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Characterization of adhesive joints under high-speed normal impact: Part II – Numerical studies
Journal Pre-proof Characterization of Adhesive Joints Under High-Speed Normal Impact: Part II – Numerical Studies Kenneth Gollins, Niell Elvin, Feridun Delale PII:
S0143-7496(19)30282-9
DOI:
https://doi.org/10.1016/j.ijadhadh.2019.102530
Reference:
JAAD 102530
To appear in:
International Journal of Adhesion and Adhesives
Please cite this article as: Gollins K, Elvin N, Delale F, Characterization of Adhesive Joints Under HighSpeed Normal Impact: Part II – Numerical Studies, International Journal of Adhesion and Adhesives, https://doi.org/10.1016/j.ijadhadh.2019.102530. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. 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. © 2019 Elsevier Ltd. All rights reserved.
Characterization of Adhesive Joints Under High-Speed Normal Impact: Part II – Numerical Studies
Kenneth Gollinsa, Niell Elvina†, Feridun Delalea a. Department of Mechanical Engineering, City College of New York, New York, NY 10031, USA † Corresponding Author, email: [email protected]
Abstract This two-part article presents an experimental method with numerical verification of the characterization of adhesive bonds under normal impact loading. Part I presented the experimentally measured performance of static and impact behavior of two adhesives (a methacrylate and an epoxy). Part II presents the numerical modeling of the static and dynamic tests performed in Part I. The finite element simulations are used to determine approximate dynamic material modeling parameters. The simulations are also useful in estimating certain critical performance aspects of the adhesive for the proposed impact test such as the estimation of contact force time histories, the dynamic stress time history profiles within the adhesive, and to the various energy components in the adhesives and adherends.
Keywords D. mechanical properties of adhesives; D. impact; C. dynamic mechanical analysis; C. finite element stress analysis
1
Introduction The pioneering work on the calculation of stresses in lap-joints under static loading using finite
element analysis (FEA) was first performed by Peppiatt and Adams in 1974 [1]. The use of numerical simulation to study the performance of various joints, various loading conditions (both static and 1
dynamic), and various damage models have been undertaken by many researchers. The relatively recent review article by He [2] provides an extensive summary of these various finite element studies, and provides an excellent overview of the current state of the art in FEA modeling of adhesive joints. The finite element work to-date has concentrated on experimental verification of a myriad of different tests but the performance under transverse impact has only been mostly studied at relatively low velocities using drop-weight type tests. For high velocity impact of adhesive joints, Park and Kim [3] numerically and experimentally studied a composite joint under ice-impact.
Later work by Hazimeh et al. [4]
presented finite element modeling results of a lap joint under in-plane, high-velocity impact loading using a split-Hopkinson bar type test with no damage of either in the adhesive or adherends. It should be noted that the present work differs significantly from the previous high velocity impact work in the following aspects. The previous work concentrated on composite material joints in which the failure is often complex (occurring as a combination of damage in the adherends and adhesives). This makes an estimation of the dynamic performance of the adhesive itself very difficult to isolate from the performance of the adherends. In fact, the impact damage mechanics of composites is still an unresolved research question. In contrast, the dynamic behavior of the steel used in the present study has been well characterized over a wide range of strain-rates, and well validated models (such as the Johnson-Cook model [5]) exist. The Park and Kim study [3] also used ice spheres as the impactor. The dynamic failure behavior of ice is not well understood and also currently an open research question. Again, in contrast the present study’s impactor, steel, is well characterized using the Johnson Cook model. Furthermore, normal-impact studies of adhesives (including the Park and Kim study [3]) have the impactor directly strike over the adhesive. This greatly complicates the combined stress distribution in the adhesive as compared to the present study where the goal of the test is to create relatively simple stress conditions (at least as far as possible for dynamic loading) in the adhesive. Thus following the approach of Bezemer et al. [6], the focus of the proposed high velocity impact test was to try create a dynamic analogous to the static blister test. The static blister test consists of uniform pressure or concentrated load applied to a circular plate bonded by a concentric adhesive layer. It has been shown [7] that the adhesive in the blister test is subjected to both normal and shear stresses, but that the strain energy contribution of the normal stresses typically is four times greater than the shear energy contribution. In the present study, we numerically analyze the experimental setup for normal-impact testing detailed in Part I of this two-part paper and show that under dynamic impact, the proposed experiment is analogous to the static blister test. This joint can be viewed as an axisymmetric lap joint for the purpose of transverse loading, which greatly reduces the computational cost of analyzing it.
The numerically calculated impact force time history,
detailed energy balance for all parts of the experiment (including adherends, adhesive and impactor), and derived dynamic mechanical parameters are also presented. 2
2
Finite Element Analysis Finite element analysis (FEA) was implemented to simulate and assist the explanation of the
experimental results presented in Part I of this two-part article. The bending joint under static loading was simulated to verify model parameters before analyzing the more complex dynamic cases. With this simulation, the dynamic material properties were found and a full energy balance was conducted. All simulations were performed using ABAQUS© 6.13 on a Dell Precision T55000 with a 3.07 GHz Intel Xeon processor and 24.0 GB of RAM.
2.1 Static Analysis: Bending The ABAQUS© - Implicit FEA package was used to perform a 2D, quasi-static simulation of the adhesively bonded bending joint (shown in Figure 2-2 in Part I [8]). The objective of this finite element study was (1) validate the FEA damage model, (2) determine the stress profile along the bond line and to (3) calculate the energy absorption of the bond under quasi-static loading so that it could be compared with the dynamic loading case. The material properties were chosen so as to match the experimental stress-strain curves results shown in Figure 3.2 in Part I [8]. The aluminum adherends were assumed to be linear elastic. The material properties for Young's modulus (E), Poisson's ratio (ߥ), yield strength (ߪ) ݀ݕ, ultimate strength
(ߪ) ݐ݈ݑ, hardening modulus (H), and the fracture strain (ߝ݂ ) ܿܽݎare listed in Error! Reference source not found.. Table 2-1 Material models for aluminum adherends, methacrylate and epoxy
Material Aluminum Methacrylate Epoxy
E [GPa] 68.9 1.5 3.3
ࣇ 0.33 0.30 0.30
࣌࢟ࢊ [MPa] 19.0 30.0
࢛࢚࣌ [MPa] 33.0 41.0
H [MPa] 99.7 101.8
ࢿࢌ࢘ࢇࢉ 0.123 0.117
To reduce computational complexity, a 2D axisymmetric approximation was used in the FEA analysis. The indentor and steel base plate of the experimental set-up were included in this model to match the compliance of the experimental set-up as shown in Figure 2-2 in Part I [8]. Permanent deformations in the patch were observed in the experimental specimens, thus plastic models were incorporated for the adherends. The Johnson-Cook plasticity models were used [5]. The Johnson-Cook model is not necessary to represent the plasticity in this quasi-static simulation, but it was incorporated for continuity with the impact simulation in the following section. The adherends were AISI 1018 steel; the
3
indentor was AISI O-1 tool steel. The base plate that the bending specimen was mounted on was modeled as a linear-elastic steel 1018. The properties of these materials and the constants used in the JohnsonCook model are listed in Table 2-2. Table 2-2 Johnson-Cook parameters for AISI 1018 steel [9] and AISI O1 steel [10]
Material AISI 1018 Steel AISI O-1 Steel
E [GPa] 200.0 200.0
࢜ [-] 0.29 0.29
A [MPa] 615.8 391.3
B [MPa] 567.7 723.9
C [-] 0.0134 0.1144
n [-] 0.2550 0.3067
ࢿሶ [࢙ି ] 1.0 1.0
The axisymmetric model approximation and boundary conditions are shown in
Figure 2-1. The steel base plate was partially restrained (ܷ௭ = 0) by rollers along the bottom surface. The two contact sets between (1) indentor-specimen and (2) specimen-backing plate was allowed to slide with a coefficient of friction (ߤ=0.2 to 0.6 for steel-on-steel) and with ABAQUS© standard "hard" normal contact. The adhesive was modeled to include the spew, on both the inner and outer diameter of the joint, which has been shown to remove the singularity at the sharp edge [11]. The model uses CAX4R 4-node bilinear second order axisymmetric quadrilateral elements. An advanced front mesh mapping method was used to create the refined free mesh around the adhesive. The mesh refinement in the adhesive region is shown Figure 2-1 (B). To validate the mesh, the strain energy within the adhesive, patch, and plate was compared for each mesh size until energy convergence to within ~5% was achieved. 9 elements are used to discretize the adhesive through the thickness. The total mesh contained 22,000 elements. The initial displacement was set to 2mm. The simulation run time was 120 seconds to replicate the 2mm/min experiment. The force-displacement curves for three experimental tests and the FEA result are shown in Figure 2-2. As comparison of these results show, the FEA results are in good agreement with the experimental results.
