A three phase model charactering low ‐ velocity impact 2 response of SMA reinforced composites under 3 vibrating boundary condition

Structural vibration induced by dynamic load or natural vibration is a nonnegligible factor in failure analysis. Based on vibrating boundary condition, impact resistance of shape memory alloy reinforced composites is investigated. In this investigation, modified Hashin’s failure criterion, Brinson’s model and visco-hyperelastic model are implemented into the numerical model to charactering the mechanical behavior of glass fiber/epoxy resin laminates, SMAs and interphase, respectively. First, fixed boundary condition is maintained in simulation to verify the accuracy of material parameters and procedures by comparing with experimental data. Then, a series of vibrating boundaries with different frequencies and amplitudes are applied during the simulation process to reveals the effect on impact resistances. The statistics of absorbed energy and contact force indicate that impact resistance of the composite under high frequency and large amplitude is lower than that under low frequency and small amplitude, and summarized by a mathematical expression.


Introduction
Fiber reinforced composite materials have been widely used and investigated in recent years due to their unique properties, such as high stiffness, high strength and low density [1].However, the applications of the composites are limited by their weak impact resistance, especially for the unidirectional cross-ply fiber/matrix laminates.Obviously, the reason for this disadvantage can be traced to the low strength, stiffness between the adjacent layers which will result in delamination and fiber breakage [2].This disadvantage can be overcome by changing the composition of material: for example, using short fiber instead of long fiber.The mechanical properties of short fiber reinforced composites are strongly influenced by the manufacturing process, such as injection position, sample geometry, pressure and temperature during the molding process [3][4].Other factors, such as fiber location, length, diameter and orientation that affect the mechanical properties, have been studied by Thomason [5][6][7], Wang [8] and Mallick [9] investigate the tensile properties of composite machined with different angles to mould flow direction (MFD).The results indicate that tensile strength (and elastic modulus) of samples machined perpendicular to MFD are nearly 40% lower than that of samples machined parallel to MFD.Functional materials and new structures have been used to improve the mechanical property of the composites.Shape memory alloys (SMAs) have been embedded between the adjacent layers for the reinforcement purpose considering their unique properties: changing shape (elastic modulus) in accordance with temperature and stress [10][11].In our previous research, the effect of SMAs positions on damage behavior and impact response (including peak force, displacement and energy) of laminates subjected to low-velocity impact have been investigated [12].In Khalili's research, the effect of SMAs type (wires, plates, strips, tubes or layers) on dissipation of the impact energies has been studied [13][14].According to Shariyat's research, a higher order global-local hyperbolic plate theory aimed at studying the asymmetric displacement fields has proposed, the related results indicate that SMAs have the ability to change shape, to repair damage and to improve impact property of composites [15][16].The new structure bounding of the conventional fiber reinforced polymer with metal layer is also found has the advantage when subjecting to impact loading [17].
We note that the fixed boundary is widely used in the experimental analysis.Except for low and high velocity impact, the mechanical properties of composite under eccentric impact [24], multi-impact [25] and compression-after-impact [26] has received much attention due to engineering practices.However, the vibrating boundary condition, which is unavoidable when membrane structures subjected to impact [27] or wind flow [28], has rarely discussed due to the difficulties associated with real experimental condition.Only vibration response during impact and after impact can be observed to evaluate the impact resistance and damage state [29][30].In Pérez MA's work, damage of CFRP induced by low velocity impact and its effect on vibration response is investigated by a micro-mechanical approach [31].
The interfacial debonding between SMA and the matrix, one common failure model, is still a key problem in the composites [32].As reported in our previous work, a three phase model-matrix, reinforcer and interphase is introduced to evaluating the mechanical behavior of fiber reinforced composite [33].In this paper, this model is further developed to matching the SMA reinforced composites.Based on this model, the effect of vibration on impact resistance of SMAs reinforced composites is mainly investigated through a series of frequencies and amplitudes.

