Failure modes of glass panels subjected to soft missile impact

Transcription

1 Failure modes of glass panels subjected to soft missile impact L. R. Dharani & J. Yu Dept. of Mech. and Aerospace Engineering and Engineering Mechanics, University of Missouri-Rolla, U.S.A. Abstract Damage to architectural glazing by the windborne debris during severe windstorms is well documented. In this study, the stress response of and fracture initiation in glass panels impacted by a large soft missile such as collapsed trees and ceiling wood is investigated using the finite element method. For the crack initiation, the energy release rate criterion is developed to determine the crack initiation time and location. The effect of various geometric, material and loading parameters on the maximum principal stress and failure modes is examined. Keywords: large soft missile, laminated glass panels, impact, energy release rate, crack initiation. 1 Introduction A hurricane s turbulent winds and strong gusts often carry windborne debris that can slam into glass windows and doors. This debris can be categorized into two types by its mass and elastic modulus compared to those of glass: small hard missiles and large soft missiles. Small hard missiles include heavy pieces of rock, tar and gravel from the building roof. Large soft missiles such as pieces of wood and large logs blown from the collapsed house roofs and trees are also often observed in hurricanes and severe windstorms. From the industrial standards, both types of missiles need to be considered when glass panels are designed [1]. The fracture of brittle solids, such as glass plates, under indentation and impact has been widely investigated. Experiments and analysis show that Hertzian cone cracking is the main failure mode. Lawn [2] gave a detailed

2 164 Damage and Fracture Mechanics VIII description of the Hertzian cone crack in review of research work on the nature and mechanics of damage induced in ceramic by a spherical indenter. Glathart and Preston [3] noted that there are two different ways in which the glass will fail under impact of a hard missile, i.e., by flexural (bending) stresses or by bearing stresses. If the thickness of glass is large, and therefore the flexibility is low, a hard missile will initiate a crack on the upper surface outside of the contact area. When the thickness of the glass plate becomes small, the glass will break on the lower surface under the contact centre. Tsai and Chen [4] showed the same fracture mode as [3] related to the ratio of plate thickness to the support span. Flocker and Dharani [5] studied the effects of various geometric and material parameters in minimizing stress wave propagation in the inside glass ply using the finite element code DYNA-2D. Their study explicitly showed that Hertzian cone is the main concern for laminated architectural glass under a small hard missile impact [6]. There has been neither analytical nor experimental work published on laminated glass subjected to large soft missile impact. In this paper, a finite element code based on the energy release rate criteria is developed to predict the failure mode of glass panel impacted by large soft missile. The effect of material and geometric parameters of laminated glass units and missiles on the maximum principal stress in the inside glass ply are presented. 2 Formulation of the impact problem 2.1 Model idealization The idealized impact problems under consideration are shown in fig. 1. A spherical-ended wooden cylinder or a flat-ended wooden cylinder with an initial velocity of V 0 normally impacts a typical three-layer laminated glass. The laminated glass unit consists of an outside glass ply of thickness h o, an inside glass ply of thickness h i and a polyvinyl butyral (PVB) interlayer of thickness h PVB which bonds the glass plies together. The glass plies are soda-lime glass, and the interlayer bond is assumed perfect with no debonding or slipping between layers during impacts. Cylindrical coordinates (r, θ, z) with a Lagrangian description of motion are used to formulate the problem, where z is the axis of the symmetry, r represents the radial direction and the origin of coordinates is located at the centre of the lower surface of the inside glass ply. The laminated glass unit is simply supported at the periphery of the innermost surface. Due to the axisymmetry, only half of the systems are modelled. 2.2 Material models As with most impact formulations, the stress tensor is computed as the sum of deviatoric and volumetric components σ = S pδ (1)

3 Damage and Fracture Mechanics VIII 165 Outside Glass Ply, h o Spherical-ended Wooden Cylinder Flat-ended Wooden Cylinder Inside Glass Ply, h i PVB Interlayer, h pvb Figure 1: Schematic of laminated glass subjected to low velocity impact. Large soft missile with spherical impacting end; large soft missile with flat impacting end. where σ is the stress tensor, S is the deviatoric stress tensor, p ( = -σ kk /3) is the pressure and δ is the Kronecker delta. The glass plies and the steel ball are modelled as linear elastic materials. The deviatoric and volumetric behaviours are given by Eνε Eε v S = + p δ + ( ν )( ν ) (2) ν Eε v p = (3) 3( 1 2ν ) where E is the Young s modulus, ν is Poisson s ratio, ε ν ( = ε kk ) is the volumetric strain and ε is the strain tensor. The PVB interlayer is modelled as linear viscoelastic material. The deviatoric stresses are given by = 2 t G( t ) e 0 S ( t) τ ( τ ) dτ (4) where t represents time, e is the deviatoric strain rate and G(t) is the stress relaxation modulus which is assumed to be of the form β t G( t) = G + ( G0 G ) e (5) where G is the long-time shear modulus, G 0 is the short-time shear modulus and β is the decay factor. The volumetric behaviour of PVB is linear, so the pressure p can be obtained from the equation p = Kε v (6) where K is the bulk modulus of elasticity and is related to the shear modulus and Poisson s ratio by the relation 2µ (1 + ν ) K = (7) 3(1 2ν ) where µ is the shear modulus and ν is Poisson s ratio. The large soft missile (Douglas fir) is modelled as linear orthotropic elastic material. It has unique and independent mechanical properties in the directions of

