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1. a Negative 0 middle surface Fig 3 Laminate plate geometry and ply numbering system Longitudinal stress Ok T lt lt Transverse stress lt gt mie LESS eae Fig 4 Infinite band model of microbuckling 15 t a b c CEN SEN HOLE Fig 5 Geometry of specimens a centre notch b single edge notch c central hole and the definitions of a R b and w 16 4 2 Details for each form More detailed information on each of the separate forms are grouped together in this section with associated tutorials 4 2 1 Laminate elastic properties forms In this section further features of the Geometry and Elastic Analysis and More Elastic Laminate Properties forms are explained Introduction The laminate geometry and lamina elastic properties are input in this form Boxes in which to input data have a white background First enter the number of plies in the laminate Click on the appropriate option button if the laminate is unsymmetric Then enter the elastic properties thickness and ply angle of the first ply and click Save Ply Data It is assumed by default that all plies have the same properties and ply thickness although this can subsequently be overwritten by typing in data for each ply and saving the ply data A composite name and comments are optional and can be entered at any time The database button accesses a database of lamina elastic properties Use the ply arrange
2. Fig A5 Infinite band model of microbuckling The Input Ply Strengths option provides an alternative to the micromechanics model for the prediction of unnotched laminate strength This model as suggested by Soutis and Edge 1997 requires as inputs the lamina compressive and shear strengths Sz and Szy such as might be obtained from unidirectional tests The strength of each ply under combined stresses 0 62 and T12 is estimated by linear interpolation neglecting the transverse stress 62 so that failure occurs when 44 o z ah el A 35 Only failure of the 0 plies is considered for both the Micromechanical and Ply Strength failure criteria A 7 Laminate failure analysis The previous two sections discuss several multiaxial failure criteria for estimating the strength of individual lamina under in plane stresses Such criteria can be used on a ply by ply basis for a laminate to determine which ply fails first under in plane loads Interlaminar stresses leading to delamination are not considered in CCSM For the BFS compressive analysis it is assumed that failure will occur first in the microbuckling 0 plies The failure of these 0 lamina is examined using equation A 34 taking the stresses in the 0 plies arising from the laminate loading The laminate strength is given when the stresses on the laminate generates the lamina failure stresses ina 0 ply The prediction of first ply failure due to in plane stresses is a straig
3. Z _ distance from middle surface to inner surface of the k th lamina Z distance from middle surface to outer surface of the k th lamina as shown in Fig A4 Upon substituting the lamina stress strain relationships from Eqs A 18 in Eqs A 19 and A 20 respectively the following relationship is obtained Ny An A2 A6 B Bir Bio Ny A2 An Ax Biz Bor Bae Nxy _ Aio 426 466 Bio B26 B66 My B Bo Be Di D2 Die My B Boy B Dir Dn D 3 xy Bio Bo Boo Dis Dre Dew where the laminate extensional stiffness A t 2__ J Q kdz Set Z 1 t 2 ij 2 the laminate coupling stiffnesses are given by t 2_ Ni 2 2 Bij Qi kzdaz 5 Qik Zk Zk 1 t 2 k 1 and the laminate bending W are given by t 2 j Qi kZ 247 4 5 DG A 1 t 2 with the subscripts 1 j 1 2 or 6 Equation A 21 may be written in partitioned form as NI A Bile M B Dij x for convenience 39 SORSD x go Ky Ky Kyy are given by A 21 A 22 A 23 A 24 A 25 In CCSM the stiffness matrix in A 21 for the laminate is shown in the More Elastic properties form by clicking the Laminate Stiffness button A 3 Laminate compliances The inverse of the stiffness matrix A 21 or A 25 gives the compliances of the laminate e A BTHN N p Sh A 26 xl IB p m M The above relation is used to calculate the lamina stresses and strains associated with prescri
4. as Ey E Vp EpVy E 1 Vp Eg Vin Em Vi2 V Vf VmVm 1 Vp Gp Vm Gm En Gi A8 33 A 1 2 The generally orthotropic lamina 2 Y 1 Y NZ coordinates such as x and y in Fig Al Consider a lamina which is rotated ee a e 0 with respect to the 1 2 axes as shown in Fig A2 The sign convention for the lamina orientation angle O is given in ok A2 The relationships are found by combining the equations for transformation of stress and strain components from the 12 axes to the xy axes and the final results are os Q1 Q2 Ae Oy Q2 Qo Q Ey A 9 Try Q6 Q 266 xy where the Qj are the components of the transformed lamina stiffness matrix which are defined as follows di Q11C QS Qiz 2066 S C Q12 Q11 Q22 4066 S7C Q C 8 Ooo 01184 Q22C XQ12 2066 S7 C O16 011 012 2066 C S Q22 Q12 2066 CS Q26 Q11 Q12 2066 CS Q22 Q12 2066 C S Q66 Q11 Q22 202 2066 C7S 056 S C A 10 with C cos0 and S sin It should be noted that the number of independent coefficients in A 10 is still four In CCSM the matrix Qi is called Qbar The Qbar matrix for each ply can be viewed in the 34 More elastic properties form The strains can be expressed in terms of the stresses as Ex S S2 St6 ox Ey p Si2 S2 S fr oy A 11 xy Sig S26 Soo xy where the Si are the components of the transformed lamina compliance matrix which
5. consisting of 0 25 mm thick AS 3501 carbon fibre epoxy laminae is subjected to a tensile uniaxial loading along the x direction The ply moduli are E 138 GPa E 9 GPa vj2 0 3 Gj2 6 9 GPa Using both the Tsai Hill and Tsai Wu criterion find the loads corresponding to first ply failure The material failure strength data are as follows longitudinal tensile strength SL 1448 MPa longitudinal compressive strength SL 1172 MPa transverse tensile strength ST 48 3 MPa transverse compressive strength ST 248 MPa in plane shear SLT 62 1 MPa Step 1 Elastic properties Follow the procedures described in Tutorial 1 to input the laminate material properties and geometry Click Calculate to calculate the elastic properties of the laminate Step 2 Failure Analysis input Select the Tsai Hill Failure Criterion option Either type in the strength data for SL SL ST ST and SLT or use the Database button to input the strength data for AS 3501 clicking on this material in the material list and then clicking on Take record as Input Use the Save Data All Plies button to store this data Now enter the force pattern Remember that the absolute numbers are irrelevant it is the ratio of forces that matters In the present case the only non zero force is Nx therefore an appropriate force pattern would be Force pattern applied Nx Mx to laminate E A d C Step 3 Now click Calculate to perform the failure analysis The ap
6. displacement u x across the microbuckle by a functional relationship For the current situation a simple linear softening law is used o jes for O0 lt u lt ue Ue A 40 Oun jo for u gt uc where u is a critical microbuckle overlap displacement The appropriate value of u is found from the measured toughness Gyc using the assumed variation of microbuckle load with displacement u Gic k 20 u du O nic A 41 To solve for the remote stress the cohesive zone is divided into N elements of equal length The traction distribution within the cohesive zone is taken to be piecewise linear with triangular distributions of crack face traction with peak value centred on the i th node The integrals in equations A 36 and A 37 are calculated numerically at the node points for triangular distributions of traction of unit magnitude at the N node points N 1 simultaneous equations concerning the non zero displacements at the 1 to N 1 discretised nodes except the node at the tip where the displacement is zero are then obtained An additional equation is that the stress intensity factor at the tip of the microbuckle K due to the remote stress and microbuckle traction is zero Km 0 therefore the system governing equations are fully decided and solved numerically The calculations give a relationship between the microbuckle length and the applied remote stress The failure stress is given by the maximum applied stress the c
7. for the failure analysis have been input click the Calculate button The output boxes below the input force pattern show the failure loads Forces to give failure M xy coon S so IIo ap toe I ton I fo telling us that the force to cause the first ply failure when the Tsai Wu failure criterion is applicable will be N O 327 MN m with Ny N xy M FM M y The magnitude of the input load Nx 0 2 MN m will not affect the actual a of the failure load N The relative magnitudes of the load components are determined by the input values of Nx Ny Nxy Mx My Mxy the failure loads scales with this load pattern Observe also that in the ply arrangement grid on the left side the fourth column marked Fails first shows which plies have failed For the present case all the 45 45 degree plies fail at the same time Hence the grid appears as Angle Thick Fails first In this section we have covered the Tsai Wu failure criterion for unnotched strength The advanced tutorials in Section 4 2 give further information on the failure analysis including an illustration using the Budiansky Fleck Soutis model for compressive failure and the prediction of both the unnotched and the notched strength of laminates 12 4 Detailed guide and advanced tutorials This section is a detailed guide to the use of CCSM Section 4 1 contains information about the nomenclature used in CCSM Section 4 2 contains further information and ad
8. helpful advice from Dr C Soutis Ms V Hawyes Mr P Schwarzel and Mr I Turner for additional programming help from Mr A Curtis and for financial support from the Procurement Executive of the Ministry of Defence contract 2029 267 and from the US Office of Naval Research grant 0014 91 J 1916 Disclaimer Although the calculations and implementation in this program are believed to be reliable the authors cannot guarantee the accuracy of the results produced by this program and shall not be responsible for errors omissions or damages arising out of use of this program Composite Compressive Strength Modeller User s Manual 1 Introduction Welcome to the Composite Compressive Strength Modeller CCSM a design tool for deformation analysis and failure prediction of composite materials CCSM incorporates the following features 1 Classical laminate theory for the prediction of laminate elastic properties 2 Stress and strain analysis when in plane forces and or bending moments are applied 3 Unnotched failure prediction by conventional failure criteria and the Budiansky Fleck compressive failure criterion 4 Compressive failure prediction for notched composite plates 5 A user expandable database to store material and geometrical properties The program is a tool to predict laminate failure once the loads on a section of the laminate are known For simple geometries it may be clear what the loading is while for more complica
