CONCRETE MATERIAL MODELING IN EXPLICIT COMPUTATIONS ABSTRACT

Transcription

1 Worksho on Recent Advances in Comutational Structural Dynamics and High Performance Comuting USAE Waterways Exeriment Station Aril CONCRETE MATERIAL MODELING IN EXPLICIT COMPUTATIONS L. Javier Malvar 1 and Don Simons 2 1 Karagozian & Case, 620 North Brand Boulevard, Suite 300, Glendale, CA LOGICON RDA, 6053 West Century Boulevard, Los Angeles, CA ABSTRACT Lagrangian finite element codes with exlicit time integration are extensively used for the analysis of structures subjected to exlosive loading. Within these codes, numerous material models have been imlemented. However, the develoment of a realistic but efficient concrete material model has roven comlex and challenging. The lasticity concrete material model in the Lagrangian finite element code DYNA3D was assessed and enhanced. In the first hase, the main modifications included the imlementation of a third, indeendent yield failure surface; removal of the tensile cutoff and extension of the lasticity model in tension; shift of the ressure cutoff; imlementation of a three invariant formulation for the failure surfaces; determination of the triaxial extension to triaxial comression ratio as a function of ressure; shear modulus correction; and imlementation of a radial ath strain rate enhancement. These modifications insure that the resonse follows exerimental observations for standard uniaxial, biaxial and triaxial tests in both tension and comression, as shown via single element analyses. The radial ath strain rate enhancement insures constant enhancement for all those tests. As a full scale examle, a standard dividing wall subjected to a blast load is analyzed and the effects of the modifications assessed. Concrete subjected to shear stresses has been observed to dilate in the direction transverse to the shear stress lane. For reinforced concrete walls or slabs with in-lane restraints this can result in significant increases in load resistance due to arching. In a second hase of the material model develoment shear dilation was imlemented. This latest model was used to model concrete cylinders wraed with comosite materials. INTRODUCTION In the analysis of comlex structures subjected to blast loading and large deformations, Lagrangian finite element codes with exlicit time integration have become a necessary and efficient tool [1]. In these codes a limited element library including trusses, beams, shells and solids has roven sufficient. However extensive material libraries have been required for 1

2 reresentation of the vast range of material behaviors. In the case of reinforced concrete structures, imlementation of a realistic but efficient concrete material model has roven comlex. Numerous analyses for rediction of small and full scale blast tests of reinforced concrete structures sonsored by the Defense Nuclear Agency have rovided an oortunity to revisit the existing material models in the finite element code DYNA3D [1,2,3]. The models otentially suitable for reresenting concrete s constitutive behavior were assessed over the full range from elastic resonse to failure. The most robust one, material model 16, still contains several shortcomings. In this reort those deficiencies and the corresonding corrections are described. ORIGINAL MATERIAL MODEL The Lagrangian finite element code DYNA3D was originally develoed by the Lawrence Livermore National Laboratory (LLNL) [1]. Within DYNA3D, several material models have been used in the ast to reresent concrete, namely material models 5 (Soil and Crushable Foam), 16 (Concrete/Geological Material), 17 (Isotroic Elastic-Plastic with Oriented Cracks), 25 (Extended Two Invariant Geologic Ca). Materials 5, 17 and 25 have exhibited significant limitations in modeling concrete behavior [4]. Material model 16, however, aeared more aroriate and resented some attractive features which could be easily enhanced. OVERVIEW The original material model 16 (subroutine f3dm16.f) decoules the volumetric and deviatoric resonses. An equation of state gives the current ressure as a function of current and revious minimum (most comressive) volumetric strain. Once the ressure is known, a moveable surface - herein denominated a yield or failure surface - limits the second invariant of the deviatoric stress tensor. The volumetric resonse is easily catured via a tabulated inut such as the one in equation of state 8. No changes were deemed necessary for this art of the resonse. However, the deviatoric resonse did resent some shortcomings which were addressed. For examle, due to the decouling of volumetric and deviatoric resonses, this original model has the limitation of not incororating shear dilation which is observed with concrete. For the case of significant structural lateral restraints and low damage levels this may result in resonses softer than exected. During initial loading or reloading, the deviatoric stresses remain elastic until the stress oint reaches the initial yield surface. The deviatoric stresses can then further increase until the maximum yield surface is reached. Beyond this stage the resonse can be erfectly lastic or soften to the residual yield surface (see Figure 1). Whenever the stress oint is on the yield surface and the stress increment corresonds to loading on that surface, lastic flow occurs in accordance with a Prandtl-Reuss (volume reserving) flow rule, imlemented by the well known radial return algorithm. The model also incororates a tensile cutoff and a ressure cutoff. ORIGINAL DEVIATORIC RESPONSE Stress Limits 2

3 The function σ which limits the deviatoric stresses is defined as a linear combination of two fixed three-arameter functions of ressure: σ = η σm + ( 1 η ) σr σm = a 0 + ( maximum stress difference), a 1+ a 2 where σr = a 0 f + (residual stress difference), a 1 + a f 2 and where = ( σxx + σyy + σ zz) /3 is the ressure (stresses are ositive in tension, ressure is ositive in comression). The arameter η is a user-defined function of a modified effective lastic strain measure λ. The function η(λ) is intended to first increase from some initial value u to unity, then decrease to zero reresenting softening. Hence, the yield surface migrates between σ r, reresenting the minimum or residual strength, and σ m, the maximum strength. The initial yield surface is given by σy = η y σm + ( 1 η y) σr where η y = η(0) is the initial value of η. Available triaxial comression concrete data indicate that for the initial yield surface the rincial stress difference σ y should be about 45% of the maximum stress difference. On the other hand, the residual strength should vanish for the unconfined comression test. Furthermore, because the two fixed surfaces become arallel for large values of, they cannot roerly reresent the brittle-ductile transition oint. The original formulation, with the constraint that the initial, maximum, and residual yield surfaces be linearly related, cannot roerly cature the exerimental data. This suggests the need for a third fixed yield surface indeendent from the other two. Comressive Meridian Figure 1. Three failure surfaces. Data for the comressive meridian are usually obtained from an unconfined comression test and triaxial comression tests with various levels of confinement. For the original model 16, a minimum of two nonzero levels of confinement are needed since three arameters define the 3