Material property of glass fiber reinforced composites
Glass fiber and the matrix used in this paper are regarded as homogeneous isotropic materials on microscale.The constitutive model can be expressed as: where ij  and ij  are nominal strain and stress in case of i j  , shear strain and shear stress in case of i j  ( , x, y and z i j  are the reference coordinates X), respectively.= is the shear modulus, E and  are the elastic modulus and Poisson's ratio,  is the Kronecker delta ( . The relationship between displacement field i u and strain field ij  defined as: However, it's quite unrealistic modeling each glass fiber in ABAQUS considering the complexity.A ply including glass fiber and epoxy resin is modeled as anisotropic material.The stacking sequence is [0°, 90°]8, and the total thickness of the sample is about 3.2 mm (0.2 mm for each ply).The related material parameters are obtained according to the experimental work of our research group [12], as shown in Table 1.For the undamaged state, the stress strain relationship can be rewritten as: where cij are the stiffness coefficients which can be derived from G and  .At elastic state, the specified damage variables i d are equal to 0.
The 3D Hashin's failure criterion accounting for fiber failure and matrix failure is embedded in ABAQUS using subroutine-VUMAT.In this part, two failure models, tensile failure and compressive failure, are mainly considered for the fiber with related damage variables which can be expressed as [34][35]: Tensile failure of fiber, Compressive failure of fiber, where ft d and fc d are the damage variables for evaluating tensile failure and compressive damage of fiber, respectively.
Similarly, tensile failure and compressive failure of the matrix can also be obtained [34][35].
Tensile failure of matrix, 1 mt d  : 1 where mt d and mc d are the damage variables for evaluating tensile failure and compressive damage of matrix, respectively.
Using the damage variables and related parameters, the stress is decreased linearly to zero once the failure criterion is reached.The stiffness coefficients obtained from Equation 3 should be re-calculated during damage process, as: where mt s and mc s are the factors that control the reduction in shear stiffness according to tensile and compressive failure, respectively.Parameters are the global damage variables charactering fiber and matrix, respectively.

Material property of SMA
A similar stress-strain relationship as Equation 1 can be observed if the reinforced material is isotropic elastic, i.e. glass fiber or carbon fiber.Differently, the constitutive model of SMA is sensitive to temperature and stress state.Brinson's model [36][37] has been referred to most often in subsequent studies due to its high accuracy.Here, the constitutive law of SMA based on energy balance equations is derived considering the effect of temperature T and phase conversion between martensite and austensite can be denoted as: where s  and s  are the Piola-Kirchhoff stress and Green strain of SMA;  is the martensite fraction charactering the phase conversion; also, s E ,  and  are the elastic modulus, transformation coefficient and thermal coefficient, respectively.Some assumptions are taken to simplify the application, as follows: the elastic modulus D is a function of martensite fraction  ; the transformation coefficient is also connected with elastic modulus: where m E and a E are elastic moduli for martensite and austenite state, L  is the maximum residual strain.The equations of the phase conersion between martensite and austenite can be expressed as (determined by temperature and stress): (a) Phase conversion to martensite In Equation 11and 12, parameters

Material property of interphase
According to our research in glass fiber reinforced polymer composite, the time (or strain rate) depended stress-strain relationship has been investigated [33].In this model, the time affect is existing in interphase part and can be expressed as stress

 
, in ij t  to strain relationship experiencing some continuous strain history given by the function

 
, where g is the relaxation modulus, t is new time variable, is strain rate.The strain history   in t  can be obtained using Boltzmann superposition principle.Also, the relaxation modulus g can be expressed using discrete relaxation spectrum as follows: where 0 g , i g and i t are the related parameters which can be obtained from relaxation tests.The parameters in interphase part can be obtained by fitting with tensile or pull-put tests with different loading speed [33].

Boundary condition
For a sample with dimension , n=16 for our experiment, the fixed boundary is employed to investigate the accuracy of numerical model.
Frequency and amplitude of the four boundaries 0 x  , As for vibrating boundary condition (Figure 1b), the movement of the boundaries can be expressed as: where A and f are the amplitude and the frequency of the vibration, respectively.

Effect of fixed boundary condition on impact resistance
The fixed boundary condition is kept in the experiment process.

Composite laminates
Simulation The SMA wires (diameter 0.2mm) have been embedded in the middle layer laminate (between layer 8 and 9) with distance 5mm (total 21 wires for a model).In summary, four types of experiments have been conducted, as shown in Table 2.

Simulation result: damage state during impact process
The model has been built in ABAQUS according to the details mentioned before, as shown in

