Simulation of the mechanical behavior and damage in components made of strain softening cellulose fiber reinforced gypsum materials

Transcription

1 Computational Materials Science 9 (7) Simulation of the mechanical behavior and damage in components made of strain softening cellulose fiber reinforced gypsum materials T. Rahman a, W. Lutz a, *, R. Finn b, S. Schmauder a, S. Aicher b a Institute for Materials Testing, Materials Science and Strength of Materials (IMWF), University of Stuttgart, Pfaffenwaldring, 7569 Stuttgart, Germany b Materialprüfungsanstalt Universität Stuttgart (MPA), Abt. Holzbau, Pfaffenwaldring 4, 7569 Stuttgart, Germany Received 4 October 5; received in revised form December 5; accepted 7 January 6 Abstract Cellulose fiber reinforced gypsum based materials are gaining increasing importance in the building industry. The non-combustible panel material is produced in thicknesses of 4 mm and with a fiber content of about vol.%. The fiber orientation in the composite is predominantly random planar. A major application for the panels is sheathing and bracing of timber frame wall elements. The material exhibits a macroscopic response that resembles that of a ductile material with pronounced strain softening. These material characteristics, which deliver high energy dissipation especially during reversed cyclic loading, are advantageous for seismically loaded structures. In this paper the homogenized fracture behavior for this material is described for the first time for static and quasi-static cyclic loading using a Plastic-Damage model proposed by [J. Lubliner, J. Oliver, S. Oller, E. Oñate, Int. J. Solids and Structures 5, (989), 9 6, J. Lee, G. L. Fenves, J. Eng. Mech. 4, 8 (998), 89 9]. This model is primarily used to simulate quasi-brittle materials such as concrete, rock, mortar and ceramics. The model input parameters such as tensile strength and fracture energy are obtained from uniaxial test results. The numerical simulations have been performed with ABAQUS. The different features and parameters of the applied Plastic-Damage model are discussed with respect to their capability to describe the behavior of cellulose fiber reinforced gypsum. Ó 6 Elsevier B.V. All rights reserved. PACS: 6..Mk Keywords: Cellulose fiber reinforced gypsum; Plastic-Damage model; Fracture; Damage parameter; Effective stress; Stiffness degradation/recovery. Introduction Cellulose fiber reinforced gypsum based material is an important structural component in the building industry. The material is produced as panels or boards, which are used for sheathing and bracing wall elements in timber frame structures. The material shows high energy dissipation under cyclic loading conditions and is well suited for earthquake resistant building constructions []. The randomly distributed short fibers are predominantly oriented * Corresponding author. address: wolfgang.lutz@mpa.uni-stuttgart.de (W. Lutz). in the plane of the panel. The fiber volume fraction ranges from 7% to %. During failure the surfaces of the matrix cracks are initially bridged by fibers. Further damage is mostly caused by fiber pull-out, however, some of the fibers also break. Macroscopically, strain softening is observed. Damage is localized in a softening zone perpendicular to the loading direction. This softening zone is also referred to as the crack band, which is a zone of distributed micro-cracks, which consequently link up during strain softening. The key macroscopic features of the material behavior are the development of a strain softening yield surface, permanent (plastic) deformation, as well as stiffness degradation and recovery /$ - see front matter Ó 6 Elsevier B.V. All rights reserved. doi:.6/j.commatsci.6..

