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1 AFRL-ML-WP-TP TERMALLY INDUCED STRESS INTENSITY IN A OMOGENEOUS PLATE CONTAINING A FINITE LENGT CRACK (PREPRINT) George Jefferson MAY 7 Approved for publi relese; distribution unlimited. STINFO COPY This work ws funded in whole or in prt by Deprtment of the Air Fore ontrt FA865-4-D-5. The U.S. Government hs for itself nd others ting on its behlf n unlimited, pid-up, nonexlusive, irrevoble worldwide liense to use, modify, reprodue, relese, perform, disply, or dislose the work by or on behlf of the U.S. Government. MATERIALS AND MANUFACTURING DIRECTORATE AIR FORCE RESEARC LABORATORY AIR FORCE MATERIEL COMMAND WRIGT-PATTERSON AIR FORCE BASE, O

2 REPORT DOCUMENTATION PAGE Form Approved OMB No The publi reporting burden for this olletion of informtion is estimted to verge hour per response, inluding the time for reviewing instrutions, serhing existing dt soures, serhing existing dt soures, gthering nd mintining the dt needed, nd ompleting nd reviewing the olletion of informtion. Send omments regrding this burden estimte or ny other spet of this olletion of informtion, inluding suggestions for reduing this burden, to Deprtment of Defense, Wshington edqurters Servies, Diretorte for Informtion Opertions nd Reports (74-88), 5 Jefferson Dvis ighwy, Suite 4, Arlington, VA -4. Respondents should be wre tht notwithstnding ny other provision of lw, no person shll be subjet to ny penlty for filing to omply with olletion of informtion if it does not disply urrently vlid OMB ontrol number. PLEASE DO NOT RETURN YOUR FORM TO TE ABOVE ADDRESS.. REPORT DATE (DD-MM-YY). REPORT TYPE. DATES COVERED (From - To) My 7 Journl Artile Preprint 4. TITLE AND SUBTITLE TERMALLY INDUCED STRESS INTENSITY IN A OMOGENEOUS PLATE CONTAINING A FINITE LENGT CRACK (PREPRINT) 6. AUTOR(S) George Jefferson 5. CONTRACT NUMBER FA865-4-D-5 5b. GRANT NUMBER 5. PROGRAM ELEMENT NUMBER 6F 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION REPORT NUMBER UES In. 44 Dyton-Xeni Rod Dyton, O SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES). SPONSORING/MONITORING AGENCY ACRONYM(S) AFRL-ML-WP Mterils nd Mnufturing Diretorte Air Fore Reserh Lbortory Air Fore Mteriel Commnd Wright-Ptterson AFB, O DISTRIBUTION/AVAILABILITY STATEMENT Approved for publi relese; distribution unlimited.. SPONSORING/MONITORING AGENCY REPORT NUMBER(S) AFRL-ML-WP-TP SUPPLEMENTARY NOTES Journl rtile submitted to the ASME Journl of Engineering Mterils nd Tehnology. This work ws funded in whole or in prt by Deprtment of the Air Fore ontrt FA865-4-D-5. The U.S. Government hs for itself nd others ting on its behlf n unlimited, pid-up, nonexlusive, irrevoble worldwide liense to use, modify, reprodue, relese, perform, disply, or dislose the work by or on behlf of the U.S. Government. PAO Cse Number: AFRL/WS 7-, 5 Apr 7. Pper ontins olor ontent. 4. ABSTRACT The delmintion of orthotropi lmintes ontining finite-length rks nd subjet to therml grdients is exmined. The ext limiting se solutions for infinitesiml- nd infinite-length rks re known, nd re equl to eh other when the rk length is pproximtely equl to the plte thikness. owever, in the trnsition region of rk length from bout to 5 times the plte thikness, both limit solutions overestimte the energy relese by -%. ene, n nlysis ws developed to better predit the energy relese rte for suh finite-length rks. The model is modifition of the infinite-rk nlysis of uthinson nd Lu (995, ASME J. Eng. Mt. Teh., 7 (4) pp. 86-9) nd provides losed form expression for the elsti energy relese rte in plne-strin orthotropi flt plte tht grees well with numeril vlues for rks of length pproximtely hlf of the plte thikness nd lrger. The nlyti result is shown to gree well with finite element results over wide rnge of rk lengths, depths nd interfe ondutivity, both for isotropi nd orthotropi mterils. 5. SUBJECT TERMS omogeneous Plte, Finite, Orthotropi Lmintes 6. SECURITY CLASSIFICATION OF: 7. LIMITATION 8. NUMBER 9. NAME OF RESPONSIBLE PERSON (Monitor) OF ABSTRACT: OF PAGES SAR. REPORT Unlssified b. ABSTRACT Unlssified. TIS PAGE Unlssified Rndll S. y 9b. TELEPONE NUMBER (Inlude Are Code) N/A Stndrd Form 98 (Rev. 8-98) Presribed by ANSI Std. Z9-8 i

