Fracture problems with ANSYS A (very) brief introduction
Contents
1. EE War Za 60 a arava Sk aera ONA K Ave AYAV ca A Figure 4 Boundary conditions 3 2 Solution The solution should take just a few seconds 4 Post processing General Postproc gt 4 1 Deformed shape and stress concentration Contours of the o stress distribution should clearly indicate the presence of a stress concentration at the crack tip Figures 5 and 6 ANSYS 5 5 2 MAR 12 2001 10 30 04 NODAL SOLUTION STEP 1 SUB 1 TIME 1 SY AVG RSYS 0 PowerGraphics EFACET 1 SMK 1 147 096483 EE oaie76 E E 456154 E soa3ta E 732473 Ea m 870632 1 009 a 1 147 Figure 5 Deformed shape and stress distribution ANSYS 5 5 2 MAR 12 2001 10 31 04 NODAL SOLUTION STEP 1 SUB 1 TIME 1 SY AVG RSYS 0 PowerGraphics EFACET 1 AVRES Mat DIX 004614 SMN 096483 SMH 1 147 096483 041676 179836 317995 456154 594314 732473 870632 1 009 1 147 Figure 6 Stress distribution in the crack tip region D004 e l 3 4 2 Extraction of the SIF As explained in the appendix we must first define a path in the vicinity of the crack tip But before doing so we must define a new coordinate system pointing ahead of the crack using the following command Work Plane Local Coordinate System Create Local CS By 3 Nodes pick the origin first crack tip node then pick a node along the new x axis i e a point along the plane of sy2. axis Oxnx OxyDy ty traction vector along y axis OyDy OxyDx O component stress n unit outer normal vector to path I u displacement vector s distance along the path I Figure 3 9 18 J integral contour path surrounding a crack tip The steps required to calculate J for a 2 D model are described below SET 1 Read in the desired set of results SET store the volume and strain energy per laa element ETABLE and calculate the strain energy density per element SEXP LPATH 2 Define a path for the line integral LPATH Figure 3 9 9 shows examples of such paths Figure 3 9 19 Examples of paths for J integral calculation Volume I Procedures 3 161 Chapter 3 Structural Analyses PDEF 3 Map the strain energy density which was stored in the element table in step 1 onto PCALC the path PDEF integrate it with respect to global Y PCALC and assign the er final value of the integral to a parameter GET Name PATH LAST This gives us the first term of equation 3 9 1 4 Map the component stresses SX SY and SXY onto the path PDEF define the path unit normal vector PVECT and calculate TX and TY PCALC using the expressions shown on with equation 3 9 1 5 Shift the path a small distance in the positive and negative X directions to calculate the derivatives of the displacement vector ux x and duy dy The following steps are involved see Figure 3 9 10 e Calculate the distance by which the path
3. circular crack front e All element edges should be straight including the edge on the crack front Calculating Fracture Parameters Once the static analysis is completed you can use POST1 the general postprocessor to calculate fracture parameters As mentioned earlier typical fracture parameters of interest are stress intensity factors the J integral and the energy release rate Stress Intensity Factors The POST1 KCALC command calculates the mixed mode stress intensity factors Ky Ky and Km This command is limited to linear elastic problems with a homogeneous isotropic material near the crack region To use KCALC properly take the following steps in POST1 1 Define a local crack tip or crack front coordinate system using the LOCAL command or CLOCAL CS CSKP etc with X parallel to the crack face perpendicular to the crack front in 3 D models and Y perpendicular to the crack face as shown in the following figure This coordinate system must be the active model coordinate system CSYS and results coordinate system RSYS when KCALC is issued yV y V r crack front a b Figure 3 9 16 Crack coordinate systems for a 2 D models and b 3 D models Volume 1 Procedures 3 159 Chapter 3 Structural Analyses LPATH KCALC 3 160 2 Define a path along the crack face using the LPATH command The first node on the path should be the crack tip node For a half crack model two additional nodes are required
