Crack Initiation and Propagation in Laminated Composite Materials

Xu, Jun; Zheng, Yanting

doi:10.1007/978-3-319-58729-5_24

Jun Xu^2,3 &
Yanting Zheng⁴

1718 Accesses

Abstract

PVB laminated glass is a kind of typical laminated composite material and its crack characteristics are of great interest to vehicle manufacturers, safety engineers, and accident investigators. Because crack morphology on laminated windshield contains important information on energy mitigation, pedestrian protection, and accident reconstruction. In this chapter, we investigated the propagation characteristics for both radial and circular cracks in PVB laminated glasses by theoretical constitutive equations analysis, numerical simulation, experiments, and tests of impact. A damage-modified nonlinear viscoelastic constitutive relations model of PVB laminated glass were developed and implemented into FEA software to simulate the pedestrian head impact with vehicle windshield. Results showed that shear stress, compressive stress, and tensile stress were main causes of plastic deformation, radial cracks, and circumferential cracks for the laminated glass subject to impactor. In addition, the extended finite element method (XFEM) was adopted to study the multiple crack propagation in brittle plates. The effects of various impact conditions and sensitivity to initial flaw were discussed. For experiment analysis, crack branching was investigated and an explicit expression describing the crack velocity and number of crack branching is proposed under quasi-static Split Hopkinson Pressure Bar (SHPB) compression experiments. And the radial crack propagation behavior of PVB laminated glass subjected to dynamic out of - plane loading was investigated. The steady-state cracking speed of PVB laminated glass was lower pure glass, and it increased with higher impactor speed and mass. The supported glass layer would always initiate before the loaded layer and the final morphologies of radial cracks on both sides are completely overlapped. Two different mechanisms of crack propagation on different glass layers explained the phenomenon above. Then further parametric dynamic experiments study on two dominant factors, i.e., impact velocity and PVB thickness are investigated: Firstly, a semiphysical model describing the relationship between the maximum cracking velocity and influential factors was established; Then the Weibull statistical model was suggested considering various factors to describe the macroscopic crack pattern in this chapter; Finally, the relation between radial crack velocity and crack numbers on the backing glass layer and the relation between the crack length and the capability of energy absorption on the impacted glass layer were proposed.

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Crack Initiation and Propagation in Laminated Composite Materials

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Keywords

Introduction

As an typical laminated composite material, the standard PVB laminated glass comprises of a PVB interlayer sandwiched by two mono soda-lime and it has extensive applications in architecture, automobile industry, as well as the structural parts during the past few decades where they usually play an important role in human protection and structural integrity due to its excellent energy-absorbing and fragment-holding ability. However, mechanical behavior especially for the cracks of PVB laminated glass subject to loading has not been fully explored yet due to the complicity caused by the composite material, i.e., polymer material sandwiched by two pieces of brittle material. As a result, it becomes a pressing need to investigate the mechanical loading information contained in the macroscopic crack initiation and propagation which enables improved design or prevention for laminated PVB glass as a structural part.

In this chapter, there will be three methods mainly applied to investigate PVB laminated glass damage mechanism and crack growth: theoretical constitutive equations describing the failure criteria of PVB laminated glass under load, numerical simulations of crack initiation and growing on PVB laminated glass, experiments and tests of impact between impactor and laminated glass plate. For theoretical investigation, a damage-modified nonlinear viscoelastic constitutive equation is developed based on the updated Lagrangian approach. For numerical simulation investigation, a FEA model with proper material parameters verified by a classical example is established to describing the impact between pedestrian head and PVB laminated windshield. In addition, the extended finite element method (XFEM) is employed to study the low-speed impact-induced cracking in brittle plates in the present chapter. Upon head impact, the interaction between the stress field and the initiation and propagation of the radial and circumferential cracks are computed using XFEM. The effects of various impact conditions and sensitivity to initial flaw are discussed. The experiment analysis of PVB laminated glass is mainly based on quasi-static loading and dynamic impact. We carried out quasi-static Split Hopkinson Bar (SHPB) experiments to study the crack velocity and number of crack branching. For the dynamic out-of-plane impact tests carried out to study the crack propagation behavior of PVB laminated glass subjected to dynamic impact by using high-speed photography, the time histories of the averaged radial crack tip position, propagation velocity, and acceleration are recorded and the parametric study of impactor speed and mass, impact velocity, PVB thickness for PVB laminated glass are investigated. Then two different mechanisms of crack propagation on different glass layers are explained. Besides, a semiphysical model describing the relationship between the maximum cracking velocity and influential factors is established. In addition, the Weibull statistical model is suggested considering various factors to describe the macroscopic crack pattern thus providing a theoretical evidence for engineering practice. Finally, the relation between radial crack velocity and crack numbers on the backing glass layer and the relation between the crack length and the capability of energy absorption on the impacted glass layer are investigated.

Constitutive Relations

The constitutive relation of laminated composite material can well reflect the mechanical behavior of that. In order to study the crack in laminated composite material, the constitutive relation should be established firstly.

PVB windshield used in automotive industry are a kind of laminated composite material which comprise of a PVB interlayer sandwiched by two momo soda-lime glass sheet. Thus, pedestrian head impacts on the vehicle PVB windshield can be regarded as a scenario where a composite plate is subjected to a concentrated force by a spherical indenter on its top surface shown in Fig. 1. The composite plate consists of n = 2 individual brittle soda-lime glass layers of thickness t_g and modulus E_g, sandwiched with a PVB interlayer of thickness t_p and modulus E_p.

By studying the constitutive relations of the PVB windshield, the crack in PVB windshield can be analyzed thoroughly. Lili et al. (Wang et al. 2003; Lili et al. 1991) study the mechanical behavior of polymer with performing dynamic impact and quasi-static tensile experiments. According to the experimental results, the behavior of the polymer can be described as the following constitutive relation:

$$ \sigma ={\sigma}_e+{\sigma}_{t_1}+{\sigma}_{t_2} $$

(1)

σ_t stand for the tensile stress.

$$ {\sigma}_e={\sigma}_m\left[1-\operatorname{exp}\left(-{\sum}_{t=1}^n\frac{{\left( m\varepsilon \right)}^t}{t}\right)\right] $$

(2)

$$ {\sigma}_{t_1}={E}_1{\int}_0^t\dot{\varepsilon}\left(\tau \right)\operatorname{exp}\left(-\frac{t-\tau }{\theta_1}\right) d\tau $$

(3)

$$ {\sigma}_{t_2}={E}_2{\int}_0^t\dot{\varepsilon}\left(\tau \right)\operatorname{exp}\left(-\frac{t-\tau }{\theta_2}\right) d\tau $$

(4)

where m, τ, E_1,E₂ are a nondimensional relative initial elastic modulus, relaxation time as defined in Maxwell body, elastic constants of first Maxwell body and elastic constants of second Maxwell body, respectively. θ_1,θ₂, are constants as well, standing for the relaxation times of first and second Maxwell body. Where σ_e stands for the stress with the damage material. σ_t1, σ_t2 standing for tensile stress of first and second Maxwell body. D is a “damage” variable.