4
Indentor (ܷ௭ )
Contact Set Adhesive
Plate Base Plate
z ߠ
r Patch Roller (ܷ = ݖ0)
(A)
(B)
Figure 2-1 (A) Diagram of the axisymmetric static bending FEA model and boundary conditions (dimensions shown in mm) (red solid line indicates a contact set between two bodies) and (B) the adhesive layer mesh and element transition to the adherends
1 2
2 3 3 1
5
(A)
(B)
Figure 2-2 Force - Displacement curve of experimental results with the FEA simulation for (A) steelmethacrylate and (B) steel-epoxy patch-bending models under quasi-static loading. For clarity the various tests are labeled as 1 to 3 on each graph.
The three axisymmetric stress components along the bond line for the static bending tests are shown in Figure 2-3.
σz
τrz σr
Figure 2-3: Radial (σr ), normal (σz ) and shear (τrz) stress distributions along the adhesive bond line for the static bending test.
2.2 Dynamic: Impact Analysis The ABAQUS© - Explicit finite element analysis (FEA) package is used for the ballistic impact studies. To reduce computational effort, an axisymmetric model was used. Even though the back plate is square in the experiment, the circular axisymmetric approximation is valid because only the first pass of the stress wave through the adhesive is if interest i.e. wave reflections are not of interest in this study and as such, the far edge boundary conditions are not of concern. The objectives of this study were to (1) compare the Taylor impact tests (i.e. the projectile force and deformed shape) with the FEA, (2) create a pseudo-constitutive model for the adhesives at high strain rate, (3) calculate the time dependent stress profiles along the bond line, and (4) perform an energy balance for the dynamic system.
2.2.1
Boundary Conditions and Meshing
A schematic of the axisymmetric model and boundary conditions are seen below in Figure 2-4 . 6
Fixed B.C.
Gas Gun
Plate
Adhesive (A) Patch
Projectile
r Velocity
z
(B)
ߠ
Figure 2-4 Schematic of (A) the three-dimensional representation of the experiment, (B) the axisymmetric approximation including the boundary conditions (fixed) and initial conditions (velocity).
The finite element mesh used in the impact analysis was the same as for the static-case as shown in Figure 2-1 (B). The fixed boundary condition of the back plate in the FEA was 76.2 mm from the axisymmetric center-line of the model. Another finite element convergence study was performed on the strain energy within the adhesive, patch, plate and impactor which showed that the mesh was adequate.
2.2.2
Material Models: Projectile and Adherends
The Johnson-Cook plasticity model was implemented for the AISI 1018 steel adherends and AISI O1-tool steel for the projectile. The Johnson-cook plasticity model is a type of von-Mises model with analytical forms of the hardening law and rate dependence, which is suitable for high-strain-rate deformation for most metals. The material properties were previously listed in Table 2-2.
2.2.3
Impact Validation
The energy imparted into the joint is transferred through an energy exchange of the kinetic energy from the projectile to the energy in the joint. This energy input was validated by comparing the FEA calculations with experimental data. Validation parameters are the dynamic yield stress and the deformed shape the contact force. The dynamic yield stress was experimentally calculated using Taylor's [12] method. The dynamic yield stress of the projectile, ߪ ݀ݕ, was found from equation 3.12, which was detailed previously in section 3.3 in Gollins et al. [8]. 7
The experimental values are shown with the FEA results below in Figure 2-5. The projectile deformation shapes are shown in Figure 2-6. As seen, there is good agreement between the experimental results and the FEA calculations.
vi = 170 m/s vi = 209 m/s
ܮଵ /ܮ
vi = 254 m/s vi = 294 m/s vi = 327 m/s
ߩݒ ଶ /ߪ௬ௗ Figure 2-5 Taylor impact test for approximating the dynamic yield stress of the projectile.
170 m/s
209 m/s
254 m/s
294 m/s
327 m/s
25.4 mm
Figure 2-6 Experimental and FEA projectile deformation shape comparison with respect to the initial length, ࡸ =25.4mm (experimental in grey and FEA in blue) at various projectile impact velocities.
2.2.4
Adhesives Under High Strain Rate
Goglio et. al [13] found that epoxy adhesive is very sensitive to strain rate. At an applied strain rate on the order of 103 s−1 , the adhesive strength, with respect to the static value, was observed to have 8
increased up to three times under compression and five times under tension. The elastic modulus was reportedly unchanged. Hence, static values for the adhesive are not valid inputs for the dynamic strength and dynamic fracture strain. Unfortunately, no rate dependent material parameters exist for the adhesives used in this study. Thus, we have performed a FEA parametric study to estimate these dynamic values. A continuum damage model was used to model the constitutive behavior of the adhesive materials. Unlike cohesive zone modeling, continuum damage models do not require the knowledge of where the onset of damage occurs a-priori. In a continuum model, the material property degradation occurs inside of the solid element and does not require the special use of interfacial elements. However, this method is highly mesh sensitive and proper validation is necessary [14]. The dynamic yield stress for the adhesive, ߪ ܣ݀ݕand dynamic fracture plastic strain, ߝ݂ ܣ, were found by implementing two algorithms similar to the procedure outlined by Yildiz et al. [15]. The dynamic material properties of the adhesive were assumed to be elastic perfectly-plastic (i.e. without hardening). The first algorithm iterated through user defined values for ߪ ܣ݀ݕand ߝ݂ ܣand ran the simulation at a velocity 10% below and 10% above the CFV for the adhesive-adherend material combination. If the result was seen to have survived an impact below the experimentally determined CFV at rupture and failed above the CFV (in a go-no-go fashion), then it was saved in a database for processing in the second algorithm. After the running through the values for ߪ ܣ݀ݕand ߝ݂ ܣ, the remaining material groupings were loaded into the second algorithm. The incident projectile velocity was varied in ABAQUS© for the experimental range of each adhesive-adherend combination. If the calculated free-flying patch energy after the adhesive ruptured in the ABAQUS© simulation matched the experimental results, it was saved into a database. Often, a few inputs had similar results to the experiments, so the reported values of ߪ ݀ݕand ߝ݂ ܣwere a range of suitable values. The results for the second algorithm are shown in Figure 2-7 (A) and
Figure 2-8 (A). The final material inputs are tabulated in Table 2-3.
