peel and shear stress in the adhesive layer. In addition, many scholars have applied the inverse method by combining the static experiment, the optimization method, the full-field optical measurement technique, and FEM models to identify the material properties and fracture parameters. Wang et al. [14] presented a global approach of mechanical identification for composite materials and obtained the elastic constants of the carbon fiber-reinforced composites through a single open-hole tensile test. The identified values were similar to values from classical mechanical tests. Valoroso et al. [15] presented an approach to obtain the mode-I fracture parameters of the cohesive zone model (CZM) through inversion between a double cantilever beam (DCB) test and a FEM numerical model. The methodology is quite easy to implement and requires minor modifications for mixed-mode situations and other specimen geometries. Lecompte et al. [16] obtained four in-plane orthotropic engineering constants for a composite plate through the inverse analysis, combining the DIC method and FEM analysis. The obtained material parameters were very well in agreement with the values obtained by traditional uniaxial tensile tests. Kim et al. [17] established a CZM criterion by field projection inversion using the J-integral and the M-integral, and the numerical experiments
showed that the accuracy of the presented approach decays with increasing mesh size, particularly when the crack-tip cohesive zone has an intricate traction–separation relationship. Kang et al. [18] solved an inverse approximation problem by a combination of finite element method and genetic algorithms for identifying the interfacial parameters. The results indicated that the proposed method can achieve a good prediction with relatively high accuracy. Until now, there is no suitable measurement method for the full-field stress measurement of the adhesive layer inside the single-lap joint. For example, it will cause discontinuities of the adhesive layer by embedding fiber optic sensors in the adhesive layer. Moreover, strain gauge can only measure the point by point in-plane strains of the outer surface of the lapped plates, and it is impossible to detect the debonding failure of the internal adhesive layer directly. Optical measurement methods are used to measure the stress and strain at the structure surface, but it is still very difficult to measure the stress and strain of the inner layer. At present, FEM analysis is still the predominant method to investigate the full-field stress distribution in the adhesive layer at the bonded joint. In the ideal numerical simulation, the input material parameters, the loading method, and the boundary condition are not exactly the same as the actual structure, and various unknown manufacturing defects and damages commonly existing in actual composite structures are always neglected in FEM analysis. This will inevitably lead to a deviation between the predicted results and the actual results. In this paper, the full-field displacement was measured through DIC method, which was interpolated to the FEM model nodes directly by combining the displacement polynomial relationship between the inner and outer surface of the lapped plates, and was converted to the boundary condition of the inverse model of adhesive layer. Finally, the full-field stress of the adhesive layer was obtained inversely. It is a new experimental-numerical hybrid method for measuring adhesive stress indirectly, and the validity has been proved by the work.
2. Hybrid inversion analysis method 2.1 Theoretical analysis For a thin composite plate, according to the Mindlin plate theory considering the first-order shear deformation, the displacement of the laminated plate can be expressed by the displacement in the middle plane as follows u x, y, z u0 x, y z y x, y v x, y, z v0 x, y z x x, y , w x, y, z w0 x, y
(1)
where, (u0, v0, w0) is the displacement in the middle plane along the direction of x, y, z, θx and θy are the rotation angles of the normal plane around x and y axis, respectively. For a thick composite plate, the corresponding relations of displacements between the inner and outer surface of the lapped plate can be obtained in different ways. As shown in Fig. 1, two composite plates in a single-lap joint are bonded together. The stress at the adhesive layer is derived from the relative motion between the plates. The three main stress components are σz, τxz, and τyz, which can be expressed as follows z Ea X z (x, y) (2) yz GaVyz (x, y) , xz GaVxz (x, y)
where Ea and Ga are the elastic modulus and the shear modulus of the adhesive respectively, and Xz, Vyz, and Vxz are three strain components.
Adhesive Layer (a) 4
3 7
8 2
z
y
1 x 6
5 (b)
Fig. 1 Schematic representation of (a) a single-lap adhesively bonded joint, and (b) an eight-node adhesive element of the adhesive layer
An eight-node adhesive element is shown in Fig. 1(b), where Node 1 to Node 4 and Node 5 to Node 8 are located at the top and bottom surfaces of the adhesive layer respectively, corresponding to the four nodes of the top and bottom elements. Then, we have X z w10 w02 ta Vxz u10 u02 ta , Vyz v10 v02 ta
(3)
where superscripts 1 and 2 denote the top and bottom surface of the adhesive layer respectively, ta is the thickness of the adhesive layer, and u, v, and w are the three displacement components along the x-, y-, and z-directions, respectively. With regard to a linear elastic adhesive layer material, the stress calculation of the adhesive layer is sufficiently accurate, as long as the exact displacements of the top and bottom surfaces of the adhesive layer are obtained. 2.2 Hybrid solution The adhesive layer in the single-lap adhesive joint is located between the upper and lower overlapped composite plates. Therefore, it is difficult to measure the stress at the adhesive layer directly. The displacement field and strain field at the outer surfaces of the joint at different times can be obtained from optical measurements, but the displacement field of the upper and lower surfaces of the adhesive layer cannot be measured. To resolve this problem, it is necessary to calculate the displacement field at the upper and lower surfaces of the adhesive layer; this can be calculated from the displacement field at the outer surfaces of the overlapped plates measured via the optical method. Then, the stress field of the adhesive layer can be accurately calculated using the interface stress formula. The hybrid solution is shown in Fig. 2.
