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Simulation of stress wave attenuation in plain weave fabric composites during in-plane ballistic impact
Composite Structures 160 (2017) 748–757
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Simulation of stress wave attenuation in plain weave fabric composites during in-plane ballistic impact Kedar S. Pandya a,b,⇑, Mandar D. Kulkarni a, Anirudh Warman a, N.K. Naik a a b
Aerospace Engineering Department, Indian Institute of Technology Bombay, Powai, Mumbai 400 076, India Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, UK
a r t i c l e
i n f o
Article history: Received 10 May 2016 Accepted 27 October 2016 Available online 28 October 2016 Keywords: In-plane impact Plain weave fabric composite Stress wave attenuation
a b s t r a c t An analytical method is presented for the simulation of longitudinal stress wave attenuation in typical plain weave fabric composites made of E-glass/epoxy and T300 carbon/epoxy under in-plane ballistic impact. The stress wave propagation/attenuation simulation studies are carried out for the case of in-plane impact of projectiles on composite beam targets. Geometrical modeling of plain weave fabric composites and an algorithm for stress wave propagation/attenuation are presented. Studies are carried out for both free edge and rigid boundary conditions for different beams. Based on the analytical simulation, for typical plain weave E-glass/epoxy composite beams studied, it is observed that the stress intensity is about 5% of the incident value when the longitudinal stress wave reaches a distance of 110 mm from the point of in-plane impact at the representative section. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction When a projectile impact takes place onto a target, different kinds of stress waves emanate and propagate outward from the point of impact [1]. In general, stress waves propagate along the planar as well as through the thickness directions. When a planar longitudinal stress wave propagates through a periodically layered material system with different physical and mechanical properties, stress wave intensity attenuates over a distance, i.e., the magnitude of stress decreases as the wave propagates further and further. When a stress wave encounters a material boundary, part of the incident wave is transmitted into the neighboring material and part of it is reflected back into the incident material. As a result, the intensity of the stress wave decreases over a distance. A typical example demonstrating the above mentioned phenomenon is that of longitudinal stress wave propagation in plain weave fabric composite beams under in-plane impact. Woven fabrics are made by the process of weaving by interlacing warp and fill yarns in a regular up and down sequence. For composite beams made of such fabrics, longitudinal stress waves generated during in-plane impact encounter many interfaces. At such interfaces, material physical and mechanical properties change leading to stress wave attenuation. ⇑ Corresponding author at: Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, UK. E-mail address: [email protected] (K.S. Pandya). http://dx.doi.org/10.1016/j.compstruct.2016.10.115 0263-8223/Ó 2016 Elsevier Ltd. All rights reserved.
Mechanisms like transmission and reflection at the material interfaces, geometric wave dispersion, crush-up of porosity, dissipation and viscous losses are responsible for attenuation of stress waves in composites. Several experimental [2–8] and analytical [9–15] studies are reported in literature on stress wave attenuation in composites. Stress wave attenuation has also been observed in other layered material systems [16,17] and ceramic plates [18,19]. Pandya et al. [20] presented experimental studies on stress wave attenuation in composites during ballistic impact. Stress wave propagation studies have been carried out in order to predict stress and strain distributions as well as fracture and energy absorbing mechanisms in composite structures under dynamic and impact loading conditions [21–24]. During in-plane impact of a projectile onto a beam target, longitudinal and transverse stress waves are generated in the target along the length of the beam. In the present study, longitudinal stress wave propagation/attenuation is considered. After a review of the literature, it is evident that the effect of reflection and transmission of stress waves at the interfaces on attenuation is not fully understood. The objective of the present study is to investigate longitudinal stress wave attenuation in plain weave fabric composite beams under in-plane impact. Impulse loading (unit step load of infinitesimal duration) is considered in the present study. Stress wave attenuation is attributed to reflection and transmission of stress waves at material interfaces.
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Nomenclature a A c d dp E h l mp P t V50
yarn size/width cross-sectional area wave velocity distance diameter of projectile elastic modulus beam height beam length mass of projectile longitudinal unit step pressure pulse time ballistic limit velocity
2. In-plane impact onto a composite beam target Composite structures undergo various loading conditions during their service life. One of the typical loading conditions is the in-plane impact of a rigid projectile (Fig. 1a). For woven fabric composite beams, a typical representative section along warp/fill adjacent to the point of impact is analyzed (Fig. 1b). This representative section is effectively a one-dimensional composite. When an inplane impact by a projectile onto a composite beam takes place, two waves namely, longitudinal and transverse, propagate from the point of impact [1,25,26]. For woven fabric composites, the typical representative section, i.e., the one-dimensional composite can be treated as a layered composite along the warp/fill direction. The mechanical properties would be varying along the warp/fill direction. This is because of undulation of the yarns in the composite. The longitudinal stress wave propagates along the layered composite beam. The incident wave is partly reflected and partly trans-
Vf
C
q r
Subscripts I incident R reflected T transmitted 1 material 1/warp 2 material 2/fill
mitted as and when it encounters the material interfaces. For the typical representative section (Fig. 1b), cross-sectional areas of the adjacent regions would be the same. However, material impedances would be different for the adjacent regions. The problem formulation for wave propagation/attenuation in plain weave fabric composite beams is based on the following assumptions: The plain weave fabric composite beam can be idealized as a mosaic model of two materials. One-dimensional wave propagation theory can be used for the analysis. Target is a beam structure. 3. Stress wave attenuation modeling When a longitudinal wave crosses the interface between two materials with different properties, the wave undergoes reflection and transmission. Wave velocities in two adjacent materials with different impedances can be calculated as:
c1 ¼
Fig. 1. In-plane ballistic impact of a projectile on to a composite beam target: (a) geometry, (b) representative section along warp, (c) longitudinal unit step pressure pulse.
fiber volume fraction impedance ratio density stress
qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi E1 =q1 and c2 ¼ E2 =q2
Fig. 2. Geometrical modeling of plain weave fabric composite: (a) a typical single layer cross-section, (b) idealized modeling, (c) sublayers 1 and 2, (d) equivalent sublayers 1 and 2.
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where c is wave velocity, E is elastic modulus and q is density. Subscripts 1 and 2 refer to materials 1 and 2, respectively. An incident stress wave with stress intensity rI approaching an interface in a slender bar gives rise to reflected and transmitted waves with stress intensities rR and rT respectively. The magnitude of the reflected and transmitted components is given by [27],
rR C 1 ¼ rI C þ 1
ð1Þ
rT 2 C A1 ¼ rI C þ 1 A2
ð2Þ
where rI, rR and rT are incident, reflected and transmitted stresses, respectively. A is cross-sectional area. Subscripts 1 and 2 refer to materials 1 and 2, respectively. Here, C is known as impedance ratio and is given below,
C¼
A2 q2 c2 A1 q1 c1
ð3Þ
For the cases when A1 = A2, it can be seen that, if C is less than 1, then both the reflected and transmitted waves have a smaller amplitude than the incident wave, and the reflected wave changes its sign. On the other hand, if C is more than 1, both the reflected
and transmitted waves maintain the sign of the incident wave. The transmitted component attains a larger magnitude, while the reflected wave magnitude is smaller. In the present analysis, longitudinal wave propagation/attenuation is considered in the impregnated yarn consisting of two sublayers as shown in Figs. 2 and 3. The impregnated yarn is treated as a one-dimensional element. Sample calculation for stress wave attenuation in a layered medium (Figs. 2 and 3) is given in Appendix A. 4. Modeling of plain weave fabric composite beams for stress wave attenuation Woven fabric composites are made by impregnating woven fabrics with polymer resins. One of the fundamental woven fabrics is the plain weave, where the warp and fill strands are interlaced in a regular sequence of one under and one over. The main geometrical parameters are: undulated length and length of the straight portion of the strand, strand thickness, strand width and interstrand gap [28]. An idealized plain weave fabric composite can be represented using a geometrical repetitive unit [28]. For composites with balanced fabrics, geometrical and material properties are the same
Fig. 3. Schematic arrangement of in-plane ballistic impact loading and one-dimensional stress wave propagation: (a) representative section, (b) sublayer 1, (c) sublayer 2, (d) typical elements, (e) schematic representation of free edge condition, (f) schematic representation of rigid boundary condition.
