Delamination Prediction and Non-local Averaging using a Composite Micro-Mechanical Model

Transcription

1 Delamination Preiction an Non-local Averaging using a Composite Micro-Mechanical Moel Ala Tabiei *, Saneep Meikona * * Department o Mechanical an Materials Engineering, University o Cincinnati, Cincinnati, Ohio 45221, USA Abstract Inter-laminar elamination in laminate composites has been stuie with the help o thickness-stretch shell elements using a 3-D material moel an compare against the traitional plane-stress shell elements. A strain-rate an pressure epenent micromechanical material moel using ply-level progressive ailure criteria has been use to simulate the initiation an propagation o elamination. The material parameters o the non-linear resin have been etermine using LS-OPT. The numerical elamination growth has been qualitatively analyze against the experimental C-scan images or multiple impact events on ierent composite plates. In aition, a non-local moel with an isotropic weight unction has been implemente to work in conjunction with the composite micro-mechanical material moel to alleviate strain sotening typically seen in composite materials. Keywors: Uniirectional composites; Micro-mechanical moel; Continuum amage mechanics; Delamination; Non-local averaging; LS-DYNA ; LS-OPT 1. Introuction Delamination, the principal moel o ailure o layere composites is the separation along interaces, an is oten consiere to be one o the most ominant amage mechanism in the ailure o composite laminates. Simulating an preicting elamination in composite materials is a challenging task, however it is critical in characterizing the overall response o Uni-Directional Composites (UDC). Typically, elamination ailure is moelle in commercial inite element programs in the ollowing ways: 1. Moel a layer o shell or soli elements or each composite layer an bon the layers with a tiebreak contact. 2. Moel a layer o shell o soli elements or each composite layer an bon the layers with cohesive elements. Both these methos have their limitations as the use o soli element layers makes the simulations computationally expensive an the use o traitional shell elements oesn t quite represent the true mechanics o elamination accurately. Mainly because traitional shell element ormulations oten use a plane stress ormulation (i.e., they o not account or the out-o-plane stress o σ33). In aition, using tiebreak contacts with shell layers also triggers an un-realistic chain reaction o ailure. Arguably the astest, most robust an eicient shell element ormulation, which is usually the eault element type in most commercial inite element coes, is base on the work o Belytschko, Lin an Tsay [1]. At its core this element ormulation epens on the Reissner-Minlin kinematic assumption which states that a plane section, originally normal to the mi-surace, remains plane an unstretche while allowing or shear eormations to occur. However, in aition to the zero thru-thickness stress limitation iscusse above, these element types also nee a moiie constitutive law which corrects or the strain in the thickness irection. Since the early 1990 s, there have been numerous attempts to ormulate shell elements accounting or throughthickness eormation. Works o Simo et al. [2], Parish [3], Hauptmann an Schweizero [4] an Doll et al. [5] June 10-12,

