Toward A Fracture Mechanics Based Design Approach For .
Toward a Fracture Mechanics–Based Design Approach forUnbonded Concrete Overlay PavementsMinmao Liao, Ph.D., S.M.ASCE1; and Roberto Ballarini, Ph.D., P.E., F.ASCE2Abstract: An illustrative fracture mechanics–based design paradigm is proposed for unbonded concrete overlays (UBCOs), an increasinglypopular pavement rehabilitation system, with the ultimate goal of establishing a more rational design procedure than those currently available.To illustrate the advantages of the fracture mechanics–based approach to design, specific attention is paid to one type of failure associated withpavement structures, reflection cracking. The design formulas derived from the results of a large number of crack propagation simulations ofboth the UBCO composite and a reference single-layer new pavement quantify the dependence of the required overlay thickness and loadcarrying capacity on all relevant material and geometric parameters. Preliminary comparisons of the results with field observations suggestthat the fracture mechanics paradigm offers a promising procedure for improved design of UBCOs. DOI: 10.1061/(ASCE)EM.19437889.0000412. 2012 American Society of Civil Engineers.CE Database subject headings: Concrete pavements; Pavement overlays; Design; Composite structures; Cracking; Finite elementmethod; Equivalence.Author keywords: Pavement overlays; Composite structures; Cracking; Finite-element method; Equivalence.IntroductionLarge increases in traffic and the end of the service life of a significant number of existing pavements in the United States have produced a growing demand for highway pavement rehabilitation.Among various rehabilitation techniques, unbonded concreteoverlays (UBCOs) are likely to become increasingly popular because numerous states have found that they perform well whenproperly designed. UBCOs are cost effective and durable, mitigatereflection cracking, require minimal preoverlay preparation, can beplaced quickly and efficiently, and are recyclable. UBCOs have beenused since the 1910s to restore ride quality, provide an appropriatesurface texture, restore or increase load-carrying capacity, and extend the life of existing pavements. The current design proceduresfor UBCOs are based on empirical equations or highly simplifiedmechanistic models. In fact, the design procedures used for UBCOsfor highway pavements have mirrored those developed for airfieldpavements that are subjected to qualitatively and quantitativelydifferent loading. The concern is that the current designs may requireoverly conservative overlay thicknesses and, thus, reduce costeffectiveness.An UBCO system, as shown in Fig. 1(a), consists of the existingdamaged portland cement concrete (PCC) pavement, a thin asphaltconcrete (AC) interlayer, and a new PCC overlay. It is assumed(however, it has not been demonstrated) that the 2.54–5.08-cm-thick1Graduate Student, Dept. of Civil Engineering, Univ. of Minnesota,Minneapolis, MN 55455. E-mail: liaox069@umn.edu2Professor, Dept. of Civil Engineering, Univ. of Minnesota, Minneapolis, MN 55455 (corresponding author). E-mail: broberto@umn.eduNote. This manuscript was submitted on October 27, 2011; approvedon February 10, 2012; published online on February 13, 2012. Discussionperiod open until February 1, 2013; separate discussions must be submittedfor individual papers. This paper is part of the Journal of EngineeringMechanics, Vol. 138, No. 9, September 1, 2012. ASCE, ISSN 0733-9399/2012/9-1195–1204/ 25.00.interlayer allows relative deformation between the overlay and theexisting pavement, and serves to prevent the reflection cracking ofthe overlay that is associated with bonded concrete overlays (BCOs).Consequently, UBCOs can be used for badly damaged existingpavements, thereby minimizing preoverlay repairs and reducingconstruction costs. An additional advantage of UBCOs in applications involving severely damaged foundations is that (unlikeBCOs) their joints are not required to line up with those of theexisting pavement. In fact, specific joint mismatching is