Copyright @ 2011 Birkhäuser BostonA nonlinear Abel-type equation is obtained in this paper to model creep crack time-dependent propagation in the infinite viscoelastic plane. A finite time when the integral equation solution becomes unbounded is obtained analytically as well as the equation parameters when solution blows up for all times. A modification to the Nyström method is introduced to numerically solve the equation and some computational results are presented

G.N. Savin

K.E. Atkinson

R. Gorenflo

R. Penny

S.E. Mikhailov

Y.N. Rabotnov

Yu.N. Rabotnov

English

L. Hakim

S. E. Mikhailov

Crossref

Nonlinear Abel-Type Integral Equation in Modeling Creep Crack Propagation

Hakim, L

Mikhailov, SE

Brunel University Research Archive

1Nonlinear Abel-Type Integral Equation inModelling Creep Crak PropagationL. Hakim and S.E. MikhailovBrunel University West London, UK;layal.hakimbrunel.a.uk, sergey.mikhailovbrunel.a.ukIn: Integral Methods in Siene and Engineering: Computational and Analyti Aspets. (eds.C. Constanda, P. Harris), Springer (Birkhäuser), Boston, ISBN 978-0-8176-8237-8, 2011, 191-201.A non-linear Abel-type equation is obtained in the paper to model reeprak time-dependent propagation in the innite viso-elasti plane. A nitetime when the integral equation solution beomes unbounded is obtained an-alytially as well as the equation parameters when solution blows up for alltimes. A modiation to the Nyström method is introdued to numeriallysolve the equation and some omputational results are presented.1.1 IntrodutionThe strength of materials under reep onditions depends not only on theinstant value of the load but also on the load duration and generally on thetemporal load history. This eet is essential, e.g. for onrete and some plas-tis under room temperature, and for strutural metals under elevated tem-peratures. The temporal strength ondition under onstant uniaxial stress,starting at time t = 0, takes the form|σ| < σ∗(t) i.e.,|σ|σ∗(t)< 1, (1.1)where the dependene of the temporal strength on time has a form shown inFig. 1.1.A popular approximation of the temporal strength σ∗(t) is the power-typefuntion, given by σ∗(t) = σ0t−1/b that we will all (similar to fatigue) theBasquin diagram, where σ0 and b are onstant material parameters, see e.g.[Ra69, PeMa71℄.When the load is not onstant but varies with time, dierent generaliza-tions of relation (1.1), often alled aumulation rules, are possible. A novelaumulation rule for life time under variable loading, given in [MiNa09℄ hasthe following form,2 L. Hakim, S.E. MikhailovtΣ*Fig. 1.1. Durability diagram: material strength σ∗(t) under onstant uniaxial stressas funtion of life time.Λ(σ; t) :=[ddt∫ t0∣∣∣∣ σ(τ)σ∗(t− τ)∣∣∣∣βdτ]1/β< 1, (1.2)where σ∗(t) is the durability under onstant loading and β 6= 0 is anothermaterial onstant.Partiularly for the Basquin diagram σ∗(t) = σ0t−1/b, temporal strengthondition (1.2) takes the form,Λ(σ; t) =[ddt∫ t0∣∣∣∣σ(τ)σ0∣∣∣∣β(t− τ)β/bdτ]1/β(1.3)=[βbσβ0∫ t0|σ(τ)|β(t− τ)βb−1dτ]1/β< 1.As remarked in [MiNa09, MiNa11℄, for β = b the above strength ondition isequivalent to the Robinson partial life time linear aumulation rule.Further generalizations are needed for the multiaxial loading. One of them,that we will further employ, is substituting the maximal prinipal stress for σin (1.2) and (1.3).Let us apply temporal strength ondition (1.3) to predit rak propaga-tion in a viso-elasti isotropi innite plane loaded at innity at time t = 0by a tration q onstant in spae and time, see