3 Thermlly Indued Stress Intensity in omogeneous Plte Contining Finite Length Crk George Jefferson Air Fore Reserh Lbortory, Mterils nd Mnufturing Diretorte Wright Ptterson Air Fore Bse, Ohio, USA, 454 UES, In., Dyton O 454 The delmintion of orthotropi lmintes ontining finite-length rks nd subjet to therml grdients is exmined. The ext limiting se solutions for infinitesiml- nd infinite-length rks re known, nd re equl to eh other when the rk length is pproximtely equl to the plte thikness. owever, in the trnsition region of rk length from bout to 5 times the plte thikness, both limit solutions overestimte the energy relese by -%. ene, n nlysis ws developed to better predit the energy relese rte for suh finite-length rks. The model is modifition of the infiniterk nlysis of uthinson nd Lu (995, ASME J. Eng. Mt. Teh., 7 (4) pp. 86-9) nd provides losed form expression for the elsti energy relese rte in plne-strin orthotropi flt plte tht grees well with numeril vlues for rks of length pproximtely hlf of the plte thikness nd lrger. The nlyti result is shown to gree well with finite element results over wide rnge of rk lengths, depths nd interfe ondutivity, both for isotropi nd orthotropi mterils. Introdution Advned high temperture mterils suh s ermi omposites re under tive investigtion for use in hot shell strutures suh s ombustion liners. These mterils re typilly hrterized by low therml ondutivity nd lminr struture with reltively low interlminr strength. Consequently, the hot strutures will be subjet to lrge therml grdients nd therml stress indued delmintion initited t internl interlminr flws re of signifint onern. The therml delmintion problem hs been studied extensively in the literture. The stress onentrtion due to n infinitesiml rk in n elsti body subjet to therml grdient is given by Sih []. The omplementry symptoti liner elsti frture solution for n infinitely long rk in n isotropi flt plte is given by uthinson nd Lu [] (-L). These bsi solutions re followed in the literture by nlysis of more omplex systems. Sorensen, Srrute, Jorgensen nd orsewell [] nlyzed the long-rk problem for lminte with dissimilr elsti lmine, nd show tht the interfil energy relese rte my be mitigted by seletive lyering of mterils. Gu nd Asro [4] onsidered delmintion frture in ontinuously grded mterils. The elsti-plsti problem ws onsidered by Aoki, S., Kishimoto, K., nd Skt [5]. The present work fouses on the homogeneous orthotropi plte problem. Numeril investigtion revels tht, with inresing rk length, the trnsition from the Sih infinitesiml rk result to the - L infinitely long rk, or stedy-stte, solution is less brupt thn estimted by uthinson nd Lu.

4 ere, we present modifition to the -L result tht provides substntil improvement for finite length rks (i.e. on the order of the plte thikness), while retining the ext limiting result for rbitrrily long rks. The numeril work tht motivted the present nlysis ws strightforwrd -d plne strin ABAQUS finite element model of entrl rk in flt plte subjet to presribed temperture differentil between the surfes s shown in Figure. A foused mesh with seond order qurter point elements t the rk tip, nd redued integrtion elements throughout ws employed[6]. Energy relese rtes were lulted using the thermoelsti J-integrl formultion built into the ommeril ode [7,8]. For verifition, the lulted energy relese rtes were ompred with the ext longrk -L solution nd were found to be 5% below the symptoti predition for rks s long s times the plte thikness, with onvergene to the symptoti result onfirmed only by modeling rks - times the plte thikness. Thus it seemed pproprite to develop n nlyti model to better predit the frture behvior for suh intermedite length rks. The -L model is bsed on long-rk isotherml frture model of uthinson nd Suo [9] long with severl infinite rk pproximtions for the temperture distribution nd resulting deformtion due to therml grdient. Numeril simultion of the isotherml model shows tht it performs remrkbly well for rks s short s twie the plte thikness. There re two key symptoti kinemti ssumptions in the thermoelsti nlysis. ) It is ssumed tht the upper nd lower setions of the plte should hve equl urvtures. ) The in-plne strin prllel to the rk fes t the rk enter is ssumed to be equl on the opposed rk fes. By inspetion of the numeril simultion, the equl urvture ssumption proves to be quite good for reltively short rks. The equl strin ondition is, however, pprohed rther slowly with inresing rk length, nd here n improvement is suggested. Bsi nlysis summry The essentil lultion is the sme here s in -L, nd so here we will highlight the key points s briefly s possible. The therml lod results in set of resultnt fores, P, nd moments M, M per unit out-of-plne thikness t the enter of the rk s shown in Figure. The stress intensity is then lulted in terms of those end lods using the isotherml frture formultion set forth in [9]. Referring to Figure, equilibrium requires M + M P. For long rk the upper nd lower plte setions re modeled s bems nd the urvtures re sserted to be equl fr from the rk tip, leding to the ondition M, thus M Abqus, In., Pwtuket R.I.