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A typical single layer plain weave fabric composite beam crosssection is shown in Fig. 2a. Fig. 2b indicates the idealized modeling for plain weave fabric composites. In this figure, the fibers along warp and fill and the resin are separated into layers, with resin occupying the middle layer. In Fig. 2c, the idealized model is split into two sublayers, namely sublayer 1 and sublayer 2. Resin is equally distributed in both the sublayers. In Fig. 2d, resin is evenly redistributed within the sublayers. Such sublayers are referred to as equivalent sublayers. Stress wave attenuation studies are carried out on a large number of equivalent sublayers placed one after another. For brevity, equivalent sublayers are referred to as sublayers for further discussion. It may be noted that the sublayers consist of simplified representation of warp and fill. Here, warp is represented as a unidirectional composite with fiber direction along the longitudinal axis whereas fill is represented as a unidirectional composite with fiber direction perpendicular to the longitudinal axis. Element 1 represents warp whereas elements 2 and 3 represent fill (Figs. 2 and 3).
Schematic arrangement of in-plane impact loading and longitudinal stress wave propagation is shown in Fig. 3. Longitudinal stress waves would be propagating along the +x direction. For sublayer 1, the stress wave propagates first in warp and then in fill towards +x direction (Fig. 3b). For sublayer 2, the stress wave propagates first in fill and then in warp towards +x direction (Fig. 3c). In-plane impact takes place at the left face of the beam at ‘O’ and longitudinal stress waves start propagating along the +x direction. Whenever warp-fill or fill-warp interface is encountered, wave reflection and transmission would take place. Because of wave reflection and transmission, stress wave attenuation takes place in all the elements. A typical ‘element 1’ is shown in Fig. 3d. As the sequence of reflection and transmission continues, the magnitude of the stress wave would be decreasing and tending towards a very small value compared to the magnitude of the incident longitudinal stress wave. At the end of the representative section of the composite beam target, the boundary condition or interface properties are different. Fig. 3e represents free edge condition whereas Fig. 3f represents rigid boundary condition. In the present study, longitudinal unit step loading at the point of in-plane impact onto a plain weave fabric composite beam is considered for ease of calculations. Results are presented for idealized plain weave fabric lamina representative section consisting of both sublayers 1 and 2. The lamina considered is the top layer of the composite beam target.
Fig. 4a. Stress wave propagation in plain weave E-glass/epoxy target sublayers under in-plane ballistic impact unit step loading, l = 200 mm, t = 0.05 ls, a = 2 mm: (a) sublayer 1, (b) sublayer 2.
Fig. 4b. Stress wave attenuation in plain weave E-glass/epoxy target sublayers under in-plane ballistic impact unit step loading, l = 200 mm, t = 0.5 ls, a = 2 mm: (a) sublayer 1, (b) sublayer 2.
along warp and fill. For such composites, geometrical repetitive unit would be of square shape with the dimension of the side equal to the sum of strand width and inter-strand gap. This dimension is referred to as ‘a’. 4.1. Geometrical modeling
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Fig. 4d. Stress wave attenuation in plain weave E-glass/epoxy target under in-plane ballistic impact unit step loading, l = 200 mm, t = 1 ls, a = 2 mm.
Fig. 4c. Stress wave attenuation in plain weave E-glass/epoxy target sublayers under in-plane ballistic impact unit step loading, l = 200 mm, t = 1 ls, a = 2 mm: (a) sublayer 1, (b) sublayer 2.
4.2. Algorithm for stress wave propagation/attenuation As the projectile strikes the beam, the beam resists penetration of the projectile. This builds up the contact force between the target and the projectile. During this event, a longitudinal stress wave is generated and propagates from the point of impact along the beam length. The contact force induces shear and normal stresses within the target. As the induced stresses exceed the permissible stresses, cylindrical cracks would form within the target around the periphery of the projectile ahead of the tip and would propagate through the thickness. The longitudinal stress waves would not propagate through these cracks since the cracks form a free boundary. The algorithm for simulation of longitudinal unit step stress wave propagation/attenuation in plain weave fabric composite beam targets is discussed below:
The plain weave fabric composite is divided into sublayers 1 and 2. Further, sublayers are divided into smaller geometrical segments, i.e., into elements. Dimensions and material properties are assigned to the elements. A longitudinal incident unit step stress wave is introduced into sublayer 1 in the +x direction from the location of impact, i.e., at point ‘O’. The stress wave velocity in each element is governed by the material properties. Wave propagation along +x direction is tracked. As and when the stress wave encounters a material boundary, it leads to reflected and transmitted components. The amplitudes of reflected and transmitted components are evaluated. The transmitted stress wave becomes an incident stress wave for the adjacent element and its data is saved for later tracking. The reflected stress wave is further tracked within the same element till it encounters an interface. After it reaches the interface, it further leads to the next level of reflected and transmitted components. The amplitudes of these components are also evaluated and saved. The process of reflection and transmission of stress waves continues at both the interfaces of the element under consideration. Saving and tracking the transmitted and reflected wave data continues until the magnitude of the stress waves is reduced to a predefined threshold value. Next, the transmitted stress waves and their corresponding reflected components are tracked and saved in a similar manner in the adjacent elements. The initial incident stress wave encounters many interfaces as it propagates, and transmission and reflection take place at all the interfaces as well as at free edge or rigid boundary of the target. At a given time interval a number of stress waves are present at all points at which the stress front has reached. The sum of all the stress wave magnitudes at each point of the element gives the resultant stress wave magnitude at that point. Similar studies are carried out for sublayer 2. Based on the stress wave propagation and attenuation studies carried out for sublayers 1 and 2, the resultant magnitude for representative section consisting of both the sublayers is computed for the duration of impact.
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The wave tracking algorithm developed allows the wave to be tracked at a predefined sampling time. The results correspond to the longitudinal stress wave attenuation behavior for an idealized plain weave fabric composite beam under in-plane impact loading. The lamina considered is the top layer of the composite beam target. Based on the above algorithm the normalized stresses are plotted as a function of time and distance from the point of impact for the plain weave fabric composite beam. 4.3. Input details The input to the algorithm consists of yarn size and composite elastic moduli and densities. Studies are carried out for balanced plain weave fabric composites with yarn size varying from 1 mm to 4 mm. Here, the yarn size refers to the sum of strand width and inter-strand gap [28].