Single-lap specimen tension test
FEM simulation of single-lap joint
Load vs. displacement curve obtained
Simulation model and parameter selection Displacement function fitting between the
from tensile test machine Full-field displacement of outer surface of lapped plate measured via DIC
inner and outer surface of the lapped plate from numerical results
Inverse FEM simulation of adhesive layer Full-field displacement of inner surface is calculated from the DIC results of the outer surface via the displacement function obtained from the FEM model of the single-lap joint Displacement of inner surface relocates to the nodes of the adhesive layer FEM model as boundary conditions Adhesive stress is simulated by the adhesive layer FEM model Fig. 2 Hybrid solution for stress analysis of the adhesive layer in single-lap adhesively bonded joint
For explaining the hybrid solution, the tension test of the adhesively bonded joint is presented as an example. The specific steps of the hybrid solution are described as follows. First, the single-lap bonded joint was subjected to a tensile test, and the full-field displacement of the outer surfaces of the composite plates was measured via the DIC optical system. The displacement information of the key points was extracted from the experimental data; thus, the load-displacement curves of these points were obtained. Second, a FEM model of the single-lap adhesively bonded joint was constructed to provide the different node displacements along the thickness direction of the lapped plate, which were fitted into the polynomial functions. Third, the displacement data of the corresponding FEM nodes was extracted from the optical measurement data for the outer surface of the lapped plates. Based on the aforementioned polynomial relationship, the displacement of the relevant nodes on the inner surface of the lapped plates was obtained. Finally, the node displacements of the inner surface of the lapped plates were applied as displacement boundary conditions to the nodes of the FEM inverse model of the adhesive layer, and the stress distribution at the adhesive layer was calculated. For finite element simulations, it is easy to obtain the stress distribution of the adhesive layer. However, it is easy to appear various unknowns manufacturing defects and damages in the actual composite bonded structures, which are always neglected in FE simulation. In addition, the input material parameters, the loading method, and the boundary condition are not exactly the same as the actual structure in the ideal numerical simulation. By comparison, the hybrid method presented in this paper is based on the actual displacement field information obtained by experiment. The nodal displacement in the inner surfaces of the lapped joint is derived by the displacement fitting function. The stress field of the adhesive layer is calculated by applying the nodal displacement to the inverse model. The inverse solution combining with the real experimental data can obtain the adhesive layer stress, which reflects the true stress distribution of the adhesive layer better than the pure finite element simulation. 3. Experiments
3.1 Specimen Bonded plates made of carbon fiber-reinforced epoxy resin composite material (T300/QY8911) were used in the experiment. As shown in Fig. 3, the dimensions of the specimen are L = 100 mm, L0 = 12 mm, L1= 38 mm, B = 25 mm, and t = 0.2 mm, which were determined according to Chinese Standard GB/T 7124. The stacking sequence is [02/902]2S, and the thickness of each layer is 0.125 mm. Table 1 lists the material parameters of the composite laminates. Shear zone Clamping end
B
L0
L1
L Fig. 3 Specimen geometry
Table 1 Material constants of T300/QY8911 Poisson’s ratio
Elastic Parameters（GPa） E1
E2 = E3
G12 = G13
G23
ν12 = ν13
ν23
126
10.7
4.47
3.57
0.33
0.38
25
0.6
20
0.5
Transverse strain Longitudinal strain
0.4
15 10
Strain (%)
Stress (Mpa)
First, the bonding interface of the composite laminates was cleaned with anhydrous alcohol and dried; then, an epoxy resin adhesive was evenly spread over the surface of the treated specimen. The adhesive used in the experiment was brittle epoxy resin adhesive, which was produced by a mixed solution of epoxy resin, curing agent and plastic agent with the ratio of 100: 8: 5 in the Optical Laboratory of the Dalian University of Technology. A stress-strain curve obtained by the experiment is shown in Fig. 4(a), which shows that the tensile property of the epoxy resin adhesive basically conforms to a linear elastic form and the elastic modulus is about 3.81 GPa. Moreover, the curves of transverse strain and longitudinal strain versus the load are shown in Fig. 4(b), and Poisson’s ratio is about 0.48.
E=3.81GPa
5
0.3
v=0.48
0.2 0.1
0
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Strain (%)
0
300
600
900
1200 1500 1800
P (N)
Fig. 4 (a) The stress-strain curve of the epoxy resin and (b) the transverse and longitudinal strains for determining Poisson’s ratio
To achieve a certain thickness of the adhesive layer, two copper wires with a diameter of 0.2 mm were placed on the plate surface along the force direction. After bonding, the specimen was slightly pressed and fixed in order to prevent sliding and subsequently cured.
Then, two glass fiber-reinforced plastic sheets were fixed to the ends of the specimen to serve as the holding ends of the testing machine. The specimen’s surface was painted with artificial speckles on the shear zone to analyze the surface deformation of the lapped plates through the DIC method. 3.2 Experimental setup The experimental setup for the tensile shear test of the composite single-lap joint is shown in Fig. 5. The basic optical setup of the 2D-DIC method is presented in Fig. 5(a), where two cameras were used to collect the speckle images of the front and rear surfaces of the specimen in real time. The camera (Guppy F080b) has an image resolution of 1024 × 768 pixels, and the frame rate is 2 fps. The basic principle of the DIC method is to obtain the displacement vector of the pixel by tracking or matching the position of the same pixel in two speckle images before and after deformation. The 2D-DIC method is suitable for studying full-field deformations in whole loading history.
Holder Light CCD
Specimen
(b)
(a)
Fig. 5(a) Schematic representation of the optical measurement, and (b) experimental setup for tensile shear test
The tensile shear test was performed using a single-arm testing machine (Instron 3345), as shown in Fig. 5(b), and the loading speed was 0.2 mm/min. Prior to loading, the centerline of the specimen was adjusted to coincide with the loading line to avoid the effects of eccentricity on the experiment. The load and displacement curve recorded by the testing machine is shown in Fig. 6. It can be observed that the load increases almost linearly with the displacement.