5 η P P M, M () η ( + η ) ( + ) where η. The nlysis is restrited to <η, ie. if the rk is off enter it must be nerer to the hot side of the plte to ensure n open mode rk []. Eqution () is s-given in [], however here we retin the resultnt fore P s n unknown. The orthotropi thermo-elsti onstitutive reltion is of the form, ε ε xz σ / Ex vyxσ σ μ xz xz yy E y v σ zx zz E z + α xδt () where ε ij, σ ij re the omponents of strin nd stress nd expressions for the remining strin omponents re obtined by suitble hnge of subsripts. E, μ, v re the orthotropi elsti onstnts whih stisfy the symmetry requirement, v ije j v jiei, αi re orthotropi therml expnsion oeffiients nd ΔT is hnge in temperture. Using Eq. (), nd ssuming plne strin ( ε ε ) yy yz yx i ij ij ε the stress intensity ftors re found diretly using the nlysis given in [9]. In the entered rk ( η ) se the rk tip field is pure mode II, with stress intensity ftor, where η K II 8 Λ + ρ Λ λ P () (4) λ nd ρ re the orthotropi mteril onstnts defined in[9] nd in Lu, Xi nd uthinson [], λ E z E (5) E ν xz + ν xyν ρ λ μxz ν xyν yx yz (6) where E ( ν ν ) nd E ( ν ν ) E x xy yx re the effetive plne strin moduli for in-plne z E z zy yz 4 unixil stressing in eh of the priniple diretions. See lso Krenk [], using ρ κ nd λ δ. Note tht λ, ρ, nd Λ re ll unity for n isotropi mteril. In the generl se for < η the rk is mixed mode with the energy relese rte nd stress intensity ftors,

6 Λ G E K K K II I λ 4 P ( ) ) ( + η) K I λ + K II η II 4 Γ ( GE Λ K ) II Eη 5 ( + η ) Where Γ is the rtio of mode II stress intensity to the η stress intensity, + η Γ η ( η + ) sinω η osω + γ + U 4 V ( + ) η (7) (7) nd U, V, w, γ re the dimensionless funtions of η, U / V sin γ 6η ω ( + η) ( + η ( + η) ) ( + η ) ( + η) ( η) + Kinemti onstrint UV sin 7 (8) In ple of the -L ondition of strin equlity t the rk enter, the kinemti ondition we will impose is tht the deformed lengths of the upper nd lower rk fes must be equl. This leds to the ondition, ( + ( ) ( ε ε )) dr (9) where ε is the strin prllel to the rk fe, r is the oordinte long the rk mesured from the tip, nd supersripts (+) nd (-) denote the upper nd lower fes respetively. The orthotropi plne strin onstitutive reltion on the trtion-free fes is, σ ( ± ε + αt ), () E whereσ is the stress prllel to the rk fe. α α x + ν yxα y, is the plne strin therml expnsion oeffiient, nd T is the rk fe temperture The nottion is used to indite tht this is pir of equtions seprtely relting the quntities on eh fe. Combining Eq. (9) nd Eq. () yields, ( + ( ) ( σ σ )) dr + Eαφ( T T ) () wheret,t re the tempertures t the plte surfes nd we define the verge temperture jump φ, ( + ) ( φ ) ( T T ) dr () T T 4

