Chapter 5 Weight function method

Transcription

1 Chpter 5 Weight function method The weight functions re powerful method in liner elstic frcture mechnics (Anderson, 1995; Td, Pris & rwin, 2). nitilly they were used for clculting the. The underlying hypothesis of the method is the principle of superposition, which llows for the clcultion of the due to vrious forces cting on the specimen or structure (see Figure 5.1) nd is vlid in the liner elstic rnge of the mteril s behvior. n this section, the weight function method is developed in some detil for the clcultions of crck opening displcements for Mode plne problems with through-the-thickness crck. Figure 5.1 Principle of superposition pplied on centrlly crcked specimen. The specimen is loded by σ ( x) nd by continuous bridging trction ϕ (x). 5.1 Stress intensity fctor For the geometry nd loding configurtion shown in Figure 5.2 the is, K (,) 2 F, π x (5.1) b x b B Figure 5.2 Specimen with n edge crck loded with dipole force. Mrch

2 ( /,/b) where is force per unit length (long the thickness, B), Fx is known nondimensionl function of specimen geometry (see Appendix nd Td, Pris & rwin, 2). When the force (clled dipole force) is unity, we define the following function, 2 π x b g x, F, 1 (5.2) s the weight function (or Green s or influence function). The definition implies tht g is the for unit dipole force. There exist weight functions for other configurtions. They re obtined by nlyticl or numericl techniques [6]. For crck in n infinite plte shown in Figure 5.3, nd unit dipole force, the closed form solution for the weight functions re, Figure 5.3 nfinitely lrge plte with centrl crck of length 2. g g A B ( x,) ( x,) + x π -x 1 -x π + x 1 (5.3) (5.3b) When it is not possible to obtin such functions nlyticlly, numericl metho bsed on finite elements re often used. An exmple of determining using such method is presented in Appendix V. Knowledge of such functions for body with crck, llows for the determintion of due to vrious lo tht pply nywhere on the specimen. n either cse shown erlier, if concentrted dipole force P is pplied on the crck fces t distnce x from the reference system of coordintes, the cn be expressed s, Mrch

3 K P, Pg x, (5.4) f distributed lod is pplied on the crck fces s shown in Figure 5.4, the is obtined by setting P(x) ϕ dx in (5.4) (lod per unit thickness) nd integrting to obtin, ( x) c 2 ( ϕ ) ϕ K, x g x, dx c1 5.5) f the forces, concentrted or distributed, pply nywhere on the specimen, nd not on the crck fces, we use the principle of superposition to bring the problem to n equivlent one where surfce trctions pply. Therefore, we solve the equivlent problem using the influence or weight function method. y c 2 ϕ (x) x c 1 ϕ (x) b Figure 5.4 nfinitely lrge plte with crck loded with distributed surfce trction ϕ. 5.2 Crck opening displcements Consider crcked body loded s shown in Figure 5.5. For simplicity we consider only mode. The other modes, cn be treted similrly. Assuming liner elstic behvior, the potentil energy of the solid is given by, ( x) Π P P P P + P P U where U is the elstic strin energy (see lso chpter 3). t consists of two prts, tht of the bulk mteril U no crck nd the portion due to the crck. The elstic energy relese rte (ERR) G, per unit specimen thickness, under controlled lo is, G ( P,,) Π U P, (5.6) Mrch

4 ntegrting (5.6) we obtin, no crck U(P,, ) U(P,,) U + d no crck U + G (P,, )d (5.7) To obtin displcements t ny point on the body we use the theorem of Cstiglino (Del Pedro, Gmür & Botsis, 21). According to this theorem, the displcement of point in the direction of the pplied lod t tht point is given by the derivtive of U with respect to the lod pplied t tht point. Figure 5.5 Crcked body loded with force P nd dipole. P nd re the displcements in the direction of the forces nd d is the distnce before deformtion. f no lod pplies in the direction of the sought displcement, virtul dipole force is pplied in the direction of the displcement to be determined nd U is clculted using stndrd procedures. Once U is known, due to the ctul lo nd the virtul force, we tke the derivtive of U with respect to the virtul force t zero virtul force. The derivtive defines the displcement t the specified point in the direction of the virtul force. The procedure outlined bove is used to clculte CODs in the mnner described in the following sections. Suppose tht the displcements P nd (Figure 5.5) re sought. According to the theorem of Cstiglino we hve, no crck G P,, U P + d P (5.8) P no crck G P,, U + d (5.8b) Mrch