9
Patch Kinetic Energy [J]
6 5 4
Experimental Jonas-Lambert Fit = 115 MPa fA = 0.08 ydA = 135 MPa = 0.06 ydA fA
3 2 1 0
100
120 140 160 Projectile Kinetic Energy [J]
180
(A)
10
(B) Figure 2-7 Experimental and finite element results for steel-methacrylate (A), and energy absorption of methacrylate with steel adherends (B).
11
Patch Kinetic Energy [J]
25 Experimental Jonas-Lambert Fit = 205 MPa = 0.09 ydA fA = 225 MPa = 0.08 ydA fA
20 15 10 5 0
0
200 400 600 Projectile Kinetic Energy [J]
800
(A)
12
(B) Figure 2-8 Experimental and finite element results for steel-epoxy (A), and energy absorption of epoxy with steel adherends (B).
Table 2-3 Dynamic material properties for the adhesives under impact loading.
Adherend-Adhesive
Young's Modulus [GPa]
Steel-Methacrylate Steel-Epoxy
1.5 3.3
Dynamic Yield Strength [MPa] 115-135 205-225
Dynamic Fracture Plastic Strain 0.06-0.08 0.08-0.09
The transient stress through the bond line was extracted from the FEA along the midline of the bond as shown. The stresses are captured at 5 ߤ ݏtime increments from when the stress wave arrives until fracture. The normalized stresses along the bond line at various time steps are shown in
Figure 2-9 for (A) normal and (B) shear stresses. As seen, the bond line has two distinct zones: with the side closer to the impact (i.e. x/L close to zero) in tension, and the other side (i.e. x/L closer to one) in compression, where x is the distance along the bond and L is the total length of the bond. Changing the boundary conditions from fixed to free in the FEA did not show any significant changes in the results once more confirming that the boundary of the model (and experiment) does not play a role in the failure mechanism of the joint.
13
20µs 0µs 15µs
(A)
0µs
20µs
(B) Figure 2-9 Transient stress along the adhesive bond midline until just before fracture for (A) normal stress and (B) shear stress. Here the adhesive is epoxy with the Young’s modulus =3.3 GPa, the dynamic yield strength σyd = 210MPa and the failure strain = 0.08. The projectile impact velocity is 316m/s.
14
The stress distributions and propagation of the fracture in the bond after fracture initiation for the steelmethacrylate bond at the CFV of ~ 170m/s are shown in the sequence of plots in Figure 2-10(A) to (D). The fracture starts at the inner diameter of the joint where the normal tensile stress (σz) is highest at approximately 25µs after impact. The facture progresses towards the outer diameter of the joint until approximately 30µs at which time the outer-diameter of the joint begins to facture. When the two sides of the fracture coalesce the patch completely detaches which occurs at approximately 50µs after impact. These results are similar for the steel-epoxy cases which are not presented for conciseness.
Tension Compression
A
15
B
C
16
D Figure 2-10: FEM results showing the normal stress (σ σz ) distribution at the joint and the sequence of crack propagation through the joint at (A) just before fracture initiation at 24 µs after impact, (B) just after fracture initiation at 26 µs after impact, (C) 32 µs after impact and (D) 50 µs after impact when the patch dethatches i.e. the crack has fully propagated through the adhesive. The results shown here are for the steelmethacrylate case at the CFV of approximately 170 m/s.
Note that the FEA confirms that the failure of the adhesive occurs before the stress wave reflects from the boundary of the model.
2.2.5
Contact Force
The finite element contact force was calculated by taking the summation of the contact force of each node along the projectile surface. The FEA derived peak force and the average force can be found with this method. The average force is calculated by numerically integrating the force time history (i.e. the impulse of the projectile) divided by the contact time (i.e. ~ 29 µs for the methacrylate and ~38 µs for the epoxy). Here, contact time is defined as the time from initial force application of the impactor (~ 2 µs) until the time when the force returns to a value < 0.5% of the peak force. The results for the material combinations are shown in Figure 2-11 (A) and listed in Table 2-4. The contact forces are calculated at the CFV at rupture for each case.
17
Contact Time Epoxy
Epoxy
Methacrylate Contact Time Methacrylate
(A)
(B) Figure 2-11 (A) Force-time history calculated from FEA for methacrylate and epoxy joints at their CFV and (B) comparison of FEA calculated average impact forces to the experimentally determined impact forces using the Taylor method and camera estimates for contact time and conservation of momentum.
18
The projectile impact force from the experimental results for steel on steel can now be estimated using the dynamic stress taken from Taylor’s approximation (see Gollins et al. [8]). The average impact forces of the projectiles at the CFVs are listed in Table 2-4. This table also shows a comparison between the dynamic and static strength of the bonds. The strength of the bond is calculated as the failure load divided by the bond area. As such, the strength is an average for the full bond length. Furthermore, Table 2-4 clearly illustrates the dynamic effect within the bond. The dynamic strength of the epoxy is shown to be 3 times higher than that of the static strength, while the dynamic strength of the methacrylate is double for steel adherends. Table 2-4 Average impact force and bond strength at CFV
Incident Speed (m/s)
Adherend-Adhesive Steel-Methacrylate Steel-Epoxy
2.2.6
~170 ~320
Avg. Contact Force, Exp. (kN) 78 119
Projectile Impact Force, FEA (kN) 76.1 113.8
Static Strength, FEA (MPa) 14.6 13.2
Dynamic Strength, FEA (MPa) 30.0 44.8
Energy Balance
The experiment outlined in section 3.5 in Gollins et al. [8] i.e. the energy dissipation of an unbonded patch, was replicated using FEA to demonstrate the direct energy extraction approach. A similar model to the one shown in Figure 2-4 without the adhesive was used to simulate only the transfer of momentum from the projectile to the patch, and the kinetic energy of the patch after the impact was compared to the experimental results. The FEA calculated kinetic energy of the patch was consistently higher than the experimental data, which was as expected since the experiment was unable to measure any energy contribution due to vibration. Furthermore, the FEA possibly under-predicts some energy loss mechanism such as from damping. The un-bonded patch energies are plotted in Figure 2-7 and listed in Table 2-5. The energies can also be directly calculated. ABAQUS© resolves the energies of the components through the conservation of energy, thus the energies can be directly extracted. Implied by the first law of thermodynamics, the law of conservation of energy states that the time rate of change of kinetic energy and internal energy for a fixed body of material is equal to the sum of the rate of work done by the surface and body forces: ݀
݀ݐ
1
න ߩ ܸ݀ ݒ ⋅ ݒ+ න ߩܷ ܸ݀൨ = න ܵ݀ ߬ ⋅ ݒ+ න ݂ ⋅ ܸ݀ ݒ 2
(2.2)
19
where ߩ is the density, ݒis the velocity, ܷ is the internal energy per unit volume, ߬ is the surface traction
and ݂ is the body force. We can extract each of these energy components from the ABAQUS© calculations to find where the bulk of kinetic energy supplied by the projectile is distributed into the system after the impact. From, the energy equation, the kinetic energy is defined as: 1 = ܧܭන ߩܸ݀ ݒ ⋅ ݒ 2
(2.3)
= ܧܫන ߩܷ ܸ݀
(2.4)
and the internal energy:
where, U, is the summation of the recoverable strain energy, the plastic dissipation, the damage dissipation and any artificial energy from hourglassing. For all cases, the plastic dissipation is the largest contributor to the internal energy. The artificial energy was less than 1% of the strain energy. Initially, the kinetic energy of the projectile is imparted to the system. Upon impact, energy is spent to plastically deform the projectile and the patch. These are the two major energy sinks of the system, the plastic dissipation in the projectile and patch. Following the impact, some of the energy is transferred back into the projectile in the form of kinetic energy. The remaining energy is used to break the adhesive and transferred to the patch as kinetic energy after it becomes detached from the system. The summation of all the components in their final state is equal to the input energy. Alternatively, equation (2.) can be rewritten in term of the system components: ܧܭ − ܧܫ − ܧܭ − ܧܫ௧ − ܧܭ௧ − ܧܫ௧ − ܧܫௗ௦௩ = 0
(2.5)
where each energy component corresponds to its respective specimen part. The energies for the material combinations are shown in
Figure 2-12. The energy absorbed by the adhesive is represented by two cases with case 1 being the lower bound and case 2 being the upper bound for the dynamic yield strength and strain at failure given in Table 2-3. As seen in Figure 2-12 there is a negligible difference in absorbed energy between the upper and lower bounds. The energy absorbed in the adhesive is listed below in Table 2-5. Note that the energy absorbed by the adhesive as calculated directly by the FEA is similar for the case of the methacrylate (i.e. 5.2 J compared with 4.6 J as seen in Table 2-5) but is significantly lower for the epoxy (11.1 J compared to 16.9 J). This is largely due to the fact that the epoxy is significantly stronger than the methacrylate under impact. Analysis of the FEA results shows that the difference in energy is transferred as internal energy into the back-plate. (This is not shown in
20
Figure 2-12 for clarity). This in turn means that, not unexpectedly, the underlying assumptions that the adhesive does not contribute to the impact dynamics of the adherends and projectile break down as the adhesive becomes stronger.
