Numerical Modeling of Woven Carbon Composite Failure

Transcription

1 8 th Internatonal LS-DYNA Users Conference Smulaton Technology (3) Numercal Modelng of Woven Carbon Composte Falure Paul F. Deslaurers, Duane S. Cronn Unversty of Waterloo Alex Duquette Multmatc Techncal Centre Abstract Ths paper presents applcaton of a MLT-based (Matzenmller, Lublner, Taylor) approach to model damage n woven carbon composte materals. The MLT formulaton has been adapted to shell elements to model ndvdual composte ples. The mplementaton of the model s dscussed along wth smple test cases to demonstrate the materal response and lmtatons wthn the orgnal MLT model. One of these lmtatons has been addressed through mplementaton of dfferent damage parameters for tensle and compressve loadng. In addton, ths damage-based approach has been modfed by the use of a non-local damage treatment to dstrbute accumulated damage across element boundares. Applcaton of ths model to smple test cases ndcates that the model demonstrates expected behavour. Introducton Numercal modelng of damage n composte structures s of sgnfcant nterest as these materals are now commonly used for structural and energy absorpton applcatons. However, the consttutve descrpton of these materals s not trval due to the varous damage mechansms whch contrbute to the materal response. A model based on the contnuum damage mechancs (CDM) approach frst proposed by Matzenmller et al. (1995) to descrbe the accumulaton of damage n composte materals has been consdered. Several authors have nvestgated ths approach. In partcular, Wllams et al. (2000) thoroughly dscusses the orgns of CDM and ts applcaton to numercal analyss of composte materals, ncludng the MLT (Matzenmller, Lublner, Taylor) approach. A verson of ths model was prevously mplemented n LS-Dyna as a user materal model for sold elements, and was successful n the smulaton of ballstc composte response and damage to mpact (van Hoof, 1999, Gower, 2003). In ths stuaton, the transverse or through-thckness response of the materal s mportant and requres the use of sold elements. Ths level of detal s acceptable when the area of nterest wthn the composte s small. However, when consderng real structures, such as composte crush structures for energy absorpton, ths level of detal s not feasble and more computatonally effcent shell elements must be consdered. It s mportant to note that the level of detal wthn the fnte element model must also be represented n the consttutve model. For example, the use of multple sold elements through the thckness of each composte ply allows for a model to descrbe response and damage at the sub-ply level. In contrast, a sngle shell element could be used to represent multple composte ples. In practce t has been found that the former s computatonally too expensve and the latter overly smplfes the complex materal response of a damaged composte. As such, an mplementaton of the MLT approach for a shell element representng a sngle composte ply has been undertaken. It s antcpated that delamnaton n the composte wll be modeled through tes between adjacent ples

2 Smulaton Technology (3) 8 th Internatonal LS-DYNA Users Conference Consttutve Model Implementaton for Shell Elements The MLT damage approach s based on the premse that damage s accumulated wthn a materal based on deformaton and loadng n varous drectons. In the case of shell elements, the damage s calculated n the longtudnal (tensle/compressve), transverse (tensle/compressve), and n-plane shear drectons. Van Hoof et al. (1999) developed equatons (1) and (2) based on the work of Matzenmller et al. (1995) and Wllams et. al. (1995) to descrbe the onset of damage for a partcular damage mode. The subscrpt corresponds to the loadng drecton (1 = longtudnal, 2 = transverse, and 4 = n-plane shear). It s mportant to note that the damage threshold f s calculated n the current tme step (t) whle r s calculated n the prevous tme step (t- t). f r ε m = ε falure, t Λt ε = ε falure, m r t t (1) (2) When the damage threshold (f ) of an element s greater than zero, the element begns to accumulate damage. The damage accumulaton rate of the element for a partcular damage mode s gven by equatons (3) and (4) below, also developed by Matzenmller et al. (1995) and later mplemented by van Hoof et al. (1999). g (1 ω ) ε ε rate, = e εfalure, εfalure, ( m 1) (3) ω = ( g + g )( t ) (longtudnal) ω = ( g + g )( t ) (transverse) ω = ( g + g + g )( t ) (n-plane shear) (4) The couplng of damage n the longtudnal, transverse, and n-plane shear drectons s evdent n equaton (4). In ths manner, an element undergong longtudnal stran wll accumulate damage n both the 1- (ω 1 ) and 4-drectons (ω 4 ). It s mportant to note that n equatons (1) to (4) damage s treated dentcally n the longtudnal and transverse drectons, for the purposes of smulatng a woven composte lamnate. Ths treatment dffers from that used prevously by Matzenmller et al. (1995) and Wllams et al. (2000) n whch longtudnal damage was not coupled wth any other damage modes, and transverse damage was coupled wth shear damage. The representatve damage varables, ω, are then used to reduce the materal stffness n the correspondng drectons. Equaton (5), frst proposed by Matzenmller et al. (1995) and later mplemented by Wllams et al. (1995) and van Hoof et al. (1999) for sold elements, defnes the reduced stffness matrx for a shell element and shows that the element stffness s predcted by the conventonal elastc constants scaled by the damage varables