4. definition to automatically change the specimen size w and the crack length a the KSCON command which allows to generate focused mesh at the crack tip the skewed element option used to generate singular elements at the crack tip the KCALC command used to extract the value of the stress intensity factors The section of the ANSYS Procedures Manual relative to the simulation of fracture problems is included in the appendix of this document Please spend a few minutes reviewing the information relative to 2 D fracture analyses on pages 3 156 to 3 160 The following lines describe the series of commands needed to solve the problem with ANSYS The pick commands are denoted by regular bold words while values to be entered using the keyboard are given in italic bold Problem description The structural problem to be solved is described in Figure A4 1 of the Appendix Taking advantage of symmetry and simplifying the geometry a little bit we will actually solve the following problem w 100 mm 0 25 w 0 6 w symmetry line crack tip crack face To prevent rigid body translation we will fix the x displacement of the point of application of the load P To facilitate the creation of the focused mesh at the crack tip we will place the origin of the axis system at the crack tip The material properties will be chosen as those of PMMA E 4000 N mm and v 0 3 although for this problem the material properties do not enter th
5. is to be shifted say DX A tule of thumb is to use one percent of the total length of the path You can obtain the total path length as a parameter using GET Name PATH LAST S e Shift the path a distance of DX 2 in the negative X direction PCALC ADD XG XG DX 2 and map the displacements UX and UY onto the path PDEF giving them labels UX1 and UY1 for example e Shift the path a distance of DX in the positive X direction i e DX 2 from its original position and map UX and UY onto the path giving them labels UX2 and UY2 for example e Shift the path back to its original location a distance of DX 2 and calculate the quantities UX2 UX1 DX and UY2 UY1 DX using PCALC These quantities represent du dx and du dy respectively T Ax 2 dx Ax GET DX PATH LAST S T Ax 2 DX DX 100 PCALC ADD XG XG DX 2 PDEF UX1 U X PDEF UY1 U Y PCALC ADD XG XG DX PDEF UX2 U X uw u PDEF UY2 U Y i PCALC ADD XG XG DX 2 C 1 DX r Ax 2 PCALC ADD C1 UX2 UX1 C C T Ax 2 PCALC ADD C2 UY2 UY1 C C Figure 3 9 20 Calculating derivatives of the displacement vector 6 Using the quantities calculated in steps 4 and 5 calculate the integrand in the second term of J PCALC and integrate it with respect to the path distance S PCALC This gives the second term of 3 9 1 3 162 ANSYS User s Manual 3 9 Fracture Mechanics 7 Calculate J according to equation 3 9 1 using the
6. C SQRT UZ UZC SQRT LIMITS AS RADIUS R l x k KI 0 83296 KII 0 00000E 00 KIII 0 00000E 00 The value of the SIF is listed at the end of the file Compare your solution with the table provided in the ASTM standard Finally create a database log file containing the list of the commands you have used To run a different crack length case open the log file with your favorite editor and just change the definition of the parameter a Restart ANSYS and read the input from the log file The whole problem will be run automatically including the definition of the three node path used to compute the SIF Try it with a 0 55 w and compare your solution to the tabulated one h E 399 maximum load that the specimen was able to Pmax sustain B thickness of specimen as determined in 8 2 1 wW width depth of specimen as determined in A3 4 1 f a crack length as determined in 8 2 2 and Sys yield strength in tension offset 0 2 see Test Methods E 8 A3 5 5 Calculation of Crack Mouth Opening Compliance Using Crack Length Measurements For bend specimens calculate the crack mouth opening compliance V P in units of m N in Ib as follows see Note A3 2 V P SIE BW q a W A3 4 where qlaiW 6 a W 0 76 2 28 a W 3 87 a W 2 04 a W 0 66 1 a W and where V crack mouth opening displacement m in P applied load kN kibf E Effective Y