2 emonstrate some o many contributions in the area o soli-shell element ormulations. Büchter an Ramm [6], [7], El-Abbasi an Megui [8], Betsch et al. [9] an Bischo [10] have urther enhance the work to be applie or thin shells by solving the problem o strong thickness locking. Caroso s [11] work in particular overcomes the limitations o the Belytschko, Lin an Tsay ormulation by relaxing the unstretche Reissner-Minlin assumption an allowing or a linear strain variation through the thickness as shown in Figure 1. This creates aitional egrees o reeom which allows or loaing on the surace o a shell element an woul require a 3-D constitutive law in aition to preserving the avantages o the Belytschko, Lin an Tsay ormulation. These type o shell elements have conventionally been use in sheet metal orming applications, where the presence o normal stresses in the thickness irection has been observe to improve the accuracy o the solution. Figure 1: Traitional shell vs Thru-thickness shell It is generally acknowlege that inite elements moels with a ine mesh (smaller element size) yiel more accurate results [12], [13]. However, this is no longer the case or strain-sotening materials. Numerical stuies have shown that the results in strain-sotening materials are essentially epenent on the inite element mesh [14] [16]. This complication is well known (Belytschko et al. [17], Larsy et al. [18]), an any type o strain localization phenomena (ailure, in-elasticity, amage), will ten to localize in the smallest element o the inite element mesh. In other wors, the smallest element in the mesh will ten to ail/eroe beore other elements. Also, iner the mesh, the energy issipate by the numerical moel ecreases an tens to extremely low values, sometimes even to zero. Hence, the uniqueness o the solution with respect to the mesh size is lost, which is quite troubling rom a numerical stanpoint. Dierent remeies aressing this problem have been presente in the open literature an can be classiie into ierent categories/approaches. Cohesive crack moels/cohesive zone moels (CZM) [19] [21], Crack Ban Moel [22] [24], Regularize ierential (graient-enriche) ormulations [25] [27] an Regularize integral ormulations [28]. The current work ocuses on stuying the eect o integral nonlocal ormulations on micromechanical composite moels. These ormulations abanon the classical assumption o locality an eine stress at a point to be epenent not only on the state variable (usually strain, amage) at that point but also on a istribution o the state variable in a vanishing region aroun the point [29] an are airly easier to work with in comparison to the ierential ormulations. 2. Micro-mechanics o the uni-irectional composite (UDC) The representative volume cell (RVC) use to evelop the micro-mechanical relations is shown in igure (1). This RVC is the same as the one originally propose by Pecknol an Rahman [30] an urther use in various micro-mechanical moels by Tabiei et al. in [31], [32], [33] an Meikona et al. [34], [35]. However, or completeness the micro-mechanics relations are briely iscusse here. The unit cell is ivie into three subcells: one iber sub-cell, enote as, an two matrix sub-cells, enote as MA an MB respectively. The June 10-12,

3 eective stresses in the RVC are etermine rom the sub-cell values by combining 2 material parts: material part A consists o the iber sub-cell an the matrix sub-cell MA, an material part B consists o the remaining matrix MB using the iso-strain bounary conitions or all the irections. The imensions o the unit cell are 1 1 unit square. The imensions o the iber an matrix sub-cells are enote by W an Wm respectively as shown in Figure 2 an eine as shown below: W = V ; Wm W = 1.. (1) where, V is the iber volume raction. Figure 2: A representative volume cell o uniirectional iber reinorce polymer composite Visco-plastic constitutive relations base on moiie Boner-Partom state variable moel, initially propose by Golberg et al. [36] an urther enhance by Zheng et al. [37] have been use to represent the matrix subcells MA an MB. The ull etails o the moel can be oun in these reerences, however or completeness only the incremental orm o those equations are given below. 2n I 1 Z S ij εij = 2D0 exp + αδ ij t 2 σ e 2 J.. (2) 2 I 2 I I ee = eij eij ; 3.. (3) where, e = ε ε & ε = (ε + ε + ε ) / 3 I I I I I I I e ij m m Z qz ( 1 Z)e I e α q( α1 α)e I e =.. (4) =.. (5) Stability against high strain increments has been ensure by implementing a 4-step Runge-Kutta integration scheme. The constitutive relations o the iber are initially assume to behave as an elastic transversely isotropic material however these relations become orthotropic with amage evolution. The irect an shear stress stiness matrices in terms o the properties o the ibers are: June 10-12,