5
Load (kN)
4 3 2 1 0 0.00
0.05
0.10
0.15
0.20
0.25
0.30
Displacement (mm)
Fig. 6 Curve of load vs. displacement
0.35
4. Results and discussion 4.1 Displacement field The full-field displacement components along x (along the tensile direction) and y (perpendicular to the tensile direction) directions of the outer surface of two overlapped plates under a load of 4 kN were measured via DIC, as shown in Fig. 7. It can be seen that the displacements are continuous. However, local noise can cause local displacement discontinuity, which can be reduced through median filtering. In calculating the full-field displacement of the outer surface of the overlapped plates with DIC, the calculated area is smaller than the actual shear zone of the overlapped plate; therefore, the displacement outside the calculation area needs to be obtained by linear interpolation.
mm 0.0011
mm 0.0523
y-Displacement
-0.0011
0.0516
(a) mm 0.0011
mm 0.0634
y-Displacement
-0.0011
0.0564
(b) Fig. 7 Full-field displacement components of (a) upper and (b) lower plates under a 4 kN load
4.2 FEM analysis of joint A three-dimensional FEM model of a single-lap joint was constructed using the commercial software ABAQUS, as shown in Fig. 8. A right-handed coordinate system is established, and the length and width directions of the plates are defined as x-axis and y-axis, respectively. The adhesive layer was simplified as a homogeneous, isotropic solid with an elastic modulus of E = 3.81 GPa and a Poisson’s ratio of υ = 0.48. The clamping end was fixed, while the other end constrained the normal displacement and was subjected to a uniform displacement load of 1 mm in 100 steps.
(a)
(b)
Fig. 8 (a) FEM model and (b) local mesh of single-lap joint
To model the contact between the composite laminates and the adhesive layer, the tie constraint was employed, which coupled the adhesive layer nodes with their corresponding composite material nodes at the contact surface for each degree of freedom. The laminated plate was divided into a grid every 4 layers in the thickness (z-direction) direction and modeled using solid elements. The adhesive layer was also divided into 4 layers along the thickness direction (z-direction). In addition, the laminate plates and the adhesive layer were meshed using an evenly spaced grid with a spacing of 0.5 mm and were modeled using reduced-integration 8-node linear hexahedral elements (C3D8R). It is worth noting that the two copper lines used in experiment to achieve a certain thickness of the adhesive layer were ignored in the FEM model. It has been verified that the result without considering the two copper wires is very close to that considering the two copper wires, and the error is within 0.6‰.
Fig. 9 X-displacement of lower plate simulated by the FEM joint model
As shown in Fig. 9, the x-displacement at each node of the lower plate was obtained using the FEM simulation under a displacement load of 0.575mm. It can be seen that the x-displacement of the lower plate is uniform. 4.3. Displacement function of overlapped plates along the thickness direction The least-squares fitting polynomial function was derived from the displacement of the plate
in the thickness direction. It can be employed for the calculation of the displacement of the inner surface using the measured displacement of the outer surface of the plate. The calculated displacements of all nodes for each loading step can be obtained by means of numerical simulation. An example of the 60th loading step (the displacement load is 0.575 mm) is shown in Fig. 10. The x and y displacements of five nodes (14984, 35849, 34722, 33595, and 13857 along the thickness direction) were selected and fitted with cubic polynomials. Additionally, the displacement functions relate node 14984 with node 13857, which correspond to the outer and inner surfaces of the plate, respectively. Therefore, the fitting curves for the outer position (Node 14984) can be obtained at different load steps, as shown in Fig. 11.
Upper plate Outer surface 14984 35849 34722 33595 13857 Adhesive layer
y displacement
x displacement
Fig. 10 Polynomial fitting of displacements from five nodes along the thickness direction
(a) z direction
(b) z direction
Fig. 11 Fitted curves for (a) x-displacement and (b) y-displacement at node 14984 of the outer plate surface under different displacement loads
Furthermore, a similar fitting procedure was applied to all the nodes of the outer surface positions of the upper and lower plates so that the displacement functions of the plates at any positions under arbitrary loads were obtained. Therefore, the displacements of the outer surfaces of the overlapped plates, measured through the DIC method, were interpolated to the corresponding FEM model nodes and could be converted to the corresponding nodal displacements on the inner surface of the overlapped plates using the aforementioned displacement functions. This conversion can be automated through a C language program. 4.4 Stress inverse analysis of adhesive layer To calculate the stress of the adhesive layer, an inverse FEM model of the adhesive layer was
established, and the displacement data of the inner surface of the overlapped plates obtained from the above section was used as the boundary condition of the inversion layer model. For example, it was assumed that the 60th loading step (the displacement loading is 0.575 mm) provides the known conditions; the displacement data of the outer surface of the plates was measured through DIC and that of the inner surface of the plates was derived from the aforementioned displacement function. These data were applied to the nodes of the FEM inversion model in the form of displacement loading, as shown in Fig. 12(a). The boundary conditions of this model are compared to that of the previous model in Fig. 8(a). The displacement load on the right end of the joint in the inverse model is replaced by the nodal displacement of the outer and inner surfaces of the upper and lower plates. The main difference between the inverse model and the FE model is the difference of the load application methods. In the inversion model, the nodal displacement field obtained by experiment and the displacement fitting function, including the true displacement information and reflecting the actual deformation of the structure, is applied to each node of the inverse model, which provides a guarantee for the inversion of adhesive stress. In order to obtain the distribution law of displacement along thickness direction, a reference load is applied on the initial FE model. Then, it is used as the basis for calculating the displacement of the inner surface of the lapped plate from the displacement of the outer surface obtained by experiment.