Fig. 4e. Stress wave attenuation in plain weave E-glass/epoxy target sublayers under in-plane ballistic impact unit step loading, l = 200 mm, t = 40 ls, a = 2 mm: (a) sublayer 1, (b) sublayer 2.
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For the typical plain weave fabric E-glass/epoxy composite beam considered, Equivalent warp elastic modulus, E1 = 31 GPa Equivalent fill elastic modulus, E2 = 8.4 GPa Density for warp and fill, q = 1750 kg/m3 For the typical plain weave fabric T300 carbon/epoxy composite beam considered, Equivalent warp elastic modulus, E1 = 94.5 GPa Equivalent fill elastic modulus, E2 = 8.2 GPa Density for warp and fill, q = 1406 kg/m3 5. Results and discussion Longitudinal stress wave propagation/attenuation studies are carried out for a typical plain weave fabric composite beam under longitudinal in-plane unit step loading at different time intervals. The results presented are for the top layer of the composite beam
Fig. 4f. Stress wave attenuation in plain weave E-glass/epoxy lamina under inplane ballistic impact unit step loading, l = 200 mm, a = 2 mm: (a) t = 20 ls, (b) t = 40 ls.
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Fig. 5a. Stress wave attenuation in plain weave E-glass/epoxy lamina under in-plane ballistic impact unit step loading, t = 40 ls, a = 2 mm, free edge: (a) l = 100 mm, (b) l = 50 mm, (c) l = 40 mm.
Fig. 5b. Stress wave attenuation in plain weave E-glass/epoxy lamina under in-plane ballistic impact unit step loading, t = 40 ls, a = 2 mm, rigid boundary: (a) l = 100 mm, (b) l = 50 mm, (c) l = 40 mm.
Table 1 Normalized stress with applied in-plane ballistic impact unit step loading on beams (length in mm), plain weave fabric E-glass/epoxy, t = 40 ls, a = 2 mm. Position from origin, x (mm)
0 10 20 30 45 60 90
200
1.2205 0.9908 0.9309 0.4607 0.2860 0.1743 0.0876
120
1.2205 0.9908 0.9309 0.4607 0.2860 0.1743 0.0876
Free edge
Rigid boundary
100
50
40
100
50
40
1.2205 0.9908 0.9309 0.4607 0.2860 0.1743 0.0482
0.8137 0.8887 1.2156 0.3271 0.0818 – –
1.0026 0.7802 0.6859 0.1055 – – –
1.2205 0.9908 0.9309 0.4607 0.2860 0.1743 0.0985
0.8137 0.8887 1.2156 0.3271 0.0818 – –
1.4385 1.2013 1.1748 0.8160 – – –
target. Studies have been carried out for both free edge (case 1) and rigid boundary (case 2) conditions (Appendix A). In the analytical studies, an impulse load of unit magnitude is assumed for simplicity. For a practical impact loading scenario, the analysis can be carried out using the actual contact force and corresponding planar longitudinal stress wave intensity. 5.1. Mechanical behavior of composite beam targets under in-plane impact During in-plane impact event, the beam target offers resistance to penetration/perforation of the projectile into it. Incident kinetic
energy of the projectile would be absorbed by the target through various damage and energy absorbing mechanisms. It should be noted that the beam does not undergo vibration during the inplane impact event. The target has a tendency to rebound leading to possible vibration only after the impact event. During in-plane impact, longitudinal and transverse stress waves propagate from the point of impact along the beam length. Based on the length of the beam, these stress waves may or may not reach the boundary of the beam. Depending upon the boundary conditions, the extent of transmission and reflection of the stress waves at the boundary can be evaluated. The reflected waves can affect the intensity of the stress waves within the target.
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However, they would not influence vibration characteristics of the target. 5.2. Stress wave propagation as a function of time Stress wave propagation/attenuation studies for a typical plain weave fabric E-glass / epoxy beam of 200 mm length are presented in Figs. 4a–4f. The yarn size, i.e., the element dimension is 2 mm. At time interval of t = 0.05 ls, the stress wave does not reach the warp-fill or fill-warp interface from the point of impact (Fig. 4a). The distance of wave propagation is more for sublayer 1 compared to sublayer 2 along +x direction. This is because warp is towards +x direction for sublayer 1 while fill is towards +x direction for sublayer 2. Warp elastic modulus is higher than fill elastic modulus. As time progresses, the stress wave encounters the interfaces and the reflection and transmission process starts as shown in Figs. 4b, 4c and 4e for both the sublayers. Figs. 4d and 4f are for representative section of idealized plain weave fabric lamina. It may be noted that the lamina consists of both sublayers 1 and 2. The algorithm presented in Section 4.2 is used until the longitudinal unit step stress wave attenuates to 5% of its incident magnitude. The corresponding time duration and distance up to which the stress wave propagates are recorded. Studies were carried out to find the extent of longitudinal stress wave attenuation at different time intervals. Specifically, as shown in Fig. 4f(a), the stress wave attenuates to 15% of its incident value at time interval of t = 20 ls. At this stage, the wave reaches a distance of 68 mm from the point of impact along the beam length. As shown in Fig. 4f(b), the stress wave attenuates to about 5% of its incident value at time interval of t = 40 ls. At this stage, the wave reaches a distance of 110 mm from the point of impact along the beam length. 5.3. Effect of beam length Stress wave attenuation for a typical plain weave fabric E-glass/ epoxy beam target of different lengths is presented in Figs. 5a and 5b and Table 1 at time interval of t = 40 ls. For the present study, if the stress wave magnitude at any location is equal to or less than 5% of its incident value, its effect is not considered for further analysis. If this occurs within the beam target, it is assumed that the stress wave does not reach the target boundary. On the other hand, if the stress wave magnitude is more than 5% of its original value at the target boundary, effect of target boundary on stress wave propagation/attenuation is considered.
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At time interval of t = 40 ls, the stress wave reaches a distance of x = 110 mm from the impact point along the beam length (Fig. 4f (b)). Hence, if the beam length is more than 110 mm, the stress wave does not reach the boundary of the target. Normalized stress wave distribution would be the same for all beams with length larger than 110 mm. Results presented in Figs. 5a and 5b are for beam lengths of 100 mm, 50 mm and 40 mm at time interval of t = 40 ls. Since these dimensions are less than 110 mm, the stress wave would reach the boundary and reflection and transmission would take place at the boundary also. Fig. 5a is for free edge case whereas Fig. 5b is for rigid boundary case. It may be noted from Table 1, for the case of beam length of 100 mm, stress state is different only near the boundary compared to the case of beam length greater than 110 mm. For the cases of beam lengths of 100 mm and 120 mm, normalized stress is 0.0876 at a distance of x = 90 mm. For a beam of length 100 mm, normalized stress is 0.0482 for free edge case whereas it is 0.0985 for rigid boundary case at a distance of x = 90 mm. This is because rR depends on target boundary condition. For the rigid boundary case, the normalized stress is more than that for the free edge case. Even for the case of beam length of 50 mm, normalized stress distribution is the same even up to the distance of x = 45 mm for both free edge and rigid boundary conditions. But for the case of beam length of 40 mm, normalized stress is significantly higher for rigid boundary case compared to free edge case. This is because the reflected stresses from the boundary get added to the incident stresses in the case of rigid boundary condition. Because of wave reflection and transmission at every interface, normalized stress distribution as a function of distance is not a smooth curve. The normalized stresses presented in Table 1 are at particular distances from the origin, i.e., from the point of impact. Figs. 4a–5b and Table 1 are based on the data obtained at each point by analytical simulation. Polynomial curve fits are also given in Figs. 4e–5b. From these figures, stress wave attenuation as a function of distance from the point of impact can be clearly seen. 5.4. Effect of fabric geometry Fig. 6 presents variation of stress wave attenuation with yarn width ‘a’. With lower value of ‘a’, faster attenuation takes place and the stress is localized near the point of impact. This is because,
Fig. 6. Stress wave attenuation in plain weave E-glass/epoxy lamina under in-plane ballistic impact unit step loading, rigid boundary, l = 50 mm: (a) a = 1 mm, t = 40 ls, (b) a = 2 mm, t = 40 ls, (c) a = 4 mm, t = 80 ls.