4 comressive meridian. The usual tests rovide no data for ressures below f c / 3 (failure in an unconfined comression test). The three-arameter maximum failure surface just described will usually overestimate the strength when extraolated to ressures below f c / 3. Similarly this formulation would overestimate the rincial stress difference for the biaxial tension test. Tensile Meridian The tensile or extension meridian of the failure surface for concrete is usually lower (closer to the hydrostat at the same ressure) than the comressive meridian. Exerimental data suggest that the ratio of the tensile to comressive meridian, herein denoted ψ, varies from about 0.5 at negative (tensile) ressures to unity at high confinements. Using equal meridians at low ressures will yield erroneous results (see Figure 2). Tensile Cutoff In an attemt to alleviate the reviously noted shortcoming at low ressures, the original material model incororates a tensile cutoff which limits the maximum rincial stress to the tensile strength f t (see Figure 2). For intermediate ressures ( 0< < f c / 3) this does not solve the roblem. In addition the tensile cutoff algorithm reduces the current stress state to zero in 20 stes. This arbitrary and abrut stress decrease contrasts with the smooth decay offered by the lasticity model when transitioning between the maximum and residual failure surfaces. Pressure Cutoff The original model also incororates a ressure cutoff which revents the ressure from going below f t / 3 (Figure 2). Although this does not affect the uniaxial tensile test, it does limit the rincial stress difference to f t / 2 for a biaxial tensile test, and to f t / 3 for a triaxial tensile test. These limits disagree with exerimental data showing that in both cases the rincial stress difference should reach aroximately f t. In addition, whenever the ressure cutoff is Figure 2. Original LLNL-DYNA3D failure surfaces. 4

5 reached in the original model, the current state of stress is maintained and no stress decay takes lace uon further straining. Rate Enhancement In the original model, at any given ressure, the failure surfaces are exanded by a rate enhancement factor which deends on the effective deviatoric strain rate. Enhancing strength at a given ressure is inconvenient, because the rate enhancement factors available in the literature aly to the uniaxial unconfined comression and extension aths, not to a ure shear ath. It is ossible to derive the following formula to relate the test data to the inut data in comression: 3a1fc rf + a2fc 2 rf 2 rc = 3aa 0 1+ ( 1+ aa 0 2) fc rf where r c = inut to DYNA3D (rate enhancement factor at fixed ressure), and r f = exerimental rate enhancement factor from an unconfined uniaxial comression test. However the original rogram uses the same factor for enhancing stress states at negative ressures. When calibrated to unconfined comression, this results in almost no enhancement for the uniaxial tension test. Elastic Behavior The original LLNL material model 16 has two otions for the elastic resonse, both isotroic. Both use the bulk modulus from the ressure-volume relation to comute a second elastic constant. One assumes a constant Poisson s ratio, the other a constant shear modulus. Although a constant shear modulus absolutely guarantees that no elastic energy can be generated, that otion was droed due to its inadequacy to reresent known data. In the other otion, the user secifies a value of Poisson s ratio. When used with equation of state 8, the model derives a shear modulus from the current unloading bulk modulus. This method easily leads to inconsistencies such as negative Poisson ratios uon initial loading [5]. The assumtion of constant Poisson s ratio was retained, but the comutation of the shear modulus was modified. ENHANCED PRANDTL-REUSS MATERIAL MODEL The original material model 16 was significantly modified to correct most of the shortcomings noted in the revious section. NEW PRESSURE CUTOFF The ressure cutoff c now has an initial value of -f t (see Figure 3). Together with changes in the maximum failure surface described below, both the biaxial and triaxial tensile tests can now reach a rincial stress difference of f t. Uon failure in the negative ressure range, the arameter η is used not only to reduce the current failure surface from the maximum to the residual, but also to increase the ressure cutoff from -f t to zero in a smooth fashion. This is done by checking the ressure returned by the equation of state subroutine, and resetting it to c if it violates c, where 5

6 c = f t if the maximum failure surface has not been reached (hardening), ηf otherwise (softening). t Note that although imlemented in the concrete material model subroutine, this modification can override the ressure calculated in the equation of state. This ressure cutoff is necessary as otherwise the equation of state would calculate very large negative ressures for large volumetric extensions beyond cracking, which would be hysically incorrect. COMPRESSIVE MERIDIANS OF THE FIXED FAILURE SURFACES A third, indeendent, fixed surface has been imlemented with three new arameters (a 0y, a 1y, a 2y ). This surface reresents initial yielding and is given by σ y = a0 y +. a1y + a2y Since for concrete the residual strength in tension is zero, the ressure indeendent arameter in the formulation of the residual surface is not needed, i.e., a 0f = 0. To ermit the residual and the maximum failure surfaces to intersect at a oint reresenting the brittle-ductile transition, a new arameter a 2f has been added. The residual surface now takes the form = a a σ r f + f 1 2 In the new model, after reaching the initial yield surface but before the maximum failure surface, the current surface is obtained as a linear interolation between the two: σ = η ( σm σ y) + σ y where η varies from 0 to 1 deending on the accumulated effective lastic strain arameter λ. After reaching the maximum surface the current failure surface is similarly interolated between maximum and residual: σ = η ( σm σr ) + σr The function η ( λ) is inut by the user as a series of ( η, λ ) airs. This function would normally begin at 0 at λ=0, increase to 1 at some value λ=λ m, and then decrease to 0 at some larger value of λ. Since λ is nondecreasing, this would ermit σ sequentially to take on the values Figure 3. Willam Warnke failure surface. 6