7 For omprison with the -L formultion, for suffiiently long rk, fr from the rk tip, the het ondution through the plte is D nd the jump in fe tempertures wy from the rk tip will be determined by the ondution ross the rk s, ( + ) ( ) T T T T + B x () where the Biot number, perfetly insulting rk, nd B, hrterizes the het flow ross the open rk, with B for B for perfet ondution[]. Ner to the tip, het flux round the tip redues the temperture jump ross the fes. owever in the limit >> the influene of the rk tip field beomes vnishingly smll, so tht φ ( + B ) φ < ( + B ). <. For ny finite length rk By inspetion of numeril results, for rks tht re pproximtely equl to the plte thikness nd lrger, the stress long the rk fes is well pproximted by mthing the symptoti rk tip (Pris nd Sih, []) nd fr-field (bem theory) solutions, * σ r r σ (4) * s K IIΛ r r π r where σ is the vlue of the tngentil stress t the plte enter, nd the sign inditor ( + ) ( ) s ; s, is introdued beuse the stress is ompressive on the upper fe nd tensile on the * lower. The trnsition rdius, ( ± r ) is the distne from the rk tip (generlly different on eh fe) where the two limiting forms re equl, whih is simply, *( ) ± Λ K r II (5) π σ Figure shows the pieewise stress pproximtion ompred with numeril results. It should be noted * tht the piee-wise funtion is somewht lower thn the tul distribution ner ( ± r ), whih ultimtely results in n over (i.e. onservtive) estimte of the energy relese rte. The rk fe stress fr from the rk tip, σ where,, is found diretly from bem theory (s in -L), 8P ( ± σ ) s g ( η) (6) ( + η) 5 ( ) ( ) ( ) g η + s (7) 6η + η + η ( η) ( 4η + η ) so tht, 5

8 r * Λ Γ 4π g ± ( ) (8) relling tht Γ is the stress intensity rtio given by Eq (7). Clerly the present nlysis must be restrited to ses where > r *. owever s region we will onservtively lso restrit pplition to se if. η. The mximum vlue of * r represents the size of the rk tip dominted > r * r in tht rnge is *, whih is ensured for the isotropi ~ t η. 5. The rk length ondition, Eq. (), n now be integrted nd solved for P, P η ( + η ) Eαφ( T T ) 5 Λ Γ ( + η) + ( ) ( ) 4π g + g In the lrge limit, nd using φ ( + B ) η P ( + η ) Eα ( T T ) 5 ( + η) ( + ) B (9), we reover the -L result extly, () Finlly, the stress intensity nd energy relese rte re found by substituting Eq (9) into Eq (7), e.g. E G [ α ( T T )] η ( + η ) Λ 4π φ Γ g 5 ( + η) + ( ) ( ) + g ( ) ( ) + It is notble tht the quntity Γ g g is pproximtely for.5 η. The lrge limit of (9) of ourse grees with -L, hene we will use the, η, perfetly insulting ( φ ), se s bseline for omprison of solutions, α ( ) () G L E [ ] T T () For referene the smll-rk isotropi limit of Sih [] is, G GL Temperture distribution π () As will be shown, Eq (9) represents n improvement over Eq () even if the rk fe temperture distribution is pproximted by tking φ ( + B ). owever, further improvement is relized by lulting φ from Eq. () using the true temperture distribution. Unfortuntely, lthough the symptoti temperture distributions ner the tip nd fr field re known (Florene nd Goodier, []), ssuming pieewise temperture distribution similr to Eq. (4) does not prove suffiiently urte. Therefore we will rely on numerilly generted results to derive n funtionl dependene of φ on the plte geometry. The therml finite element nlysis ws performed using therml 6