5 Define, no crck U P no crck P nd no crck U no crck And recll tht 2 G K /E nd K(P,,)K (P,) + K (,). Thus, G P,, 2 K P, K, E K P, + K, + (5.9) 2 with K P, / nd where E E for plne stress nd E E/(1 ν ) for plne strin. Using eqs (5.8) (5.8b) nd (5.9) nd tking, the corresponding displcements re, ( P, ) 2 K no crck P P + K P, d E (5.1) P no crck 2 K, + K P, d E (5.11) Note tht in (5.11), K(,)/ does not depend on nd K(,) when. When we seek the displcement t the crck fces, i.e. the COD, t distnce from the origin of the coordinte reference system, the term no crck is zero nd the COD t the point where the dipole pplies is, 2 K, K P, d E (5.12) The lower limit of the integrl in (5.12) is set s becuse K(,)/ when <. Therefore, by using informtion tht is lredy vilble bout crcked body (e.g. stress intensity fctors) we cn determine the COD long the crck fce. Exmple To illustrte the method described bove, the mximum crck opening of centrl crck of length 2, in n infinite body under uniform, unixil pplied stress σ, is determined. Due to symmetry, the mximum crck opening occurs t x (Figure 5.6). Thus, we pply virtul force on the top nd bottom surfces of the crck t x. For the corresponding s, we know tht (see 4.37 nd 5.3), K ( σ,) σ π nd K (,) π We need to consider the effects of both crck tips nd write using eq. (5.12), Mrch

6 2 K (, ) 2 K (, ) K(,) d K(,) -d E σ + σ E (right hnd crck tip) (left hnd crck tip) σ σ E E π 1 1 4σ 2 π d + 2 π (-d ) π E Note: to reduce the problem we could hve used symmetry to write, 2 K (, ) 2 K (, ) d E σ, from the first step. To obtin the COD t point x, we need to pply the dipole force, t distnce x from the center (where the origin of the coordinte system my be ttched). Figure 5.6 Centrl crck in n infinite plte subjected to remote stress σ Crck opening displcement - Generl formultion We cn generlise the bove-mentioned formlism in the cse where severl forces pply on the body. f the body is subjected to set of forces P i ( i 1,,n), the displcement, is, where no crck 1 k n P k P + K k P 1,...,P k,...,p n, d E P k 2 K P,...,P,...,P, K ( P,) no crck 2 k P + K k P 1,...,P k,...,p n, d E P k K ( P,...,P,...,P,) K P, n 1 k n i i 1 is the totl for the prticulr mode of frcture. The displcement between two points nywhere in the body where no force pplies is, P k Mrch

7 no crck 2 K, + ( 1 k n ) K P,...,P,...,P, d E where is virtul dipole force pplied t the corresponding points. f the COD is sought,, nd, no crck 1 k n E 2 K, COD u K P,...,P,...,P, d (5.13) where is the distnce from the reference xis to the dipole force. Recll the expressions for the weight functions (5.2 or 5.4) nd define, ( ) g, K, (5.14) Therefore, when K (,) is known, for the COD nd (5.13) cn be written s, K, cn be looked upon s weight function 2 COD u K P,...,P,...,P, g, d E ( 1 k n ) ( ) (5.15) Strting from (5.15), we wnt to express K ( P,...,P,...,P,) t generl expression for the COD. 1 k n with weight function nd rrive y P u x x P x 1 Suppose we need the COD t point x 1 due to unit dipole force P1, t x. n this cse, the K P,...,P,...,P, tkes the form (lso see 5.4), 1 k i Figure 5.7 Mode crck subjected to dipole force P 1. Mrch

8 K P, g x, P 1 nd the COD due to the unit dipole force t is obtined from (5.15) s ( s), x 2 u x,x, g x,s g x,s g x,x, COD E mx( x,x1) (5.16) COD u g x,x 1, x1 Define s weight function for the COD t due to unit dipole force pplied on the crck fces t x estblished on the bsis of g. COD (Figure 5.7). t is importnt to notice here tht the g is Note tht the integrl in (5.16) is over the crck length nd tht the weight function(s) re zero if the crck length is smller thn x or x 1. To ccount for it, the lower limit in (5.16) is tken s the mx(x,x 1 ). Eqution (5.16) cn be used to determine the COD t ny point of the crck due to known dipole force on the crck fces. Consider the configurtion shown in Figure 5.8. The COD due to ϕ is obtined by integrting (5.16) over the length of ppliction of ϕ ( x). ( x) σ y ϕ (x) x c σ Figure 5.8 Mode crck under remote stress σ (x) nd trctions ϕ( x) Accordingly, with c 1 -c, the COD t x 1 is, on the crck fces. COD u ϕ,x, 1 g x,x, 1 ϕ x dx (5.17) c1 Mrch

