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Experimental method for the determination of material parameters of plasticity models for toughened adhesives
Author’s Accepted Manuscript EXPERIMENTAL METHOD FOR THE DETERMINATION OF MATERIAL PARAMETERS OF PLASTICITY MODELS FOR TOUGHENED ADHESIVES A. Chiminelli, R. Breto, M.A. Jiménez, F. Velasco, J. Abenojar, M.A Martínez www.elsevier.com/locate/ijadhadh
PII: DOI: Reference:
S0143-7496(16)30050-1 http://dx.doi.org/10.1016/j.ijadhadh.2016.03.004 JAAD1807
To appear in: International Journal of Adhesion and Adhesives Received date: 1 June 2015 Accepted date: 20 February 2016 Cite this article as: A. Chiminelli, R. Breto, M.A. Jiménez, F. Velasco, J. Abenojar and M.A Martínez, EXPERIMENTAL METHOD FOR THE DETERMINATION OF MATERIAL PARAMETERS OF PLASTICITY MODELS FOR TOUGHENED ADHESIVES, International Journal of Adhesion and Adhesives, http://dx.doi.org/10.1016/j.ijadhadh.2016.03.004 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.
EXPERIMENTAL METHOD FOR THE DETERMINATION OF MATERIAL PARAMETERS OF PLASTICITY MODELS FOR TOUGHENED ADHESIVES A. Chiminelli, R. Breto, M.A. Jiménez Materials and Components Division, ITAINNOVA, Aragon Institute of Technology, María de Luna 7, 50018, Zaragoza, Spain. F. Velasco, J. Abenojar, M.A.Martínez Materials Science and Engineering Department, Univ. Carlos III de Madrid, Av. Universidad 30, 28911, Leganés, Spain. Abstract Toughened adhesives commonly present complex non-linear behaviours affecting the stress/strain distribution that develops in bonded joints under loads and, consequently, influencing their maximum load capacity. Although they can be related to different phenomena, these non-linearities are generally dominated by the plastic yield of the material. For detailed analysis of adhesive joints, these behaviours need to be adequately characterised through proper material models. In this sense, models that take into account the dependency of the yield surface and the plastic potential with the hydrostatic component of the stress tensor have been widely used with certain success in the last years. The definition of such models usually requires the execution of various experimental tests for the determination of the associated material parameters, constituting a significant cost/effort in terms of characterization. In order to minimize the amount of tests required, the present work is focused on the definition of a sequential/multiaxial test that can be carried out using a single sample giving enough information to completely determine the mentioned parameters. More concretely, the experimental procedure proposed involves torsion and tensile tests that are executed consecutively. The design of the test has been carried out using the finite element method as calculation tool and considering the exponent Drucker-Prager model. The verification of the experimental procedure proposed has been carried out for a toughened epoxy adhesive that has been characterized through conventional tensile and torsion tests in previous works. Keywords Adhesives experimental testing, sequential torsion and tensile tests, toughened adhesives, non-linear behaviour, Drucker-Prager model, Finite Element Method.
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1. Introduction Adhesive joints are increasingly being used due to the advantages that their offer when compared to other joining techniques: they allow the bonding of dissimilar materials, provide more continuous loads transfer across the joint area (in some cases even acting as sealant), and offer resistance to corrosion and environmental degradation, among other aspects. The progress made in the last years in the control and automation of the bonding processes, in new surface treatments, in the improvement of physical-chemical properties of adhesives (from their chemical formulation) and in the understanding of the mechanics of failure, has increased significantly the reliability of the technique. From the point of view of the analysis and design of bonded joints, the finite element method (FEM) constitutes without a doubt one of the most useful, precise and powerful calculation tools. Through a proper definition of the models, the FEM allows the determination of accurate fields of stresses and strains in the adhesive and in the adherents even for complex joint geometries or load histories [1-6]. Moreover, the influence of the geometrical parameters (joint configuration, thicknesses, fillets, etc) on the stress and strain levels in critical regions of the adhesive can be easily evaluated. In this way, different designs can be analysed to optimize the performance and reliability of the joints [7,8]. The definition of the material models clearly constitutes one of the key aspects of the numerical analysis of bonded joints. The prediction capability of the FE simulations and the quality of the numerical results will strongly depend on the ability of the material models to reproduce the real mechanical behaviour of the adhesive and the adherents. Toughened adhesives commonly present complex non-linear behaviours that affect the stress/strain distribution that it is produced along the bond-lines [9]. Consequently, the maximum load capacity of the joints will be partially determined by these behaviours. Although they can be related to different phenomena (for example ‘crazing’ prior to failure), these non-linearities are generally dominated by the plastic yield of the material. In this sense, models that take into account the dependency of the yield surface and the plastic potential with the hydrostatic component of the stress tensor have been widely used with certain success in the last years [10]. Among them, the exponential Drucker-Prager [9,11] model and those based on the Mahnken-Schlimmer yield function [12-14] can be highlighted. The definition of these models usually requires the execution of various experimental tests for the determination of the associated material parameters, usually two of the following testing modes: tensile, compression and shear. Regardless of the characterization tests considered, this implies an effort that sometimes cannot be assumed as part of a detailed study or design of an adhesive joint. Taking this into account, the present work proposes the utilization of a sequential/multiaxial test that can be carried out using a single sample and gives enough information to completely determine the parameters required by the material models previously cited. Particularly, the test is a combination of a torsion and a tensile sequential loading (applied in this order). Controlling the strains generated during the execution of the torsion test (in real time), this allows to perform subsequently the tensile test without changing the sample. In this way, this experimental method allows 2