7 σ y, σ m, and σ f. In fact, there are no internal checks to guarantee that the user s inut takes on these secific values. Thus, at the beginning of the subroutine the value of λ m is defined simly as the value of λ corresonding to the first relative maximum of η in the inut table. Then, whenever λ λ m the current surface is interolated between the initial yield and the maximum; conversely, if λ > λ m the current surface is interolated between the maximum and the residual. In summary a total eight arameters define the three fixed surfaces, as follows: σ m = a 0 + (maximum failure surface), a 1+ a2 σ r = (residual failure surface), a 1f + a2 f σ y = a 0 y+ (yield failure surface). a 1y+ a2y At ressures above the brittle-ductile transition, σ r should be limited to σ m. In the code this is ensured by resetting σ r to σ m () if σ r from the nominal formula exceeds σ m (). The yield surface is similarly limited to σ m [2]. DAMAGE ACCUMULATION New Shear Damage Accumulation The current failure surface is interolated from the maximum failure surface and either the yield or the residual failure surface as σ = η( σ m σ min) + σ min where σ min is either σ y or σ r deending on whether λ λ m or λ > λ m, and where η is a function of λ. In the original model 16, the modified effective lastic strain λ, is defined as λ = b 0 1 ( 1 + /f t ) where the effective lastic strain increment is given by d ε = ( 2 3) ε ε ε dε ij ij /. In the new model, two changes have been imlemented. First rate effects were included, and second, the arameter b 1 is relaced by b 2 for tensile ressure (<0), as follows: λ = λ = ε 0 ε 0 r r f f dε ( + rf ) for 0 b / f t 1 1 dε ( 1+ rf ) for < 0 b2 / f t Note that at = 0, the denominator is a continuous function. In this way, the damage evolution can be different in tension and comression, if needed. Volumetric Damage 7

8 With damage accumulation as just described, if a triaxial tensile test is modelled, wherein the ressure decreases from 0 to -f t with no deviators, then no damage accumulation occurs. The arameter λ remains 0 and so does η. The equation of state decreases the ressure to -f t but kees it at that level thereafter. To imlement a ressure decay after tensile failure, a volumetric damage increment can be added to the deviatoric damage whenever the stress ath is close to the triaxial tensile test ath, i.e., the negative hydrostatic axis. The closeness to this ath is measured by the ratio 3J 2 /, which, for examle, is 1.5 for the biaxial tensile test. To limit the effects of this change to the aths close to the triaxial tensile ath, the incremental damage is multilied by a factor f d given by f d = 1 3J2 / * 10 for 0 3J2 / < 01., 0 otherwise. The modified effective lastic strain is incremented by λ = b 3 f d k d ( ε v ε v, yield ) where b 3 = inut scalar multilier, k d = internal scalar multilier, ε v = volumetric strain, and ε v,yield = volumetric strain at yield. Determination of Damage Evolution Parameters b 2 and b 3 The values of b 2 and b 3 govern the softening art of the unconfined uniaxial tension stress-strain curve as the stress oint moves from the maximum to the residual failure surfaces. It is well known that, unless such softening is governed by a localization limiter or characteristic length, the results will not be objective uon mesh refinement, i.e. they will be mesh-deendent. One way to eliminate this mesh deendency is to force the area under the stress-strain curve to be G f /h, where G f = fracture energy and h = a characteristic length, which may be associated with a localization width. The fracture energy usually varies from 40 to 175 N/m (0.23 to 1 Uniaxial Stress, MPa b2 = 4 b2 = 1 Single Element Uniaxial Extension Resonse for WSMR-5 Concrete Original Material 16 K&C Modification b2 = E+0 4.0E-4 8.0E-4 1.2E-3 1.6E-3 2.0E-3 Uniaxial Strain Triaxial Stress, MPa b3 = 10 b3 = 1 Single Element Triaxial Extension Resonse for Sac-5 Concrete Original Material 16 K&C Modification b3 = E+0 1.0E-3 2.0E-3 3.0E-3 Triaxial Strain (a) Parameter b 2 : effect on uniaxial tension strain softening. (b) Parameter b 3 : effect on triaxial tensile strain softening. Figure 4. Effects of arameters b 2 and b 3 on tension softening. 8

9 lbf/in) according to the Euroean CEB-FIP model code (Section : Fracture Energy) [6]. In a tyical analysis, a localization width (width of the localization ath transverse to the crack advance) is chosen together with G f, and b 2 and b 3 are determined by iterative calculations. An examle of the effects of b 2 and b 3 on the stress-strain resonse of a single element subjected to uniaxial and triaxial tensile tests is shown in Figure 4. If the analysis yields a different localization width than anticiated, this should be corrected and the calculation restarted. These arameters will be of imortance when the structure analyzed is lightly reinforced or is tensionor shear- critical. In dynamic analyses the localization attern may vary during the run, deending on the relative amount of daming. The localization width can vary from one to several element widths. Similar considerations should be brought to bear when selecting a value for b 1, which governs softening in comression. THREE-INVARIANT FAILURE SURFACE FORMULATION Develoment of a Three-Invariant Model The revious σ versus relationshis actually define only the comressive meridians of the failure surfaces in rincial stress sace. The original material model 16 assumes the full failure surfaces are obtained by rotating these meridians around the hydrostatic axis, thereby forming circular cross sections in the deviatoric lanes. The surfaces are functions of ressure and the second invariant of the deviatoric stress tensor, J 2, whose square root is roortional to the radius of the circle. A third invariant, such as J 3 or Lode angle θ (angular offset in the deviatoric lane of the stress oint from the image of a ositive, i.e. tensile, rincial stress axis) may be introduced to ermit more general shaes in the deviatoric lane, such as the triangular curves with smooth corners shown in Figure 5a. For concrete the deviatoric section tyically transitions from this shae at low ressures to circular at high ressures. Figures 5b and 5c show the large difference that can exist between the tensile and comressive meridians. Moreover, the difference is amlified when considering failure levels under comressive, roortional loadings, reresented by rays emanating from the origin in stress sace. These differences can be catured in the model only if a third invariant is included. To introduce the third invariant, a deendence on the Lode angle θ (Figure 6a) is sought. The shae roosed by Willam and Warnke [7] is adoted, roviding a smooth, convex triangular surface generated by ellitical segments as shown in Figure 6b. If r c is the distance from the hydrostatic axis to the failure surface at the comressive meridian, and r t the distance at the tensile meridian, then at any intermediate osition the distance r (r t < r < r c ) will be given by 2rc( r 2 c- r t 2 ) cosθ + rc( 2rt- rc) 4( r 2 c- r t 2 ) cos 2 θ +5rt 2-4rr t c r = ( r - r ) cos θ + ( r -2r ) c t c t 9