7. Fracture problems with ANSYS A very brief introduction Introduction and objective ANSYS is a commercial finite element code used in industry to solve large scale problems It can be used to solve a wide variety of problems including linear and nonlinear structural response buckling modal analysis full harmonic response transient dynamic response heat transfer electro magnetic and fluid flow problems ANSYS offers a large library of elements ranging from the simplest 1 D elastic bar element to the very complicated 3 D nonlinear elasto plastic element Information about these elements and the type of analysis available in ANSYS can be found in various sources and manuals Help is also available interactively within ANSYS The objective of this short exercise is to extract the value of the stress intensity factor for a compact tension test CTT linearly elastic specimen described in Section A4 of the E 3999 ASTM Standard see first two pages of the Appendix of this document Basically we want to perform a 2 D structural analysis with ANSYS to extract in an automatic fashion the value of the stress intensity factor for various crack lengths and reproduce the results indicated in Equation A4 1 and Table A4 5 3 1 This will allow us to assess the precision of the finite element code for fracture problems We will use the following utilities available in ANSYS log files used to run similar simulations automatically scalar parameters
8. be obtained fc testing the specimen sizes and oy E ratios given in 7 1 3 lower strength grip material is used or if substantially large specimens are required at a given o ys E ratio than those show in 7 1 3 then heavier grips will be required As indicated i Fig A4 2 the clevis corners may be cut off sufficiently t accommodate seating of the clip gage in specimens less tha 0 375 in 9 5 mm thick A4 3 1 2 Careful attention should be given to achieving lt good alignment as possible through careful machining of a auxiliary gripping fixtures A4 3 2 Displacement Gage For generally applicable de tails concerning the displacement gage see 6 3 For th compact specimen the displacements will be essentially ind pendent of the gage length up to 1 2 W A4 4 Procedure A4 4 1 Measurement For a compact specimen measur the width W and the crack length a from the plane of tt centerline of the loading holes the notched edge is a conv nient reference line but the distance from the centerline of th holes to the notched edge must be subtracted to determine and a Measure the width W to the nearest 0 001 in 0 02 i E 399 25W t 005W DIA 2 HOLES _A _A a an gt 2 6wt 0osw k gt 6W 005 W ke A 0 a 9 X lt W 1 25Wt 010OW B 5 t 010W Nore 1 A surfaces shall be perpendicular and parallel as applicable to within 0 002 W TIR Nore 2 The intersection of the crack starter notch tips wi
9. both along the crack face For a full crack model where both crack faces are included four additional nodes are required two along one crack face and two along the other The following figure illustrates the two cases for a 2 D model symmetry or anti symmetry plane a Figure 3 9 17 Typical path definitions for a a half crack model and b a full crack model Use the KCALC command to calculate Ky Kyr and Km The KPLAN field on the KCALC command specifies whether the model is plane strain or plane stress Except for the analysis of thin plates the asymptotic or near crack tip behavior of stress is usually thought to be that of plane strain The KCSYM field specifies whether the model is a half crack model with symmetry boundary conditions a half crack model with anti symmetry boundary conditions or a full crack model J integral In its simplest form the J integral can be defined as a path independent line integral that measures the strength of the singular stresses and strains near a crack tip Equation 3 9 1 shows an expression for J in its 2 D form is shown below It assumes that the crack lies in the global Cartesian XK Y plane with X parallel to the crack see Figure 3 9 8 ou duy J ves ogi yay ds 3 9 2 ANSYS User s Manual 3 9 Fracture Mechanics where T any path surrounding the crack tip W strain energy density i e strain energy per unit volume tx traction vector along x