4 (1 (1 ) (1 ) (1 ) (1 ) 1) E1 (1 ν ) 1 E1 2 E2 ν 21 1 E1 3 E2 ν 21 (1 ν 2ν 21) (1 ν 2ν 21) (1 ν 2ν 21) (1 ) (1 ) ( + ) (1 )(1 ) 2 E2 ν ν 21 ν 21 E2 2 3 [ C ] =.. (6) (1 + ν )(1 ν 2ν 21) (1 + ν )(1 ν 2ν 21) (1 ) (1 ) 3 E2 ν 21 Symm. (1 + ν )(1 ν 2ν 21) G [ C s ] = G 0.. (7) Symm. Go 12 Once the stresses in all the constituent sub-cells have been obtaine, they are then combine using the iso-strain bounary conitions to obtain the eective stresses o the RVC. RVC 2 2 R σ = σ + ( 1 W σ.. (8) σ σ σ σ σ 11 W 11 ) 11 = σ + ( 1 σ.. (9) RVC 2 2 R 22 W 22 W ) R [ W σ + ( 1 W σ ] RVC 33 z 33 ) 33 =.. (10) =.. (11) RVC R 12 W Vs4σ 12 + ( 1 W Vs4 )(1 4 ) σ 12 R [ W Vs5σ + ( 1 W Vs5 )(1 5 σ ] R [ W V σ + ( 1 W V )(1 σ ] RVC yz ) =.. (12) =.. (13) RVC 31 zx s4 31 s4 6 ) 31 Since, the use o iso-strain bounary conitions or shear isn t quite realistic in a physical sense [34], ierent a-hoc shear volume raction coeicients, Vs4 an Vs5, or the in-plane an transverse shear have been introuce an have values quite lower than the volume raction o the ibers. Damage parameters i, i = 4, 5, 6, represent the amages impose on the matrix material an aect only the shear stresses o the resin. Damage parameters i, i = 1,...,6, in the above relations ollow progressive ailure moels an are iscusse in the ollowing section. Damages z, yz an zx are introuce by the inter-laminar elamination moel use the same criteria as MAT161 in LS-DYNA. Delamination initiation, which is a consequence o the quaratic interaction between the out-o-plane stresses o a lamina an is assume to be mainly a lamina ailure is given by the ollowing relation: E ε G γ G γ S + + r = 0.. (14) S t S + S SR S + S SR It has to be note here that the elastic material parameters speciie in Equation (14) correspon to the macroproperties o the lamina, which are back calculate rom the stiness matrix assemble or the RVC (Equations (8-13)) as iscusse by Qu an Cherkaoui [38]. Once the amage threshol has been reache, elamination ailure is introuce using a Weibull amage unction in a progressive manner. ( n+ 1) 1 max 1-exp ( 1- m n lam = r ), lam.. (15) m Depening on the opening or closing o the amage suraces, Equation (15) is use to subsequently reuce z, yz an zx. Note that the presence o riction is also accounte with the help o the Coulomb-Mohr theory when June 10-12,

5 the amage/elaminate suraces are close [32], [39]. The avantage o using a CDM base ailure moel is that it can eectively simulate ailure uner all conitions such as opening, closure an sliing o ailure suraces. 3. Damage an Non-local ormulation Progressive amage moels with strain sotening behavior have traitionally been observe as goo remeies to signiicantly improve amage preictions [40] [42]. The irst well known CDM moel was evelope by Matzenmiller, Lubliner an Taylor (MLT) [43]. Works o Williams an Vaziri [44], [45] have later reviewe an suggeste improvements to the MLT moel. Damage growth in the current work is hence base on ierent variations o the Weibull istribution unctions. Fibers are assume to govern the behavior o the composite in irect loaing, while the matrix is assume to ictate the response in the shear irections. m amage k 1 1 Ek ε n+ ij n k = max 1 exp, t c k me σ ij.. (16) where, ij = 12, or 31 an k = 1,2 or 3 m s ε ( n+ 1) ij ( n) k = max 1 exp, k ε km.. (17) where, ij = 12, or 31 an k = 4,5 or 6 Where, t c enotes tension or compression. When a positive strain is etecte, the parameters or tension are u utilize otherwise the parameters provie or compression are use. σ ij t c is the unamage stress in the ibers an when the amage 1 reaches 0.01 in tension, the inite element is consiere to be totally aile. Damages 2, 3 are constraine to not all below 0.1 an the shear amages below 0.2. The primary reason or constraining the amages is to account or the numerical instabilities that arise when stress in an element goes to zero. For the non-local approach, the ormulation propose by Anrae et al. [46] has been incorporate in the urmathn subroutine to work with the user-eine material moel (UMAT) in LS-DYNA (Figure 3). The nonlocal approach consists o calculating its nonlocal counterpart obtaine by weighte averaging over a spatial neighborhoo o each point uner consieration. Hence in a omain iel V, the corresponing nonlocal amage variable is eine as: ( x) = β ( x, ξ) ( ξ) V ( ξ).. (18) k Where, β ( x, ξ) is a given nonlocal operator. In an ininite boy, the weight unction epens only on the istance between the source point, ξ, an the target point, x, an is given by the ollowing relation: α( x, ξ) β ( x, ξ) =.. (19) α( x, ξ) V ( ξ) It shoul be note that the weighing unction α ( x, ξ) is a monotonically ecreasing non-negative unction o the istance r = x ξ. Typically, a Gaussian istribution is consiere as the weight unction an is given by the ollowing relation: June 10-12, V V α( x, ξ) = exp k x ξ 2L (20)