9. re Calculate the elastic properties Step 2 Deformation analysis Click the Deformation Analysis button in the Elastic Properties form to invoke the deformation analysis form Input the applied force Nx 0 05 MN m converting from mm to m N xy M xy applied to laminate 0 05 E Now click Calculate The mid plane strain and the resulting stresses and strains in each ply will be calculated as Mid plane strain 2137 __ 1485 o OO By default the stresses in the first ply are also shown at the bottom of the form as Location through Stress state for ply No 1 MPa selected ply top 5o o IA middle 50 0 21 1614 bottom 50 fo 21 1614 Average over 50 laminate Note that data are shown at the top middle and bottom of each ply in order to consider the possibility that the stresses are not constant through the thickness Since the curvatures are zero here the stresses do not depend on the through thickness location The last row shows the laminate stresses which are the averaged stresses of the corresponding stress components over the whole laminate thickness These stresses are in global co ordinates i e running along the x and y directions of the laminate To see the ply stresses in local co ordinates i e running along and transverse to the fibre direction in each ply click on Local Coordinates in the Output Option box to give 22 Location through Stress state for ply No 1 MPa the thic
10. section 49 As noted in Sutcliffe and Fleck s 1993 paper the weight functions used are for isotropic materials they show that errors in using them for the orthotropic materials are not too large A 9 Interpolation and extrapolation of the bridging analysis data A number of bridging analyses have been carried out for three types of commonly used specimens a centre cracked panel CEN a single edge cracked panel SEN and a plate with a hole HOLE and for a range of crack and ligament sizes Sutcliffe and Fleck 1993 Results are expressed as functions of a rp and b rp where a is the half crack size for CEN crack size for SEN and hole radius for HOLE b is the ligament rp K Sun is a characteristic length scale K is the toughness and o the unnotched laminate strength Dimensionless results for the variation of the notched strength and critical microbuckle length with a 1 and b rp between 10 and 10 for the three notched geometries are stored as look up tables Use of a look up table is more robust and faster than performing the calculation in real time The look up table files are CENLENG DAT CENSTRE DAT SENLENG DAT SENSTRE DAT HOLELENG DAT and HOLESTRE DAT Data in these files should not be changed If the geometry to be analysed is located at one of the calculated data grid points the results will be retrieved directly from the look up table If the geometry to be calculated lies in between or outside t
11. suadeesunseneasscyednaavenacdsonsectenndseees 21 4 2 4 Deformation Analysis Tutorial 0 0 0 ceesecssccecsecceceeceeceeeeecseeeeceeeecseeeceeeeeeeeeees 21 425 Fail re ANNA LY 1 iil lta Sa el fhe rt tg gu aig el ose Ge ag acl Dasa la wan 23 4 2 5 1 Conventional Failure Criteria sseseeessesssessseeeseetssstesseessessersseeessseesseesseesseessees 23 42 39 27 BPS SP aire Criteria ssrin naen e a ease echo E E TRS 24 4 2 6 Conventional Failure Analysis Tutorial 0 0 0 eecceeccecesceeeseececeeeeeceeeeecseeeecneeeeeeeeees 25 4 2 7 BFS Compressive Failure Analysis Tutorial 2 0 0 0 cee eeeessecsseceseeeeeeesseeesaeeneensaes 27 4 3 Materials ACAD ASS i tei sh E E E asia eta rod ass EAE eat kde as yada aaa wean avec oes 29 43 1 DataBase Elastic Property Data siisstiyiscsisesicesunsdesatisvsssacssuesedeanteedatvnn joacteunasente lesen 29 43 2 DAA BASE Strength nesini e aa a a a A E tees gusieaadnc ened 29 4 3 3 Database for the bridging analysis ssssesssesssesesseeesseessresseesseeesseeesseesseesseeesseessees 29 APPENDIX A THEORETICAL BACKGROUND sssseeseteseereesrtsresetertessesetettessesetestessesstertessesetesreeseeseee 31 A l Lamina stress strain relationships seesssssesssesseseseesseesseesseesseessseessressresseesseeeesseessrest 31 A 1 1 The orthotropic lara aise cons catieessintncyah it jtuea aiecase sasvaensseaducgotdesteceessesnde uv nsaaaaentoeee 31 A 1 2 The generally orthotropic lamina s 222224 u
12. x can be conveniently found from the point load solution given by published results for a crack emanating from a central hole and for a single edge notch and a centre notched panel which are the three geometries dealt with in the current version of CCSM Using the weight function method the crack closing displacement u x on the microbuckle face where u is half the relative displacement of the two faces of the microbuckle due to o x is given by 1 L i ug x 5 mle atte oxy A 37 E Jo 0 where the orthotropic equivalent elastic modulus E is defined by 1 2 1 2 1 2 1 1 Eyy Eyy 7 z E zi V yy A 38 E 2E Eyy Ez 2G yy The weight function method is also used to find the stress intensity factor at the microbuckle tip K m and the displacements across the microbuckle u x due to the remote stress for the notched panels using Bueckner s rule briefly that is A 36 and A 37 are used again to calculated K and u x but the traction on the crack faces is that along the crack line for a 47 specimen containing no crack For specimens with a central hole Newman 1982 gives the stress intensity factor at the end of the microbuckle and the displacements due to a remote uniform stress The net stress intensity factor at the microbuckle tip K and the displacements along the microbuckle u x are then Km Ko Km u x Ug X u x A 39 The compressive traction o x across the microbuckle is related to the closing
13. 1993 Effect of geometry upon compressive failure of notched composites Int J Fracture 59 pp 115 132 MPF Sutcliffe and NA Fleck 1997 Microbuckle propagation in fibre composites Acta Metallurgica and Materialia pp921 932 56
14. If a panel of total width 50 mm is made of this laminate with a central hole of radius 2 5 mm find 27 the notched failure strength and corresponding microbuckle length at failure using the measured value of fracture toughness of the laminate of 42 5 MPa m Find the variation in strength with hole size for a fixed panel width of 50 mm Step 1 Elastic properties Follow the procedure described in Tutorial 1 to construct the 24 ply laminate and click on Calculate this file has the correct ply geometry and material properties Lamina elastic properties for T800 924C are stored in the property database The laminate should have a stiffness Ex 61 707 GPa Step 2 Predicting the unnotched strength Go to the Failure Analysis and click the BFS compressive option Check that the Strength Input Option is set to the Micromechanics model The data input appearance changes to a suitable layout for this failure criterion Now type the following input data Matrix shear strength k 62 35 Fibre waviness phi 3 Kink band inclination angle B Click on the Save Data All Plies Put a stress Sigma_L equal to 1 in the stress pattern panel at the top of the form then on the Calculate button to perform an unnotched strength analysis ignore the warning re notched strength predictions The output box half way down the screen on the right shows that the laminate unnotched strength Sigma_L is 566 6 MPa In this case the output is the composit
15. User s Manual Composite Compressive Strength Modeller A Windows based composite design tool for engineers Version 2 0 2013 M P F Sutcliffe X J Xin N A Fleck and P T Curtis Engineering Department Cambridge University Cambridge CB2 1PZ U K DSTL Farnborough GU14 0LW U K CONTENTS 1 INTRODUCTION sssascasteainesswalnrysistunnedaacy udpeaseabasas alates AE A begs deusbana E A A 4 2s INSTALLATION i aiie ie 2s iiss vasa A AE A E E a a A E i 3 2il System reg irementS hae a E T A N s 5 DD NCEE SNM AC Kae nonren esas cosa A e a EEAS E A E E EE ROEST 5 23 SCCM p COSM svcccisesiensateseesnetansnceeadscvaudanvousecdasaevetaaasapaawisdagausasnasaaedlecnecassnodavssoeeedeadeetacees 5 Six QUICK STAR Taner tin E E O T es E A E E oS OE ident de 5 3 1 How t use COS MMs nerenin nenene n n E R AEE A RE R RE 6 3 2 Arguick start example sanien E A E ook musa a eai i 7 4 DETAILED GUIDE AND ADVANCED TUTORIALS csssccscceeseecsseeesneesccecseeceseesseeeeccecseecsaeeeseeenee es 13 4 1 Nomenclature convention in CCSM sssssssssesssessssseesseessresseessetesseeessressresseesseeeesseesssees 13 4 2 Details foreach fotosi eruan apai i n E E decals E E 17 4 2 1 Laminate elastic properties forms sssessesssesssesessetessetsseesseesseeesseeesseesseesseeeseeesseee 17 422 Elastic Prop rties Tutoriali sim inniinn A EE 19 4 2 5 Deformation analysis i c icvsyeccivasssversasvsaesesdasaeeasecsaasalg
16. W N Assumption 5 is a result of the assumed state of plane stress in each ply whereas assumptions 5 and 6 together define the Kirchhoff deformation hypothesis that normals to the middle surface 36 remain straight and normal during deformation According to assumptions 6 and 7 the displacements can be expressed as u u x y zZF1 x y v v x y zF2 x y w w x y w x y A 12 where u and v are the tangential displacements of the middle surface along the x and y directions respectively Due to assumption 7 the transverse displacement at the middle surface w x y is the same as the transverse displacement of any point having the same x and y coordinates so w x y w x y Substituting Eqs A 12 in the strain displacement equations for the transverse shear strain and using assumption 5 we find that u Ow Ow F 3 0 Yxz J OR ue Wt OV Ow Ow Y yz z Oy 2 y ay and that Ow Fi x y 1 y a F x y Ow A 14 oy Substituting Eqs A 12 and A 14 in the strain displacement relations for the in plane strains we find that ou a eX ZK Ey sa By tZKy oy ou OV o lora og e Ty A 15 where the strains on the middle surface are o o o o and the curvatures of the middle surface are K PEE K BRE K __ ow A 17 x ax y y xy x y i Here Kx is the bending curvature associated with bending of the middle surface in the xz plane ky is the bending curvat