3 8 th Internatonal LS-DYNA Users Conference Smulaton Technology (3) σ1 (1 ω1) E1 (1 ω1)(1 ω2) ν12e2 0 ε1 1 σ = (1 ω )(1 ω ) ν E (1 ω ) E 0 ε D τ (1 ω4) G 12 γ 12 (5) where ν 21 ν 12 = E2 E1 D = 1 (1 ω )(1 ω ) ν ν > Equatons (1) through (5) detal the manner n whch an element accumulates damage and undergoes a reducton n stffness. The rate at whch damage accumulates (and the rate at whch stffness s degraded) s thus a functon of the materal parameters (most notably the falure strans) and the exponents m. Effect of the Damage Exponent m In general, the materal parameters requred to descrbe a composte are avalable from varous publcatons, ncludng materal data publshed by the manufacturers for varous fbre/matrx combnatons. Ths ncludes modulus, falure strength and Posson s rato. However, determnaton of the damage exponents m requres more attenton. The effect of the exponent m on the stress-stran response of an element s shown n Fgure 1. Ths exponent determnes the brttle/ductle response of the element. Hgh values of m cause a stress-stran response smlar to a brttle materal, wth lttle or no loss n stffness pror to falure and full damage correspondng to zero stffness shortly after falure. Low values of m descrbe a materal that absorbs more energy pror to complete damage, wth sgnfcant stffness degradaton pror to falure and a more gradual loss of stffness after falure. Fgure 1: The effect of the exponent m 11-35

4 Smulaton Technology (3) 8 th Internatonal LS-DYNA Users Conference As dscussed by Wllams et al. (2000) the selecton of m s dffcult as t has been found to be a functon of the materal, loadng rate, and element sze. These are common ssues when consderng damage-based consttutve models and are addressed n more detal below. In studes of ballstc mpact on woven Kevlar compostes, van Hoof et al. (1999) found that an exponent value of m = 8 provded a reasonable predcton of the materal stress-stran response. Specfcally, ths value of m resulted n relatvely brttle behavour (Fgure 1), whch represented the materal consdered at hgh rates of stran. In smlar studes of mpact on undrectonal CFRP lamnates, Wllams et al. (2000) acheved good correlaton wth expermental results at low to medum mpact energy usng an exponent value of m = 10. In the same study t was also found that at hgher mpact energes an exponent value of m = 20 provded better correlaton than a value of 10, hghlghtng the dependence of m on the loadng rate. Consttutve Model Implementaton The materal model outlned above has been mplemented nto a FE code as a user-defned materal model n LS-DYNA v The followng secton outlnes the model and the predcted results. Examples of the model performance are provded usng a smple sngle element test case. Defnton of Materal Propertes The materal parameters that defne the behavour of the materal are shown n Table 1. These propertes correspond to publshed values for a common 2x2 twll-weave pre-mpregnated carbon/epoxy fabrc. In ths case, the falure strans requred for equatons (1) to (4) were calculated from the strengths lsted below. The damage exponents specfed can be dfferent n each materal drecton (local 1, 2, or 4), but were set equal to 10 for the ntal studes followng the recommendatons of Wllams et al. (2000) and van Hoof et al. (1999). Table 1: Input parameters for ACG CFS003/LTM25 for shell elements. [Cruz et al., 1996] Parameter Symbol Value Descrpton E GPa Modulus of elastcty n the longtudnal (local 1) drecton. E GPa Modulus of elastcty n the transverse (local 2) drecton. G GPa Modulus of elastcty n the shear (local 4) drecton. ν Posson s rato. σ f1t MPa Tensle strength n the longtudnal (local 1) drecton. σ f1c MPa Compressve strength n the longtudnal (local 1) drecton. σ f2t MPa Tensle strength n the transverse (local 2) drecton. σ f2c Mpa Compressve strength n the transverse (local 2) drecton. τ MPa Shear strength n the n-plane (local 4) drecton m 1 10 Damage exponent n the longtudnal (local 1) drecton m 2 10 Damage exponent n the transverse (local 2) drecton m 4 10 Damage exponent n the n-plane (local 4) drecton 11-36

