Delamination Prediction of Uni-Directional Composite Laminates using Shell Elements and a Strain Rate Dependent Micro-mechanical Material Model

Transcription

1 14 th International LS-DYNA Users Conerence Session: Coposites Delaination Preiction o Uni-Directional Coposite Lainates using Shell Eleents an a Strain Rate Depenent Micro-echanical Material Moel Saneep Meikona 1, Ala Tabiei 1 1 University o Cincinnati, Cincinnati, OH, USA Abstract The eectiveness o stuying inter-lainar elaination in coposites with the help o newly orulate thickness-stretch shell eleents (ELFORM=5 as copare to the traitional plane-stress shell eleents (ELFORM= has been investigate using LS-DYNA. A strain-rate epenent icro-echanical aterial oel using ply-level progressive ailure criteria has been use to siulate the initiation an propagation o elaination. The nuerical elaination growth has been qualitatively analyze against the experiental C-scan iages or ultiple ipact events on a T800H/3900- CFRP plate. As an aition to the capability o the icro-echanical aterial oel, a ethoology o assigning physical signiicance to the choice o aage paraeters has been presente. Keywors: Uniirectional coposites; Micro-echanical oel; Continuu aage echanics; Delaination; Finite Eleent Metho; LS-DYNA 1. Introuction Delaination is an inter-lainar ailure oe that is critical in characterizing the overall response o Uni-Directional Coposites (UDC. Preicting elaination ailure accurately in LS-DYNA using stanar shell eleent orulations is a challenge as this ailure oe is epenent on the stresses in the through-thickness irection. Historically, LS-DYNA [1] has only peritte the use o stanar shell eleent orulation where the Z-Stress is zero. However, recently evelope thickness-stretch shell orulations (ELFORM = 5, 6 which utilize 3D constitutive oels are believe to siulate the elaination behavior ore accurately because o the presence o a non-zero Z-stress. In the current work, this behavior has been teste using a strain-rate epenent icro-echanical constitutive oel [3] with progressive post-ailure criteria an is ipleente as an UMAT (user-aterial oel in LS-DYNA. The nonlinear, strain-rate epenent behavior o the resin in this oel is capture using the oiie Golberg-Stouer visco-plastic constitutive relations [4]. As iscusse in [5], it is well known that the strain sotening aage paraeter which accounts or the progressive post-ailure behavior o the coposite is iicult to eterine. Hence, a proceure o characterizing the aage variables ro the strain energy release is iscusse. The current paper is organize as ollows: section iscusses the theory behin the icroechanics o the uni-irectional coposite aterial oel. Section 3 presents a paraeter estiation proceure or the aage variables. Section 4 copares the elaination results with the experiental results ro Willias an Vaziri et al. []. Lastly, section 5 conclues with the learnings ro this work. June 1-14,

2 Session: Coposites 14 th International LS-DYNA Users Conerence. Micro-echanics o the uni-irectional coposite (UDC The representative volue cell (RVC use to evelop the icro-echanical relations is shown in igure (1. The current RVC is the sae as the one iscusse by Tabiei an Babu [3]. However or copleteness the icro-echanics relations are briely iscusse here. The ibers are assue to be o square cross-section or coputational eiciency since this oel is ipleente in an explicit FE coe which uses very sall tie steps or siulations. The unit cell is ivie into three sub-cells: one iber sub-cell, enote as, an two atrix sub-cells, enote as M A an M B respectively. The three sub-cells are groupe into two parts: aterial part A consists o the iber sub-cell an the atrix sub-cell M A, an aterial part B consists o the reaining atrix M B. The iensions o the unit cell are 1 1 unit square. The iensions o the iber an atrix sub-cells are enote by W an W respectively as shown in igure (1 an eine as shown below: W V ; W 1 W.. (1 where, V is the iber volue raction. As explaine in section.5 below, eective stresses in the RVC are eterine ro the sub-cell values in two phases: irst, stresses in iber an atrix M A are cobine to obtain eective stresses in part A which are then cobine with stresses in atrix M B to obtain the eective RVC stresses. Figure (1: A representative volue cell o uniirectional iber reinorce polyer coposite.1. Viscoplastic Constitutive Relations or Matrix Material The strain rate epenent behavior o a polyer atrix coposite is ainly attribute to the viscoplastic nature o the resin coponent. Hence, strain-rate epenency is incorporate in the current oel by using the viscoplastic relationship evelope by Golberg an Stouer [4] or the atrix constituent (sub-cells M A an M B. Golberg an Stouer evelope this constitutive relationship or resins using the state variable approach an use it in their aterial oel or uniirectional coposites. They eine their state variable as an internal stress, which evolve with stress an inelastic strain an represente the average eects o the eoration 1- June 1-14, 016