Fig. 1.2.Let the rak have a length of 2a(τ). The stress distribution σ22(τ ;x1)ahead of the rak in an innite elasti isotropi (or orthotropi) plane hasthe following form (e.g. [Sa61℄), whih remains valid also for the viso-elastimaterial, see [Ra80℄,σ22(τ ;x1) =qx1√x21 − a2(τ). (1.4)Note that due to the problem symmetry, σ22 is a prinipal stress omponentat x2 = 0.Assuming that the durability of the viso-elasti material is desribed bythe durability ondition given in (1.3) with σ = σ22(τ, x1), we substitute there1 Nonlinear Abel-Type Integral Equation 3Fig. 1.2. Crak in an innite planestress distribution (1.4). Replaing the inequality with the equality at the tipof the propagating rak, x1 = a(t), we arrive at the following nonlinearVolterra integral equation and initial ondition for a(τ),∫ t0(t− τ)βb−1[a2(t)− a2(τ)]β/2dτ = ca−β(t), t > 0, (1.5)a(0) = a0, (1.6)where b, β > 0, c =bβ( qσ0)−βand the initial rak half-length a0 are knownonstants.Normalising the variables t → t/t∞ and a → a/a0, where t∞ = (σ0/q)bis the life time of the innite plane without a rak under the same load q,simplies the problem to∫ t0(t− τ)βb−1[a2(t)− a2(τ)]β2dτ =bβa−β(t). (1.7)a(0) = 1, (1.8)The funtion to be sought is a(t), whih denotes the normalized length of therak, and t denotes the normalized time.1.2 Integral Equation CharaterizationEquation (1.7) is a non-linear Volterra integral equation and more spei-ally, sine the integrand has a week singularity as τ approahes t, this is anintegral equation of the Abel type. The unknown funtion partiipates nonlin-early both in the integrand and in the out-of-integral term, and the integranddepends not only on a(τ) but also on a(t).For β = b, equation (1.7) was obtained and solved analytially in [MiNa03a,MiNa03b℄. For this ase the equation is redued to a simpler Abel-type integralequation,4 L. Hakim, S.E. Mikhailovab(t)∫ t01[a2(t)− a2(τ)]b2dτ = 1. (1.9)After the hange of dependent and independent variables (f. also [GoVe91℄)the equation beomes linear, and the solution an be written ast(a) = 1−(a0a)2−b 2 sin(pi(b−2)2 )π(b− 2)2F1(1−b2, 1−b2, 2−b2;a20a2)(1.10)where 2F1 is the hypergeometri funtion.One an onlude from (1.10) that the solution a(t) of equation (1.9) tendsto innity as t→ 1. That is, the rak propagates through the whole inniteplane in the normalized time t = 1. Another one onlusion from expression(1.10) is that the solution of (1.9) does exist when 0 < b < 2 and blows upwhen b → 2. The latter phenomenon also follows from the observation thatthe integral does not exist in (1.9) for β = 2 for any t.Let us analyse these eets for more general ases of equation (1.7). Frommehanial reasonings, we assumed that parameters b and β are positive. Atany t where the funtion a(τ) has a ontinuous derivative a′(t), the integrandin (1.7) tends to(t− τ)βb−β2−1[2a(t)a′(t)]−β2as τ → t. Thus for the integral in (1.7) to exist, we obtain the same ondition0 < b < 2 as for equation (1.9).To nd the normalized breaking time t as a(t) tends to innity, let usmultiply both sides of equation (1.7) by aβ(t) to arrive at the equation∫ t0(t− τ)βb−1aβ(t)[a2(t)− a2(τ)]β2dτ =bβ(1.11)and onsider its limit as a(t)→∞. Evidently,lima(t)→∞aβ(t)[a2(t)− a2(τ)]β2= 1.Assuming that the limit an be transferred under the integral, the limit ofequation (1.11) redues to∫ t0(t− τ)βb−1dτ =bβ.Evaluating the integral yields tβb = 1, whih gives t = 1. Therefore for anyβ > 0 and b ∈ (0, 2), the normalized breaking time is t = 1 if the assumptionon the limit transfer is orret.1 Nonlinear