Nevertheless, the above constitutive relations are only valid when strain is less than 7% (Lili et al. 1991), and it cannot express the relation completely. To further utilize the above relations, Fenghua et al. (Zhou et al. 1992) suggested the following equation to describe the constitutive relations:

$$ \sigma =\left(1-D\right){\sigma}_e+{\sigma}_{t_1}+{\sigma}_{t_2} $$

(5)

where D is a “damage” variable and is defined as:

$$ D=\left\{\begin{array}{cc}\hfill 0\hfill & \hfill \varepsilon \le {\varepsilon}_{th}\hfill \\ {}\hfill {\dot{D}}_0{\dot{\varepsilon}}^{\delta -1}\left(\varepsilon -{\varepsilon}_{th}\right)\hfill & \hfill \varepsilon >{\varepsilon}_{th}\hfill \end{array}\right. $$

(6)

And ε_th here regards as strain threshold of damage evolution. $ {\dot{D}}_0 $, δ both are material parameters determined by experiments. Then the dynamic damage criterion is put forward:

$$ {D}_f={\dot{D}}_0{\dot{\varepsilon}}^{\delta -1}\left(\varepsilon -{\varepsilon}_{th}\right) $$

(7)

Incremental Form of Constitutive Relations When the Material Is Not Damaged

When employing the constitutive relations, we should consider some conditions like whether the strain is larger than the threshold strain. Thus we need an equivalent strain to determine whether it is or not. As following is the equivalent:

$$ {\varepsilon}_{eq}=\frac{\sqrt{2}}{3}\sqrt{{\left({\varepsilon}_1-{\varepsilon}_2\right)}^2+{\left({\varepsilon}_2-{\varepsilon}_3\right)}^2+{\left({\varepsilon}_3-{\varepsilon}_1\right)}^2+\frac{3}{2}{\left({\gamma}_{12}^2+{\gamma}_{23}^2+{\gamma}_{31}^2\right)}^2} $$

(8)

where subscripts “1,” “2,” and “3” denote the directions of principal stresses.

On the basis of updated Lagrangian virtual work method, stress-strain relations of material that undergoes large deformation with the ignorance of physical nonlinear properties can be regarded as (Fischer and Washizu 1982):

$$ \left\{{S}_{ij}\right\}=\int_{0}^{t}E\left(t-\tau \right)\left[A\right]\frac{\partial \left\{{e}_{kl}\right\}}{\partial \tau } d\tau $$

(9)

{S_ij}, {e_kl} are Kirchhoff stress tensors and Green strain tensors, respectively.

$$ \begin{array}{l}\left[A\right]=\frac{1-v}{\left(1+v\right)\left(1-2v\right)}\cdot \\[6pt] {}\left[\begin{array}{cccccc}\hfill 1\hfill & \hfill \frac{v}{\left(1-v\right)}\hfill & \hfill \frac{v}{\left(1-v\right)}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill \frac{v}{\left(1-v\right)}\hfill & \hfill 1\hfill & \hfill \frac{v}{\left(1-v\right)}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill \frac{v}{\left(1-v\right)}\hfill & \hfill \frac{v}{\left(1-v\right)}\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1-2v}{2\left(1-v\right)}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1-2v}{2\left(1-v\right)}\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1-2v}{2\left(1-v\right)}\hfill \end{array}\right]\end{array} $$

(10)

Similarly, as PVB windshield is a three-dimensional structure, Equations from 1 to 6 can be rewritten as (Swanson and Christensen 2015; Wang et al. 2007; Shen et al. 1987):

$$ {S}_{ij}={S}_{ij,e}+{S}_{ij,{t}_1}+{S}_{ij.{t}_2} $$

(11)

where

$$ {S}_{ij,e}={\sigma}_m\left\{1-\operatorname{exp}\left(-{mAE}_{kl}-{\sum}_{t=1}^n\frac{{\left({mE}_{ij}\right)}^t}{t}\right)\right\} $$

(12)

$$ {S}_{ij,{t}_1}={E}_1\int_{0}^{t}A\frac{\partial {E}_{kl}}{\partial \tau} \operatorname {exp}\left(-\frac{t-\tau }{\theta_1}\right) d\tau\qquad \quad\qquad\ \ \,$$

(13)

$$ {S}_{ij,{t}_2}={E}_2\int_{0}^{t}A\frac{\partial {E}_{kl}}{\partial \tau} \operatorname {exp}\left(-\frac{t-\tau }{\theta_2}\right) d\tau\qquad \quad\qquad\ \ \, $$

(14)

On the basis of Boltzmann superposition (Feng 1995), incremental form of constitutive relations can be obtained. Also, implementation in commercial FEA software needs the incremental form of the stress tensor of Eqs. 12, 13, and 14 to be rewritten. Thus, it is a must for us to reach the incremental form first. According to proofs of Chain Rule, Eq. 12 can be rewritten as follows:

$$ \begin{aligned}{dS}_{ij,e}&={\sigma}_m\operatorname{exp}\left(-{mAE}_{kl}-{\sum}_{t=2}^n\frac{{\left({mE}_{ij}\right)}^t}{t}\right) \\ {}&\quad\cdot\left[mA\frac{\partial {E}_{kl}}{\partial \tau }+{\sum}_{t=2}^n\frac{{\left({mE}_{ij}\right)}^{t-1}}{t}\frac{\partial {E}_{kl}}{\partial \tau}\right]\cdot \Delta t\end{aligned} $$

(15)

Considering Δt is short enough yields (Feng 1995):

$$ \frac{\partial {E}_{kl}}{\partial \tau }=\frac{\partial {E}_{kl}}{\partial t}=\frac{\partial {E}_{kl}}{\Delta \tau } $$

(16)

Thus Eq. 13 becomes:

$$ \begin{array}{l}\Delta {S}_{ij,e}={\sigma}_m\operatorname{exp}\left(-{mAE}_{kl}-{\sum}_{t=2}^n\frac{{\left({mE}_{ij}\right)}^t}{t}\right)\cdot \left[\!mA\Delta {E}_{kl}+{\sum}_{t=2}^n\frac{{\left({mE}_{ij}\right)}^{t-1}}{t}\Delta {E}_{kl}\!\right]\end{array} $$

(17)

Similarly, viscoelastic stress tensors for both low strain rate and high strain rate at t are:

$$ \Delta {S}_{ij,{t}_1}={E}_1{\theta}_1A\frac{1-\operatorname{exp}\left(-\frac{\Delta t}{\theta_1}\right)}{\Delta t}\Delta {E}_{kl} -\left(1-\operatorname{exp}{\left(-\frac{\Delta t}{\theta_1}\right)}^t\right){S}_{ij,{t}_1} $$

(18)

$$ \Delta {S}_{ij,{t}_2}={E}_2{\theta}_2A\frac{1-\operatorname{exp}\left(-\frac{\Delta t}{\theta_2}\right)}{\Delta t}\Delta {E}_{kl} -\left(1-\operatorname{exp}{\left(-\frac{\Delta t}{\theta_2}\right)}^t\right){S}_{ij,{t}_2} $$

(19)

