COMPARATIVE STUDY OF LAMINATED GLASS BEAMS RESPONSE

Transcription

1 XVIII GIMC Conference Siracusa, September 1 COMPARATIVE STUDY OF LAMINATED GLASS BEAMS RESPONSE Loredana Contrafatto 1, Giuseppe Tommaso Di Venti 1 and Paola Posocco 1 Department of Civil and Environmental Engineering, University of Catania, Italy loredana.contrafatto@dica.unict.it, diventi@dica.unict.it MOSE Lab, Department of Chemical Environmental and Raw Materials Engineering, University of Trieste posocco@dicamp.units.it Keywords: Laminated Glass, Interfaces, Strong Discontinuities. Abstract. Failure of laminated glass units is characterized by the growth and propagation of interfaces that can arise in unpredictable location of the layers. Theoretical and numerical procedures can be used to predict the peak load and the stress distribution. In the paper different approaches, based on identical hypotheses, are compared with reference to a laminated glass units subjected to a static transversal load. An experimental test has been performed to validate the predictions. The possibility of establishing a correlation between the fracture energy and the work of adhesion has been investigated by means of molecular dynamics simulations of the PVB-glass interface. 1 INTRODUCTION The most recent developments in technology have made possible glass is used as major structural elements such as beams and columns. Laminated glass units for structural applications consist of two layers of glass bonded together by a thin interlayer, usually polyvinyl butyral (PVB). Under most load conditions the assembly presents some advantages over monolithic glass of the same nominal thickness with respect to impact resistance and post fracture behaviour. The failure is dominated by a lot of factors as thickness of the glass layers and of the interlayer, temperature, composition of the interlayer. Despite the increasing interest on the matter, while experimental data on laminated glass exist, theoretical models are scarce. Moreover, existing theoretical analyses are restricted by additional simplifying assumptions that the actual structural behaviour of the laminated glass beam lies somewhere between two limiting cases: the layered limit and the monolithic limit. Analytical models that predict stress development and ultimate strength of laminated glass beams and plates, able to correlate the structural performance with the geometrical and mechanical parameters, have been presented in [1,, 3,, 5]. They are useful in assessing the safety and predicting the failure strength, in order to identify the basis for rational design with glass-polymer laminates. However, the analysis of the post-critic behaviour and the prediction of crack paths are not frequent in the literature. In [] an analytical model based on cumulative damage theory is discussed for the prediction of the probabilities of inner glass ply breakage. 1

2 Taking into account that, on the whole, glass units exhibit a damaging-fracturing behaviour and discontinuities in the displacement field can arise in unpredictable locations, it seems possible that the problem can be studied in a different area. Indeed, this phenomena can be effectively described by means of mechanical models that incorporate the kinematics of strong discontinuities, obtained by an enrichment of the displacement field with a discontinuous term. A number of formulations is based on the Strong Discontinuities Approach (SDA) [7, 8], the differences originating from the numerical implementation in the Finite Element Method. Elements with Embedded Discontinuities [9] and the extended Finite Element Method [1] are the main tools for the discrete description of the problem. In this paper a comparison between theoretical and numerical predictions of the response of laminated glass beams is presented. Specifically the results in term of stress and peak load by the theoretical procedure in [] have been compared with the numerical predictions obtained by means a Finite Element discretization, both in the case of linear elasticity. Experimental results are considered too. Moreover a general model considering damaging embedded interfaces in the context of the SDA has been considered [11, 1]. The model is obtained in a consistent thermodynamic framework from a generalized variational principle incorporating internal variables, thus allowing also for nonlinear continua. The principle starts from basic energetic considerations. The strain field is decomposed into a compatible and an enhanced term. The numerical implementation of the algorithm is based on the formal analogy between the equations of the enriched continuum and the theory of classical plasticity, following recently developed strategies [13], [1],[15]. The growth or the activation of interfaces is ruled by specific activation functions. In this way it is obtained a structure of the numerical algorithm that allows the use of the procedure inside classical F.E. codes for the solution of the equilibrium problem of elastic-plastic solids. The model has been specialized for the application to laminated glass beams, introducing cohesive fracture-like criteria, and applied to investigate the flexural behaviour of a simply supported laminated glass beam and the growth and propagation of interfaces inside the glass layers and across the polymeric interlayer. The fracture path due to a static transversal load has been compared with experimental evidence from a standard three point bending test. The predictions about the initiation of cracks inside the layers are also compared with the analytical and numerical results previously obtained. Some remarks about the identification of the constitutive parameters defining the cohesive fracture-like criterion ruling the interface behaviour in the SDA model close the paper. An attempt has been made in estimating the fracture energy from the molecular work of adhesion among the layers by means molecular dynamics simulations. ANALYTICAL AND NUMERICAL MODELS VS. EXPERIMENTAL TESTS The behaviour of laminated glass beams is analyzed by means different approaches and the results are compared with experimental tests. The laminated composite beam under investigation is characterized by two layers of thin glasses and one layer of PVB. The beam is simply supported at the ends and an increasing vertical load is applied at mid-span. The geometry is shown in fig.1. The span length L of the beam is.5 m; the cross section width is 15 cm, the glass thickness h is 8 mm for both the layers and the interlayer thickness t is.38 mm. Glass elastic modulus and interlayer shear modulus are taken as.5 GPa and 187 kpa, respectively. The Poisson s ratio of glass and PVB are taken to be.3 and.9, respectively. Firstly a D finite element model has been developed and solved with ADINA code, version