Abel-Type Integral Equation 51.3 Numerial AlgorithmsNo exat or numerial solution of equation (1.7) seems to be available in theliterature for ases when β 6= b. This paper aims to solve numerially equation(1.7) in order to nd the normalized rak length, a, as the normalized time,t, inreases, for dierent values of β.There is a wide range of numerial methods for solving Fredholm andVolterra equations. One of the methods is known as the Nyström method andis explained in details e.g. in [At97℄. However, the standard Nyström methodis not well suited for integral equations with singularities. Therefore, to usethe Nyström method for nding approximate solutions to equation (1.7), wemodied it as desribed in the following.Let us introdue a mesh of n node points, ontaining equidistant timesteps, ti = ih, where h is the step size, i = 0, 1, 2, ..., n . Then, we applyolloations for equation (1.7) in the node points starting from i = 1,∫ ti0(ti − τ)β2−1[a2(ti)− a2(τ)]β2dτ =bβa−β(ti), i = 1, 2, ..., n. (1.12)To handle the singularity, we split the integral into two parts,∫ ti−10(ti − τ)βb−1[a2(ti)− a2(τ)]β2dτ +∫ titi−1(ti − τ)βb−1[a2(ti)− a2(τ)]β2dτ =bβa−β(ti)i = 1, 2, ..., n, (1.13)where the rst integral disappears if i = 1.First, we use linear interpolation to approximate the square braket termin the denominator of the seond integral,a2(ti)− a2(τ) ≈ [a2(ti)− a2(ti−1)]ti − τh, ti−1 ≤ τ ≤ ti, i = 1, 2, ..., n.(1.14)Then substituting this approximation into the seond integrand of equation(1.13), applying a quadrature rule to the rst integral and denoting ai = a(ti),we redue problem (1.7)-(1.8) to the following triangular algebrai system foraj , j = 0, ..., n,a0 = 1, (1.15)i−1∑j=0(ti − tj)βb−1[a2i − a2j ]β2wi−1,j +(ha2i − a2i−1) β2∫ titi−1(ti − τ)βb−1− β2 dτ=bβa−βi , i = 1, 2, ..., n, (1.16)where wi−1,j represent the quadrature weights. To make equation (1.16) validfor i = 1 we set w0,j = 0.6 L. Hakim, S.E. MikhailovEvaluating the integral and simplifying the result, we an rewrite equation(1.16) asi−1∑j=0(ti − tj)βb−1[a2i − a2j ]β2wi−1,j+hβb(βb −β2)[a2i − a2i−1]β2=bβa−βi i = 1, ..., n. (1.17)Sine the summation term vanishes in (1.17) for i = 1, we an nd from(1.15) and (1.17) an expliit expression for a1 in terms of h, β and b,a1 =1−[βbhβb(βb−β2)−1] 2β−12. (1.18)To nd ai for i ≥ 2 we will use a numerial algorithm. The trapeziumquadrature rule was used further in this paper, i.e., the quadrature weights inequation (1.17) have the formwi−1,j =h2for j = 0 or j = i− 1h otherwise.Substituting the weights and making simpliations, equation (1.17) anbe written asF (ai, i) :=h(ti − t0)βb−12[a2i − a20]β2+ hi−2∑j=1(ti − tj)βb−1[a2i − a2j ]β2+hβb(1 + 2(βb −β2)−1)2[a2i − a2i−1]β2−bβa−βi = 0, i = 2, 3, ..., n. (1.19)For a xed i, we will solve non-linear equation (1.19) for ai using the New-ton method. Thus, eah ai will be approximated using the following iterativerelationai,k+1 = ai,k −F (ai,k, i)F ′(ai,k, i)i = 2, ..., n. (1.20)where F ′(ai, i) denotes the derivative of F (ai, i) with respet to ai. Here, irefers to the olloation point in the mesh, while k is the iteration step in theNewton method. Dierentiating and simplifying the result yields the followingexpression for the derivative of F (ai, i),F ′(ai, i) = −hβai(ti − t0)βb−12[a2i − a20]β2+1− hi−2∑j=1βai(ti − tj)βb−1[a2i − a2j ]β2+1−hβb βai(1 + 2(βb −β2)−1)2[a2i − a2i−1]β2+ ba−β−1i , i = 2, ..., n.1 Nonlinear Abel-Type Integral Equation 7To avoid the numerial problems related with the innite growth of a(t) ast→ 1 while omputing the