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    A FINITE ELEMENT MODEL FOR ANALYZING THE DYNAMIC

    CRACKING RESPONSE OF CONCRETE

    Jeffrey W. Simons, Tarabay H. Antoun, and Donald R. CurranSRI International, Menlo Park, CA 94025

    Presented at 8th International Symposium on Interaction of the Effects of Munitions withStructures, McClean, Virginia, April 22-25, 1997

    SUMMARY

    The physical basis for the softening of concrete (often with associated localization of strain and subsequent failure) is

    the evolution of microcracking damage. A physically-based constitutive model is being developed that describes the

    nucleation and growth of tensile and shear cracks in concrete. The constitutive relations are incorporated into a

    multiplane cracking model that has been implemented in the finite element code DYNA2D. Microcracks nucleate

    and grow on several prespecified planes in each element. Local stresses are introduced to account for local stress

    variations caused by the aggregates. Time-dependence arises naturally from the finite propagation velocities of the

    microcracks and helps ensure well-posedness and stability of the numerical solution. The model has been used to

    simulate laboratory experiments including uniaxial tension and uniaxial compression at several different strain rates

    ranging from static to impact loading conditions, and a spherical explosive test in marble. The results of the

    simulations show promising results, including softening curves that result from the cracking process, the increase in

    strength at high rates of loading, and calculated cracking patterns similar to those observed in experiments.

    1. OBJECTIVE AND APPROACH

    The main objective of this work, sponsored by the Defense Special Weapons Agency*, is to develop a physics-based

    constitutive model for analyzing the deformation and failure of concrete including damage and failure due to

    cracking. It is important that the model be thermomechanically consistent, numerically stable, and easy to

    implement in a finite element code. We plan to demonstrate through numerical examples that the model reproduces

    such macroscopic response features as softening, load-induced anisotropy, and rate effects. Our approach is to

    explicitly model the nucleation and growth of microcracks in concrete. By modeling the damage processes directly,

    the degradation of concrete properties and softening result from the accumulation of damage.

    2. INTRODUCTION AND BACKGROUND

    2.1 NEED

    Much research has been conducted to develop constitutive models for concrete capable of simulating the static and

    dynamic responses of concrete structures beyond peak load and into the softening region. A good review of some of

    these models is given by Chen et al. (1991) in which descriptions are given for elasticity based [e.g., Saenz (1964),

    Popovics (1973), Carriena and Chu (1985), Tsai (1988)], plasticity-based [Ohtani and Chen (1987), Chen and Chen

    *Review of this material does not imply Dept. of Defense endorsement of factual accuracy or opinion.

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    (1975), and Murray (1979)], elasto-plastic and fracturing [Maekawa and Okamura (1983), Dougill (1976), Bazant

    and Kim (1979), and Han and Chen (1986)], and block models [Cundall (1971) and Kawai (1977)]. Although the

    models described simulate much of the highly nonlinear response of concrete including softening, Chen et al.

    conclude that a need still exists for realistic models that are rational, reliable and practical. Particular features of

    such models include the capability to capture (1) the post-elastic non-linear deformation of brittle cracked concrete,

    (2) the transition between brittle and ductile regimes; (3) the response of a concrete composite system in a non-

    uniform and non-homogeneous stress field; and (4) the post-peak stress-deformation response of fractured concrete,

    including localized deformation, strain softening, crack healing and ductile fracture.

    2.2 FAILURE PROCESS IN CONCRETE

    One recognized problem in the modeling of concrete beyond peak load is that the softening response is not a

    continuum property of the material but the result of a structural response. As described by Read and Hegemier

    (1984), strain softening observed in laboratory tests is the result of internal cracking of material caused by imperfect

    boundary conditions between the specimen and the loading plates. The resulting strain distribution in a failed

    specimen is often very nonuniform and contains regions of high local strains. The localization of deformation during

    cracking in concrete has been measured experimentally under both tension (Gopalaratnam and Shah 1985) and

    compression (Shah and Sankar 1987).

    For concrete, microcracking has long been known to be the dominant source of nonlinear inelastic behavior. The

    cracking response of concrete under uniaxial compression is well established (Mindess and Young 1981). Initially,

    microcracks are present in concrete as a result of shrinkage, thermal movement, discontinuities at aggregate

    interfaces, and voids due to incomplete compaction (Chen et al., 1991). At loads of about 30% of the peak load,

    bond cracks (or interfacial cracks) begin to propagate, primarily under shear loading (Wittmann 1979). At loads of

    about 50% of peak, cracks begin to grow into the matrix in the direction of the applied load. At about 75% of peak

    load, matrix cracks increase rapidly in size and number and begin to coalesce. Ultimately the specimen fails by

    splitting along planes parallel to the direction of the applied load. Although the response mechanism causing

    softening is well established, the shape of the softening curves are strongly dependent on the compliance of the

    testing machine (Kotsovos 1983).