to reduce the amount of samples required and the total duration of the tests, being this the objective of this research. The analysis of the test has been carried out using the FEM as calculation tool and considering, among the referenced, the exponent Drucker-Prager model. This model is implemented in FEA commercial software packages such as ABAQUS. The verification of the experimental procedure proposed has been carried out for a toughened epoxy adhesive that has been characterized through conventional tensile and torsion tests in a previous work [15]. In this study, the exponent Drucker-Prager model has demonstrated to provide accurate fits for the non-linear response of this material, and for this reason it was also selected in this work. The verification performed is focused on the corroboration of the negligible effect of the torsion test in the subsequent tensile test results if the shear plastic strains are adequately controlled. The toughened adhesive considered in the study is the SPABOND 340 (Gurit, UK, Ltd). This epoxy adhesive is being used in numerous industrial applications, including the manufacture of blades for wind turbines. In general, the procedure proposed can be used for adhesives that can be casted or injected to prepare bulk specimens and on which the bulk properties will be representative of the ones that the adhesive will have in the joints. 2. Exponent Drucker-Prager material model Toughened adhesives are generally characterized by non-linear responses. In these nonlinear zones, the total strains are constituted by the sum of elastic (fully reversible) and plastic strains (non-recoverable), and the stress-strain relations are based on three fundamental concepts: - The yield criterion, which determines the stress required for plastic deformation initiation. - A hardening rule, which specifies the variation of the initial yield condition in the process of continued plastic deformation. - A plastic flow law, showing the variation of the plastic strain in terms of stress. If the plastic potential function is assumed to be equal to the yield function the flow is called associated, but in general the use of associated flow constitutes a simplification. In most cases the behaviour for this type of materials is non-associated [16]. Different proposals exist for the yield criteria and the hardening and plastic flow rules of isotropic materials: Von Mises, Mohr-Coulomb, Raghava, Drucker-Prager, among others. As it has been previously introduced, the models that take into account the effect of the hydrostatic stresses on the plastic yield have demonstrated to be particularly adequate for toughened adhesives, being the linear and exponent Drucker-Prager and those based on the Mahnken-Schlimmer yield function the most accepted. [9-14] Within the yield criteria, the exponent Drucker-Prager model has proven to be the one that best reproduce the behaviour of toughened adhesives [9,11,16,17]. This model, that is sensitive to hydrostatic stress components even for high stress values [16,18], is particularly adequate for the analysis of bonded joints under shear and compressive loads [19]. 3
Specifically, the exponent Drucker-Prager yield criterion is defined by the following expression (eq. 1) [16,20]:
e b T2 3( 1) T m
(1)
Where e is the effective Von Mises stress, b is the exponent parameter, T is the tensile yield stress, m is the hydrostatic component of stress and is the hydrostatic stress sensitivity which relates stresses under uniaxial compression, shear and uniaxial tension. The hydrostatic stress sensitivity can be calculated using different pairs of tests: tensile-shear, compression-shear or tensile-compression [16,20]. Among them, tensile tests are usually selected since it is one of the simplest in terms of both execution and the interpretation of results. On the other hand, shear loads are one of the basic working conditions for bonded joints, so it seems that a shear test would be more appropriate than a compression test to analyze the their mechanical behaviour [1,8]. For this reason, for adhesives, traction-shear tests are usually preferred. Considering these pair of tests, the hydrostatic stress sensitivity can be calculated as indicated in eq. 2:
3 S2
T2
(2)
where S is the shear yield stress. The definition of the hardening rule is given directly by the hardening curve, relating values of stresses and plastic strains p , which can be obtained from a tensile test (eq. 3).
p
(3)
Finally, the flow rule is defined by the plastic flow potential F . Its expression for the exponent Drucker-Prager model is presented in eq. 4.
F
e T tan 2 e2 m tan
(4)
where e is the eccentricity parameter and is the flow parameter. The eccentricity defines the rate at which the function approaches the asymptote (the flow potential tends to a straight line as the eccentricity tends to zero). The flow parameter is the angle of dilation, i.e. the angle of inclination of the failure surface towards the hydrostatic axis, measured in the meridional plane.
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The way in which this material model is defined in the commercial code used in the present work, ABAQUS, is described in the next section.
2.1. Material model FE implementation. Parameters definition. The exponent Drucker-Prager yield criterion is implemented in ABAQUS as a plasticity model [18], and its definition is expressed in the following form (eq. 5): aq b p pt
(5)
where a and b are material parameters independent of plastic strain, pt is the hardening parameter that represents the strength to hydrostatic stress, q is the effective Von Mises stress and p is the hydrostatic component of stress. For polymers, it has been found [9,16] that a value of 2 for the b parameter allows high correlation levels between the model and the experimental data. Taking this value for the parameter b and considering q e and p m , pt and a can be obtained equating the coefficients of (1) and (5). a
1 3 T ( 1)
pt a T2
(6)
(7)
The last parameter needed to define the model is the flow parameter , which defines the plastic flow law and it is obtained from the following expression (eq. 8, non associated flow):
tan
3(1 2 p ) 2(1 p )
(8)
where p is the plastic component of Poisson’s ratio. For the application of this material model in FE simulations, the experimental characterization needed to obtain all these parameters is described in the following section. 2.2. Experimental characterization required. From the yield criterion (equation (1)) and the parameters required by ABAQUS for the model definition (equations (6), (7) and (8)), the final set of data needed to define the exponent Drucker-Prager model consists of:
T , S p
T
,T
Yield stress in tension and shear. p
Plastic component of Poisson’s ratio. Hardening curve 5
These data are additional to the Young’s modulus E and the shear modulus G , required for the definition of the elastic response of the material.