usuallyrecommended to provide a sleeper slab arrangement, which improvesload transfer. Finally, UBCOs can be used under any traffic level andclimate scenario.Construction of UBCOs is similar to that of conventional PCCpavements and does not require specialized equipment. However,special attention should be given to geometric constraints such asoverhead vertical clearances. According to Minnesota’s experience(Engstrom 1993), compared with US 500,000 for reconstructedPCC pavements, UBCOs cost only around US 350,000 per twolanes per mile. Therefore, with the advances in paving materialsand paving technology, UBCOs are becoming more attractive as analternative for highway pavement rehabilitation.For a given set of material properties, the required overlaythickness is determined using several empirical and simplifiedmechanistic design procedures. For example, the Minnesota Department of Transportation (MnDOT 1993) relies on the empiricalmethods developed by the Departments of the Army and the AirForce (DA/DAF 1970) and the mechanistic models proposed byTayabji and Okamoto (1985). Detailed descriptions of the twoprocedures are provided in the appendix. Because pavement engineers have developed relatively robust design procedures for newsingle-layer pavements consisting of a PCC slab on a foundation, thestructural equivalency design paradigm has been proposed to designUBCOs. A similar approach is adopted here. This proceduredemands that the thickness of the UBCO be such that it attains thesame value of a certain metric as does a single-layer pavementdesigned for the same service conditions. Some of the currentlyavailable mechanistic designs, including the Tayabji and OkamotoJOURNAL OF ENGINEERING MECHANICS ASCE / SEPTEMBER 2012 / 1195
Fig. 1. (a) Schematic of an UBCO system; (b) stress equivalency(1985) procedure, use stress equivalency [Fig. 1(b)]; the maximumnominal stress in the UBCO and the single-layer pavement should beequal. However, such stress-based material failure theories are notcapable of quantifying the loads required to initiate and propagatethe reflection cracks that emanate from singular stress-producingcracklike features in the existing PCC pavement. This paper focusesprecisely on this type of potential pavement failure through a twodimensional cohesive zone (fracture mechanics) model (CZM) thatrelates the ultimate load capacity of the UBCO structure to thefundamental material properties and geometric dimensions, capturesthe well-known size effect in quasi-brittle materials, and eliminatesthe mesh size dependence that is present in existing stress-basedmechanistic models. The comparison of the results obtained from thedesign formulas derived from the large number of fracture simulations suggest that the fracture mechanics–based paradigm offerspromise for developing more robust design procedures and determining if it is possible to reduce overlay thickness requirementsthrough selection of material properties and structural geometry.tip of a cracklike feature representing a preexisting crack or joint inthe existing pavement is illustrated on the left-hand side in Fig. 2.The load-carrying capacities of both structures are determined fromthe results of the CZM. The UBCO and the single layer are deemedstructurally equivalent if they have equal load-carrying capacity.Consequently, the proposed thickness design procedure for anUBCO is as follows. For the prescribed service requirements, a newsingle-layer PCC pavement is designed according to the currentlyavailable robust procedures. The ultimate load capacity of the designis determined from a CZM failure simulation. The thickness and/ormaterial properties of the UBCO are, in turn, determined to render itstructurally equivalent to the single-layer configuration.The remaining part of this paper is divided into four sections.The first two present the results of the CZM simulations of thesingle-layer and UBCO pavements, respectively. The third sectionderives and discusses the implications of the design formulas produced from the structural equivalency concept. The final