    2.3 PREVIOUS MODELS

    Several classes of fracture models have been developed to analyze the response of material during microcrack

    damage. Micromechanical models have been developed in which a brittle material like rock and concrete is modeled

    as a homogeneous material containing a known distribution of pre-existing cracks (Kachanov 1958, Horii and

    Nemat-Nasser 1985, Fanella and Kracjinovic 1988). Cracks then propagate in response to the stress state. Crack

    activation is monitored using either a stress intensity factor formulation (Seaman et al. 1985) or a generalized

    Griffith criterion (Margolin 1984). Espinoza (1995) has developed a micromechanically-based multiplane cracking

    model to analyze dynamic impact experiments in ceramics. Margolin (1983) describes a model for an elastic

    material with distributions of cracks in arbitrary orientations. A similar model has been presented by Krajcinovic

    and Fonseka (1981) for application to concrete fracture. Other microplane models based on the slip theory of

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    plasticity have been developed and extended to model many complex response mechanisms of concrete including

    cyclic loading (Bazant 1984, Bazant and Oh 1985 and 1983).

    In smeared crack models, cracks are not treated explicitly but are used to generate prescribed stress-strain curves.

    Such models have been developed by Rashid (1968), Hillerborg et al. (1976), Gopalaratnam and Shah (1985), Bazant

    and Oh (1983) and Gran (1985). A useful property of the cracking models is damage-induced anisotropy in whichthe stiffness properties of the concrete are reduced in the direction normal to cracking, a well-documented

    characteristic of brittle material behavior. The dependence of crack propagation on the orientation of the applied

    load introduces a directional property into the model that changes an initially isotropic response to anisotropic.

    2.4 LOCAL STRESSES

    Microcrack propagation in concrete is driven by local stresses which, in general, are different from the remotely

    applied stresses. The microstructure of concrete includes aggregates, cement paste, and voids. Local stresses arise

    as a result of the heterogeneous nature of concrete and magnitudes of local stresses are related to the mismatch in

    material properties between the aggregate particles and the mortar matrix. In particular, the stresses aroundaggregates will be higher than the average continuum stresses whereas stresses around voids will be lower. Local

    strains up to four times the average strains have been recorded experimentally in concrete specimens loaded under

    uniaxial stress conditions (Dantu, 1958). An indication of the local nature of cracking stresses is that the mode of

    failure for concrete specimens loaded under unconfined compression is cracks aligned in a direction parallel to the

    loading axis despite the fact that the applied stress is zero normal to the crack surface. Accepting that cracking is a

    stress-driven process suggests that local tensile stresses are present that drive the cracking process.

    Costin and Stone (1987) developed a model for microfractured brittle rock that includes local stress variations by

    assuming that local tensile stresses are proportional to the deviatoric stress normal to the crack face and depend on

    the average distance between cracks, and the size of the local tensile region around the crack tip. These local stress

    variations are added to the continuum stresses and the combined stress is used as the driving stress for crack

    propagation. A similar approach is taken in the present formulation to account for local stress fluctuations.

    2.5 STABILITY AND UNIQUENESS

    Softening models for concrete can have numerical problems associated with the stability and uniqueness of the

    solution and may show severe mesh dependence of results particularly if softening is treated as a rate-independent

    property of the material, as discussed by Sandler and Wright (1984), Belytschko and Bazant (1985) and Read and

    Hegemier (1984). Theoretically, the reason for mesh dependency is loss of ellipticity in the governing equations

    which causes strains to localize in regions of vanishing volume (Bicanic et al. 1991). Various methods have been

    developed to help overcome this numerical sensitivity including: basing damage on nonlocal measures of strain

    (Bazant, Belytchko and Chang, 1984), including rate effects (Needleman 1988; Sluys and de Borst 1990), or

    specifying minimum mesh dimensions (Bazant and Oh, 1983).

    In a broader sense, the framework presented by Coleman and Gurtin (1967) sets forth conditions that assure

    uniqueness and stability for thermodynamic processes and can be straightforwardly applied to analyzing the

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    mechanical response of concrete. Within the Coleman and Gurtin irreversible thermodynamics framework, the

    stress-strain relationship is derived from the Helmholtz free energy - a scalar valued function of the invariants of the

    strain tensor and the vector i containing the set of internal state variables that represent the internal rearrangements

    in the material during deformation. The energy function has the general form

    = ij,i( ) (1)where is the Helmholtz free energy density, ij are the Cartesian components of the Green-St. Venant strain

    tensor, and in the present model, the internal state variables are chosen to describe the cracking response. The stress-

    strain relationship is derived from the Helmholtz free energy function as follows:

    ij

    ij

    = 0 (2)

    where 0 is the density and ij are the components of the second Piola Kirchhoff stress tensor.