, the
The yield stress in tension T , the plastic component of the Poisson’s ratio
p
hardening curve T , T and the Young’s modulus E can be obtained directly from the tensile test. On the other hand, the shear test allows obtaining the shear modulus G and the yield stress in shear ( S ). In order to determine the plastic component of p
the Poisson’s ratio ( p ), both the longitudinal and transverse strains should be measured during the tensile test. Finally, the hydrostatic stress sensitivity parameter can be calculated from equation (4), with the values of T and S obtained respectively from each of the tests mentioned. 3. Description and analysis of the test As has been previously introduced, the testing procedure proposed is based on a sequential execution of a torsion and a tensile test over a cylindrical sample. The torsion test is proposed as a pure shear characterization, applying the Nadai method [21] to obtain the shear stresses in the external surface of the sample from the measured torque. This test has been used successfully in many studies for the in-bulk shear characterization of adhesives [22, 23]. Within the procedure proposed, the objective during the torsion test is to introduce the minimum amount of plastic strain needed to detect the initiation of the plastic yielding (i.e. to determine the shear yield stress) but ensuring that it will be low enough to have a negligible effect on the subsequent tensile test performed over the same sample. As the shear test is only required to determine the shear modulus G and the shear yield stress S , the test can be stopped just when the plastic yield is initiated. This can be done controlling the strains that are generated during the test in real time, ensuring that a low level of plastic strains have been introduced in the sample when the torsion test is ended (not higher than 0.2%). In addition, it should be noted that the distribution of stresses and strains is not uniform within the test sample in torsion tests, varying along its radius with the maximum values in the outer surface of the cylinder. Then, this test is particularly adequate for the objective of maintaining the samples with a very low level of plastic strains. If the test is stopped when the plastic yield initiates, most of the sample will still be in the elastic regime. The level of plastic strains will be low and they will be located only in the outer surface of the samples. Then, for practical purposes, the specimen will be almost unaffected by the torsion test. In order to verify the testing procedure proposed, the adequacy of the samples geometry, and to check the previous hypothesis quantifying the influence of the plastic strains that are introduced in torsion tests at the initiation of the yielding, a FE model has been developed. The model has been defined considering both geometric and material non-linearities. The simulations were carried out under static considerations 6
using linear elements with reduced integration (C3D8R in Abaqus). The adhesive was modelled as an elasto-plastic material considering the exponent Drucker-Prager model (the one to be feed with the procedure proposed in this work) with the properties and parameters (Table 1) obtained for the SPABOND 340 in the previous work already cited [15].
Figure 1. Hardening curve considered in the FE models obtained from uniaxial tensile tests[15]. Table 1. Material properties and model parameters considered[15].
E (GPa) 2.557
νe 0.4
λ 1.210
a (MPa-1) 0.0387
νp 0.27
Ψ (º) 28
σT(εTp) (FIGURE 1)
The dimensions of the sample considered are shown in figure 2.
Figure 2. Specimen geometry.
In order to simulate the test, a torque has been applied to the sample on its ends. This load has been applied in the model constraining completely all the nodes in one of the ends of the sample and linking all the nodes of the other end through MPCs (multi-point constrains) to reference node on which a rotation angle is applied (Figure 3). This node is constrained in all the rest of its degrees of freedom. 7
Figure 3 shows the distribution of the plastic strains (PE) in the sample obtained from the FE simulations at the beginning of the yielding. As expected, it can be observed in the half section that an important part of the sample remains in the elastic regime. PE23 = PEθZ
R Z
θ Zones where the boundaries and the MPCs are applied
Figure 3. FE model of the specimen. The contour plot shows the shear plastic strains produced in the torsion test.
Considering as yield criteria a value of 0.2 % of shear plastic strains in the external surface, it has been checked that the plastic/total strain energies ratio is lower than 5 %. This is shown in the next curve (Figure 4).
Total strain energy Elastic strain energy
Plastic strain energy
Figure 4. Strain energies during the torsion test.
With this level of plastic strains, the FE simulations predict that a subsequent tensile test (after the torque release) gives a stress-strain curve very similar to the one obtained 8
from samples without a previous torsion test, as shown in Figure 5. These simulations have been stopped when the longitudinal stresses reached a value near the maximum admissible for the adhesive (Figure 1). 45
TENSILE
40
TORSION + TENSILE
35
Stress (MPa)
30
VIRTUAL TESTS - RESULTS OBTAINED FROM FE SIMULATIONS
25
20
15
10
5
0 0
0.005
0.01
0.015
0.02
0.025
Strain (mm/mm)
Figure 5. Stress-strain curves obtained from virtual tensile tests with and without a previous torsion test.
The differences in terms of elastic modulus and yield stress are of approximately 3.2% and 1.6% respectively. As will be seen later, this result has been also corroborated experimentally. In summary, the experimental procedure proposed covers the following steps: 1) Performing the torsion test until a certain shear yield criteria is reached in the sample surface (measurement of the shear plastic strains produced during the test is required). For the yield criteria, values within the range of 0.1 % – 0.2 % of shear plastic strains is recommended. 2) Release of the load of the torsion test (torque). 3) Perform the tensile test over the same sample. The tests can be done sequentially using a biaxial machine, facilitating the testing procedure. The strains can be measured through a biaxial extensometer. The plastic Poisson’s ratio can be determined measuring simultaneously the transverse narrowing through, for example, a video-extensometer [15]. In order to determine the instant when the yield criterion is reached during the torsion test, a code has been developed which allows calculating the plastic strains during this step in real time. The procedure performed by the code is described in the next section. 9
4. Control of plastic strains during the torsion test The calculation procedure implemented in the code developed for the measurement of the shear plastic strain produced during the torsion test is based on the following steps: 1) Determination of the shear modulus applying initially a low torque (fully in the elastic range, Figure 6). The shear modulus is obtained through eq. 9.
G
T L J
(9)
Figure 6. Calculation of the shear modulus (G) in a torsion test.
2) Continue increasing the torque. Utilization of the Nadai method [21] to obtain the instantaneous shear stress in the external surface of the sample from the measured torque. This method is commonly accepted for the determination of the stress–strain relation for rod-shaped cylindrical samples tested in torsion [22-24], and is based on two main assumptions: -
The shear strain is proportional to the distance from the axis of the sample. The shear stress is determined only by the shear strain.