Group A1:
The simulation results of the composite laminate under impact at different time (0.0015 s < t < 0.009 s) are shown in Figure 3 and Figure 4.Under lower impact energy, 32J, a generally elastic behavior of the composite laminate can be drawn as: deformation is increased with time t from initial state (t = 0) to the maximum deformation (t  0.0045s), then the deformation is decreased.
As shown in Figure 3, it is clear that the impactor has been bounced back by the composite laminate.Three layers have been chosen to investigate the fracture morphology during impact as shown in Figure 4: layer 16, 8 and 1.A hole shape damage can be founded on several layers, especially layer 16, however the damage region on the layer 8 and layer 1 are smaller, as shown in Group A2: Different from group A1, the composite is destroyed completely under higher energy, 64J, as shown in Figure 5.During simulation process, the impactor is founded moving along the top layer to the bottom layer of the composite without bounced back (t > 0.0030s), this can also be confirmed by the experimental result, as shown in Figure 5g.The velocity of impactor is reduced in the breakdown process then remained as a constant.
From Figure 6c, a larger damage region is founded at the final state.This is different from group A1, and can be explained by delamination due to the friction between impactor and layer 1 during penetration.Group A3 and A4: Embedding SMA alloys is effective to improve the impact resistance of composite laminates.As shown in Figure 7a, SMAs was stretched to a larger strain in the case of 32J.In Figure 7b, a broken or invalid state of SMA is obtained due to the larger strain which is beyond critical value.More specific, 5 SMAs in the center region are chosen to demonstrate the working mechanism, as shown in Figure 8c and 8d.
Beyond that, the defect is obviously: the damage region of layer 8 (contact with SMA) is larger than that of group A1 and A2, as shown in Figure 8a and 8b.The positive value in Table 3 means that the moving direction of the boundary is on the contrary of the impactor's moving direction (+z direction) at initial state.The negative value means that moving directions of impactor and boundary are same at initial time.
When applying amplitude A to control the movement of the boundary, the morphologies of composite laminate after impact are shown in Figure 10 and 11.As shown in Figure 10a, the damage region is close to the size of impactor.From Figure 11b-11d, the damage regions are increased with increasing of the value of applied amplitude, however, general damage with larger area can be observed in case of A=0.016m, as shown in Figure 10d.It is interesting that the damage state is depends on the absolute value of amplitude rather that the valuence when comparing related groups, i.e. Figure 10b  Three amplitudes are shown to demonstrate the impact process of group C: A = 0.0032m, 0.008m and 0.016m, as shown in Figure 13.From Figure 13a, the layer damage along the SMA direction is founded during impact process.This is due to the weak impact resistance of laminate after separating from SMA, especially for layer 8.With increasing amplitude, a clear delamination can be observed between layer 8 and 9, t = 0.0005s, as shown in Figure 13b.As for group C4, a clearly damage is shown at early time, t = 0.0004s.The damage state is still extending even after separating,due to the vibrating of the boundary, as shown in Figure 13c.With different amplitudes, the absorbed energy and contact force are plotted against time, as shown in Figure 14.As for group B1 and C1, the maximum value of absorbed energies is 32J, however, the values at t = 0.01s are 30.04Jand 28.6J, respectively.With increasing the value of amplitude the absorbed energy is decreased.As for group B3, B5 and B7, the maximum value of absorbed energies are 9.24J, 4.24J and 0.30J, respectively.The maximum value of absorbed energies in related opposite direction groups, i.e.B4, B6 and B8, kept as same level and about 20% lower.As for related group C, the absorbed energy is 3.5 times larger.As for high amplitude, the absorbed energy is close to zero, as group B7 and B8.From t =0 to 0.004s, effect of vibrations on the energy-time curve can be observed, then, the energy is kept as constant, as shown in Figure 14a.In Figure 14b, the force is also affected by the vibration and shown more dramatic changes comparing with section 3.

Effect of frequency
In order to fully understand the influence of frequency on the impact resistance, a small amplitude A = 0.0032m is kept considering the small influence of this level.Several frequencies, f, are chosen for the study as shown in Table 3.
During the simulation f = 100 cycle/s to 500 cycle/s in group D1 to D3, the damage state is close to each other at time t = 0.01s, only a small hole can be founded according to the fracture morphology shown in Figure 15a, 16b and 16c.As for higher frequency f = 2000 cycle/s to 10000/s, the damage states show randomicity with larger area, as shown in Figure 15d and 16e.
Damage states of SMA reinforced composite laminates are shown in Figure 15f to 15j, also.
Overall, the effect of frequency on the damage state is similar with that in group D. It should be noted that the separated region in group E5 maintains a more complete shape, as shown in Figure 15j.Applying different frequencies f on the boundary, the simulation process of two groups: D5 and E5 are invested to demonstrate the details of damage morphologies of the composite laminate, as shown in Figure 16.As for D4, an hole shape damage region is gradually appear at time t = 0.0025s, then increased with impact process, at time t = 0.01s the center region is damaged completely.As for group E5, delamination is observed except the hole shape damage, this is mainly due to the global enhancement effect of SMA.
The relationship between absorbed energy and time or contact force and time can be founded in Figure 17.As shown in Figure 17a, the absorbed energy is decreased with increasing frequency generally.Considering a real low frequency f =100/s, the maximum value of absorbed energy for group D1 and E1 are same, 32J.Considering a high frequency f = 10000/s, the absorbed energy for group E5 is 2.36J, as for D5 the absorbed energy is 1.79J, which means the composites can barely bearing impact.As shown in Figure 17b, the maximum value of force in the case of f<2000 cycle/s is under the range 7N to 7.5N for group D, 9N to 12N for group E. As for group D5 and E5, a saltation is observed comparing force with the adjacent groups which is mainly due to the transient change of velocity.
More important, the maximum value of absorbed energy and contact force for SMA reinforced composites is generally 15%-30% larger that than of pure glass fiber reinforced composites under same amplitude or frequency.4, the maximum energy-Emax, energy at time 0.01s-Et=0.01,maximum force-Fmax and average force-Favg of different groups are shown.Energy Emax=32J denotes a rebound behavior of impactor.The average force is defined as:

Statistical analysis of damage state
where N is the total number of output data of force Fi within time 0.01s, more important, the average value is still in accordance with A and f.
In Figure 18, the ratio between damaged area and the whole model are plotted against time.
Generally, the damaged areas of laminates for A < 0.0032m or f <500 cycle/s is kept as < 5%.In Figure 18a, the damaged areas of top layer (layer-16) for A > 0.0032m or f >500 cycle/s is kept as 30-50%.In Figure19a, the average damage area are investigated by calculating 16 layers, the results indicate that a 4% damage for small amplitude and frequency and a nearly 50% damage for large amplitude or frequency at time 0.01s.In Figure19b, the damage state of SMA is studied, the damage of SMA can be observed at time earlier than the damage of composite laminates which means that SMA has advantages absorbing energy.

Mathematical expression-effect of amplitude and frequency
where, where ka and mi are parameters related to amplitude; kf and ni are parameters related to frequency.Inserting Equation 15 to energy equation, the relationship between absorbed energy and amplitude (or frequency ) is obtained as: where E and m is the absorbed energy and mass of the impactor, respectively.Considering that average avg F m v t    , the average force is also obtained as: The simulation results using the equations mentioned above are shown in

Conclusions
The effect of vibrating boundary on the impact resistance of SMA reinforced composite Comparison between the simulation results and our previous work with fixed boundary shows that the parameters and the simulation process are acceptable with relative error smaller than 10% (both for 32J and 64J).As for vibrating boundary condition with different amplitudes and frequencies, the absorbed energy and the fracture morphology of the composite laminate have been studied.
g Both high frequency and amplitude can weaken the impact resistance of composite laminate, extensive damage can be observed rather than impact hole.
Four important parameters are introduced to character the phase conversion temperature: s M -start temperature of martensite phase; f M -finish temperature of martensite phase; s A -start temperature of austenite phase; f A -finish temperature of austenite phase.The stress-strain relationship at arbitrary temperature T can be obtained according to Equation 9-12.Debonding behavior of interface between SMA and matrix is carried out by use of cohesive zone model.
results of the model under fixed boundary are investigated and compared with the experimental results.Stacking sequence of the laminate is [0 o ,90 o ]8, 0o and 90o are the layer angles (glass fiber) to X-direction.The sample ( 100mm×100mm×3.2mm) is impacted by a rigid half ball-cylinder on the center of the top surface, as shown in Figure 1.Diameter of the half ball of the impactor is 14mm, and the mass of the impactor is 8kg (Steel).Two impact energies are considered in the tests, 32J and 64J, and the corresponding impact velocities are 2.83m/s and 4m/s.

Figure 5 .Figure 6 .
Figure 5. Simulation results of the composite laminates at different time, group A2.

Figure 10 .Figure 11 .Figure 12 .
Figure 10.Fracture morphology of the top layer of composite laminate under different amplitudes, group B.

Figure 15 .Figure 16 .
Figure 15.Fracture morphology of the top layer with different frequencies.

PreprintsFigure 17 .
Figure 17.Analysis of the impact resistance of composite laminate in group D and E: (a) absorbed energy; (b) contact force.
laminates has been investigated.Two main factors are used to charactering the vibration of the boundary: amplitude and frequency.A 3D finite element model based on Hashin's failure criterion was employed in ABAQUS to study the destruction process.A Mathematical expression is obtained to charactering the effect of vibration on energy and force.

Table 1 .
Material parameters of unidirectional glass fiber/epoxy composite laminates.

Table 4 .
Statistics of absorbed energy and contact force.