3 14 th International LS-DYNA Users Conerence Session: Coposites echaniss. For copleteness, the Golberg-Stouer relations are iscusse briely in this section. Further etails about the relations can be oun in [4]. The total strain rate is assue to be the su o elastic an inelastic strain rates. The elastic strain rate is equal to the ratio o stress rate to Young s oulus o the aterial while the inelastic strain rate is eine to be proportional to the exponential o the overstress, the ierence between the applie stress an the tensorial internal stress state variable. It is given by the relation: where I, n 1 S I Z 0 D0 exp 3K.. ( K are the coponents o inelastic strain rate, an internal stress respectively, D 0 is a scale actor representing axiu inelastic strain rate, n is a variable which controls rate epenence o the eoration response, Z 0 represents the isotropic, initial harness o the aterial beore any loa is applie, S are coponents o the eviatoric stress tensor given by the relation: kk S.. (3 3 where, are the coponents o stress, an is Kronecker s elta. K in equation ( is eine as an eective stress given by the relation: 1 K S S.. (4 an represents the secon variant o the overstress tensor. The proceure or eterining the resin aterial constants can be oun in [4]. The internal stress rate is given by the relation: where,, an strain rate respectively an q 3 q June 1-14, I I e.. (5 I are coponents o internal stress, internal stress rate, an inelastic I e is eective inelastic strain rate given by the relation: I I I e.. (6 3 It is to be note that equations ( through (6 actually orulate one ierential equation per coponent o the tensorial paraeters involve or one irst orer tensorial ierential equation which has no close or solution. Hence, a nuerical solution is obtaine at each tie step o the explicit FE siulation by integrating using the 4 th orer Runge-Kutta etho... Constitutive Relations o Fibers The ibers are assue to be linearly elastic aterials which are initially transversely isotropic but becoe orthotropic with aage evolution. It is assue that aages to the ibers are a result o irect stresses applie on the only an that shear stresses o not cause any

4 Session: Coposites 14 th International LS-DYNA Users Conerence aages. The aages are assue to be oriente in the aterial irections o the ibers an inepenent. The constitutive relations o the ibers can be written in atrix or as: C.. (7 where, C is the stiness atrix which can be partitione into irect an shear stress stiness atrices as ollows: 1 1 S 0 C S 1.. (8 0 S s The irect stress copliance atrix, whose inverse is the irect stress stiness atrix, shoul be syetric an the ollowing relationship shoul be obeye: ji, i, j 1,,3 an i j (no suation.. (9 E E i j The irect an shear stress copliance atrices in ters o the properties o the ibers are: (1 1 E1 (1 1 E1 (1 E (1 1 E1 (1 3 E 1 3 S (1.. (10 E (1 E(1 3 E 1 Sy. (1 3 E G1 1 S s 0.. (11 G 3 1 Sy. Go 1 where E 1, E are the longituinal an transverse ouli o the ibers respectively,, i, j 1,,3 an i j, are its Poisson s ratios, G o1, G 1 are its initial an strain-rate epenent in-plane shear ouli respectively, G 3 is its transverse shear ouli, an i, i 1,, 3, are aage paraeters given in the ollowing section on progressive ailure oeling. In-plane shear is consiere as a paraeter which is strain-rate epenent. Hence, the shear oulus G 1 is given by the ollowing relation: G a s G.. (1 1 G s o1 t 1 1 s s log t.. (13 t 0 o where a G is a paraeter which expresses the strain-rate sensitivity o G 1, t is the tie elapse, G o1 is the initial in-plane shear oulus o the ibers, 1 is the in-plane shear strainrate, o is a basic strain-rate with which the current strain-rates are copare an is accepte as the strain-rate o the static loaing or a given working strain-rate range. A tie integration o the strain rates is eee necessary as they are not constant in ipact siulations an also because the stress-strain relationship or the ibers is base on their secant stiness an not the tangential stiness. 1-4 June 1-14, 016