The expression that relates the shear stresses in the external surface of the sample and the torque is (eq. 10):
i
1 2 a3
T i T i 1 i 3 T i i i 1
(10)
Ti Torque at instant i Ti-1 Torque at instant i-1 γi Shear strain measured by the extensometer at instant i γi-1 Shear strain measured by the extensometer at instant i-1 a Sample radius 10
For elasto-plastic materials, the yield starts in the outer surface of the sample, meanwhile the inner part remains in the elastic regime. The elastic and plastic shear strains can be calculated as indicated in steps 3) and 4). The validity of the method to characterize the shear stress-strain curve of different type of materials (in a non rate dependent regime) has been demonstrated in hundreds of works, among them those previously referenced. 3) Calculation of the elastic shear strain from the shear stress (eq. 11).
ei
i
(11)
G
γi e Elastic component of shear strain at instant i 4) Calculation of the plastic shear strains (eq. 12).
ip i ei
(12)
γi p Plastic component of shear strain at instant i After the first step (i.e. once the shear modulus is determined), the code developed can be run as an application during the torsion test that perform the calculations corresponding to steps 2), 3) and 4) sequentially. Then, the program is able to report the values of the plastic shear strains in real time, allowing to stop the test once the yield criteria is reached. After this, the torque can be released and the tensile test can be performed. The main advantage of the procedure proposed relies on the fact that both tests are carried out over the same sample and without releasing it from the clamps of the testing machine. This reduces not only the duration of the characterization process comparing with the standard approaches (i.e. if two tests are executed independently), also reduces the amount of samples needed. Moreover, the torsion test is executed only until obtaining the information required, so its duration is also optimized. The code described has been programmed using LabVIEW (National Instruments Corp.). 5. Experimental verification As mentioned before, the adhesive considered in the study (SPABOND 340) to check the operation of the experimental procedure proposed has been characterized in a previous work [15]. For all the tests carried out, the specimens have been fabricated by casting, using silicone moulds manufactured specifically for this study. Specimens polymerisation have been carried out in oven at 60ºC for 10h.
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The procedure has been implemented following the steps defined in section 3 and using the application previously presented in order to be able to stop the torsion test at the initiation of the plastic yielding. Three samples have been tested with the experimental method described. The tests have been carried out using a biaxial universal testing machine INSTRON available at ITA (MTS Alliance RF100). The torsion tests were performed at 0,1º/min (controlled by angle), being the measured parameters the torque (N·m) and the angular strains (obtained through a biaxial extensometer). These two parameters allow obtaining the shear stress - angular strain test curve. The tensile tests were performed at 1 mm/min (controlled by displacements) being the measured parameters the axial load (N) and the longitudinal strains (obtained through the biaxial extensometer). Both tests have been realized again under controlled temperature and humidity conditions (20-26 ºC and 4555 % humidity, range of temperatures in the laboratory).
Figure 7. Execution of the experimental procedure proposed.
One of the stress-strain curves obtained in the torsion tests is shown below (dashed line, Figure 8). The tests have been stopped once 0.2% of shear plastic strains ( p ) were reached, as it can be seen comparing with the linear-elastic curve (grey continuous line).
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Stress vs Strains - Obtained from Torsion test 18
Plastic strain
p
16 14
Stress (MPa)
12 10 8 6
Total Totalstrains strain
4
Elastic strain e Elastic
2 0 0
0,01 0.01
0,005 0.005
0,015 0.015
0,02 0.02
0,025 0.025
Strain
Figure 8. Shear stress vs. strain curve obtained in one the torsion tests (SPABOND 340) using eqs. 10, 11 and 12.
The tensile strain-stress curves obtained in this study are shown in figure 9 (samples previously tested in torsion, Tor+Tract), compared with some of the curves obtained from conventional tensile tests in the previous work cited (Tract) [15]. The FE curves (MEF) previously presented (Figure 5, Section 3) are also included. 45 40
Stress (MPa)
35 30 25 Tract_1
20
Tract_2 Tract_3
15
Tor+Tract_1 Tor+Tract_2
10
Tor+Tract_3 Tract_MEF
5
Tor+Tract_MEF
0 0
0.01
0.02
0.03
Strain (mm/mm)
Figure 9. Tensile tests with (Tor+Tract, green) and without (Tract, red) a previous torsion test performed until yield point (SPABOND 340).
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Then, the experimental results confirm that the influence of the torsion test in the tensile results is negligible, since the differences are within the deviations range of the tests. This is because the plastic strains introduced by the torsion loads have been kept low enough to avoid significant effects in the subsequent tests. This evidences the feasibility of the experimental methodology proposed for the type of adhesives studied. Finally, table 2 presents the results obtained from the characterization performed following the procedure proposed comparing with the one obtained in the previous work previously referenced. It can be seen that the results are very similar, with differences below the 12.5 %. Table 2. Main material properties and model parameters obtained, compared with the ones obtained in the previous work[15].
E (GPa) νe Previous[15] 2.557 0.4 New procedure proposed 2.59 0.45 Error (%) 1.3 12.5
λ 1.210 1.16 4.1
νp 0.27 0.29 7.4
Ψ (º) 28 26 7.1
6. Conclusions An experimental methodology is proposed to reduce the cost of the characterizations usually required to feed the material models commonly used in detailed FE simulations of adhesives and bonded joints. The methodology consists on a sequential torsion and tensile tests that can be performed using the same specimen. This approach reduces the amount of samples required and the total duration of the tests. The torsion test is performed until the shear yield point is reached in the external surface of the specimen. In these conditions, the level of plastic strains introduced is small enough to not affect the subsequent tensile test. To be able to stop the test at the initiation of the yield, the quantification of the shear plastic strains in real time is needed. For this, a specific calculation code has been developed that is a key tool for the implementation of the methodology presented. Finally, the method has been applied with success to characterize a toughened epoxy adhesive analysed in previous studies through standard testing procedures. The agreement in the tensile stress-strain curves and in the material parameters finally obtained in both characterizations has evidenced the adequacy and feasibility of the methodology proposed to characterize this type of adhesives.
Acknowledgements The authors would like to express their thanks to the Spanish Science and Innovation Ministry for financial support (projects MAT2011-29182-C02-01 and MAT201129182-C02-02).