KE: Projectile
IE: Patch
IE: Projectile KE: Patch
IE: Adhesive 1 and 2
(A)
IE: Patch
KE: Projectile IE: Projectile KE: Patch
IE: Adhesive 1 and 2
21
(B) Figure 2-12 Energy-time history for methacrylate (A) and epoxy (B) joints. Note that the internal energies (IE) for the both adhesives for Cases 1 and 2 are essentially the same and are indistinguishable on these plots.
Table 2-5 Comparison of energy absorbed in the adhesive by the direct and extraction approaches using FEA and experimental methods. The extraction method assumes the energy absorption of the adhesive is the difference between two tests: bonded and unbonded, whereas the direct approach resolved directly through FEA calculations
Adherend-Adhesive Steel-Methacrylate Steel-Epoxy
Projectile Speed [m/s] ~170
~320
Energy Absorbed EXP. [J] 4 16
Extraction Energy Absorbed FEA [J]
Direct Energy Absorbed FEA [J]
4.6 16.9
5.2 11.1
The absorbed energy calculated from the static simulation and the calculated dynamic energy absorbed are compared in Figure 2-13. The absorbed energy of the adhesive is seen to increase with the loading rate.
Dynamic
Dynamic
Static
Static
Figure 2-13 Comparison between the static and dynamic energy absorption for methacrylate and epoxy adhesives as calculated from the direct FEA calculations.
22
3
Conclusions We have presented a novel protocol for the dynamic characterization of adhesively bonded joints
under ballistic impact loading. The test is meant to be a dynamic analog to the commonly used static blister test. Finite element simulations show that similar to the static blister test, the maximum dynamic normal stresses are approximately four times greater than the dynamic shear stress. Thus, the adhesive is predominantly subjected to Mode I stresses. The finite element (FE) computational work presented in this part of this two-part article, was used to obtain stress profiles through the bond-line and to estimate peak contact forces. The FE work also validated some of the assumptions of the experimental results such as that the dynamical stress profiles are similar to the static-blister test profiles and are predominantly Mode I.
Furthermore, the Taylor
impact tests assume a rigid target, whereas visual inspection of the experimental test specimens after impact and digital video analysis during the impact indicate that there is relatively large deformation in the patch. Thus, the experimentally calculated impact forces only serve as an approximation of the actual dynamic forces experienced by the patch. Additionally, the stiffness of the adhesive is assumed to be negligible relative to the adherends. If the stiffness of the adhesive is significant, the experimentally measured energy absorption of the adhesives may not be accurate. Both the rigid target and negligible adhesive stiffness assumptions were taken into account in the present study with a validated FEA. There are several obvious extensions and further validations of the work presented here that should be addressed in future work. The first is to fully study the effect of geometry (for example adhesive bond length and thickness) on the performance of the adhesives under ballistic loading. The impact dynamics used in the FEM could be further validated by comparing the plastic deformation of the patch to the experimental results. Furthermore, environmental effects such as temperature and humidity should also be extensively studied under dynamic loading. The experimental method presented in this paper can be readily adapted to include these modifications. Furthermore, a complete dynamic constitutive model for the adhesives which would be invaluable for the design of adhesive joints should be the focus of future work.
4
Acknowledgements
23
This work was supported by U.S. Army- TARDEC (currently US-Army GVSC) under contract #69201511 through NAMC. We thank Dr. Robert Jensen of ARL and Daniel Schaffren of US Army-GVSC (previously TARDEC) who provided their expertise on surface preparation and bonding. Additionally, we thank our colleagues Dr. Bruno Zamorano and Dr. Jack Chiu for their insightful discussions and contribution to this work.
24
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[13]
[14] [15]
R. D. Adams and n. a. Peppiatt, “Stress analysis of adhesive-bonded lap joints,” The Journal of Strain Analysis for Engineering Design, vol. 9, no. 3, pp. 185–196, 1974. X. He, “A Review of Finite Element Analysis of Adhesively Bonded Joints,” International Journal of Adhesion & Adhesives, vol. 31, pp. 248–264, 2011. H. Park and H. Kim, “Damage resistance of single lap adhesive composite joints by transverse ice impact,” International Journal of Impact Engineering, vol. 37, no. 2, pp. 177–184, 2010. R. Hazimeh, G. Challita, K. Khalil, and R. Othman, “International Journal of Adhesion & Adhesives Finite element analysis of adhesively bonded composite joints subjected to impact loadings,” International Journal of Adhesion and Adhesives, pp. 1–8, 2014. G. R. Johnson and W. H. Cook, “Fracture characteristic of three metals subjected to various strains, strain rates, temperatures and pressures,” Engineering Fracture Mechanics, vol. 21, no. 1, pp. 31–48, 1985. A. A. Bezemer, C. B. Guyt, and A. Vlot, “New impact specimen for adhesives: optimization of high-speed-loaded adhesive joints,” International Journal of Adhesion and Adhesives, vol. 18, no. 4, pp. 255–260, 1998. D. P. Updike, “Effect of adhesive layer elasticity on the fracture mechanics of a blister test specimen,” International Journal of Fracture, vol. 12, no. 6, pp. 815–827, 1976. K. Gollins, N. Elvin, and F. Delale, “Characterization of Adhesive Joints Under High-Speed Normal Impact: Part I – Experimental Studies,” Submitted for Publication, 2019. T. Lee, M. Leok, and N. H. McClamroch, “Geometric numerical integration for complex dynamics of tethered spacecraft,” Proceedings of the 2011 American Control Conference, no. March, pp. 1885–1891, 2011. B. McClain, S. A. Batzer, and G. I. Maldonado, “A numeric investigation of the rake face stress distribution in orthogonal machining,” Journal of Materials Processing Technology, vol. 123, no. 1, pp. 114–119, 2002. R. D. Adams and J. A. Harris, “The influence of local geometry on the strength of adhesive joints,” International Journal of Adhesion and Adhesives, vol. 7, no. 2, pp. 69–80, 1987. G. Taylor, “The Use of Flat-Ended Projectiles for Determining Dynamic Yield Stress. I. Theoretical Considerations,” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 194, no. 1038, pp. 289–299, 1948. L. Goglio, L. Peroni, M. Peroni, and M. Rossetto, “High strain-rate compression and tension behaviour of an epoxy bi-component adhesive,” International Journal of Adhesion and Adhesives, vol. 28, no. 7, pp. 329–339, 2008. L. F. M. da Silva, Modelling of Adhesively Bonded Joints. Springer, 2008. S. Yildiz, Y. Andreopoulos, R. E. Jensen, D. Shaffren, D. Jahnke, and F. Delale, “Characterization of adhesively bonded aluminum plates subjected to shock- wave loading,” International Journal of Impact Engineering, vol. 127, no. January, pp. 86–99, 2019.