17. a fixed orientation to the direction of the 0 degree fibres Typically this B angle of propagation of the microbuckle lies between 20 and 30 Sutcliffe and Fleck 1997 give theoretical predictions of the propagation direction In practice there appears to be little variation in the value of B in composite materials and the analysis is not sensitive to this parameter so that the chosen value of B is not critical The review paper by Fleck 1997 gives further details of these microbuckling models and their inputs 43 Although the effect of shear stresses have been verified by Jelf and Fleck the effect of transverse stresses has not been and this part of the model should be used with caution While the analysis predicts the failure due to plastic microbuckling the user should also be aware that other modes of failure may occur for example elastic microbuckling splitting fibre crushing or matrix failure A check on the elastic microbuckling limit equal to G12 is carried out by CCSM but the user should be careful in checking that these other modes do not occur Refer to the work of Jelf and Fleck 1992 for further information The B F model does not allow for plate bending components and the stresses at the mid plane of each ply are used in equation A 34 See Shu and Fleck 1997 for a discussion of strain hardening effects which are not modelled here Longitudinal stress a T lt Transverse stress q T Kink band ann
18. ack faces is assumed to equal the tensile yield strength of the solid The material response elsewhere 46 in the cracked specimen or structure is assumed to be linear elastic Other sophisticated examples adopt more realistic consequently more complicated bridging laws One such example is the bridging law derived from an infinite band calculation of fibre microbuckling the crack traction T versus crack overlap 2v law is assumed to equal the remote stress versus extra remote displacement Av response of an infinite microbuckle band under remote compression the extra remote displacement is the end shortening minus the contribution to shortening associated with elastic axial straining Details about the bridging analysis which is incorporated in the current version of CCSM can be found in Sutcliffe and Fleck s 1993 paper Only a brief outline is given below Although the effect of compressive loading has been well tested experimentally e g Soutis et al 1993 Soutis and Edge 1997 recent work by Fleck Liu and Shu 1998 suggest that the predicted effects of transverse and shear loading should be treated with caution for notched geometries Considered the geometries shown in Fig A6 The stress intensity factor K due to a distribution of normal compressive stresses o x along the microbuckle at the tip of a microbuckle of length L is given by the integral Ko o x m x dx A 36 0 where the weight function m
19. ailure determined by a stress intensity factor criterion to failure given by the unnotched strength as b rp changes from being very large to very small In the extrapolation scheme it is assumed that the form of the transition is not dependent on the value of a rp in this area as long as a rp remains small and varies in the following way with b rp 54 g B A B w lim B w3 lim B A 57 where B ai 2 w 0 5 0 Seog SAA ar L 6 P where the functions lim and lim are the expressions for the stress or stress intensity factor K and wi w2 1 which apply asymptotically at either end of the boundary of the calculated grid with b rp small and large respectively in this case they would be given by equations A 54 and A 51 A which is a function of b tp expresses the form of the transition in behaviour between these extremes and is found from the data computed at the edge of the grid along a rp 0 001 in this case Similar expressions are used along any other boundaries where there is a transition in limiting behaviour B 10 References References for each of the above theory sections are given below Sections B 1 R F Gibson Principles of Composite Material Mechanics McGraw to B 5 B 7 Hill 1994 and B 10 Section B 6 B Budiansky and N A Fleck Compressive failure of fibre composites J Mech Phys Solids 41 1 pp 183 211 1993 Sections B 8 and M P F Sutcliffe and N A Fleck Effe
20. ailure due to plastic microbuckling the user should also be aware that other modes of failure may occur for example elastic microbuckling splitting fibre crushing or matrix failure This criterion should not be used if off axis plies could fail first this could be checked using conventional failure criteria Refer to the Appendix B for a more detailed explanation references and comments on the validity of these models Unnotched strength The BFS failure model for unnotched strength is used in a similar way to the conventional failure analysis The unnotched strength of the laminate can be predicted based on either a micromechanics model or using strength data for each lamina such as might be obtained from unidirectional tests The Strength Input Options are used to change this Further details of these strength inputs are given in Appendix B section B 6 After choosing the BFS failure criterion and the Input ply strengths from the Strength Input Options ply strength data should be entered in the appropriate data boxes either directly or using the Database Where the failure strengths are the same for each ply click the Save Data All Plies button For a laminate made up of different materials individual strength data for each ply can be entered using the Save Data This Ply button The strength of each ply can be examined by clicking and navigating through the ply arrangement table In the BFS criterion the stress pattern rather
21. al The composite is subjected to a remote axial stress oL parallel to the fibre direction an in plane transverse stress oT and an in plane longitudinal shear stress t The infinite band is inclined with respect to the fibre axes such that the normal to the band is rotated by an angle B with respect to the remote fibre direction as shown in Fig AS For the case where the composite displays a rigid perfectly plastic in plane response the compressive strength of the lamina is given from Slaughter et al 1993 by ak t or tan B 9 where k is the shear yield strength of the composite and 41 R tan The constant R is taken as 1 5 Jelf and Fleck 1994 loL A 34 It may be helpful to review the various input parameters to this model The dominating influences are the matrix shear strength and the fibre waviness The matrix strength k can be estimated from the yield strength of the unidirectional composite in shear Typical values for polymer matrices are in the range 30 100 MPa The estimate of fibre waviness 6 is not trivial for real composites and is the subject of current research However a typical value for would be in the range of 1 3 for standard polymer matrix composites Research suggests that a misaligned region of more than say 30 50 fibres will be needed to affect the strength a few misaligned fibres would not make a difference Experimentally it is found that microbuckles propagate across the specimen at
22. alculations also give the corresponding critical microbuckle length lc at the maximum stress CCSM presents results at failure in terms of the remote stress oL the average stress over the unnotched ligament b and the critical microbuckle length 48 The bridging analysis used in CCSM does not strictly apply with applied shear and transverse loads However an estimate of the failure load under these conditions can be made by using the notched results for purely compressive loading with the compressive strength knocked down by an amount given by a failure analysis for unnotched laminates This is the approach used in CCSM ee fk ef fe a b c CEN SEN HOLE Fig A6 Geometry of specimens a centre notch b single edge notch c central hole and the definitions of a b and w In general the bridging analysis can be described by the following functional relationship a b lt f A 42 rp Tp Tp O a b E A 43 p p Tp 2 _EGr _ Kic PeR where p is the bridging length scale Oun is the unnotched strength of the Oun Oun laminate lo is the critical microbuckle length oy is the remote stress of the notched specimen at failure ais the half crack length CEN or hole radius HOLE or total crack length SEN and b is the ligament length Equations A 42 and A 43 are the basis for the interpolation and extrapolation for the bridging analysis in CCSM described in the following
23. are defined by equations similar to but not exactly the same form as Eqs A 10 see the reference listed in A 10 for details 35 A 2 Classic laminate theory with bending Fig A3 defines the coordinate system to be used in the description of laminate theory used in CCSM The xyz coordinate system is assumed to have its origin on the middle surface of the plate so that the middle surface lies in the xy plane The displacements at a point in the x y z directions are u v and w respectively The basic assumptions are Mx N an ee Mxy i Nyx Myx Ny Fig A3 Coordinate system and stress resultants for laminates plate 1 The plane consists of orthotropic laminae bonded together with the principal material axes of the orthotropic laminae oriented along arbitrary directions with respect to the xy axes The thickness of the plate t is small compared to the lengths along the plate edges a and b The displacement u v and w are small compared with the plate thickness The in plane strains x y and yxy are small compared with unity Transverse shear strains Y xz and y y are neglected Tangential displacements u and v are linear functions of the through thickness z coordinate The transverse normal strain x is neglected Each ply is linear elastic The plate thickness t is constant 0 The transverse shear stresses Tx and ty vanish on the plate surfaces defined by z t 2 PO AeoAONDHD NM H
24. ate and ply either stiffness properties Deformation analysis Calculate stresses and strains for each ply Failure analysis Calculate unnotched and Use the database notched strength if needed A flow diagram showing the structure of CCSM 3 2 A quick start example This section goes through a simple analysis to illustrate the essential features of CCSM More advanced tutorials are given in chapter 4 The problem Determine the stiffness and compliance matrices for a 0 45 0 symmetric laminate consisting of 0 1 mm thick unidirectional AS 3501 carbon fibre epoxy laminae Also find the stresses and strains for each lamina when the laminate is subjected to a single uniaxial force per unit length Nx 200 MN mm Use the Tsai Wu failure criterion to decide the load level corresponding to first ply failure The following lamina stiffness and strength data are given longitudinal modulus Ej 1 138 GPa transverse modulus E22 9 GPa shear modulus G 12 6 9 GPa Poisson s ratio v12 0 3 longitudinal tensile strength denoted as SL 1448 MPa longitudinal compressive strength SL 1172 MPa Transverse tensile strength ST 48 3 GPa Transverse compressive strength ST 248 GPa in plane shear strength SLT 62 1 GPa taken from R F Gibson s Principles of Composite Material Mechanics Table 2 2 P 48 1994 A step by step illustration is given below Step 1 Starting CCSM After starting Windows launch the Compress