10 (a) Deviatoric sections for increasing ressure. (b) Hydrostatic section. (c) Tyical tensile and comressive meridians. Figure 5. Tyical failure surface section for concrete from [7]. 10

11 By dividing both sides by r c, then dividing the numerator and denominator of the right hand side by r c 2, we obtain r = ( ) cos ( ) ( ) cos 41- ( ψ ) cos θ+1-2 ( ψ) ψ θ+ 2ψ ψ θ +5ψ -4ψ where r = r / rc and ψ = rt / rc. Note that r deends only on ψ and θ, and that in general ψ in turn deends on. For θ = 0 the formula yields r = ψ corresonding to ure extension, and for θ = 60 it yields r = 1 corresonding to ure comression. The value of θ can be obtained from 3 s1 3 3 J3 cosθ = or cos 3θ = 2 / J 2 2 J 2 32 where s 1 = first rincial deviatoric stress = max[ σ1-, σ2-, σ 3- ], (σ 1, σ 2, σ 3 ) are the rincial stresses, and the stress invariants J 2 and J 3 are given by x xy xz J 2 = 1 s τ τ ( ) yx y yz 2 s + s + s, J3 = s1s2s3 = τ s τ. τ τ s zx zy z Once the value of r is known, the original comressive meridians are multilied by it to obtain the meridian at that location. (a) Angle of similarity θ. (b) Ellitical aroximation for 0 < θ < 60. Figure 6. Deviatoric lan section in the Willam Warnke model adoted from [7]. 11

12 Migration Between Fixed Failure Surfaces in Triaxial Extension Lacking guidance from laboratory data, the transition between fixed failure surfaces in triaxial extension was taken to be the same as in triaxial comression. This transition is given by the λ-η relationshi, which has been discussed in sections If a different transition were found for the triaxial extension cases, then a second, indeendent λ η relationshi would have to be imlemented. New Comressive Meridian U to this oint in the develoment it was imlied that the comressive meridian is known and inut to the code, and the extension meridian can be found as a fraction ψ of the comression one. In fact, deending on data availability, in some ressure ranges it is more aroriate to define the tensile meridian, and then obtain the comressive meridian from the tensile one. The secification built into the new model results from a combination of both aroaches, as described here and in the following subsections. For ressures in excess of f c /3, the inut comressive meridian (determined by the arameters a 0, a 1 and a 2 ) does serve as a basis for all the others. For ressures below f c /3 and above f t, we limit the maximum tensile stress on the extension meridian to f t. This uniquely defines the extension meridian as σ = 15. ( + f t ) which asses through both the triaxial tensile test failure oint at (, σ) = (-f t, 0) and the uniaxial tensile test oint at (, σ) = (-f t /3, f t ). At =f c / 3 the two formulations are forced to coincide by determining the aroriate value of ψ. The comressive meridian for ressures less than f c / 3 then follows as the image of the tensile meridian, i.e., the tensile meridian divided by ψ() at each ressure. The following subsections describe the determination of ψ() in detail. All three comression failure surfaces (yield, maximum, residual) have corresonding tensile images. Given this segmental failure surface formulation and the iecewise linear definition of ψ as a function of ressure, the failure surface will not be smooth. This does not violate any fundamental theoretical requirement. In fact, due to the use of a Prandtl-Reuss flow rule as imlemented with the radial return algorithm, it creates no numerical difficulties either. Definition of ψ() To comlete the imlementation of the three-invariant failure surface, the function ψ() has to be defined for the full range of ossible ressures. As mentioned earlier, for concrete ψ varies from 1/2 at negative (tensile) ressures to unity at high comressive ressures. In order to satisfy various observations for secific triaxial stress aths, the values of ψ are reset within the code for several ressures, as follows. Case 0 (tensile ressure). For 0 the tensile meridian has to include the oints (, - σ) = (-f t, 0) and (,- σ) = (-f t /3, f t ), which reresent failure in triaxial and uniaxial tensile tests resectively. At = -2f t /3 the comressive meridian is 12

13 σ 3 2ft 1 f = + ft = ψ 2 3 ψ 2ψ However, this should reresent failure in the biaxial tensile test, which test data suggest is aroximately given by σ = f t. By equating both stress differences, ψ = 1/2 at = -2f t /3. Another test of interest is the ure shear test in lane stress. If the coordinates are rotated 45 in lane, the resulting state of stress is (σ 1, σ 2, σ 3 ) = (τ, 0, -τ). Assuming that the maximum tensile stress is limited by f t, then τ = f t at failure. From (σ 1, σ 2, σ 3 ), σ can be found as: σ = 3J = 3f t. 2 For this test, the rincial stress difference is given by r times the comressive meridian, i.e. 3( + ft) 3ft σ = r = r 2ψ 2ψ with r = r ( ψ, θ = 30 o ). The two exressions are equal rovided ψ = 1/2 at = 0, because r ( 1/ 2, 30) = 1/ 3. Thus uniaxial, biaxial, and triaxial tensile failure and ure shear failure can all be lausibly reresented with ψ = 1/2 for 0. For examle, this gives the nominal maximum comressive failure surface the form σ m = 3(+f t ). t. Case =f c /3 (unconfined comression test). At =f c /3, the uniaxial unconfined comressive test yields a rincial stress difference of f c. The corresonding oint on the extension meridian is σ = ψ f c. This should be equal to the defined extension meridian (Figure 7) 3 3 fc σ = ( + f = + t ) ft hence 1 3 ft Ψ= fc This exression actually reresents an uer limit for ψ. For examle, if ft fc = 010. (tyical of concretes with fc 5000 si) then ψ = Case =2α f c / 3 (biaxial comression test). Biaxial comression tests conducted by Kufer et al. [8] have shown failure occurring at (σ 1, σ 2, σ 3 ) = (0, α fc, α fc ) with α The stress oint lies on the tensile meridian at a ressure =2α fc / fc / 3 and a stress difference σtm = α f c. The corresonding oint on the comressive Figure 7. Derivation of ψ for < 0. 13