Therefore, the incremental form of stress tensor during one unit time step Δt can be obtained:

$$ \begin{aligned}&\Delta {S}_{ij,n+1}=\Delta {S}_{ij,e}+\Delta {S}_{ij,{t}_1}+\Delta {S}_{ij,{t}_2}={\sigma}_m\!\operatorname{exp}\!\left(\!-{mAE}_{kl}-{\sum}_{t=2}^n\frac{{\left({mE}_{ij}\right)}^t}{t}\right) \\ & \cdot \left[mA\Delta {E}_{kl}+{\sum}_{t=2}^n\frac{{\left({mE}_{ij}\right)}^{t-1}}{t}\Delta {E}_{kl}\!\right]\!+{E}_1{\theta}_1A\frac{1-\operatorname{exp}\left(-\frac{\Delta t}{\theta_1}\right)}{\Delta t}\Delta {E}_{kl}\\ &-\left(1-\operatorname{exp}{\left(-\frac{\Delta t}{\theta_1}\right)}^t\right){S}_{ij,{t}_1}+{E}_2{\theta}_2A\frac{1-\operatorname{exp}\left(-\frac{\Delta t}{\theta_2}\right)}{\Delta t}\Delta {E}_{kl}\\ &-\left(1-\operatorname{exp}{\left(-\frac{\Delta t}{\theta_2}\right)}^t\right){S}_{ij,{t}_2}\end{aligned} $$

(20)

In the following way, the stress tensor can be calculated:

$$ {S}_{ij,n+1}={S}_{ij,n}+\Delta {S}_{ij,n+1}\quad \varepsilon <{\varepsilon}_{th} $$

(21)

where t_n + 1 = t_n + Δt.

Incremental Form of Constitutive Relations When the Material Is Damaged

When the material is not damaged, the incremental form of constitutive relations can be described using function relation above. However, when the strain is larger than the threshold strain, material turns into “damage” phase. As mentioned above, stress tensor in “damage” phase at time t_n + 1 can be concluded as:

$$ {\left\{{S}_{ij}\right\}}_{n+1}={\left\{{S}_{ij}\right\}}_n+{\left\{\Delta {S}_{ij}\right\}}_{n+1} $$

(22)

and Eq. 21 can be rewritten as:

$$ {S}_{ij,n+1}={S}_{ij,n}+\Delta {S}_{ij,n+1} $$

(23)

Therefore,

$$ \begin{array}{l}{S}_{ij,n+1}={S}_{ij,n}+\Delta {S}_{ij,n+1}\\[5pt] {}=\frac{1}{1-{D}_{n+1}}{S}_{ij,n+1}^d+{1}^d-\frac{1}{1-{D}_n}{S}_{ij,n}^d\\[6pt] {}=\frac{1}{1-{D}_{n+1}}\Delta {S}_{ij,n+1}^d+\frac{\Delta {D}_{n+1}{S}_{ij,n}^d}{\left(1-{D}_{n+1}\right)\left(1-{D}_n\right)}\end{array} $$

(24)

where ΔD_n + 1 = D_n + 1 − D_n is the increase of damage variable.

Rewriting Eq. 23 yields:

$$ \begin{aligned}\Delta {S}_{ij,n+1}^d &=\left(1-{D}_{n+1}\right)\Delta {S}_{ij,n+1}-\dfrac{\Delta {D}_{n+1}{S}_{ij,n}^d}{\left(1-{D}_n\right)}\\[8pt] {}&=\left(1-{D}_{n+1}\right)\Delta {S}_{ij,n+1}-\Delta {D}_{n+1}\Delta {S}_{ij,n}\end{aligned} $$

(25)

Similarly, the stress tensor under damage can be calculated as:

$$ {S}_{ij,\mathrm{n}+1}^d={S}_{ij,\mathrm{n}}^d+\Delta {S}_{ij,\mathrm{n}+1}^d\quad \varepsilon \ge {\varepsilon}_{th} $$

(26)

Verification of Constitutive Relations

The constitutive relation of PVB windshield are gotten preliminary and the accuracy of that should be studied next. In order to verify the above constitutive relations, a classical example is chosen. A thin plate with infinite length and 2b = 12 mm in width is under the uniform loadq(t) = 1 kN ⋅ m. There is a small crack in center with length 2a = 1 mm, see Fig. 2. Material properties are listed in Table 1.

Table 1 Material parameters for plate

Full size table

By implementing the above constitutive relations into FEA software numerical and adding the command of failure of material, results can be obtained. Comparing numerical results with theoretical ones, we can see that the constitutive relations results can coincide with the theoretical solutions well, shown in Fig. 3.

Numerical Simulation

PVB Windshield FEA Modeling Based on Constitutive Relations of PVB Laminated Glass

By verifying the constitutive relations above with a classical example, the relations have a sufficient accuracy. Then the FEA modeling of PVB windshield are introduced in detail following.

Contact Analysis

A nonconforming two-body contact model of a PVB windshield loaded by a headform (sphere) is illustrated in Fig. 4. The thickness of PVB windshield is 8.76 mm, the length of it is 350 mm, and the thickness of interlayer PVB is 0.76. Shell elements were chosen to mesh the two bodies in contact. Hourglass control and automatic surface- to-surface contact with dynamic friction coefficient 0.1 were employed in LS-Dyna. Initial nodal gaps between the two contact surfaces were prescribed. The contact elements were not activated until the penetration of the sphere into the composite glass occurred for the sake of saving computation time.

An extra command that allows the failure of material is added in the input file of LS-Dyna. Then the above-mentioned constitutive relations with damage variable could be fully employed in FEA. The accuracy of the contact FEA model is firstly examined by classical Hertzian pressure calculation method. Illustration of classical Hertzian theory is shown in Fig. 5 (Daphalapurkar et al. 2007). The δ and a₀ stand for mutual approach between two bodies and their contact radius.

Deviations between FEA and theoretical results are all within 1.5%. Comparison shown in Fig. 6 demonstrates the sufficient accuracy of the FEA model to stimulate the impact between pedestrian head and PVB windshield.

Material Model

The PVB laminated glass is m odeled according to the following principles: if the glass does not fail, the composite is treated as a shell; otherwise, the interlayer PVB film acts as a membrane. A Belytschko–Tsay shell element for the glass material and a membrane element for the interlayer are employed (Zhou et al. 1992; Fischer and Washizu 1982). Both types of elements are fully integrated in simulations so that no hourglass modes should be expected (Sharon and Fineberg 1999).

Glass and PVB film both can be treated as isotropic materials according to their physical and material properties. Therefore, we assume that each layer of composite windshield is isotropic with respect to its material symmetry lines and obeys Hooke’s law. It is assumed that, due to the influence of dynamic loading, there is no sliding between two contact layers. It then becomes reasonable to consider the composite windshield as integrity. As a result, an equivalent elastic modulus and Poisson’s ratio should be introduced as follows (Dwivedi and Espinosa 2003).

$$ \overline{E}=\frac{2{t}_g{E}_g+{t}_p{E}_p}{2{t}_g+{t}_p} $$

(27)

$$ \overline{v}=\frac{2{t}_g{v}_g+{t}_p{v}_p}{2{t}_g+{t}_p} $$

(28)

where $ \overline{E} $ and $ \overline{v} $ are equivalent Young’s modulus and equivalent Poisson’s ratio of windshield glazing.

Material parameters are listed in Table 2.