9 ntroducing (5.16) in (5.17) we rrive t, or ( ϕ ) ϕ 1 E c1 mx( x,x1) 2 u,x, g x,s g x,s x dx 1 2 s u ( ϕ,x 1,) g ( x,s) ( x) dx g ( x,s) E ϕ 1 mx( x,x1) c 1 (5.18) N ote tht the integrl, in the brckets is the due to ϕ( x ). This is lso evident by comprison of eq. (5.18) with eq. (5.12). The double integrl in (5.18) gives the COD t ny point on the crck fces due to trctions on the crck. To pply the sme procedure for ny lod (i.e. lod pplied nywhere on the body) we use the principle of superposition to represent the lod in the body with n equivlent trction distribution on the crck fces. f the ltter step is crried out, (5.18) is used for the COD t ny point on the crck fce. A simple exmple to illustrte the lst point is the COD due to remote stress σ ( x) principle of superposition results in n equivlent loding cse which consists of the sme specimen with σ x pplied on the crck fces. Thus, s 2,x 1,) g ( x,s) σ( x) dx g ( x 1,s) E mx( x,x1) c 1. The (5.19) where the integrl in the brckets is the of the specimen subjected to σ ( x). Referring to Figures 5.1 nd 5.8, if ϕ ( x) σ ( x) b ϕ ( x) is bridging trction distribution, i.e.,, due to the reinforcement in composite mteril, the totl COD is, s 2,x, u,x, g x,s x dx g x,s ( σ σ ) ( σ ) σ u, b 1 1 b 1 E mx( x,x1) (5.2) The COD given by (5.2) is the totl displcement of the crck fces. When the COD is mesured with respect to the middle plne of the crck, the coefficient 2 is eliminted from (5.19) nd (5.2). Note tht the function defined in (5.4), is re-written s, g x, 2g x, Mesuring the COD from the middle plne of the crck nd using the bove definition for the weight function, reltions ( 5. 19) nd (5.2) cn be expressed, for point x, s, Mrch

10 s (5.21) mx( x,x) 4,x,) g ( x,s) σ ( x ) dx g ( x,s) E nd s 4, σ,x,),x,) g ( x,s) σ ( x) dx g ( x,s) b b E mx( x,x) (5.22) Anlyticl expressions for the weight functions for certin common specimens hve been reported in the literture. For other geometries the weight function cn be generted numericlly, for instnce by creting finite element model where unit dipole force is pplied t different points on the crck fce (Td, Pris & rwin, 2; Fret & Munz, 1997). 5.3 Exmples Centrl crck in n infinite plte Consider the centrl crck specimen shown in the Figure 5.1. The specimen is loded by uniform remote stress σ, nd constnt continuous pressure p on the crck fces long its entire length. Assuming tht /b clculte the crck opening displcement due to the combined lo u(x). The weight function for crck in n infinite plte is given by, 1 1 g(x,) π 1 x / 2 2 (5.23) According to ( 5.22 ), the COD is given by, s 4,p,x,),x,) g( x ) E,s pdx g x,s () x where, s 4,x,) g ( x,s) σ dx g ( x,s) E x (b) ntegrl (b) results in, 4,x,) E s σdx πs 1 x /s πs 1 x /s x s 4σ 1 1 1,x,) dx π E s x 1 x /s 1 x /s Mrch

11 s 4σ 1 1 x 4σ 1 s π,x,) s rcsin πe s 2 2 s E s x 1 x /s π x 1 x /s s 2σ 2σ 2σ 2 2,x,) s x E 1 x /s E s x E x x x 2σ 2 2,x,) x (c) E Figure 5.9 Lrge plte with centrl crck subjected to remote stress σ, on its boundry nd pressure p, long the crck fces. n similr mnner, integrtion of the second term on the right-hnd-side of eq. () results in, 2p u p,x, x E 2 2 (d) Therefore the totl displcement is, 2( σ p) u,p,x, x E 2 2 ( σ ) (e) Edge crck in semi-infinite plte Consider the single edge notched specimen shown in Figure 5.1 t is loded by remote stress σ. The crck of length, is bridged by prticle long prt of the crck s shown in the figure. The pressure p, pplied by the prticle, is ssumed to be constnt long the bridged section. The crck opening displcement, is mesured experimentlly t x. Determine the vlue of the pressure imposed by the prticle. Mrch

