VUMAT for Fabric Reinforced Composites

Transcription

1 VUMAT or Fabric Reinorce Composites. Introuction This ocument escribes a constitutive mo or abric reinorce composites that was introuce in Abaqus/Exicit 6.8. The mo has been imemente as a built-in VUMAT user subroutine. It can be accesse by naming your material such that it begins with the string ABQ_PLY_FABRIC, e.g. ABQ_PLY_FABRIC_. The mo is currently supporte or ane-stress ements; this inclues shl (S4R an S3R, continuum shl (SC6R an SC8R, ane stress (CPS amily an membrane (M3D amily ements. User materials are currently not supporte in ABAQUS/Exicit or small-strain shl ements (S4RS. When user-eine materials are emoye to eine the material response o shl ements, ABAQUS/Exicit cannot calculate a eault value or the transverse shear stiness o the ement. Hence, you must manually eine the ement's transverse shear stiness. See Shl section behavior, Section o the Version 6.8 ABAQUS Analysis User s Manual, or guiines on choosing this stiness. This ocument escribes the basic equations o the constitutive mo an provies etaile inormation o the user interace to the VUMAT imementation in ABAQUS.. Continuum amage mo or abric reinorce composites A schematic representation o the geometry o the woven abric reinorcement consiere in the constitutive mo is shown in Figure. The iber irections are assume to be orthogonal. Figure : Schematic representation o woven abric. Fibers are aligne with irections an o a local coorinate system. The constitutive stress-strain rations are ormulate in a local Cartesian coorinate system with base vectors aligne with the iber irections, as shown in Figure. The abric-reinorce y is moe as a homogeneous orthotropic astic material with the potential to sustain progressive stiness egraation ue to iber/matrix cracking, an astic eormation uner shear loaing. The ierent aspects o the mo are iscusse next. Copyright Dassault Systèmes, 8 Page o 9

2 VUMAT or Fabric Reinorce Composites.. Elastic stress-strain rations It is assume that the astic stress-strain rations are given by orthotropic amage asticity. Reerre to a local coorinate system aligne with the iber irections (Figure the astic rations take the orm: ν ( E E σ ν σ E ( E σ ( ( The amage variables an are associate with iber racture along the an irections respectivy, whereas is rate to matrix micro-cracking ue to shear eormation. The mo ierentiates between tensile an compressive iber ailure moes by activating the corresponing amage variable epening on the stress state in the iber irections. Thus: < σ < σ < σ < σ + + ; + ( σ σ σ σ In orer to incorporate ierent initial (unamage stiness in tension an compression, the values o the astic constants E, E an ν are assume to take their tensile or compressive values epening on the sign o tr ( +... Fiber response The material response along the iber irections is characterize with amage asticity. It is assume that the iber amage variables are a unction o the corresponing eective stress, that is: The eective stresses are eine as: ( ( < σ ( < σ ( ( ( < σ ( < σ ( (3 (4 In orer to simiy notation, an inex will be use in subsequent iscussions, such that it takes the value (+/, (+/, epening on the sign o the corresponing stresses. Thus, the above our equations or the amage variables are rewritten as: (5 (

3 VUMAT or Fabric Reinorce Composites It is note that the eective stresses are irectly rate to the thermoynamic orces, Y, that are work conjugate to the amage variables, through the rationship E Y. Thereore the above equation states that the iber amage variables epen only on the corresponing thermoynamic orce. At any given time the astic omain is eine in terms o the amage activation unctions, F, as F φ r (6 The unctions φ provie a criterion or iber ailure an are assume to take the orm σ φ ; ( +,, +, X (7 where X are the tensile/compressive strengths or uniaxial loaing along the iber irections. The amage threshols, r, are initially set to one. Ater amage activation ( φ they increase with increasing amage accoring to: r ( t φ ( τ (8 τ t The einition ensures that the amage threshols are non-ecreasing quantities ( r ( t. The amage threshols are assume to obey the Kuhn-Tucker comementary conitions: an the consistency conition: F r r F (9 r F ( Note that the ormulation can be easily enhance to take the eects o amage upon loa reversal into account. For exame, compressive amage will usually egrae the tensile response i the loaing is reverse rom compression to tension. On the other han, tensile cracks close uner compressive loaing an have little eect on the compressive response. The evolution o the amage variables are a unction o the amage threshols an the racture energy per unit area uner uniaxial tensile/compressive loaing,. The ormulation o the amage evolution law ensures that the amage variables are monotonically increasing quantities. It also ensures that the correct amount o energy is issipate when the lamina is subjecte to uniaxial loaing conitions along the iber irections. The evolution o the amage variables is given by the equation: exp( A ( r ;, ( r where g L A ( L c g c 3