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Corresponding author Agustín Chiminelli Instituto Tecnológico de Aragón, María de Luna 7-8, 50018, Zaragoza, Spain. Tel: (34) 976 01 1098 Fax: (34) 976 01 1888 e-mail: [email protected]
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PII: DOI: Reference:
S0143-7496(16)30050-1 http://dx.doi.org/10.1016/j.ijadhadh.2016.03.004 JAAD1807
To appear in: International Journal of Adhesion and Adhesives Received date: 1 June 2015 Accepted date: 20 February 2016 Cite this article as: A. Chiminelli, R. Breto, M.A. Jiménez, F. Velasco, J. Abenojar and M.A Martínez, EXPERIMENTAL METHOD FOR THE DETERMINATION OF MATERIAL PARAMETERS OF PLASTICITY MODELS FOR TOUGHENED ADHESIVES, International Journal of Adhesion and Adhesives, http://dx.doi.org/10.1016/j.ijadhadh.2016.03.004 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.
EXPERIMENTAL METHOD FOR THE DETERMINATION OF MATERIAL PARAMETERS OF PLASTICITY MODELS FOR TOUGHENED ADHESIVES A. Chiminelli, R. Breto, M.A. Jiménez Materials and Components Division, ITAINNOVA, Aragon Institute of Technology, María de Luna 7, 50018, Zaragoza, Spain. F. Velasco, J. Abenojar, M.A.Martínez Materials Science and Engineering Department, Univ. Carlos III de Madrid, Av. Universidad 30, 28911, Leganés, Spain. Abstract Toughened adhesives commonly present complex non-linear behaviours affecting the stress/strain distribution that develops in bonded joints under loads and, consequently, influencing their maximum load capacity. Although they can be related to different phenomena, these non-linearities are generally dominated by the plastic yield of the material. For detailed analysis of adhesive joints, these behaviours need to be adequately characterised through proper material models. In this sense, models that take into account the dependency of the yield surface and the plastic potential with the hydrostatic component of the stress tensor have been widely used with certain success in the last years. The definition of such models usually requires the execution of various experimental tests for the determination of the associated material parameters, constituting a significant cost/effort in terms of characterization. In order to minimize the amount of tests required, the present work is focused on the definition of a sequential/multiaxial test that can be carried out using a single sample giving enough information to completely determine the mentioned parameters. More concretely, the experimental procedure proposed involves torsion and tensile tests that are executed consecutively. The design of the test has been carried out using the finite element method as calculation tool and considering the exponent Drucker-Prager model. The verification of the experimental procedure proposed has been carried out for a toughened epoxy adhesive that has been characterized through conventional tensile and torsion tests in previous works. Keywords Adhesives experimental testing, sequential torsion and tensile tests, toughened adhesives, non-linear behaviour, Drucker-Prager model, Finite Element Method.
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1. Introduction Adhesive joints are increasingly being used due to the advantages that their offer when compared to other joining techniques: they allow the bonding of dissimilar materials, provide more continuous loads transfer across the joint area (in some cases even acting as sealant), and offer resistance to corrosion and environmental degradation, among other aspects. The progress made in the last years in the control and automation of the bonding processes, in new surface treatments, in the improvement of physical-chemical properties of adhesives (from their chemical formulation) and in the understanding of the mechanics of failure, has increased significantly the reliability of the technique. From the point of view of the analysis and design of bonded joints, the finite element method (FEM) constitutes without a doubt one of the most useful, precise and powerful calculation tools. Through a proper definition of the models, the FEM allows the determination of accurate fields of stresses and strains in the adhesive and in the adherents even for complex joint geometries or load histories [1-6]. Moreover, the influence of the geometrical parameters (joint configuration, thicknesses, fillets, etc) on the stress and strain levels in critical regions of the adhesive can be easily evaluated. In this way, different designs can be analysed to optimize the performance and reliability of the joints [7,8]. The definition of the material models clearly constitutes one of the key aspects of the numerical analysis of bonded joints. The prediction capability of the FE simulations and the quality of the numerical results will strongly depend on the ability of the material models to reproduce the real mechanical behaviour of the adhesive and the adherents. Toughened adhesives commonly present complex non-linear behaviours that affect the stress/strain distribution that it is produced along the bond-lines [9]. Consequently, the maximum load capacity of the joints will be partially determined by these behaviours. Although they can be related to different phenomena (for example ‘crazing’ prior to failure), these non-linearities are generally dominated by the plastic yield of the material. In this sense, models that take into account the dependency of the yield surface and the plastic potential with the hydrostatic component of the stress tensor have been widely used with certain success in the last years [10]. Among them, the exponential Drucker-Prager [9,11] model and those based on the Mahnken-Schlimmer yield function [12-14] can be highlighted. The definition of these models usually requires the execution of various experimental tests for the determination of the associated material parameters, usually two of the following testing modes: tensile, compression and shear. Regardless of the characterization tests considered, this implies an effort that sometimes cannot be assumed as part of a detailed study or design of an adhesive joint. Taking this into account, the present work proposes the utilization of a sequential/multiaxial test that can be carried out using a single sample and gives enough information to completely determine the parameters required by the material models previously cited. Particularly, the test is a combination of a torsion and a tensile sequential loading (applied in this order). Controlling the strains generated during the execution of the torsion test (in real time), this allows to perform subsequently the tensile test without changing the sample. In this way, this experimental method allows 2