iterations in the Newton method solving equation(1.16), we introdued a new variable,Ai =1a2i. (1.21)Thus, by substituting (1.21) in (1.16) and dividing it by Aβ/2i , we arrive atthe equationG(Ai, i) =h(ti − t0)βb−12[1− AiA0]β/2 +hβ2(1 + 2(βb −β2)−1)2[1− AiAi−1]β2+i−2∑j=1h(ti − tj)βb−1[1− AiAj]β2−bβ= 0, i = 2, 3, ..., nCalulating the derivative of G with respet to Ai givesG′(Ai, i) =βh(ti − t0)βb−14A0[1− AiA0] β2+1+βhβb(1 + 2(βb −β2)−1)4Ai−1[1− AiAi−1]β2+1+i−2∑j=1βh(ti − tj)βb−12Aj[1− AiAj]β2+1,and we an nd values of Ai by using the Newton methodAi,k+1 = Ai,k −G(Ai,k, i)G′(Ai,k, i)where i = 2, 3, ..., nand then obtain ai = 1/√Ai.This algorithm modiation an be assoiated with solving numeriallythe equation in the form of equation (1.11) instead of (1.7).1.4 Numerial ResultsThe desribed algorithms were implemented using MATLAB as a program-ming language. As was already disussed, we are interested in the parameterranges β > 0, 0 < b < 2. The numerial examples given in this paper are forb = 1.5 and several values of β.First, we onsider the ase β = b, for whih exat solution (1.10) is avail-able, and plot on the same graph, Fig. 1.3, the exat and numerial solutions8 L. Hakim, S.E. Mikhailov0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.900.511.522.533.544.5tln(a/a 0)  Exactn=50n=100n=200n=400Fig. 1.3. Creep rak length vs. normalized time for dierent n, β = b = 1.5.0.71 0.715 0.72 0.725 0.732.342.362.382.42.422.442.462.48tln(a/a 0)  Exactn=50n=100n=200n=400Fig. 1.4. Zoomed in version of the graphs from Fig. 1.3.for a verses t for dierent numbers of steps, n, on the interval 0 < t < 1. Thezoomed in part of the plot is given in Fig. 1.4. The graphs show the numerialsolution onvergene to the exat solution, as n inreases.To quantify the onvergene, let us onsider the relative errorǫ =maxi(| Ai,approx −Ai,exat |)maxi(| Ai,exat |). (1.22)Note thatmaxi(| Ai,exat |) = A0 = a−20 = 1 in our ase. For b = 1.5, the graphin Fig. 1.5 represents the error verses n, where both axes are in logarithmisale.One an expet that the onvergene at other values of the parameter β,where no analytial solution is available, will have the same harater as forβ = b and, partiularly, for the number of steps, n = 500, the error in A willnot exeed 1%.1 Nonlinear Abel-Type Integral Equation 9101 102 10310−310−210−1nErrorFig. 1.5. Error for A vs. n for b = β = 1.5Figs. 1.6, 1.7 present numerial results for the ases β = 2b, β = b/2, andβ = 1 with n = 500 on the graphs of ln[a(t)/a0] verses t, and the graphs ofA(t)/A0 verses t.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1012345678tln(a/a 0)  β=b=1.5β=2b=3β=b/2=0.75 β=1Fig. 1.6. Creep rak length vs. normalized time for b = 1.5Graphs representing A(t) are partiularly useful for analysing the breakingtime; this is when the urve rosses the time axes, sine the rak lengtha = 1/A2 tends to innity when A tends to zero. From the graph in Fig. 1.7,we an see that when β = 2b the normalized breaking time is t = 1. For othervalues of β, it is not ompletely lear from the graphs but we ome to thesame onlusions under the assumption made in Setion 1.2.10 L. Hakim, S.E. Mikhailov0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.20.40.60.81tA/A 0  β=2b=3β=b=1.5β=1β=b/2=0.75Fig. 1.7. A vs. normalized time for b = 1.51.5 ConlusionA non-linear Abel-type equation, whih kernel depends on the unknown fun-tion at several values of the variable, was obtained in the paper to model reeprak propagation in the innite viso-elasti plane.A modiation to the Nyström