25
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Please see the attached WORD documents. All the best Niell Elvin ________________________________________ From: [email protected] Sent: Tuesday, December 10, 2019 6:15 AM To: Niell Elvin Subject: [EXTERNAL] Problem [JAAD_102530] Our reference: JAAD 102530 Re: Characterization of Adhesive Joints Under High-Speed Normal Impact: Part II – Numerical Studies Dear Dr. Elvin, We recently received your accepted manuscript for publication; however, our typesetters are unable to use PDF files for typesetting and production purposes. An editable text file that exactly matches the final, accepted version of your manuscript is required for publication. Please e-mail me the electronic version of your accepted manuscript so that we may proceed with the publication of your article. Acceptable text file formats include MS Word, Word Perfect, RTF, TEX and plain ASCII text. Thank you in advance for your assistance and prompt attention to this matter. Yours sincerely, Sajitha Sridhar Journal Manager Elsevier
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DOI:
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Reference:
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To appear in:
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Please cite this article as: Gollins K, Elvin N, Delale F, Characterization of Adhesive Joints Under HighSpeed Normal Impact: Part II – Numerical Studies, International Journal of Adhesion and Adhesives, https://doi.org/10.1016/j.ijadhadh.2019.102530. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. 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. © 2019 Elsevier Ltd. All rights reserved.
Characterization of Adhesive Joints Under High-Speed Normal Impact: Part II – Numerical Studies
Kenneth Gollinsa, Niell Elvina†, Feridun Delalea a. Department of Mechanical Engineering, City College of New York, New York, NY 10031, USA † Corresponding Author, email: [email protected]
Abstract This two-part article presents an experimental method with numerical verification of the characterization of adhesive bonds under normal impact loading. Part I presented the experimentally measured performance of static and impact behavior of two adhesives (a methacrylate and an epoxy). Part II presents the numerical modeling of the static and dynamic tests performed in Part I. The finite element simulations are used to determine approximate dynamic material modeling parameters. The simulations are also useful in estimating certain critical performance aspects of the adhesive for the proposed impact test such as the estimation of contact force time histories, the dynamic stress time history profiles within the adhesive, and to the various energy components in the adhesives and adherends.
Keywords D. mechanical properties of adhesives; D. impact; C. dynamic mechanical analysis; C. finite element stress analysis
1
Introduction The pioneering work on the calculation of stresses in lap-joints under static loading using finite
element analysis (FEA) was first performed by Peppiatt and Adams in 1974 [1]. The use of numerical simulation to study the performance of various joints, various loading conditions (both static and 1
dynamic), and various damage models have been undertaken by many researchers. The relatively recent review article by He [2] provides an extensive summary of these various finite element studies, and provides an excellent overview of the current state of the art in FEA modeling of adhesive joints. The finite element work to-date has concentrated on experimental verification of a myriad of different tests but the performance under transverse impact has only been mostly studied at relatively low velocities using drop-weight type tests. For high velocity impact of adhesive joints, Park and Kim [3] numerically and experimentally studied a composite joint under ice-impact.
Later work by Hazimeh et al. [4]
presented finite element modeling results of a lap joint under in-plane, high-velocity impact loading using a split-Hopkinson bar type test with no damage of either in the adhesive or adherends. It should be noted that the present work differs significantly from the previous high velocity impact work in the following aspects. The previous work concentrated on composite material joints in which the failure is often complex (occurring as a combination of damage in the adherends and adhesives). This makes an estimation of the dynamic performance of the adhesive itself very difficult to isolate from the performance of the adherends. In fact, the impact damage mechanics of composites is still an unresolved research question. In contrast, the dynamic behavior of the steel used in the present study has been well characterized over a wide range of strain-rates, and well validated models (such as the Johnson-Cook model [5]) exist. The Park and Kim study [3] also used ice spheres as the impactor. The dynamic failure behavior of ice is not well understood and also currently an open research question. Again, in contrast the present study’s impactor, steel, is well characterized using the Johnson Cook model. Furthermore, normal-impact studies of adhesives (including the Park and Kim study [3]) have the impactor directly strike over the adhesive. This greatly complicates the combined stress distribution in the adhesive as compared to the present study where the goal of the test is to create relatively simple stress conditions (at least as far as possible for dynamic loading) in the adhesive. Thus following the approach of Bezemer et al. [6], the focus of the proposed high velocity impact test was to try create a dynamic analogous to the static blister test. The static blister test consists of uniform pressure or concentrated load applied to a circular plate bonded by a concentric adhesive layer. It has been shown [7] that the adhesive in the blister test is subjected to both normal and shear stresses, but that the strain energy contribution of the normal stresses typically is four times greater than the shear energy contribution. In the present study, we numerically analyze the experimental setup for normal-impact testing detailed in Part I of this two-part paper and show that under dynamic impact, the proposed experiment is analogous to the static blister test. This joint can be viewed as an axisymmetric lap joint for the purpose of transverse loading, which greatly reduces the computational cost of analyzing it.
The numerically calculated impact force time history,
detailed energy balance for all parts of the experiment (including adherends, adhesive and impactor), and derived dynamic mechanical parameters are also presented. 2
2
Finite Element Analysis Finite element analysis (FEA) was implemented to simulate and assist the explanation of the
experimental results presented in Part I of this two-part article. The bending joint under static loading was simulated to verify model parameters before analyzing the more complex dynamic cases. With this simulation, the dynamic material properties were found and a full energy balance was conducted. All simulations were performed using ABAQUS© 6.13 on a Dell Precision T55000 with a 3.07 GHz Intel Xeon processor and 24.0 GB of RAM.
