Analyss of 3D Cracks n an Arbtrary Geometry wth Weld Resdual Stress Greg Thorwald, Ph.D. Ted L. Anderson, Ph.D. Structural Relablty Technology, Boulder, CO Abstract Materals contanng flaws lke nclusons and lack of weld fuson can cause cracks to form and grow; a crtcal sze crack can cause a catastrophc fracture falure, even at low stress. Fracture mechancs allows cracks to be evaluated as bengn or requrng repar. Modelng the actual crack locaton n a complcated geometry s necessary to obtan accurate crack stress ntensty values, crucal n a thorough crack evaluaton. When exstng stress ntensty solutons are not avalable, FEA of 3D cracks provdes a way to compute the stress ntensty. A method for quckly generatng 3D crack meshes wthn an arbtrary shape volume s needed to effcently compute the stress ntensty. Ths method uses a mesh of brck elements to defne the arbtrary shape volume around the crack n the structure. The 3D crack mesh s generated wthn the defnton mesh and nserted nto the larger model; the meshes are connected by bonded contact. For a crack n a weld, the resdual stresses can be ncluded by mappng all stress components from an uncracked model onto the crack mesh as an ntal stress. The weld resdual stress ncreases the stress ntensty. The stress ntensty s computed usng ANSYS results durng post-processng. Introducton Snce all engneerng materals contan flaws, such as nclusons, porosty, lack of weld fuson, and pttng, these defects can cause cracks to form and grow over tme n many types of structures. Crack evaluaton s mportant n petroleum, chemcal, power generaton, aerospace, mechancal, and cvl structures. A crtcal sze crack can cause a catastrophc fracture falure, even at low stresses below the yeld strength. Usng fracture mechancs methods, a crack can be evaluated usng the stress ntensty at the crack front to determne f t s bengn or requres repar, and to compute how quckly the crack wll grow. Computng the crack fracture condton and fatgue lfe allows for an effcent nspecton and repar schedule, reducng rsk and cost. Computng the crtcal crack sze also verfes that nspecton methods can fnd the crack whle t s stll smaller than the crtcal sze to cause fracture. Accurate crack stress ntensty values, K I, are crucal for a thorough crack evaluaton. Stress ntensty solutons are avalable from handbooks and the lterature for many basc geometres and crack locatons; however, modelng the actual crack locaton and orentaton n a complcated geometry s an mportant mprovement for obtanng accurate crack stress ntensty values. When an exstng stress ntensty soluton that matches the structure geometry and crack locaton s not readly avalable, fnte element analyss of 3D cracks provdes a way to compute the crack front stress ntensty. Some of the dffcult and tmeconsumng tasks to create a 3D crack mesh nclude generatng the collapsed brck elements along the crack front and the concentrc rngs of elements around the crack front for the spder-web mesh pattern, cracks followng curved surfaces n more complcated geometres, lstng the node sets along the crack front correctly for the J-ntegral calculaton, applyng crack plane symmetry constrants, applyng crack face loads, and extractng the J-ntegral and stress ntensty values from the results. When a varety of crack szes and locatons are examned, the effort to generate each new crack mesh must be repeated. More complcated geometres wth numerous possble crack locatons prohbt tables of stress ntensty values to be computed for all possble cases; nstead the stress ntensty needs to be computed for each gven crack locaton and sze. These tme consumng modelng dffcultes led to the development of FEA-Crack to generate the 3D crack meshes quckly and easly, and allows cracks at any locaton to be routnely analyzed. Havng an easy-to-use method for quckly generatng 3D crack mesh nput fles wthn an arbtrary shape volume s needed to effcently compute the crack front stress ntensty at any locaton wthn complcated structures. Ths method uses a grd mesh of brck elements extracted from the larger structure model to