Table 2 Parameters used in both constitutive relations and finite element analysis

Full size table

Results and Discussions

Internal Stress Analysis

The cracks in PVB windshield are demonstrations of internal and external stresses. Thus, the FEA model above is used to analyze the internal stresses. Only through investigating the internal stresses can we know the cracks on windshields better.

During the dynamic crack evolution shown in FEA analysis, stages of damage evolution could be sketchily drawn as Fig. 7. Three kinds of cracks are: circumferential cracks, web crack, and plastic crack. The circumferential cracks are mainly caused by pressure stress while the web cracks are mainly caused by tensor stress. Plastic cracks are the last stage that the windshield deforms plastically. In Fig. 8, three kinds of internal stresses, tensile stress σ_t, compressive stress σ_c, and shear stress σ_s, are plotted versus impact time. We can find out that the value of stresses sorted from large to small is σ_s, σ_c and σ_t. Lawn and Wilshaw (1975) stated that in addition to the Hertzian cone crack and the median vent crack, there is also an inelastic deformation zone where shear and hydrostatic compression is greatest. Quite similarly, it is obvious to see in Fig. 8 that shear stress is the greatest among three stresses, causing the windshield to deform plastically.

Internal Stress Variation Among Different Point on Windshield Plate

Internal stresses of different points on windshield plate are far different and crack information can prove it. The cracks become weaker as the radius grows larger for circumferential cracks, as do the radial cracks. In Fig. 9, maximum stresses appear at the very point that impacted by pedestrian head. Stresses decrease as the distances from the central point increase. The decreasing rates of three stresses are nearly the same according to the computational results.

Effects of Poisson’s Ratio on Crack Angle

Poisson’s ratio plays an important role of cone crack angle. A parametric study is performed to manifest the effects on crack angle. Poisson’s ratio is ranged from 0.2 to 0.4, with a step-width of 0.05. Table 3 shows the results of FEA analysis compared to that of stress intensity factor analysis (SIF). Maximum difference between the results is below 7%. With the increase in Poisson’s ratio, the conical crack angle also increases though the increase pattern is still unknown.

Table 3 Comparison results of two models on Poisson’s ratio effect to crack angle

Full size table

Effects of Impact Velocity

Vehicle speed is a critical factor in pedestrian-vehicle traffic accident. Extends and degrees of injuries largely depend on the impact velocity between pedestrian and vehicle. Therefore, studies of cracks on windshields under different head impact velocity would provide the essential foundations in pedestrian protection. According to the census data in NTADTU, in most cases, vehicles impact pedestrians at the speed of about 40–70 km/h. Thus, we vary the impact velocity of the head from 0 to 20 m/s and make impact velocity V0 as an independent variable. On the other hand, we choose the maximum radius of circumferential cracks R_c, the maximum length of radial cracks L_r, and the maximum radius of plastically deformed area R_p as dependent variables. Three dependent variables are illustrated and defined in Fig. 10.

In Fig. 11, the impact velocity has very great influence to three chosen variables. At the impact velocity of 20 m/s, R_c can reach up to about 642 mm, much larger than R_c ≈ 89 mm under the impact velocity of 5 m/s. Vehicle velocity plays an extremely important role in both pedestrian head injury and damage of windshield. In addition, it is reasonable to infer the extend of injuries of pedestrian head and impacting velocity of vehicle roughly through R_c, L_r, and R_p.

Numerical Simulation Based on the Extended Finite Element Method

After the head impacts on the windshield material, cracks including radial crack and circumferential crack appear in the glass material. The fracture characteristics of the windshield are widely recognized as one of the most important factors in automotive crashworthiness. The crack profiles (e.g., length, pattern, etc.) on the windshield material contain critical information for impact speed (which is extremely useful for accident reconstruction), vehicle crashworthiness, and insights for improving pedestrian and passenger protection. This motivates us to study the characteristics of low-speed head impact-induced crack propagation on the windshield glass material. In the past studies of impact on windshield glass, due to the intrinsic complexity of numerical analyses of cracking, researchers in automotive engineering often avoid explicit simulation of crack propagation. The following study aims to study the crack propagation characteristics (including both radial and circumferential cracks) when a model windshield undergoes low-speed impact and the numerical investigation of multiple crack propagation is based on the extended finite element method (XFEM) in this study.

Fundamentals of XFEM

XFEM incorporates a discontinuous displacement field across the crack facing away from the crack tip, in the form of:

$$ {u}^h=\sum\limits_{i=1}^n{N}_i(x)\left({u}_i+{b}_iH(x)+\sum\limits_{l=1}^4\sum\limits_{j=1}^2{c}_{il}{F}_l^i(x)\right) $$

(29)

where n is the number of nodes in the mesh, N_i(x) is the shape function of node i, u_i are the classical DOFs of node i. b_i and c_il are the DOFs associated with the Heaviside “jump” function H(x), with value 1 above the crack and below the crack. The crack-tip function$ {F}_l^i(x) $ is:

$$ {F}_l\left(r,\theta \right)=\left(\sqrt{r} \operatorname {sin}\left(\frac{\theta }{2}\right),\sqrt{r} \operatorname {cos}\left(\frac{\theta }{2}\right),\sqrt{r} \operatorname {sin}\left(\frac{\theta }{2}\right)\operatorname{sin}\theta, \sqrt{r}\operatorname{cos}\left(\frac{\theta }{2}\right)\operatorname{sin}\theta \right) $$

(30)

where (r, θ) are the local polar coordinates at the crack tip.

Figure 12 illustrates the enriched nodes near a crack tip, where H(x) is discontinuous across the crack surface. Under general mixed mode-loadings, the asymptotic near-tip hoop and shear stress components are

$$ \left\{\begin{array}{c}\hfill {\sigma}_{\theta \theta}=\frac{1}{4\sqrt{2\pi r}}\left({k}_1\left(3 \operatorname {cos}\left(\theta /2\right)\right.+\operatorname{cos}\left(3\theta /2\right)+{k}_{II}\left(-3 \operatorname {sin}\left(\theta /2\right)-3 \operatorname {sin}\left(3\theta /2\right)\right.\right)\hfill \\[7pt] {}\hfill {\sigma}_{\theta \theta}=\frac{1}{4\sqrt{2\pi r}}\left({k}_1\left(\operatorname{sin}\right.\left(\theta /2\right)+\operatorname{sin}\left(3\theta /2\right)+{k}_{II}\left(\operatorname{cos}\right.\left(\theta /2\right)-3 \operatorname {cos}\left(3\theta /2\right)\right)\hfill \\[7pt] {}\hfill {\sigma}_{\theta \theta}=\frac{1}{4\sqrt{2\pi r}}\left({k}_1\left(5 \operatorname {cos}\right.\left(\theta /2\right)-\operatorname{cos}\left(3\theta /2\right)+{k}_{II}\left(5 \operatorname {sin}\right.\left(\theta /2\right)-3 \operatorname {sin}\left(3\theta /2\right)\right)\hfill \end{array}\right. $$

(31)

For fracture propagation, we use the popular criterion of maximum local hoop stress, where the crack propagation direction is determined from Eq. 31 as

$$ {\theta}_c=arc \operatorname {cos}\left(\frac{3{k}_{II}^2+\sqrt{K_I^4+8{k}_I^2{k}_{II}^2}}{K_I^2+9{k}_{II}^2}\ \right) $$

(32)

Here, θ_c is measured with respect to a local polar coordinate system with its origin at the crack tip and aligned with the direction of the existing crack. Numerical simulations based on the above XFEM algorithm is carried out using commercial code ABAQUS.