2.1 Static Analysis: Bending The ABAQUS© - Implicit FEA package was used to perform a 2D, quasi-static simulation of the adhesively bonded bending joint (shown in Figure 2-2 in Part I [8]). The objective of this finite element study was (1) validate the FEA damage model, (2) determine the stress profile along the bond line and to (3) calculate the energy absorption of the bond under quasi-static loading so that it could be compared with the dynamic loading case. The material properties were chosen so as to match the experimental stress-strain curves results shown in Figure 3.2 in Part I [8]. The aluminum adherends were assumed to be linear elastic. The material properties for Young's modulus (E), Poisson's ratio (ߥ), yield strength (ߪ) ݀ݕ, ultimate strength
(ߪ) ݐ݈ݑ, hardening modulus (H), and the fracture strain (ߝ݂ ) ܿܽݎare listed in Error! Reference source not found.. Table 2-1 Material models for aluminum adherends, methacrylate and epoxy
Material Aluminum Methacrylate Epoxy
E [GPa] 68.9 1.5 3.3
ࣇ 0.33 0.30 0.30
࣌࢟ࢊ [MPa] 19.0 30.0
࢛࢚࣌ [MPa] 33.0 41.0
H [MPa] 99.7 101.8
ࢿࢌ࢘ࢇࢉ 0.123 0.117
To reduce computational complexity, a 2D axisymmetric approximation was used in the FEA analysis. The indentor and steel base plate of the experimental set-up were included in this model to match the compliance of the experimental set-up as shown in Figure 2-2 in Part I [8]. Permanent deformations in the patch were observed in the experimental specimens, thus plastic models were incorporated for the adherends. The Johnson-Cook plasticity models were used [5]. The Johnson-Cook model is not necessary to represent the plasticity in this quasi-static simulation, but it was incorporated for continuity with the impact simulation in the following section. The adherends were AISI 1018 steel; the
3
indentor was AISI O-1 tool steel. The base plate that the bending specimen was mounted on was modeled as a linear-elastic steel 1018. The properties of these materials and the constants used in the JohnsonCook model are listed in Table 2-2. Table 2-2 Johnson-Cook parameters for AISI 1018 steel [9] and AISI O1 steel [10]
Material AISI 1018 Steel AISI O-1 Steel
E [GPa] 200.0 200.0
࢜ [-] 0.29 0.29
A [MPa] 615.8 391.3
B [MPa] 567.7 723.9
C [-] 0.0134 0.1144
n [-] 0.2550 0.3067
ࢿሶ [࢙ି ] 1.0 1.0
The axisymmetric model approximation and boundary conditions are shown in
Figure 2-1. The steel base plate was partially restrained (ܷ௭ = 0) by rollers along the bottom surface. The two contact sets between (1) indentor-specimen and (2) specimen-backing plate was allowed to slide with a coefficient of friction (ߤ=0.2 to 0.6 for steel-on-steel) and with ABAQUS© standard "hard" normal contact. The adhesive was modeled to include the spew, on both the inner and outer diameter of the joint, which has been shown to remove the singularity at the sharp edge [11]. The model uses CAX4R 4-node bilinear second order axisymmetric quadrilateral elements. An advanced front mesh mapping method was used to create the refined free mesh around the adhesive. The mesh refinement in the adhesive region is shown Figure 2-1 (B). To validate the mesh, the strain energy within the adhesive, patch, and plate was compared for each mesh size until energy convergence to within ~5% was achieved. 9 elements are used to discretize the adhesive through the thickness. The total mesh contained 22,000 elements. The initial displacement was set to 2mm. The simulation run time was 120 seconds to replicate the 2mm/min experiment. The force-displacement curves for three experimental tests and the FEA result are shown in Figure 2-2. As comparison of these results show, the FEA results are in good agreement with the experimental results.
4
Indentor (ܷ௭ )
Contact Set Adhesive
Plate Base Plate
z ߠ
r Patch Roller (ܷ = ݖ0)
(A)
(B)
Figure 2-1 (A) Diagram of the axisymmetric static bending FEA model and boundary conditions (dimensions shown in mm) (red solid line indicates a contact set between two bodies) and (B) the adhesive layer mesh and element transition to the adherends
1 2
2 3 3 1
5
(A)
(B)
Figure 2-2 Force - Displacement curve of experimental results with the FEA simulation for (A) steelmethacrylate and (B) steel-epoxy patch-bending models under quasi-static loading. For clarity the various tests are labeled as 1 to 3 on each graph.
The three axisymmetric stress components along the bond line for the static bending tests are shown in Figure 2-3.
σz
τrz σr
Figure 2-3: Radial (σr ), normal (σz ) and shear (τrz) stress distributions along the adhesive bond line for the static bending test.
2.2 Dynamic: Impact Analysis The ABAQUS© - Explicit finite element analysis (FEA) package is used for the ballistic impact studies. To reduce computational effort, an axisymmetric model was used. Even though the back plate is square in the experiment, the circular axisymmetric approximation is valid because only the first pass of the stress wave through the adhesive is if interest i.e. wave reflections are not of interest in this study and as such, the far edge boundary conditions are not of concern. The objectives of this study were to (1) compare the Taylor impact tests (i.e. the projectile force and deformed shape) with the FEA, (2) create a pseudo-constitutive model for the adhesives at high strain rate, (3) calculate the time dependent stress profiles along the bond line, and (4) perform an energy balance for the dynamic system.
2.2.1
Boundary Conditions and Meshing
A schematic of the axisymmetric model and boundary conditions are seen below in Figure 2-4 . 6
Fixed B.C.
Gas Gun
Plate
Adhesive (A) Patch
Projectile
r Velocity
z
(B)
ߠ
Figure 2-4 Schematic of (A) the three-dimensional representation of the experiment, (B) the axisymmetric approximation including the boundary conditions (fixed) and initial conditions (velocity).
The finite element mesh used in the impact analysis was the same as for the static-case as shown in Figure 2-1 (B). The fixed boundary condition of the back plate in the FEA was 76.2 mm from the axisymmetric center-line of the model. Another finite element convergence study was performed on the strain energy within the adhesive, patch, plate and impactor which showed that the mesh was adequate.
2.2.2
Material Models: Projectile and Adherends
The Johnson-Cook plasticity model was implemented for the AISI 1018 steel adherends and AISI O1-tool steel for the projectile. The Johnson-cook plasticity model is a type of von-Mises model with analytical forms of the hardening law and rate dependence, which is suitable for high-strain-rate deformation for most metals. The material properties were previously listed in Table 2-2.
2.2.3
Impact Validation
The energy imparted into the joint is transferred through an energy exchange of the kinetic energy from the projectile to the energy in the joint. This energy input was validated by comparing the FEA calculations with experimental data. Validation parameters are the dynamic yield stress and the deformed shape the contact force. The dynamic yield stress was experimentally calculated using Taylor's [12] method. The dynamic yield stress of the projectile, ߪ ݀ݕ, was found from equation 3.12, which was detailed previously in section 3.3 in Gollins et al. [8]. 7
The experimental values are shown with the FEA results below in Figure 2-5. The projectile deformation shapes are shown in Figure 2-6. As seen, there is good agreement between the experimental results and the FEA calculations.
vi = 170 m/s vi = 209 m/s
ܮଵ /ܮ
vi = 254 m/s vi = 294 m/s vi = 327 m/s
ߩݒ ଶ /ߪ௬ௗ Figure 2-5 Taylor impact test for approximating the dynamic yield stress of the projectile.
170 m/s
209 m/s
254 m/s
294 m/s
327 m/s
25.4 mm
Figure 2-6 Experimental and FEA projectile deformation shape comparison with respect to the initial length, ࡸ =25.4mm (experimental in grey and FEA in blue) at various projectile impact velocities.