sectioncompares the implications with observations of UBCOs tested at theMnROAD test facility in Minnesota.Fracture Mechanics–Based Structural EquivalencyApproachSingle-Layer PCC PavementThe metric chosen here for structural equivalency is not a nominalstress within the pavement, it is the maximum load achieved duringthe failure simulation of the two structures shown in Fig. 2. A singlelayer pavement resting on a foundation that, under the action ofa monotonically increasing point force, experiences the initiationand propagation of a crack at the location of maximum nominalstress in the initially uncracked configuration is illustrated on theright-hand side in Fig. 2. A three-layer UBCO system that, undersimilar loading, fails as a result of a reflection crack that forms at theThe plane strain model of a single-layer PCC pavement of length,L, depth, h, and thickness, b, resting on a Winkler foundation(Westergaard 1947) with stiffness, k, is shown in Fig. 3. For theillustrative purposes of this paper, the pavement is assumed to carrya concentrated vertical force at midspan and to be fully bonded withthe foundation. Thus, potential separation between the pavementand the foundation is not simulated. In addition, pavement jointload transfer is not considered in this paper; the ends of the pavement are free to rotate. Failure is assumed to result from the initiation1196 / JOURNAL OF ENGINEERING MECHANICS ASCE / SEPTEMBER 2012
Fig. 2. Proposed structurally equivalent structuresFig. 3. CZM of a single-layer PCC pavementand subsequent propagation of a cohesive edge crack at the point ofmaximum tensile stress. The CZM assumes that the crack openingdisplacement at each point along the crack surfaces, d, is resisted bya conjugate traction, s. As shown by Petersson (1981), Wittmannet al. (1988), Hilsdorf and Brameshuber (1991), and Roesler et al.(2007), concrete is best described using a bilinear softening tractionseparation relationship. A bilinear relationship similar to the oneproposed by Wittmann et al. (1988) (shown in Fig. 4) is adopted inthis paper. The fracture energy, GF , is defined by the tensile strength,ft , and the critical crack opening displacement, dc . The coordinates ofthe kink point are 0:125dc , and 0:25ft . According to Ba zant (2002),the initial fracture energy (the area under the first descending slope ofthe softening curve) controls the maximum load of ordinary concretestructures. The remaining portion of the fracture energy determinesthe postpeak behavior. As discussed in detail in Ba zant and Planas(1998) and Ba zant and Novak (2001), the ft in the cohesive law isa fundamental property of the material, and should not be confusedwith the modulus of rupture (MOR) that the pavement communitydetermines using the ASTM standard for flexural strength. Ba zantand Planas (1998) provide a review of numerous experiments thatdrive home the point that the MOR is strongly dependent onspecimen size. Therefore, all the results presented in this papershould be understood in terms of an ft that at some point needs to bemeasured independently or be experimentally correlated with a sizedependent flexural strength. It is also important to note that in theillustrative examples presented in this paper a constant dc 5 0:2 mmis used. This is justified by the fact that only the ultimate capacity isof interest. Moreover, it is noted that the change in abscissa of thekink point, which quantifies the first portion of the fracture energythat controls capacity, is equal to only one-eighth of the change inabscissa of dc . Maintaining a constant dc renders the total fractureenergy GF a function only of ft .The cohesive zone is implemented using the concrete damagedplasticity material properties in ABAQUS [Dassault Systèmes SimuliaCorporation (SIMULIA) 2010], for which the s d relationship isinput in tabular form. The material model involves a scalar tensiondamage parameter indicating the extent of fracture varying from 0 to 1that is also input as a tabular function of d. The damage parameter is setequal to zero when the crack opening