4 166 Damage and Fracture Mechanics VIII three mutually perpendicular axes: longitudinal (L), radial (R) and tangential (T). E r, E θ, and E z are the Young s moduli in the R, T, and L direction, respectively. Only three Poisson s ratios, ν zr, ν θr and ν θz are independent and G rz is the only shear modulus needed. 3 Finite element computation 3.1 Stress response The stress response of laminated glass subjected to large soft missile was studied first using the nonlinear dynamic explicit finite element code, DYNA-2D. A four-node solid element with one-point Gauss quadrature integration is employed to simulate this torsionless, axisymmetric problem. The mesh for the glass ply and the PVB interlayer is made finely near the impact site. Similarly the impacting end of the missile where contact occurs is meshed finely. The following baseline parameters are used in the analysis: initial velocity V 0 = m/s, radius of soft cylindrical missile R = mm, radius of laminate r = 800 mm, h o = mm, h PVB = mm, h i = 6.35 mm; for PVB, K = 2 GPa, G 0 = 1 GPa, G = 0.69 MPa, β = 12.6 s -1 and ρ 0 = 1100 kg/m 3 ; for glass, ν = 0.25, E = 72 GPa, ρ 0 = 2500 kg/m 3 ; for wood, E r = 884 MPa, E θ = 650 MPa, E z = 13 GPa, G rz = 832 MPa, ν zr = 0.29, ν θr = 0.37, ν θz = 0.03, ρ 0 = 505 kg/m Crack initiation A global model of the whole system with geometric nonlinearity was first run using ABAQUS/Explicit without introduction of any surface flaw to get the displacement and stress response of laminated glass due to impact of a large soft missile. Then a refined submodel with the surface crack was run through V 0 Region A Outside Glass Ply PVB Interlayer Surface Flaw Region B Inside Glass Ply Region C Figure 2: Schematic plot of regions where submodels are located.

5 Damage and Fracture Mechanics VIII 167 ABAQUS/Standard to obtain the J-integral on the local regions of the global model at different time steps. The connection between the global model and submodel is that the displacements at the boundary of the submodel are interpolated from the results of global model. For a laminated glass plate, there are three regions where the maximum tensile stress may cause the surface flaws to extend. These regions are shown in fig. 2. Region A is located just outside of the contact area, region B is located close to the centre of the inner surface of the outside glass ply where it is bonded to the PVB interlayer and Region C is near the centre of the supported surface of the inside glass ply. One of the submodels used to calculate the J-integral are shown in fig. 3. The mesh is made finer gradually towards the surface flaw. It is used to study crack initiation on the contact surface (Region A). A similar one with the notch at the lower left corner of the submodel is used to study crack initiation on the bonded surface of the outside glass ply (Region B) and the innermost surface (Region C). In the submodel, three rings of collapsed quadrilateral elements completely surrounding the crack tip are used to create the strain singularity of1 / r, where r is the distance from the crack tip. In the remaining area of the submodel, fournode bilinear, axisymmetric solid elements are used. The mesh around the crack front is highly refined, and the J-integral estimates from the three contours are very close, showing an accurate J-integral value has been reached. If the J- integral obtained from the submodel reaches the critical energy release rate, the crack will initiate at that particular time and location. In this paper, the critical energy release rate (J IC ) is taken to be two times of the fracture surface energy, which is 3.9 J/m 2 for soda-lime glass [7]. The surface flaw is modelled as an annular gap with a sharp end of 5 µm long normal to the glass surface and 1 µm width in radius. Figure 3: Submodel with the notch at the upper centre and its magnified view near crack tip. The dark area represents very fine mesh.