(a)
(b)
(c)
(d) Fig. 12 (a) Inversion model, shear stress S13 distributions of the adhesive layer (b) at the x-y interface (z = 0.2 mm), (c) in the x-y cross-section (half thickness of z = 0.1 mm) and (d) in the x-z cross-section (y = 12.5 mm)
As shown in Figs. 12(b) and 12(c), the shear stress S13 at the interface and in the middle of the adhesive layer obtained using the inverse model has a continuous distribution. As shown in Fig. 13, the shear stress data on the horizontal x line from Figs. 12(b) and 12(c) is compared with the results of the FEM joint model in Fig. 8(a). It can be seen that the two results are mostly identical with an error of less than 1%; hence, the effectiveness of the inverse method is validated. Moreover, the gradient of the shear stress is larger near the edge than in the middle of the adhesive layer. The shear stress S13 in the x-z cross-section (y = 12.5 mm) of the adhesive layer is obtained using the inverse model, as shown in Fig. 12(d). It indicates that the shear stress remains essentially uniform except a few adhesive thicknesses from the adherend, where the stress gradient
is high. Crocombe and Adams [19] gave a possible explanation for this stress variation across the adhesive thickness near the adherend is the discontinuity caused by the unloaded adherend. In the process of actual bearing of the structure, the edge of the adhesive layer is destroyed first and the crack extends to the middle of adhesive layer gradually, leading to structural failure eventually. Bogy [20] derived the characteristic equation of stress singularity at the end of bi-material with arbitrary bonding angles by means of Mellin transform. The shear stress S13 curves along x direction (y = 12.5mm) are extracted at the x-y interface of the adhesive layer, as shown in Fig. 13. It can be seen that the shear stress gradually decreases from the outer surface to the inner surface of the plate; it is asymmetric at the edges because solid elements are used in the adhesive layer, and the distribution is anti-symmetrical in the thickness direction. The shear stress concentration effect is clearly observed near the adhesive interface at the adherend.
0 -20
S13 (MPa)
-40 -60 -80 Inverse result at the interface Inverse relslt in the middle FEM result at the interface FEM relslt in the middle
-100 -120 0
2
4
6
8
10
12
x (mm)
Fig. 13 Shear stress distributions at the interface and in the middle of the adhesive layer obtained by the inverse model and FEM model
The real-time deformation characterization and damage evolution behavior of composite bonding structures is of great importance to the assessment of their safety and reliability. The further goal of this work is the application of the hybrid solution to defect detection and parameter identification of adhesive layer in single-lapped composite joint. In the previous inverse model, the adhesive layer was modeled by solid element, without considering crack initiation and expansion. Therefore, the next work is to obtain the damage degree inversely through the displacement field of the outer surface of the lapped plates measured by DIC method.
5. Conclusions In this paper, an inverse hybrid method combining the optical measurement and FEM numerical analysis was proposed to obtain the stress distribution at the adhesive layer in a composite single-lap adhesively bonded joint. The results showed that the displacement of the outer and inner surfaces of composite joint plates can be fitted well using a cubic polynomial, and there is no limit regarding the thickness of the plate. The displacement field of the outer surface of the plates measured through DIC can be converted to the displacement of the inner surface of the plates using the cubic polynomials. The displacements of the outer and inner surfaces of composite joint plates were applied to the inverse model of the adhesive layer as the surface
displacement loading condition, and the shear stress was calculated inversely. The error was less than 1%, and the effectiveness of the inverse method was validated.
Acknowledgments The authors thank the National Basic Research Program of China (2014CB046506), the National Natural Science Foundation of China (Nos. 11572070, 11472070, 11772081), the Natural Science Foundation of Liaoning Province of China (No. 2015020145).
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PII: DOI: Reference:
S0143-7496(17)30214-2 https://doi.org/10.1016/j.ijadhadh.2017.11.011 JAAD2086
To appear in: International Journal of Adhesion and Adhesives Received date: 10 June 2017 Accepted date: 4 November 2017 Cite this article as: Ruixiang Bai, Shuanghua Bao, Zhenkun Lei, Cheng Yan and Xiao Han, Finite element inversion method for interfacial stress analysis of composite single-lap adhesively bonded joint based on full-field deformation, International Journal of Adhesion and Adhesives, https://doi.org/10.1016/j.ijadhadh.2017.11.011 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Finite element inversion method for interfacial stress analysis of composite single-lap adhesively bonded joint based on full-field deformation Ruixiang Bai1, Shuanghua Bao1, Zhenkun Lei1,* , Cheng Yan1,2, Xiao Han1 1 State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, China 2 School of Chemistry, Physics and Mechanical Engineering, Science and Engineering Faculty, Queensland University of Technology, Brisbane, Queensland, Australia *Correspondent email: [email protected]
Abstract: A hybrid inversion method combining the finite element method and full-field displacement data was proposed to analyze the adhesive interface stress at a composite single-lap bonded joint. The displacement field distribution for the outer surface of the overlapped composite plate was measured through the two-dimensional digital image correlation method at different times by performing a tensile test. At the same time, the finite element analysis of the single-lap joint was carried out, which provided the data on structural deformation characteristics. Then, a polynomial interpolation method was used to establish the relation between the internal and external surface displacements. The displacements of the inner surface of the plate can be calculated from the measured displacement data of the outer surface of the plate by using the polynomial function. The displacements of the inner surface of the plate are regarded as the displacement load conditions for the outer and inner surfaces of the plate to establish the finite element model of the adhesive layer. The validity of the hybrid inversion method was proved by FEM analysis and experiments. Keywords: Composites; Finite element stress analysis; Lap-shear; Mechanical properties of adhesives
1. Introduction Fiber-reinforced composites are being widely used in aviation, aerospace, and other engineering fields because of their light weight, high performance, multifunctionality, and low cost. To reduce the weight of structures further, adhesive joint connections are widely adopted in composite structures. Compared to welding, riveting, and other traditional bonding methods, adhesive bonding ensures a more uniform stress distribution at the bonding interface so that the load transfer is continuous in plane. Therefore, it ensures the continuity of the reinforcing fiber and reduces plastic deformation caused by stress concentration at the edge of holes. Many experiments showed that the composite bonding interface is prone to manufacturing defects, and adhesive joints can be easily damaged owing to local stress concentration. Chai [1] says that these defects, along with damage propagation at the bonded interface, are likely to cause the early failure of adhesive joints. Therefore, the reliability of the bonding interface influences the overall mechanical properties of a composite structure, and it must be ensured that the bonding interface is not damaged first. Understanding stress distribution in adhesive joints, particularly near the adhesive interface, is very helpful to design composite adhesive joints for increased load capacity with reasonable accuracy. Many researches have been focus on the stress distribution of the adhesive joints in