25. bed laminate loads In CCSM the compliance matrix in A 26 for laminate is shown in the More Elastic properties form by clicking the Laminate Compliance button For a balanced symmetric laminate the B sub matrix is zero indicating that there is no coupling between in plane and bending terms The upper left quarter of equation A 26 is now in the same form as the equivalent equation A3 for an orthotropic lamina Hence it is appropriate to define laminate engineering constants Ex Ey Gxy Vxy and vyx using the compliance matrix S in equation A 26 with equivalent expressions to those given in equation A 4 and converting from forces to stresses via the thickness t E E fe a GS tS 14 ISa Vo Vy Sy tS t533 For an unsymmetric matrix there is coupling between the in plane and bending terms so that this decomposition is no longer valid However the compliance matrix S can still give an effective stiffness where there is only loading in the relevant direction for example 1 tS gives an effective value for Ex where there is only an Nx load term These are the values that are quoted A 4 Determination of lamina stresses and strains Calculation of lamina stresses and strains is a straightforward procedure Making use of Eqs A 9 the stresses in the k th lamina when written in abbreviated matrix notation are given by fot olle h zt A 28 40 where e and x are the midplane strains and the curvatures respectively He
26. can then be backed out from the K value by the LEFM formula For the HOLE specimen failure is determined by the stress concentration factor i e Kio Oyn while the 4 tends to 0 and becomes unstable The limits in summary are when alr gt gt 1 and bir gt gt 1 CEN K gt K le gt 0 75r o K IYJ 7a A 48 SEN KaK le gt 0 75r o K IY vra A 49 HOLE Le 0 unstable O O n K Kis the stress concentration factor A 50 53 Region 2 when a fp is small and b fp large the situation is summarised below when alr lt lt 1 and b r gt gt 1 CEN Le gt 0 45rp O gt O b w K Yo Jma A 51 SEN le UAT O gt 0 b wW K Yo vaa A 52 HOLE Le gt 0 55r O gt Oynd w A 53 Region 3 when b r is very small regardless of the value of a r we find when b r lt lt 1 CEN le gt b O gt Oy b w K Yo za A 54 SEN le gt b o gt On b w K Yo za A 55 HOLE le gt b O gt O b w A 56 In the actual calculation the above asymptotic solutions are assumed to be reached when log a rp or log b rp 3 While there are asymptotic solutions in certain areas outside the interpolation grid in other areas there is a transition between two types of asymptotic behaviour The way in which CCSM handles data in this area is explained by way of example Consider a centre notched specimen with small a rp there is a transition from LEFM with f
27. ct of geometry upon B 9 compressive failure of notched composites Int J Fracture 59 pp 115 132 1993 B N Cox and D B Marshall Acta Metall et Mater 39 59 589 1991 J C Newman in Proceedings of the AGARD Conference on the Behaviour of Short Cracks in Airframe Components France 1982 Section B 9 W H Press S A Teukolsky W T Vetterling and B P Flannery Numerical Recipes in FORTRAN Second Edition Cambridge University Press 1994 55 Other references and further reading B Budiansky and N A Fleck 1994 Compressive kinking of fiber composites a topical review Appl Mech Rev 47 6 part 2 S246 S250W B Budiansky and N A Fleck 1993 Compressive failure of fibre composites J Mech Phys Solids 41 1 pp 183 211 B N Cox and D B Marshall 1991 Acta Metall et Mater 39 59 589 N A Fleck L Deng and B Budiansky 1995 Prediction of kind band width in fiber composites J Applied Mech 62 pp329 337 NA Fleck 1997 Compressive Failure of Fibre Composites Advances in Applied Mechanics Vol 34 Academic Press ed JW Hutchinson and TY Wu pp 43 118 NA Fleck D Liu and JY Shu 1998 Microbuckle initiation from a hole and from the free edge of a fibre composite submitted to Int J Solids and Structures R F Gibson 1994 Principles of Composite Material Mechanics McGraw Hill P M Jelf and N A Fleck 1992 Compression failure mechanisms in unidirectional composites Journal of Com
28. ctice the lamina is often assumed to be in a simple two dimensional state of stress In this case the orthotropic stress strain relationships in Eq A 1 can be simplified to Si S2 0 2 S21 S2 0 hoz A 3 712 0 0 Se 72 where the compliances Sj and the engineering constants are related by 1 1 V21 V12 1 S155 a OS a ee OS A 4 E E2 E2 E Gio Thus there are five non zero compliances and only four independent compliances for the specially orthotropic lamina The lamina stresses are given in terms of strains by oaj Qoi Q2 0 l 027 Q21 Qz 0 E2 A 5 712 0O 0 2O66 712 2 where the Q j are the components of the lamina stiffness matrix which are related to the compliances and the engineering constants by 32 _ __ S27 Qi 7 1 S11822 Sj2 Vy2Va1 652 S12 __YnpE _ S1822 Sf I vy2V01 S E Oe u z 2 S118522 Sf 1 v12v21 1 Q66 Pan VE A 6 66 The lamina properties are calculated using the above formulae with the engineering properties E 1l E etc Where the Micromechanics input option is used the elastic properties of the lamina are calculated from the Elastic modulus Poisson s ratios of the fibres and matrix Er vt Em and Vm and the fibre volume fraction Vp as follows The volume fraction of matrix Vm and the shear moduli of fibres and matrix Er and Em are given by Vm 1 V grai f a l v Em E A7 HL ap and the elastic moduli and Poisson s ratio according to the law of mixtures
29. d and tabulated in data files These data files are CENLENG DAT f and its derivatives for CEN specimen CENSTRE DAT g and its derivatives for CEN specimen SENLENG DAT f and its derivatives for SEN specimen SENSTRE DAT g and its derivatives for SEN specimen HOLELENG DAT f and its derivatives for HOLE specimen HOLESTRE DAT g and its derivatives for HOLE specimen In the CCSM bridging analysis if the input data point log a ry log b r is within the lower and upper bounds of the macro grid the program will find the right unit grid square which encloses the input data point a r b r and the function values for the remote stress oy and the critical microbuckle length at the input point will be interpolated from the unit grid using 52 the bicubic interpolation A 44 A 9 2 For points outside the macro grid of the look up table When the input data point falls outside the macro grid an extrapolation technique has to be used CCSM relies on the asymptotic nature of the solution in these areas to obtain the correct values Details are given in the references quoted Three distinct regions are noted Region 1 when a r b r are both very large loading is in the so called LEFM situation where the remote applied stress intensity factor at failure will be equal to the fracture toughness of the composite i e K K and the microbuckle length is found to be le 0 75rp for both CEN and SEN specimens The remote stress
30. d ia die aetna 34 A 2 Classic laminate theory with bending csceseceeeceeseccesseeconteccenteccessencontescetteneensers 36 A3 Laminate COMPHANCES lt 0c vs5 chicevnuntegeaeadessaseeulny cute tayebevseeagean eaa a a e tiea wens 40 A 4 Determination of lamina stresses and strains ssseessseesseesseessessseessseessessresseresseesssees 40 A 5 Conventional lamina failure criteria sssessesseeeseeeesseessetsseesseeessetessressresseesseeesseeessrest 41 AS l Mazim m stresse e a e artes E ae 41 A 5 2 Maximum SU ain ssccise pcaccaesacasgiscevaacacysdacdade denedaseseaadasdusaded spavesea TE NKEA EE pEi 41 ACD Tsa Ail raeng aa eaaa eeek 42 ASA TSE Wie apie E E TE E E E E A TETA 42 A 6 BFS compressive failure criterion sssssssssseseseeessseessetssressresseeesseeessressresseesseeeesseesseest 43 A7 Laminate failure analysis ssee i iien ie a E D E Sahi 45 A 8 Failure of notched laminates sisiiv csscdiss ciacnsavavaseissgeacianssvadsaes saaceaswasaeeasedoecseavevaceesdeceeensvoces 45 A 9 Interpolation and extrapolation of the bridging analysis data e ce eeeceeeseeeeeteeeeeneeeees 50 A 9 1 For points inside the macro grid of the look up table ooo eee eeeeeeseeeeneeeeeeeees 51 A 9 2 For points outside the macro grid of the look up table 0 0 0 eee eeeceeeeseeenteeeeneeees 53 B10 RETERE NCES 53 cacedts 2 o5ns cee an a ands aaah ccoas ac a EES 55 Acknowledgements The authors are grateful for
31. e materials database file for CCSM is called CCSM MDB this resides in the CCSM directory To access the appropriate database click on the Database button in the relevant forms 4 3 1 DataBase Elastic Property Data The two input options Engineering or Micromechanics are explained in section 4 2 1 There is only a database for the Engineering option The name list on the left side of the form lists the all the relevant materials data stored in the database Navigate through the database using a mouse or arrow keys The current record is highlighted Click Take record as input to fill in the appropriate input boxes in the Elastic Analysis form from this record and to go back to this form To add a new data set click the Add Record button first then type in the material name which should be unique and the related properties data Finally click the Save Data button if you want to store this record permanently in the database If the save fails for example if you don t have write access to the database file CCSM mdb then this will be indicated by a pop up window Click the Delete button to delete the current record in the database To change an existing data record edit the appropriate cells then save the data Once completing editing of the database either take the current record as input to the elastic form or quit without taking the record 4 3 2 DataBase Strength The DataBase Strength form is very similar to the DataBase Elastic P
32. e based on limited data and should be used only as guidelines The equivalent hole is used to model post impact compressive strength as described by Soutis and Curtis 1996 They observe that when a panel is loaded in uniaxial compression damage propagates from regions of delamination e g arising from impacts in a similar way to that observed with open holes Further details in particular describing how to estimate the diameter of this equivalent hole are given in this reference The length of the microbuckle at peak load for the equivalent filled and countersunk holes may be estimated from the equivalent calculation for an open hole Carpet plots which give the effect of ply mix on notched strength for a symmetric laminate are calculated from the data in the failure form for strength toughness and geometry type It is assumed that the composite is composed only of 0 90 and 45 plies Several curves each corresponding to a constant proportion of 0 plies are produced showing the variation of strength with the change in 45 plies The proportion of 90 plies is found using the fact that the total proportion of plies sums to 100 The axial compressive stress is plotted on the graph The stress pattern in the failure form is used for these plots The other stress components will be in the proportions specified by the load pattern specified here and shown in the Plots form The unnotched strength for each ply mix is predicted based on