Model and Methods

Model Setup

A featureless spherical headform is used to model the human head; The model head has a radius R = 90 mm and mass 4.5 kg. Note that effective head weights are changeable in different accident cases because human body may also get involved due to inertia, causing the effective impact mass larger than the mass of the pure head, especially during higher speed impact. Thus, the effective head mass M is also varied from 4.5 kg up to 90 kg by varying the density (from 1,500 to 30,000 kg/m³) and keeping volume the same in this study. According to 151 pedestrian–vehicle accident cases from the National Traffic Accident Database of Tsinghua University (NTADTU), most of the impact speeds of the vehicle in accident fall into the range of 40–80 km/h, i.e. 11.1–22.2 m/s, and that is the range of low-speed impact we consider. We focus on normal impact and assume that impact occurs at the center of the plate, we neglect the small windshield curvature and focus on a flat rectangular plate with dimensions a × b = 700 × 300 mm and thickness t = 4.76 mm as Fig. 13 shows.

In terms of the material properties, typical material parameters of glass are assigned to the plate, with density ρ = 2,500 kg/m³, Young’s modulus to the plate, Possion’s ratio v = 0.22, mode I energy release rate G₁ = 10 J/m², mode II and III energy release rate G_II = G_III = 50 J/m². The properties are assumed to be rate independent. To take advantage of symmetry, only one quarter of the model is needed for simulation. The boundary condition of the windshield is assumed to be fixed. The plate and head are modeled with eight-node linear brick elements. The contact between the head and plate obeys the Coulomb’s friction law with coefficient f = 0.1 (since according to our accident scene investigation, a small slip exists between head and windshield).

Computational Method

Suppose there is no crack or da mage, we first compute the deformation response with response to impact. For a given effective head mass M and impact speed v, the maximum displacement D is computed and presented in a dimensionless form in Fig. 14. For the range of impact speed and head mass considered in this study, D/t can be fitted as a dimensionless function of v/v₀ and M/M₀, as

$$ D/t=\left(135.3{\left(\nu /{\nu}_0\right)}^{0.7115}-{0.0305}^{518.5\left(\nu /{\nu}_0\right)}\right){\left(M/{M}_0\right)}^{0.2669} $$

(33)

where $ {v}_0=\sqrt{E/\rho } $ and M₀ = abt ρ.

Since the impact speed is relatively low (much smaller than the wave speed), our simulation shows that the dynamic effect is relatively minor for the range of parameters investigated in this chapter. That is, for a given head mass, the computed stress field from dynamic impact (with incident speed v) is very close to that of quasi-static indentation (with maximum indentation depth D/t) as long as they induce the same deformation curvature of the plate (i.e., when v and D/t satisfy Eq. 33); see an example in Fig. 15 for the stress field. This enables us to simplify the dynamic impact problem into a quasi-static indentation one, where the normalized indentation depth D/t becomes the governing variable to indicate the effective “impact condition.”

Our previous study showed that for windshield fracture, circumferential cracks always grow after the radial crack has developed, except near the vicinity of impact where extensive damage can be found (Xu et al. 2009). This is consistent with Fig. 15 that the hoop stress field is more prominent than radial stress. Thus, the crack simulation can be decoupled into two steps. First, we embed a number of initial flaws at the center of the plate (impact location); this is consistent with the fact that extensive damage is initiated in the contact zone and that facilitates subsequent crack initiation and propagation. The size of the initial flaw is a₀ = 3 mm with variable directions. When these initial flaws are very small, only the radial flaws will grow to become radial cracks upon impact, so as to release strain energy. For a given indentation depth D, the radial crack grows gradually until a certain length; with the increase of D, the final length of radial crack also increases. In the second step, based on the radially cracked specimen, new circumferential flaws with size a₀ = 3 mm are introduced (the sensitivity of the flaw geometry and distribution is discussed below), and without further increasing the load, with the residual stress and strain fields inherited from the first step, the development of circumferential cracks is simulated by XFEM.

Rack Propagation Characteristics

Radial Crack Propagation Characteristics

Overall Crack Feature

In Fig. 16, for a typical displacement of D/t = 0.62, the evolved radial crack is shown along with the hoop stress field. Note that with the fully developed crack, this stress field is already much relaxed compared to its counterpart without cracking (see for example the difference between Figs. 15 and 16); moreover, the current hoop stress field is specified for the global coordinate, whereas during crack propagation, it is the local hoop stress that determines the crack trajectory. When the local maximum hoop stress exceeds a critical threshold, referred to as the damage stress σ_d, the crack grows incrementally toward such a local principal direction.

In the beginning stage of impact, since the growth of radial crack can release more energy than the circumferential crack (due to the more prominent hoop stress field, Fig. 15), radial crack grows with the increase of the indentation depth D/t. It is noticeable that the crack growth direction keeps changing slightly with growth direction 22° approximately. This is mainly caused by the rectangular shape of the windshield plate, which makes the overall strain in the y-direction larger than that in the x-direction. When the crack grows further, the boundary constraint changes and the crack direction approaches about 45°.

During its growing process, the crack seeks to release as much energy as possible through mode I fracture. The released strain energy E_strain is computed as the difference between the strain energies of two specimens, one cracked and one without crack (at the same D/t). In Fig. 17, E_strain is normalized by the initial impact kinetic energy of the head E_kinetic (with M = 4.5 kg, the impact speed can be deduced from Eq. 33). It is also illustrated that the released strain energy is relatively small compared with the overall system energy, and it increases as D/t enlarges (although when it is normalized by the incident kinetic energy, which also increases as D/t gets larger, the dimensionless number E_strain/E_kinetic fluctuates somewhat).

Sensitivity of Initial Flaw and Fracture Criterion

The damage stressesσ_d of different soda-lime glass specimens are different due to its intrinsic stochastic flaws, usually from 10 to 60 MPa. Under quasi-static indentation, the effect of σ_d on crack evolution is shown in Fig. 18. The results indicate that as the critical stress for crack initiation σ_d becomes larger, the crack angle is smaller (the boundary constraint is more prominent) and the crack is shorter. In other words, if the impact condition is known, one can roughly estimate the damage stress threshold value according to the crack path and length. In what follows, σ_d = 10 MPa is employed.

The model I energy release rate is another very important parameter governing the brittleness of the material. Consider typical material candidates for windshield such as soda-lime glass and PMMA, the former is a brittle material whereas the later has certain degree of ductility, their mode I energy release rates are about G_I = 10 J/m² and G_I = 400 J/m², respectively. Without loss of generality, we normalized the crack length C with plate size a. In Fig. 19, the effect on the resulting length of crack is examined by varying G_I from 10 to 400 J/m². Numerical study demonstrates that if the G_I value is within the brittle material domain, there is little effect on the crack length; otherwise, if the windshield is made by the more ductile material, the crack length is shorter under the same plate deformation. In what follows, G_I = 10 J/m² is employed.