defne an arbtrary shape volume wth sx surfaces around the crack locaton. The defnton mesh volume has sx surfaces to match the shape of the prelmnary 3D crack mesh. The 3D crack mesh ANSYS nput fle s generated by FEA-Crack wthn the defnton mesh volume and s then nserted back nto the larger model. The meshes can easly be connected by bonded contact n ANSYS [reference 1], whch permts a dfferent mesh pattern between the crack mesh and larger structure mesh. Welds have regons of tensle resdual stresses that ncreases the crack stress ntensty and may adversely affect the crtcal fracture condton. When the crack s n or near a weld, the weld resdual stresses can be ncluded n the crack analyss by mappng all the stress components from the uncracked model resdual stress analyss results onto the crack mesh as an ntal stress. ANSYS uses an ntal stress fle [reference 2] to nclude the resdual stresses n a crack analyss along wth other boundary condtons. Includng both the weld resdual stress and other loadng n the crack analyss gves a more thorough and accurate calculaton of the crack front stress ntensty. J-ntegral Post Processng After completng the ANSYS analyss, the crack front J-ntegral and stress ntensty values are computed as an extra post processng calculaton usng the dsplacement, stress, and stran results. The J-ntegral, an energy release rate, s a preferred method to compute the crack front stress ntensty snce the calculaton s defned as a contour ntegral, but can be converted by the dvergence theorem to a volume ntegraton around the crack front. Integraton of volume values s a straghtforward task usng fnte element data. Several concentrc rngs of elements around the crack front allow for comparson of several J-ntegral contours to check for acceptable result convergence. The J-ntegral s computed usng element gauss ntegraton pont results n the equaton [reference 3]: u q x u J = w det w q w (1) m j j j σj δ1 p σ2 j Volume p= 1 x1 x ξk crack x p 1 faces Where m s the number of Gauss ntegraton ponts n the element, typcally 8 n a reduced ntegraton brck element, and w p and w are Gauss ntegraton weghtng factors. The summaton of the Gauss pont p terms gves the element volume ntegraton. The crack face summaton term can be omtted when there are no tractons on the crack face. In an elastc analyss, the J-ntegral values along the crack front can be converted to stress ntensty, K I, values usng the equaton [reference 4]: K I = JE ( 1 ν ) 2 (2) Where E s the modulus of elastcty, and ν s the Posson rato. The stress ntensty values along the crack front are then avalable for use n a crack evaluaton. As a partcular example of a 3D crack n an arbtrary geometry, a set-n nozzle s used to demonstrate the method. Example As an example of nsertng a 3D crack mesh wthn a larger and more complcated structure, a pressure vessel wth a set-n nozzle and external renforcng pad s used. The nternal surface crack s located at the shell to nozzle weld, whch follows a saddle-shaped 3D surface; see Fgure 1.
Fgure 1. Uncracked set-n nozzle mesh Nozzle Data For ths example, the set-n nozzle dmensons have generc values, whch are gven as follows. The shell nsde radus s 20 n and the shell thckness s 1.5 n. The nozzle nsde radus s 10 n, and the nozzle thckness s 2.0 n. The flange at the top of the nozzle s 2.0 n thck, and has an outsde radus of 18 n from the nozzle centerlne. The nozzle length s 20 n above the shell, or a total of 40 n from the shell centerlne to the top of the flange. The shell to nozzle and nozzle to pad welds are 1.5 by 1.5 n. The renforcng pad thckness s 1.5 n, and the pad length s 6.0 n from the outsde of the nozzle along the outsde surface of the shell; the pad to shell fllet weld s also 1.5 n. The renforcng pad has a unform length as t wraps around the nozzle on top of the shell. The total crack length, 2c, s 8 n, and the crack depth, a, s 1 n. A large surface crack was used to ad vsualzaton of the results. In a typcal crack assessment a range of small to large crack szes would be examned. An nternal pressure of 1000 ps s appled to the nsde surfaces of the shell and nozzle. The equvalent axal pressure thrust s appled to the rght end of the shell as a unform tensle tracton of 6425.7 ps. The top of the nozzle flange s constraned n the vertcal drecton. Symmetry constrants are appled to the nodes n the x-y and y-z planes.
