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Notching and bolting are commonly utilised in connecting fibrereinforced polymer (FRP) laminates. These mechanical methods are usually superior to other connections, particularly when joining thick composite laminates. Stress distributions, damage modes and ultimate strengths in notched or bolted FRP laminate designs are of particular interest to the industrial community. To predict the ultimate strengths and the failure processes of notched or bolted composite laminates, nonlinear progressive damage analyses (PDA) based on the finite element method (FEM) at the mesoscale level are performed in this paper. A threedimensional strength criterion in terms of strains, which can distinguish different damage modes, was developed and adopted in the analysis model to detect damage initiation in the laminates. Different material degradation methods and the influence of cohesive layers were discussed and compared with results of verification experiments. The results showed that the analysis model that used the succinct strength criterion proposed in this paper could properly predict the damage initiation and the ultimate strengths of notched or bolted FRP laminates. The errors between the numerical results and experimental data were small. The material degradation method with continuum damage mechanics (CDM)based exponential damage factors using the damage index as the independent variable achieved greater accuracy and convergence than the method with CDMbased exponential damage factors using the square index as the independent variable or than the method with constant damage factors. Adding cohesive layers in the model had negligible influence on the final results, largely because the succinct analysis model proposed in this paper is sufficiently accurate in cases of small delamination.
Fibrereinforced polymers (FRPs) are composed of highstrength fibres embedded in a polymer matrix. They were first used in military applications approximately 50 years ago [
Connections involving FRP laminate structures will inevitably occur in application. There are three main types of connections: bolting, bonding and clamping [
Although research regarding the failure behaviours and strengths in FRP laminates has existed worldwide for decades, there remains no method that can perfectly predict the failure characteristics of FRP laminates, particularly in complex cases [
Based on previous studies, the threedimensional finite element progressive damage analyses for notched or bolted FRP laminates were performed in this paper. A threedimensional strain strength criterion, similar to that described by Linde [
Damage mechanics (DM) forms the theoretical foundation of progressive damage analysis on solid materials [
Main progressive damage analysis procedure.
A succinct threedimensional strain strength criterion was developed and used in the progressive damage analysis model. This strength criterion is based on research from Linde [
Inequalities of the strength criterion according to different damage modes.
Damage mode  Graphic representation  Strength criterion inequality 

Fibre damage  
Matrix damage  
Delamination damage 
As shown in
In this strength criterion, strain is used to evaluate the damage. Compared with other stressbased criteria, this can facilitate the nonlinear iteration calculations because most nonlinear mechanics calculations apply the strain increment for iteration [
According to the assumption of the progressive damage analysis, material damage can be held equivalent to the degradation of corresponding material properties [
To establish the constitutive relation in the damaged orthotropic material, the damage tensor
In the threedimensional case, the damage tensor
Introducing the concept of effective stress
The complementary energy
Based on the assumption of energy equivalence, that is, the form of the complementary energy of the damaged material,
Substituting (3) into (6) results in:
Comparing (4) and (7) gives:
Therefore, the stiffness matrix of the damaged material is:
Finally, at the damaged material points, the constitutive model expressed in terms of stressstrain relation can be written as:
Because the analytical stress solution is usually difficult to obtain, numerical methods are commonly used to solve the stress field [
The specimen studied in case study 1 was a glass fibrereinforced polymer (GFRP) laminate containing a central hole. Its diagrammatic sketch and geometry information are shown in
Diagrammatic sketch and geometry of the GFRP laminate specimen.
The fibre was S2 glass fibre and the polymer was epoxybased resin. The property data of the specimen came from O’Higgins [
Material properties of the specimen (each ply).
Elastic property  Value  Strength property *  Value 


52,000 MPa  1,840 MPa  
8,000 MPa  1,580 MPa  
3,000 MPa  44 MPa  

2,900 MPa  172 MPa  
0.28  39 MPa  

0.34 

32 MPa 
Note:
This specimen was a quasiisotropic laminate with the stacking sequence [45 / 0 / −45/ 90]
Schematic of the simplified calculation model with the fibre orientation coordinate, load conditions and boundary conditions.
Based on the above simplified calculation model, the FE model was established in the general FEM software ABAQUS. The threedimensional solid element C3D8 was adopted, which is an 8node linear brick [
Finite element model of the GFRP laminate specimen with the global coordinate system.
After establishing the material constitutive relation and the finite element model, the damage factors in the damage tensor were determined so as to carry out the degradation of the damaged material. The commonly used forms of the damage factor are classified into two categories: the constant form and the continuum damage mechanics (CDM)based variable form. The latter can also be divided into two classes: the linear form and the exponential form. In this paper, constant damage factors and two types of CDMbased exponential damage factors were adopted in calculations and compared for accuracy and convergence.
As the name implies, the constant damage factors
Specific values of corresponding damage factors.
Damage mode  Damage factor  Value 