5 14 th International LS-DYNA Users Conerence Session: Coposites.3. Daage Evolution in Constituents In the current oel, aage growth is base on a Weibull istribution o strengths which is coonly associate with the strength o ibers. The evolution unction or aage escribing iber breakage at tie step n + 1, is expresse as ollows: 1 ( 1 1 E1 n 11 ( n 1 in 1 exp, 1.. (14 1e 1t c where, t c enotes tension or copression. When 0, the paraeters or tension are 11 utilize otherwise the paraeters provie or copression are use. When the aage 1 reaches 0.01 in tension, the inite eleent is consiere totally aile. is not siply the strength o the ibers but is a reuce value given by the relation: X t c 1t c.. (15 b where, t c X t c is the tensile/copressive strength o the pure ibers, an t c 1t c b is a reuction actor. The physical justiication or using such a actor is the act that ibers generally isplay reuce strengths in uniirectional coposites as evience by the lower strength o the latter copare to the orer in uni-axial longituinal tension. In the transverse irections, the aage evolution unctions (equations 16 an 17 are consiere to inepenent o the strain-rate. The properties o the ibers in both transverse irections are the sae, thereore the evolution unctions as well as their paraeters are the sae an only the history o the loaing is ierent. 1 ( 1 1 E n ( n in 1 exp,.. (16 1e t c 1 ( 1 1 E n ( n 3 in 1 exp, 3.. (17 1e t c The aages in transverse irections o the ibers are constraine to not excee Daages are ipose on the atrix aterial, but they aect only the shear stresses o the resin, which is consiere to be the ain contributor to the shear stresses o the RVC. A single Weibull istribution unction is accepte again as an evolution unction o the aages but it involves the ultiate strain or the aage evelopent rather than the ultiate stress. The aage evolution unction or in-plane shear is as ollows: s ( n 1 1 ( n 4 in 1 exp, 4.. (18 4 Strain-rate epenency is consiere only or the ultiate strain, 4, an the strain-rate sensitivity actor is the sae as or the shear eect on the iber breakage: 4 as4ss o4.. (19 The other aage evolution unctions or the atrix aterial are not strain-rate epenent. These aages or tie step n + 1 are calculate as ollows: June 1-14,

6 Session: Coposites 14 th International LS-DYNA Users Conerence in 1 exp 5 ( n 1 3 ( n 5, 5 s.. (0 s ( n 1 ( n 6 in 1 exp, 6.. (1 o4 The aages o the atrix aterial are constraine to not excee 0.0. These aages are applie on the atrix aterial shear stresses when the stress response o the sub-cells is calculate. The concept o the eective stress is accepte here or the atrix aterial, rather than the concept o the eective elastic ouli, because the atrix aterial oel is isotropic while the aages in the aterial are not..4. Delaination Delaination is a ailure oe which is ue to the quaratic interaction between the throughthe-thickness stresses o a laina an is assue to be ainly a atrix ailure. The loaing criterion or this ailure oe has the ollowing or: E3 G3 3 G S r 0.. ( S t S S SR S S SR where are Macaulay brackets, E 3 is the noral tensile oulus o the laina, G 3 an G are the transverse shear ouli o the laina, S 3 t is the through-the-thickness tensile strength o the laina, S 30an S 0 are the transverse shear strengths o the laina or tensile, r is the aage threshol, an S is a scale actor introuce to provie better correlation o elaination area with experients which can be eterine by itting analytical preiction to experiental ata or the elaination area. Uner copressive through-the-thickness strain, 0, the aage surace (elaination is consiere to be close, an the aage strengths are assue to epen on the copressive noral strain z siilar to Coulob-Mohr theory, i.e., SSR E3 tan z.. (3 where, is the Coulob s riction angle. The noral tensile oulus o the laina is copute at the irst tie step an store as a aterial property. ( n 1 la is the aage variable associate with this ailure oe an its evolution is given by the relation: ( n1 1 n la in 1 exp 1 r, la. (4 where, r is the aage threshol as given in equation ( an is the aage exponent or elaination. Delaination aage is constraine at 0.10 to avoi nuerical iiculties an when this axiu value is reache in an eleent, it is consiere to be ully elainate. 1-6 June 1-14, 016