2 66 T. Rahman et al. / Computational Materials Science 9 (7) For successful use of this material in the building industry a fundamental understanding of the material behavior is necessary. This can be obtained by numerical investigations. This paper focuses on the modeling of this material using averaged material properties obtained from uniaxial loading experiments. Simulations have been performed both for static and quasi-static cyclic loading conditions as anticipated in regular and earthquake situations. The determination of material parameters from experiments as required for the simulations is also discussed. The behavior of cellulose fiber reinforced gypsum is similar to that of other quasi-brittle materials. Therefore, material models developed for quasi-brittle materials are considered. The chosen material model should be applicable for static and cyclic loading conditions. Furthermore, the material parameters should be obtainable from uniaxial experiments. Based on these requirements, the Plastic- Damage model proposed by Lubliner et al. [] and extended by Lee and Fenves [] has been chosen for the present study. This model was developed for quasi-brittle materials like concrete, rock and ceramics. It captures the material behavior using both classical plasticity and continuum damage mechanics. Therefore, this model is supposed to have a wide range of applicability and can serve as an appropriate material model for the present material. The modeling of the material behavior has been performed with the finite element software ABAQUS where an implementation of the proposed Plastic-Damage model is available [4]. Simulation of the material behavior under static loading has been done successfully using the existing model in the course of this work. Simulation of the hysteresis loops occurring under quasi-static cyclic loading has been achieved with modifications of the existing model that account for varying elastic stiffness degradation and recovery inside the elastic domain. In Section, performed uniaxial experiments are described which serve as the source of the material parameters. In Section, the original material model and the necessary modifications are discussed briefly. The material parameter determination process, finite element modeling and the simulation results under static and quasi-static cyclic conditions are discussed in Sections 4 and 5, respectively. The conclusions and the outlook of the present work are mentioned in Section 6.. Experiments Fig.. (a) Specimen and (b) experimental set-up for the static and quasistatic cyclic investigations. Displacement controlled experiments were performed using unnotched specimens under uniaxial static and quasi-static cyclic loading. Contact-free optical strain measurements were obtained with laser extensometry. The same test set up (Fig. (b)) with different specimen dimensions (Fig. (a) and Fig. (a)) was used for the static and quasi-static cyclic experiments [5]. For the purpose of optical strain measurement, a grid was glued on one side of the specimen. The grid consisted of a set of black lines of mm width arranged with a mm center to center distance. During the test, the transition of light and black lines was scanned by the laser extensometer with a frequency of 5 Hz. The strains between the individual lines on the grid were determined by differentiation of the relative changes of the grid distances. Therefore, the minimum possible gauge length for strain measurement was mm. In the present experiments the chosen gauge lengths for the strain measurement were 5 mm (whole grid length),, 7 (softening zone) and mm (minimum possible gauge length). The damage localization was always inside the respective gauge length. The stress was computed as the measured force at the clamp divided by the cross-sectional area of the center of the specimen. The static experiments were monotonic uniaxial tension tests with a displacement rate of. mm/min. Quasi-static cyclic tests were carried out, to identify the energy dissipation capacity, damage progress and decrease of strength and stiffness during alternating loading as occurs during seismic action. Due to the slow loading rate, inertia forces can be neglected. The quasi-static cyclic tests were performed with two different load schemes. The experiment with the first load scheme is referred to as the tension threshold test (Fig. (b)). In this case, the applied displacement was varied between zero and a prescribed positive value. The experiment with the second load scheme is termed as alternating tension compression test (Fig. (c)). In this test the applied displacement was varied between the same magnitude of positive and negative displacement within the individual displacement levels. The displacement amplitudes of the quasi-static cyclic tests were determined as multiples of the ultimate force (F u ) obtained from the static experiments. The yield point was evaluated with the..4.9 F u method [6], which is based upon initial stiffness and an averaging of slope between.4 F u and.9 F u. In this method, the intersection of two lines is specified. The first line is the straight line through. F u and.4 F u. The second line is the tangent to the load displacement curve which arises from parallel translation of the secant through.4 F u and.9 F u. For

