International Journal o Composite Materials 5, 5(6): 77-8 DOI:.593/j.cmaterials.556.6 Notched Strength Estimation o Graphite/Epoxy Laminated Composite with Central Crack under Uniaxial Tensile Loading Rakesh Singh *, V. K. Srivastava Department o Mechanical Engineering Indian Institute o Technology (BHU), Varanasi, India Abstract Notched strength o composite material with dierent type o irregularities has been estimated analytically by many researchers and scholars. Some models are extremely complicated and leads to a cumbersome calculation process and others are o limited use. This intrigued authors to seek an estimation which is simple and accurate. This paper represents statistical approach or prediction o notch strength o composite s o dierent lay-ups containing a central crack using point stress criterion and average stress criterion. For calculation o characteristic lengths or point stress criterion and average stress criterion, dierent expressions are used which are very simple and accurate. Final equations or notched strength o d composites are in the simple polynomial orm and can simpliy urther calculations. Results are compared with experimental data o graphite/epoxy composite and data obtain rom improved inherent law model. The notched strength estimations are ound to be within range o tested and improved inherent law model values. Keywords Laminated composite, Notched strength, Characteristic length, Central crack. Introduction In the development o racture mechanics or composite materials many theories and models have been suggested. In which most popular approaches are linear elastic racture mechanics (LEFM) model suggested by Waddoups, Eisenmann and Kaminski (WEK) []. WEK model o estimation o notched strength was based on LEFM which assumes some intense energy region around a hole and known as crack in traverse direction o loading. Later on Whitney and Nuismer [, 3], suggested two ailure criterion called point stress criterion and average stress criterion. According to PSC ailure o a composite occur when applied uniaxial tensile stress at some distance (d ) rom discontinuity, perpendicular to loading direction, is equal to or greater than unnotched strength o s i.e. σ y (d, ) = () σ y is normal stress, applied along y direction and is unnotched strength o d composite. Another model ASC o Whitney-Nuismer [, 3] which assumes the ailure occur when average stress σ y over some * Corresponding author: Rakesh.singh.mec4@itbhu.ac.in (Rakesh Singh) Published online at http://journal.sapub.org/cmaterials Copyright 5 Scientiic & Academic Publishing. All Rights Reserved distance a is equal to or more than the unnotched strength, i. e. a a a+a σ y (x, )dx = () a is hal crack length. In Whintney-Nuisimer s both models characteristic length d and a are assumed to be material constant but later on experimental results have shown that d and a are not material constants. As the ailure process o composite depends not only type o material but also notch shape and type o lay-ups. PSC has been modiied by Pipes et al [4] and Kim et al [5] which gave a three parameter expression o characteristic length d or a hole type notch, i.e. d = k R W n (3) k and n are constants or particular material and related to width (W) and radius (R) o hole. An expression o characteristic lengths in terms o racture parameters, notched and unnotched strength or a crack notch presented by Potti et al [6] gives accurate and precise results and quite simple in calculation. In this paper racture strength o graphite/epoxy composite o dierent lay-ups is calculated with PSC and ASC where characteristic lengths are calculated rom expression presented by Potti et at [6]. Values obtained are compared with experimental values rom Morries and Hahn [7] and values obtained rom inherent law model (IFM) presented by Potti et al [8].