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reflection and transmission at the warp-fill and fill-warp interfaces as well as target boundaries. Studies are carried out for targets with free edge and rigid boundary conditions. For the plain weave fabric E-glass/epoxy composite beams with a yarn size of 2 mm and corresponding material properties considered, the longitudinal stress wave reaches up to 110 mm on either side of the point of impact at the representative section before complete attenuation of the stress wave takes place. Stress wave attenuation is faster with lower yarn size. With the in-plane impact conditions and target geometrical and material properties remaining the same, stress wave attenuation is greater for plain weave T300 carbon/epoxy than for plain weave E-glass/epoxy.
Appendix A. Sample calculation for stress wave attenuation in plain weave fabric composite beams At the end of the representative section or composite beam target, the boundary condition or the interface properties are different. There are two limiting cases as given by Meyers [1]: Case 1: Wave is incident on a free edge In this case, E = 0. At free edge, c = 0. rR/rI = 1 and rT/rI = 0. Case 2: Wave is incident on a rigid boundary In this case, E = 1 leading to c = 1. rR/rI = 1 and rT/rI = 2. In the present analysis, longitudinal stress wave propagation/ attenuation is considered in the impregnated yarn consisting of two sublayers as shown in Figs. 2 and 3. The impregnated yarn is treated as a one-dimensional element. A.1. Equivalent sublayer 1
Fig. 7. Stress wave attenuation in plain weave fabric composite lamina under inplane ballistic impact unit step loading, l = 100 mm, t = 40 ls, a = 2 mm, free edge: (a) E-glass/epoxy, (b) T300 carbon/epoxy.
with lower value of ‘a’, reflection and transmission takes place at more number of interfaces. 5.5. Effect of target material Fig. 7 presents stress wave attenuation for plain weave fabric Eglass/epoxy and T300 carbon/epoxy with all the in-plane impact conditions remaining the same. For E-glass/epoxy, stress wave reaches the beam target boundary whereas it does not reach the target boundary for T300 carbon/epoxy. In the case of T300 carbon/epoxy, complete stress wave attenuation takes place within the target whereas complete stress wave attenuation does not take place for E-glass/epoxy. This is because of higher elastic property and lower density of T300 carbon/epoxy compared to E-glass/ epoxy. 6. Conclusions An analytical method is presented for the simulation of longitudinal stress wave propagation/attenuation in plain weave fabric composite beam targets. The analytical simulation is based on
Consider a single unit of equivalent sublayer 1 consisting of warp element 1 and fill element 2, as shown in Figs. 2 and 3. Because of in-plane impact on plain weave fabric composite beams, a longitudinal stress wave of magnitude rI is generated at the point of impact. Stress wave propagation away from the point of impact towards the right end of the equivalent sublayer 1 is considered. When the stress wave reaches material interface, wave reflection and transmission takes place. At the interface, A1 = A2 and q1 = q2, i.e., the cross-sectional areas and material densities are the same for warp and fill. Therefore,
C¼
A2 q2 c2 c2 ¼ A1 q1 c1 c1
ðA:1Þ
or,
sffiffiffiffiffi E2 C¼ E1
ðA:2Þ
In the present study, for E-glass/epoxy, E1 = 31 GPa and E2 = 8.4 GPa. As a result, C = 0.52. Hence,
rR ¼ 0:315 rI rT ¼ 0:685 rI
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For simulating stress wave attenuation in a plain weave fabric composite beam, a large number of units of equivalent sublayer 1 are placed one after another and the studies are carried out on the entire representative section from the point of impact. A.2. Equivalent sublayer 2 The calculations shown above are repeated for the equivalent sublayer 2. In this case, C is given by:
sffiffiffiffiffi E1 C¼ E2
ðA:3Þ
As a result, C = 1.92. Hence,
rR ¼ 0:32 rI rT ¼ 1:32 rI References [1] Meyers MA. Dynamic behavior of materials. New York: John Wiley & Sons, Inc.; 1994. p. 27–31. p. 45–50. [2] Tauchert TR. Propagation of stress waves in woven-fabric composites. J Compos Mater 1971;5:456–65. [3] Lundergan CD, Drumheller DS. Propagation of stress waves in a laminated plate composite. J Appl Phys 1971;42:669–75. [4] Barker LM. A model for stress wave propagation in composite materials. J Compos Mater 1971;5:140–62. [5] Barker LM, Lundergan CD, Chen PJ, Gurtin ME. Nonlinear viscoelasticity and the evolution of stress waves in laminated composites: a comparison of theory and experiment. J Appl Mech 1974;41:1025–30. [6] Sve C. Stress wave attenuation in composite materials. J Appl Mech 1972;39:1151–3. [7] Daniel IM, Liber T, LaBedz RH. Wave propagation in transversely impacted composite laminates. Exp Mech 1979;19:9–16. [8] Roylance D. Stress wave damage in graphite/epoxy laminates. J Compos Mater 1980;14:111–9. [9] Ting TCT. The effects of dispersion and dissipation on wave propagation in viscoelastic layered composites. Int J Solids Struct 1980;16:903–11.