Creatng The Analyss Mesh When buldng the structure mesh, create a regon around the crack locaton for the defnton mesh that gves the desred shape of the crack mesh. The crack regon s left empty n the structure mesh and wll be flled by the crack mesh. For ths example the defnton mesh volume ncludes part of the bottom of the nozzle cylnder and all of the nternal nozzle to shell weld. Fgure 2 shows the defnton mesh beng removed from the nozzle mesh. The defnton mesh, shown n Fgure 3, has sx surfaces to match the crack mesh ntal shape. The defnton mesh brck element shape functons are used drectly for the crack mesh shape transformaton; more elements n the defnton mesh along the curved surfaces gve a more accurate transformed crack mesh shape. Fgure 2. Remove the defnton mesh from nozzle mesh Fgure 3. The defnton mesh requres a grd pattern of brck elements wth 6 surfaces
To generate the 3D crack mesh, the defnton mesh s mported nto FEA-Crack from an ANSYS fle, and the crack s located and orented wthn the defnton mesh. The defnton mesh corner node ID numbers are used as reference ponts to locate the crack and to select the boundary condtons on each of the sx mesh surfaces. Boundary condtons and contact surfaces are appled to selected surfaces, and the crack mesh ANSYS nput fle s created, all wthn a few mnutes. Snce the crack mesh s located on the nsde of the vessel, the bottom and left surfaces of the crack mesh have the vessel nternal pressure appled to them. The crack faces should also have the nternal pressure appled snce the crack opens to the nsde surface of the vessel. The top and rght surfaces of the crack mesh are selected for bonded contact; these are the mesh surfaces that connect to the larger vessel mesh. The other two crack mesh surfaces are located on the two symmetry planes. The front crack mesh face s n the x-y plane and a z-constrant s appled for symmetry; the back-left crack mesh face s n the y-z plane and an x-constrant s appled for symmetry. The 3D crack mesh s shown n Fgure 4 wth an offset from the bottom of the nozzle where t s nserted nto the nozzle mesh. Fgure 4. Insert the 3D crack mesh nto the nozzle mesh Wthout a 3D crack mesh generator and the defnton mesh method, the crack mesh modelng tasks would take many days of effort for a sngle crack mesh; ths crack mesh example was completed n just a few hours. For ths example, the half-symmetrc crack s located on the front symmetry plane for easer vsualzaton; the x-y symmetry plane passes through the center of the crack length leavng the back half of the crack n the fnal mesh. The surface crack could be located anywhere wthn the defnton mesh and have other orentatons, such as a short radal crack. For cracks at other locatons n the nozzle geometry, another defnton mesh can be extracted from the model and replaced wth another crack mesh; multple cracks could be present n the vessel analyss. Next, the crack and nozzle meshes are combned wthn a sngle ANSYS nput fle, and bonded contact s used to connect the two meshes. The element szes are dfferent along the connected surfaces, especally near the crack where the mesh refnement n the crack mesh s hgher than n the surroundng nozzle and shell, makng bonded contact a useful method to connect the two meshes. The crack mesh generator
automatcally provdes the contact surface data on the selected surfaces to ad n combnng the mesh nput fles. The matchng surfaces n the larger mesh must also have the contact surface data defned. In ANSYS the contact surface s defned by selectng the nodes on each surface. One surface s defned as the target surface and the other as the contact surface. Of the several avalable types of contact, bonded contact s selected usng the KEYOPT command so that the two meshes reman connected throughout the analyss. Complete the combnng of the two nput fles by ncludng the nternal pressure, equvalent axal pressure thrust, and symmetry constrants. Includng The Weld Resdual Stress To nclude the effect of weld resdual stress, an uncracked mesh s used to obtan the resdual stresses by applyng a thermal stran to only the weld materal, as a basc way of smulatng the weldng process. For a post-weld-heat-treated structure, the resdual stress s typcally assumed to be 20% of the yeld strength for fracture evaluatons. For ths example a temperature change of 43.6 o F was mposed on the weld elements (usng a coeffcent of thermal expanson of 6.5x10-6 