    A critical element of the Coleman and Gurtin approach is that the formulation is inherently time-dependent because

    the dependence of the internal state variables on the strain field is specified using evolution equations of the form

    iiji f ,= (3)

    This is an important consideration in modeling the rate-dependent cracking response of concrete subjected to

    dynamic loading.

    The final element of the formulation is that the Helmholtz free energy function, and the state variable evolution

    equations must satisfy the dissipation inequality

    0

    i

    i

    (4)

    which insures that the entropy production rate during irreversible thermodynamic processes is non-negative.

    The concrete model presented in this paper can be cast in the form of Eqs. (1-3) and can be shown to unconditionally

    satisfy the dissipation inequality represented by Eq. (4). The model therefore satisfies the conditions for uniqueness

    and stability derived by Coleman and Gurtin (1967).

    3. DESCRIPTION OF THE MODEL

    In the model, concrete is assumed to be an elastic-cracking material. The mechanical behavior of concrete is

    described using a multiplane cracking model similar in concept to some of the microplane models described above,

    but is more explicit in describing the cracking process. Cracking can occur on a discrete number of predetermined

    planes that may be randomly or preferentially oriented, depending on the nature and distribution of pre-existing flaws

    in the material. The number of cracking planes is arbitrary, limited only by practical considerations related to the

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    speed and efficiency with which calculations can be performed, and physical considerations related to the ability of

    the model to realistically capture the cracking patterns in the material for arbitrary loading conditions.

    We assume that the cracks are non-interacting, penny-shaped, and uniformly distributed. On an arbitrarily chosen

    plane with normaln , the microcrack distribution can therefore be characterized by a microcrack density parameter,

    N, and a characteristic microcrack radius, a . While the crack orientation,

    n , and the crack density, N, are fixedparameters, the crack size, a evolves as a function of applied loading in accordance with the laws of dynamic

    fracture mechanics. Cracks on different planes evolve independently, but their combined effect on the overall

    response of the material is accounted for within the framework of a compliance-based formulation. In describing

    this formulation, we begin by presenting the model equations for a single cracking plane; we then generalize the

    formulation for an arbitrary number of planes.

    3.1 SINGLE CRACK PLANE LOADED IN TENSION

    Consider an elastic body loaded in uniaxial tension with a single set of tension cracks oriented normal to the load.

    We assume that the total strain can be decomposed into an elastic component and a cracking component. In a localcoordinate system, along an axis normal to the cracking plane we have,

    et = ee + ec (5)

    where etis the total strain, e

    eis the elastic strain and e

    cis the cracking strain. The cracking strain for a set of

    noninteracting tensile cracks can be related to the remotely applied stress, s , as follows, (Sneddon and Lowengrub

    1969, Kachanov 1993)

    ect =16 1 2( )

    3ENa3s s > 0

    (6)

    where Eis the Youngs modulus and is the Poissons ratio for the elastic material and ect

    identifies cracking

    strain for tensile cracks. Here, we can define the compliance of the set of cracks in tension, Cct

    as,

    Cct =

    16 1 2( )3E

    Na3

    (7)

    From Hooks law, the elastic response of the material can be expressed as,

    ee =

    s

    E

    = Ces (8)

    where Ce

    is the elastic compliance. We can combine Eqs. (5-8) to obtain,

    et = Ce + Cct( )s (9)

    Expressing Eq. (9) in rate form, and realizing that the cracking compliance is a function of crack radius (i.e., Eq. (7))

    gives,

    ( ) sCsCCe ctctet ++= (10)

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    Combining this with Eq. (7) and rearranging gives,

    ( )

    +=

    s

    a

    aCeCCs

    cttcte

    31

    (11)

    This equation is very instructive because it reveals the mechanisms for strain softening and rate dependence in themodel. The second strain rate term, ( )saaCct3 , is the cracking strain rate associated with an increase in crackradii at a given stress level. Because the crack growth rate is always positive, the effect of this term is to reduce the

    stress whenever the crack growth rate is non-zero. If this term is greater than the applied strain rate,t

    e , the stress

    rate will become negative for a positive applied strain rate, i.e., softening will occur. Thus, the mechanism for

    softening is the transfer of elastic strain into cracking strain.

    Rate dependence is a result of the competition between the rate of applied strain,t

    e , and the crack growth rate, a .

    At low loading rates, the increase in stress due to the applied loading will be small compared to the stress drop due to

    increased cracking. At high rates of loading, the overall stress will continue to rise after cracking begins as long as

    the applied strain rate is greater than the rate of cracking strain due to crack growth. Thus, at high rates the concretewill appear stronger and more ductile than at low rates.