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[10] Clements BE, Johnson JN, Hixson RS. Stress waves in composite materials. Phys Rev E 1996;54:6876–88. [11] Chen B, Chou TW. The propagation of one-dimensional transient elastic waves in woven fabric composites. Compos Sci Technol 1998;58:1385–96. [12] Chen B, Chou TW, Hsiao GC. Theoretical analysis of wave propagation in woven fabric composites. J Compos Mater 1999;33:1119–40. [13] Parga-Landa B, Vlegels S, Hernandez-Olivares F, Clark SD. Analytical simulation of stress wave propagation in composite materials. Compos Struct 1999;45:125–9. [14] Han C, Sun CT. Attenuation of stress wave propagation in periodically layered elastic media. J Sound Vib 2001;243:727–61. [15] Biwa S, Watanabe Y, Ohno N. Analysis of wave attenuation in unidirectional viscoelastic composites by a differential scheme. Compos Sci Technol 2003;63:237–47. [16] Chen X, Chandra N, Rajendran AM. Analytical solution to the plate impact problem of layered heterogeneous material systems. Int J Solids Struct 2004;41:4635–59. [17] Aggelis DG, Polyzos D, Philippidis TP. Wave dispersion and attenuation in fresh mortar: theoretical predictions vs. experimental results. J Mech Phys Solids 2005;53:857–83. [18] Goel R, Kulkarni MD, Naik NK. Stress wave attenuation during ballistic impact on a ceramic target. In: Kageyama K, Ishikawa T, Takeda N, Hojo M, Sugimoto S, Ogasawara T, editors. Book of abstract. Sixteenth international conference on composite materials, vol. 2. Kyoto, Japan: Japan Society of Composite Materials; 2007. p. 986–7. [19] Naik NK, Goel R, Kulkarni MD. Stress wave attenuation in ceramic plates. J Appl Phys 2008;103:103504. [20] Pandya KS, Dharmane L, Pothnis JR, Ravikumar G, Naik NK. Stress wave attenuation in composites during ballistic impact. Polym Test 2012;31:261–6. [21] Naik NK, Pandya KS, Pothnis JR, Gelu TA. Revisiting Kolsky bar data evaluation method. Compos Struct 2014;111:446–52. [22] Pasquali M, Terra C, Gaudenzi P. Analytical modelling of high-velocity impacts on thin woven fabric composite targets. Compos Struct 2015;131:951–65. [23] Bedsole RW, Bogert PB, Tippur HV. An experimental investigation of interlaminar and intralaminar dynamic fracture of CFRPs: effect of matrix modification using carbon nanotubes. Compos Struct 2015;132:1043–55. [24] Li R, Natsuki T, Ni Q-Q. A novel dynamic stress analysis in bimaterial composite with defect using ultrasonic wave propagation. Compos Struct 2015;132:255–64. [25] Cheeseman BA, Bogetti TA. Ballistic impact into fabric and compliant composite laminates. Compos Struct 2003;61:161–73. [26] Naik NK, Shrirao P. Composite structures under ballistic impact. Compos Struct 2004;66:577–90. [27] Stronge WJ. Impact mechanics. Cambridge: Cambridge University Press; 2000. p. 150–3. [28] Ganesh VK, Naik NK. Failure behavior of plain weave fabric laminates under on-axis uniaxial tensile loading: I – laminate geometry. J Compos Mater 1996;30:1748–78.
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Composite Structures journal homepage: www.elsevier.com/locate/compstruct
Simulation of stress wave attenuation in plain weave fabric composites during in-plane ballistic impact Kedar S. Pandya a,b,⇑, Mandar D. Kulkarni a, Anirudh Warman a, N.K. Naik a a b
Aerospace Engineering Department, Indian Institute of Technology Bombay, Powai, Mumbai 400 076, India Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, UK
a r t i c l e
i n f o
Article history: Received 10 May 2016 Accepted 27 October 2016 Available online 28 October 2016 Keywords: In-plane impact Plain weave fabric composite Stress wave attenuation
a b s t r a c t An analytical method is presented for the simulation of longitudinal stress wave attenuation in typical plain weave fabric composites made of E-glass/epoxy and T300 carbon/epoxy under in-plane ballistic impact. The stress wave propagation/attenuation simulation studies are carried out for the case of in-plane impact of projectiles on composite beam targets. Geometrical modeling of plain weave fabric composites and an algorithm for stress wave propagation/attenuation are presented. Studies are carried out for both free edge and rigid boundary conditions for different beams. Based on the analytical simulation, for typical plain weave E-glass/epoxy composite beams studied, it is observed that the stress intensity is about 5% of the incident value when the longitudinal stress wave reaches a distance of 110 mm from the point of in-plane impact at the representative section. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction When a projectile impact takes place onto a target, different kinds of stress waves emanate and propagate outward from the point of impact [1]. In general, stress waves propagate along the planar as well as through the thickness directions. When a planar longitudinal stress wave propagates through a periodically layered material system with different physical and mechanical properties, stress wave intensity attenuates over a distance, i.e., the magnitude of stress decreases as the wave propagates further and further. When a stress wave encounters a material boundary, part of the incident wave is transmitted into the neighboring material and part of it is reflected back into the incident material. As a result, the intensity of the stress wave decreases over a distance. A typical example demonstrating the above mentioned phenomenon is that of longitudinal stress wave propagation in plain weave fabric composite beams under in-plane impact. Woven fabrics are made by the process of weaving by interlacing warp and fill yarns in a regular up and down sequence. For composite beams made of such fabrics, longitudinal stress waves generated during in-plane impact encounter many interfaces. At such interfaces, material physical and mechanical properties change leading to stress wave attenuation. ⇑ Corresponding author at: Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, UK. E-mail address: [email protected] (K.S. Pandya). http://dx.doi.org/10.1016/j.compstruct.2016.10.115 0263-8223/Ó 2016 Elsevier Ltd. All rights reserved.
Mechanisms like transmission and reflection at the material interfaces, geometric wave dispersion, crush-up of porosity, dissipation and viscous losses are responsible for attenuation of stress waves in composites. Several experimental [2–8] and analytical [9–15] studies are reported in literature on stress wave attenuation in composites. Stress wave attenuation has also been observed in other layered material systems [16,17] and ceramic plates [18,19]. Pandya et al. [20] presented experimental studies on stress wave attenuation in composites during ballistic impact. Stress wave propagation studies have been carried out in order to predict stress and strain distributions as well as fracture and energy absorbing mechanisms in composite structures under dynamic and impact loading conditions [21–24]. During in-plane impact of a projectile onto a beam target, longitudinal and transverse stress waves are generated in the target along the length of the beam. In the present study, longitudinal stress wave propagation/attenuation is considered. After a review of the literature, it is evident that the effect of reflection and transmission of stress waves at the interfaces on attenuation is not fully understood. The objective of the present study is to investigate longitudinal stress wave attenuation in plain weave fabric composite beams under in-plane impact. Impulse loading (unit step load of infinitesimal duration) is considered in the present study. Stress wave attenuation is attributed to reflection and transmission of stress waves at material interfaces.
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Nomenclature a A c d dp E h l mp P t V50
yarn size/width cross-sectional area wave velocity distance diameter of projectile elastic modulus beam height beam length mass of projectile longitudinal unit step pressure pulse time ballistic limit velocity
2. In-plane impact onto a composite beam target Composite structures undergo various loading conditions during their service life. One of the typical loading conditions is the in-plane impact of a rigid projectile (Fig. 1a). For woven fabric composite beams, a typical representative section along warp/fill adjacent to the point of impact is analyzed (Fig. 1b). This representative section is effectively a one-dimensional composite. When an inplane impact by a projectile onto a composite beam takes place, two waves namely, longitudinal and transverse, propagate from the point of impact [1,25,26]. For woven fabric composites, the typical representative section, i.e., the one-dimensional composite can be treated as a layered composite along the warp/fill direction. The mechanical properties would be varying along the warp/fill direction. This is because of undulation of the yarns in the composite. The longitudinal stress wave propagates along the layered composite beam. The incident wave is partly reflected and partly trans-
Vf
C
q r
Subscripts I incident R reflected T transmitted 1 material 1/warp 2 material 2/fill
mitted as and when it encounters the material interfaces. For the typical representative section (Fig. 1b), cross-sectional areas of the adjacent regions would be the same. However, material impedances would be different for the adjacent regions. The problem formulation for wave propagation/attenuation in plain weave fabric composite beams is based on the following assumptions: The plain weave fabric composite beam can be idealized as a mosaic model of two materials. One-dimensional wave propagation theory can be used for the analysis. Target is a beam structure. 3. Stress wave attenuation modeling When a longitudinal wave crosses the interface between two materials with different properties, the wave undergoes reflection and transmission. Wave velocities in two adjacent materials with different impedances can be calculated as:
c1 ¼
Fig. 1. In-plane ballistic impact of a projectile on to a composite beam target: (a) geometry, (b) representative section along warp, (c) longitudinal unit step pressure pulse.