Fibre damage  0.99  
Matrix damage  0.95  
Delamination damage  0.95 
In contrast with the constant damage factors, the variable damage factors can change with increasing damage. They are based on the theory of continuum damage mechanics and their values are directly related to the damage degree. Therefore, it is important to evaluate the damage degree. According to the damage mechanics, the damage degree relates to the external load and is positively correlated with the ratio between the applied stress (or strain) and the material strength [
Letting every
Substituting
Solving the quadratic equation, where
Getting the positive quadratic root:
Therefore:
With a clear physical concept, the damage index
Tsai [
Comparison between the damage index
As shown in
Based on the above analysis, the damage index
Assuming that the damage evolution was also controlled by the fracture energy dissipated during material damage [
In the above equations,
Despite the lack of theoretical basis and being evaluated by Tsai as a bad index to use for composite materials, the square index
Substituting
The experimental data for verification came from O’Higgins [
Experimental
Comparison of OHT strengths from numerical and experimental results.
Damage factor  Numerical result  Experimental result  Error 

Constant  385.23 MPa  350 MPa  10.07% 
Exponential (square index)  376.94 MPa  7.70%  
Exponential (damage index)  370.60 MPa  5.89% 
Compared with the degradation method using constant damage factors, the method with the exponential damage factors using the damage index as the independent variable achieved higher accuracy and a smoother stressstrain curve, along with a considerably shorter iteration convergence time. When compared with adopting similar exponential damage factors but using the square index as the independent variable, adopting the exponential damage factors with the damage index yielded a more accurate result in addition to a clear physical meaning.
Because the numerical results from the three degradation methods were similar, and because the method with exponential damage factors that used the damage index generated results closer to the experimental outcomes, all of the following shown numerical simulations are based on this method.
The deformation and damage diagram of the specimen in the final failure state, which is defined as the termination of the numerical calculation, is shown in
Deformation and damage distribution at the specimen in the final failure state with the fibre orientation coordinate.
As shown in
The fibre, matrix and delamination damage evolutions at the laminate specimen are shown in
Illustration of fibre damage evolution at four plies of the second sublaminate in different loading states.
Illustration of matrix damage evolution at four plies of the second sublaminate in different loading states.
Illustration of delamination damage evolution at four plies of the second sublaminate in different loading states.
As shown in
From the illustrations of the three types of damage evolutions above, one can see that the FRP laminate could still bear the increasing load after damage initiation, even in the fibre damage mode case. In the final failure state, the three types of damages were fully extended at the crosssections near the notch. In general, the damage evolutions of the laminate obtained by numerical modelling coincided well with the observation results of the verification experiment.
The specimen studied in this case was the carbon fibrereinforced polymer (CFRP) laminate with a single bolted joint. Its diagrammatic sketch and geometry information are shown in
Diagrammatic sketch and geometry of the CFRP laminate specimen.
The laminate was made of CFRP prepregs from the SGL Group in an autoclave process. The fibre used was the SGL C30 carbon fibre, and the polymer was the epoxybased resin. The specimen had a fibre content of 52% and a stacking sequence of
Material properties of the specimen (each ply).
Elastic property  Value  Strength property *  Value 

116,000 MPa  2,160 MPa  
7,500 MPa  1,900 MPa  
3,000 MPa  80 MPa  
2,800 MPa  210 MPa  
0.3  110 MPa  
0.33  90 MPa 
Note:
The corresponding experiment was conducted in a laboratory at the Technical University of Berlin. Two CFRP laminate specimens were tested. They were clamped at one end and a single bolt joint was positioned at the other end. The details of the specimens and the experimental setup are shown in
As shown in
Part section view of the specimen and the experimental setup.
The experimental model was simplified for the finite element calculations. The simplified calculation model that includes fibre orientation coordinates and load and boundary conditions is shown in
Schematic of the simplified calculation model with fibre orientation coordinates, load conditions and boundary conditions.
Based on the above simplified calculation model, the FE model was created in the general FEM software ABAQUS. Considering both the calculation accuracy and the speed, the steel part of the single bolt joint was simplified to a model including only the bolt and the washer. For both CFRP and steel, a threedimensional solid 8node linear element C3D8 was adopted. Surfacetosurface contact technology with a friction coefficient of 0.1 was adopted to simulate the interaction between the CFRP part and the steel part. To conduct the progressive damage analysis, the same degradation method with the CDMbased exponential damage factors using the damage index as the key independent variable introduced in
Additionally, another similar FE model was established to investigate the influence of cohesive layers. In this model, delamination was considered and controlled by the cohesive layers, which were composed of cohesive elements. The geometrical thickness of these layers was set to 1% of that for the single ply: 0.0025 mm. Therefore, the thickness of CFRP plies was slightly reduced. Specifically, the outer ply and the inner plies were set to 99.5% and 99% of their original thicknesses, respectively.
FE model of the bolted CFRP laminate specimen with the global coordinate system.
The ABAQUS builtin cohesive element was adopted in this paper, which complied with the tractionseparation law. This tractionseparation law before damage initiation can be written as [
In the above equation,
The abovementioned tractionseparation law can be uncoupled or coupled. When uncoupled, only
Some studies have demonstrated that coupling is important in this tractionseparation model [
In this paper, the research focus was on the composite laminate instead of on the cohesive zone. Therefore, cohesive elements with uncoupled tractionseparation law were adopted. Based on (30), the constitutive behaviour of the cohesive elements used can be written as:
The real thicknesses of the cohesive layers
The failure stresses of the cohesive material were also derived from the matrix property,
After damage initiation was reached, the material properties of the cohesive elements would also be exponentially degraded by the ABAQUS builtin exponential damage factors. Mixedmode behaviour was also accounted for during damage evolution. A complete description of the cohesive element used in this paper is given in [
The FE model with cohesive layers is shown in
FE model of the bolted CFRP laminate specimen containing cohesive layers with the global coordinate system.
The numerical and experimental bearing stress–bearing strain results are shown in
Experimental
Comparison of bearing strengths obtained by tests and numerical calculations.
FE model  Numerical result  Test 1 result  Error 1  Test 2 result  Error 2 