6 Where L, is a parameter relecting the internal length o the nonlocal continuum an shoul be experimentally etermine. From a numerical implementation point o view, the non-local value o the amage variable can be calculate by using the amage rom the previous time step n (Equation (16)) an the Gaussian quarature integration rule. npg i n n k j jβij k j= 1 = wj.. (21) Where, βij is the nonlocal operator that relates the Gauss points i an j locate at global coorinates x an ξ respectively. In aitions the quantities w j an J j are the Gaussian weights an Jacobian evaluate at Gauss point j. Lastly, npg is the number o Gauss points that lie insie the nonlocal volume o interaction rom point i i. It shoul be note that the actors w j, β ij an J j are merely geometrical in nature an they epen on the inite element mesh itsel rather that the constitutive moel (UMAT). Hence, these actors only nee to be calculate once, at the start o a simulation. The key part o the implementation lies in calculation o the nl nonlocal penalty actor K : K nl n k n k =.. (22) which is then use to calculate the nonlocal value o the amage variable at the current time step ( n 1)* + k ( n+ 1)* nl n+ 1 = K () k k Lastly, instea o the local value o the amage the upate nonlocal value can be use in reucing the stiness o lamina. amage ( n+ 1)* un amage E = (1 ) E.. (24) k k k The reuce values o the stiness are then use to calculate the eective stresses o the representative volume cell iscusse in the previous section. Note that, the rawback o accessing the neighboring integration points at once is overcome by aopting a strategy that saves an uses inormation o the amage variable rom a previous time step. The isavantage o such as assumption is that it necessitates small time steps or enough accuracy. However, since the explicit time integration scheme o LS-DYNA naturally requires a very small-time step (less than the critical time step ( t 2 ω ) to guarantee stable solutions this conition is easily met. It is worth max mentioning that LS-DYNA oes oer the option o using nonlocal ormulations through the keywor *MAT_NONLOCAL, however this option is limite to the use o very ew elastoplastic moels, nonetheless user-eine material moels. June 10-12,