5 8 th Internatonal LS-DYNA Users Conference Smulaton Technology (3) Consttutve Model Response A sngle element was used to verfy the behavour of the user-defned materal model. The predcted stress/stran and damage/stran curves of the element subjected to monotonc tenson and compresson (separately) are shown n Fgure 2. It s evdent that damage rapdly accumulates after approxmately 1% stran, and softenng of the materal s evdent at slghtly hgher strans. It s mportant to note the dfferent falure stresses (and strans) n tenson and compresson correspondng to the values n Table 1. The equvalence of the damage varables ω 1 and ω 4 s a consequence of equaton (4). The behavour of the 1-element model matches the expected response from the consttutve equatons for unaxal tenson and compresson. Fgure 2a: Stress vs. stran output of 1-element model wth exponent m = 10. Fgure 2b: Damage vs. stran output of 1-element model wth exponent m =

6 Smulaton Technology (3) 8 th Internatonal LS-DYNA Users Conference Lmtatons of the MLT Consttutve Model The most sgnfcant lmtaton of the MLT consttutve model s the couplng of ntal modulus wth post-falure deformaton. The use of a hgh value for the exponent m provdes an essentally lnear elastc materal that fals n a brttle manner, wth lttle or no post-falure stffness (Fgure 1, m = 100). In ths case, the ntal modulus of the materal s accurately represented, but no allowance exsts to ncorporate post-falure load-carryng capablty. A low value of the exponent m represents a more ductle response that may undergo sgnfcant deformaton after the onset of damage whle stll sustanng load. However, the elastc modulus devates grossly from the nput modulus pror to falure (Fgure 1, m = 1). A related ssue s the use of dentcal values of the exponent m for tensle and compressve calculatons. As shown n Fgure 2, the materal response s essentally dentcal n tenson and compresson (wth the excepton of falure stress and stran). For many composte materals such a response s not realstc. A thrd ssue wth the MLT approach s related to unloadng of the materal. Some examples of the stress/stran response to a partal load (and subsequent removal of load) are shown n Fgure 3. Pror to the accumulaton of damage, the model may be loaded and unloaded elastcally. However, once damage has been generated n the model, the element wll unload lnearly (at the reduced stffness) along a lne that ntersects the orgn on a stress vs. stran dagram. Due to ths behavour, the model cannot accurately predct the permanent deformaton of a partally damaged composte materal. Ths s an ssue f unloadng of the materal s encountered and f ths unloadng contrbutes to the overall response of the structure. However, n many mpact stuatons, the deformaton s monotonc and ncreasng such that the unloadng phase does not affect the general response of the model Fgure 3: Stress vs. stran output of 1-element model showng (1) monotonc tenson and (2,3) partal tenson and subsequent unloadng. Fnally, although t s not evdent n the presented 1-element model, localzaton of damage s a known problem assocated wth most CDM approaches. As dscussed by Wllams et al. (2000),

7 8 th Internatonal LS-DYNA Users Conference Smulaton Technology (3) localzaton of damage leads to a dependence of the numercal soluton on the mesh densty, usually wthout convergence to a unque soluton. Ths s an mportant aspect from a numercal modelng perspectve snce there s a desre to mnmze computaton tme by maxmzng element sze. Further, real structures are necessarly complcated n geometry, whch leads to non-unform element sze n varous regons. Whle the couplng of pre- and post-falure response and reduced unloadng stffness are fundamental characterstcs of the consttutve model, some of the ssues wth the MLT model can be addressed drectly. The focus of the current research s to address the lmtatons n the orgnal MLT model and mprove ths model for applcaton to predct the response and damage of composte structures. Asymmetrc Tensle and Compressve Behavour It s well known that, at a ply level, the tensle and compressve damage response of a materal may dffer. Ths leads to the need for asymmetrc (n tenson and compresson) values of the exponent m, whch has been ncorporated nto the consttutve descrpton. An example of the modfed model s shown n Fgure 4. In ths case, the exponent m = 10 n tensle loadng whle n compressve loadng m = 5. In ths manner any combnaton of exponents could be used to better characterze the behavour of a gven materal. Fgure 4: Stress vs. stran response of 1-element model usng asymmetrc values of the exponent m. Non-Local Damage Dstrbuton As ndcated above, damage-based models have an nherent element sze dependency. Ths s mportant from a practcal standpont snce t requres a fnte element mesh of constant sze be used to model a component. Theoretcally, ths also mples that fcttous boundares (element edges) exst whch contan or confne the materal damage. In order to reduce the effects of localzaton of damage (and mesh dependency), a non-local damage dstrbuton functon has been mplemented wth the composte model. The functon, *MAT_NONLOCAL n LS-DYNA 11-39