to reduce the amount of samples required and the total duration of the tests, being this the objective of this research. The analysis of the test has been carried out using the FEM as calculation tool and considering, among the referenced, the exponent Drucker-Prager model. This model is implemented in FEA commercial software packages such as ABAQUS. The verification of the experimental procedure proposed has been carried out for a toughened epoxy adhesive that has been characterized through conventional tensile and torsion tests in a previous work [15]. In this study, the exponent Drucker-Prager model has demonstrated to provide accurate fits for the non-linear response of this material, and for this reason it was also selected in this work. The verification performed is focused on the corroboration of the negligible effect of the torsion test in the subsequent tensile test results if the shear plastic strains are adequately controlled. The toughened adhesive considered in the study is the SPABOND 340 (Gurit, UK, Ltd). This epoxy adhesive is being used in numerous industrial applications, including the manufacture of blades for wind turbines. In general, the procedure proposed can be used for adhesives that can be casted or injected to prepare bulk specimens and on which the bulk properties will be representative of the ones that the adhesive will have in the joints. 2. Exponent Drucker-Prager material model Toughened adhesives are generally characterized by non-linear responses. In these nonlinear zones, the total strains are constituted by the sum of elastic (fully reversible) and plastic strains (non-recoverable), and the stress-strain relations are based on three fundamental concepts: - The yield criterion, which determines the stress required for plastic deformation initiation. - A hardening rule, which specifies the variation of the initial yield condition in the process of continued plastic deformation. - A plastic flow law, showing the variation of the plastic strain in terms of stress. If the plastic potential function is assumed to be equal to the yield function the flow is called associated, but in general the use of associated flow constitutes a simplification. In most cases the behaviour for this type of materials is non-associated [16]. Different proposals exist for the yield criteria and the hardening and plastic flow rules of isotropic materials: Von Mises, Mohr-Coulomb, Raghava, Drucker-Prager, among others. As it has been previously introduced, the models that take into account the effect of the hydrostatic stresses on the plastic yield have demonstrated to be particularly adequate for toughened adhesives, being the linear and exponent Drucker-Prager and those based on the Mahnken-Schlimmer yield function the most accepted. [9-14] Within the yield criteria, the exponent Drucker-Prager model has proven to be the one that best reproduce the behaviour of toughened adhesives [9,11,16,17]. This model, that is sensitive to hydrostatic stress components even for high stress values [16,18], is particularly adequate for the analysis of bonded joints under shear and compressive loads [19]. 3
Specifically, the exponent Drucker-Prager yield criterion is defined by the following expression (eq. 1) [16,20]:
e b T2 3( 1) T m
(1)
Where e is the effective Von Mises stress, b is the exponent parameter, T is the tensile yield stress, m is the hydrostatic component of stress and is the hydrostatic stress sensitivity which relates stresses under uniaxial compression, shear and uniaxial tension. The hydrostatic stress sensitivity can be calculated using different pairs of tests: tensile-shear, compression-shear or tensile-compression [16,20]. Among them, tensile tests are usually selected since it is one of the simplest in terms of both execution and the interpretation of results. On the other hand, shear loads are one of the basic working conditions for bonded joints, so it seems that a shear test would be more appropriate than a compression test to analyze the their mechanical behaviour [1,8]. For this reason, for adhesives, traction-shear tests are usually preferred. Considering these pair of tests, the hydrostatic stress sensitivity can be calculated as indicated in eq. 2:
3 S2
T2
(2)
where S is the shear yield stress. The definition of the hardening rule is given directly by the hardening curve, relating values of stresses and plastic strains p , which can be obtained from a tensile test (eq. 3).
p
(3)
Finally, the flow rule is defined by the plastic flow potential F . Its expression for the exponent Drucker-Prager model is presented in eq. 4.
F
e T tan 2 e2 m tan
(4)
where e is the eccentricity parameter and is the flow parameter. The eccentricity defines the rate at which the function approaches the asymptote (the flow potential tends to a straight line as the eccentricity tends to zero). The flow parameter is the angle of dilation, i.e. the angle of inclination of the failure surface towards the hydrostatic axis, measured in the meridional plane.
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The way in which this material model is defined in the commercial code used in the present work, ABAQUS, is described in the next section.
2.1. Material model FE implementation. Parameters definition. The exponent Drucker-Prager yield criterion is implemented in ABAQUS as a plasticity model [18], and its definition is expressed in the following form (eq. 5): aq b p pt
(5)
where a and b are material parameters independent of plastic strain, pt is the hardening parameter that represents the strength to hydrostatic stress, q is the effective Von Mises stress and p is the hydrostatic component of stress. For polymers, it has been found [9,16] that a value of 2 for the b parameter allows high correlation levels between the model and the experimental data. Taking this value for the parameter b and considering q e and p m , pt and a can be obtained equating the coefficients of (1) and (5). a
1 3 T ( 1)
pt a T2
(6)
(7)
The last parameter needed to define the model is the flow parameter , which defines the plastic flow law and it is obtained from the following expression (eq. 8, non associated flow):
tan
3(1 2 p ) 2(1 p )
(8)
where p is the plastic component of Poisson’s ratio. For the application of this material model in FE simulations, the experimental characterization needed to obtain all these parameters is described in the following section. 2.2. Experimental characterization required. From the yield criterion (equation (1)) and the parameters required by ABAQUS for the model definition (equations (6), (7) and (8)), the final set of data needed to define the exponent Drucker-Prager model consists of:
T , S p
T
,T
Yield stress in tension and shear. p
Plastic component of Poisson’s ratio. Hardening curve 5
These data are additional to the Young’s modulus E and the shear modulus G , required for the definition of the elastic response of the material.