12 Figure 5.1 Single edge notched specimen loded by σ nd bridged by constnt pressure p. Assuming lrge width of the plte (i.e, Appendix ), /b ), the weight function for the problem is (see g x, (x/) F(x,) 2 π π 1 (x/) 3/2 (5.24) The trctions on the crck fce re, σ (x ) p for / 2 x 3 / 4 b σ (x ) for x / 2 nd x 3 / 4 b () The generl expression for the crck opening is given by (5.22), s 4, σ,x,),x,) g ( x,s) σ ( x) dx g ( x,s) b b E mx( x,x) Due to the loding s shown by (), the integrl becomes, for x /2, 3/4 4p,p,x,),x,) g( x,s) dx g ( x,s) E /2 / 2 (b) with s 4σ,x,) g ( x,s) dx g ( x,s) E x (c) Mrch

13 To obtin p, we need to tke the COD given theoreticlly by (b) t x, multiply it by 2 nd set it equl to the mesured one ( ), 3/4 4p 2,p,,) 2,,) g ( x,s) dx g ( x,s) E /2 / 2 x Therefore, the pressure p is given by, ( σ ) E 2u,, p 3/4 8 g( x,s) dx g( x,s) /2 / 2 x (d) nserting (5.24) in (d) nd integrting we obtin the pressure in terms of the specimen geometry, pplied externl stress nd Young s modulus. Often, in such problems numericl integrtion is necessry to rrive to numericl vlue. 5.4 An ppliction of the weight function method to composites There re severl mechnisms tht cn slow or even stop the propgtion of crck through mteril. Figure 5.11 shows severl exmples of such mechnisms. Some re due to the presence of two or more mterils (i.e. in composite) ner the crck, s in crck deflection by obstcles, crck bridging by prticles or fibers, etc. Other mechnisms tht slow crck propgtion re due to trnsformtion in the mteril ner the crck tip, s oxide-induced closure, plsticity-induced closure or crck shielding by mcrocrcks. Such mechnisms often dominte the crck behvior in structure. We consider the mechnism of crck bridging by fibers becuse it plys one of the most importnt roles in the strength of fiber-reinforced composites. Typiclly, reinforcing fibers hve higher stress-to-filure thn the mtrix, thus the crck propgtes initilly through the mtrix leving the fibers intct. Figure 5.12 () nd (b) re photogrphs of mtrix crcks in two different composites. The first is titnium mtrix with long ligned SiC fibers nd the second is cermic mtrix reinforced with SiC whiskers. n both cses the crck hs completely trversed the mtrix, however, the fibers re undmged. Reinforcing fibers in composites ct to slow the propgtion of mtrix crcks through two simultneous effects, The intct fibers bridging the crck pply bridging trctions to the crck fces which ct to close the crck. The fibers hed of the crck tip increse the stiffness of the mteril close to the crck tip which reduces the deformtion ner the crck tip. These effects chnge significntly the stress-strin reltion of fiber reinforced composite once mtrix crcking hs begun, s shown qulittively in Figure Below the stedy stte crcking stress σc, the stress strin behvior is liner with n effective elstic modulus Ec. Mrch

14 When the lod is incresed to σ c mtrix crck propgtes, cusing sudden increse in strin under the sme lod. Since the fibers re still intct, the composite continues to support the full lod. However, s the lod is further incresed the fibers rech their yield limit nd brek. σ c Note tht due to the sttisticl vribility in the strength of the fibers, not ll fibers brek t the sme time, explining the slight descent of the stress-strin curve before the complete filure of the composite. The gol of this ppliction is to use the weight function technique to describe n effective stress intensity fctor t the crck tip, tking into ccount the fiber bridging. n concrete terms, we need to define the effective K in terms of the crck length, the COD profile u(x), nd the fiber spcing Λ, which re ll quntities tht we cn mesure experimentlly. These vribles re shown on schemtic of bridging zone in Figure The problem is considered to be 2D. Such simplifiction is relistic in the cse of lrge fiber volume frctions. The influences of 3D effects (such s fiber geometry nd specimen thickness) re still the subject of current reserch in this re. Figure 5.11 Common mechnisms tht slow crck propgtion (Suresh, 1991). Mrch