fiber volume fraction impedance ratio density stress
qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi E1 =q1 and c2 ¼ E2 =q2
Fig. 2. Geometrical modeling of plain weave fabric composite: (a) a typical single layer cross-section, (b) idealized modeling, (c) sublayers 1 and 2, (d) equivalent sublayers 1 and 2.
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where c is wave velocity, E is elastic modulus and q is density. Subscripts 1 and 2 refer to materials 1 and 2, respectively. An incident stress wave with stress intensity rI approaching an interface in a slender bar gives rise to reflected and transmitted waves with stress intensities rR and rT respectively. The magnitude of the reflected and transmitted components is given by [27],
rR C 1 ¼ rI C þ 1
ð1Þ
rT 2 C A1 ¼ rI C þ 1 A2
ð2Þ
where rI, rR and rT are incident, reflected and transmitted stresses, respectively. A is cross-sectional area. Subscripts 1 and 2 refer to materials 1 and 2, respectively. Here, C is known as impedance ratio and is given below,
C¼
A2 q2 c2 A1 q1 c1
ð3Þ
For the cases when A1 = A2, it can be seen that, if C is less than 1, then both the reflected and transmitted waves have a smaller amplitude than the incident wave, and the reflected wave changes its sign. On the other hand, if C is more than 1, both the reflected
and transmitted waves maintain the sign of the incident wave. The transmitted component attains a larger magnitude, while the reflected wave magnitude is smaller. In the present analysis, longitudinal wave propagation/attenuation is considered in the impregnated yarn consisting of two sublayers as shown in Figs. 2 and 3. The impregnated yarn is treated as a one-dimensional element. Sample calculation for stress wave attenuation in a layered medium (Figs. 2 and 3) is given in Appendix A. 4. Modeling of plain weave fabric composite beams for stress wave attenuation Woven fabric composites are made by impregnating woven fabrics with polymer resins. One of the fundamental woven fabrics is the plain weave, where the warp and fill strands are interlaced in a regular sequence of one under and one over. The main geometrical parameters are: undulated length and length of the straight portion of the strand, strand thickness, strand width and interstrand gap [28]. An idealized plain weave fabric composite can be represented using a geometrical repetitive unit [28]. For composites with balanced fabrics, geometrical and material properties are the same
Fig. 3. Schematic arrangement of in-plane ballistic impact loading and one-dimensional stress wave propagation: (a) representative section, (b) sublayer 1, (c) sublayer 2, (d) typical elements, (e) schematic representation of free edge condition, (f) schematic representation of rigid boundary condition.
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A typical single layer plain weave fabric composite beam crosssection is shown in Fig. 2a. Fig. 2b indicates the idealized modeling for plain weave fabric composites. In this figure, the fibers along warp and fill and the resin are separated into layers, with resin occupying the middle layer. In Fig. 2c, the idealized model is split into two sublayers, namely sublayer 1 and sublayer 2. Resin is equally distributed in both the sublayers. In Fig. 2d, resin is evenly redistributed within the sublayers. Such sublayers are referred to as equivalent sublayers. Stress wave attenuation studies are carried out on a large number of equivalent sublayers placed one after another. For brevity, equivalent sublayers are referred to as sublayers for further discussion. It may be noted that the sublayers consist of simplified representation of warp and fill. Here, warp is represented as a unidirectional composite with fiber direction along the longitudinal axis whereas fill is represented as a unidirectional composite with fiber direction perpendicular to the longitudinal axis. Element 1 represents warp whereas elements 2 and 3 represent fill (Figs. 2 and 3).
Schematic arrangement of in-plane impact loading and longitudinal stress wave propagation is shown in Fig. 3. Longitudinal stress waves would be propagating along the +x direction. For sublayer 1, the stress wave propagates first in warp and then in fill towards +x direction (Fig. 3b). For sublayer 2, the stress wave propagates first in fill and then in warp towards +x direction (Fig. 3c). In-plane impact takes place at the left face of the beam at ‘O’ and longitudinal stress waves start propagating along the +x direction. Whenever warp-fill or fill-warp interface is encountered, wave reflection and transmission would take place. Because of wave reflection and transmission, stress wave attenuation takes place in all the elements. A typical ‘element 1’ is shown in Fig. 3d. As the sequence of reflection and transmission continues, the magnitude of the stress wave would be decreasing and tending towards a very small value compared to the magnitude of the incident longitudinal stress wave. At the end of the representative section of the composite beam target, the boundary condition or interface properties are different. Fig. 3e represents free edge condition whereas Fig. 3f represents rigid boundary condition. In the present study, longitudinal unit step loading at the point of in-plane impact onto a plain weave fabric composite beam is considered for ease of calculations. Results are presented for idealized plain weave fabric lamina representative section consisting of both sublayers 1 and 2. The lamina considered is the top layer of the composite beam target.
Fig. 4a. Stress wave propagation in plain weave E-glass/epoxy target sublayers under in-plane ballistic impact unit step loading, l = 200 mm, t = 0.05 ls, a = 2 mm: (a) sublayer 1, (b) sublayer 2.
Fig. 4b. Stress wave attenuation in plain weave E-glass/epoxy target sublayers under in-plane ballistic impact unit step loading, l = 200 mm, t = 0.5 ls, a = 2 mm: (a) sublayer 1, (b) sublayer 2.
along warp and fill. For such composites, geometrical repetitive unit would be of square shape with the dimension of the side equal to the sum of strand width and inter-strand gap. This dimension is referred to as ‘a’. 4.1. Geometrical modeling
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Fig. 4d. Stress wave attenuation in plain weave E-glass/epoxy target under in-plane ballistic impact unit step loading, l = 200 mm, t = 1 ls, a = 2 mm.
Fig. 4c. Stress wave attenuation in plain weave E-glass/epoxy target sublayers under in-plane ballistic impact unit step loading, l = 200 mm, t = 1 ls, a = 2 mm: (a) sublayer 1, (b) sublayer 2.