, the
The yield stress in tension T , the plastic component of the Poisson’s ratio
p
hardening curve T , T and the Young’s modulus E can be obtained directly from the tensile test. On the other hand, the shear test allows obtaining the shear modulus G and the yield stress in shear ( S ). In order to determine the plastic component of p
the Poisson’s ratio ( p ), both the longitudinal and transverse strains should be measured during the tensile test. Finally, the hydrostatic stress sensitivity parameter can be calculated from equation (4), with the values of T and S obtained respectively from each of the tests mentioned. 3. Description and analysis of the test As has been previously introduced, the testing procedure proposed is based on a sequential execution of a torsion and a tensile test over a cylindrical sample. The torsion test is proposed as a pure shear characterization, applying the Nadai method [21] to obtain the shear stresses in the external surface of the sample from the measured torque. This test has been used successfully in many studies for the in-bulk shear characterization of adhesives [22, 23]. Within the procedure proposed, the objective during the torsion test is to introduce the minimum amount of plastic strain needed to detect the initiation of the plastic yielding (i.e. to determine the shear yield stress) but ensuring that it will be low enough to have a negligible effect on the subsequent tensile test performed over the same sample. As the shear test is only required to determine the shear modulus G and the shear yield stress S , the test can be stopped just when the plastic yield is initiated. This can be done controlling the strains that are generated during the test in real time, ensuring that a low level of plastic strains have been introduced in the sample when the torsion test is ended (not higher than 0.2%). In addition, it should be noted that the distribution of stresses and strains is not uniform within the test sample in torsion tests, varying along its radius with the maximum values in the outer surface of the cylinder. Then, this test is particularly adequate for the objective of maintaining the samples with a very low level of plastic strains. If the test is stopped when the plastic yield initiates, most of the sample will still be in the elastic regime. The level of plastic strains will be low and they will be located only in the outer surface of the samples. Then, for practical purposes, the specimen will be almost unaffected by the torsion test. In order to verify the testing procedure proposed, the adequacy of the samples geometry, and to check the previous hypothesis quantifying the influence of the plastic strains that are introduced in torsion tests at the initiation of the yielding, a FE model has been developed. The model has been defined considering both geometric and material non-linearities. The simulations were carried out under static considerations 6
using linear elements with reduced integration (C3D8R in Abaqus). The adhesive was modelled as an elasto-plastic material considering the exponent Drucker-Prager model (the one to be feed with the procedure proposed in this work) with the properties and parameters (Table 1) obtained for the SPABOND 340 in the previous work already cited [15].
Figure 1. Hardening curve considered in the FE models obtained from uniaxial tensile tests[15]. Table 1. Material properties and model parameters considered[15].
E (GPa) 2.557
νe 0.4
λ 1.210
a (MPa-1) 0.0387
νp 0.27
Ψ (º) 28
σT(εTp) (FIGURE 1)
The dimensions of the sample considered are shown in figure 2.
Figure 2. Specimen geometry.
In order to simulate the test, a torque has been applied to the sample on its ends. This load has been applied in the model constraining completely all the nodes in one of the ends of the sample and linking all the nodes of the other end through MPCs (multi-point constrains) to reference node on which a rotation angle is applied (Figure 3). This node is constrained in all the rest of its degrees of freedom. 7
Figure 3 shows the distribution of the plastic strains (PE) in the sample obtained from the FE simulations at the beginning of the yielding. As expected, it can be observed in the half section that an important part of the sample remains in the elastic regime. PE23 = PEθZ
R Z
θ Zones where the boundaries and the MPCs are applied
Figure 3. FE model of the specimen. The contour plot shows the shear plastic strains produced in the torsion test.
Considering as yield criteria a value of 0.2 % of shear plastic strains in the external surface, it has been checked that the plastic/total strain energies ratio is lower than 5 %. This is shown in the next curve (Figure 4).
Total strain energy Elastic strain energy
Plastic strain energy
Figure 4. Strain energies during the torsion test.
With this level of plastic strains, the FE simulations predict that a subsequent tensile test (after the torque release) gives a stress-strain curve very similar to the one obtained 8
from samples without a previous torsion test, as shown in Figure 5. These simulations have been stopped when the longitudinal stresses reached a value near the maximum admissible for the adhesive (Figure 1). 45
TENSILE
40
TORSION + TENSILE
35
Stress (MPa)
30
VIRTUAL TESTS - RESULTS OBTAINED FROM FE SIMULATIONS
25
20
15
10
5
0 0
0.005
0.01
0.015
0.02
0.025
Strain (mm/mm)
Figure 5. Stress-strain curves obtained from virtual tensile tests with and without a previous torsion test.
The differences in terms of elastic modulus and yield stress are of approximately 3.2% and 1.6% respectively. As will be seen later, this result has been also corroborated experimentally. In summary, the experimental procedure proposed covers the following steps: 1) Performing the torsion test until a certain shear yield criteria is reached in the sample surface (measurement of the shear plastic strains produced during the test is required). For the yield criteria, values within the range of 0.1 % – 0.2 % of shear plastic strains is recommended. 2) Release of the load of the torsion test (torque). 3) Perform the tensile test over the same sample. The tests can be done sequentially using a biaxial machine, facilitating the testing procedure. The strains can be measured through a biaxial extensometer. The plastic Poisson’s ratio can be determined measuring simultaneously the transverse narrowing through, for example, a video-extensometer [15]. In order to determine the instant when the yield criterion is reached during the torsion test, a code has been developed which allows calculating the plastic strains during this step in real time. The procedure performed by the code is described in the next section. 9
4. Control of plastic strains during the torsion test The calculation procedure implemented in the code developed for the measurement of the shear plastic strain produced during the torsion test is based on the following steps: 1) Determination of the shear modulus applying initially a low torque (fully in the elastic range, Figure 6). The shear modulus is obtained through eq. 9.
G
T L J
(9)
Figure 6. Calculation of the shear modulus (G) in a torsion test.
2) Continue increasing the torque. Utilization of the Nadai method [21] to obtain the instantaneous shear stress in the external surface of the sample from the measured torque. This method is commonly accepted for the determination of the stress–strain relation for rod-shaped cylindrical samples tested in torsion [22-24], and is based on two main assumptions: -
The shear strain is proportional to the distance from the axis of the sample. The shear stress is determined only by the shear strain.