15 () (b) Figure 5.12 Exmples of crck fiber bridging, () SiC fibers bridging crck in titnium mtrix, (b) whisker reinforced cermic (Anderson, 1995). Figure 5.13 Stress-strin behvior of fiber-reinforced composite experiencing mtrix crcking: () liner behvior, (1) mtrix crcking, (2) fiber filure Effective stress intensity fctor A first pproch to determine the effective stress intensity fctor is to model the fiber bridging trctions s series of point lo, ech pplied t the center of the fiber. This loding scheme is shown in Figure ncluded in this ssumption is perfect interfce between the fiber nd the mtrix (since the force is only pplied t one point of the fiber). For n ctul crck in composite mteril, prticulrly ftigue crck in which considerble fiber slipping nd fiber/mtrix debonding occurs, this ssumption is not vlid. However this technique gives us first estimte of K. Lter, we tret distributed bridging trctions model, in which we cn ssume the effects of fiber/mtrix debonding. Mrch

16 Figure 5.14 Schemtic of bridging zone in fiber reinforced composite, 2D representtion. The bridging trctions re defined s the series of point lo {p i, c i } (i 1,...n), where p i is the mgnitude of ech point lod nd c i is the loction long the x-xis where it is pplied. The problem is considered to be symmetric bout x, thus p i cts on both the upper nd lower surfce of crck. The remote loding σ, cn be uniform or ny other distribution for which the stress intensity fctor cn be clculted for the homogeneous problem (i.e. without the presence of bridging fibers). The COD t the loction ci is defined s i. Figure 5.15 Prmeters of discrete bridging fibers model. One cn use the principle of superposition to divide the problem into tht of the remote loding nd tht of the bridging trctions pplied to n unloded body. The loding cn be further broken down into the remote stress plus ech individul fiber bridging trction pplied to n Mrch

17 unloded body. This scheme is shown in Figure Thus, the K, is given by the following eqution, i i 1 n K K ( σ,) + K (p,) (5.25) where K( σ,) is the due to the remote lod nd K (p i,) is the stress intensity fctor due to single bridging force pi which cn be determined using eq (5.1), i i K( p i,) 2p F, π c b (5.25b) When the bridging forces in ech fiber re known, K cn be esily clculted. However, this is not s simple becuse it is not possible to experimentlly mesure the bridging forces directly in ech fiber. Thus, it is necessry to clculte the forces bsed upon quntities tht we cn mesure. For instnce, we cn mesure the COD reltively esily using opticl techniques such s Electronic Speckle Pttern nterferometry (ESP) nd derive the bridging forces from the COD. While such clcultions re not trivil, n efficient numericl solution cn be developed bsed upon the weight function method. For ech fiber (i 1,..., n) we consider its loction c i, nd the corresponding COD i, s known nd the force in the fiber p i s unknown. Define the COD t point c i due to unit dipole lod pplied t c j, s. As seen erlier with the weight function method, the totl displcement t ij point ci, is given by the sum of the ij plus the displcement t ci due to the remote lod i, + i i n j1 ij (5.26) The vlues trctions. i cn be clculted nlyticlly for ech point nd re independent of the bridging The displcements ij, re expressed in terms of weight function, 8p ij j E mx(c i,c j) i j g c,s g c,s (5.27) For exmple, the weight function for centrl crck in n infinite plte nd for single edge notched specimen when / b re given by eqs (5.23) nd (5.24), respectively, i.e., 1 1 g(x,) π 1 x / 2 2 (5.23) nd, Mrch

18 g x, 3/ (x/) π 1 (x/) 2 (5.24) Figure 5.16 Principle of superposition pplied to crck under remote lod nd bridging forces. Note tht g( x,) is strictly function of the specimen geometry which becomes importnt when we consider the speed of the clcultions. Figure 5.17 shows the effect of two bridging fibers on the COD of single edge crck. The weight function for this exmple is tht given by eq. (5.24). The grph shows the COD, clculted t 5 points long the crck length, for severl mgnitudes of pplied trctions, with p 1 2p 2 for ll cses. The solid line is plot of the COD due to the remote lod σ, tken s uniform. Note tht the COD t the point where force pplies, due to tht sme force, is not defined. Thus, few points round the bridging forces re excluded from the plots in Figure The COD in Figure 5.17 is obtined from know bridging trctions. However, wht is of greter interest is the inverse problem. Tht is, the clcultion of the trctions bsed on known COD profile. Mrch