10. de Shearing mode Tearing mode Kp Kn Km Figure 3 9 11 The three basic modes of fracture Volume Procedures 3 155 Chapter 3 Structural Analyses 3 9 4 Howto Solve Fracture Mechanics Problems Solving a fracture mechanics problem involves performing a linear elastic or elastic plastic static analysis and then using specialized postprocessing commands or macros to calculate desired fracture parameters In this section we will concentrate on two main aspects of this procedure e Modeling the Crack Region e Calculating Fracture Parameters See Section 3 2 for details about the general static analysis procedure See also Section 3 8 for a discussion of structural nonlinearities a Modeling the Crack Region The most important region in a fracture model is the region around the edge of the crack We will refer to the edge of the crack as a crack tip in a 2 D model and crack front in a 3 D model This is illustrated in Figure 3 9 2 Crack front Crack tip Figure 3 9 12 Crack tip and crack front In linear elastic problems it has been shown that the displacements near the crack tip or crack front vary as vr where r is the distance from the crack tip The stresses and strains are singular at the crack tip varying as 1 vr To pick up the singularity in the strain the elements around the crack tip or crack front should be quadratic with the midside nodes placed at the quarter points Such elements are called singular el
11. e expression of the stress intensity factor The load P will be chosen as unity We will perform the analysis in plane strain ANSYS analysis This session assumes some familiarity with ANSYS Only the steps specific to the fracture analysis will be described here The other steps are identical to those of a conventional plane strain structural analysis 1 Preliminary steps Specify new job name optional and new title optional Under the heading Parameters define the two scalar parameters w 100 a 0 45 w Select Structural under Preferences optional 2 Preprocessing ste Preprocessor gt 2 1 Element and material definition We will use 6 node triangular elements Plane2 Make sure to select the plane strain option No real constant definition is needed for this type of element Define a material set Constant Isotropic with the appropriate properties stiffness and Poisson s ratio 2 2 Geometry definition The easiest way to define the geometry is to define asemi circle of radius a 5 centered at the origin a large rectangle xmin a w ymin 0 xmax a w 4 ymax 0 6 w asmall rectangle xmin a xmax a w 4 ymin 0 275 w ymax 0 6 w Then use the Operate Overlap Areas action to subtract the two smaller surfaces from the large rectangle Figure 1 Compact tension test specimen Figure 1 Definition of areas lines and keypoints 2 3 Mesh generation First create the mesh in the semi circle F
12. e used they must t reversed or inset to provide the same measurement point location A4 SPECIAL REQUIREMENTS FOR THE TESTING OF COMPACT SPECIMENS A4 1 Specimen A4 1 1 The standard compact specimen is a single edge notched and fatigue cracked plate loaded in tension The general proportions of this specimen configuration are shown in Fig A4 1 A4 1 2 Alternative specimens may have 2 W B 4 but with no change in other proportions A4 2 Specimen Preparation A4 2 1 For generally applicable specifications concerning specimen size and preparation see Section 7 A4 3 Apparatus A4 3 1 Tension Testing Clevis aA loading clevis suitable for testing compact specimens is shown in Fig A4 2 Both ends of the specimen are held in such a clevis and loaded through pins in order to allow rotation of the specimen during testing In order to provide rolling contact between the loading pins and the clevis holes these holes are provided with small flats on the loading surfaces 4 Other clevis designs may be used if it can be demonstrated that they will accomplish the same result as the design showr A4 3 1 1 The critical tolerances and suggested proportions of the clevis and pins are given in Fig A4 2 These proportions are based on specimens having W B 2 for B gt 0 5 in 12 7 436 mm and W B 4 for B 0 5 in 12 7 mm If 280 000 psi 1930 MPa yield strength maraging steel is use for the clevis and pins adequate strength will