displacement is zero and 0.9 whend reaches dc . The aspect ratio of the elements within the cohesive zone ismade equal to 1.0 to mitigate mesh sensitivity. Details of the modelingapproach, including the choice of the value of the damage parametersand mesh density that guaranteed convergence of the peak loads towithin a few percent, can be recovered in Liao (2011). It is noted that thesymmetry conditions were not applied because the large number ofgeometry discretizations that were constructed as part of the parameter study will be used in the near future for nonsymmetric loadings.The fracture energy introduces a characteristic length, which isproportional to the length of the process zone in the vicinity of thecrack front, defined bylch ¼EGFft2ð1Þwhere E 5 Young’s modulus. In general, a quasi-brittle structureis expected to behave in a brittle manner if h/lch is large and inJOURNAL OF ENGINEERING MECHANICS ASCE / SEPTEMBER 2012 / 1197
a ductile manner if it is small. However, in subsequent simulationsthe magnitude of the fracture energy is increased through an increase in tensile strength, which concomitantly increases themagnitude of the slope in the first portion of the traction-separationlaw shown in Fig. 4.A 2.54-mm-thick cohesive zone was placed at the midspan ofthe slab through the whole depth to simulate crack initiationand propagation under displacement control. First-order quadrilateral plane strain elements, designated as CPE4 in ABAQUS, wereassigned to all elements. The elements outside the cohesive zonewere linear elastic. The algorithm used to solve the finite-elementequations relied on the Riks method because it is capable ofcapturing the snap-back instabilities associated with relatively highvalues of h/lch .Fig. 5 illustrates the CZM’s ability to capture the localized deformation in the region that initiates the crack for the illustrativeparameters, L 5 3.66 m, h 5 0.25 m, E 5 27.58 GPa, ft 5 3.10 MPa,GF 5 118.21 N/m, k 5 27.14 MPa/m, and Poisson’s ratio n 5 0:20.These plots show the 2,0003-amplified mesh deformation, uponwhich the horizontal stress component contours (in units of psi), sxx ,are superimposed. Four instants of the loading history are shown.Fig. 5(a) corresponds to the unloaded reference state. Fig. 5(b) showsthe point at which the tensile stress at the trail end of the cohesivezone reaches the tensile strength, ft . At this point, where the reactionforce at the loaded nodes is approximately 74% of the maximumload, Pult , the cohesive zone starts to unzip. In Fig. 5(c) at 0:9Pult thecohesive elements are clearly stretched and in Fig. 5(d) the loadcapacity Pult is reached.The structural system of the single-layer pavement involves thephysical parameters Pult , L, b, h, E, ft , GF , and k. Because GF isa function of only ft , the other seven physical parameters are independent. They are expressible in terms of three independent fundamental physical quantities: mass, length, and time. Buckingham’s ptheorem (Buckingham 1914) states that if a system involves nphysical parameters that are expressible in terms of m independent fundamental physical quantities, then n 2 m dimensionlessparameters are required to fully describe the system. Therefore,four dimensionless parameters are required for the single-layerpavement. The following normalization choices reduce subsequentparameter studies. All dimensions are normalized with respect toL, lch , and the radius of relative stiffnessrffiffiffiffi4 Dl ¼kFig. 4. Bilinear traction-separation relationship of concretewhereFig. 5. Deformation and tensile stress distribution along the cohesive zone1198 / JOURNAL OF ENGINEERING MECHANICS ASCE / SEPTEMBER 2012ð2Þ