3 Glass Beam PVB h 1 t h b L L Figure 1: Laminated glass beam. 8., under the hypothesis of perfect elasticity for both the materials. A 8-nodes discretization of 5 x (1++1) plane stress elements has been used. The results are then commented with reference to the prediction of the mathematical model for the behaviour of laminated glass beams of Aşik and Tezcan []. It predicts that simply supported laminated glass beams behave close to monolithic glass beams. Their behaviour is bounded by two limiting cases which are monolithic and layered behaviour. The model, derived by using large deflection theory, shows nonlinear behaviour in the case of a fixed supported beam but gives linear results for a simply supported laminated glass beam. In this latter case it is based on the following differential equation: where ( ) λ = Gb A 1 +A Et A 1 A + h t I d N 1 dx λ d N 1 = βq (1) dx β = Gb h t h Et I t = h 1 + h + t () N 1 being the axial force in the top glass layer, E the elastic modulus of glass, G the shear modulus of PVB and I the sum of the moment of inertia of the two glass layers. h 1,h and t are the glass and PVB thickness respectively; b is the unit width. It results N 1 = N The analytic solution gives the following stresses at the surfaces of the plies: σ top 1 = M I σ bottom 1 = M I h 1 + N 1 A 1 h 1 + N 1 A 1 σ top = M I σ bottom = M I h + N A h + N A (3) The behaviour for monolithic, laminated and layered beam is illustrated in fig. 3, the transition being influenced by the thickness of the interlayer and its stiffness, and compared with the ADINA simulation, denoted in the pictures legend as Laminated ELD. Moreover the results from experimental tests (denoted in the legend Laminated test ) are depicted, showing a good agreement between experimental, theoretical and numerical predictions. The experimental results were obtained performing a standard three point bending test. The axial strain in the section at mid-span and at two symmetrically located sections with respect to the mid-span were measured in both the glass layers. The geometrical position of the strain gauges is shown in fig.. 3

4 Figure : Geometrical position of the strain gauges. Vertical displacement 1 Normal stress midspan; x=5mm 8 1 w [mm] 3 5 Laminated ELD Laminated test x [mm] 8 Laminated ELD Laminated test σ x [MPa] (a) (b) 1 Normal stress; x=3mm 1 Normal stress; x=7mm Laminated ELD Laminated test σ [MPa] x 8 Laminated ELD Laminated test σ [MPa] x (c) (d) Figure 3: Aşik and Tezcan model [] and ADINA solution. (a) Vertical displacement at midspan. (b) Normal stress at midspan. (c) Normal stress at x=3 mm. (d) Normal stress at x=7 mm.