13. ements Figure 3 9 3 shows examples of singular elements for 2 D and 3 D models 3 156 ANSYS User s Manual 3 9 Fracture Mechanics L N J PLANE2 M K LPL M o J PLANE82 N a Figure 3 9 13 Examples of singular elements for a 2 D models and b 3 D models 2 D Fracture Models KSCON The recommended element type for a two dimensional fracture model is PLANE2 the six node triangular solid The first row of elements around the crack tip should be singular as illustrated in Figure 3 9 3 a The PREP7 KSCON command which assigns element division sizes around a keypoint is particularly useful in a fracture model It automatically generates singular elements around the specified keypoint Other fields on the command allow you to control the radius of the first row of elements number of elements in the circumferential direction etc Figure 3 9 4 shows a fracture model generated with the help of KSCON Volume Procedures 3 157 Chapter 3 Structural Analyses VN L N AVA ER 22220 S ZAZE SAZ LARS ZS v aS A N Z WN DA e VAN DKI a SY 4 N i Figure 3 9 14 A fracture specimen and its 2 D F E model Other modeling guidelines for 2 D models are as follows e Take advantage of symmetry where possible In many cases you need to model only one half of the crack region with symmetry or anti symmetry boundary conditions as shown below Symmetry b
14. ginal fabrication had enlarged by crack propagation over the years until the aluminum skin simply tore apart This is an example of loss of structural integrity by fracture The engineering field of fracture mechanics was established to develop a basic understanding of such crack propagation problems 3 9 3 What is Fracture Mechanics Fracture mechanics deals with the study of how a crack or flaw in a structure propagates under applied loads It involves correlating analytical predictions of crack propagation and failure with experimental results The analytical predictions are made by calculating fracture parameters such as stress intensity factors in the crack region which you can use to estimate crack growth rate Typically the crack length increases with each application of some cyclic load such as cabin pressurization depressurization in an airplane Further environmental conditions such as temperature or extensive exposure to irradiation can affect the fracture propensity of a given material Some typical fracture parameters of interest are e stress intensity factors Kj Ky Km associated with the three basic modes of fracture see Figure 3 9 1 e J integral which may be defined as a path independent line integral that measures the strength of the singular stresses and strains near a crack tip energy release rate G which represents the amount of work associated with a crack opening or closure lt i rd Opening mo
15. irst create a concentrated mesh at the crack tip with Mesh Size Control Concentrated Keypoint Pick a 20 as the radius of the first circle Pick 1 5 for the radius ratio 2 row 1 row Use 5 or 6 for the number of elements around the circumference Use the Skewed 1 4 pt option for the midside node position Then use size control along the radial lines emanating from the crack tip using for example 8 elements with a spacing ratio of 2 5 and along the circumference of the semi circle say 12 equal size elements Then mesh the semi circle with triangular elements Figure 2 Figure 2 Mesh in crack tip region Then create a mesh in the remainder of the domain using the mesh tool and various levels of refinement A typical mesh should look like that presented in Figure 3 7 V7 TIX DIAS ETE VV RA A 3 Solution step Solution gt 3 1 Boundary conditions P T T T T SVAW A CANINE NY Pe A VAVAVAVAVAVAVAVAVAVAVAVAVAVAVAND 9 a AU AV ACA aaa A VATA VA WW FAA S Da ses ZRS MW PERE A VV AANA DORE VAT wi Figure 3 Full mesh Apply symmetry bc along the line ahead of the crack tip zero x displacement on the point of application of the load and the vertical load P at the keypoint corresponding to the lower left corner of the small rectangle Figure 4 Se Ag Se a 4 We T R A V 2 aS pa ree in SO AY IIX SZ Me AX IX TS