33. e effective laminate engineering constants are calculated and the More Elastic Properties Button is enabled To view the stiffness and compliance of the laminate click on this button The stiffness matrix is viewed by default as 43 497 29 697 29 697 ai ka o l s 2 eo a Seg e o o a p fe 3 6266 247E6 1 89E 6 24TE 6 _ 3 62E 6 1 89E 6 1 89E 6 1 89E 6 2 8686 with units MPa m MPa m and MPa m for extensional coupling and bending terms respectively 20 Click the Laminate Compliance button to view the compliance matrix as Ee a EE 57465 2 9685__ 1 84E5 296E5_ 5 74E5__ 1 84E5 1 84E5 1 84E5_ 5 9585 where again units are in MPa and m In this table 0 represents a very small value in the order of 1 E 17 1 E 20 which is due to numerical rounding and should be practically taken as zero The default output format is to have up to three digits after the decimal point Very small or large numbers can be displayed in scientific notation In the Elastic Properties form choose the Data format Scientific4 1 2345E00 option to change the output data format 4 2 3 Deformation analysis Further details of the deformation analysis form are investigated in this section In this form the deformation of the laminate and stresses and strains in the individual plies of the laminate are calculated You must enter the load applied to the laminate either in terms of applied line loads and bendin
34. e failure stress rather than the force per unit length This compares with the measured value by Soutis et al 1993 quoted in the references in section B11 of 568 MPa Step 3 Predicting the notched strength To proceed to the calculation of the notched strength data about the geometry of the notched panel must be input together with the toughness of the laminate First check that the Notched strength by clicking on Open equivalent hole in the Geometry option To input the geometry in a convenient form select R and w from the Geometry Input Options box of the notched strength part of the form Input the radius 2 5 mm and the semi width 25 mm in the input data boxes as required for the problem To input the known fracture toughness select Ke given in the Toughness Option box and type in the required value of 42 5 in the Ke input data box 28 To perform the notched analysis click the Calculate button The remote compressive stress oL and the critical microbuckle length at failure Ic are given by 361 928 Critical microbuckle length Ic 3 7619 Note that the results show us that the remote stress of 362 MPa is substantially less than the unnotched strength of 568 MPa but that the microbuckle can grow to a length of 3 7 mm longer than the hole radius in this case before the maximum load is reached and failure occurs 4 3 Materials Databases One of the powerful features of CCSM is its connection to a user maintainable database Th
35. es are allowed The equivalent hole model suggested by Soutis and Curtis 1996 is used to model post impact compressive strength The format for defining the lengths of the specimen is changed by clicking on the appropriate Geometry input option The notch size is defined by the notch length or semi length a or the hole radius R depending on the geometry option The panel size is given by the panel width or semi width w or the unnotched ligament length b These dimensions are illustrated in section 4 1 and further details of the analysis are given in Appendix B 6 The toughness of the laminate can either be input directly in terms of Ge or Ke This choice is governed by clicking on the apporpriate Toughness option Prediction of notched strength Once the notch geometry and toughness have been input the notched failure analysis can be performed by clicking on the Calculate button Where required the laminate unnotched strength will be predicted at the same time The notched strength of the panel is given in the output box at the bottom of the form For the centre and edge notched geometries and for the open hole the length of the microbuckle at peak load is also given This length can be estimated for the equivalent filled and countersunk holes from the corresponding calculation for an open hole 4 2 6 Conventional Failure Analysis Tutorial 25 In this tutorial a conventional failure analysis is worked through Problem A 90 0 90 laminate
36. g moments or in terms of applied micro strains and curvatures Use the input option box to switch between these Deformation data for each ply is displayed either in the global x y co ordinates of the laminate or in terms of the local 1 2 co ordinates running along the fibre direction for each ply and in terms of stresses or strains Again these options are controlled by output option boxes The ply arrangement table is used to display data for each of the plies navigating using the mouse or arrow keys When changing input options the equivalent load for the new input option is automatically calculated This may cause small loads of the order of rounding errors being put into the input boxes these can safely be ignored 4 2 4 Deformation Analysis Tutorial Problem A 45 45 45 45 symmetric angle ply laminate consisting of 0 25 mm thick AS 3501 carbon fibre epoxy laminae is subjected to a single uniaxial force per unit length Nx 50 MN mm Determine the mid plane strain and the resulting stresses along the x and y axes in each lamina 21 Step 1 Elastic properties The calculation of elastic properties is as described in the previous section and will not be repeated here Navigate back to the Elastic Form To change the material properties use the Database button to put the properties for AS 3501 in the material input boxes and then use the Elastic Properties button in the Change All tool to give the required laminate Now
37. he grid points an interpolation or extrapolation scheme is used to calculate results from data at the nearest grid points The rest of this section explains in detail the procedures used Data from the bridging analysis for the hole are only available for b w lt 0 25 and a lc w lt 0 75 Values for geometries outside this range have been estimated by comparison with the asymptotic values and the centre notch calculations 50 A 9 1 For points inside the macro grid of the look up table X1L X2U X1U X2U grid point 4 grid point 3 X1 X2 input point to be interpolated grid point 1 grid point 2 X1L X2L X1U X2L Fig A7 Schematic figure of a unit grid for the bicubic interpolation A bicubic interpolation is adopted in CCSM To do a bicubic interpolation within a unit grid square XiL Xz Xiu X21 Xiu Xu Xi X2u see Fig A7 The function Y and NT Oe ee es pY Past ical a cane e derivatives f Tao a ee a ge ere look up table The numbering of the corner points starts at the lower left corner and counts anti clock wise There are two steps to the interpolation described in detail in the reference listed in A 10 first obtain the sixteen quantities Cj based on the given functions and its derivatives at the four grid corners i j l 4 Next substitute the C s into the following bicubic interpolation formula for the point of interest or the input point Xj Xp 4 4 aL a
38. htforward application of the appropriate multiaxial lamina strength criterion in combination with the lamina stress analysis from the classical lamination theory Since a laminate generally has plies at several orientations the ultimate load carrying capacity of the laminate may be higher than the first ply failure The analysis of subsequent ply failure is not implemented in CCSM A 8 Failure of notched laminates In this section we describe in detail the Fleck Soutis model used to estimate the notched strength of laminates The model assumes that a microbuckle and associated delamination damage grow from the edge of a sharp notch or hole The resistance to damage can be modelled using the unnotched strength and a compressive fracture toughness of the laminate The unnotched strength may either be input directly or the Budiansky Fleck or ply strength failure criteria described in section A 6 can be used to predict this The laminate compressive fracture toughness Ke which is derived from the failure load for a panel with a sharp notch may either be taken directly from experiments or it may be predicted from data on laminates of the same material as that under consideration but with other lay ups Typical values for CFRP composites are in the range 40 50 MPavm 45 For countersunk and filled holes a simple knockdown or strengthening factors of 0 85 and 1 21 respectively are applied as suggested by Soutis and Edge 1997 These factors ar
39. inate are valid for other plies as long as they are for the same material Different properties for each lamina are allowed in CCSM you just type in the corresponding data for each click Save Data click Data Ply Nos 5 8 Ply data for plies 5 8 have been automatically filled in because the laminate type option is by default symmetric The ply geometry and property data are symmetrical about the centre line Step 3 Calculating the stiffness of the laminate At this point data for all laminae have been entered Now click the Calculate button to calculate the laminate stiffness The first 5 components represent Ex Ey Gyy Vxy and vyx for the laminate using standard notation for an orthotropic laminate The sixth component E is the appropriate elastic modulus for an orthotropic material such that G K E where G is the elastic energy release rate and K is the mode I stress intensity factor Further details of the meanings of these symbols are given in the Theory section Appendix B Section 4 2 1 covers in more detail the various controls on the Elastic analysis form In particular it explains in detail how the Ply Input Ply Editing and Change All tools can be used to speed up the input and modifications to the laminate geometry In brief the different columns in the Ply Input tool refer to different angles required for the Previous Current and Next ply Click Save Ply Data to store changes after clic