7 Figure 3: Schematic lowchart illustrating the implementation o nonlocal strategy in LS-DYNA [46] 4. Numerical Results an Discussion o Inter-laminar elamination Impact events on T800H/ CFRP plates have been chosen or veriication. Since the current material moel has been base on micro-mechanics, there is a nee to characterize the visco-plastic material parameters o the resin. However, it must be note that the stress-strain responses that are necessary or the calculation o the viscoplastic parameters are not available irectly in the literature. For the resin, these have been back-calculate base on the work Tabiei an Babu [47]. In their work, Tabiei an Babu presente the viscoplastic parameters or a resin material moel base on the Golberg-Stouer relations. A stanalone material moel base on these relations an the material parameters has been evelope as a VUMAT in LS-DYNA an stress-strain curves have been generate or 3 strain rates in the tensile (0.1/s, 1.4/s an 510/s) an shear (0.1/s, 1.76/s an 420/s) irections. The generate ata has then been it to the current resin material moel using the LS-OPT sotware, the low chart o which is shown in Figure 4. LS-OPT [48] is a stanalone optimization sotware that can be linke to any inite element coe. It is particularly useul or the current case since it provies a simple interace to work with LS-DYNA. One popular use o LS-OPT is or calibrating material parameters. Parameter ientiication problems are non-linear inverse problems solve using optimization, in other wors, the compute curve rom LS-DYNA (epenent on parameters) is matche to an input curve. The two essential components involve in parameter ientiication are the optimization algorithm an the curve matching metric. The Sequential Response Surace Metho (SRSM) is the recommene/eault optimization algorithm use in parameter ientiication problems, the reaer is irecte to Staner et al. [48] or urther etails on this algorithm. To calculate the mismatch between the target an the compute curve, an orinate-base Mean Square Error (MSE) curve matching metric has been selecte. For completeness, the unerlying principle o this metric is June 10-12,

8 iscusse here. Once the target stress-strain curves have been inputte as ile histories, the mean square resiual error between the input ata an the numerically generate ata (base on a tensile an shear responses or various strain rates) has been calculate base on the equation given in Equation (25) an subsequently minimize. 2 2 N N i x G i ei x i i i= 1 i i= 1 i 1 ( ) 1 ( ) MSE( x) = W = W min. N s N s where, ( x): simulation response as a unction o variable vector x i G : target value i W : weighting actor i s : normalization actor (absolute max. value o each curve) i N: no. o points e ( x) : error at each point i.. (25) Figure 4: LS-OPT Flowchart or resin material parameter calculation It must be note that the backen solver has been built using intel compilers an the UMAT (User MATerial) subroutine in LS-DYNA, base on the constitutive relations iscusse in section 2. As inputs or LS-OPT, parameterize single element input ecks with tensile an shear bounary conitions have been use with the respective strain rate speciie. As it can be observe rom Figure 6, the itte response o the resin is quite close to the target ata speciie. Table 1 lists all the visco-plastic parameters obtaine or the resin rom LS-OPT. The above iscussion an the use o an optimization tool (LS-OPT) presents a simpliie way o characterizing the visco-plastic material once the experimental ata is available. June 10-12,

9 Resin E ( GPa ) 0 ν D (1 / s ) m 0 n Z ( MPa ) Z ( MPa ) q 0 1 α 0 α * Table 1: Material Constants or the Resin The impact event on CFRP plates mae o T800H/ iber/resin system with a laminate stacking sequence o [ 45/ 90 / 45 / 0] 3S an total thickness o 4.65 mm has been simulate using the current material moel in LS-DYNA. These experimental results were originally obtaine by an extensive investigation o out-o-plane impact loaing o composite test coupons by Delosse [49] an were use by Williams an Vaziri et al. [45] to evaluate the preictive capability o a plane-stress CDM base moel or composite materials that they implemente in LS-DYNA. The goal o this stuy is to qualitatively preict the elamination an compare it with the experiments as reporte in Williams et al. [45]. The test coupon consists o a simply supporte 76.2 mm by 127 mm plate impacte by a hemispherical steel impactor ( 25.4 mm in iameter), which in the numerical computation is treate as rigi boy. The FE moel is shown in Figure 5. Figure 5: A ull moel view o the T800H/ CFRP laminate June 10-12,