8 Smulaton Technology (3) 8 th Internatonal LS-DYNA Users Conference v. 970 has been used to dstrbute the materal damage over a representatve volume of materal, specfed by the user. Although ths method s meant to be appled to elements wth szes approachng the length scale of the materal (e.g. materal gran sze n metals, or repeatng unt cell sze n woven compostes), t can also be used practcally at larger element szes to reduce mesh sze dependency. A detaled descrpton of the equatons and typcal parameter values can be found n the LS-DYNA Keyword User s Manual (2003). In the current model the non-local treatment s appled to the damage varables ω and the prmary nput of nterest s the radus, L, whch defnes the area over whch the functon s appled. A two-element model can be used to llustrate the effect of the non-local treatment, shown n Fgure 5. The dsplacement of all nodes s prescrbed to create a sub-falure stran n the lower element and a super-falure stran n the upper element, and to prevent any nteracton of the strans that mght affect the accumulaton of damage. Fgure 5: Schematc dagram of 2-element model. Fgure 6 shows the damage n the 1 (longtudnal) and 2 (transverse) drectons wthout the use of non-local damage treatment. Fgure 7 shows the same results wth non-local damage treatment enabled, usng a value of L such that the rato L/L e = 1 (where L e = element length). In both cases the shear damage s not shown, as t s dentcal to the longtudnal damage due to the damage couplng descrbed prevously. Comparng Fgures 6 and 7, t can be seen that damage s dstrbuted from the upper element to the lower element n all drectons, reducng the effect of element boundares wthn the fnte element mesh. Note that the ncreased damage n the longtudnal drecton (and the resultant reducton n stffness) causes a reducton n the Posson s contracton of the element, whch leads to a reducton of the transverse stran and damage

9 8 th Internatonal LS-DYNA Users Conference Smulaton Technology (3) Fgure 6: Damage vs. tme for 2-element analyss wthout non-local damage treatment. Fgure 7: Damage vs. tme for 2-element analyss wth non-local damage treatment (L/L e =1). Summary The consttutve model frst proposed by Matzenmller, Lublner, and Taylor has been adapted to a shell formulaton and mplemented nto LS-DYNA as a user-defned materal model. The man lmtatons of the consttutve model are the couplng of pre- and post-falure response, the couplng of tensle and compressve propertes, the elastc unloadng of the materal followng partal damage, and the mesh senstvty cause by the tendency for localzaton of damage. Through modfcaton of the consttutve relatonshp, a modfed composte model allows for the use of separate damage exponents m n tenson and compresson, effectvely de-couplng these responses. In addton, the use of non-local damage treatments avalable wthn LS-DYNA provdes a means of reducng mesh senstvty. Acknowledgements The authors fully acknowledge the fnancal and techncal support provded by the Natural Scences and Engneerng Research Councl of Canada and Multmatc Techncal Centre

10 Smulaton Technology (3) 8 th Internatonal LS-DYNA Users Conference References [1] Cruz JR, Shah, CH, Postyn AS. Propertes of Two Carbon Composte Materals Usng LTM25 Epoxy Resn. NASA Techncal Memorandum NASA, Langly Research Centre, Vrgna. November [2] Gower H L. Ballstc Impact Response of Woven Kevlar Compostes. Masters Thess, Unversty of Waterloo, Waterloo, Ontaro, Canada [3] Lvermore Software Technology Corporaton. *MAT_NONLOCAL. LS-DYNA Keword User s Manual, Verson 970. Aprl 2003; [4] Matzenmller A, Lublner J, Taylor RL. A consttutve model for ansotropc damage n fber-compostes. Mechancs of Materals 20 (1995); [5] Wllams KV, Vazr R, Floyd, AM, Poursartp A. Smulaton of Damage Progresson n Lamnated Composte Plates Usng LS-DYNA. Proceedngs of the 5 th Internatonal LS-DYNA Conference. Southfeld, Mchgan. September [6] Wllams KV, Vazr R. Applcaton of a damage mechancs model for predctng the mpact response of composte materals. Computers & Structures 79 (2001); [7] van Hoof J. Modellng of Impact Induced Delamnaton n Composte Materals. Doctoral Thess, Carleton Unversty, Ottawa, Ontaro, Canada. October