40. ive Composite Strength Modeller program The Geometry and elastic analysis form is then loaded automatically and the cursor will be blinking in the Composite name text box Step 2 Entering elastic properties data In this step laminate data and elastic properties are entered All white text boxes are input boxes and the light yellow boxes are output or information boxes Now type in the following data according to the problem AS 3501 ite O Total number of plies 8 Ply No 1 Type the current ply number into this box Angie __ Type the angle into this box in degrees In mm The total thickness box displays the total thickness of the laminate based on the entered thicknesses for each ply Lamina s Young s modulus in first fibre direction E in local lamina co ordinates in GPa a a Lamina s Young s modulus in second transverse to fibre direction E22 Nul2 Poisson s ratio v12 Nuz fs Poissonis ratio vig O c2 6 9 in plane shear modulus G12 click Save Ply At this point all necessary data for ply No Data 1 have been input Click the Save Ply Data button to save the input The ply arrangement grid is filled for each ply where data has been saved For ply Nos 2 to No 4 the input procedures will be similar Save Ply Notice that after inputting data for ply No 1 the Lamina properties and thickness text boxes still hold the data for Ply No 1 Those data because expressed in local lamina co ord
41. king on the appropriate button Step 4 Deformation analysis In our example problem we wish to find the stresses and strains in each ply This is done after calculating the laminate stiffness using the Deformation Analysis clicking on the appropriate button under the GoTo tool On the Deformation analysis form first input the applied load in this case 0 2 in the Force resultant N text box converting from MN mm to MN m The other text boxes can be left empty as these components are zero The input text boxes should be Force resultants onlaminate o2 This is the only input needed for the deformation analysis in this problem Now click the Calculate button The mid plane strains and curvatures will be calculated and shown as Mid plane strains and curvatures 3075 1934 o o o o where 0 is a very small value of the order of rounding errors The mid plane shear strain yxy Gamma xy and all curvatures are zero because the laminate is symmetric and there are no bending components Though the stresses for all plies have been calculated only stresses in one ply will be shown in the Stress State grid the first ply by default Stresses at the top middle and bottom of each selected ply are shown this takes into account the possibility of linear variation in the stress in the presence of bending At the present example the stresses in each ply is constant The bottom row sho
42. kness t of selected ply middle 46 1614 3 8386 bottom 46 1614 3 8386 Average over laminate To display the ply stresses of other plies navigate through the ply arrangement grid either clicking with a mouse or using cursor key The current ply is highlighted in this grid To view strains in local coordinates click on Strain in the Output Option box to give the microstrains in the first ply Location through Strain state for ply No 1 thickness t of selected pl top 326 1583 326 1583 3623 1884 midde 326 1583 326 1583 3623 1884 bottom 326 1583 326 1583 3623 1884 Average over laminate 4 2 5 Failure analysis Further details of the laminate failure analysis form are described in this section As well as conventional failure criteria the Budiansky Fleck Soutis criterion for compressive failure of unnotched and notched laminates is included Sections 4 2 6 and 4 2 7 give Tutorial examples for conventional and compressive failure analysis 4 2 5 1 Conventional Failure Criteria The prediction of first ply failure due to in plane stresses is a straightforward application of the appropriate multiaxial lamina strength criterion in combination with the lamina stress analysis from the classical lamination theory Details of the various failure criteria are described in Appendix B Since a laminate generally has plies at several orientations the ultimate load carrying capacity of the laminate may be higher
43. ment box to navigate through the plies either by clicking on the relevant ply or using cursor keys to see the saved material properties When all the ply data have been saved the laminate properties are calculated using the Calculate button The total thickness box displays the total laminate thickness based on the ply thicknesses entered Materials Data Input Option There are two choices for defining the elastic data for each ply The Engineering option requires conventional stiffness moduli for the lamina such as might be obtained from tests on a unidirectional laminate The stiffnesses are defined in local co ordinates i e along and transverse to the fibre direction so do not change with the ply orientation The Micromechanics option requires data stiffness about the constituent fibres and matrix and the fibre volume fraction These are used to estimate laminate properties using standard equations as detailed in Appendix B 1 1 and in Gibson 1994 Laminate Type By default when starting CCSM it is assumed that the laminate is symmetric For a symmetric option only data for plies at or above the centreline can be directly input If an unsymmetric laminate is required then the appropriate laminate type option should be chosen Subsequently 17 clicking on the symmetric button will cause all data below the centre line to be overwritten Fast input Once data for the first ply has been filled in the lamina elastic properties and
44. mpliance and stiffness matrix are given in the format detailed in Appendix B 1 1 In addition the transformed stiffness matrix Q for each ply see Appendix B 1 2 can be viewed by clicking on the corresponding row in the ply arrangement grid The selected ply is highlighted in this grid 4 2 2 Elastic Properties Tutorial Problem Determine the stiffness and compliance matrices for a 45 45 45 45 symmetric angle ply laminate consisting of 0 25 mm thick T300 934 carbon fibre epoxy laminae Find out also the transformed lamina stiffness for each ply Step 1 Activate CCSM by double clicking the CCSM icon in the Windows environment Step 2 Input data into the text boxes as following 300 934 Total number of 4 plies 19 Click the Database button See Section 4 3 for details on Click on T300 934 in the name changing database entries list to the left of the form Click the Take record as Input button automatically filled i automatically filled es automatically filled click Save Ply Data Note the change in the ply arrangement grid Ply No 2 click 45 button in the Next column Angle Automatically filled Uses the same data as for the previous ply Automatically filled a Click on Save Ply Data This should be the default action when Enter is hit on the keyboard Ply Nos 3 4 Automatically filled in for a symmetric laminate Step 3 Calculate the stiffness by clicking the Calculate button Th
45. n introductory example of the use of CCSM The chapter is intended to be followed at a computer running CCSM commands and data to type in to CCSM are listed CCSM is written in Visual Basic VB a package designed to produce especially user friendly graphical interfaces The user should work through the various forms in CCSM by following the logic of a problem For example in order to calculate the stiffness of the laminate sufficient information about each lamina should be provided first Similarly analysis of stress and strain would be meaningless without previously calculating the laminate properties and specifying the applied loads CCSM contains a number of forms for each stage of the analysis Details at each stage are filled in using text boxes containing data option buttons and command buttons 3 1 How to use CCSM The following flow chart shows what to do in CCSM In each form in CCSM corresponding to each box in the flow chart there is an information box providing information about what to do next Boxes in which to input data have a white background Further help can be obtained from the buttons and from the manual Details of the program authors are included using the About button Start CCSM by clicking the CCSM icon from within Windows Use the database lt gt if needed Elastic properties input ply properties calculate laminate stiffness Use the More Elastic Properties form to look at further lamin
46. or 010 os T 1 172 52 12 gt 2 2 2 R2 SE SE ST SIT A 32 The Tsai Hill criterion assumes that the material has equal strengths in tension and compression When tensile and compressive strengths are different modification can be made by using the appropriate value of S and Sy for the corresponding stress components For example if oj is positive and is negative the values of S 9 and S a would be used in A 32 A 5 4 Tsai Wu The Tsai Wu criterion states that failure occurs when the following relation satisfies Fo Fy05 Feet jy Fo F405 2F 70 02 gt 1 A 33 where 1 1 1 1 biis ae Loa ey 1 aa o o gt SE SE Sr Sr SE Sz 1 1 1 FiF ae ea a a F So se Sir 2 42 A 6 BFS compressive failure criterion The unnotched strength of the laminate can be predicted based on either a micromechanics model or using strength data for each lamina The micromechanics model is based on the Budiansky Fleck compressive failure analysis In this section we describe its application to a 0 lamina Section A 7 explains how this information is used to find the laminate unnotched strength This criterion assumes that the unnotched compressive strength of the lamina is governed by imperfection sensitive plastic microbuckling with the imperfection in the form of fibre misalignment Consider microbuckling from an infinite band of uniform fibre misalignment as shown in Fig A5 in a unidirectional materi
47. plied force pattern is scaled up or down to give the applied loads at failure in the output grid at the bottom of the form In this case we have Applied loads at failure 26 and the failed plies will also be marked by Fail in the ply arrangement grid al Yes oo a E 3 ozs Ye s0 02s yes s p ps s foo ozs yes Therefore according to the Tsai Hill criterion first ply failure will occur in the 90 plies with Nx 0 422 MN m Step 4 Now choose the Tsai Wu failure criterion option and re Calculate the failure loads as Applied loads at failure oso 0 T According to the Tsai Wu criterion first ply failure will occur at Nx 0 420 MN m close to the result obtained using the Tsai Hill criterion in this case 4 2 7 BFS Compressive Failure Analysis Tutorial Details of the Budiansky Fleck Soutis failure analysis are investigated in this tutorial Problem A 45 0 90 laminate consisting of 0 125 mm thick T800 924C carbon fibre epoxy laminae is subjected to a compressive uniaxial loading along the x direction Find the compressive failure stress corresponding to 0 ply failure by the Budiansky Fleck compressive failure criterion using the lamina elastic properties from the data base of E 1 161 GPa E22 9 25 GPa vj 2 0 34 G12 6 GPa and assuming the following material properties matrix shear strength k 62 35 MPa initial fibre waviness 3 microbuckle band inclined angle B 25