3 T. Rahman et al. / Computational Materials Science 9 (7) Fig.. (a) Boundary conditions for cyclic simulation, (b) load scheme for the tension threshold test and (c) load scheme for the tension compression test. both quasi-static cyclic experiments the applied displacement levels were.5,.5,.75,.,.5,.5,.75,. and.5 mm. The displacement levels were comprised of three cycles each, except the first two displacement levels which were comprised of one cycle each.. Material model.. Plastic-Damage model In the Plastic-Damage model proposed by Lubliner et al. [] and extended by Lee and Fenves [], stiffness degradation due to damage is embedded in the plasticity part of the model. Damage is represented by two independent scalar damage parameters: one for tension (d t ) and one for compression (d c ). This is done because quasi-brittle materials show different damage mechanisms in tension and compression. In tension the damage is associated with cracking while in compression it is associated with crushing. The initial undamaged state and final damaged state of the material under tension and compression are indicated by d t = d c = and d t = d c = respectively. Any intermediate value indicates a partially damaged state. Apart from this, a stiffness recovery scheme is used for simulating the effect of micro-crack opening and closing. The effect of damage is embedded in the plasticity theory and all stress definitions (true stress) are reduced to the effective stress [7]. This enables the decoupling of the constitutive relations for the elastic plastic response from stiffness degradation (damage) response. Consequently, the numerical implementation of the model becomes much simpler. For the plasticity part of the model, a non-associated plasticity scheme is used. Different yield strengths in tension and compression are considered. The yield surface proposed by Lubliner et al. [] is based on modifications of the classical Mohr Coulomb plasticity (Eq. ()). In the following equations, underlined symbols indicate vector or tensor quantities, a line above the stress expressions indicates effective stress. Symbols without an underline are scalar quantities. All strain symbols with a tilde are equivalent strains. In Eq. (), Macauley brackets hi have been used which are defined as: hxi = x if x>, otherwise hxi =. F ðr;~e pl Þ¼ a q ap þ bð~epl Þ ^r max c ^r max rc ð~e pl c Þ ðþ r Stress tensor, r c Uniaxial compressive stress, p Effective hydrostatic pressure, a, c Material constants, q Equivalent effective deviatoric stress, ^r max Max. principal stress, ~e pl Equivalent plastic strain. The material constants a =. and c =, which are typical values for quasi-brittle materials according to [], have been assumed. Since the Plastic-Damage model assumes non-associated flow, a separate flow potential is necessary to determine the direction of plastic flow in the principal stress space. The flow potential chosen for this model is the Drucker Prager hyperbolic function G (Eq. (6)). When high confining stress is present, the function asymptotically approaches the linear Drucker Prager flow potential in the deviatoric plane and intersects the hydrostatic pressure axis at 9 [4]. In Fig. the yield surface and the flow potential function are illustrated in the D principal stress space. The material modeling was performed using an existing implementation of the Plastic-Damage model in ABAQUS.

4 68 T. Rahman et al. / Computational Materials Science 9 (7) flow potential (ψ = 5 ) yield surface ε pl ( 4 ) flow potential (ψ = 4 ) (a) σ ˆ ε pl - -4 ( 5 ) σˆ eccentricity εσt q (b) direction of plastic flow pl ε ψ dilation angle 6 p Fig.. Illustration of: (a) yield surface and flow potentials and (b) dilation angle. The details of the mathematical formulation of the model are given in [,,8] and the ABAQUS theory and analysis manual [4]... Modified Plastic-Damage model The simulation results of the static behavior of the investigated material obtained using the existing Plastic-Damage model in ABAQUS is close to that of the experiment (Section 4). When applying the existing model to cyclic loading, however, considerable model limitations are observed. Referring to the stress strain curve in Fig. 6(a), the existing model is valid up to point. Subsequent to this point, where unloading starts, the material behavior shows different stiffnesses at different stages of unloading and reloading. Possible reasons for this behavior include the reorientation of fibers, the presence of debris inside the micro-cracks and other complex fiber matrix interactions. It is not possible to handle the varying unloading and reloading stiffnesses with the available stiffness recovery effects implemented in the Plastic-Damage model provided in ABAQUS. The existing model in ABAQUS was originally developed for concrete and other quasi-brittle materials. The present material is similar to quasi-brittle materials in a general sense but differs significantly with respect to its unloading reloading behavior. Hence, modifications of the existing model are necessary for the present