16. mmetry finally pick any node in the new x y plane 1 e any node off the plane of symmetry After defining the new CS let us define a path with Path Operators gt Define Path gt By Nodes Then pick successively three nodes behind the crack tip 1 e along the crack face with the first one at the crack tip the second close to the crack tip and the third one a little further away you can pick the first three nodes if you want Finally extract the stress intensity factor with Nodal Calcs gt Stress Intensity Factor which will create a separate window with the following information x CALCULATE MIXED MODE STRESS INTENSITY FACTORS ASSUME PLANE STRAIN CONDITIONS ASSUME A HALF CRACK MODEL WITH SYMMETRY BOUNDARY CONDITIONS USE 3 NODES EXTRAPOLATION PATH IS DEFINED BY NODES 26 36 37 WITH NODE 26 AS THE CRACK TIP NODE USE MATERIAL PROPERTIES FOR MATERIAL NUMBER 1 EX 4000 0 NUXY 0 35000 AT TEMP 0 00000E 00 PRINT THE LOCAL CRACK TIP DISPLACEMENTS CRACK TIP DISPLACEMENTS UXC 0 11685E 02 UYC 0 00000E 00 UZC 0 78886E 30 NODE CRACK FACE RADIUS UX UXC UY UYC UZ UZC 26 LLP 0 00000E 00 0 00000E 00 0 00000E 00 0 00000E 00 6 36 TOP 0 22500 0 21686E 05 0 13854E 03 0 00000E 00 37 TOP 0 90000 0 75442E 05 0 27840E 03 0 00000E A Fl 1 1 APPROACHES 0 0 TOP FACE ARE R 0 34449E 05 UY UYC SORT R 0 29160E 03 R 0 00000E 00 UX UX
17. oundary Anti symmetry conditions boundary conditions Figure 3 9 15 Taking advantage of symmetry e For reasonable results the first row of elements around the crack tip should have a radius of approximately a 8 or smaller where a is the crack length In the circumferential direction roughly one element every 30 or 40 degrees is recommended e The crack tip elements should not be distorted and should take the shape of isosceles triangles 3 D Fracture Models The recommended element type for three dimensional models is SOLID95 the 20 node brick element As shown in Figure 3 9 3 b the first row of elements around the crack front should be singular elements Notice that the element is wedge shaped with the KLPO face collapsed into the line KO 3 158 ANSYS User s Manual POST1 LOCAL CLOCAL cS CSKP etc RSYS 3 9 Fracture Mechanics Generating a 3 D fracture model is considerably more involved than a 2 D model The KSCON command is not available and you need to make sure that the crack front is along edge IM of the elements Other meshing guidelines for 3 D models are as follows e Element size recommendations are the same as for 2 D models In addition aspect ratios should not exceed approximately 4 to 1 in all directions e For curved crack fronts the element size along the crack front will depend on the amount of local curvature As a rough guide you should have at least one element every 15 to 30 degrees along a
18. oung s Modulus E for plane stress Pa psi E 1 v for plane strain Pa psi v Poisson s Ratio and S B W and a are as defined in A3 5 3 Note A3 2 This expression is considered to be accurate to within 1 0 for any a W 23 This expression is valid only for crack mouth displacements measured at the location of the integral knife edges shawn in Fig 5 If attachable knife edges are used they must be reversed or inset to provide the same measurement point location i A3 5 5 1 To facilitate the calculation of crack mouth opet ing compliances values of q a W are given in the followir table for specific values of a W aW AAM a W AAM 0 450 6 79 0 500 8 92 0 455 6 97 0 505 9 17 0 460 7 16 0 510 9 43 0 465 7 38 0 515 9 70 0 470 756 0 520 9 98 0 475 7 77 0 525 10 27 0 480 7 98 0 530 10 57 0 485 8 21 0 535 10 88 0 490 8 44 0 540 11 19 0 495 8 67 0 545 11 53 0 550 11 87 A3 5 6 Calculation of Crack Lengths Using Crack Mou Opening Compliance Measurements For bend specimen calculate the normalized crack length as follows see Not A3 3 alW 0 9997 3 95U 2 982U7 3 214U 51 52U 113 0 where U 1 1 E BV P 4W S 4 Note A3 3 This expression fits the equation in A3 5 5 with 0 01 of W for 0 3 a W 0 9 2A This expression is valid only fc crack mouth displacements measured at the location of the integral kni edges shown in Fig 5 If attachable knife edges ar