48. ply If strength properties have already been defined for the laminate these properties are associated with each ply and are cut and paste with the plies The total number of plies is automatically updated when plies are cut or pasted The number of plies can also be changed by entering a new number in the Total Number of Plies box Change All The materials properties and thickness of all plies can be changed at once by entering the new data in the input data box and then clicking on the appropriate button Laminate Elastic Properties Laminate stiffnesses Ex etc have the normal definitions for an elastic orthotropic laminate when the laminate is symmetric see Appendix B1 1 and B 3 for details For unsymmetric matrices due to coupling between different loads it is not possible to define stiffnesses such as Ex for the laminate in the normal sense However the inverse of the 18 appropriate element of the compliance matrix can give an effective stiffness where there is only loading in the relevant direction These are the values that are quoted Refer to Appendix B 3 for more details E is an effective elastic modulus for use in the relation GE K between the strain energy release rate G and the stress intensity factor K where mode one loading is considered and a crack runs in the y direction More Elastic Laminate Properties Form In this form further derived elastic properties of the laminate are output The laminate co
49. posite Materials 26 2706 2726 P M Jelf and N A Fleck 1994 The failure of composite tubes due to combined compression and torsion J Materials Science 29 3080 3086 J C Newman 1982 in Proceedings of the AGARD Conference on the Behaviour of Short Cracks in Airframe Components France W H Press S A Teukolsky W T Vetterling and B P Flannery 1994 Numerical Recipes in FORTRAN Second Edition Cambridge University Press JY Shu and NA Fleck 1997 Microbuckle initiation in fibre composites under multiaxial loading Proc Roy Soc A453 2063 2083 S Slaughter N A Fleck and B Budiansky 1992 Microbuckling of fibre composites the role of multi axial loading and creep J Eng Mater Tech 115 5 pp 308 313 C Soutis N A Fleck and P A Smith 1991 Failure prediction technique for compression loaded carbon fibre epoxy laminate with open holes J Comp Materials 25 1476 1498 C Soutis PT Curtis and NA Fleck 1993 Compressive failure of notched carbon fibre composites Proc R Soc London A Vol 440 pp 241 256 C Soutis and PT Curtis 1996 Prediction of the post impact compressive strength of CFRP laminated composites Composites Science and Technology 56 pp 677 684 C Soutis and EC Edge 1997 A method for the production of carpet plots for notched compression strength of carbon fibre reinforced plastic multidirectional laminates Proc Instn Mech Engrs 211 G pp 251 261 M P F Sutcliffe and N A Fleck
50. r ue A 44 i l j l xX X X X where Ue RE A 45 1U AE Xa Xz As discussed in the previous section the bridging analysis can be generally described by the following functional relationship 51 f A 46 Tp lp p R A47 Tp Tp p The bicubic interpolation formula A 44 is valid for any grid square Since the precision of the interpolation will be better for smaller grid sizes the area of interpolation is divided into smaller grids and A 44 is applied to each of these sub divided grids To avoid confusion in the description we refer to the sub divided grid as the unit grid while the total area is the macro grid The smoother the function over the grid the better is the precision of the bicubic interpolation When the functions A 46 and A 47 are plotted with log axes for a 1 and b rp they are smooth Therefore the interpolation is carried out in the log a r log b r space rather than as a function of a Ip and b Tp For the three types of specimens the macro grid covers the following unit grid points log a rp and log b rp 3 to 3 Function values of A 46 and A 47 f and g together with their corresponding first order ae Henk Of O f partial derivatives and cross derivative LE SSS Olog a rp log b rp af g g a g and at the unit log a lp P log b rp Olog a r log b r log a lp log b Tp grid points have been calculate
51. re five failure analysis criteria available in CCSM the maximum stress the maximum strain the Tsai Hill the Tsai Wu and the Budiansky Fleck Soutis BFS compressive failure criteria All these criteria are lamina failure criteria Select the Tsai Wu option for the present problem Now type in the following strength data for the material AS 3501 Which input box What you type Longitudinal tensile strength SL 1448 In plane shear strength SLT Click on the Save Data All Plies button to store this data Instead of typing the above data in each input text box you can make use of the Database menu Click on the Database button All relevant data which have been stored in the database will appear in the list box on the left side of the form Click on the required material s name in this list which is AS 3501 in the present example then click the Take record as input button You will return to the Failure analysis form with the selected strength data displayed in the appropriate input text boxes Now click on the Save Data All Plies button as before Finally we need to input the Force pattern which is applied to the laminate Enter the appropriate data in the force input boxes e g applied to laminate a P a ee es 11 Note that it is only the ratio of forces that is required here so that a pattern Force pattern applied to laminate would convey exactly the same information When the necessary data
52. re the subscript k refers to the k th ply A 5 Conventional lamina failure criteria Five lamina strengths are relevant in the lamina failure analysis They are se the longitudinal tensile strength s the longitudinal compressive strength si the transverse tensile strength si the transverse tensile strength S_ Tt the in plane shear strength A 5 1 Maximum stress This criterion predicts failure when any principal material axis stress component exceeds the corresponding strength i e failure occurs whenever one of the following holds o lt s or 0 2 sh or o2 lt S s0 or 072 si or fa Spr A 29 The maximum stress in each ply is used in equation A 29 and in corresponding equations for the other conventional failure criteria or maximum strain where appropriate This maximum stress or strain need not be at the centre of the ply A 5 2 Maximum strain This criterion predicts failure when any principal material axis strain component exceeds the corresponding ultimate strain i e failure occurs whenever one of the following holds amp lt a or E amp 2 et or E S e or E2 2 en or 12 2 err A 30 4 Assuming linear elastic behaviour the ultimate strains can be calculated by 4 SL GSi _ Sr _ Sr _ Spr er A S er a er gt er gt ELT A 30 E Ey Ey Ey G12 A 5 3 Tsai Hill The Tsai Hill criterion states that failure occurs when the following relation satisfies
53. roperties form described above The text boxes now require strength data for the material of course The database is only available for the conventional failure criteria 4 3 3 Database for the bridging analysis The large scale bridging analysis see Appendix B 8 is the underlying theory used by CCSM to 29 predict the notched composite compressive strength and microbuckle length at failure To ensure robustness and run time efficiency CCSM uses the strategy of interpolating from look up tables rather than carrying out a real time bridging analysis Details about the interpolation extrapolation are explained in sections B 8 and B 9 The look up tables for the bridging analysis are stored in text files which are accessed by CCSM when needed in a way which is transparent to the user The look up table files are CENLENG DAT CENSTRE DAT SENLENG DAT SENSTRE DAT HOLELENG DAT and HOLESTRE DAT These files should not be changed or deleted 30 Appendix A Theoretical Background Appendix A describes the theoretical background behind the CCSM First the classical laminate theory is introduced A 1 A 4 This theory is used for the calculation of the stresses and strains of the laminated composite Four conventional failure criteria for the orthotropic lamina A 5 and the Budiansky Fleck compressive failure criteria A 6 are then described which when combined with the laminate theory leads to the failure analysis of a laminate on the ply b
54. ted geometries the program may be used as part of a larger calculation to check for failure at critical points in the structure CCSM is written in Microsoft Visual Basic language using Visual Studio 2012and it runs in the Microsoft Windows operating system It is structured so that the user with a basic knowledge of composite mechanics can use it with little reference to the manual However the user would benefit from reading through this manual in particular the Quickstart section chapter 3 and the detailed guide chapter 4 CCSM 2 0 updates and simplifies Version 1 4 This manual consists of the following chapters 1 Introduction 2 Installation Instructions on how to install CCSM 3 Quick Start A quick start self sufficient chapter illustrating how to use the CCSM package 4 Detailed guide Detailed guide to all aspects of the program with tutorials Appendix A Theoretical Background underlying principles of CCSM 2 Installation 2 1 System requirements CCSM uses Microsoft NET Framework 4 or above which needs to be installed before running the application 2 2 The CCSM package The CCSM deployment file contains the following items User s manual A Quickstart manual Application and data files 2 5 Setting up CCSM To install CCSM run the setup exe program If you want to change the database then you will need write permission for CCSM MDB 3 Quick Start This chapter contains a