7 14 th International LS-DYNA Users Conerence Session: Coposites When elaination ailure given by equation ( occurs in an eleent, epening on the ( 1 opening or closing o the aage surace, the aages variables ( n ( n1 ( 1, an n speciie in equations (5 an (6 are applie on the eective RVC stresses. For tensile oe, 0 : ( n1 ( n1 ( n1 ( n1 z, yz, zx la.. (5 For copressive oe, 0 : ( n1 ( n1 ( n1 yz, zx la.. (6 For tensile oe, all the through-the-thickness stress coponents, 3 an o RVC are reuce while or copressive oe, the aage surace is consiere to be close, an thus, is assue to be elastic an only 3 an are reuce. z yz zx.5. RVC Stress Calculations The eective stresses in the RVC are eterine ro the sub-cell values in two phases: irst, stresses in the iber an atrix M A are use to eterine the eective stresses o part A; then these stresses an the stresses in atrix M B are use to eterine the eective stresses in the RVC. As state earlier, iso-strain bounary conitions are assue or all the three sub-cells o the RVC. This iplies the rule o ixture or the stress calculations. The siple rule o ixture applie on all coponents o the iber an the atrix aterial stresses eans physically that the iber an atrix aterials act in parallel in all irections uner loaing, which is einitely not realistic. However, this assuption is ae in orer to sipliy the icro-echanical relations. The irect stresses o part A are calculate ro the irect stresses o the iber sub-cell an the atrix sub-cell M A using the ollowing relations: A ( 1W A.. (7 11 W A W ( 1W A.. (8 A W ( 1W A.. (9 The behavior o uniirectional coposites uner shear is oinate by the behavior o the atrix aterial. The contribution o the ibers to the shear stress is very low copare to the contribution o the atrix aterial. Hence, a hoc volue raction coeicients are ipleente or shear an a rule o ixture involving the is applie. Then, the shear stress o part A is eterine, applying the aages o the atrix aterial introuce in the previous section, as ollows: A A 1 Vs4 1 ( 1Vs4 ( (30 A ( 1V (1 A.. ( 3 Vs5 3 s5 5 3 A A Vs4 ( 1Vs4 (1 6.. (3 The shear volue raction coeicients, V s4 an V s5, are ierent or the in-plane an transverse shear. They have values quite lower than the volue raction o the ibers. Since the atrix aterial is oele as viscoplastic an the ibers are oele as elastic, ater the saturation o the plasticity in the atrix aterial, the contribution o the ibers to the shear stress o the sub-cells plays a role o strain harening. Finally, the eective stresses in the RVC are obtaine by applying the rule o ixtures again which yiels the ollowing relations incluing the sotening o through-the-thickness coponents ue to elaination ailure as ollows: June 1-14,