14 meridian is given in terms of the inut arameters as σ cm = a0 + a = a0 + + a a 1 2 2α fc / 3 + a ( 2α f / 3) 1 2 c, so the ratio ψ = σ / σ is tm cm ψ = a 0 + a α fc 2α fc / 3 + 2a α f / c with α = 1.15 Comletion of definition of ψ. For comutational uroses, the function ψ() is iecewise linear, using the reviously defined values. For higher comression ressures, two additional data oints from existing databases were chosen, as follows: ψ = at =3f c, ψ = 1 for 8.45f c. The last entry reresents the transition oint beyond which the tension and comression meridians are equal, and the failure surface becomes a circle in the deviatoric lane. In summary, 1/ 2, 0, 1/ 2+ 3ft / 2fc, = fc / 3, αfc ψ ( ) =, = 2αf /, α / 3 2 f 3 c c a0 + a1+ 2a2αfc / , = 3fc, 1, 845. f, and the function is linear between the secified oints. Comarison with reviously reorted values of ψ. Based on various exerimental data, Ahmad and Shah have roosed the following values for ψ [9]: ψ = if 1/ 3 / f c < 175., ψ = f c if 175. / f c < The roosed values of ψ for / f c 3 are based on this and additional data for SAC5 concrete. For / f c = 1/ 3, using ψ = imlies that, in some cases and while unloading uniaxially from an isotroic comression state, the failure surface would only be reached for σ 1 = 0.124f c. This would robably only haen for concretes with low comressive strengths c 14

15 ( fc 4000 si), where ft / fc Based on several data sets, Chen [7] suggests that ψ = 0.5 for / 0, ψ = 0.8 for / f c 7. f c RADIAL RATE ENHANCEMENT Since in tyical exeriments rate enhancements are obtained along radial aths from the origin in the rincial stress difference versus ressure lane (via unconfined comressive and tensile tests), strength enhancement was imlemented in general along radial stress aths. This is accomlished as follows. Let r f be the enhancement factor and the ressure after calling the equation of state subroutine. The enhanced value σ e of the failure surface at ressure is desired, assuming the enhancement factor is alied radially. To get σ e an unenhanced ressure /r f is first obtained, then the unenhanced strength σ(/r f ) is calculated for the secified failure surface. Finally the unenhanced strength is multilied by the enhancement factor to give σ e = rf σ( / rf ) This simle formulation resents the following advantages: (1) the code inut is obtained directly from the test data at the same strain rate, and (2) strength is equally enhanced along any radial stress ath, including uniaxial and biaxial tension and comression. This is far more consistent with data than the earlier formulation. COMPRESSIVE MERIDIAN IN THE SOFTENING REGIME Comressive Meridian For Negative Pressures With the modifications discussed so far, if < 0 and softening is underway, there will be a vertical segment in the current failure surface (in the versus σ lane,) due to the reduction in minimum ressure c.. In other words, the current failure surface is given by σ = η ( σ m - σ r ) + σ r for > c and a vertical segment at = c.. To avoid this vertical segment but maintain the reduction in magnitude of c, a modified maximum failure surface Y 1 (,η) can be defined as follows when ressure is negative and softening is under way (λ > λ m ): where f Y1 (, η) = Ym ( ) Ym( c)= 3 + f t ( η) η c Y m = nominal maximum failure surface in comression = 3(+f t ), f = intersection of the residual surface with the ressure axis = 0 (for concrete), ( η ) = η + ( 1 η ) = η, c m f m m = intersection of the maximum surface with the ressure axis = -f t. c 15

16 As defined, Y 1 is continuous with Y m at = f = 0. The current failure surface in the softening range can then be defined as follows: η σ m ( ) + ( 1 η) σ f( ), > f, Y(, η) = f ηy1 (, η) = ηym Y m( c)= 3( + ηf ) t f. f m The formula above is uncorrected for rate enhancement. The correction follows as outlined in Section 3.5. Given an udated ressure (which imlicitly includes effects of rate enhancement), the corresonding unenhanced state is denoted by u = /r f where r f is the enhancement factor. The current unenhanced failure surface for negative ressures can be written as Yu( u, η) = ηy1 ( / rf, η) = 3 + ηft for / rf < f ( f 0 for concrete) r = f The corresonding enhanced failure surface follows by multilying by r f : f / rf Ye(, η) = rf ηym( / rf ) Ym( c) = 3( + ηrff t ). f m Corrections to the Flow Rule The foregoing modification has the undesirable effect of comlicating the deendence of the failure surface on and η. The exressions for the udated stresses and the increment dλ of the damage arameter must be also modified. The derivation, which is based on the assumtion of Prandtl-Reuss (volume-reserving) lastic flow, is resented in Reference [2]. SHEAR MODULUS CORRECTION With the constant Poisson s ratio otion and equation of state 8, the original model 16 comutes the elastic shear modulus from the secified constant Poisson s ratio and the current unload/reload bulk modulus. This can easily lead to a negative effective Poisson s ratio on loading whenever there is a large enough disarity between loading and unload/reload bulk moduli. In a first attemt at correcting this deficiency, the shear modulus was made deendent on whichever bulk modulus was currently in effect. However, this method failed because even infinitesimal ressure oscillations, for examle during an unconfined comressive loading, led to large shear modulus oscillations which did not reflect the nominally continuously increasing load. In addition, these oscillations were encouraged by the fact that elastic energy could be generated whenever ressure increased while shear stress decreased. A better aroach is to comute the shear modulus based on a scaled bulk modulus, one which varies from the loading to the unload/reload value deending on how far the ressure is below the virgin curve. A scaling factor which varies from zero to unity as ressure dros from the virgin loading curve to f is given by ε ϕ = ε + ( )/ K f U 16