theoretical analysis and FEM. Banea et al. [2] and Budhe et al. [3] have reviewed the typical mechanical problems of adhesively bonded joints in composite materials. The main factors affecting the performance of adhesive layer were described, the analytical and numerical methods for analyzing stress in joints were discussed, and several methods for predicting failure adhesive layer were presented. Adams et al. [4] have made great efforts at the aspect of the stress distribution of the adhesive joints by Finite element method (FEM) and experimental test. The results showed that the maximum shear stress in the adhesive layer appeared at the end of the overlapped plate. Many measures have been adopted to reduce the peak shear stress of the adhesive layer and increase the bearing capacity of adhesive joints. Sahoo et al. [5] have presented a non-linear analysis of adhesively bonded lap joints and drew a conclusion that the material non-linearity leads to beneficial effect by decreasing the stress concentration factors at the ends of the lap length. Li et al. [6] have performed a 2-D finite element analysis to determine the stress and strain distributions across the adhesive bond thickness of composite single-lap joints. The results showed that the peak shear and peel stresses increase with the bond thickness and elastic modulus. The oblique treatment is to cut the end of the lapping plate into an oblique angle. da Silva et al. [7] and Oterkus et al. [8] said that when the notch oblique angle changes, the stress singularity at the interface will correspondingly change. In particular, it is possible to design notch angles within a given range so that there is no singularity at the interface. Therefore, the optimization of end geometry can eliminate and reduce the stress concentration at the end of the interface. The experimental test method plays an important role in the establishment of the mechanical model. Nunes [9] investigated the shear deformation of single-lap joints using the digital image correlation (DIC) method, and the shear modulus was estimated. Tsai and Morton [10,11] demonstrated that there was a nonlinear response in aluminum single-lap joints by means of surface strain gauges and Moiré interferometry. Lee et al. [12] also used a surface strain gauge technique to measure the strain of double-lap joints and found that there was a linear relationship between surface strain and load. Cheng et al. [13] detected surface strain of the joint from the change of the electric signal from the piezoelectric material, and demonstrated that appropriate piezoelectric patches can be used to effectively reduce the concentration of peel and shear stress in the adhesive layer. In addition, many scholars have applied the inverse method by combining the static experiment, the optimization method, the full-field optical measurement technique, and FEM models to identify the material properties and fracture parameters. Wang et al. [14] presented a global approach of mechanical identification for composite materials and obtained the elastic constants of the carbon fiber-reinforced composites through a single open-hole tensile test. The identified values were similar to values from classical mechanical tests. Valoroso et al. [15] presented an approach to obtain the mode-I fracture parameters of the cohesive zone model (CZM) through inversion between a double cantilever beam (DCB) test and a FEM numerical model. The methodology is quite easy to implement and requires minor modifications for mixed-mode situations and other specimen geometries. Lecompte et al. [16] obtained four in-plane orthotropic engineering constants for a composite plate through the inverse analysis, combining the DIC method and FEM analysis. The obtained material parameters were very well in agreement with the values obtained by traditional uniaxial tensile tests. Kim et al. [17] established a CZM criterion by field projection inversion using the J-integral and the M-integral, and the numerical experiments
showed that the accuracy of the presented approach decays with increasing mesh size, particularly when the crack-tip cohesive zone has an intricate traction–separation relationship. Kang et al. [18] solved an inverse approximation problem by a combination of finite element method and genetic algorithms for identifying the interfacial parameters. The results indicated that the proposed method can achieve a good prediction with relatively high accuracy. Until now, there is no suitable measurement method for the full-field stress measurement of the adhesive layer inside the single-lap joint. For example, it will cause discontinuities of the adhesive layer by embedding fiber optic sensors in the adhesive layer. Moreover, strain gauge can only measure the point by point in-plane strains of the outer surface of the lapped plates, and it is impossible to detect the debonding failure of the internal adhesive layer directly. Optical measurement methods are used to measure the stress and strain at the structure surface, but it is still very difficult to measure the stress and strain of the inner layer. At present, FEM analysis is still the predominant method to investigate the full-field stress distribution in the adhesive layer at the bonded joint. In the ideal numerical simulation, the input material parameters, the loading method, and the boundary condition are not exactly the same as the actual structure, and various unknown manufacturing defects and damages commonly existing in actual composite structures are always neglected in FEM analysis. This will inevitably lead to a deviation between the predicted results and the actual results. In this paper, the full-field displacement was measured through DIC method, which was interpolated to the FEM model nodes directly by combining the displacement polynomial relationship between the inner and outer surface of the lapped plates, and was converted to the boundary condition of the inverse model of adhesive layer. Finally, the full-field stress of the adhesive layer was obtained inversely. It is a new experimental-numerical hybrid method for measuring adhesive stress indirectly, and the validity has been proved by the work.
2. Hybrid inversion analysis method 2.1 Theoretical analysis For a thin composite plate, according to the Mindlin plate theory considering the first-order shear deformation, the displacement of the laminated plate can be expressed by the displacement in the middle plane as follows u x, y, z u0 x, y z y x, y v x, y, z v0 x, y z x x, y , w x, y, z w0 x, y
(1)
where, (u0, v0, w0) is the displacement in the middle plane along the direction of x, y, z, θx and θy are the rotation angles of the normal plane around x and y axis, respectively. For a thick composite plate, the corresponding relations of displacements between the inner and outer surface of the lapped plate can be obtained in different ways. As shown in Fig. 1, two composite plates in a single-lap joint are bonded together. The stress at the adhesive layer is derived from the relative motion between the plates. The three main stress components are σz, τxz, and τyz, which can be expressed as follows z Ea X z (x, y) (2) yz GaVyz (x, y) , xz GaVxz (x, y)
where Ea and Ga are the elastic modulus and the shear modulus of the adhesive respectively, and Xz, Vyz, and Vxz are three strain components.