material. Therefore, the Plastic-Damage model has been implemented and extended using the user-defined material subroutine UMAT in ABAQUS. In the quasi-static cyclic experiments the yield point in compression is not reached and compression damage is absent (d c = ). Damage occurs due to tensile loading which is represented by the scalar tension damage parameter d t (Section.). The total damage parameter d in the modified Plastic-Damage model is correlated with the tension damage parameter d t as d ¼ d t s ðþ s ¼ w ðþ where, s is the stiffness recovery factor and w is the weight factor that controls the stiffness recovery. A value w = implies complete stiffness recovery corresponding to d = whereas w = implies no stiffness recovery corresponding to d = d t. On the yield surface, d is obtained from Eqs. () and (). The evolution of yield stress, tension damage, d t and weight factor, w are functions of plastic tensile strain, e p t. The material subroutine UMAT requires strain softening, damage evolution and stiffness recovery curves. During unloading and reloading in the elastic domain, d is redefined in the material subroutine UMAT based on rules derived by observing the unloading/reloading slope (E) in the uniaxial quasi-static cyclic stress strain curves. The corresponding damage parameter d is obtained from the varying slope (E) and the initial stiffness (E )as d ¼ E ð4þ E The determination process of the rules controlling d in the elastic domain and strain softening, damage evolution and stiffness recovery curves are discussed in Section Modeling: Static loading conditions 4.. Determination of material parameters Modeling of the material behavior under static loading conditions has been performed using the finite element software ABAQUS with its existing Plastic-Damage material model. The material parameters required for the model can be categorized into three types, namely elasticity, plasticity and damage. The elasticity parameters, Young s modulus E = 87 MPa and Poisson s ratio m =.9, were obtained from the static experiments. For plasticity and damage, strain softening and damage evolution curves are required. In addition to these parameters, a further material constant called dilation angle w is necessary. In the following the determination process of the above mentioned curves and of the dilation angle w are discussed Strain softening curve The strain softening curve is provided in the form of a yield stress versus inelastic strain relation. To avoid mesh sensitivity this curve is converted to a yield stress versus

5 T. Rahman et al. / Computational Materials Science 9 (7) Damage Displacement [mm] (a) Displacement [mm] (b) Fig. 4. Material input curves for the static simulation: (a) strain softening and (b) damage evolution curve. displacement relation (according to [4]). The inelastic strain is identified by subtracting the elastic strain from total strain as given in Eq. (5). e in ¼ e e el ) e in ¼ e r ð5þ E Therefore, using Eq. (5), the inelastic strain e in can be computed from the experimental stress strain curve. The experimental stress strain curve in the softening zone of the specimen is used for this purpose. Furthermore the inelastic strain e in is converted to displacement by multiplying with the length of the softening zone. In this way, the yield stress versus displacement relation is determined, Fig. 4(a) Damage evolution curve The evolution of damage has been assumed to be the same as in quasi-brittle materials such as concrete. The damage evolution is given as a damage versus displacement relation (Fig. 4(b)). From the damage evolution curve shown in Fig. 4(b), pronounced damage increase occurs initially and later gradually approaches. This is consistent with the experimental observation that the matrix in fiber reinforced gypsum cracks near the tension yield point and most of the load carrying area is lost. At a later state of loading in the post-peak regime, fiber breaking and pull-out occur gradually and the rate of damage development is decreased Dilation angle The dilation angle w controls the orientation of the flow potential function G, which is defined as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G ¼ ðer t tan wþ þ q p tan w ð6þ where, the eccentricity (see Fig. (b) and [4] for further explanations), e =. (a typical value for quasi-brittle materials []) and r t =.9 MPa is the yield strength in tension derived from the experiment, Fig. 5(a). The equivalent effective deviatoric stress q and the hydrostatic stress p are defined as p ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi r : I; q ¼ r S : S where I is the unit matrix and S the effective stress deviator. It is not possible to obtain w directly from the results of the static experiment. Therefore, the dilation angle was determined by applying inverse modeling by comparing the simulated and experimentally measured stress strain curves. Experiment Simulation L = mm L = 5 mm 4 (a) d ψ = 5, mesh size = mm ψ = 5, mesh size = 4 mm ψ = 4, mesh size = mm experiment, L = 5 mm d (b) (c) Fig. 5. Static simulation results: (a) at length scales of mm and 5 mm, (b) comparison of dilation angle w and mesh size, and (c) contour plot of damage variable d.