The characteristics of initial flaw have some minor effects on crack propagation. We first keep the initial flaw length to be 3 mm and let the flaw angle θ_i to be 30°, 45°, and 60° with respect to the x-axis, and the final crack pattern is shown in Fig. 20. It is observed that θ_i does have some influence on the initial crack trajectory; however, eventually, all crack paths converge with similar lengths, indicating both the robustness of the current model. The length of the initial flaw has negligible effect on both the crack propagation pattern and crack length (as long as the initial flaw is short). When the initial flaw length is taken to be 0.5 mm, 1 mm, 3 mm, and 5 mm, the resulting radial crack lengths in these four scenarios are exactly the same; if the initial flaw is 10 mm, the crack propagation pattern is slightly different. Therefore, the 3 mm initial flaw used in the present study is validated and it is a relatively insensitive parameter in the present XFEM simulation.

Radial Crack Length

The length of the radial crack can be represented as a function of the quasi-indentation depth D/t, which is in turn related to the impact speed v for a given effective head mass (see Eq. 34). Figure 19 also shows the normalized crack lengths, i.e., C/a versus the normalized head displacement D/t. Initially, the crack length is almost proportional to head displacement; however, after a certain length owing to the fixed boundary, the crack length almost remains a constant. The relationship between C/a and D/t is nonlinear and may be fitted as

$$ \begin{array}{l}C/a=-0.4046+0.9231\left(D/t\right)-0.05074{\left(D/t\right)}^3\\ {}+0.03803\operatorname{exp}\left(D/t\right)-0.4243\sqrt{\left(D/t\right)}\operatorname{ln}\left(D/t\right)\end{array} $$

(34)

By combining Eqs. 33 and 34, a new method can be derived to estimate the impact speed directly based on the knowledge of radial crack length (assuming the effective head mass is known) using Newton–Raphson method, thus useful for traffic accident reconstruction. This is illustrated in the next section.

Equation 34 shows that if the head mass is fixed, the crack length increases nonlinearly with impact speed. In order to estimate the influence of the effective head mass, we can combine Figs. 14 and 19, along with Eqs. 33 and 34. For instance, from Fig. 14 (or Eq. 33), if the impact speed is fixed at 5 m/s, for different effective head masses, the variation of crack length (as a function of the corresponding D/t) can be obtained from either Fig. 19 or Eq. 34. Compared with the crack length of M = 22.5 kg, M = 45 kg, and M = 90 kg its 193.27%, 241.35%, and 294.23% for M = 4.5 kg, respectively. That is to say, the effective head mass has a strong influence over the crack length, indicating that during a low-speed impact accident, the shape of vehicle front end and pedestrian–vehicle contact point is critical in impact speed estimation since these factors will largely determine the pedestrian revolution angle. In addition, the influence of M becomes relatively smaller when M is large, implying that the influence of the effective mass starts to decay if the effective head mass is much larger than the windshield mass.

Circumferential Crack Propagation Characteristics

On the basis of the radially cracked geo metry, in the second step, we introduce circumferential flaws of length a₀ = 3 mm along the radial crack, with a spacing of 50 mm (the spacing is found to be a relatively minor factor on the final circumferential crack pattern). The stress field calculated from the end of first step is transferred into the second step. Since the first step has relaxed most of the hoop stresses, the remaining excessive radial stress needs to be relaxed via the development of circumferential crack, and the most efficient way of releasing the radial stress is to develop the circumferential crack with maximum radius, i.e., on the same order of C. This is verified from Fig. 21: Cracks also grow near the contact zone because of the high stress. This is consistent with the observation that intensive cracks are identified within the contact area while the crack density is lower outside, and the largest circumferential crack almost bounds the radial cracks. The strain energy released by circumferential crack is given in Fig. 17, where the total released energy is less than 50% than that of radial crack. This evidence strongly supports the argument that crack is prone to grow in a radial way rather than a circumferential way, since more energy can be released. From Fig. 17, we can see that less than about 5% of the total kinetic impact energy is consumed by both radial cracks and circumferential cracks and thus there is enough room for improving the automotive windshield for enhanced energy mitigation efficiency and pedestrian protection (for example, using transparent energy absorption materials).

Experiment Analysis

The mechanical behavior of PVB laminated glass is quite complex, especially under dynamic loadings due to the combination of rate-dependent effect of PVB and brittle-polymer composite structure. Therefore, experimental data are very important for the proper suggestion of constitution model and FE model to further investigation mechanical behavior of PVB laminated glass under both static and dynamic loadings.

Typically, one can categorize experiments for studying mechanical response of such a composite material into three regimes in terms of different strain rates: quasi-static, low impact speed, and high impact speed (Stout et al. 1999).

Experiment of Quasi-Static Loading

Because of the difficulties in c onducting a quasi-static tension experiment with a slippery surface and the windshield undergoes compression subject to human head impact, we adopted a compression experiment rather than the bending test which the shear effect will influence the mechanical property of the specimen to a large extent.

Experimental Setup

PVB laminated glass circular flake is introduced to be the testing specimen, with the dimension of u = 10 mm, where 0.76 mm-thick PVB interlayer is sandwiched by two pieces of 2 mm-thick glass, shown in Fig. 22. Note that the glass, as a typical rate-independent brittle material, has a density of ρ = 2,500 kg/m³, Young’s modulus of E = 70 GPa, and Poisson’s ratio of m = 0.22 (Forquin and Hild 2010). It is worthy to point out that PVB laminated glass is made by putting two glass sheets into an autoclave and compressed with 0.76 mm-thick of PVB interlayer under specific cycle of pressure (10 bar) and heat (120 °C). Thus, we can regard there is no slide between three layers (Wang et al. 2007). To make the experiment more engineering applicable, the thickness of specimen used here has the same thickness as that in most passenger cars. In experiment, specimen is put in a cylinder container which has a steel pin with a diameter u = 38 mm inside for compression and the experiments are carried out on Material Testing System (MTS). The load is applied perpendicularly through the pin to the sample. A clip-gage type extensometer is installed between the gaps to measure the displacement of the pin as a function of load during the experiment, shown in Fig. 23. 1 mm/min, 0.1 mm/min, and 0.01 mm/min are three loading rates, corresponding to the strain rate on the upper and lower sides of the specimen of $ \dot{\varepsilon}\approx 1\times {10}^{-3}{s}^{-1} $, $ \dot{\varepsilon}\approx 1\times {10}^{-4}{s}^{-1} $, $ \dot{\varepsilon}\approx 1\times {10}^{-5}{s}^{-1} $ chosen to measure the quasi-static mechanical property of PVB laminated glass. During tests, unloading threshold value is set to be 80% of the maximum loading value and thus complete load–displacement history is recorded. The temperature and relative humidity during the experiments remain in the range of 25–26 °C and 40–45%, respectively.

Experiment Results

It is commonly accepted that the failure process of composite subject to quasi-static compression loading usually involves a sequential damage accumulation process (Xu et al. 2009). As aforementioned, we set the unloading threshold value to be 80% of the maximum load value due to the major load decrease occurred during the experiment when major crack could be observed clearly. Thus, we define the stress and strain at this moment as major failure on-set (MFO) stress and strain. Figure 24 shows us the “load decrease” phenomenon and the morphology of the cracked samples.