78 Rakesh Singh et al.: Notched Strength Estimation o Graphite/Epoxy Laminated Composite with Central Crack under Uniaxial Tensile Loading. Analytical Work.. Point Stress Criterion Based on the point stress racture criterion it is assumed that will ail when the stress at some distance d away rom the opening edge and on the axis normal to the loading, reaches unnotched strength [, 3], i.e. σ y (d, ) = (From equation ) Fracture strength o ininite width plate under uniaxial tensile loading according to PSC is given by σ N = ξ (4) σ N is notched strength o ininite width d composite and ξ = a a+d (5) The characteristic dimension d is initially expressed in terms o racture parameters (k and m) as [6]: d = γ π (6) γ = k m σ N (7) From equations (5) and (6) we obtain ξ = + γ πa (8) Now rom equation (4), (7) and (8) we get a ollowing equation: z = z = σ N, α = k m (αz +βz+η) (9) πa η = k, β = m πa +, k πa and Equation (9) can be simpliied assuming dierent constants, and inally we get a polynomial equation o sixth degree. Az 6 + Bz 5 + Cz 4 + Dz 3 + Ez + Fz + G = () A = α, B = αβ, C = β α + ηα, D = β(η α), E = η β ηα, F = βη and G = η. Equation () can be solved with help o Bisection method or determining roots with initial approximation between z = and z =... Average Stress Criterion Based on the average stress racture criterion it is assumed that when the normal stress averaged over some distance (a ), away rom the opening edge and on the axis normal to the applied load reaches or greater than the unnotched strength o the, the will ail [, 3] i.e. a+a σ y (x, )dx = (From equation ) a a According to ASC σ N = φ +φ φ = () a a+ a () a is hal crack length, a is average characteristic length, which can be expressed as [6] Putting a = γ π a = γ π in equation () we get an expression (3) φ = + γ πa (4) γ = k m σ N (5) From equation (), (4) and (5) we get λ z 4 + λ z 3 + λ 3 z + λ 4 z + λ 5 = (6) z = σ N, λ σ = mk, λ σ = m k, λ 3 = k + πa mk, λ 4 = m k and λ 5 = k. Equation (6) can be solved with Bisection method with initial approximation between z = and z =. Table. Unnotched strength o s with standard deviation and coeicient o variation [7] Material Lay-up Average unnotched strength ( ) (MPa) Standard deviation (MPa) Coeicient o variation (%) Thornel3/N58 [/±45] s [/±45] s [/9/±45] s 54.7 54.7 46. 5.4 5.4 9. 9.5 9.5
International Journal o Composite Materials 5, 5(6): 77-8 79 Table. Fracture parameters [8] Material Lay-up k (MPa m) m Thornel3/N58 [/±45] s [/±45] s [/9/±45] s 47.9 4.7 47..559.358.5 Crack a Table 3. Notched strength or lay-up [±45/] s Experimental[7] IFM [8] Notched strength (σ N ) (Mpa).7 4.9 3.5 34. 34.54 93.33.7.8 64.8 43. 5. 55..7 4.99 7. 7.5.9.33.69.5 85.6 78.6 99.38 6.8.7 5.5 5.4 54.4 83.9 8.48 Crack a Table 4. Notched strength or lay-up [±45/] s Experimental [7] IFM [8] Notched strength σ N (MPa).88 4.95 89. 3.6 35.8 34.8.88.6 38. 36.7 43.64 47.9.88 5. 3.6 99..88 9.5.88.9 8.9 7. 89.84 94.3.86 5.7 4. 45.7 73.78 8.49.84 3. 9.6 4.6 6.88 65.7 Crack a Table 5. Notched strength or lay-up [/9/±45] s Experimental [7] IFM [8] Notched strength σ N (MPa).7 5.8 36.3 37.9 39.9 334.6.6.3 64. 59.9 67.43 73.8.8 5.4 6.6 9.7 3.7 35.8..6 9. 9.6.3 7.79.9 5.5 64. 63.9 94.39.9 3. Results and Discussion The Whitney-Nuismer [, 3] racture model (viz., point stress criterion, average stress criterion) are applied to correlate the experimental data [7] o graphite/epoxy, Thornel3/N58 d composite containing centre crack under uniaxial tensile loading. Width o each is 5.8 mm and lay-ups are [ /±45] s, [/ ±45 ] s and [/9/±45] s. The properties o Thornel3/N58 d composite are mentioned in Table and. The comparative results o PSC and ASC analysis o unnotched strength or lay-up orders [ /±45 ] s, [/ ±45 ] s and [/9/ ±45] s are presented in Table 3, 4 and 5 respectively. Dierent s o graphite/epoxy composite are considered and good amount o racture data generated. It is observed that procedure described in the preceding section has increased the eectiveness o the suggested models. Morries and Hans [7] have presented racture data on Thornel 3/Narmo 58 epoxy. Total 35 centre cracked tension specimen were tested. Three specimen or each crack length o [/±45] s, two specimen or each crack length o [/ ±45 ] s and two specimen or each crack length o [/9/±45] s were tested. Width o specimen was 5.8 mm and length 34.8 mm with crack length 5.8 mm to 5.4 mm. Figure, and 3 shows the variation o strength ratio σ N with crack size or [ /±45 ] s, [/ ±45 ] s and [/9/±45] s respectively.