6 7 T. Rahman et al. / Computational Materials Science 9 (7) FE Modeling The geometry and boundary conditions applied in the FE model conformed to the experimental conditions. The D simulation was performed with a 4-noded quadrilateral plane stress elements (CPS4R). Fig. (a) shows that the length of the minimum crosssection area of the specimen with constant width is 5 mm. Assuming stochastically distributed micro-defects, an equal likelihood of failure occurrence exists throughout this length. Therefore, a weak section has been inserted at the center of the specimen to enforce strain softening in that area. The width of the weak section corresponds to the width of the crack band during the experiment. It consists of a row of finite elements across the center of the specimen. A slightly lower yield stress (.9 MPa) is assigned to the weak section compared to the rest of the specimen (. MPa). The results of the static simulation are presented in Fig. 5(a) (c). 4.. Discussion of static simulation results The simulation results for the stress strain relationship at the gauge lengths (Section ) of and 5 mm show good agreement with the experiment (Fig. 5(a)). Material models for strain softening are usually subject to mesh sensitivity effects [9]. The Plastic-Damage model takes care of the mesh sensitivity using the fracture energy criterion of Hilleborg [9]. Fig. 5(b) shows that the effect of mesh size ( mm versus 4 mm) on the simulated stress strain behavior is very small. Therefore, it is concluded that the measure to reduce mesh sensitivity effects in the Plastic- Damage model works well with the present material. Fig. 5(b) also shows the effect of the dilation angle on the stress strain behavior. A dilation angle of 5 was determined applying inverse modeling (see Section 4.). The stress strain curve with a dilation angle of 5 has been compared with the result obtained with a dilation angle of 4. It is observed that the stress strain curve obtained with a dilation angle of 4 remains much higher than that with 5. The dilation angle controls the shape and orientation of the plastic flow potential and consequently the direction of plastic flow (Eq. (6)). Referring to Fig. (a) it is observed that in the case of a dilation angle of 4, the plastic flow possesses components in the directions of the principal stresses ^r and ^r, although the direction of loading is in the direction of ^r. The component of plastic flow in the direction of ^r consumes additional energy and the resulting stress strain curve is, therefore, higher in the case of w =4. For the dilation angle of 5 the direction of plastic flow is almost identical to the direction of loading and thus, shows rather good agreement with experimental results. Fig. 5(c) shows the distribution of the damage variable d at the almost fully damaged material state (d ) at.5% strain. This distribution is expected because the softening zone is supposed to occur at the weakest location. Thus the contour plot of damage parameter d is capable to visualize the damaged state of the material. 5. Modeling: Quasi-static cyclic loading conditions 5.. Determination of material parameters To explain how the material parameters required by the modified material model (Section.) are determined, three intermediate load cycles of the alternating tension compression test have been chosen (Fig. 6(a)). The determination of the input curves namely strain softening (Fig. 7(a)), damage evolution (Fig. 7(b)) and stiffness recovery (Fig. 7(c)) are discussed. Additionally, the determination of the rules governing the varying stiffness degradation/recovery which control the damage parameter d during unloading and reloading are also discussed. Further.5 % Experiment (a) Experiment Simulation % (b) 4 Fig. 6. Illustration of the determination of the material parameters for the cyclic simulation: (a) experimental data, (b) enlarged view of experimental data compared to the simulation.

7 T. Rahman et al. / Computational Materials Science 9 (7) Yield stress [MPa] B A B A C.8 D.75 B.6 C C.5 D.4 D.5 A Plastic strain [%] Plastic strain [%] Plastic strain [%] (a) (b) (c) Tension damage d t Fig. 7. Material input curves for cyclic simulation: (a) strain softening, (b) damage evolution and (c) stiffness recovery. Weight factor w details are available in []. Relevant points in the stress strain curve shown in Fig. 6(a) and (b) have been designated as,, etc Strain softening curve The strain softening behavior of the material can be extracted from the envelop of the stress strain curve obtained from quasi-static cyclic experiments (Fig. 6(a)). The strain softening behavior is provided in the material model in the form of a yield stress versus plastic strain curve. Point in Fig. 6(a) provides a yield stress of.4 MPa. This value defines point A of the strain softening curve in Fig. 7(a). From points, 7 and 9 yield stresses of three more points of the strain softening curve are obtained with respective values of.7,.4 and. MPa. When full unloading occurs from these points, the points 4, 8 and are reached, respectively. The total strains at points 4, 5 and correspond to the plastic strains at points, 7 and 9. The values