6 168 Damage and Fracture Mechanics VIII Figure 4: Time history plots of various stress components (σ r, σ θ, σ z, τ rz ) and the maximum principal stress (σ 1 ) at the centre of supported surface for the base line case: spherical end, flat end. 4 Results 4.1 Stress response Fig. 4 shows the responses of four stress components (σ r, σ θ, σ z and τ rz ) and the maximum principal stress (σ 1 ) at the centre of the supported surface using baseline parameters. For both types of impacting ends, the maximum principal stresses reach their peaks at about 0.5 ms following initial impact, which is much longer than that for the small hard missile case (approximately 18 µs) [5]. The missile with a spherical impacting end produces much higher maximum principal stress as compared to that of a flat-ended missile. The r, θ and z directions are very nearly the principal directions and the maximum principal tensile stress corresponds to Mode I crack opening stress for a surface crack located normal to the surface. Thus any parameter, geometric or material, that results in a lower maximum principal stress will yield a more impact resistant glazing. Fig. 5 shows the effect of varying projectile mass density (ρ) and initial projectile velocity (V 0 ) from the baseline value on the maximum principal stress. Power law fits (dashed line for the spherical-ended missile and dotted line for the flat-ended missile) are shown. For reference, a solid line from the results of the small hard missile impact [5] is also shown in this figure. From the Hertzian equations, when one considers a completely elastic impact and quasi-static assumption, the maximum interface force, and therefore the maximum expected stress is proportional to ρ 3/5 and V 0 6/5. The results of small steel ball impact follows these relations very well, but they cannot apply to large soft missile impact on glass. That means Hertzian cone cracking is not the failure mode for large soft missile impact.

7 Damage and Fracture Mechanics VIII 169 Figure 5: Effect of projectile mass density and initial velocity on the maximum principal stress in the inside layer. 4.2 Fracture initiation To study the J-integral response, the submodel (fig. 3) is placed on the impacted surface at different distances from the contact circle, until the location where J- integral first reaches its critical or maximum value is obtained. For the other surfaces, the submodel with the notch located at the lower left corner is always located at the centre of the surfaces being studied.

8 170 Damage and Fracture Mechanics VIII Figure 6: J-integral of laminated glass impacted by large soft missile. Spherical impacting end; flat impacting end. Fig. 6 shows the variation of the J-integral with time for the sphericalended wooden cylinder impact. The J-integral reaches its critical value at the centre of bonded surface of the outside glass ply (Region B) at 35 µs after impact. The variation of the J-integral with time for the flat-ended wooden cylinder impact is plotted in fig. 6. The J-integral value reaches it critical value at the supported surface of the inside ply (Region C) at 103 µs. The ratio of the initiation radius, r, to the cylinder radius, 32.81mm, is 0.70.

9 Damage and Fracture Mechanics VIII Conclusions The finite element stress analysis indicates that the spherical-ended wooden cylinder is more dangerous than the flat end due to the higher tensile principal stress at the central bottom point of the laminated glass plate. The energy release rate criterion is also developed to predict the failure mode of glass panel impacted by large soft missile. For the laminated glass, the numerical results have shown that failure mode depends upon the missile types. If the missile is large and soft, a higher bending stress field is produced on the surface opposite to the impact site. In this case, crack is prone to open there. This means that Hertzian theory is not applicable for the large soft missile. Acknowledgments This work is being supported by the National Science Foundation (Grant No. CMS ), the Missouri Department of Economic Development through the Manufacturing Research Training Center, the E. I. dupont de Nemours and Company, and Solutia In. References [1] Standard Specification for Performance of Exterior Windows, Glazing Curtain Walls, Doors and Storm Shutters Impacted by Windborne Debris in Hurricanes; American Society for Testing & Materials, ASTM E [2] Lawn, B.R., Indentation of ceramics with spheres: A century after Hertz, J Am Ceram Soc, 81(8), pp , [3] Glathart, J.L. & Preston, F.W., Behavior of glass under impact: theoretical considerations, J Glass Tech, 9(4), pp , [4] Tsai, Y.M. & Chen, Y.T., Transition of Hertzian fracture to flexure produced in glass plates by impact, Eng. Fracture Mech., 18(6), pp [5] Flocker F.W. & Dharani L.R., Stress in laminated glass subject to low velocity impact, J Engng Struct, 19(10), pp , [6] Flocker, F.W. & Dharani, L.R., Modeling fracture in laminated architectural glass subject to low velocity impact, J. Mater. Sci., 32(10), pp , [7] Lawn, B.R., Wiederhorn, S.M. & Johnson, H. H., Strength degradation of brittle surfaces: blunt indenters, J. Am. Ceramic Soc., 58(8-10), pp , 1975.