4 VUMAT or Fabric Reinorce Composites Here, L c is the characteristic length o the ement, tensile/compressive loaing, an amage initiation: is the racture energy per unit area uner uniaxial g is the astic energy ensity (i.e. per unit volume at the point o X g (3 E The ormulation o the amage evolution law ensures that the amage variables are monotonically increasing quantities. It also ensures that the correct amount o energy is issipate when the lamina is subjecte to uniaxial loaing conitions along the iber irections. For instance, uner uniaxial tensile loaing in the iber irection, the issipate energy per unit area is equal to the racture energy. This hols true provie that: (4 g g Lc Lc < L The ormulation thereore imposes a restriction on the imum ement size that can be use to accuraty capture the right amount o energy issipation uring racture. I the characteristic ement size o the FE mesh is greater than L, the analysis will over-preict the energy issipation. Note that, rom Abaqus 6.-EF onwars, the critical ement length or each o the ABQ_PLY_FABRIC materials an a representative list o ements that excee this criterion are printe in the.sta ile..3. Shear response As mentione earlier, the shear response is ominate by the non-linear behavior o the matrix, which inclues both asticity an stiness egraation ue to matrix microcracking. The main ingreients o the shear response are iscusse bow..3. Elasticity The astic rations give the eective (unamage stress in terms o astic strain:.3. Plasticity Yi unction: σ ( (5 ( F σ σ ( (6 The harening unction is assume to be o the orm: ~ + Flow rule: Assuming associate low, then p ( σ y C ( (7 F ( ~ ~ sign σ (8 σ 4

5 VUMAT or Fabric Reinorce Composites The evolution o the astic work uring yiing is given as U σ ( ( (9.3.3 Damage The astic omain is eine in terms o the amage activation unction, F, as F φ r ( The unction φ provies the criterion or initiation o shear amage o the matrix, which is assume to be o the orm ~σ φ ( S Here σ σ /( is the eective shear stress, an S is the shear stress or initiation o matrix amage. The amage threshol, r, is initially set to one an increases ater amage activation ( φ accoring to ( τ τ t r t φ ( ( Finally, base on [] it is assume that the shear amage variable increases with the logarithm o r until a imum value is reache. Thus: min( ln( r, (3 where, an are material properties..4. Element Detion The VUMAT provies two options to ete ements: The ement is ete when any one tensile/compressive amage variable along the iber irections reaches a imum speciie value, or, or when the astic strain ue to shear eormation reaches a imum speciie value,. This option is activate by setting the lag ldflag. The ement is ete when the amage variables along both iber irections reach a imum speciie value,, or when the astic strain ue to shear eormation reaches a imum speciie value,. This option is activate by setting the lag ldflag. These two options can be combine with a eormation-base ement etion criterion base on the values o the imum ( ˆ an minimum ( ˆmin < principal logarithmic strains that the ement can sustain beore it gets ete. 5

6 VUMAT or Fabric Reinorce Composites 3. Calibration proceure The astic constants an the iber tension/compression strengths, X, are easily measure rom stanar coupon tests in uniaxial tension/compression loaing o /9 laminates. The calibration o amage evolution in the iber ailure moes is base on the racture energy per unit area o the material,, which can be measure experimentally. The shear response is usually calibrate with a cyclic tensile test on a ±45 laminate, where the strains along the iber irections can be neglecte. Figure shows the typical shear response o a abric reinorce composite. It is note that the unloaing/roaing paths in this igure correspon to an iealization o the actual response, which usually exhibits hysteretic behavior. The igure will serve as the starting point or the iscussion o a general calibration proceure or the parameters that enter the amage an asticity equations. σ σ y S ( Figure : Schematic representation o typical shear response o a abric reinorce composite. The lev o amage can be measure rom the ratio o the unloaing stiness to the initial (unamage astic stiness. This allows us to compute pairs o stress-amage values, ( σ,, or each unloaing curve. This ata can be represente in the space o versus ln(, where σ /(. A linear it o the ata provies the values o (slope o the line an S (intersection with the horizontal axis as shown in Figure 3. Sometimes the amage ata shows inication o a saturation value, which woul be use to etermine. Otherwise a value o shoul be use. Figure 3: Calibration o the shear amage parameters an S ln(s ln( 6