17 where ε = ε v,min ε v, ε vis volumetric strain, and K U is the unload/reload bulk modulus from equation of state 8. If K L is the corresonding loading modulus, the scaled bulk modulus is K = (K L - K U ) e ϕ + K U where the constant 5.55 is chosen so that K will increase half way to the unload/reload value when has droed 1/8 of the way from the virgin curve to f. The shear modulus is then calculated as G = (1.5-3ν) K / (1+ ν) APPLICATION EXAMPLE: SUBSTANTIAL DIVIDING WALLS Substantial dividing walls (SDW s) in munitions roduction, maintenance, and storage facilities are used to subdivide exlosives to revent symathetic detonation and to rovide oerational shields for ersonnel. They are 12 inch thick concrete walls with #4 reinforcing bars at 12 inch sacings on each face and in each direction, and without any shear reinforcing. Current Army and Air Force safety regulations assume that the 12 inch SDW s will revent roagation for u to 425 ounds of Class/Division 1.1 exlosives. This examle was initially analyzed to rovide a verification of the roagation revention limits of the 12 inch SDW [10]. Although both the mass and velocity of secondary fragments are used for their caability of detonating the accetor charge, only their velocity is estimated in this examle. This is due to the difficulty for current analytical models to rovide reliable estimates of fragment sizes. Established criteria indicates exected velocities of 400 to 500 feet er second for tyical fragment sizes. LOAD DEFINITION Definition of airblast loadings was erformed using two widely used Navy codes: SHOCK [11] to roduce the shock loading (early time airblast) resulting from the incident blast wave and FRANG [12] to comute the gas ressure (late time airblast) resulting from exansion of the detonation roducts and heating of the air within the room. The rocess adoted was to comute loads indeendently of the resonse of the wall, i.e., the walls were assumed to be rigid. The assumtion of rigid boundary conditions is considered reasonable for this set of roblems because the shock ressure ulse lasts less than a millisecond, in which time the wall has not yet moved significantly. TEST DESCRIPTION A descrition of the selected test (C6) and the relevant design data was obtained from Reference [13]. This exeriment was conducted in the early 1960 s at the Naval Ordnance Test Station (NOTS) in China Lake, California and consisted of a cased donor munition laced within a cubicle with three side walls (no roof) with numerous accetor charges laced immediately outside the dividing walls. The charge (272 lbs) was detonated, resulting in comlete destruction 17

18 of the dividing wall but no symathetic detonations. Data from Reference [13] indicates that C-6 fragments were measured at a velocity of 500 ft/sec. Table 1. Summary of steel roerties. Yield Stress (ksi) Ultimate Stress (ksi) Fracture Strain Steel Grade Static High Rate Static High Rate (%) STRUCTURAL MODEL The discretization used to reresent a tyical SDW in the arametric calculations is shown in Figure 8, which illustrates a model with three suorted sides (one side wall is frangible). The model for a wall with two suorted sides (roof is also frangible) is similar, excet that the stub along the roof line is omitted. In both cases no significant in-lane restraints exist for the walls, hence no significant effects from shear dilation are exected. The selected mesh has 6 brick elements through the thickness for the concrete, one for the cover on each side and four inside the rebar cage. Reinforcing steel was modeled using truss elements at 12 inch sacings in each direction. The model used for steel, identified in DYNA3D as Material 19, has similar features to the concrete model: inclusion of strain rate effects, non-linear ost-yield hardening, and failure uon reaching a re-defined level of strain. This last roerty is essential as bar failure is observed in each one of the runs erformed, and without accurate reresentation Figure 8. View of DYNA3D concrete and steel meshes (3-sided suort). 18

19 of the breakage of reinforcing bars, the resulting secondary fragment velocity could not be adequately redicted. An examle of the stress strain curves required as inut for this material is resented in Figure 9: one for static roerties, one for infinite rate (limit case), and one for an intermediate value at which roerties have been measured (see also Table 1). This allows indeendent scaling of yield and ultimate strengths as a function of strain rate. Stress ε = ε = ε = 0 ANALYTICAL PREDICTIONS VERSUS TEST RESULTS FOR TEST C-6 Strain Fracture Strain Figure 9. Inuts to material 19 (steel). Two runs were erformed for this case to assess the desirability of using finer meshes, one with a 2-inch element size (fine) and one with a 4-inch element (coarse). The resulting deformations were quite similar in shae, and the general tye of failure was localized breaching in the immediate area of the charge, with shear failure along the floor and side wall. Figure 10 shows the deformed shaes for the original and current versions. The original model s sudden stress release uon uniaxial tensile fracture exlains the backface salling and large boundary deformations. The original model excessive energy dissiation in biaxial tension exlains the (a) Original version. (b) Current version Figure 10. Comarative results for original and current versions. 19

20 reduced deformations at the load level. 500 Figure 11 shows the horizontal velocity of a node on the front face of ENHANCED MODEL the slab located aroximately one foot 400 above the intersection of the dividing wall with the floor, in line with the charge. This reresents the location of 300 maximum velocity and greatest damage. The time histories indicate that the slab has clearly failed since 200 there is little or no late time reduction ORIGINAL MODEL in velocity due to failure of the concrete and reinforcing steel. The 100 calculations redicted fragment velocities 470 ft/s for the fine mesh. This comares well with the exected ft/s and is a clear imrovement on the 170 ft/s rediction with the original model. In conclusion, the tests results TIME (SEC) Figure 11. Velocity time histories. aear to confirm the validity of the analytical models, both with regard to the redicted secondary fragment velocity as well as the level and mode of damage incurred by the dividing walls. Significant imrovements were obtained with the enhanced version of the concrete material model. VELOCITY (FT/SEC) IMPLEMENTATION OF SHEAR DILATION EXPERIMENTAL OBSERVATIONS Concrete subjected to shear stresses exhibits dilation in the direction transverse to the shear stress lane. Uon cracking the dilation is exected to continue due to aggregate interlock until the crack oening is large enough to clear the aggregates on both sides. This imlies there is a limit in the amount of dilatancy. In the resence of a constraint normal to the shearing lane, such as an external force or steel reinforcement across the lane, the shear caacity across the lane increases. This can be observed in tests by K&C [14] (Figure 12) and Reinhardt and Walraven [15] (Figure 13) among others [16]. In the tests by K&C the joint was tested both monolithically (uncracked) and recracked. 20