    3.2 SINGLE CRACK PLANE UNDER GENERAL LOADING

    The above relations apply to the case of crack opening under tension, where the only cracking strain component

    occurs in the direction normal to the crack surface. Similar relations have been developed for shear loading of

    tension cracks and for compression and shear loading of shear cracks. The shearing strain due to shear loading of an

    open tension crack, ecs

    , is related to the applied shear stress, by an equations similar to Eq. (6),

    ecs = 16 1 2

    ( )3E Na3 (12)

    For shear cracks, assuming a frictional relationship on the crack surface, the applied shear stress in Eq. (12) is

    replaced by the effective shear stress, eff , given by

    eff = +s s < 0 (13)

    where is the coefficient of friction.

    3.3 CRACK EVOLUTION

    In accordance with experimental observations (e.g., Mindess and Young, 1981), concrete is viewed as an initially

    elastic and isotropic material permeated by an array of pre-existing defects. Experimental evidence also suggests

    that under the action of applied loads, the pre-existing defects grow into crack-like features that primarily populate

    the aggregate-matrix interfaces (e.g. Hsu et al., 1963 and Gopalaratnam and Shah, 1985). In our model, this

    interfacial crack formation phase of the damage evolution process is termed the crack nucleation phase. Penny-

    shaped cracks with a size proportional to the maximum aggregate size are assumed to nucleate once the applied

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    stress reaches a threshold value. Cracks nucleate under both shear and tension. A tension crack nucleates if the

    applied stress normal to the crack surface reaches thnuc

    , the threshold stress for tensile crack nucleation. Likewise,

    a shear crack nucleates if the shear stress acting on the crack surface reaches thnuc

    , the threshold stress for shear

    crack nucleation. Both thnuc

    and thnuc

    are model parameters determined based on experimental data.

    Increasing the applied stress beyond the crack nucleation threshold causes crack growth. Cracks can grow either in

    mode I (tension) or mode II (shear). Two different crack size variables are used to distinguish between the two

    different modes of crack propagation. Thus atdesignates the radius of a tension crack and as designates the radius

    of a shear crack.

    Crack activation is monitored using a fracture-mechanics-based stress intensity factor approach. With this approach,

    a tension crack with radius atis activated if the tensile stress normal to the crack plane reaches a threshold value

    given by the relation

    thgrow =

    4a

    t

    KIc (14)

    where thgrow

    is the threshold tensile stress for crack growth and KIc is the critical stress intensity factor (or fracture

    toughness) for crack growth under mode I. Similarly, a shear crack with radius as is activated if the shear stress

    tangent to the crack plane reaches a threshold value given by the relation

    thgrow =

    4as

    KIIc (15)

    where thgrow

    is the threshold shear stress for crack growth and IIc is the critical stress intensity factor (or fracture

    toughness) for crack growth under mode II.

    Once a crack is activated, its propagation is governed by a stress- and crack-size-dependent viscous growth law

    derived based on the principles of dynamic fracture mechanics. Freund (1973) analyzed the response of a semi-

    infinite crack to an incident tensile stress wave and found that the instantaneous value of the mode I stress intensity

    factor for a moving crack is equal to the product of the stress intensity factor for an equivalent stationary crack times

    a universal function of the crack tip velocity. This is represented by the relation

    =

    ten

    t

    s

    I

    d

    I

    c

    ag

    K

    K

    max

    (16)

    where KIs

    and KId

    are the static and dynamic mode I stress intensity factors, respectively, and cmaxten

    is the

    maximum crack velocity in mode I (equal to a fraction of the Rayleigh wave velocity). Tsai (1973) derived the

    universal function for a penny-shaped crack moving at a constant velocity in an infinite elastic solid. This function is

    shown in Figure 1 along with the second order polynomial approximation used by Gran and Seaman (1986). A more

    useful form of Eq. (16), and of the functions plotted in Figure 1 can be obtained by inverting the function g and

    realizing that the stress intensity factor for a propagating crack should always be equal to the fracture toughness.

    This yields,

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    =

    s

    Ic

    d

    Icten

    tK

    Kgca

    1

    max (17)

    Using the stress intensity factor equation for a penny-shaped crack, one may express Eq. (17) in terms of the driving

    stress for tensile cracking, dr, and the threshold stress for crack growth, thgrow

    as follows,

    =

    dr

    grow

    thten

    t fca

    max

    (18)

    The crack growth law as expressed by Eq. (18) is used in the present model. Numerically, this growth law is

    implemented using the polynomial approximation of Gran and Seaman (1986).