are.54,.5 and.9, respectively. Thus points B, C and D of the strain softening curve are obtained. The complete strain softening curve shown in Fig. 7(a) is constructed in the same manner from further cycles Damage evolution curve The damage evolution curve can be obtained by computing the tension damage parameter d t and the corresponding plastic strain at certain points of the uniaxial stress strain curve (Fig. 6(a)). The tension damage d t can be computed from the degraded unloading stiffness E using Eq. (4) (here, d is replaced by d t ). The first d t value of the damage evolution curve can be obtained from the unloading stiffness beyond point. In fiber reinforced gypsum, a varying stiffness recovery is observed through the various unloading/reloading stages. The most damaged state in this first cycle is reached in segment 4 5 where the stiffness is reduced to only.9 MPa. With the initial elastic stiffness E = 87 MPa the corresponding tension damage from Eq. (4) is d t =.994. The corresponding plastic strain can be identified in the same way as for the strain softening curve. Thus, a plastic strain of.54 can be associated with point. Hence, corresponding to point in Fig. 6(a), point B in Fig. 7(b) can be identified. The points C and D can be identified in the same way from points 7 and 9. From the next cycles it is possible to get additional points of the damage evolution curve Stiffness recovery curve According to Eqs. () and (), the damage parameter d is controlled by the stiffness recovery s, which in turn is expressed in terms of the weight factor w. As opposed to the existing model in ABAQUS [4], w is not assumed to be constant, but rather to vary with plastic strain in tension e p t. This variable w with plastic strain is given as a material input to the modified model to account for stiffness recovery. The damage parameter d corresponding to point in Fig. 6(a), can be found from the slope of the segment 4. Hence, from Eq. () and (), w is identified as w = d/d t. Thus from point in Fig. 6(a) the point B in Fig. 7(c) is derived. The corresponding plastic strains e p t are evaluated in the same way as described for the strain softening curve. Similarly, the points from the next cycles can be derived and finally the stiffness recovery curve as shown in Fig. 7(c) can be constructed. Although d could be identified directly from Fig. 6(a) using Eq. (4), to maintain the framework of the existing Plastic-Damage model, d is expressed in terms of d t and w Rules governing the stiffness degradation/recovery in the elastic domain In order to describe the rules governing the stiffness degradation and recovery in the elastic domain, three intermediate cycles from the alternating tension compression test are compared with the simulated ones. The simulated curve has been divided accordingly into several segments (Fig. 6(b)). Segment belongs to the initial loading in the undamaged state (d = ). In segment (yield surface), d is controlled by Eqs. () and () in accordance with the strain softening, damage evolution and stiffness recovery curves. The next segments starting from 4 5 up to 8 9 lie in the elastic domain and show varying stiffness degradation and recovery effects. In the following, the rules governing the varying unloading/reloading stiffness (consequently, d from Eq. (4)) are derived by analyzing the experimental data from the first loading cycle. Later it will be shown along with the simulations performed (Figs. 8 and 9) that the presented rules are applicable for the further cycles as well.

8 7 T. Rahman et al. / Computational Materials Science 9 (7) Displ. level =.5 mm Displ. level =.5 mm Displ. level =.75 mm Displ. level =. mm Experiment Simulation Fig. 8. Cyclic simulation results of the tension threshold test at all displacement levels Displ. level =.5 mm Displ. level =.75 mm Displ. level =. mm Experiment Simulation Fig. 9. Cyclic simulation results of the tension compression test at all displacement levels. Segment 4 5 (Unloading in tension): The starting point of segment 4 5 (Fig. 6(b)) is the point of first unloading. From this point, the modified model deviates from the existing one. According to the existing model, unloading should follow path 5. The experimental results, however, show an unloading path with higher stiffness. Hence, in the modified material model the path 4 is followed instead of path 5. In the developed subroutine UMAT the slope of segment 4 is set three times higher than that of segment 5. To determine the position of point 4, it is assumed that this point stays at a stress level of % of that of point according to the observations in the experiment. The position of point 5 is already known, because the x-distance of point 5 from the origin along the horizontal axis is the plastic strain accumulated along the path. Segment (Reloading in compression): The unloading path 5 6 (Fig. 6(b)) is followed in accordance with the accumulated tension damage d t. Stiffness recovery does not play a role here and the slope of 5 6 is related to d t through Eq. (4). In the modified model, the point 6 is assumed to be the position from where the material starts to regain its stiffness due to crack