D ¼Eh312ð1 2 n2 Þð3ÞIn the absence of the Winkler foundation, three dimensionlessparameters are required: h/L, h/lch , and the normalized load capacity.The normalized load capacity is defined as the ratio of the MOR to ftand is written in terms of the maximum bending moment at themidspan, Mult , as6Mult3Pult L¼2bh2 ftbh2 ftBecause the influence of h/L is insignificant for relatively slenderbeams (Ioannides and Sengupta 2003), the normalized load capacity is plotted as a function of h/lch in Fig. 6. This plot illustratesthe well-known transition from ductile to brittle structural responseof quasi-brittle materials. The curve can be approximated by theequation 4 3 2 3Pult Lhhhh¼3:7629:33þ8:7423:842bh2 ftlchlchlchlchþ 2:04(4)In the presence of the Winkler foundation, the maximum bendingmoment at the midspan (Boresi and Schmidt 2003) is written �ffiffiffi44ð1 2 n2 ÞPult lFð5ÞMult ffiffiffiffiffiffiffiffiffi2 cos pcosh p442l4ð1 2 n Þ4ð1 2 n2 Þ"#"F �ffiffiffiffiffiffisinh pþ sin p444ð1 2 n2 Þ l4ð1 2 n2 Þ"#Ll#Llð6Þaccounts for the finite length of the slab. Consequently, the naturalchoice for the normalized load capacity �ffiffiffi6Mult3 4 4ð1 2 n2 ÞPult lF¼2bh2 ftbh2 ftwhich is plotted in Fig. 6 for practical values of h/l 5 0.21, 0.27, and0.32, respectively. In addition, Fig. 6 shows the sensitivity of thenormalized load capacity on h/L for constant values of h/l. Theresults suggest that for the practical range of the relevant parameters,the various curves of this choice of normalized capacity can becollapsed into the following ��ffiffiffiffiffi 4 3 23 4 4ð1 2 n2 ÞPult lFhhh¼ 5:412 13:44þ 12:562bh2 ftlchlchlch h2 5:40þ 2:35(7)lchUBCO PavementThe configuration and finite-element model of the UBCO is shownin Fig. 7. The PCC overlay, AC interlayer, and existing PCCpavement have thicknesses ho , hi , and he , and material properties Eo ,Ei , Ee , fto , fti , GFo , and GFi , respectively. The ends of each layer arefree to rotate and, thus, are representative of pavements with zerojoint load transfer efficiency. Between each layer, the interfaces arefully bonded to represent the cohesion of the AC interlayer. Theseparation between the foundation and the existing pavement is notconsidered. It is important to note that additional types of loadings onthe single-layer PCC pavement and on the UBCO system may havesignificant effects and need to be considered in future analyses.These include geometric imperfections, such as vertical offsetsbetween both sides of the existing pavement in the UBCO system,and the magnitude and distribution across the thickness of the residual stresses produced by shrinkage and changes in temperature.Fig. 6. Normalized load-carrying capacity versus h lchJOURNAL OF ENGINEERING MECHANICS ASCE / SEPTEMBER 2012 / 1199
Fig. 7. CZM of an UBCO pavementThe cohesive laws of the overlay and the interlayer have equalshapes as the one used for the single-layer pavement. However,a larger critical crack opening displacement dc 5 0:5 mm is used forAC. It is noted once again that the simulations and approach presented subsequently are illustrative and quantitatively correct onlyfor the assumed values of critical crack opening displacements of theasphalt and concrete layers.Reflection cracking within the overlay is assumed to initiate fromone preexisting 2.54-mm-thick crack at the midspan of the existingPCC pavement. Note that in the simulations cracking could initiateand propagate through the interlayer before cracking initiated in theoverlay, or cracking could initiate in the overlay before cracking inthe interlayer propagated through the interlayer, depending on therelative values of fracture toughness.The UBCO composite involves many more physical parametersthan the single-layer PCC pavement; i.e., Pult , b, L, k, ho , hi , he ,Eo , Ei , Ee , fto , fti , GFo , and GFi . Because GFo and GFi are functions offto and fti , respectively, 12 physical parameters are independent.According to the Buckingham’s p theorem, nine dimensionlessparameters are necessary to fully describe the system. The normalizedload capacity is Pult /bho fto , and it is a function of the following eightdimensionless quantities: ho /lcho , hi /lchi , Eo /Ei , Eo /Ee , ho /hi , ho /he ,ho /L, and kho /Eo , where lcho and lchi are the characteristic lengths ofthe PCC overlay and the AC interlayer, respectively.The results of the large number of fracture simulations demonstrated that for fixed values of overlay thickness, ho , a combination of power-law and