5 Loredana Contrafatto, Giuseppe Tommaso Di Venti and Paola Posocco A number of samples were tested and mean values are considered in fig. 3. Fig. reports the laboratory apparatus and the cracked beam. For instance for sample N. 1 the peak value of the load was 133 N and the corresponding vertical displacement at mid-span 3. mm. (a) (b) (c) Figure : Laboratory apparatus (a) Testing machine. (b) Stain gauges. (c) Cracked sample. Strain Strain gauges [µ m/m] Stress [MPa] Table 1: Vertical load vs measured axial strain and calculated axial stress in the glass layers. Table 1 reports the values of the measured strain at the peak load and the value of the axial stress evaluated under the hypothesis of elastic behaviour of the glass, assuming the Young modulus equal to.5 GPa. In fig. 5 the load-axial strain curve is represented. From fig. 3 derives that the simplificative hypothesis of elasticity for all the materials of the assemblage leads to a good estimation of the stress state at the peak load. 3 STRONG DISCONTINUITIES APPROACH In order to study the post-critic behaviour of simply supported two-ply laminated glass beams, a model based on the Strong Discontinuities Approach [11] has been applied to simulate the growth and propagation of fracture in the composite unit under investigation. The hypothesis of elastic behaviour for both the materials of the composite beam has been formulated, until interface occur, where an elastic-softening behaviour is introduced. The interfaces growth is assimilated to flexural cracks opening in the glass and to shear slip in the interlayer. The numerical analysis of model in fig. 1 has been performed. According to geometrical and material data from [], the span length L of the beam is.8 m; the cross section width is 1 cm, 5

6 18 Load Strain Load [N] 1 8 SG1 SG SG3 SG SG5 SG SG7 SG8 SG9 SG1 8 Strain [µm/m] Figure 5: Load-strain curves. the glass thickness is 5 mm for both the layers and the interlayer thickness is.38 mm. Glass elastic modulus and interlayer shear modulus are taken as.5 GPa and 187 kpa, respectively. The Poisson s ratio of glass and PVB are taken to be.3 and.9, respectively. An increasing prescribed displacement is applied at mid-span. The activation functions are recovered in table where t S,t Sn and t Sm are respectively the stress on the surface S and its normal and tangential component and f tu is the tensile strength of the material. χ S is an internal scalar force, defined on the surface S, ruling the evolution of the softening behaviour of the interface; it is conjugated to an internal kinematic variable α Se, whose constitutive law is such that the cohesive fracture dissipation energy is equal to the fracture energy of the material. glass f(t S, χ S ) = t Sn f tu + χ S f tu = PVB f(t S, χ S ) = [t S m ].5 f tu + χ S f tu = 1 Table : Unit materials activation function. For instance a linear or an exponential softening can be assumed: χ S = f tu G f α Se f tu α S e χ S = f tu (1 e G f ) () G f being the fracture energy, so that the softening modulus is given by H = f tu G f. In fig. the prediction of the initial crack and its evolution in the layers is shown. Specifically the zoom in the mid-span of the internal softening variable α Sp is depicted. No interpolation inside the element is introduced in the graph so that the actual value at the Gauss points is represented. The interface begins at the midpoint of the bottom beam and propagate along the vertical direction in the top beam. At a certain stage the shear force in the interlayer exceeds its limit value and the irreversible mutual shear displacement takes place. In order to verify the correct prediction of the initial location of interfaces in the glass layers and in the interlayer, the results are ones again compared with those of a D finite element model developed and solved with ADINA code. The 8-nodes discretization of 8 x (1++1) plane stress elements has been used. The hypothesis of initial perfect elasticity has been assumed for the glass and PVB bulks. Indeed, the material has an elastic behaviour until interfaces occur, so