10 150 Tensile Stress VS Tensile Strain Tensile Stress (MPa) Input Data: 0.1/sec Input Data: 1.4/sec Input Data: 510/sec Fitte Data: 0.1/sec Fitte Data: 1.4/sec Fitte Data: 510/sec Tensile Strain 120 Shear Stress VS Shear Strain Shear Stress (MPa) Input Data: 0.1/sec Input Data: 1.76/sec 20 Input Data: 420/sec Fitte Data: 0.1/sec Fitte Data: 1.76/sec Fitte Data: 420/sec Shear Strain Figure 6: Tensile an Shear stress itte curves or resin June 10-12,

11 The CFRP plate consists o 24-thru thickness integration points with each integration point representing a layer o the laminate stacking sequence [ 45/ 90 / 45/ 0] 3S. In aition to the visco-plastic properties o the resin speciie in Table 1, the remaining material properties neee to carry out the simulations have been speciie in Table 2. 1 V ε o s ( ) E ( GPa ) 1 ν 12 X t ( Mpa ) X c( Mpa ) E ( GPa ) 2 ν σ ( ) 2 t Mpa σ ( ) 2 c Mpa G ( ) o12 GPa G ( GPa ) s4 a ε 4m ε 5m s V V s5 ag ( GPa ) b t b S ( ) 3 MPa S ( MPa ) S ( MPa ) ϕ (eg.) c t S m Table 2: Fiber an amage properties or the T800H/ lamina Figure 7 qualitatively compares the preictions o projecte inter-laminar elamination to the C-scan images o elamination growth or the low mass impact events provie in [45]. The box rawn aroun the numerical results highlights the location o the plate bounaries relative to the part o the plate moelle. Observing the results presente in Figure 7, the ollowing comments can be mae: a) The total elamination area looks smaller as compare to the experiments when both sets o images in each igure (numerical an C-scan) are set to the same scale. The reasons or these can be attribute to two possibilities. i. First, the C-scan images inicate matrix ailure an elamination in the complete laminate. In other wors, they show the cumulative elamination in all layers. The numerical results on the other han are shown only or the interace that experiences the maximum amount o elamination (which is the bottom most interace or the current cases), this is ue to a limitation in the post-processing capabilities or shell elements. ii. Seconly, the elamination moel use in the current stuy uses a stress-base criterion, where the inter-laminar an out-o-plane stresses were use to preict the initiation an growth o elamination. These moels have traitionally been proven to be eective in capturing the initiation o elamination, but aren t as eective in capturing the scale eects. Davies an Zhang [50] an subsequently Tabiei an Babu [32] have shown that the stress-base elamination criteria uner preict the elamination area. b) The shape o the elamination however looks quite ientical in both the cases as compare to the experiments. c) The elamination areas preicte by the thickness-stretch element moels seem to be ientical as compare to the areas preicte by the plane-stress shell element moels. June 10-12,

12 Figure 7: Comparison o the elamination amage an experimental C-scan images on a T800H/ CFRP plate. Numerical results obtaine using plane-stress shell elements have been shown on the let an the thickness-stretch shell elements on the right. To urther unerstan the eect o Z-Stress on elamination behavior, it is important to consier the eect o each term on the let-han sie o equation (14). In this equation, the irst term accounts or the contributions June 10-12,

13 rom Z-Stresses, the secon term or the YZ-Stresses an the last term or the ZX-Stresses o the lamina. The contributions o each o these stresses can be better unerstoo rom Figure 8-Figure 10. The ollowing comments can be mae on the results presente in these igures. a) A contribution o the Z-Stresses has been seen in the moel run using thickness-stretch shell elements, however this is small compare to the contribution rom the YZ an ZX-Stresses, i.e., the number o re-spots are ewer. Which explains the ientical shapes observe in Figure 7. It must be note that as the total value o these terms goes beyon a value o 1, amage is introuce into the moel an the loa bearing capacity o the lamina in Z, YZ an ZX irections is reuce. b) As expecte, or the moel run with traitional plane-stress shell elements, the Z-Stress contribution is zero an the total amage is completely ominate by the YZ an ZX-Stress contributions in the elamination criteria. c) Despite being small, the contribution o Z-Stresses in preicting elamination cannot be completely ignore. These stresses have been observe to have contribute nearly 6% o the total elamination amage at the point o initiation. June 10-12,