polynomial relationships existed betweenthe remaining dimensionless variables and the load-carrying capacity. The results presented subsequently, for ho 5 10.16, 12.70,15.24, 17.78, and 20.32 cm, respectively, were calculated byconsecutively sweeping through a practical range of one independent variable while keeping the rest constant. The ranges ofthe parameters were the following: L 5 3.66–6.10 m, k 5 27.14–81.43 MPa/m, hi 5 1.27–5.08 cm, he 5 15.24–30.48 cm, Eo 527.58 GPa, Ei 5 3.45–10.34 GPa, Ee 5 34.47–62.05 GPa, fto 52.76–4.14 MPa, fti 5 2.76–4.14 MPa, GFo 5 105.08–157.61 N/m,and GFi 5 262.69–394.04 N/m. Using a least-squares analysisgave the following the relationships.For ho 5 10.16 cm: 20:60 0:49 20:30 0:05 0:13PulthohiEoEoho¼ 104bho ftolch olch iEiEehi 2 0:01 0:54 "hokhoho8:4667 heEoL# ho2 0:4860þ 0:0101L(8a)For ho 5 12.70 cm: Pultho 20:62 hi 0:50 Eo 20:32 Eo 0:06 ho 0:13¼ 104bho ftolch olch iEiEehi 2 0:04 0:48 "hokhoho2:8756 heEoL# hoþ 0:0050 (8b)2 0:1911LFor ho 5 15.24 cm: Pultho 20:60 hi 0:51 Eo 20:33 Eo 0:07 ho 0:12¼ 104bho ftolch olch iEiEehi 0:05 0:41 " 2hokhoho 1:0627heEoL# hoþ 0:0025 (8c)2 0:0785L1200 / JOURNAL OF ENGINEERING MECHANICS ASCE / SEPTEMBER 2012
For ho 5 17.78 cm:Structural Equivalency Design of UBCOs 20:59 0:48 20:33 0:07 0:11PulthiEoEoho3 ho¼ 10bho ftolch olch iEiEehi 0:07 0:34 " 2hokhoho 3:8264heEoL# ho2 0:3043þ 0:0118 (8d)LFor ho 5 20.32 cm: Pultho 20:53 hi 0:51 Eo 20:32 Eo 0:07 ho 0:11¼ 103bho ftolch olch iEiEehi 2 0:07 0:30 "hokhoho2:1634 heEoL# hoþ 0:0082 (8e)2 0:1798LThese equations are plotted in Fig. 8 together with the results ofthe simulations and with the results of simulations performed forrandom choices of the parameters within the stated ranges. Theagreement between the regressions and the simulations is deemedacceptable for all intents and purposes. It is noted the exponent ofeach dimensionless variable is not a strong function of the overlaythickness. However, the polynomial functions of the pavement aspectratio differ significantly from thickness to thickness.The ultimate capacity equations can be further simplified forthe practical range considered in this paper by noting that theexponents associated with Eo /Ee and ho /he are nearly zero. Theweak dependence of the ultimate load capacity on the stiffness andthickness of the existing pavement is in contrast with the currentlyavailable design formulas, which state that thicker existing pavements require thinner UBCOs. It is noted that fracture mechanics–based design equations that may eventually be developed should beconservative. Therefore, equations similar to those listed previously will be used that bound all of the simulations listedsubsequently.Eq. (8) also sheds insights on how to improve the ultimate capacityof UBCOs that fail as a result of reflection cracking. It suggests that theload capacity is most sensitive to ho /lcho and hi /lch i . Therefore, the mosteffective ways of increasing capacity are to increase the toughness(actually the tensile strength because the critical crack opening displacement is assumed to be a constant value) of the overlay and/or theinterlayer. Increasing the foundation stiffness k, and/or increasingthe thickness of the interlayer hi , and/or decreasing the stiffness ofthe interlayer Ei also result in increased ultimate capacity.The influence of the overlay length L is not monotonic because theseparation between the UBCO pavement and the foundation is notconsidered. When ho /L is relatively large, there is no separation andthe foundation is in full compression. The load capacity decreases as Lincreases. However, when L increases beyond a certain length, thefoundation at the ends of the pavement is required to carry relativelysmall tension forces to maintain contact, and these result in an increasein the ultimate capacity. If there is sufficient adhesion between thepavement and the foundation, then this increase can be achieved. Ifnot, then a more complex moving boundary problem simulation isrequired to determine the effects of the layer aspect ratio.The following procedure is proposed as