Adhesive Layer (a) 4
3 7
8 2
z
y
1 x 6
5 (b)
Fig. 1 Schematic representation of (a) a single-lap adhesively bonded joint, and (b) an eight-node adhesive element of the adhesive layer
An eight-node adhesive element is shown in Fig. 1(b), where Node 1 to Node 4 and Node 5 to Node 8 are located at the top and bottom surfaces of the adhesive layer respectively, corresponding to the four nodes of the top and bottom elements. Then, we have X z w10 w02 ta Vxz u10 u02 ta , Vyz v10 v02 ta
(3)
where superscripts 1 and 2 denote the top and bottom surface of the adhesive layer respectively, ta is the thickness of the adhesive layer, and u, v, and w are the three displacement components along the x-, y-, and z-directions, respectively. With regard to a linear elastic adhesive layer material, the stress calculation of the adhesive layer is sufficiently accurate, as long as the exact displacements of the top and bottom surfaces of the adhesive layer are obtained. 2.2 Hybrid solution The adhesive layer in the single-lap adhesive joint is located between the upper and lower overlapped composite plates. Therefore, it is difficult to measure the stress at the adhesive layer directly. The displacement field and strain field at the outer surfaces of the joint at different times can be obtained from optical measurements, but the displacement field of the upper and lower surfaces of the adhesive layer cannot be measured. To resolve this problem, it is necessary to calculate the displacement field at the upper and lower surfaces of the adhesive layer; this can be calculated from the displacement field at the outer surfaces of the overlapped plates measured via the optical method. Then, the stress field of the adhesive layer can be accurately calculated using the interface stress formula. The hybrid solution is shown in Fig. 2.
Single-lap specimen tension test
FEM simulation of single-lap joint
Load vs. displacement curve obtained
Simulation model and parameter selection Displacement function fitting between the
from tensile test machine Full-field displacement of outer surface of lapped plate measured via DIC
inner and outer surface of the lapped plate from numerical results
Inverse FEM simulation of adhesive layer Full-field displacement of inner surface is calculated from the DIC results of the outer surface via the displacement function obtained from the FEM model of the single-lap joint Displacement of inner surface relocates to the nodes of the adhesive layer FEM model as boundary conditions Adhesive stress is simulated by the adhesive layer FEM model Fig. 2 Hybrid solution for stress analysis of the adhesive layer in single-lap adhesively bonded joint
For explaining the hybrid solution, the tension test of the adhesively bonded joint is presented as an example. The specific steps of the hybrid solution are described as follows. First, the single-lap bonded joint was subjected to a tensile test, and the full-field displacement of the outer surfaces of the composite plates was measured via the DIC optical system. The displacement information of the key points was extracted from the experimental data; thus, the load-displacement curves of these points were obtained. Second, a FEM model of the single-lap adhesively bonded joint was constructed to provide the different node displacements along the thickness direction of the lapped plate, which were fitted into the polynomial functions. Third, the displacement data of the corresponding FEM nodes was extracted from the optical measurement data for the outer surface of the lapped plates. Based on the aforementioned polynomial relationship, the displacement of the relevant nodes on the inner surface of the lapped plates was obtained. Finally, the node displacements of the inner surface of the lapped plates were applied as displacement boundary conditions to the nodes of the FEM inverse model of the adhesive layer, and the stress distribution at the adhesive layer was calculated. For finite element simulations, it is easy to obtain the stress distribution of the adhesive layer. However, it is easy to appear various unknowns manufacturing defects and damages in the actual composite bonded structures, which are always neglected in FE simulation. In addition, the input material parameters, the loading method, and the boundary condition are not exactly the same as the actual structure in the ideal numerical simulation. By comparison, the hybrid method presented in this paper is based on the actual displacement field information obtained by experiment. The nodal displacement in the inner surfaces of the lapped joint is derived by the displacement fitting function. The stress field of the adhesive layer is calculated by applying the nodal displacement to the inverse model. The inverse solution combining with the real experimental data can obtain the adhesive layer stress, which reflects the true stress distribution of the adhesive layer better than the pure finite element simulation. 3. Experiments
3.1 Specimen Bonded plates made of carbon fiber-reinforced epoxy resin composite material (T300/QY8911) were used in the experiment. As shown in Fig. 3, the dimensions of the specimen are L = 100 mm, L0 = 12 mm, L1= 38 mm, B = 25 mm, and t = 0.2 mm, which were determined according to Chinese Standard GB/T 7124. The stacking sequence is [02/902]2S, and the thickness of each layer is 0.125 mm. Table 1 lists the material parameters of the composite laminates. Shear zone Clamping end
B
L0
L1
L Fig. 3 Specimen geometry
Table 1 Material constants of T300/QY8911 Poisson’s ratio
Elastic Parameters（GPa） E1
E2 = E3
G12 = G13
G23
ν12 = ν13
ν23
126
10.7
4.47
3.57
0.33
0.38
25
0.6
20
0.5
Transverse strain Longitudinal strain
0.4
15 10
Strain (%)
Stress (Mpa)
First, the bonding interface of the composite laminates was cleaned with anhydrous alcohol and dried; then, an epoxy resin adhesive was evenly spread over the surface of the treated specimen. The adhesive used in the experiment was brittle epoxy resin adhesive, which was produced by a mixed solution of epoxy resin, curing agent and plastic agent with the ratio of 100: 8: 5 in the Optical Laboratory of the Dalian University of Technology. A stress-strain curve obtained by the experiment is shown in Fig. 4(a), which shows that the tensile property of the epoxy resin adhesive basically conforms to a linear elastic form and the elastic modulus is about 3.81 GPa. Moreover, the curves of transverse strain and longitudinal strain versus the load are shown in Fig. 4(b), and Poisson’s ratio is about 0.48.