2.2.4
Adhesives Under High Strain Rate
Goglio et. al [13] found that epoxy adhesive is very sensitive to strain rate. At an applied strain rate on the order of 103 s−1 , the adhesive strength, with respect to the static value, was observed to have 8
increased up to three times under compression and five times under tension. The elastic modulus was reportedly unchanged. Hence, static values for the adhesive are not valid inputs for the dynamic strength and dynamic fracture strain. Unfortunately, no rate dependent material parameters exist for the adhesives used in this study. Thus, we have performed a FEA parametric study to estimate these dynamic values. A continuum damage model was used to model the constitutive behavior of the adhesive materials. Unlike cohesive zone modeling, continuum damage models do not require the knowledge of where the onset of damage occurs a-priori. In a continuum model, the material property degradation occurs inside of the solid element and does not require the special use of interfacial elements. However, this method is highly mesh sensitive and proper validation is necessary [14]. The dynamic yield stress for the adhesive, ߪ ܣ݀ݕand dynamic fracture plastic strain, ߝ݂ ܣ, were found by implementing two algorithms similar to the procedure outlined by Yildiz et al. [15]. The dynamic material properties of the adhesive were assumed to be elastic perfectly-plastic (i.e. without hardening). The first algorithm iterated through user defined values for ߪ ܣ݀ݕand ߝ݂ ܣand ran the simulation at a velocity 10% below and 10% above the CFV for the adhesive-adherend material combination. If the result was seen to have survived an impact below the experimentally determined CFV at rupture and failed above the CFV (in a go-no-go fashion), then it was saved in a database for processing in the second algorithm. After the running through the values for ߪ ܣ݀ݕand ߝ݂ ܣ, the remaining material groupings were loaded into the second algorithm. The incident projectile velocity was varied in ABAQUS© for the experimental range of each adhesive-adherend combination. If the calculated free-flying patch energy after the adhesive ruptured in the ABAQUS© simulation matched the experimental results, it was saved into a database. Often, a few inputs had similar results to the experiments, so the reported values of ߪ ݀ݕand ߝ݂ ܣwere a range of suitable values. The results for the second algorithm are shown in Figure 2-7 (A) and
Figure 2-8 (A). The final material inputs are tabulated in Table 2-3.
9
Patch Kinetic Energy [J]
6 5 4
Experimental Jonas-Lambert Fit = 115 MPa fA = 0.08 ydA = 135 MPa = 0.06 ydA fA
3 2 1 0
100
120 140 160 Projectile Kinetic Energy [J]
180
(A)
10
(B) Figure 2-7 Experimental and finite element results for steel-methacrylate (A), and energy absorption of methacrylate with steel adherends (B).
11
Patch Kinetic Energy [J]
25 Experimental Jonas-Lambert Fit = 205 MPa = 0.09 ydA fA = 225 MPa = 0.08 ydA fA
20 15 10 5 0
0
200 400 600 Projectile Kinetic Energy [J]
800
(A)
12
(B) Figure 2-8 Experimental and finite element results for steel-epoxy (A), and energy absorption of epoxy with steel adherends (B).
Table 2-3 Dynamic material properties for the adhesives under impact loading.
Adherend-Adhesive
Young's Modulus [GPa]
Steel-Methacrylate Steel-Epoxy
1.5 3.3
Dynamic Yield Strength [MPa] 115-135 205-225
Dynamic Fracture Plastic Strain 0.06-0.08 0.08-0.09
The transient stress through the bond line was extracted from the FEA along the midline of the bond as shown. The stresses are captured at 5 ߤ ݏtime increments from when the stress wave arrives until fracture. The normalized stresses along the bond line at various time steps are shown in
Figure 2-9 for (A) normal and (B) shear stresses. As seen, the bond line has two distinct zones: with the side closer to the impact (i.e. x/L close to zero) in tension, and the other side (i.e. x/L closer to one) in compression, where x is the distance along the bond and L is the total length of the bond. Changing the boundary conditions from fixed to free in the FEA did not show any significant changes in the results once more confirming that the boundary of the model (and experiment) does not play a role in the failure mechanism of the joint.
13
20µs 0µs 15µs
(A)
0µs
20µs
(B) Figure 2-9 Transient stress along the adhesive bond midline until just before fracture for (A) normal stress and (B) shear stress. Here the adhesive is epoxy with the Young’s modulus =3.3 GPa, the dynamic yield strength σyd = 210MPa and the failure strain = 0.08. The projectile impact velocity is 316m/s.
14
The stress distributions and propagation of the fracture in the bond after fracture initiation for the steelmethacrylate bond at the CFV of ~ 170m/s are shown in the sequence of plots in Figure 2-10(A) to (D). The fracture starts at the inner diameter of the joint where the normal tensile stress (σz) is highest at approximately 25µs after impact. The facture progresses towards the outer diameter of the joint until approximately 30µs at which time the outer-diameter of the joint begins to facture. When the two sides of the fracture coalesce the patch completely detaches which occurs at approximately 50µs after impact. These results are similar for the steel-epoxy cases which are not presented for conciseness.
Tension Compression
A
15
B
C
16
D Figure 2-10: FEM results showing the normal stress (σ σz ) distribution at the joint and the sequence of crack propagation through the joint at (A) just before fracture initiation at 24 µs after impact, (B) just after fracture initiation at 26 µs after impact, (C) 32 µs after impact and (D) 50 µs after impact when the patch dethatches i.e. the crack has fully propagated through the adhesive. The results shown here are for the steelmethacrylate case at the CFV of approximately 170 m/s.
Note that the FEA confirms that the failure of the adhesive occurs before the stress wave reflects from the boundary of the model.
2.2.5
Contact Force
The finite element contact force was calculated by taking the summation of the contact force of each node along the projectile surface. The FEA derived peak force and the average force can be found with this method. The average force is calculated by numerically integrating the force time history (i.e. the impulse of the projectile) divided by the contact time (i.e. ~ 29 µs for the methacrylate and ~38 µs for the epoxy). Here, contact time is defined as the time from initial force application of the impactor (~ 2 µs) until the time when the force returns to a value < 0.5% of the peak force. The results for the material combinations are shown in Figure 2-11 (A) and listed in Table 2-4. The contact forces are calculated at the CFV at rupture for each case.
17
Contact Time Epoxy
Epoxy
Methacrylate Contact Time Methacrylate
(A)
(B) Figure 2-11 (A) Force-time history calculated from FEA for methacrylate and epoxy joints at their CFV and (B) comparison of FEA calculated average impact forces to the experimentally determined impact forces using the Taylor method and camera estimates for contact time and conservation of momentum.