9 ( T T ) + ontt reltion over the rk surfe suh tht the het flux ross the rk is q B k ( ) ( ), where k is the (isotropi) therml ondutivity of the plte. Note tht the het trnsfer ross physil rk is, in generl, strongly dependnt on the rk opening, however here we hve ssumed tht lol vrition of the rk opening over the rk length my be negleted nd tht B is n effetive vlue whih ounts for the verge rk opening. The onsequenes of this ssumption will be disussed in subsequent setion. A high qulity generl fit to the numeril results is found by rguing s follows. For suffiiently long rks, inresing the rk length does not signifintly ffet the non-uniform field ner the tip, further, for n infinite rk we require φ ( + B ), therefore we fit our numeril simultion results to the form, C + B φ (4) where C is funtion of B nd η. C is determined from numeril solution of the het trnsfer problem s the lrge limit of the quntity ( ( + B ) φ )( ), whih onverges to onstnt quikly with inresing rk length ( ( ) is suffiient). We find tht for rks longer thn ( ) ( ) ξ η φ (5) + B + B, where the dependene on B is empiril. The vlue of ξ is only wekly dependnt on η over the rnge.5 < < η, hving vlues of ξ. t η nd. 88 exellent fit to numerilly derived vlues for ll B nd for ξ t η. As Figure 4 shows, this is n >. Eqution (5) is lerly not vlid for ξ ( + B ), but this restrition is in line with other spets of the nlysis. Results nd Disussion Figure 5 shows the energy relese rte lulted using Eq (), with φ from Eq (5) s well s the pproximtion, φ ( + B ), for omprison. The full lultion shows good (nd s expeted slightly onservtive) greement with the finite element results. Ignoring the ner-tip temperture vrition results in substntilly higher energy relese rte predition, lthough it still provides some improvement over the symptoti result. Figure 6 shows the effetiveness of the nlysis for vrition of both η nd B. The greement with numeril results is best for η nd somewht dereses, beoming more onservtive, for off-enter rks. As noted previously this nlysis is not intended for signifintly smller vlues of η (s for exmple subsurfe rks), where the differene undoubtedly inreses. 7

10 It is worthwhile to refer to Figure 6 in disussing the effet of the onstnt shows slightly inresing energy relese rte for deresing η t fixed vlue of B ssumption. The nlysis B. owever, relistilly, the het trnsfer oeffiient itself dereses substntilly (potentilly by orders of mgnitude) s the rk opens nd the mehnism of trnsfer trnsitions from ontt ondution to gs trnsport nd rdition (-L). As η is deresed, the stress intensity beomes inresingly mode I, therefore the rk opens further. Consequently if the dependene of het ondution on the rk opening is ounted for then the dependene of G on η my be fr more pronouned thn suggested by Figure 6. This spet of the problem is disussed in detil in -L nd is not qulittively ffeted by the present nlysis. Closure In losing we note tht lthough the exmples given hve been for isotropi mterils the formultion holds quite well for orthotropi mterils s well. Eq. (9) predits tht the energy relese rte will be redued for lrge vlues of the prmeter Λ, while the -L result is independent of ny orthotropy (the orthotropy ffets only the mode mixity). Lrge Λ orresponds to smll vlues of through-thikness moduli, E z nd μ xz, whih is notbly typil of mny omposite systems. As n exmple we onsider in-plne isotropy, nd fix ν ν., ν ν. 5 nd λ. Figure 7. xy yx xz yz shows the theory long with finite element results for η nd η s the in-plne sher modulus is vried. The model orretly predits the redution in relese rte for low sher, however the effet is signifint in this exmple only for very smll vlues. Aknowledgment This work ws performed while the uthor held Ntionl Reserh Counil Reserh Assoiteship Awrd t the Air Fore Reserh Lbortory Mterils nd Mnufturing Diretorte. Nomenlture rk hlf-length,, totl, upper, nd lower thikness η thikness rtio x,y,z orthogonl oordintes r oordinte from tip long fe T temperture. T,T tempertures, plte surfes P, M, M resultnt fore nd moments ε ij, σ ij omponents of strin nd stress E i, μ ij, vij elsti onstnts α i therml expnsion oeffiients E, E z effetive moduli α effetive therml expnsion K, stress intensity ftors I K II 8

11 η K II stress intensity ftor for η G energy relese rte. G,G L symptoti limits of G. λ, ρ,λ orthotropi mteril onstnts φ verge temperture jump B Biot number σ fr field rk fe stress s sign inditor * r trnsition rdius k therml ondutivity q het flux dimensionless funtions of η U, V, w,γ stress intensity deomposition g Eq (7), g ( η ) η Γ rtio K II ( η) / K II, Γ( η ) ξ fit prmeter supersripts (+) (-) upper nd lower rk fe (±) quntity in eqution pplied seprtely to eh fe. 9