8 Rakesh Singh et al.: Notched Strength Estimation o Graphite/Epoxy Laminated Composite with Central Crack under Uniaxial Tensile Loading.6 average values o characteristic length or PSC and ASC or dierent lay-ups. Strength ration (z).5.4.3.. 4.9.8 4.99.5 5.5 Crack size (a) present analysis present analysis Characteristic length d.6.4..8.6.4. 5 5 5 Crack size a [±45/]s [±45/]s [/9/±45]s Figure. Variation o strength ratio (z) with crack size (a) or [±45/] s.7.6 Figure 4. Variation o characteristic length (d ) with crack size (a) (point stress criterion) 7 Strength ratio(z).5.4.3.. 4.95.6 5..9 5.7 3. present analysis present analysis Characteristic length a 6 5 4 3 [±45/]s [±45/]s [/9/±45]s Crack size a Figure. Variation o strength ratio (z) with crack size (a) or [±45/] s 5 5 5 Crack size a Strength ratio (z).8.7.6.5.4.3.. 5.8.3 5.4.6 5.5 Crack size a present analysis present analysis Figure 3. Variation o strength ratio (z) with crack size (a) or [/9/±45] s All the curves are giving approximately same results. Figure 4 and 5 shows variation o characteristic length with crack size or PSC and ASC respectively. Figure 6 shows the Figure 5. Variation o characteristic length (a ) with crack size (a) (average stress criterion) In general it is seen that the characteristic length estimation based on the PSC and ASC are not very close to each other or all s. The average stress criterion values are nearly our times higher than the point stress criterion (see igure 6). This represent that the characteristic length depends on the surace racture energy. The crack length is generally dominated by the progress o debonding iber racture and increasing crack length [7]. However, the results indicate that the strength ratio decreases with increase in crack size due to concentration o stress at crack tip. Figure 6 shows the average values o characteristic length or PSC and ASC or dierent lay-ups. A inite width correction actor [6] Y = sec πa W o a centre crack can be considered to obtain σ N by multiplying σ N. Since here width o plate is large compared to crack length (a) so σ N = σ N is assumed.
International Journal o Composite Materials 5, 5(6): 77-8 8 Chartaristic length(d, a) 6 5 4 3 point stress criterion Figure 6. Average values or characteristic lengths (d, a ) 4. Conclusions Stress criterion average stress criterion [±45/]s [±45/]s [/9/±45]s All values are ound in an acceptable region and as the crack size is increasing the deviation o analytical values rom experimental is also increasing. This is because at large crack length the validity o the assumption o ininite width or large crack is ineective on strength. This can be improved by considering a suitable actor or inite width s. Finally, this analysis has reduced the cumbersome analytical work required or notched strength estimation. The analysis has been improved by considering just a polynomial equation o six and our degree or PSC and ASC respectively. ACKNOWLEDEMENTS The authors would like to thank Department o Mechanical Engineering, Indian Institute o Technology (BHU), Varanasi-5, India or their support. REFERENCES [] Waddoups M.E, Eisenmann, J.R and Kaminski B.E. Macroscopic racture mechanics o advance composite material. J Composite Material 97; 5: 446-454. [] R.J. Nuismer, and J.M Whitney, Uniaxial ailure o composite s containing stress concentration. ASTM STP 975; 593: 7-4. [3] Whitney J.M and Nuismer R.J. Stress racture criteria or d composites containing stress concentrations. J Composite Materials 974; 8: 53-65. [4] Pipes R.B, Wetherhold R.C and Gillespie J.W. Macroscopic racture o ibrous composites, Material Science and Engineering 98; 45: 47-53. [5] Kim J.K, Kim D.S and Takeda N. Notched strength and racture criterion in abric composite plates containing a circular hole. J Composite Material 995; 9: 98-998. [6] Potti P.K.G, Rao B.N and Srivastava V.K. Fracture strength o centre notched graphite/epoxy tensile stripes. Journal o Material Science Letters ; 9: 9-94. [7] Morries D.H, and Hahn H.T. Fracture resistance characteristic o graphite/epoxy composites. ASTM STP 977; 67: 5-7. [8] Potti P.K.G, Rao B.N and Srivastava V.K. Notched strength estimation o graphite/epoxy composite s containing central holes and crack: A statistical approach, The Aeronautical Journal 4;738: 63-69.