19. quantities calculated in steps 3 and 6 You can simplify the J integral calculations by writing a macro that performs the above operations Macros are described in Section 15 3 page 15 7 Energy Release Rate Energy release rate is a concept used to determine the amount of work change of energy associated with a crack opening or closure One method to calculate the energy release rate is the virtual crack extension method outlined below In the virtual crack extension method you perform two analyses one with crack length a and the other with crack length a Aa If the potential energy U strain energy for both cases is stored the energy release rate can be calculated from Ua Aa T U oe BAa where B isthe thickness of the fracture model Extending the crack length by Aa for the second analysis is quite simple select all nodes in the vicinity of the crack and scale them in the X direction NSCALE by the factor Aa Note If you used solid modeling you will first need to detach the solid model from the finite element model MODMSH DETACH before scaling the nodes The vicinity of the crack is usually taken to mean all nodes within a radius of a 2 from the crack tip Also the factor Aa for node scaling is usually in the range of 1 2 to 2 percent of the crack length l Volume Procedures 3 163
20. th the two specimen surfaces shall be equally distant from the top and bottom edges of the specimen within 0 005 W Nore 3 Integral or attachable knife edges for clip gage attachment to the crack mouth may be used see Fig 5 and Fig 6 Note 4 For starter notch and fatigue crack configuration see Fig 7 FIG A4 1 Compact Specimen C T Standard Proportions and Tolerances mm or 0 1 whichever is larger at not less than three Ko Po Bw HAW A4 1 positions near the notch location and record the average value where A4 4 1 1 For general requirements conceming specimen measurement see 8 2 A4 4 2 Compact Specimen Testing When assembling the 13 32a7 7W 14 7205 W 5 6a W loading train clevises and their attachments to the tensile LO ae machine care should be taken to minimize eccentricity of loading due to misalignments external to the clevises To obtain where satisfactory alignment keep the centerline of the upper and Po load as determined in 9 1 1 klbf kN 2 a W 0 886 4 64a W A4 2 lower loading rods coincident within 0 03 in 0 76 mm during B specimen thickness as determined in 8 2 1 in cm the test and center the specimen with respect to the clevis W specimen width as determined in A4 4 1 in cm opening within 0 03 in 0 76 mm and oS 4 4 2 1 Load the compact specimen at such a rate thatthe 99 crack length as determined in 8 2 2 and A4 4 1 in rate of increase of stress intensi
21. ty is within the range 30 to 150 cm ksi in min 0 55 to 2 75 MPa m s corresponding toa Nore A4 1 This expression is considered to be accurate within loading rate for a standard W B 2 l in thick specimen 0 5 over the range of a W from 0 2 to 1 12 13 between 4500 and 22 500 Ibf min 0 34 to 1 7 KN s A45 3 1 To facilitate calculation of Kp values of f a W Q 2 Oa For details concerning recording of the test record are tabulated below for specific values of a W aie Compact Specimens A4 5 Calculations Ry a _Sl_ _____ fal A pi 0 500 9 66 A4 5 1 For general requirements and procedures in interpre 0 455 8 46 0s a tation of the test record see 9 1 pee 8 58 0 510 9 96 Be a f F 8 70 0 515 10 12 A4 5 2 For a description of the validity requirements in 0470 883 0 520 1029 terms of limitations on P Pp and the specimen size regu 0 475 8 96 0 525 10 45 ments see 9 1 2 and 9 1 3 0 480 9 09 0 530 10 63 0 485 i A4 5 3 Calculation of Ko For the compact specimen cal 0 490 Ti PERA o culate Koi in units of ksi in MPa m from the following 0 495 9 51 0 545 11 17 0 550 11 36 expression Note A4 1 437 3 9 Fracture Mechanics Cracks and flaws occur in many structures and components sometimes leading to disastrous results Several years ago a commercial airliner flying near the Hawaiian islands suddenly lost the top part of its fuselage Apparently microscopic defects introduced in the ori
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