7 VUMAT or Fabric Reinorce Composites Finally, or each unloaing curve in Figure, the astic strain at the onset o unloaing is etermine rom the value o resiual eormation in the unloae state. The values o ~ ( σ, at the onset o unloaing are then use to it the parameters o the harening curve, as illustrate in Figure 4. ~σ ~ σ y + p y C ( Figure 4: Calibration o the shear harening curve 4. User interace To activate the mo or abric reinorce composites the material name must start with the string ABQ_PLY_FABRIC, e.g. ABQ_PLY_FABRIC_. A synopsis o the interace is shown bow. The number o solution epenent variables (uner keywor option *DEPVAR is 6, an the DELETE parameter is equal to 6. Reer to Table or a etaile escription o each material constant speciie in the keywor interace. *MATERIAL, NAME ABQ_PLY_FABRIC *DENSITY ρ *USER MATERIAL, CONSTANTS4 ** Line : E, E, ν +,, E, E, ν ** Line : X, X, X, X, S ** Line 3:,,, ** Line 4: y, C, p ** Line 5: ldflag,, *DEPVAR, DELETE6 6,,, ˆ, min ˆ Table : User material constants or the abric reinorce composite material mo. 7

8 VUMAT or Fabric Reinorce Composites LINE: Pos. Symbol Description 3 + E Young s moulus along iber irection when tr ( E Young s moulus along iber irection when tr ( ν Poisson ratio when tr ( 4 Shear moulus E Young s moulus along iber irection when tr ( < E Young s moulus along iber irection when tr ( < ν Poisson ratio when tr ( < 8 Not use LINE: Damage initiation coeicients Pos. Symbol Description X Tensile strength along iber irection X Compressive strength along iber irection 3 X Tensile strength along iber irection 4 X Compressive strength along iber irection 5 S Shear stress at the onset o shear amage 6-8 Not use LINE: 3 Damage evolution coeicients Pos. Symbol Description 3 4 Energy per unit area or tensile racture along iber irection Energy per unit area or compressive racture along iber irection Energy per unit area or tensile racture along iber irection Energy per unit area or compressive racture along iber irection 5 Parameter in the equation o shear amage 6 Maximum shear amage 7-8 Not use 8

9 VUMAT or Fabric Reinorce Composites LINE: 4 Shear asticity coeicients: ( + C ( Pos. Symbol Description y y Initial eective shear yi stress C Coeicient in harening equation 3 p Power term in harening equation 4-8 Not use p LINE: 5 Controls or material point ailure Pos. Symbol Description ldflag Element etion lag: ldflag: Element is not ete (eault ldflag: Element is ete when either iber ails, or, or when ldflag: Element is ete when both ibers ail,, or when. Maximum value o amage variable use in ement etion criterion Maximum value o equivalent astic strain or ement etion criterion. 3 (A value o zero means that is not use as criterion or ement etion Maximum (positive principal logarithmic strain beyon which the 4 ˆ ement will get ete. Ignore i zero, not speciie, or ldflag. Minimum (negative principal logarithmic strain beyon which the 5 min ˆ ement will get ete. Ignore i zero, not speciie, or ldflag. 6-8 Not use 9

10 VUMAT or Fabric Reinorce Composites 5. Output In aition to the stanar (material-inepenent output variables in Abaqus/Exicit or stressisacement ements (such as stress, S, strain, LE, ement STATUS, etc. the ollowing output variables have a special meaning or the user material or abric-reinorce composites: Output Variable Symbol Description SDV Tensile amage along iber irection SDV Compressive amage along iber irection SDV3 Tensile amage along iber irection SDV4 Compressive amage along iber irection SDV5 Shear amage SDV6 r Tensile amage threshol along iber irection SDV7 r Compressive amage threshol along iber irection SDV8 r Tensile amage threshol along iber irection SDV9 SDV r r Compressive amage threshol along iber irection Shear amage threshol SDV SDV SDV3 SDV4 SDV5 SDV6 Equivalent astic strain Elastic strain component Elastic strain component Not use Elastic strain component MpStatus Material point status: i active, i aile. 6. Reerences [] Alastair F. Johnson an Jose Simon, Moing Fabric Reinorce Composites uner Impact Loas. In EUROMECH 4: Impact an Damage Tolerance o Composite Materials an Structures. Imperial College o Science Technology & Meicine, Lonon 7-9 September 999.