    An analytic solution similar to the one expressed by Eq. (18) is not yet available for cracking in mode II. However,

    existing solutions for the crack velocity function for semi-infinite cracks propagating in mode I and mode III

    (antiplane shear) exhibit the same trend of decreasing stress intensity factor with increasing crack speed (Sih and

    Chen, 1977). Extrapolating this observation to mode II, it is assumed that, to a good approximation, the same

    functional form used to account for the rate dependence of mode I crack growth under tension can be used to account

    for the rate dependence of mode II crack growth under shear. Mathematically, the crack growth equation for shear

    cracks can therefore be expressed as follows

    =

    grow

    th

    drshear

    s fca

    max

    (19)

    where cmaxshear

    is the maximum mode II crack velocity (equal to a fraction of the shear wave velocity) and dris the

    driving stress for shear cracking.

    The driving stresses for tensile and shear cracking, drand dr, eluded to in Eqs. (18) and (19) are local stresseswhich, in general, are different from the remotely applied continuum stresses. These local stresses arise as a result of

    inhomogeneities in the material and mismatches in material properties between the aggregates and the cement paste.

    The effect of local stresses on the cracking response is included in our present formulation in a manner similar to that

    used by Costin (1987) in his model for rock fracture. To graphically illustrate the local stress concept, let us begin by

    examining the Mohr diagram of Figure 2. The large circle represents the continuum stress state, and the four smaller

    circles represent the local stresses on each of four cracking planes labeled 1 through 4 in the figure. Mathematically,

    the local stress is derived from the continuum stress using the simple relation

    loc = ij ij (20)

    where loc is the local stress and is a material constant that reflects the effect of microstructure on the local

    stress field. As evident from Eq. (20) the local stress at a point (or in a finite element) has the same magnitude on all

    the cracking planes. This is also illustrated in Figure 2 where the Mohr circles representing the local stresses on the

    various cracking planes all have the same size.

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    Cracking on a plane occurs if the driving stress on the plane exceeds the threshold for either tensile or shear cracking.

    The thresholds are shown as straight lines in Figure 2 where it is also shown that the local stress field exceeds the

    1.0

    0.8

    0.6

    0.4

    0.2

    0.0

    KI

    d /KI

    s

    1.00.80.60.40.20.0

    CRACK VELOCITY/MAXIMUM CRACK VELOCITY

    Exact Solution (Tsai, 1973)Polynomial fit (Gran and Seaman, 1986)

    Figure 1.Dependence of the ratio of the dynamic to static stress intensity factor on crack tip velocity

    for a propagating mode I penny-shaped crack.

    thresholds (i.e., the cracking criteria are satisfied) on two of the four cracking planes. The local stress state on

    cracking plane No. 3 satisfies the criterion for crack growth under tension; whereas the local stress state on cracking

    plane No. 2 satisfies the criterion for crack growth under shear. On these two cracking planes, cracks will grow

    according to the crack growth laws (Eqs. 18 and 19) described earlier. Crack growth modifies both the stress state

    and the cracking thresholds, and the cracks will continue to propagate until the local stress states on all the cracking

    planes cease to satisfy the crack growth criteria.

    Mathematically, the cracking process described above is implemented as follows. The driving stress for tension

    cracks is calculated as the algebraic sum of the applied stress normal to the cracking plane, n , and the local stress,loc ,

    dr

    = n

    + loc

    (21)

    In Figure 2, the resulting stress state is represented by Point A on cracking plane No. 3.

    The driving stress for shear cracks is calculated as the sum of the applied stress tangent to the cracking plane, n ,and the projection of the local stress, loc , in a direction normal to the shear cracking threshold surface. When the

    normal component of the stress is compressive, the frictional resistance of the material on the cracking plane is also

    taken into account to yield the following relation for the driving stress:

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    dr =

    n + loccos( )+ n locsin( ) for n < 0

    n + loccos( ) for n 0

    (22)

    where the friction angle, ,and the friction coefficient, , are related by the standard relation, = tan (23)In Figure 2, the stress state computed based on Eq. (22) is represented by Point B on cracking plane No. 2.

    Mohrs circle representingapplied stress state

    Mohrs circle representinglocal stress fluctuations

    1

    2

    3

    4

    1

    1, 2, 3 and 4 represent the stressstates on the designated planes.

    TensileThreshold

    Shear Threshold(Coulomb friction)

    A

    B

    Figure 2. Mohr diagram illustrating the local stress concept and themethodology used to incorporate local stresses into thecrack growth equations.