closure in compression. The strain difference De 6 7 between point 6 and 7 is derived from the experiment as.6% (alternating tension compression) and.5% (tension threshold) and given as a material parameter to the material UMAT subroutine. Since the stress strain state of point 7 cannot be calculated within the material model, it is also given as a material input. In the present case it is 4 MPa at.8% strain. The unloading in compression begins at point 7. Once the position of this point is given as a material input, this point and the subsequent unloading points are stored as state variables in the material UMAT subroutine. Therefore, defining the position of the first unloading point in compression (here point 7) as a material parameter is enough to predict the behavior of the subsequent loading cycles. Segment 7 8 (Unloading in compression): From point 7 (Fig. 6(b)), compression unloading occurs. Since the material regains its initial stiffness during this step, the unloading path is very steep. According to the experimental data the slope is identical to that for the initial elastic stiffness (87 MPa). Segment 8 9 (Reloading in tension): At the beginning of this segment, the cracks reopen. As a result, the loading path 8 9 (Fig. 6(b)) remains parallel to the path 5 6. From point 6, the stiffness starts to increase due to reorientation of the fibers along the tensile loading direction. The strain difference De 9 between point 9 and point is observed to be.5% and is taken as a material parameter. Since the UMAT material subroutine stores the stress and strain level at the previous unloading point as a state variable, the position of point 9 can be determined. Since the positions of both of the points 9 and are known, the slope of path 9 can be determined. Furthermore, this slope has to be reduced by 5% according to the experimental results. From point 9, the simulated loading path continues to point. To summarize: To simulate the quasi-static cyclic stress strain curve, in addition to the Young s modulus and the Poisson s ratio, the input of a strain softening curve, a damage evolution curve and a stiffness recovery curve are required. Further material parameters are the stress strain state of the first unloading point in compression (here point 7) and the strain difference in tension (here De 9 ) and compression (here De 6 7 ). 5.. FE Modeling The material response in the softening zone of the specimen has been modeled with the modified material model, using a single element (Fig. ). A 8-noded, linear interpolating, hexahedral solid element (CDSR) was used for the quasi-static cyclic simulations. Because the structural response of the entire specimen cannot be captured with a single element, the experimental stress strain behavior obtained from the softening zone based on the minimum possible gauge length ( mm) has been simulated. Approximately the same level of strain as derived from the experimentally obtained stress strain response (Fig. (b) and (c)) has been applied to the single element. All the dimensions of the single element have been chosen as unity. Therefore,

9 T. Rahman et al. / Computational Materials Science 9 (7) the magnitude of the applied strain was the same as the displacement applied in the experiment. The geometry and the boundary conditions applied to the single element are shown in Fig. (a) (c). The same values for Young s modulus, E = 87 MPa and Poisson s ratio m =.9 have been used as for the simulation under static loading conditions. Other necessary material parameters and their determination process have been discussed in Section 5. and visualized in Figs. 6 and 7. The simulation of the material response in the softening zone during the tension threshold test has been performed at four applied displacement levels (.5,.5,.75 and. mm). Fig. 8 shows the simulation results at these displacement levels. The simulation results for the alternating tension compression tests have been presented in a similar way (Fig. 9). Simulations are shown for three displacement levels (.5,.75 and. mm). 5.. Discussion on cyclic simulation results The basic reason for the modification of the existing Plastic-Damage model was to quantitatively reproduce the varying unloading reloading stiffnesses observed in the quasi-static cyclic experiments (Section 5.). Comparison of the simulation results in Figs. 8 and 9 show that the basic changes of stiffness during unloading and reloading have been captured. In particular, it is observed that the derived rules from the first load cycle governing the stiffness degradation and recovery in the elastic domain apply for the subsequent load cycles. The non-linear stress strain response with continuously varying unloading reloading stiffnesses has been approximated by a multi-linear stress strain response. In the modified model, the close representation of the dissipated energy inside the hysteresis loops is emphasized and the simulation results show good agreement with the experiment on this aspect. For example the energy dissipation of the mid-cycle for the tension threshold test (displacement level of.5 mm, Fig. 8) is.6 MJ/m (experiment) and.8 MJ/m (simulation), respectively. In