21 Figure 12. Effect of lateral ressure on shear transfer across a joint, from [14]. Figure 13. Influence of shear reinforcement ratio ρ on shear stress and on crack oening [15]. 21

22 THEORY AND IMPLEMENTATION Figure 14. Associative and non-associative flow rule. arameter ω. In the original version, a constant volume Prandtl-Reuss model had been imlemented. This is a articular case of non-associative flow rule (θ=θ N in Figure 14). That model has the shortcoming of not being able to reresent the dilatancy due to shearing. Other models have imlemented associative flow, where the lastic strain increment is normal to the failure surface (Figure 14). However it is known that this yields excessive dilation due to shear. In the current version a general non-associative flow rule is imlemented [17,18] where the amount of artial associativity is indicated by an inut PRANDTL-REUSS (NON-DILATANT) FLOW RULE To set the stage for the artially associated flow rule, the non-dilatant one is first derived. In matrix symbolic notation, the decomosition of strain increments into elastic and lastic arts is e dε = dε + dε = σ dµ + Cdσ where σ is the deviatoric stress and C the (fourth order) elastic tensor, and where we have assumed Prandtl-Reuss (volume-reserving) lastic flow. Premultilying by C -1, 1 1 C dε = C σ dµ + dσ. (1) 1 The left-hand side of (1) is the trial elastic stress increment dσ *, and C σ = 2Gσ, where G is the shear modulus, so (1) simlifies to dσ* = 2 Gσ dµ + dσ. (2) (In this discussion we regard the ressure change as art of the trial elastic increment, since with Prandtl-Reuss flow it can be comuted immediately and comletely once the strain is known.) Multilying (2) by σ f, the gradient with resect to stress of the failure function f = 3J 2 Y( σλ, ), ( σf) dσ* = 2 G( σf) σ dµ + ( σf) dσ. (3) Here, Y is roortional to the radius of the failure surface measured normal to the hydrostatic axis in rincial stress sace, and λ is the hardening arameter. Now by differentiating the consistency equation f ( σ, λ ) = 0 (which ensures that the stress oint remains on the failure surface during lastic flow), we have ( f) dσ+ f dλ = 0 ( f) dσ = f dλ = Y dλ (4) σ, λ σ, λ, λ 22

23 where the comma denotes a artial derivative. Using (4) in (3) gives ( σf) dσ* = 2 G( σf) σ dµ + Y, λdλ. (5) On the other hand, the increment in the hardening arameter λ will be related to the lastic strain increment: dλ = h( σ) dε = h( σ) ( 2/ 3) dε dε = h( σ) ( 2/ 3 ) σ : σ dµ = g( σ ) dµ (6) ij where we have defined g( σ) = h( σ) ( 2/ 3 ) σ : σ. In this model we further define b 1 1 [ 1+ ( s/ 100)( rf 1)] ( 1+ / rfft ), > 0, h( σ ) = 1 b2 [ 1+ ( s/ 100)( rf 1)] ( 1+ / rfft ), < 0, ij where the arameters are defined in Section 4.2. By substituting dλ from (6) into (5) and solving for dµ, ( σ f) dσ * dµ = 2G( σ f) σ + Y, λg( σ). (7) Equation (7) can be made more exlicit by rewriting the failure function in terms of stress invariants. The case of interest has f( σλ, ) = 3 J Y$ 2 (, θλ, ) (8) where θ is the Lode angle, which satisfies 3 3J3 cos( 3θ ) = 32 2( J ) / 2. By the chain rule, the comonents of the gradient with resect to stress of (8) can be written f 3 J Y$ Y$ 2 (cos 3θ) = σij 2 3J2 σij σij (cos 3θ) σij. (9) The required derivatives are ( σ δ mm / 3) ij = = σij σij 3, (cos 3θ) (cos 3θ) J2 (cos 3θ) J3 = +, σij J2 σij J3 σij J2 σ ( mnσ mn / 2) = = K = σ ij, σij σij (10) J3 σ ( klσ lmσ mn / 3) σ σ σ σ δ ij = = K = im jm nm nm σ σ 3. ij ij Now (9) and (10) can be used to write the first term of the denominator of (7) in the form 2G f σ ij = 2G 3J 2. (11) σij It is very interesting to note that the gradient terms involving the Lode angle and ressure derivatives vanish. There is a geometrical exlanation for this: In rincial stress sace, (11) is a dot roduct between two vectors, the gradient and the deviatoric stress. At the current stress 23