12 Referenes [] Sih, G.C., 96, "On the Singulr Chrter of Therml Stresses Ner Crk Tip," ASME Journl of Applied Mehnis Vol 7 pp [] uthinson, J.W. nd Lu, T.J. 995, "Lminte Delmintion Due to Therml Grdients," ASME Journl of Engineering Mterils nd Tehnology, Vol 7 (4) pp [] Sorensen, B.F., Srrute, S., Jorgensen, O. nd orsewell, A., 998, "Thermlly Indued Delmintion of Multilyers," At Mterili Vol. 46 (8) pp [4] Gu, P. nd Asro, R.J., 997, "Crks in Funtionlly Grded Mterils," Interntionl Journl of Solids nd Strutures. Vol 4 () pp. -7. [5] Aoki, S., Kishimoto, K., nd Skt, M., 98, "Elsti-plsti Anlysis of Crk in Thermlly-loded Strutures," Engineering Frture Mehnis, Vol 6 () pp [6] Brsoum, R.S., 976, "On the Use of Isoprmetri Finite Elements in Liner Frture Mehnis," Interntionl Journl for Numeril Methods in Engineering. Vol () pp [7] ABAQUS Theory Mnul, Version 6.4,, Setion.6., ABAQUS, In, Pwtuket, RI. [8] Kishimoto, K., Aoki, S. nd Skt, M. 98, "On the Pth Independent Integrl-J," Engineering Frture Mehnis, Vol (4) pp [9] uthinson, J.W. nd Suo, Z. 99, "Mixed Mode Crking in Lyered Mterils," in Advnes in Applied Mehnis Vol 9, edited by J. W. uthinson nd T. Y. Wu. (Ademi Press, Boston) pp [] Lu, T.J., Xi, Z.C., nd uthinson, J.W., 994, "Delmintion of Bems Under Trnsverse Sher nd Bending," Mterils Siene nd Engineering A 88 (-) pp. -. [] Krenk, S., 979,. "On the Elsti Constnts of Plne Orthotropi Elstiity," Journl of Composite Mterils. Vol () pp [] Pris P. C. nd Sih, G.C. 965, "Stress Anlysis of Crks" in Frture Toughness Testing nd Its Applitions, ASTM STP 8, Amerin Soiety for Testing nd Mterils, Phildelphi. [] Florene, A.L. nd Goodier, J.N., 96, "Therml Stresses Due to Disturbne of Uniform et Flow by n Insulted Ovloid ole," ASME Journl of Applied Mehnis Vol. 7 (4) pp

13 Figures T T Figure. Geometry of n infinite flt plte with finite length entrl rk. Configurtion shown is, / Contour shding shows the numerilly lulted temperture distribution for prtly onduting rk ( ). B. P,M P,M z y x ( T + ) ( x) ( T ) ( x) T T symmetry plne Figure. Plne geometry of thermlly loded enter rked plte showing the resultnt fore nd moments t the entrl symmetry plne. Surfe tempertures T,T re fixed with ( ) ( ) T > T, while rk fe tempertures T +, T re dependnt on position long the rk fe.

14 ( ) T T σ E α 4.5. * r / 4π theory numeril K II, σ theory lrge limit FEM / FEM / r Figure. Pieewise stress distribution funtion. Exmples shown re for η, Λ nd B. For the symmetri se the bsolute vlue of the stress (shown) is the sme on upper nd lower fes. The theory lines re lulted using Eq. (4) with K II nd σ ) extrted from the finite element solution (solid lines) nd b) bsed in the lrge limit, σ Eα ( T T ), K II Eα ( T T )( ) 8 (dshed line). theory B B B Figure 4. Averge temperture jump ross rk fe for entered rk ( η ). Mrkers re bsed on finite element results, lulted using Eq. (). Theory lines re Eq. (5) with fit prmeterξ.. The fit is remrkbly good for wide rnge of B nd.

15 smll limit -L, stedy stte theory, φ theory with lulted φ FEM Figure 5. Energy relese rte, for η, Λ nd B. Theory lines re Eq. () using φ, for fully nlyti result tht demonstrtes n improvement over the infinite rk length solution. Using the improved vlue of φ from Eq. (5) yields n exellent fit to the numeril results. The lultion is remrkbly well behved for shorter rks thn one might expet (dshed region), it does however beome non-onservtive go to zero t nonzero. theory.9. FEM, B η.5 FEM, B η FEM, B η.5 FEM, B η..5 Λ, for η,, nd B,. The theory, Eq's. () nd (5), fits best for η, nd provides onservtive estimte for off-enter rks. The numeri vlues to the right re the ext Figure 6. Energy relese rte for the isotropi se ( ) symptoti vlues for the four ses from Eqn's (7), nd ().

16 theory η η λ B 4 μ xz E x λ B showing the vrition with Figure 7. Orthotropi exmple. Normlized energy relese rte for 4,, sher modulus. Redued energy relese rte is predited by both theory nd numeril simultion, but only for very smll vlues of μ. xz 4