4.2. Algorithm for stress wave propagation/attenuation As the projectile strikes the beam, the beam resists penetration of the projectile. This builds up the contact force between the target and the projectile. During this event, a longitudinal stress wave is generated and propagates from the point of impact along the beam length. The contact force induces shear and normal stresses within the target. As the induced stresses exceed the permissible stresses, cylindrical cracks would form within the target around the periphery of the projectile ahead of the tip and would propagate through the thickness. The longitudinal stress waves would not propagate through these cracks since the cracks form a free boundary. The algorithm for simulation of longitudinal unit step stress wave propagation/attenuation in plain weave fabric composite beam targets is discussed below:
The plain weave fabric composite is divided into sublayers 1 and 2. Further, sublayers are divided into smaller geometrical segments, i.e., into elements. Dimensions and material properties are assigned to the elements. A longitudinal incident unit step stress wave is introduced into sublayer 1 in the +x direction from the location of impact, i.e., at point ‘O’. The stress wave velocity in each element is governed by the material properties. Wave propagation along +x direction is tracked. As and when the stress wave encounters a material boundary, it leads to reflected and transmitted components. The amplitudes of reflected and transmitted components are evaluated. The transmitted stress wave becomes an incident stress wave for the adjacent element and its data is saved for later tracking. The reflected stress wave is further tracked within the same element till it encounters an interface. After it reaches the interface, it further leads to the next level of reflected and transmitted components. The amplitudes of these components are also evaluated and saved. The process of reflection and transmission of stress waves continues at both the interfaces of the element under consideration. Saving and tracking the transmitted and reflected wave data continues until the magnitude of the stress waves is reduced to a predefined threshold value. Next, the transmitted stress waves and their corresponding reflected components are tracked and saved in a similar manner in the adjacent elements. The initial incident stress wave encounters many interfaces as it propagates, and transmission and reflection take place at all the interfaces as well as at free edge or rigid boundary of the target. At a given time interval a number of stress waves are present at all points at which the stress front has reached. The sum of all the stress wave magnitudes at each point of the element gives the resultant stress wave magnitude at that point. Similar studies are carried out for sublayer 2. Based on the stress wave propagation and attenuation studies carried out for sublayers 1 and 2, the resultant magnitude for representative section consisting of both the sublayers is computed for the duration of impact.
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The wave tracking algorithm developed allows the wave to be tracked at a predefined sampling time. The results correspond to the longitudinal stress wave attenuation behavior for an idealized plain weave fabric composite beam under in-plane impact loading. The lamina considered is the top layer of the composite beam target. Based on the above algorithm the normalized stresses are plotted as a function of time and distance from the point of impact for the plain weave fabric composite beam. 4.3. Input details The input to the algorithm consists of yarn size and composite elastic moduli and densities. Studies are carried out for balanced plain weave fabric composites with yarn size varying from 1 mm to 4 mm. Here, the yarn size refers to the sum of strand width and inter-strand gap [28].
Fig. 4e. Stress wave attenuation in plain weave E-glass/epoxy target sublayers under in-plane ballistic impact unit step loading, l = 200 mm, t = 40 ls, a = 2 mm: (a) sublayer 1, (b) sublayer 2.
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For the typical plain weave fabric E-glass/epoxy composite beam considered, Equivalent warp elastic modulus, E1 = 31 GPa Equivalent fill elastic modulus, E2 = 8.4 GPa Density for warp and fill, q = 1750 kg/m3 For the typical plain weave fabric T300 carbon/epoxy composite beam considered, Equivalent warp elastic modulus, E1 = 94.5 GPa Equivalent fill elastic modulus, E2 = 8.2 GPa Density for warp and fill, q = 1406 kg/m3 5. Results and discussion Longitudinal stress wave propagation/attenuation studies are carried out for a typical plain weave fabric composite beam under longitudinal in-plane unit step loading at different time intervals. The results presented are for the top layer of the composite beam
Fig. 4f. Stress wave attenuation in plain weave E-glass/epoxy lamina under inplane ballistic impact unit step loading, l = 200 mm, a = 2 mm: (a) t = 20 ls, (b) t = 40 ls.
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Fig. 5a. Stress wave attenuation in plain weave E-glass/epoxy lamina under in-plane ballistic impact unit step loading, t = 40 ls, a = 2 mm, free edge: (a) l = 100 mm, (b) l = 50 mm, (c) l = 40 mm.
Fig. 5b. Stress wave attenuation in plain weave E-glass/epoxy lamina under in-plane ballistic impact unit step loading, t = 40 ls, a = 2 mm, rigid boundary: (a) l = 100 mm, (b) l = 50 mm, (c) l = 40 mm.
Table 1 Normalized stress with applied in-plane ballistic impact unit step loading on beams (length in mm), plain weave fabric E-glass/epoxy, t = 40 ls, a = 2 mm. Position from origin, x (mm)
0 10 20 30 45 60 90
200
1.2205 0.9908 0.9309 0.4607 0.2860 0.1743 0.0876
120
1.2205 0.9908 0.9309 0.4607 0.2860 0.1743 0.0876
Free edge
Rigid boundary
100
50
40
100
50
40
1.2205 0.9908 0.9309 0.4607 0.2860 0.1743 0.0482
0.8137 0.8887 1.2156 0.3271 0.0818 – –
1.0026 0.7802 0.6859 0.1055 – – –
1.2205 0.9908 0.9309 0.4607 0.2860 0.1743 0.0985
0.8137 0.8887 1.2156 0.3271 0.0818 – –
1.4385 1.2013 1.1748 0.8160 – – –
target. Studies have been carried out for both free edge (case 1) and rigid boundary (case 2) conditions (Appendix A). In the analytical studies, an impulse load of unit magnitude is assumed for simplicity. For a practical impact loading scenario, the analysis can be carried out using the actual contact force and corresponding planar longitudinal stress wave intensity. 5.1. Mechanical behavior of composite beam targets under in-plane impact During in-plane impact event, the beam target offers resistance to penetration/perforation of the projectile into it. Incident kinetic
energy of the projectile would be absorbed by the target through various damage and energy absorbing mechanisms. It should be noted that the beam does not undergo vibration during the inplane impact event. The target has a tendency to rebound leading to possible vibration only after the impact event. During in-plane impact, longitudinal and transverse stress waves propagate from the point of impact along the beam length. Based on the length of the beam, these stress waves may or may not reach the boundary of the beam. Depending upon the boundary conditions, the extent of transmission and reflection of the stress waves at the boundary can be evaluated. The reflected waves can affect the intensity of the stress waves within the target.
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However, they would not influence vibration characteristics of the target. 5.2. Stress wave propagation as a function of time Stress wave propagation/attenuation studies for a typical plain weave fabric E-glass / epoxy beam of 200 mm length are presented in Figs. 4a–4f. The yarn size, i.e., the element dimension is 2 mm. At time interval of t = 0.05 ls, the stress wave does not reach the warp-fill or fill-warp interface from the point of impact (Fig. 4a). The distance of wave propagation is more for sublayer 1 compared to sublayer 2 along +x direction. This is because warp is towards +x direction for sublayer 1 while fill is towards +x direction for sublayer 2. Warp elastic modulus is higher than fill elastic modulus. As time progresses, the stress wave encounters the interfaces and the reflection and transmission process starts as shown in Figs. 4b, 4c and 4e for both the sublayers. Figs. 4d and 4f are for representative section of idealized plain weave fabric lamina. It may be noted that the lamina consists of both sublayers 1 and 2. The algorithm presented in Section 4.2 is used until the longitudinal unit step stress wave attenuates to 5% of its incident magnitude. The corresponding time duration and distance up to which the stress wave propagates are recorded. Studies were carried out to find the extent of longitudinal stress wave attenuation at different time intervals. Specifically, as shown in Fig. 4f(a), the stress wave attenuates to 15% of its incident value at time interval of t = 20 ls. At this stage, the wave reaches a distance of 68 mm from the point of impact along the beam length. As shown in Fig. 4f(b), the stress wave attenuates to about 5% of its incident value at time interval of t = 40 ls. At this stage, the wave reaches a distance of 110 mm from the point of impact along the beam length. 5.3. Effect of beam length Stress wave attenuation for a typical plain weave fabric E-glass/ epoxy beam target of different lengths is presented in Figs. 5a and 5b and Table 1 at time interval of t = 40 ls. For the present study, if the stress wave magnitude at any location is equal to or less than 5% of its incident value, its effect is not considered for further analysis. If this occurs within the beam target, it is assumed that the stress wave does not reach the target boundary. On the other hand, if the stress wave magnitude is more than 5% of its original value at the target boundary, effect of target boundary on stress wave propagation/attenuation is considered.