14 Figure 8: Contribution o Z-Stresses in the calculation o elamination or the bottom layer. Numerical results obtaine using plane-stress shell elements have been shown on the let an the thickness-stretch shell elements on the right. June 10-12,

15 Figure 9: Contribution o YZ-Stresses in the calculation o elamination or the bottom layer. Numerical results obtaine using plane-stress shell elements have been shown on the let an the thickness-stretch shell elements on the right. June 10-12,

16 Figure 10: Contribution o ZX-Stresses in the calculation o elamination or the bottom layer. Numerical results obtaine using plane-stress shell elements have been shown on the let an the thickness-stretch shell elements on the right. June 10-12,

17 5. Non-local Results an Discussion To test the non-local technique, it has been implemente along with the micro-mechanical material moel iscusse in the previous sections an applie on a tensile og-bone specimen, commonly use in the experimental etermination o the properties o composites. The imensions o the specimen are shown in Figure 11. Figure 11: Tensile Dog-bone Specimen [46] The FE moels have been built or 3 mesh sizes. It shoul be note that the mesh size has been only signiicant increase or the curve part o the specimen since this is the primary area o interest. The symmetry in the specimen has been taken into consieration an hence only a quarter o the moel has been moele to reuce computational eort. The material properties o E-glass/Epoxy iscusse in Meikona et al. [35] have been use here. Note that one en o the specimen has been ixe an the other en o the specimen is being pulle by 1.2 mm. Initially, the moels have been run without calling the non-local routine an the results o local amage in the pull irection in the en eorme state have been presente in Figure 12. As expecte, strain tens to localize in a ew elements or these cases an as a result the material amages completely in those regions. In aition, note that the maximum allowable amage in the iber irection ( ) has been constraine to 0.01 to account or 1 numerical instabilities an element eletion has been turne o. June 10-12,

18 Results shown in Figure 13 correspon to the tensile tests carrie out with the non-local ormulation activate. Figure 12: Local Damage in the Tensile Dog-bone Specimen or ierent mesh sizes It is clearly seen that the non-local ormulation prevents strain-localization an shows a more smeare eect o amage. It is important to note that the maximum amage (lower value shown in the igure) experience by each moel is ierent an or consistent comparison the ringe limits have been manually ajuste to correspon to the maximum amage experience by the moels in the igure (which is the moel with most elements in this case). As the no. o elements in the moel increases, the amage istribution tens to become similar an the variation in the maximum amount o amage observe in each moel, becomes quite less as well. Also, as expecte with an increase in the mesh ensity, a smoother variation o amage has been observe. June 10-12,

19 Figure 13: Non-local amage in the Tensile Dog-bone Speciment or ierent mesh sizes 6. Conclusions A strain-rate epenent micro-mechanical material moel has been evelope in LS-DYNA (as a user-eine material UMAT), to stuy elamination amage growth in uni-irectional composites using thickness-stretch shell elements an plane-stress shell elements uner impact loaing conitions. This is mainly one as, throughthickness stresses resulting rom out o plane loaings, such as contact orces, are expecte to either promote or inhibit elamination growth. The ability o thickness-stretch shell elements in using a 3-D constitutive law makes them an interesting option in stuying elamination growth, especially when use in conjunction with a stress-base approach. The strain-rate epenency is accounte with the help o the moiie Boner-Partom viscoplastic relations in the resin. A signiicant avantage o these relations is that they account or the contribution o the hyrostatic stresses in preicting the non-linear response, which is a well-known an crucial characteristic o polymers. The material parameters o the resin have been characterize by minimizing the mean square error in an optimization sotware (LS-OPT). June 10-12,

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