    3.4 ASSEMBLING THE GLOBAL EQUATIONS

    The development of the equations in three dimensions follows along similar lines to the one-dimensional

    development. In global coordinates, the total strain vector is decomposed into the elastic and the cracking strain,

    t = e + c (24)

    where tis the total strain, e is the elastic strain and c is the cracking strain vector. In the multiplane model,

    cracking occurs on a set of planes of prespecified orientations. In the two-dimensional case, we specify a set of five

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    discrete planes oriented as shown in Figure 3. The relative orientation of one plane relative to another within the set

    of five planes is fixed; however, the set of planes is assigned an initial global orientation, , that can be either a

    random number between +22.5 and - 22.5, or any prespecified angle. This feature makes it possible to represent

    randomly oriented as well as preferentially oriented cracking patterns. On each plane, k, the stresses acting across

    the planes are related to the global stresses through the transformation matrix Qk,

    = Qksk (25)

    where,

    Qk =cos

    2k sinkcosksin

    2k sinkcosk2sinkcosk cos

    2k sin2k

    (26)

    and kis the angle between the normal to the crack and

    the global x-axis. The cracking strains on the planes are

    assumed to be independent of one another, and they are

    all added together as they are transformed to the global

    coordinate system.

    c = Qkekc

    k=1

    nplanes

    (27)

    4 5

    1

    23

    Figure 3. Orientation of the cracking

    planes in two dimensions

    The cracking compliance from the planes can also be assembled into the global coordinate system as,

    Cc = Qk

    TCk

    cQk

    k=1

    nplanes

    (28)

    Combining Eqs. (24) and (29) and using the elastic compliance matrix, Ce

    , gives,

    t = Ce + Cc (29)

    The rate form of this equation is

    ssse cct CCCe ++= (30)

    wherecC can be assembled from the derivatives of the cracking compliance matrices on the planes and are explicit

    functions of the crack tip velocities on the planes. We can integrate this equation with respect to time to make it

    suitable for implementation into a finite element code. The equations are solved numerically for the stress increment

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    using the semi-implicit algorithm described by Espinoza (1993) in which the crack tip velocity is updated explicitly

    and held constant during the time step, and the stress increment is solved for implicitly. The solution for the stress

    increment is then given by,

    ( ){ } ( )01

    t ctcce CCCC ++=

    (31)

    where tis the time step.

    4. SIMULATIONS OF LABORATORY EXPERIMENTS

    Standard materials properties tests, including uniaxial tension, uniaxial compression, and confined compression tests,

    are commonly used to measure the key mechanical properties of concrete. The response mechanisms for these

    standard experiments are fairly well understood, and the cracking patterns and associated failure modes are well

    documented. To illustrate the ability of our newly developed model to capture the important features of the response

    of concrete, we performed calculations of the standard tests mentioned.

    4.1 UNIAXIAL TENSION TEST

    We modeled uniaxial tension tests performed by Gopalaratnem and Shah (1985). In these experiments the concrete

    specimen was a notched rectangular prism 30.5 cm (12 in.) tall, 7.6 cm (3 in) wide and 1.9 cm (3/4 in) thick. The

    loading was applied by shear grips on either end of the specimen. The specimen was notched to predetermine the

    location of failure. The notch depth resulted in a 5.1 cm (2 in.) effective loaded dimension across the width. Local

    strains were measured at several locations on the specimen with strain gages over 10 mm (0.25 in.) gage lengths and

    overall specimen displacement was measured with displacement transducers over a 8.2 cm (3.25 in.) gage length.

    The specimens were tested quasistatically to failure.

    The finite element model for the tension specimen is shown in Figure 4a. Material properties for the model are listed

    in Table 1. The elastic properties were chosen to match the initial response of the specimen. The simulated load-

    displacement curve for the specimen loaded in uniaxial tension is shown in Figure 4b along with records from two

    experiments. The applied stress as reported is the applied load averaged over the notched cross section. The

    displacement is that measured over a 8.2 cm gage length. The specimen reaches a peak average stress of about 3.6

    MPa, at a displacement of about 400 in.

    The resulting cracking patterns are shown in Figure 4c at two strain levels. Here the damage in the rock is

    represented graphically by a line oriented along each cracking plane in each element. For each plane orientation, the

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    4

    3

    2

    1

    0

    Stress(MPa)

    120010008006004002000

    Displacement (in)

    MON*MSNModel simulation

    *Golpalaratnam and Shah (1985)

    Applied displacement

    0.3

    10.2

    5.1

    7.6

    dimensions in cm

    (a) Finite element mesh

    Damage at A Damage at B

    (c) Calculated damage patterns

    Figure 4. Calculated response of notched tension test.

    (b) Load-displacement curve

    A

    B

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    length of the line is proportional to the amount of damage, defined as a function of the crack density and crack

    length. A damage of one is displayed with a line that reaches the element. Thus, the graphics displays tensile and

    shear "cracks" that represent damage on specific orientations. At point A, at peak load, small cracks have nucleated

    throughout the specimen, above and below the material inside the notches, and cracks have started to grow from the

    notches into the specimen. At point B, after the load has dropped to about 1/3 of the peak, tension cracks have

    developed over much of the material between the notches.