18
The projectile impact force from the experimental results for steel on steel can now be estimated using the dynamic stress taken from Taylor’s approximation (see Gollins et al. [8]). The average impact forces of the projectiles at the CFVs are listed in Table 2-4. This table also shows a comparison between the dynamic and static strength of the bonds. The strength of the bond is calculated as the failure load divided by the bond area. As such, the strength is an average for the full bond length. Furthermore, Table 2-4 clearly illustrates the dynamic effect within the bond. The dynamic strength of the epoxy is shown to be 3 times higher than that of the static strength, while the dynamic strength of the methacrylate is double for steel adherends. Table 2-4 Average impact force and bond strength at CFV
Incident Speed (m/s)
Adherend-Adhesive Steel-Methacrylate Steel-Epoxy
2.2.6
~170 ~320
Avg. Contact Force, Exp. (kN) 78 119
Projectile Impact Force, FEA (kN) 76.1 113.8
Static Strength, FEA (MPa) 14.6 13.2
Dynamic Strength, FEA (MPa) 30.0 44.8
Energy Balance
The experiment outlined in section 3.5 in Gollins et al. [8] i.e. the energy dissipation of an unbonded patch, was replicated using FEA to demonstrate the direct energy extraction approach. A similar model to the one shown in Figure 2-4 without the adhesive was used to simulate only the transfer of momentum from the projectile to the patch, and the kinetic energy of the patch after the impact was compared to the experimental results. The FEA calculated kinetic energy of the patch was consistently higher than the experimental data, which was as expected since the experiment was unable to measure any energy contribution due to vibration. Furthermore, the FEA possibly under-predicts some energy loss mechanism such as from damping. The un-bonded patch energies are plotted in Figure 2-7 and listed in Table 2-5. The energies can also be directly calculated. ABAQUS© resolves the energies of the components through the conservation of energy, thus the energies can be directly extracted. Implied by the first law of thermodynamics, the law of conservation of energy states that the time rate of change of kinetic energy and internal energy for a fixed body of material is equal to the sum of the rate of work done by the surface and body forces: ݀
݀ݐ
1
න ߩ ܸ݀ ݒ ⋅ ݒ+ න ߩܷ ܸ݀൨ = න ܵ݀ ߬ ⋅ ݒ+ න ݂ ⋅ ܸ݀ ݒ 2
(2.2)
19
where ߩ is the density, ݒis the velocity, ܷ is the internal energy per unit volume, ߬ is the surface traction
and ݂ is the body force. We can extract each of these energy components from the ABAQUS© calculations to find where the bulk of kinetic energy supplied by the projectile is distributed into the system after the impact. From, the energy equation, the kinetic energy is defined as: 1 = ܧܭන ߩܸ݀ ݒ ⋅ ݒ 2
(2.3)
= ܧܫන ߩܷ ܸ݀
(2.4)
and the internal energy:
where, U, is the summation of the recoverable strain energy, the plastic dissipation, the damage dissipation and any artificial energy from hourglassing. For all cases, the plastic dissipation is the largest contributor to the internal energy. The artificial energy was less than 1% of the strain energy. Initially, the kinetic energy of the projectile is imparted to the system. Upon impact, energy is spent to plastically deform the projectile and the patch. These are the two major energy sinks of the system, the plastic dissipation in the projectile and patch. Following the impact, some of the energy is transferred back into the projectile in the form of kinetic energy. The remaining energy is used to break the adhesive and transferred to the patch as kinetic energy after it becomes detached from the system. The summation of all the components in their final state is equal to the input energy. Alternatively, equation (2.) can be rewritten in term of the system components: ܧܭ − ܧܫ − ܧܭ − ܧܫ௧ − ܧܭ௧ − ܧܫ௧ − ܧܫௗ௦௩ = 0
(2.5)
where each energy component corresponds to its respective specimen part. The energies for the material combinations are shown in
Figure 2-12. The energy absorbed by the adhesive is represented by two cases with case 1 being the lower bound and case 2 being the upper bound for the dynamic yield strength and strain at failure given in Table 2-3. As seen in Figure 2-12 there is a negligible difference in absorbed energy between the upper and lower bounds. The energy absorbed in the adhesive is listed below in Table 2-5. Note that the energy absorbed by the adhesive as calculated directly by the FEA is similar for the case of the methacrylate (i.e. 5.2 J compared with 4.6 J as seen in Table 2-5) but is significantly lower for the epoxy (11.1 J compared to 16.9 J). This is largely due to the fact that the epoxy is significantly stronger than the methacrylate under impact. Analysis of the FEA results shows that the difference in energy is transferred as internal energy into the back-plate. (This is not shown in
20
Figure 2-12 for clarity). This in turn means that, not unexpectedly, the underlying assumptions that the adhesive does not contribute to the impact dynamics of the adherends and projectile break down as the adhesive becomes stronger.
KE: Projectile
IE: Patch
IE: Projectile KE: Patch
IE: Adhesive 1 and 2
(A)
IE: Patch
KE: Projectile IE: Projectile KE: Patch
IE: Adhesive 1 and 2
21
(B) Figure 2-12 Energy-time history for methacrylate (A) and epoxy (B) joints. Note that the internal energies (IE) for the both adhesives for Cases 1 and 2 are essentially the same and are indistinguishable on these plots.
Table 2-5 Comparison of energy absorbed in the adhesive by the direct and extraction approaches using FEA and experimental methods. The extraction method assumes the energy absorption of the adhesive is the difference between two tests: bonded and unbonded, whereas the direct approach resolved directly through FEA calculations
Adherend-Adhesive Steel-Methacrylate Steel-Epoxy
Projectile Speed [m/s] ~170
~320
Energy Absorbed EXP. [J] 4 16
Extraction Energy Absorbed FEA [J]
Direct Energy Absorbed FEA [J]
4.6 16.9
5.2 11.1
The absorbed energy calculated from the static simulation and the calculated dynamic energy absorbed are compared in Figure 2-13. The absorbed energy of the adhesive is seen to increase with the loading rate.
Dynamic
Dynamic
Static
Static
Figure 2-13 Comparison between the static and dynamic energy absorption for methacrylate and epoxy adhesives as calculated from the direct FEA calculations.
22
3
Conclusions We have presented a novel protocol for the dynamic characterization of adhesively bonded joints
under ballistic impact loading. The test is meant to be a dynamic analog to the commonly used static blister test. Finite element simulations show that similar to the static blister test, the maximum dynamic normal stresses are approximately four times greater than the dynamic shear stress. Thus, the adhesive is predominantly subjected to Mode I stresses. The finite element (FE) computational work presented in this part of this two-part article, was used to obtain stress profiles through the bond-line and to estimate peak contact forces. The FE work also validated some of the assumptions of the experimental results such as that the dynamical stress profiles are similar to the static-blister test profiles and are predominantly Mode I.
Furthermore, the Taylor
impact tests assume a rigid target, whereas visual inspection of the experimental test specimens after impact and digital video analysis during the impact indicate that there is relatively large deformation in the patch. Thus, the experimentally calculated impact forces only serve as an approximation of the actual dynamic forces experienced by the patch. Additionally, the stiffness of the adhesive is assumed to be negligible relative to the adherends. If the stiffness of the adhesive is significant, the experimentally measured energy absorption of the adhesives may not be accurate. Both the rigid target and negligible adhesive stiffness assumptions were taken into account in the present study with a validated FEA. There are several obvious extensions and further validations of the work presented here that should be addressed in future work. The first is to fully study the effect of geometry (for example adhesive bond length and thickness) on the performance of the adhesives under ballistic loading. The impact dynamics used in the FEM could be further validated by comparing the plastic deformation of the patch to the experimental results. Furthermore, environmental effects such as temperature and humidity should also be extensively studied under dynamic loading. The experimental method presented in this paper can be readily adapted to include these modifications. Furthermore, a complete dynamic constitutive model for the adhesives which would be invaluable for the design of adhesive joints should be the focus of future work.
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Acknowledgements
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This work was supported by U.S. Army- TARDEC (currently US-Army GVSC) under contract #69201511 through NAMC. We thank Dr. Robert Jensen of ARL and Daniel Schaffren of US Army-GVSC (previously TARDEC) who provided their expertise on surface preparation and bonding. Additionally, we thank our colleagues Dr. Bruno Zamorano and Dr. Jack Chiu for their insightful discussions and contribution to this work.
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