an illustration of howfracture mechanics can be used as the basis of UBCO designprocedures.1. For a chosen set of material and foundation properties, determine the thickness of a new single-layer pavement that isrequired to meet the service requirements. The requirementsmay include various thermomechanical loadings and crackingscenarios.2. Using Eq. (7), determine the load capacity of the single-layerpavement.3. Using Eq. (8), select a combination of material properties andgeometric configurations, then determine the load capacities ofthe UBCO systems for 10.16- to 20.32-cm-thick overlays.4. Select the required UBCO thickness to render the UBCOstructurally equivalent to the single-layer pavement.This procedure is illustrated for L 5 6.10 m, k 5 27.14 MPa/m,hi 5 2.54 cm, he 5 20.32 cm, Eo 5 27.58 GPa, Ee 5 34.47 GPa, fto 52.76 MPa, and GFo 5 105.08 N/m. Assuming the requiredthickness of a new single-layer pavement is 20.32 cm and itsmaterial properties are the same as those of the overlay, Eq. (7)predicts its capacity as 2.10 kN (the ABAQUS simulation of this casepredicts a capacity equal to 2.09 kN). Assuming AC interlayerproperties Ei 5 3.45 GPa and fti 5 4.14 MPa, Eq. (8) predicts theload capacities for 10.16-, 12.70-, 15.24-, 17.78-, and 20.32-cmthick overlays are 1.71, 1.85, 2.04, 2.24, and 2.53 kN, respectively.Therefore, the 20.32-cm-thick single-layer pavement is equivalentto an UBCO with an overlay thickness approximately equal to 16.51cm (by interpolation). If the properties are changed to Ei 5 3.45 GPaand fti 5 3.45 MPa, then the required overlay thickness becomes18.29 cm; thus, illustrating that the interaction between the overlayand the interlayer is significant and cannot be neglected as is doneby the currently used MnDOT (1993) procedures. The two currentlyavailable procedures described in the appendix require, respectively, a constant 16.51- or 17.27-cm-thick overlay for poor existingpavement conditions regardless of the properties of the interlayer.The previous examples drive home the point that the usefulnessof the fracture mechanics modeling lies in its ability to enable thedesigner to explore the effects of all of the material and geometricparameters on the required thickness of the overlay. If the values ofthe material and geometric parameters fall outside the ranges investigated in this paper, the same process carried out here can berepeated to develop similar design equations. Eventually, fracturemechanics-based design guidelines could be developed that accountfor all possible loadings, temperature conditions, etc. With properchoices of material properties and geometric parameters, thinneroverlay thicknesses may be achieved.Comparison with Field StudyIn July 2010, the writers observed the condition of the four UBCOtest sections built in 2008 at the MnROAD test facility locatedparallel to westbound Interstate Highway I-94 near Albertville,Minnesota. The dimensions of the cells, numbered 105, 205, 305,and 405, are shown in the schematic in Fig. 9; the thickness of theAC interlayer of all sections was 2.54 cm. Also shown in Fig. 9 is animage representative of the cracking that was observed in all sections25–50 cm from the cell-separating joints. These cracks were notreflective. Instead they were attributed to temperature effects such ascurling or warping.Most importantly, no other cracks were observed within theoverlays. This suggests that the thicknesses of the overlays wereJOURNAL OF ENGINEERING MECHANICS ASCE / SEPTEMBER 2012 / 1201
Fig. 8. Derived relationship between the capacity and the variables: (a) ho 5 10.16 cm; (b) ho 5 12.70 cm; (c) ho 5 15.24 cm; (d) ho 5 17.78 cm;(e) ho 5 20.32 cm1202 / JOURNAL OF ENGINEERING MECHANICS ASCE / SEPTEMBER 2012