8 Session: Coposites 14 th International LS-DYNA Users Conerence ( n1 z W W.. ( ( n1 sy 3 yz ( n1 RVC zx A M B The total strains o the RVC, the total stresses in the atrix aterial, the internal state variables o the atrix aterial, the aage variables, an the tie average strain-rate logariths, S an S s, are kept as history variables at each tie step o the explicit tie integration process or the next tie step calculations. 3. Daage Paraeter Estiation The aage paraeter that has been use in the Weibull unctions above (equations 14, 16, 17 etc. has been observe to be very proble epenent an iicult to characterize. A sall value o the aage exponent (or ex. 1 in equation 14 akes the aterial behave in a very uctile anner an the behavior becoes increasingly brittle as this value increases. Hence, it is iicult to obtain the sotening response o ost quasi-brittle aterials. The sotening response heavily epens on the set-up an test achines, which can lea to very ierent results. The choice o aage paraeters or each oe has been ebate by Vaziri et al. [5] an Tabiei et al. [3]. A proceure or the calculation o sotening paraeter use in the aage evolution unction has been iscusse below. In their work, Pinho et al. [6] have calculate the axiu strain, as a unction o the 0 energy per unit area o the surace create, the aterial strength an a one eleent iension, L. *.. (34 0 * L This axiu strain calculate above has been in turn use in the calculation o the aage variable. 0 ( n1 ( n in 1 *, 0.. (35 ( Now, apart ro the unction itsel this orulation o aage is siilar to what we have use above in section (.3. Hence, the area uner the aage unctions can be been equate (equations (35 an (0 an we can iteratively solve or the aage paraeter by iniizing the error. As a preliinary test, this concept has been ipleente in MATLAB or both the aage unctions entione above or a wie range o eleent lengths an the results can be observe in igure below. Each curve on the right in igure represents the aage paraeter calculate or a speciie eleent length an it can be clearly seen that the aage curves look ientical in both the igures. 1-8 June 1-14, 016

9 14 th International LS-DYNA Users Conerence Session: Coposites Figure (: Daage unction coparision ater iteratively solving or the aage paraeter. 4. Nuerical Result an Discussion o Inter-lainar elaination As a veriication exaple, an ipact event on CFRP plates ae o T800H/3900- iber/resin syste with a lainate stacking sequence o 45/ 90/ 45/ 0 3S an total thickness o 4.65 is siulate using the current aterial oel in LS-DYNA. These experiental results were originally obtaine by an extensive investigation o out-o-plane ipact loaing o coposite test coupons by Delosse (8 an were use by Willias an Vaziri et al. [] to evaluate the preictive capability o a plane-stress CDM base oel or coposite aterials that they ipleente in LS-DYNA. The goal o this stuy is to preict the elaination easureent ae by the experients an reporte in Willias et al []. The test coupon consists o a siply supporte 76. by 17 plate ipacte by a heispherical steel ipactor ( 5.4 in iaeter, which in the nuerical coputation is treate as rigi boy. The FE oel is shown in igure 3. Figure (3: A ull oel view o the T800H/3900- CFRP lainate The CFRP plate in itsel consists o 4-thru thickness integration points with each integration point representing a layer o the lainate stacking sequence45/ 90/ 45/ 0. 3S June 1-14,

10 Session: Coposites 14 th International LS-DYNA Users Conerence Figure (4: Coparison o the elaination aage an experiental C-scan iages on a T800H/3900- CFRP plate. Nuerical results obtaine using Shell-5 eleent orulation in LS-DYNA June 1-14, 016

11 14 th International LS-DYNA Users Conerence Session: Coposites Figure 4 qualitatively copares the preictions o projecte inter-lainar elaination to the C-scan iages o elaination growth or the low ass ipact events provie in []. It is to be note that the nuerical results preicte in Figure 4 use the thickness-stretch shell eleent orulation.i.e., ELFORM=5. The box rawn aroun the nuerical results highlights the location o the plate bounaries relative to the part o the plate oelle. The sae set o tests have been carrie out with the stanar shell eleents i.e., ELFORM= in LSDYNA an the results can be observe in Figure 5. June 1-14,

12 Session: Coposites 14 th International LS-DYNA Users Conerence Figure (5: Coparison o the elaination aage an experiental C-scan iages on a T800H/3900- CFRP plate. Nuerical results obtaine using Shell- eleent orulation in LS-DYNA. Observing the results presente in Figures 4 an 5, the ollowing coents can be ae: 1-1 June 1-14, 016