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At time interval of t = 40 ls, the stress wave reaches a distance of x = 110 mm from the impact point along the beam length (Fig. 4f (b)). Hence, if the beam length is more than 110 mm, the stress wave does not reach the boundary of the target. Normalized stress wave distribution would be the same for all beams with length larger than 110 mm. Results presented in Figs. 5a and 5b are for beam lengths of 100 mm, 50 mm and 40 mm at time interval of t = 40 ls. Since these dimensions are less than 110 mm, the stress wave would reach the boundary and reflection and transmission would take place at the boundary also. Fig. 5a is for free edge case whereas Fig. 5b is for rigid boundary case. It may be noted from Table 1, for the case of beam length of 100 mm, stress state is different only near the boundary compared to the case of beam length greater than 110 mm. For the cases of beam lengths of 100 mm and 120 mm, normalized stress is 0.0876 at a distance of x = 90 mm. For a beam of length 100 mm, normalized stress is 0.0482 for free edge case whereas it is 0.0985 for rigid boundary case at a distance of x = 90 mm. This is because rR depends on target boundary condition. For the rigid boundary case, the normalized stress is more than that for the free edge case. Even for the case of beam length of 50 mm, normalized stress distribution is the same even up to the distance of x = 45 mm for both free edge and rigid boundary conditions. But for the case of beam length of 40 mm, normalized stress is significantly higher for rigid boundary case compared to free edge case. This is because the reflected stresses from the boundary get added to the incident stresses in the case of rigid boundary condition. Because of wave reflection and transmission at every interface, normalized stress distribution as a function of distance is not a smooth curve. The normalized stresses presented in Table 1 are at particular distances from the origin, i.e., from the point of impact. Figs. 4a–5b and Table 1 are based on the data obtained at each point by analytical simulation. Polynomial curve fits are also given in Figs. 4e–5b. From these figures, stress wave attenuation as a function of distance from the point of impact can be clearly seen. 5.4. Effect of fabric geometry Fig. 6 presents variation of stress wave attenuation with yarn width ‘a’. With lower value of ‘a’, faster attenuation takes place and the stress is localized near the point of impact. This is because,
Fig. 6. Stress wave attenuation in plain weave E-glass/epoxy lamina under in-plane ballistic impact unit step loading, rigid boundary, l = 50 mm: (a) a = 1 mm, t = 40 ls, (b) a = 2 mm, t = 40 ls, (c) a = 4 mm, t = 80 ls.
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reflection and transmission at the warp-fill and fill-warp interfaces as well as target boundaries. Studies are carried out for targets with free edge and rigid boundary conditions. For the plain weave fabric E-glass/epoxy composite beams with a yarn size of 2 mm and corresponding material properties considered, the longitudinal stress wave reaches up to 110 mm on either side of the point of impact at the representative section before complete attenuation of the stress wave takes place. Stress wave attenuation is faster with lower yarn size. With the in-plane impact conditions and target geometrical and material properties remaining the same, stress wave attenuation is greater for plain weave T300 carbon/epoxy than for plain weave E-glass/epoxy.
Appendix A. Sample calculation for stress wave attenuation in plain weave fabric composite beams At the end of the representative section or composite beam target, the boundary condition or the interface properties are different. There are two limiting cases as given by Meyers [1]: Case 1: Wave is incident on a free edge In this case, E = 0. At free edge, c = 0. rR/rI = 1 and rT/rI = 0. Case 2: Wave is incident on a rigid boundary In this case, E = 1 leading to c = 1. rR/rI = 1 and rT/rI = 2. In the present analysis, longitudinal stress wave propagation/ attenuation is considered in the impregnated yarn consisting of two sublayers as shown in Figs. 2 and 3. The impregnated yarn is treated as a one-dimensional element. A.1. Equivalent sublayer 1
Fig. 7. Stress wave attenuation in plain weave fabric composite lamina under inplane ballistic impact unit step loading, l = 100 mm, t = 40 ls, a = 2 mm, free edge: (a) E-glass/epoxy, (b) T300 carbon/epoxy.
with lower value of ‘a’, reflection and transmission takes place at more number of interfaces. 5.5. Effect of target material Fig. 7 presents stress wave attenuation for plain weave fabric Eglass/epoxy and T300 carbon/epoxy with all the in-plane impact conditions remaining the same. For E-glass/epoxy, stress wave reaches the beam target boundary whereas it does not reach the target boundary for T300 carbon/epoxy. In the case of T300 carbon/epoxy, complete stress wave attenuation takes place within the target whereas complete stress wave attenuation does not take place for E-glass/epoxy. This is because of higher elastic property and lower density of T300 carbon/epoxy compared to E-glass/ epoxy. 6. Conclusions An analytical method is presented for the simulation of longitudinal stress wave propagation/attenuation in plain weave fabric composite beam targets. The analytical simulation is based on
Consider a single unit of equivalent sublayer 1 consisting of warp element 1 and fill element 2, as shown in Figs. 2 and 3. Because of in-plane impact on plain weave fabric composite beams, a longitudinal stress wave of magnitude rI is generated at the point of impact. Stress wave propagation away from the point of impact towards the right end of the equivalent sublayer 1 is considered. When the stress wave reaches material interface, wave reflection and transmission takes place. At the interface, A1 = A2 and q1 = q2, i.e., the cross-sectional areas and material densities are the same for warp and fill. Therefore,
C¼
A2 q2 c2 c2 ¼ A1 q1 c1 c1
ðA:1Þ
or,
sffiffiffiffiffi E2 C¼ E1
ðA:2Þ
In the present study, for E-glass/epoxy, E1 = 31 GPa and E2 = 8.4 GPa. As a result, C = 0.52. Hence,
rR ¼ 0:315 rI rT ¼ 0:685 rI
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For simulating stress wave attenuation in a plain weave fabric composite beam, a large number of units of equivalent sublayer 1 are placed one after another and the studies are carried out on the entire representative section from the point of impact. A.2. Equivalent sublayer 2 The calculations shown above are repeated for the equivalent sublayer 2. In this case, C is given by:
sffiffiffiffiffi E1 C¼ E2
ðA:3Þ
As a result, C = 1.92. Hence,
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