7 .E+ 1.88E- 3.77E- 5.5E- 7.53E- 9.E- 1.13E-5 1.3E E-5 1.7E E-5.7E-5.E-5 Min =.E+ Max =.E-5 Time = 3.E-.E+ 3.89E E E- 1.55E- 1.9E-.33E-.7E- 3.11E- 3.5E- 3.89E-.7E-.E- Min =.E+ Max =.E- Time =.7E-1 (a) (b).e+ 8.5E E-.5E- 3.E-.7E- 5.13E- 5.98E-.83E- 7.9E- 8.5E- 9.E- 1.3E-1 Min =.E+ Max = 1.3E-1 Time =.E-1 (c) Figure : - SDA prediction of the post post-critic behaviour. (a) Initial crack at midspan of the bottom glass beam. (b) Diffusion of cracks in the bottom glass beam. (c) Diffusion in the top glass beam. that, using the activation function in table, that at the first stage when χ = is a Rankine-like criterion for glass, cracks rise where the maximum value of the tensile stress is achieved. This first happens at the bottom surface of the bottom glass beam. The results are also commented with reference to the prediction of the mathematical model of Aşik and Tezcan [], described in the section. In the case of coupled response both the analytical model and the ADINA numerical simulation predict that the activation candidate point is the midpoint of the bottom surface, as it is predicted by the SDA simulation of fig.. Model [] predicts the same curvature for both the glass beams. This result derives from the kinematic hypothesis and it is independent on the interlayer stiffness and beam slenderness. Actually, the top beam exhibits a greater curvature than the bottom beam, the difference becoming larger with the decrease of the slenderness and of the interlayer stiffness. In any case, the behaviour is strongly influenced by the stiffness of the interlayer, ranging from layered to monolithic for increasing values of the shear modulus. In fig. 7 the comparison between the prediction of model [], denoted in the pictures legend as Laminated, and the ADINA simulation ( Laminated ELD ) is reported for two different slenderness ratio ρ = (h 1+h +t), namely ρ L 1 = ( ) =.15 1/8 and 8 ρ = ( ) 3 =.3 1/3 and three different value of the PVB shear modulus G. In the case of deep beam and low interlayer shear modulus the behaviour tends to the layered one and the maximum of the tensile stress is reached at the bottom surface of the top beams. (fig. 7(b)). In this situation the crack propagation starts from the upper beam, as it can be 7

8 Normal stress midspan; x=l Normal stress midspan; x=l Laminated ELD Laminated ELD σ [MPa] x σ [MPa] x (a) (b) Normal stress midspan; x=l Normal stress midspan; x=l Laminated ELD Laminated ELD σ x [MPa] σ x [MPa] (c) (d) Normal stress midspan; x=l Normal stress midspan; x=l Laminated ELD Laminated ELD σ [MPa] x σ [MPa] x (e) (f) Figure 7: Uniaxial stress. Aşik and Tezcan model [] and ADINA solution. (a) ρ 1 1/8 - G =.18 (b) ρ 1/3 - G =.18 (c) ρ 1 1/8 - G = 1.87 (d) ρ 1/3 - G = 1.87 (e) ρ 1 1/8 - G = 1.87 (f) ρ 1/3 - G =

9 observed in fig. 8 in which the prediction of the SDA model is reported. The same results is predicted by the ADINA simulation. Successively cracks also arise in the bottom beam and follows the same evolution in both the beams..e+.3e-5.8e-5 7.1E E E- 1.E- 1.E- 1.87E-.1E-.3E-.57E-.81E- Min =.E+ Max = 3.38E- Time =.E-.E+.93E-5 9.8E-5 1.8E- 1.97E-.E-.9E- 3.5E- 3.9E-.E-.93E- 5.E- 5.9E- Min =.E+ Max =.79E- Time = 3.E- (a) (b).e+ 3.3E-.85E- 1.3E E E-3.E-3.E-3.7E-3 3.8E-3 3.3E E-3.11E-3 Min =.E+ Max =.3E-3 Time = 8.E-.E+ 1.5E-3 3.3E-3.55E-3.7E E-3 9.1E-3 1.E- 1.1E- 1.3E- 1.5E- 1.7E- 1.8E- Min =.E+ Max = 1.89E- Time = 1.7E-1 (c) (d) Figure 8: Laminated deep beam - ρ = ρ 1/3 - SDA prediction of the post post-critic behaviour. (a) Initial crack at mid-span of the top glass beam. (b) Diffusion of cracks in the bottom glass beam. (c) Diffusion in the top glass beam. REMARK ABOUT THE IDENTIFICATION OF INTERFACE PARAMETERS In the previous section the constitutive behaviour of the interface, representing the crack opening into the glass and PVB layers, has been defined. It is given by a cohesive tractionseparation law sketched in fig. 9. Two parameters identify the model: the tensile strength of the material f tu and the fracture energy G f. They are usually determined by means of experimental tests. The fracture energy can be macroscopically obtained by means experimental tests, i.e. the CSS test [1] and the TCT test [17, 18]. Specifically the goal of the TCT test is the measure of the force arising at the interface under prescribed displacement (fig. 1). The macroscopic energy balance gives the partition of the external work into the two components: 1) the energy responsible for the deformation in the polymer, that, if all the irreversible phenomena develop in the fractured zone, can be assumed equal to elastic energy; ) the energy 9