55. than the first ply failure The analysis of subsequent ply failure is not implemented in CCSM After choosing your failure criterion ply data should first be entered in the appropriate data boxes or using the Database Where the failure strengths are the same for each ply click the Save Data All Plies button For a laminate made up of different materials individual strength 23 data for each ply can be entered using the Save Data This Ply button The strength of each ply can be examined by clicking and navigating through the ply arrangement table You must specify the load pattern the ratio of all non zero force components The absolute values of these components are not important The failure analysis is performed by clicking Calculate CCSM will find out the proportionality factor for failure scaling the forces accordingly The failure loads are given below the input force pattern The ply grid identifies plies which fail first 1 e at the failure load 4 2 5 2 BFS Failure Criteria The model of unnotched strength used in CCSM assumes that failure occurs in the 0 plies by plastic microbuckling The notched strength of the composite is then predicted using the Fleck Soutis model of microbuckle growth giving the longitudinal axial stress or strain of the laminate at failure and the length of the microbuckled region emanating from the end of the notch at this critical peak failure load While the analysis predicts the f
56. than the force pattern is used as input Because the BFS criterion is a compressive one the axial stress Sigma_L must be negative Again the 24 absolute values of these components are not important A typical input pattern would be Sigma_L 1 Sigma_T 0 Tau 0 Components of bending are not modelled in the BFS compressive failure theory Failure can be output in terms of stresses or micro strains using the Output Option The failure analysis is performed by clicking Calculate The laminate strength is given when the stresses in a 0 ply exceed the lamina failure stresses The laminate unnotched strength is printed on the right in the middle of the form The ply grid identifies the plies which fail first Notched Strength The Fleck Soutis model of notched strength assumes that a microbuckle and associated delamination damage grows from the edge of a sharp notch or hole The resistance to damage can be modelled using the unnotched strength and a compressive fracture toughness The laminate unnotched strength can be input directly after changing the Strength Input Options or predicted as described in the previous section The fracture toughness K is measured using centre notched coupon specimens Typical values for CFRP composites are in the range 40 50 MPavm Notched strength inputs The notch geometry type is chosen using the Geometries option Centre or single edge notches and open equivalent countersunk or filled hol
57. the lamina strength option and geometry type chosen in the failure form As for the failure analysis only failure in the 0 degree plies is considered The Soutis Fleck bridging analysis is used to calculate the notched strength The toughness is assumed to be independent of ply mix and is taken from the Failure Analysis form The percentage of 0 degree plies is varied from 10 to 90 Elastic and strength data for the first ply are used throughout the laminate As with the standard BFS compressive failure analysis care should be taken in interpreting results at extremes of ply mix as the BFS compressive failure criterion may be inappropriate Soutis and Edge 1997 give further details of carpet plot calculations In the rest of this section the details of the method used to calculate the notched strength are outlined The method involves crack bridging models which have had notable success in the prediction of damage from notches in engineering materials under remote tension They have also been used to estimate the development of microbuckling from a hole in a composite under compression The usual strategy is to concentrate the inelastic deformation associated with plasticity cracking microbuckling and so on within a crack and to assume some form of traction displacement bridging law across the crack faces As a simple example in Dugdale s analysis of plastic yielding in metals from the root of a notch the bridging normal traction across the cr
58. thickness become the default for subsequent plies However the ply angle needs to be entered for each ply and all the data needs to be saved The Ply Input buttons give a fast method of inputting this data The columns refer to the ply position relative to the current one as highlighted in the ply arrangement grid The row denotes the ply angle Thus clicking on the Next column and the 90 row increases the number in the Ply Number data box by one and puts 90 in the Ply Angle data box Now the data can be saved either by clicking the Save Ply Data button or by hitting the Enter key on the keyboard the default action for the Enter key in this case the Save Ply Data button is highlighted on the screen The data boxes for ply number and angle can speedily be changed using the Previous current or Next columns repeatedly clicking on the appropriate column to increment or decrease the ply number as required and keeping track of the current ply using the ply arrangement highlighting When the required ply number and angle have been put in the input data boxes this information plus the materials and thickness data can be saved using the Save Ply Data button Ply Editing Use the ply arrangement box to navigate through the plies using either the mouse or cursor keys The saved material properties for the highlighted ply are displayed Several plies can be selected for cutting or deleting by dragging with the mouse Plies are pasted above the selected
59. ure associated with bending of the middle surface in the yz plane and 37 Kxy s the twisting curvature associated with out of plane twisting of the middle surface which lies in the xy plane before deformation Since Eqs A 15 give the strains at any distance z from the middle surface the stresses along arbitrary xy axes in the k th lamina of a laminate may be found by substituting Eqs A 13 into the lamina stress strain relationships from Eqs A 9 as follows Fy Yr Fy oO Oy Qi Q2 De amp 2x n pag ra oO Sy Q12 O22 Dre A ETRE A 18 Try G16 226 6 e ZKyy Fig A4 Laminated plate geometry and ply numbering system It is convenient to use forces and moments per unit length in the laminated plate analysis The magnitude of these forces may be clear where the geometry of the component is relatively simple Where the component is more complex CCSM should be used as part of a larger calculation which finds the forces throughout the structure to assess failure of critical sections The forces and moments per unit length shown in Fig A3 are referred to as stress resultants The force per unit length in the i th direction N is given by i x y zZ t 2 Nn 2 Nj foidz 2 f oi kdz A 19 t 2 k 1 eye and the moment per unit length M is given by t 2 N zy oi kM Jojzdz 4 f oi zdz A 20 t 2 k 1 es 38 where t laminate thickness 0 i th stress component in the k th lamina
60. vanced tutorials for each form in CCSM To help use the full capability of CCSM it is suggested that you go through these tutorials Finally section 4 3 contains details of the databases used by CCSM 4 1 Nomenclature convention in CCSM This section describes in detail the nomenclature used in CCSM Global co ordinate the two co ordinates of the global laminate system are denoted by x and y The first axis of the global co ordinate system the x axis coincides with the fibre direction of the 0 degree plies Local co ordinate the two axial directions of the local co ordinates of a lamina are denoted by 1 and 2 The first axis of the local lamina co ordinate system the 1 axis coincides with the fibre direction of the lamina The sign convention for lamina orientation with relation to the global co ordinate is illustrated in Fig 1 The stress resultants used for the laminate analysis are defined in Fig 2 The laminated plate geometry and ply numbering system is illustrated in Fig 3 The nomenclature for the Budiansky Fleck Soutis compressive failure theory is illustrated in Fig 4 For the compressive failure case the stress in the longitudinal fibre direction 6 will be negative The definitions of a or R b and w for the three types of specimens for the bridging analysis are shown in Fig 5 Note that for the centre notch panels a and w are the semi notch length and the semi width of the specimen 13 0 NY
61. ws the average stress through the full thickness of the laminate The results are shown as Through Stress state for ply No 1 thickness t of selected Sigma_x Sigma_y Tau_xy pl 421 697 si o O 421 697 sis o O 421 697 sss o O Laminate 250 stresses These stresses are in global coordinates i e running along the x and y directions of the laminate To see the ply stresses in local coordinates i e running along and transverse to the fibre direction in each ply click on the appropriate Output Option To display the ply stresses of other plies navigate through the ply arrangement grid either clicking with a mouse or using cursor keys The current ply is highlighted in this grid Notice that in the Output Option box the stress option is set as the default Clicking the strain option will display strain data giving Through Strain state for ply No 1 thickness t of selected Epsilon_x Epsilon_y Gamma_xy ply top 3075 1934 3075 1934 3075 1934 Laminate strains 10 Note that strains are output in microstrain It can be seen that there are stress discontinuities at the ply interfaces while the strains are continuous across the interface this reflects the basic assumption of the classical laminate theory Step 5 Failure analysis To predict laminate failure click on the Failure Analysis button once the laminate properties have been entered and the laminate stiffness calculated There a
62. y ply basis A 7 The bridging analysis which deals with the failure analysis for the notched laminate or laminate with a hole is described in section A 8 The numerical implementation of the bridging analysis results is described in A 9 A list of the references applicable to each of the theory sections is given in A 10 In particular most of the material about the laminate theory comes from Gibson s book Principles of Composite Material Mechanics referenced in A 10 A 1 Lamina stress strain relationships A 1 1 The orthotropic lamina As shown in Fig Al a unidirectional composite lamina has three orthogonal planes of symmetry i e the 12 23 and 13 planes and is called an orthotropic material The coordinate axes in Fig Al are referred to as the principal material coordinates since they are associated with the reinforcement directions Expressed in terms of engineering constants the stress strain relationship for a three dimensional state of stress is amp MVE v2 E v31 E3 0 0 0 Ilo E2 v25 1 E V39 E3 0 0 0 o2 amp 3 _ v3 E Vv23 E 1 E3 0 0 0 o yzl 0 0 0 1 Go3 0 0 Ilr Gp 731 0 0 0 0 1 G3 0 tz n2 0 0 0 0 0 1 G pj la2 31 Fig Al Orthotropic lamina with principal and non principal coordinate system where E E2 and E are the elastic moduli and v j ej g are the Poisson ratios Note that the following relationship holds V V Z no sum on i j A 2 i Ej In pra

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