13 14 th International LS-DYNA Users Conerence Session: Coposites a The total elaination area looks saller as copare to the experients when both sets o iages in each igure (nuerical an C-scan are set to the sae scale. b The shape o the elaination however looks quite ientical in both the cases as copare to the experients. c It can be sai that the elaination behavior s when using ELFORM= or 5 is ientical. However, in orer to unerstan the eect o Z-Stress on elaination behavior it is iportant to consier the eect o each ter on the let han sie o equation (. In this equation, the irst ter accounts or the contributions ro Z-Stresses, the secon ter or the YZ-Stresses an the last ter or the ZX-Stresses o the laina. Further, a single eleent insie the elaination zone has been selecte or the oels run with ierent eleent types (ELFORM= an ELFORM=5 an the contribution o each iniviual ter in equation ( has been analyze. The results o which are shown in Figures 6 an 7. The ollowing coents can be ae on the results presente in Figures 6 an 7. a A contribution o the Z-Stresses is seen in the oel run using Shell ELFORM=5, however this is signiicantly sall copare to the contribution ro the YZ-Stresses. It is to be note that as the total value o these ters goes beyon a value o 1, aage is introuce into the oel an the loa bearing capacity o the laina in Z, YZ an ZX irection is reuce. b As expecte or the oel run with Shell ELFORM=, the Z-Stress contribution is zero an the total aage is oinate by the YZ-Stress contributions in the elaination criteria. c The saller contribution o the Z-Stresses in preicting elaination explains why we observe near ientical results or the results presente in igures 4 an 5. Figure (6: Eect o each ter in the calculation o elaination, Moel run with Shell ELFORM = 5 June 1-14,

14 Session: Coposites 14 th International LS-DYNA Users Conerence Figure (7: Eect o each ter in the calculation o elaination, Moel run with Shell ELFORM = 5. Conclusions In the current work, using LS-DYNA an a icro-echanical aterial oel [3] a qualitative elaination stuy has been perore or ipact events with various noral incient energy levels (9.4J, J,.4J an 56.4J on CFRP plates ae o T800H/3900- iber/resin syste. These analyses have been carrie out using shell eleent orulations (which uses a plane-stress orulation an 5 (which requires a ull 3D constitutive oel, priarily to stuy the eect o Z-Stresses in the elaination preiction. The results inicate that espite a saller eect o Z-Stresses in the overall elaination preiction, the contribution resulting ro this ter cannot be ignore an hence it can be sai that the thickness-stretch orulations (Shell ELFORM=5 can be consiere ore reliable in preicting elaination in coposites. Though not copletely accurate, both the eleent orulations have been able to preict quite realistic elaination results or ipact siulations consiere. In aition, a physically quantiiable etho o eterining the aage paraeters base on the energy per unit area o the surace create has been presente. A 1-D case eonstrating the proceure has been ipleente in MATLAB an has been oun to be viable. As a uture aspect o this work it is intene to ipleent this proceure into the icro-echanical aterial oel an enhance its capability. Reerences [1]. LS-DYNA Keywor User s Manual, Volue I & II, LS-DYNA R8.0, Lawrence Liverore Sotware Technology Corportation(LSTC, June 1-14, 016

15 14 th International LS-DYNA Users Conerence Session: Coposites []. Willias, Kevin V., Reza Vaziri, an Anoush Poursartip. "A physically base continuu aage echanics oel or thin lainate coposite structures." International Journal o Solis an Structures 40.9 (003: [3]. Tabiei, Ala, an S. Babu Ainjikarai. "A strain-rate epenent icro-echanical oel with progressive post-ailure behavior or preicting ipact response o uniirectional coposite lainates." Coposite Structures 88.1 (009: [4]. Golberg, Robert K., an Donal C. Stouer. "Strain rate epenent analysis o a polyer atrix coposite utilizing a icroechanics approach." Journal o coposite aterials 36.7 (00: [5]. Willias, Kevin V., an Reza Vaziri. "Application o a aage echanics oel or preicting the ipact response o coposite aterials." Coputers & Structures (001: [6]. Pinho, S. T., L. Iannucci, an P. Robinson. "Physically base ailure oels an criteria or lainate ibre-reinorce coposites with ephasis on ibre kinking. Part II: FE ipleentation." Coposites Part A: Applie Science an Manuacturing 37.5 (006: June 1-14,