The expression that relates the shear stresses in the external surface of the sample and the torque is (eq. 10):
i
1 2 a3
T i T i 1 i 3 T i i i 1
(10)
Ti Torque at instant i Ti-1 Torque at instant i-1 γi Shear strain measured by the extensometer at instant i γi-1 Shear strain measured by the extensometer at instant i-1 a Sample radius 10
For elasto-plastic materials, the yield starts in the outer surface of the sample, meanwhile the inner part remains in the elastic regime. The elastic and plastic shear strains can be calculated as indicated in steps 3) and 4). The validity of the method to characterize the shear stress-strain curve of different type of materials (in a non rate dependent regime) has been demonstrated in hundreds of works, among them those previously referenced. 3) Calculation of the elastic shear strain from the shear stress (eq. 11).
ei
i
(11)
G
γi e Elastic component of shear strain at instant i 4) Calculation of the plastic shear strains (eq. 12).
ip i ei
(12)
γi p Plastic component of shear strain at instant i After the first step (i.e. once the shear modulus is determined), the code developed can be run as an application during the torsion test that perform the calculations corresponding to steps 2), 3) and 4) sequentially. Then, the program is able to report the values of the plastic shear strains in real time, allowing to stop the test once the yield criteria is reached. After this, the torque can be released and the tensile test can be performed. The main advantage of the procedure proposed relies on the fact that both tests are carried out over the same sample and without releasing it from the clamps of the testing machine. This reduces not only the duration of the characterization process comparing with the standard approaches (i.e. if two tests are executed independently), also reduces the amount of samples needed. Moreover, the torsion test is executed only until obtaining the information required, so its duration is also optimized. The code described has been programmed using LabVIEW (National Instruments Corp.). 5. Experimental verification As mentioned before, the adhesive considered in the study (SPABOND 340) to check the operation of the experimental procedure proposed has been characterized in a previous work [15]. For all the tests carried out, the specimens have been fabricated by casting, using silicone moulds manufactured specifically for this study. Specimens polymerisation have been carried out in oven at 60ºC for 10h.
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The procedure has been implemented following the steps defined in section 3 and using the application previously presented in order to be able to stop the torsion test at the initiation of the plastic yielding. Three samples have been tested with the experimental method described. The tests have been carried out using a biaxial universal testing machine INSTRON available at ITA (MTS Alliance RF100). The torsion tests were performed at 0,1º/min (controlled by angle), being the measured parameters the torque (N·m) and the angular strains (obtained through a biaxial extensometer). These two parameters allow obtaining the shear stress - angular strain test curve. The tensile tests were performed at 1 mm/min (controlled by displacements) being the measured parameters the axial load (N) and the longitudinal strains (obtained through the biaxial extensometer). Both tests have been realized again under controlled temperature and humidity conditions (20-26 ºC and 4555 % humidity, range of temperatures in the laboratory).
Figure 7. Execution of the experimental procedure proposed.
One of the stress-strain curves obtained in the torsion tests is shown below (dashed line, Figure 8). The tests have been stopped once 0.2% of shear plastic strains ( p ) were reached, as it can be seen comparing with the linear-elastic curve (grey continuous line).
12
Stress vs Strains - Obtained from Torsion test 18
Plastic strain
p
16 14
Stress (MPa)
12 10 8 6
Total Totalstrains strain
4
Elastic strain e Elastic
2 0 0
0,01 0.01
0,005 0.005
0,015 0.015
0,02 0.02
0,025 0.025
Strain
Figure 8. Shear stress vs. strain curve obtained in one the torsion tests (SPABOND 340) using eqs. 10, 11 and 12.
The tensile strain-stress curves obtained in this study are shown in figure 9 (samples previously tested in torsion, Tor+Tract), compared with some of the curves obtained from conventional tensile tests in the previous work cited (Tract) [15]. The FE curves (MEF) previously presented (Figure 5, Section 3) are also included. 45 40
Stress (MPa)
35 30 25 Tract_1
20
Tract_2 Tract_3
15
Tor+Tract_1 Tor+Tract_2
10
Tor+Tract_3 Tract_MEF
5
Tor+Tract_MEF
0 0
0.01
0.02
0.03
Strain (mm/mm)
Figure 9. Tensile tests with (Tor+Tract, green) and without (Tract, red) a previous torsion test performed until yield point (SPABOND 340).
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Then, the experimental results confirm that the influence of the torsion test in the tensile results is negligible, since the differences are within the deviations range of the tests. This is because the plastic strains introduced by the torsion loads have been kept low enough to avoid significant effects in the subsequent tests. This evidences the feasibility of the experimental methodology proposed for the type of adhesives studied. Finally, table 2 presents the results obtained from the characterization performed following the procedure proposed comparing with the one obtained in the previous work previously referenced. It can be seen that the results are very similar, with differences below the 12.5 %. Table 2. Main material properties and model parameters obtained, compared with the ones obtained in the previous work[15].
E (GPa) νe Previous[15] 2.557 0.4 New procedure proposed 2.59 0.45 Error (%) 1.3 12.5
λ 1.210 1.16 4.1
νp 0.27 0.29 7.4
Ψ (º) 28 26 7.1
6. Conclusions An experimental methodology is proposed to reduce the cost of the characterizations usually required to feed the material models commonly used in detailed FE simulations of adhesives and bonded joints. The methodology consists on a sequential torsion and tensile tests that can be performed using the same specimen. This approach reduces the amount of samples required and the total duration of the tests. The torsion test is performed until the shear yield point is reached in the external surface of the specimen. In these conditions, the level of plastic strains introduced is small enough to not affect the subsequent tensile test. To be able to stop the test at the initiation of the yield, the quantification of the shear plastic strains in real time is needed. For this, a specific calculation code has been developed that is a key tool for the implementation of the methodology presented. Finally, the method has been applied with success to characterize a toughened epoxy adhesive analysed in previous studies through standard testing procedures. The agreement in the tensile stress-strain curves and in the material parameters finally obtained in both characterizations has evidenced the adequacy and feasibility of the methodology proposed to characterize this type of adhesives.
Acknowledgements The authors would like to express their thanks to the Spanish Science and Innovation Ministry for financial support (projects MAT2011-29182-C02-01 and MAT201129182-C02-02).
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Corresponding author Agustín Chiminelli Instituto Tecnológico de Aragón, María de Luna 7-8, 50018, Zaragoza, Spain. Tel: (34) 976 01 1098 Fax: (34) 976 01 1888 e-mail: [email protected]
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