19 n the following, method for the determintion of the bridging trctions from the COD is outlined bsed on the formlism presented bove Discrete bridging model For ese of clcultion, we cn lso write eq. (5.26) in mtrix form. Setting eq. (5.27) in the form, ij ij p j (no summtion over j) eq. (5.26) is written s, + i i n j1 ij p j where n is the number of bridging forces (Figure 5.16). n vector form the lst reltion tkes the form, or { { } { } + { p} b where } { p} (5.28) is n nxn mtrix nd { p }, { b } re vector relted by { b } { p }. The chrcteristic tht mkes this technique powerful is its efficiency. The mtrix vector { } nd the depend only upon the specimen geometry which mens tht one cn clculte nd { } once for given specimen type nd store them for future clcultions. This is by fr the most time consuming clcultion in this procedure. Then, for given set of dt, one cn quickly clculte the vector { b }, nd solve eq. (5.28) for { p }. Unfortuntely, the mtrix is ill defined when we consider the COD t the points where the bridging forces pply. This is due to singulrity of the weight function when we consider the COD t the point of the pplied bridging force. To void this problem, we cn replce the concentrted forces with n equivlent stress distribution over the fiber dimeter. Another wy to void the singulrity is to mesure CODs t points other thn those where the concentrted forces re pplied nd use them s input to the bove equtions. A concern tht rises next is with regrd to the loction of the COD mesurements used in the method. Since the number of the COD mesurements should be t lest equl to the number of fibers in the bridging zone, the points of mesurements should be chosen so tht the error in the trctions is minimized. Mrch

20 Figure 5.17 Effects of two bridging fibers on the COD of single edge notched specimen with p 1 2p 2 for ech loding condition nd c 1.3, c Distributed bridging trctions model The requirements of the discrete model (i.e. the bridging trctions re pplied s point lo nd the mtrix is perfectly bonded to the fibers) re difficult to meet. To simplify the problem, the lrge number of bridging fibers in typicl composite, is replced by distributed closing trctions (the bridging trctions) on the crck fces. The procedure followed for this model is the sme s for the discrete point lo described in the previous section. Once gin we use the principle of superposition. Thus, we cn write the COD in terms of the sme weight functions s before, however, now the effect due to the bridging trctions is integrted long the crck length nd weighted by the function x. The function u(x) is the COD due to the remote lod (see eq. 5.22). ϕ s 4 u( x) u ( x) g( x,s) ϕ( x) dx g( x,s) E (5.22) x Figure 5.18 presents n exmple of the COD clcultion. The exmple is once gin single edge crck under uniform remote lod with pplied bridging trctions due to two fibers. However, in this exmple the trctions re pplied s uniform lod distributed over the fiber dimeter, s shown. The clculted COD is plotted for severl vlues of pplied bridging trctions. Note the difference between the COD in Figure 5.18 nd tht in Figure Unfortuntely, the solution of the inverse problem (i.e. determine ϕ( x) from mesured u(x)) is not solvble directly s tht for the discrete fiber force model presented before. nsted we hve n inverse integrl eqution to solve. Two techniques to solve this eqution cn be used. First: solve the eq. (5.22) using n itertive procedure. This procedure cn require lot of computing time, lthough it cn be reduced by choosing n initil guess for the bridging trctions tht is plusible bsed on previous experience. This technique is not Mrch

21 prcticl when there re mny sets of dt to tret becuse the computtion must be restrted from the beginning for ech new set of dt. Second: pproximte ϕ x by piecewise continuous function. Suppose tht the COD dt re given t discrete points N. Next, we divide the entire intervl of points into smller intervls nd define the function, N 1 x x x ϕ ( x) ϕ h ( x ) h ( x) j1 j j j j (5.29) j1 x < x j1, x > x j which is piecewise-continuous pproximtion of the function ϕ( x). By choosing / N «D, where D is the fiber dimeter, the function ϕ ( x) cn be well pproximted by this piecewise function. This pproximtion lso permits discontinues in ϕ ( x). Substituting into eq. (5.22) for ϕ( x) nd performing some lgebr, we cn rewrite eq. (5.22) in mtrix form similr to eq. (5.28). The detils re not presented here, however, using lest squres optimiztion, one cn reduce esily the solution 1. As in the cse of the discrete model, the mtrices depend only upon the geometry nd thus need only be clculted once. The solution is thus much more rpid thn the itertive procedure if there re mny sets of dt to tret. Figure 5.18 Effects of two bridging fibers on the COD. The ttrctions re distributed over the fiber dimeter, D.1. c 1.35, c The specimen is loded with σ (not shown for clrity). 1 M. Studer, J. Pietrzyk, K. Peters, J. Botsis nd P. Giccri (22), nterntionl Journl of Frcture, Vol. 114, pp Mrch