Figure 25 illustrates three different stress–strain relation curves under three loading rates. The stress–strain curves show “nonlinearity” characteristic in mechanical behavior of PVB laminated glass and this “nonlinearity” phenomenon is probably caused by both nonlinear mechanical response of PVB interlayer and progressive microcrack growth in sample. As the loading rate increasing, the MFO strain increases while the MFO stress remains nearly the same. The major responsible reasons are: (i) in extremely low strain rate (quasi-static) situation, the outer glass panel plays a critical role in mechanical response and (ii) glass is a rate-independent material whereas PVB is a rate-dependent one.

Number of Crack Branching

The numbers of cracks in testing samples according to different quasi-static loading rates are illustrated in Fig. 26. The number of complete cracks (or major cracked branching) increases with the loading rate. As aforementioned, in quasi-static situation, the stress remains the same whereas the strain rises with the increasing loading rate. Therefore, the number of cracked branching, namely the “damage degree,” depends on the value of strain, not the stress, in quasi-static circumstance that is in accordance with Wang’s (Wang et al. 2006) study. Further, if a nonlinear fitting is tried on the number of crack branching versus loading rate, we will harvest the following explicit formula with fitting coefficient R = 0.999.

$$ N=\left[8.975{W}^{0.339}\right] $$

(35)

where [X] is Round function, e.g. [5.2] = 5. N is the number of crack branching and W is loading rate with unit of mm/min. From Eq. 35, we can infer the numbers of crack zones under different loading rates in the quasi-static domain.

Mandelbrot (1977) pointed out that crack branching contained critical information of crack propagation speed and complexity. Bouchaud et al. (2012) further investigated the relationship between crack length and crack branching mode and concluded

$$ {D}_f=\frac{\operatorname{lg}\;b}{\operatorname{lg}\;k} $$

(36)

where $ {D}_f=\frac{\operatorname{lg}\;b}{\operatorname{lg}\;k} $ is degree of branching, b is the number of basic branching pattern, and K ≈ 0.4 in brittle material.

Assuming that the initial crack branching length is a₀ and the length becomes a₁ = a₀ + ka₀ (Bouchaud et al. 2012), thus after the initial crack branches for i times, the length finally comes to

$$ {a}_i=\left(1+\cdots +{K}^i\right){a}_0 $$

(37)

Since the number of branching can be written as N ¼ biþ1, we are able to know that

$$ i={\mathrm{log}}_b^N-1 $$

(38)

According to Freund’s (Michel and Freund 1990) theory , the dynamic crack propagation velocity in a finite body obeys the following relation:

$$ v={C}_r\left(1-\frac{a_0}{a_i}\right) $$

(39)

where C_r is the Rayleigh wave velocity and v is the crack propagation velocity.

By combination of Eqs. 36, 37, 38, and 39, the relation between crack velocity and crack branching number is

$$ v={C}_r\frac{1+{b}^{-1/{D}_f}-2{N}^{-1/{D}_f}}{2\left(1-{N}^{-1/{D}_f}\right)} $$

(40)

Further, we substitute Eq. 35 into Eq. 40 and get

$$ v={C}_r\frac{1+{b}^{-1/{D}_f}-2{\left[8.975{W}^{0.339}\right]}^{-1/{D}_f}}{2\left(1-{\left[8.975{W}^{0.339}\right]}^{-1/{D}_f}\right)} $$

(41)

Equation 40 may help us to clarify the crack propagation speeds in different loading rates. We may also conclude that crack propagation speeds are different during the three loading rates in quasi-static experiments based on Eq. 41.

Experiments of Dynamic Out-of-Plane Loading

By starting from a fundamental point, quasi-static mechanical behavior of PVB laminated glass has been widely investigated experimentally above (Xu et al. 2011a). Unlike quasi-static cracking problem, dynamic fracture is more complex and challenging but there are few reported investigations that focused on the dynamic fracture of PVB laminated glass except for some numerical simulations those based on probabilistic damage mathematical model (Forquin and Hild 2010), continuum damage mechanics (Zhao et al. 2005, 2006a, b; Sun et al. 2005), explicit finite element method (Bois et al. 2003; Timmel et al. 2007; Sun et al. 2009), and extended finite element method (Xu et al. 2010) that often neglected the important PVB layer and most of them did not involve explicit crack growth and pattern analysis. The dynamic experimental investigation is pressing need.

System Setup of Dynamic Out-of-Plane Loading

The impact system setup is sketched in Fig. 27, where a weight block can slide freely along the two standing poles. At the end of the drop-weight, there is a tip which is used to hit the force direction converter on the testing sample and it can be referred in Fig. 28. It is a necessity to convert the vertical impact direction into a horizontal one, which would facilitate the capture of the film.

Upon impact high speed photography is employed for capturing the dynamic crack growth in the plate in situ. The entire system consists of a high-voltage charging controller, multispark box, two field lenses, and an array camera with films (see in Fig. 28). In addition, a multispark high-speed camera is adopted to provide enough spot light source. The multispark box provides 16 independently spot lights triggered by a high-voltage charging controller by a preset adjustable interval time from 1 to 9,999 μs whose charging voltage is able to reach up to 30 kV. Meanwhile, a 4 × 4 array camera with 16 films is set correspondingly on the other side to receive the light and therefore record the crack image information. In order to make sure that the images recorded on the films are in the time series of the ignition of the spot lights, two field lenses are used to make up the optical path (see Fig. 28b). Moreover, the delay controller is introduced to control the time interval between each spot light. In addition, to minute the exact ignition time for each spot light, the electric-light receiver along with the oscillograph monitor are used together where the electric-light receiver is employed to collect the electric-light signals and send them to the oscillograph monitor. Reference control experiments of impact-induced fracture are carried out using pure glass sheets, with results in good agreement with the literature (Nielsen et al. 2009; Sharon and Fineberg 1999) (see in Fig. 29), thus validating the current test system.

The Preliminarily Experiment to Investigate the Radial Crack Propagation

Experiment Condition

In the preliminarily impact fracture experiment to investigate the radial crack propagation behavior of PVB laminated glass , the minimum mass of the drop-weight is 0.5 kg and the maximum height of the drop-weight is 1.84 m (corresponds to the maximum impact velocity of 6 m/s); both the mass and height of the drop-weight are also varied in this experimental investigation. Prior to impact, the drop-weight is suspended and fixed through electromagnet. The temperature and relative humidity during the experiments remain in the range of 20–21 and 20–25% respectively.

Specimen Preparation

The standard PVB layer has a Young’s modulus E_PVB = 100 MPa, Poisson’s ratio υ_PVB = 0.48, and mass density ρ_PVB = 870 kg/m³ (Xu and Li 2009). To make close connection to the automotive industry, the PVB thickness is t_PVB = 0.76 mm for passenger cars. In addition to the middle PVB layer, for the upper and lower soda-lime glass sheets in laminated glass, the thickness t_glass = 2 mm, with ρ_glass = 2,500 kg/m³, E_glass = 70 GPa, and υ_glass = 0.22. They are bonded together at 10 bar and 120 °C following standard procedures, and the effective in-plane dimension is 200 × 150 mm.