Without cohesive layers  509.42 MPa  498.98 MPa  2.09%  540.32 MPa  −5.72% 
With cohesive layers  502.22 MPa  0.65%  −7.05% 
Because the numerical results from these two FE models were similar, and because omitting the cohesive layers would considerably simplify the preprocessing and the postprocessing in the finite element analysis, the following numerical results are based on the FE model without cohesive layers.
The deformation and damage distribution from the numerical calculation in the final failure state are compared with the test results in
Deformation and damage distribution at the specimen in the final failure state with the fibre orientation coordinate.
As shown in
The fibre, matrix and delamination damage evolutions of the laminate are shown in
Illustration of fibre damage evolution at seven plies (sublaminate [−45 / 0 / 45 / 90], [45 / 90] and middle 0° ply) in different loading states.
Illustration of matrix damage evolution at seven plies (sublaminate [−45 / 0 / 45 / 90], [45 / 90] and middle 0° ply) in different loading states.
Illustration of delamination damage evolution at seven plies (sublaminate [−45 / 0 / 45 / 90], [45 / 90] and middle 0° ply) in different loading states.
The above progressive damage analysis has demonstrated that the single bolted CFRP laminate has good ductility. The deformation of the laminate from the initial damage, even in the fibre damage mode, to the final failure was longer than its elastic deformation. The washer played a considerable role in controlling the delamination damage by preventing the delamination from fully extending in an early stage. The ultimate strength and the failure process of the bolted laminate were in good agreement with the results of the verification experiment.
Based on the damage mechanics, a threedimensional FE model was created in the general FEM software ABAQUS to perform the nonlinear progressive damage analyses of notched or bolted FRP laminates. A threedimensional strain strength criterion was proposed and adopted in this model to predict damage initiation in the laminates. A damage tensor constituted by damage factors was applied to degrade material properties after damage had occurred. The results from using various damage factors were compared for accuracy and convergence. These factors included constant damage factors, CDMbased exponential damage factors that used the damage index as the key independent variable and CDMbased exponential damage factors that used the square index. The influence caused by the absence or presence of cohesive layers in the model was also explored. Throughout this paper, the progressive damage analysis of composite materials is presented and the selection of some key parameters in the analysis is discussed. Key conclusions drawn from this work are:
The numerical results based on the threedimensional strain strength criterion proposed in this paper are in good agreement with the experimental results. This shows that this succinct strain strength criterion is suitable for using in industry and research to predict the ultimate strengths of notched or bolted FRP laminates.
The damage tensor used in this paper, which is strictly derived from the basic theory of damage mechanics, consists of damage factors. It is suitable for degrading the FRP material properties according to different damage modes. It was also found that the CDMbased exponential damage factors using the damage index as the independent variable are preferable to the similar CDMbased exponential damage factors using the square index as the independent variable and the conventional constant damage factors. This preferred type of exponential damage factors was controlled by the damage index
Many existing studies demonstrate that the calculation model for composite materials with cohesive layers probably performs better than the model without cohesive layers, particularly in some complex cases. However, in the case of inplane FRP laminate loading, such as notched or bolted FRP laminates under unidirectional tension, the threedimensional strain strength criterion proposed in this paper shows sufficient accuracy. Adding cohesive layers to the model to simulate the delamination damage will considerably increase the amount of work necessary for preprocessing and postprocessing, with limited increase in calculation accuracy.
The authors would like to thank the German SGL Group for its generous donation of materials. We would also like to acknowledge the Technical University of Berlin for its financial support of this research project.
The work presented here was carried out in collaboration between all authors. Yue Liu, Bernd Zwingmann and Mike Schlaich defined the research theme. Yue Liu designed methods and carried out numerical analyses. Yue Liu and Bernd Zwingmann carried out the laboratory verification experiments and analyzed the data. Then the paper was written by Liu Yue and Bernd Zwingmann. Mike Schlaich codesigned experiments, discussed analyses, interpretation and revision.
The authors declare no conflict of interest.