E=3.81GPa
5
0.3
v=0.48
0.2 0.1
0
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Strain (%)
0
300
600
900
1200 1500 1800
P (N)
Fig. 4 (a) The stress-strain curve of the epoxy resin and (b) the transverse and longitudinal strains for determining Poisson’s ratio
To achieve a certain thickness of the adhesive layer, two copper wires with a diameter of 0.2 mm were placed on the plate surface along the force direction. After bonding, the specimen was slightly pressed and fixed in order to prevent sliding and subsequently cured.
Then, two glass fiber-reinforced plastic sheets were fixed to the ends of the specimen to serve as the holding ends of the testing machine. The specimen’s surface was painted with artificial speckles on the shear zone to analyze the surface deformation of the lapped plates through the DIC method. 3.2 Experimental setup The experimental setup for the tensile shear test of the composite single-lap joint is shown in Fig. 5. The basic optical setup of the 2D-DIC method is presented in Fig. 5(a), where two cameras were used to collect the speckle images of the front and rear surfaces of the specimen in real time. The camera (Guppy F080b) has an image resolution of 1024 × 768 pixels, and the frame rate is 2 fps. The basic principle of the DIC method is to obtain the displacement vector of the pixel by tracking or matching the position of the same pixel in two speckle images before and after deformation. The 2D-DIC method is suitable for studying full-field deformations in whole loading history.
Holder Light CCD
Specimen
(b)
(a)
Fig. 5(a) Schematic representation of the optical measurement, and (b) experimental setup for tensile shear test
The tensile shear test was performed using a single-arm testing machine (Instron 3345), as shown in Fig. 5(b), and the loading speed was 0.2 mm/min. Prior to loading, the centerline of the specimen was adjusted to coincide with the loading line to avoid the effects of eccentricity on the experiment. The load and displacement curve recorded by the testing machine is shown in Fig. 6. It can be observed that the load increases almost linearly with the displacement.
5
Load (kN)
4 3 2 1 0 0.00
0.05
0.10
0.15
0.20
0.25
0.30
Displacement (mm)
Fig. 6 Curve of load vs. displacement
0.35
4. Results and discussion 4.1 Displacement field The full-field displacement components along x (along the tensile direction) and y (perpendicular to the tensile direction) directions of the outer surface of two overlapped plates under a load of 4 kN were measured via DIC, as shown in Fig. 7. It can be seen that the displacements are continuous. However, local noise can cause local displacement discontinuity, which can be reduced through median filtering. In calculating the full-field displacement of the outer surface of the overlapped plates with DIC, the calculated area is smaller than the actual shear zone of the overlapped plate; therefore, the displacement outside the calculation area needs to be obtained by linear interpolation.
mm 0.0011
mm 0.0523
y-Displacement
-0.0011
0.0516
(a) mm 0.0011
mm 0.0634
y-Displacement
-0.0011
0.0564
(b) Fig. 7 Full-field displacement components of (a) upper and (b) lower plates under a 4 kN load
4.2 FEM analysis of joint A three-dimensional FEM model of a single-lap joint was constructed using the commercial software ABAQUS, as shown in Fig. 8. A right-handed coordinate system is established, and the length and width directions of the plates are defined as x-axis and y-axis, respectively. The adhesive layer was simplified as a homogeneous, isotropic solid with an elastic modulus of E = 3.81 GPa and a Poisson’s ratio of υ = 0.48. The clamping end was fixed, while the other end constrained the normal displacement and was subjected to a uniform displacement load of 1 mm in 100 steps.
(a)
(b)
Fig. 8 (a) FEM model and (b) local mesh of single-lap joint
To model the contact between the composite laminates and the adhesive layer, the tie constraint was employed, which coupled the adhesive layer nodes with their corresponding composite material nodes at the contact surface for each degree of freedom. The laminated plate was divided into a grid every 4 layers in the thickness (z-direction) direction and modeled using solid elements. The adhesive layer was also divided into 4 layers along the thickness direction (z-direction). In addition, the laminate plates and the adhesive layer were meshed using an evenly spaced grid with a spacing of 0.5 mm and were modeled using reduced-integration 8-node linear hexahedral elements (C3D8R). It is worth noting that the two copper lines used in experiment to achieve a certain thickness of the adhesive layer were ignored in the FEM model. It has been verified that the result without considering the two copper wires is very close to that considering the two copper wires, and the error is within 0.6‰.
Fig. 9 X-displacement of lower plate simulated by the FEM joint model
As shown in Fig. 9, the x-displacement at each node of the lower plate was obtained using the FEM simulation under a displacement load of 0.575mm. It can be seen that the x-displacement of the lower plate is uniform. 4.3. Displacement function of overlapped plates along the thickness direction The least-squares fitting polynomial function was derived from the displacement of the plate
in the thickness direction. It can be employed for the calculation of the displacement of the inner surface using the measured displacement of the outer surface of the plate. The calculated displacements of all nodes for each loading step can be obtained by means of numerical simulation. An example of the 60th loading step (the displacement load is 0.575 mm) is shown in Fig. 10. The x and y displacements of five nodes (14984, 35849, 34722, 33595, and 13857 along the thickness direction) were selected and fitted with cubic polynomials. Additionally, the displacement functions relate node 14984 with node 13857, which correspond to the outer and inner surfaces of the plate, respectively. Therefore, the fitting curves for the outer position (Node 14984) can be obtained at different load steps, as shown in Fig. 11.