Experimental Results and Discussion

Crack Morphology

A series of experiments under different loading conditions, i.e., with different height and mass of the drop-weight are recorded. Figure 30 shows the typical dynamic crack propagation snapshots of the PVB laminated glass sheet, in response to a 1 kg weight dropped from 1 m height. Extensive radial fracture is observed, with no appearance of circumferential crack, which is consistent with numerical simulation that when a foreign object impacts a large windshield, the radial crack always appears before the circumferential crack (Xu et al. 2010). It is seen that there are about ten radial cracks on the composite sheet, and the development and evolution of each crack are quite different due to the inevitable stochastic flaw in glass. Here, the average crack length is employed to discuss the behavior of radial crack propagation. By using nonlinear fitting method to the experimental points obtained from the experiments, we finally got the crack tip position in time history. Then the averaged time evolution history of the radial crack tip velocity and acceleration can be obtained from the time history of the radial position of the crack tip as showed in Fig. 31.

The measured crack velocity first increases and then fluctuates slightly, which goes accordance with the experiment results of cracking speeds of pure soda-lime glass at various cracking modes in Nielsen et al. (2009) and Sharon and Fineberg (1999), as well as being consistent with previous numerical simulation results of several brittle materials (Song et al. 2008; Grégoire et al. 2009) and composites (Daphalapurkar et al. 2007; Dwivedi and Espinosa 2003).

As proposed by Ravichandar and Knauss (1984), near the vicinity of crack tip, there is a microcrack zone which impedes the further propagation of crack. Within the impact area, the spacing between the initial cracks is so small and such an intensively damaged zone may reduce the cracking speed substantially. With the advancement of cracks, the retardment effect will be greatly lessened until it reaches an almost constant influence, when the crack speed becomes steadier in the later stage. Thus, the crack evolution may be qualitatively divided into two stages: in stage I, the cracking speed keeps rising, and in stage II the crack speed becomes more stable but with some fluctuations. The transition occurs roughly at about time (further discussion will be made later). During stage I, the crack propagation acceleration is decreasing until stage II begins; that is, as the stress in the crack tip accumulates and develops, finally the crack tip will be able to break through the energy barrier surrounding material to reach the equilibrium.

In this example, the averaged steady-state cracking speed during stage II is about $ {\overline{v}}_{PVB}\approx 811\ m/s $. Comparing with the stable cracking speed of Polymethyl Methacrylate (PMMA)$ {\overline{v}}_{PMMA}\approx 600\;m/s $ (Sharon and Fineberg 1999) and tempered soda-lime glass $ {\overline{v}}_{GLASS}\approx 1470\;m/s $ (Nielsen et al. 2009) (One may refer to Fig. 32 for detail comparison. Note that although the impact energy is a little bit different in (Xu and Li 2009) and this experiment, the comparison is still valid), the PVB laminated glass may be regarded a composite material whose overall effective “fracture toughness” is between that of PMMA and pure glass. In Fig. 32, we can also see that the general variation trend of crack speeds in two different materials are almost the same. One may use the Rayleigh wave speed v_R of the glass sheet to normalize the crack speed (since cracking occurs in the glass sheet), i.e., v_R = 3,098 m/s. Thus, the stable cracking speed of PVB laminated glass is about 0.26 v_R in this particular case, which is lower than the typically measured maximum speed of about 0.4–0.5 v_R in brittle materials and we can clearly see that the PVB interlayer serves to slow down the crack speed, which would also make the material as well as the brittle-polymer sandwiched structure a candidate for energy absorption/shielding.

Different Driving Mechanisms of In-Plane Cracking on Two Glass Layer

Figure 33 shows a typical crack propagation comparison on supported and loaded glass layers at the loading speed with PV B thickness h = 0.76 m. Here, time t = 0 μs is used to denote crack tip initiation for the radial cracks on supported glass layer. It is clear that the radial cracks (Fig. 33a–d) initiated firstly on the “supported glass layer,” then the radial cracks on the “loaded glass layer” generated after about t = 600 μs as shown in Fig. 33f. On the other hand, it is also interesting to observe that the cracks on the two glass layers are completely overlapped even though they propagate at totally different times.

In fact, the stress wave dominated inertial effects during impact loading on the PVB laminated glass play an important role in crack propagation (Singh and Parameswaran 2003; Clements et al. 1996). Due to the high modulus of glass material and the fixed mode of the specimen, the plate is small enough to ignore the bending deformation during the impact. For the convenience of analysis, a schematic diagram in a layer system consisting of two glass plates bonded by a PVB interlayer is shown in Fig. 34. Impact occurs on the surface of the “loaded glass layer” of the PVB laminates, the compressive waves caused by the impact travel down the depth direction (Fig. 34). Due to its viscoelasticity, the PVB layer plays an important role in reducing the amplitude and the speed of waves that eventually pass through by internal friction (Thom 2005), compared with the glass layer. Finally, the compressive waves reflect from the boundaries as tensile waves, which continuously arrive at the supported plate thus the tensile dominated cracks initiate while reaching the limit of dynamic fracture toughness KID of the material (Lambros and Rosakis 1997). Considering that glass plate is more sustainable in compressive stress wave but prone to fail in tensile stress wave, the radial cracks initiate first on supported glass layer.

Effect of Drop Weight and Height

Next, we investigate the influence of drop weight and height on the crack propagation. The drop mass ranges from 0.5 to 2 kg, and the height varies from 0.6 to 1.4 m. Table 1 shows the detail combination of the parametric experiments. Figures 7 and 8 show that with the increase of drop weight and height, the basic trend of crack would remain almost the same while the value would increase, i.e., larger mass and higher impact speed will cause faster crack propagation. Zhang et al. (2010) investigated the effect of loading rate on crack velocities in concrete and found with the increase of loading rate, the crack velocity increase proportionally; in quasi-static indentation, it was also found that with larger indentation load, the radial and lateral cracks in ceramic also became longer. All these conclusions are qualitatively consistent with our experimental findings. Particularly, $ {\overline{v}}_{PVB} $ increases 17.70% and 12.21% when the impact speed changes from 3.43 to 4.43 m/s and 5.24 m/s, respectively, and $ {\overline{v}}_{PVB} $ increases 26.72% and 14.92% when the drop weight is increased from 0.5 to 1 kg and 2 kg, respectively. Furthermore, if we combine the effect of impact mass and velocity together into the term of impact energy E, and normalize it by a reference potential energy E_n = 9.81 J, then the normalized steady-state cracking speed, i.e., $ {\overline{v}}_{PVB}/{v}_R $ can be fitted as

$$ {\overline{v}}_{PVB}/{v}_R=0.167\pm 0.137/\left(1\pm 10 \operatorname {exp}\left(1.50\right.\left(0.739-E/{E}_0\right)\right) $$

(42)

for the range of parameters in this study. From Fig. 35, it can also be found that the stable crack velocity will not likely to increase indefinitely as the external impact energy gets higher, and it has a theoretical limit, i.e., Rayleigh wave speed v_R, but not reaching it.

Regarding the transition time between stages I and II, the general trend is that the smaller the external energy, the faster the transition time (although the difference is relatively small and still on the same order of magnitude). That is to say, if the impact energy is higher, it is easier for the crack speed to achieve steady-state. The detailed physical mechanism of the variation of such a transition time will be explored in future.

The Further Experiment Investigating the Radial and Circular Crack