Fig. 9. MnROAD test sections of UBCOssufficient to resist the potential reflection cracking considered inthis paper. The developed models cannot be directly applied tointerpret the MnROAD section because the material parameters ofthe section were not known to the writers. Furthermore, the temperature, moisture, and traffic effects were not considered in themodel. Further research is suggested to consider these effects inthe future. Nevertheless, assuming the practical values of theparameters—Eo5 27.58 GPa, Ei 5 3.45–10.34 GPa, Ee 5 34.47–62.05 GPa, fto 5 2.76–4.14 MPa, fti 5 2.76–4.14 MPa, GFo 5105.08–157.61 N/m, GFi 5 262.69–394.04 N/m, and k 5 27.14MPa/m—Eqs. (7) and (8) state that the originally designed 17.78-cmthick single-layer pavement is structurally equivalent to a minimum overlay thickness of 12.19 cm. Therefore, Overlays 105 and205 tested at MnROAD (even though they were 2 cm thinner thanthe lower bound suggested by the fracture mechanics–based designapproach) appear to be of sufficient thickness to mitigate reflectioncracking. These observations reinforce the notion that the simplified assumptions, such as plane strain deformation and zero jointload transfer efficiency, may lead to conservative values of required overlay thickness. Moreover, these observations suggestthat experiments be performed to determine whether currently useddesign formulas are conservative [a minimum overlay thickness of15.24 cm is required by MnDOT (1993)], and whether thinneroverlays may prove to be sufficiently robust
used since the 1910s to restore ride quality, provide an appropriate surface texture, restore or increase load-carrying capacity, and ex- . 2012; published online on February 13, 2012. Discussion . the kink point are 0:125d c, and 0:25f t. According to Bazant (2002), .
A.2 ASTM fracture toughness values 76 A.3 HDPE fracture toughness results by razor cut depth 77 A.4 PC fracture toughness results by razor cut depth 78 A.5 Fracture toughness values, with 4-point bend fixture and toughness tool. . 79 A.6 Fracture toughness values by fracture surface, .020" RC 80 A.7 Fracture toughness values by fracture surface .
Fracture Liaison/ investigation, treatment and follow-up- prevents further fracture Glasgow FLS 2000-2010 Patients with fragility fracture assessed 50,000 Hip fracture rates -7.3% England hip fracture rates 17% Effective Secondary Prevention of Fragility Fractures: Clinical Standards for Fracture Liaison Services: National Osteoporosis .
This article shows how the fracture energy of concrete, as well as other fracture parameters such as the effective length of the fracture process zone, critical crack-tip opening displacement and the fracture toughness, can be approximately predicted from the standard . Asymptotic analysis further showed that the fracture model based on the .
Fracture is defined as the separation of a material into pieces due to an applied stress. Based on the ability of materials to undergo plastic deformation before the fracture, two types of fracture can be observed: ductile and brittle fracture.1,2 In ductile fracture, materials have extensive plastic
hand, extra-articular fracture along metaphyseal region, fracture can be immobilized in plaster of Paris cast after closed reduction [6, 7]. Pin and plaster technique wherein, the K-wire provides additional stability after closed reduction of fracture while treating this fracture involving distal radius fracture.
6.4 Fracture of zinc 166 6.5 River lines on calcite 171 6.6 Interpretation of interference patterns on fracture surfaces 175 6.6.1 Interference at blisters and wedges 176 6.6.2 Interference at fracture surfaces of polymers that have crazed 178 6.6.3 Transient fracture surface features 180 6.7 Block fracture of gallium arsenide 180
Fracture control/Fracture Propagation in Pipelines . Fracture control is an integral part of the design of a pipeline, and is required to minimise both the likelihood of failures occurring (fracture initiation control) and to prevent or arrest long running brittle or ductile fractures (fracture propagation control).
the Brittle Fracture Problem Fracture is the separation of a solid body into two or more pieces under the action of stress. Fracture can be classified into two broad categories: ductile fracture and brittle fracture. As shown in the Fig. 2 comparison, ductile fractures are characterized by extensive plastic deformation prior to and during crack