24 oint, decomose the gradient into three orthogonal comonents, in the directions of, σ, and θ. Only the comonent in the direction of σ, corresonding to the first term on the r.h.s. of (9), contributes to the dot roduct. Using (11) in (7), ( σ f) dσ * dµ = 2G 3J + Y$ 2, λg( σ). (12) From an analytical standoint, equation (12) comletes the incremental solution to the roblem, since it could be used in (2) and (6) to give dσ and dλ in terms of dε. RADIAL RETURN IN THE NUMERICAL IMPLEMENTATION In rearation for numerical imlementation, (12) can be simlified somewhat by observing that if σ n is the stress at the beginning of the current time ste, then f ( σ n) = 0, so a Taylor series exansion gives f( σ*) = f( σn + dσ*) ( σ f) dσ*. But by definition, f( σ *) = 3J 2 * Y*. Thus 3J 2* Y* dµ =. (13) 2G 3J2 + Y, λg( σ) For an accurate, efficient numerical solution, more work is required, in art because using just the first-order equations (2,6,12) would not leave the stress oint recisely on the failure surface at the end of the time ste. Therefore the idea of radial return is introduced. Equation (2) shows that the stress increment dσ can be regarded as consisting of two arts: the elastic trial increment and a second art which is collinear with the current deviatoric stress itself. In rincial stress sace, any stress vector arallel to a urely deviatoric one is normal to the hydrostatic axis. Since at the end of the time ste the stress oint must be on the failure surface, the second art of the stress increment can be considered a radial return to the failure surface in a lane normal to the hydrostat. The difficulty is to know where the failure surface should be at the end of the time ste, because it will have moved during the time ste. Our function Y $ deends on ressure and Lode angle as well as hardening arameter. However, neither of the first two changes during the radial return, so their effect on Y $ will be fully accounted for by the elastic trial stress increment, and it is only necessary to consider the hardening arameter s effect on the failure surface location once the trial elastic increment is alied. The rocedure is therefore to locate the failure surface according to the trial elastic stresses, then to use essentially a first order aroximation for the increment in the hardening arameter to further move the failure surface due to strain hardening, and finally to radially scale the deviatoric stress comonents back from the trial elastic oint so that they lie exactly on the new surface. In fact the only reason for comuting dµ is to use it in the first order aroximation for the new failure surface location. The stress changes during the last ste would of course agree to first order with those corresonding to the negative of the first term on the right-hand side of (2), but the radial return scheme guarantees that the stress oint will be on the failure surface, to within machine accuracy, at the end of each finite time ste. 24

25 Now recall that Y* denotes the failure surface corresonding to the udated ressure and deviatoric stresses including the trial elastic stress increment, but the rior value of λ. If Y n+1 denotes the fully udated surface, the increment due to λ alone is aroximated as Y$ * g J Y Yn+ Y = Y$ d = Y$, ( *)( 3 2* λ σ *) 1, λ λ, λg( σ *) dµ =. (14) 2G 3J 2* + Y$, λg( σ*) This is the udate to the failure surface after the trial elastic art. It only remains to scale back the trial stress by the factor that follows from (14): Y$ λg σ J, ( *) 3 2* σn+ 1 = σ *. (15) 2 G 3 J 2* + Y$, λg( σ*) Y * FRACTIONALLY ASSOCIATED FLOW If the lastic flow were fully associated, we would have, assoc dε = ( σ f) dµ ~ where d µ ~ is a roortionality constant. Again, consider the comonents of the gradient in rincial stress sace, in the directions of, σ, and θ. Assume the θ-comonent does not roduce lastic flow, and that the -comonent roduces only a fraction ω of that which would occur if fully associated. With reference to (8,9,10), the lastic strain comonents are then, ω 3 J Y ij Y ij dεij ω $ d µ ~ 3σ ω$, δ = dµ ~ ( σij mδ ij ) dµ J σij σ = ij J + 2 = or, symbolically, dε = ( σ + mi) dµ, (16) where m = 2ω 3J Y$ 2, / 9, I is the identity tensor, and where the roortionality constant dµ is just a re-scaled version of d µ ~. Proceeding analogously with the non-dilatant case, the full strain increment is dε = ( σ + mi) dµ + Cdσ. (17) 1 Noting that C I = 3 KI where K is the bulk modulus, (17) becomes dσ* = ( 2Gσ + 3 mki) dµ + dσ. (18) Multilying by the stress gradient, using (4) and (6), and solving for dµ, ( σ f) dσ * dµ =.(19) 2G( σf) σ + 3mK( σf) I+ Y$, λg( σ) Noting that f ( ) $ = = $ σ f I Y, = Y,, σkk σkk and using (11), the definition of m following (16), and the comments receding (13), equation (19) simlifies to 25

26 3J2 * Y* dµ =. (20) ωky$ 2, 2G 3J2 1+ Yλg σ 3G + $, ( ) Now assume the radius of the failure surface takes the form Y $ (, θλ, ) = r$[ ψ ( ), θ ] Y $ c (, λ ), where Y $ c (, λ ) is the comressive meridian (i.e., the failure surface in conventional triaxial comression, at θ=60 ), ψ( ) is the ratio of the failure surface radius in extension to comression, and r$[ ψ( ), θ ] is a dimensionless function giving the current radius of the failure surface as a fraction of the comressive meridian. In the current K&C/LRDA version of DYNA3D model 16, ψ( ) varies from 1/2 to unity as ressure varies from the minimum allowable (set to the negative of the tensile strength) to ositive infinity; while the function r$( ψθ, ) is based on the Willam-Warnke failure surface and takes the form ( ψ )cos θ+ ( 2ψ 1) 41 ( ψ )cos θ+ 5ψ 4ψ r$( ψθ, ) = ( ψ )cos θ+ ( 1 2ψ) One effect of this formulation is to comlicate the algebra needed to comute dµ (and subsequently through equation (16) the lastic volumetric strain increment). The - derivative in (19) will involve not only the variation of the comressive meridian Y $ c(, λ ), but also the variation of r$[ ψ( ), θ ]. IMPLEMENTATION OF FRACTIONAL ASSOCIATIVITY Here, in order, is an outline of the numerical rocedure for udating the stresses. On entering the subroutine (f3dm16), the total strains will already have been udated according to the equation of motion. Three successive stress states are involved: current, denoted with the subscrit n, where the stresses are those from the revious time ste; trial elastic, denoted by suerscrit *, where the stresses have been udated according to certain elastic moduli and strain increments to be detailed below; and final, denoted by subscrit n+1, the fully udated state. 1. Comute the bulk modulus K and trial elastic ressure * from the EOS based on the difference between the total volume strain and the current lastic volume strain. The latter must be stored and udated for each element throughout the course of the calculation. The array ex5 is used for this urose. It had been used for the tensile cutoff in the original model 16, but is no longer needed for that urose since now the tensile cutoff is incororated in the failure surface. The structure of DYNA3D requires that the strain difference be comuted in subroutine sueos8, called from within f3dm16, so a minor change to the former is required. 2. Comute the shear modulus based on bulk modulus, Poisson s ratio, and the extent of unloading in the ressure-vs-elastic volume relationshi 26