    TABLE 1. Tension Properties for concrete model simulation.

    Youngs Modulus 19.5 GPa

    Poissons ratio 0.20

    Tensile strength 3.8 MPa

    Nucleated crack length 1. mm

    Crack density 1/cc

    Maximum crack growth rate 5% Rayleigh wave velocity

    Density 2.18 g/ccLocalized stress factor 0.12

    4.2 RATE EFFECT IN TENSION

    Rate effects in tension are shown in Figure 5. Figure 5a shows measured effects on peak stress of strain rate. The

    measured points are the results of several experimental studies. The general trend is that there is a small effect of

    strain rate up to rates of about 1/s. At the lowest strain rates effects are probably due to creep. At strain rates greater

    than 1/s the strength increases significantly with strain rate. Between strain rates of 10 and 100/s the strength

    increases dramatically. There is significant scatter in the measured data, some of which is attributable to different

    testing techniques (Antoun 1991).

    Also shown in Figure 5a is the calculated effect of finite crack velocity on peak strength as a function of strain rate.

    The rate effect was calculated using a single element pulled in uniaxial tension. The maximum crack tip velocity

    was taken as 5% of the Rayleigh wave velocity. No rate effect is seen for strain rates up to 0.1/s. The strength

    increases at rates above 1/s. At 10/s the strength is about twice the static strength and at 50/s the strength is over 5

    times the static strength. The calculated strain rate effect on strength agrees well with the measured data.

    The calculated effect of strain rate on the stress-strain curve and the damage distribution is shown in Figures 5(b) and

    (c). For these simulations a planar concrete specimen 40 cm (16 in) tall and 20 cm (8 in.) wide was pulled in tension.

    The specimen was loaded by steel plates above and below the specimen moving at a constant velocity. These

    simulations thus include effects of both finite wave velocity and also some wave propagation effects. To minimizeinitial dynamic transients, the rock was given an initial velocity field consistent with the elastic response. The

    interface between the rock and the plates was assumed to be frictionless. The nominal stress-stress curves at 10-2/s

    and 10/s are shown in Figure 5(b). At the low rate, the response is relatively brittle; the stress reaches a peak

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    7

    6

    5

    4

    3

    2

    1

    0

    NORMALIZEDPEAKTEN

    10-7

    10-6

    10-5

    10-4

    10-3

    10-2

    10-1

    100

    101

    102

    STRAIN RATE (s-1

    )

    Antoun, 1991Cowell, 1966aCowell, 1966bGran et al., 1987Johns et al., 1991aJohns et al., 1991bMellin er, 1966Ross et al., 1989, concrete, splittin tensionRoss et al., 1989, concrete, direct tension

    Model Simulations

    (a) Rate effect on tensile strength

    c) Calculated rate effect on damage mechanism

    (b) Calculated rate effect on stress-strain curve

    Figure 5. Calculated rate effects on concrete response in tension.

    strain rate = 10-2/s strain rate = 10/s

    8

    6

    4

    2

    0

    NOMINALSTRESS(MPa

    0.0200.0150.0100.0050.000NOMINAL STRAIN

    10-2/s

    10/s

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    of about 3.2 MPa at a strain of about 0.02% and drops quickly after the peak. At the higher rate the response is

    significantly stronger and more ductile; a peak stress above 8 MPa is reached at a strain of about 0.06%.

    As shown is Figure 5(c) the calculated cracking patterns at the two rates show a very different mechanism. At the

    low rate the damage is localized, a single crack develops across the specimen. At 10/s the damage is distributed

    throughout the specimen and cracking occurs on more than just a single plane. The wide distribution in damage isbecause the stress relief from the cracks happens slowly compared to the applied loads.

    4.3 UNCONFINED COMPRESSION TEST

    We performed simulations of unconfined compression tests of concrete presented by Shah and Sanker (1987). The

    specimens tested were nominally 6 in tall, 3 in diameter cylindrical specimens. To minimize the effects of the end

    conditions, through cuts were made in the specimens spaced at 3.25 in. The specimens were tested quasistatically in

    unconfined compression well beyond the peak load. Specimens were tested to various levels of axial strain then

    unloaded. These specimens were sliced and prepared to allow mapping of the cracks for both radial and longitudinal

    slices.