the case of the alternating tension compression test (displacement level of.75 mm, Fig. 9) the mid-cycle energy dissipations are found as.477 MJ/m (experiment) respectively.455 MJ/m (simulation). Therefore, for these cycles approximately 6% deviation is observed regarding dissipated energy in the tension threshold test while it is only about 5% for the alternating tension compression test. 6. Conclusion and summary This work deals with the finite element analysis of the material behavior of a cellulose fiber reinforced gypsum matrix composite which is used as a panel material in the building industry. The material is characterized by pronounced energy dissipation capacity which makes it suitable for seismically loaded structures. The finite element program ABAQUS was used. For the simulation of the material behavior the Plastic-Damage model of Lubliner et al. [] and Lee and Fenves [] has been used. The simulations were performed for static tensile and quasi-static cycling loading of necked specimens of a cellulose fiber reinforced gypsum composite. The static simulation results show good agreement with the experiment for different gauge lengths ( and 5 mm) with near mesh independence in comparison to experiments with a unique set of model parameters. The introduction of a slightly pre-weakened section at the center of the tensile specimen which reflects the heterogeneities in the material led to a close representation of the material behavior of the static experiment. Plotting the distribution of the damage parameter d, showed a realistic localization of damage. The dilation angle w, which is a parameter of the Plastic- Damage model, has been explored using inverse modeling. In the Plastic-Damage model the dilation angle w controls the direction of the plastic flow. A choice of the dilation angle w = 5 made the direction of plastic flow identical to the direction of loading, leading to less energy dissipation compared to a dilation angle of 4 and the resulting stress strain curve showed close agreement with the experiment. Choices of dilation angle other than w = 5 (e.g. w = 4 ) involved components of plastic flow perpendicular to the direction of loading, which consumed extra energy giving a higher calculated stress than measured. Due to the complex behavior of the material in the unloading and reloading regime during quasi-static cyclic loading, the existing model in ABAQUS was modified within the elastic domain. In the unloading and reloading regions of the stress strain curve the material behavior shows multiple slopes. An ABAQUS material routine (UMAT) has been developed based on the formulation of the existing Plastic-Damage model, but taking into account the complex behavior in the unloading and reloading. With this newly developed material routine it is possible to describe the loading cycles in close agreement with the experiment. The approximation of the energy dissipation, which is the area below the loading curve, was improved considerably. The simulations were performed for a single element in the region of strain localization of the specimen. Based on the achievements in this work, further studies could extend the material model to the structural level, which means in this case the whole specimen with and without mechanical fasteners. A further goal of future investigations will be the simulation of a seismically loaded component composed of fiber reinforced gypsum plates which fixed to a timber frame by dowels. Acknowledgements The authors gratefully acknowledge the financial support by the German Research Foundation (DFG) within the collaborative research center (SFB) 8 Characterisation of Damage Evolution in Fibre Reinforced Composites by Nondestructive Testing Methods (projects A and C5) [].

10 74 T. Rahman et al. / Computational Materials Science 9 (7) References [] S. Aicher, W. Klöck, in: Proceed. Int. Conf. on Wood and Wood fiber Composites, Otto-Graf-Institute, Stuttgart,, pp [] J. Lubliner, J. Oliver, S. Oller, E. Oñate, Int. J. Solids Struct. 5 () (989) 9 6. [] J. Lee, G.L. Fenves, J. Eng. Mech. 4 (8) (998) [4] N.N., ABAQUS Theory manual, version 6.4,, Chapter 4.5.,, pp.. [5] S. Aicher, R. Finn, Otto-Graf J. 5 (4) 9. [6] B. Dujič, R. Žarnič, Int. Council for Research and Innovation in Building and Construction. Working Commission W8. in: Proceedings Meeting 6, Colorado, paper CIB-W8/ [7] J. Lemaitre, A Course on Damage Mechanics, second ed., Springer, Paris, 996. [8] W. Lutz, T. Rahman, S. Schmauder, R. Finn, S. Aicher, Simulation cellulosefaserverstärkter Gips-Verbundwerkstoffe, in: F. Dehn, K. Holschemacher, N.V. Tue (Eds.), Faserverbundwerkstoffe, Bauwerk- Verlag, Berlin, 5, pp [9] A. Hilleborg, M. Modeer, P.E. Petersson, Cement Concr. Res. 6 (976) [] T. Rahman, Simulation of the damage development in components made of cellulose fiber reinforced gypsum material, Master Thesis, Institut für Materialprüfung, Werkstoffkunde und Festigkeitslehre (IMWF), Universität Stuttgart, 5. [] G. Busse, Chracterisation of damage evolution in fiber reinforced composites by nondestructive testing methods, Final report of the collaborative research center, SFB 8, University of Stuttgart,.