n/n/ o F) around the nozzle; the temperature s unchanged n the rest of the vessel. The stress result components from the uncracked structure results are then mapped onto the crack mesh as ntal stress usng the nverse dstance weghted (IDW) 3D nterpolaton method (Shepard s method) [reference 5], gven by the followng equatons: F( x, y, z) = w f (3) n = 1 w = d p n p d j j= 1 (4) ( ) ( ) ( ) 2 2 2 = + + d x x y y z z (5) Where the value, F(x,y,z), s nterpolated from the gven n data values, f ; for the weld resdual stress the f values are the stress components from the uncracked analyss, and the F values are the nterpolated stress components appled to the element Gauss ntegraton ponts n the crack mesh. The w weghtng values are computed usng the dstance from the nterpolaton pont poston to the surroundng data ponts. The exponent value, p, s a postve number, typcally 2. A local subset of data ponts can be used to lmt the stress nterpolaton to a smaller volume around each nterpolaton pont and speed up the nterpolaton calculatons. Only elements near the shell to nozzle nternal weld wll have nonzero ntal stress. The nterpolated resdual stresses are ncluded n the ANSYS analyss by referencng the ntal stress fle from wthn the combned nput fle. Analyss Results & Dscusson After runnng the combned crack mesh fnte element analyss, the ANSYS results are used to compute the crack front J-ntegral and stress ntensty durng post-processng. Stress results for the whole vessel are shown n Fgure 5, and a close up of the nternal surface crack s shown n Fgure 6; the dsplacement scale s 500 tmes the actual amount to verfy that the crack s openng due to the appled loads. The hgher stress n the nearby welds, due to the resdual stress, above the crack can also be seen n Fgure 6. A closer zoom-n of the crack depth s shown n Fgure 7 (agan wth 500x dsplacement), whch also gves a better vew of the crack front spder-web focused mesh pattern.
Fgure 5. Stress results for nternal pressure plus weld resdual stress Fgure 6. Stress results, close up of the crack openng, 500x dsplacement scale
Fgure 7. Stress results, zoom-n at the crack depth, 500x dsplacement scale The crack front stress ntensty due to the nternal pressure wth and wthout the weld resdual stress s compared n the plot n Fgure 8 to show how the weld resdual stress ncreases the crack front stress ntensty. If the weld s not post weld heat treated, the resdual stresses would be much hgher, and the stress ntensty would ncrease even more. To descrbe the poston of the crack front nodes, the plot x-axs uses the crack ph angle. The crack front ph angle s zero at the crack tp and π /2 at the crack depth, and usng the ph angle allows dfferent sze cracks to be compared on the same plot. For ths example, the stress ntensty s greatest near the deepest pont of the crack (ph of π /2), but the poston of maxmum stress ntensty can change dependng on the crack sze, crack length-to-depth aspect rato, crack locaton, and appled loadng. The next step n a crack evaluaton s to use the crack front stress ntensty values to determne the crtcal crack sze or compute crack growth rates n a fatgue analyss. Concluson Usng a 3D crack mesh generator and the defnton mesh method to nsert cracks nto more complcated geometres, more accurate crack front stress ntensty values can be computed qucker and easer for use n crtcal crack sze evaluaton and fatgue crack growth analyss. Weld resdual stress s ncluded n the crack analyss by nterpolatng the stress components from an uncracked analyss onto the crack mesh. Usng these two methods makes more thorough and accurate crack evaluatons possble by supportng routne use of 3D fnte element crack analyss. An example of a set-n nozzle was used to demonstrate the 3D crack mesh nserted nto the nozzle geometry, and the effect of the weld resdual stress ncreasng the crack front stress ntensty.
Fgure 8. Comparson of crack front stress ntensty results References [1] ANSYS Release 9.0 Documentaton, HTML onlne format, Element Reference, Part I Element Lbrary, CONTA174 and TARGE170 element descrpton. [2] ANSYS, Commands Reference, I Commands, ISFILE. [3] T. L. Anderson, Fracture Mechancs, Fundamentals and Applcatons, 3rd ed., CRC Press, Taylor & Francs Group, 2005, p. 570. [4] Anderson, p. 110. [5] D. Shepard, A two-dmensonal nterpolaton functon for rregularly-spaced data, Proceedngs of the 23 rd Natonal Conference ACM, ACM 517-524.