method was introdued to numeriallysolve the equation. The numerial results obtained for the speial ase β =b, and the analysis of the error with respet to the mesh size, demonstrateonvergene of the modied Nyström method and indiate that this shemean be applied to solve the integral equation with other values of the parameterβ and also more general nonlinear Abel-type equations.From the results on the reep rak growth presented in the paper, onean onlude that an inrease of the parameter β dereases the rak growthrate. The theoretial analysis of the integral equation shows that its solutionblows up at b ≥ 2 for any β > 0. On the other hand, for 0 < b < 2 and anyβ > 0, the normalized breaking time for an innite plane with a rak is t = 1.Under some fatigue models, the analysis of fatigue rak propagation in anelasti material an be similarly redued to equation (1.7), where the time, t,should be however replaed with the number of load yles, n, f. [MiNa03a℄.One an use this similarity to make orresponding onlusions about the fa-tigue rak propagation as well.Referenes[At97℄ Atkinson, K.E.: The Numerial Solution of Integral Equations of theSeond Kind, Cambridge University Press, New York-Melbourne (1997).[GoVe91℄ Goreno, R., Vessella, S.: Abel Integral Equations Analysis and Aplia-tions. Springer Verlag, Berlin-New York (1991).1 Nonlinear Abel-Type Integral Equation 11[MiNa03a℄ Mikhailov S.E., Namestnikova I.V.: Loal and Non-Loal Approahesto fatigue Crak Initiation and Propagation. In: Movhan A.B. (ed)IUTAM Symposium on Asymptotis, Singularities and Homogenisationin Problems of Mehanis. Kluwer, The Netherlands, 285294 (2003).[MiNa03b℄ Mikhailov, S.E., Namestnikova, I.V.: Loal and Non-Loal Approahesto Creep Crak Initiation and Propagation. In: Proeedings of the 9thInternational Conferene on the Mehanial Behaviour of Materials.Geneva, Switzerland (2003).[MiNa09℄ Mikhailov, S.E., Namestnikova, I.V.: About aumulation rules for life-time predition under variable loading. In: Proeedings of the 12th Inter-national Conferene on Frature, Ottawa, Canada. Natural ResouresCanada & National Researh Counil of Canada (2009).[MiNa11℄ Mikhailov, S. E., Namestnikova I. V.: History-sensitive aumulationrules for life-time predition under variable loading. Arhive of AppliedMehanis, DOI: 10.1007/s00419-011-0511-6, (2011).[PeMa71℄ Penny, R., Marriott, D.: Design for Creep. MGraw-Hill, London (1971).[Ra69℄ Rabotnov, Y.N.: Creep Problems in Strutural Members. North-HollandPubl., Amsterdam-London (1969).[Ra80℄ Rabotnov, Yu. N.: Elements of Hereditary Solid Mehanis, Mir Pub-lishers, Mosow (1980).[Sa61℄ Savin G.N.: Stress onentration around holes, Oxford, Pergamon Press(1961).

Nonlinear Abel type integral equation in modelling creep crack propagation

This is the author accepted manuscript. The final version is available from the publisher via the DOI in this record.A nonlinear Abel-type equation is obtained in this paper to model creep crack timedependent
propagation in the infinite viscoelastic plane. A finite time when the integral
equation solution becomes unbounded is obtained analytically as well as the
equation parameters when solution blows up for all times. A modification to the
Nyström method is introduced to numerically solve the equation and some computational
results are presented

Open Research Exeter

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Nonlinear Abel type integral equation in modelling creep crack propagation

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