10 f tu χ S G f 1 = f tu H α S p Figure 9: Interface constitutive model Figure 1: TCT test scheme responsible for the fracture opening. The energy balance is written: P dδ = dw e + bγ da (5) where dδ is the fracture length increment, P the applied load, W e the polymer elastic energy, b the sample width and Γ the fracture energy at the interface (see fig. 1). The displacement increment dδ can be expressed as a function of the nominal strain dδ = εda. Assuming an elastic linear behaviour for the polymer we obtain: P = b(γ Eh) 1/ ) ε = (Γ /Eh) 1/ ) a = δ/ε () where E is the Young modulus and h the polymeric layer thickness. Therefore the fracture energy can be estimated by the measurement of P and ε. In the present case its value is in the range 1 - J/m [17]. Numerical simulations based on molecular dynamics (MD) calculations have been performed to evaluate the interfacial energies between the PVB polymer layer and the glass surface, taking into account the effect of water moisture, ions, and plasticizer on the adhesion strength. Molecular bonding is the most widely accepted mechanism for explaining adhesion between two surfaces in close contact. It entails intermolecular forces between adhesive and substrate such as dipole-dipole interactions, van der Waals forces and chemical interactions (i.e. ionic, covalent, and metallic bonding) [19]. The relevance of each contribution and the resulting adhesion strength depends mostly on the interface chemistry and the atomic-scale properties of the inorganic and organic components. The effects of molecular and chemical interactions at 1

11 material interfaces are not accounted for in many analytical models, and are difficult to ascertain experimentally. In this context MD simulations can provide information on interfacial strength at a more fundamental level. The thermodynamic concept used to quantify adhesion at a molecular level is the work of adhesion W A. In the present case, we estimated a value of W A of approximately 5 mj/m, that is in line with the expected order of magnitude for comparable polymeric materials []. Figure 11: Molecular model of the polymer-glass interface. According to the work of Andrews and Kinloch [1], the measured fracture energy G f can be related to the work of adhesion W A by G f = W A ϕ (7) where ϕ is a temperature- and rate-dependent viscoelastic term. This multiplicative relationship shows that, although W A is generally much smaller than G f, small changes in the work of adhesion can result in considerable changes in the measured adhesion strength. In this context, the prediction of W A and of the interfacial characteristics through molecular simulation techniques can efficiently assist the development of materials with improved interfacial properties. In other terms, tailoring the interfacial characteristics through a priori design of the interface can lead to the desired value of the adhesion strength in terms of G f. A correlation between the thermodynamic work of adhesion W A from MD and the fracture energy G f for this system is currently under investigation by our group and will be described in details in a more extended paper. 5 CONCLUSIONS In the analysis of the flexural behaviour of laminated glass units the hypothesis of linear elasticity for the glass and the interlayer seems to be useful in the predictions of the stress state before the cracks. The comparisons among different procedures is positive and a good agreement can be found with numerical results. However, the post-critic behaviour can not be predicted and different models have to be used. For instance in the paper a procedure based on the SDA has been applied and some considerations about the ability in the predictions of the crack initiation have been made. In SDA models the main difficulty stays in the identification of the constitutive parameters defining the properties of the interface, for which a cohesive fracture criterion has been used. The possibility of establishing a correlation between these parameters, directly related to the fracture energy and the ultimate strength of the materials, and the work of adhesion has been investigated. Molecular dynamics simulations of the PVB-glass interface 11

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