Upper plate Outer surface 14984 35849 34722 33595 13857 Adhesive layer
y displacement
x displacement
Fig. 10 Polynomial fitting of displacements from five nodes along the thickness direction
(a) z direction
(b) z direction
Fig. 11 Fitted curves for (a) x-displacement and (b) y-displacement at node 14984 of the outer plate surface under different displacement loads
Furthermore, a similar fitting procedure was applied to all the nodes of the outer surface positions of the upper and lower plates so that the displacement functions of the plates at any positions under arbitrary loads were obtained. Therefore, the displacements of the outer surfaces of the overlapped plates, measured through the DIC method, were interpolated to the corresponding FEM model nodes and could be converted to the corresponding nodal displacements on the inner surface of the overlapped plates using the aforementioned displacement functions. This conversion can be automated through a C language program. 4.4 Stress inverse analysis of adhesive layer To calculate the stress of the adhesive layer, an inverse FEM model of the adhesive layer was
established, and the displacement data of the inner surface of the overlapped plates obtained from the above section was used as the boundary condition of the inversion layer model. For example, it was assumed that the 60th loading step (the displacement loading is 0.575 mm) provides the known conditions; the displacement data of the outer surface of the plates was measured through DIC and that of the inner surface of the plates was derived from the aforementioned displacement function. These data were applied to the nodes of the FEM inversion model in the form of displacement loading, as shown in Fig. 12(a). The boundary conditions of this model are compared to that of the previous model in Fig. 8(a). The displacement load on the right end of the joint in the inverse model is replaced by the nodal displacement of the outer and inner surfaces of the upper and lower plates. The main difference between the inverse model and the FE model is the difference of the load application methods. In the inversion model, the nodal displacement field obtained by experiment and the displacement fitting function, including the true displacement information and reflecting the actual deformation of the structure, is applied to each node of the inverse model, which provides a guarantee for the inversion of adhesive stress. In order to obtain the distribution law of displacement along thickness direction, a reference load is applied on the initial FE model. Then, it is used as the basis for calculating the displacement of the inner surface of the lapped plate from the displacement of the outer surface obtained by experiment.
(a)
(b)
(c)
(d) Fig. 12 (a) Inversion model, shear stress S13 distributions of the adhesive layer (b) at the x-y interface (z = 0.2 mm), (c) in the x-y cross-section (half thickness of z = 0.1 mm) and (d) in the x-z cross-section (y = 12.5 mm)
As shown in Figs. 12(b) and 12(c), the shear stress S13 at the interface and in the middle of the adhesive layer obtained using the inverse model has a continuous distribution. As shown in Fig. 13, the shear stress data on the horizontal x line from Figs. 12(b) and 12(c) is compared with the results of the FEM joint model in Fig. 8(a). It can be seen that the two results are mostly identical with an error of less than 1%; hence, the effectiveness of the inverse method is validated. Moreover, the gradient of the shear stress is larger near the edge than in the middle of the adhesive layer. The shear stress S13 in the x-z cross-section (y = 12.5 mm) of the adhesive layer is obtained using the inverse model, as shown in Fig. 12(d). It indicates that the shear stress remains essentially uniform except a few adhesive thicknesses from the adherend, where the stress gradient
is high. Crocombe and Adams [19] gave a possible explanation for this stress variation across the adhesive thickness near the adherend is the discontinuity caused by the unloaded adherend. In the process of actual bearing of the structure, the edge of the adhesive layer is destroyed first and the crack extends to the middle of adhesive layer gradually, leading to structural failure eventually. Bogy [20] derived the characteristic equation of stress singularity at the end of bi-material with arbitrary bonding angles by means of Mellin transform. The shear stress S13 curves along x direction (y = 12.5mm) are extracted at the x-y interface of the adhesive layer, as shown in Fig. 13. It can be seen that the shear stress gradually decreases from the outer surface to the inner surface of the plate; it is asymmetric at the edges because solid elements are used in the adhesive layer, and the distribution is anti-symmetrical in the thickness direction. The shear stress concentration effect is clearly observed near the adhesive interface at the adherend.
0 -20
S13 (MPa)
-40 -60 -80 Inverse result at the interface Inverse relslt in the middle FEM result at the interface FEM relslt in the middle
-100 -120 0
2
4
6
8
10
12
x (mm)
Fig. 13 Shear stress distributions at the interface and in the middle of the adhesive layer obtained by the inverse model and FEM model
The real-time deformation characterization and damage evolution behavior of composite bonding structures is of great importance to the assessment of their safety and reliability. The further goal of this work is the application of the hybrid solution to defect detection and parameter identification of adhesive layer in single-lapped composite joint. In the previous inverse model, the adhesive layer was modeled by solid element, without considering crack initiation and expansion. Therefore, the next work is to obtain the damage degree inversely through the displacement field of the outer surface of the lapped plates measured by DIC method.
5. Conclusions In this paper, an inverse hybrid method combining the optical measurement and FEM numerical analysis was proposed to obtain the stress distribution at the adhesive layer in a composite single-lap adhesively bonded joint. The results showed that the displacement of the outer and inner surfaces of composite joint plates can be fitted well using a cubic polynomial, and there is no limit regarding the thickness of the plate. The displacement field of the outer surface of the plates measured through DIC can be converted to the displacement of the inner surface of the plates using the cubic polynomials. The displacements of the outer and inner surfaces of composite joint plates were applied to the inverse model of the adhesive layer as the surface
displacement loading condition, and the shear stress was calculated inversely. The error was less than 1%, and the effectiveness of the inverse method was validated.
Acknowledgments The authors thank the National Basic Research Program of China (2014CB046506), the National Natural Science Foundation of China (Nos. 11572070, 11472070, 11772081), the Natural Science Foundation of Liaoning Province of China (No. 2015020145).
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