    The finite element configuration for our simulations are shown in Figure 6(a). We modeled the specimen including

    the through cuts, the capping compound (quick drying cement) and the steel end caps. An axisymmetric analysis was

    performed. The through cuts were modeled with slidelines with no friction. The specimen was loaded by specifying

    a constant vertical velocity to the steel end caps. The calculated load-strain curve is shown in Figure 6(b) along with

    two representative measured curves from the tests. The load is normalized by the compressive strength of the

    concrete (4200 psi). The nominal strain is the normalized displacement in the specimen measured over a 3.25 in

    gage length. The calculated load curve shows a much more brittle response than the measured curves. For the

    measured curves the peak load occurs at a strain value of about 30 x 10-4and 80% of the peak load post-peak occurs

    at about 65 x 10

    -4

    . The peak calculated load occurs at a strain value of about 17 x 10

    -4

    and 80% of the peak load postpeak occurs at about 25 x 10

    -4.

    Measured and calculated cracking damage is shown in Figure 6(c). At peak load the tested specimen shows some

    cracking around aggregates and a few vertical cracks connecting aggregates are observed near the outside edge of the

    specimen. The calculated damage shows similar patterns, cracks have nucleated over much of the specimen and a

    few vertical cracks appear near the outside edges of the specimen. At 80% of peak load in the post-peak range, the

    tested specimen shows much cracking, many of the aggregates have cracks around them and there are several long

    vertical cracks connecting aggregates. The calculated damage shows similar patterns, vertical cracks have grown

    throughout the specimen.

    Spherical Wave Test

    The model was used to simulate the internal cracking for a spherical explosive test in marble. Qualitatively, the

    response of marble is similar to that of concrete; it is a frictional material that fails by cracking in tension and shear.

    In the experiment, a small spherical charge of explosive is detonated inside a block of marble (Antoun and Curran,

    1996). The marble specimen also contains a concentric set of wire loops that measure radial particle

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    Figure 6. Unconfined compression test

    (b) Nominal stress-strain curves

    (c) Cracking patterns

    CALCULATION

    EXPERIMENTAT PEAK LOAD AT 81% LOAD POST-PEAK

    Steel cap

    Concretespecimen

    Cappingcompound

    Through

    cut

    8.3

    7.6

    (a) Finite element mesh

    dimensionsin cm

    3.8

    1.2

    1.0

    0.8

    0.6

    0.4

    0.2

    0.0

    NORMALIZED

    LOAD

    6050403020100

    STRAIN (e-4)

    Measured gage 1*Measured gage 2Model calculation

    * Shah and Sanker 1987

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    Circumferentialcracks

    Explosive

    charge cavity

    (b) Model Simulation

    Figure 7. Spherical explosive charge in marble.

    (a) Experiment

    Radialcracks

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    velocity histories as the loops expand in an imposed magnetic field. In addition to velocity histories, dynamic strain

    paths can also be obtained from these records.

    The test was modeled including the detonation of the explosive. A small sphere of compactable material was

    included around the explosive to more accurately model the velocity histories for the material close in to the

    explosive. Figure 7 compares the calculated and observed fracture patterns from a test. The model calculates both

    the radial cracking pattern emanating from the blast hole and the circumferential cracks caused by reflection of the

    shock wave from the outer free surface. Combined with the measured particle velocity records, the observed

    cracking patterns can be used to help calibrate the dynamic parameters of the model for marble.

    CONCLUSIONS

    The model being developed for analyzing the response of concrete including damage and failure appears promising.

    The model is physically-based, in that the cracking process is modeled explicitly and the major input parameters are

    all physical, measurable quantities. Cracks nucleate and grow along prespecified orientations according to fracture

    mechanics laws. Softening is a result of crack growth, as cracks grow elastic strain feeds into cracking strain. Model

    simulations of notched tensile tests produce nominal stress-strain curves that agree to within about 20% of

    experimentally measured curves . Rate effects due to the finite propagation speed of crack growth give large

    strength enhancements at rates between 10 and 100/s, results that are consistent with those observed experimentally.

    At low rates the calculated cracking damage is localized to a single crack across the specimen, at high rates the

    damage is distributed throughout the specimen.

    Simulations of unconfined compression tests show damage patterns that are consistent with those observed

    experimentally, vertical cracks that near the peak load, nucleate along the outside of the specimen, then in the post

    peak region grow throughout the specimen. The calculated concrete response in unconfined compression is much

    more brittle than that measured. We believe that the laws used for crack growth, taken from linear elastic fracture

    mechanics, need to be modified especially for cracking in compression, to be more consistent with many cracks

    nucleating and coelescing around aggregates rather than cracks growing in an uncontrolled manner. Simulations of

    explosive tests in marble show patterns and extent of cracking can be modeled. Future development plans for the

    model include futher comparison with experiments and also implementing a three-dimensional version of the model

    into DYNA3D.

    ACKNOWLEDGMENTS

    This work was sponsored by the Defense Special Weapons Agency (DSWA) under the technical direction of Dr.

    Paul Senseny and Dr. Michael Giltrud. The authors gratefully acknowledge their support.

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