2.2 Studies on Mixed Mode Interface Fracture
2.2.2 Mixed Mode Experiments
Interface Characterization 2.2 Studies on Mixed Mode Interface Fracture
is equivalent to the pure Mode I curve given as input. As the tangential crack opening is increased, the response on the normal stress diminishes to a minimum. Along with the normal stress response, a diagram in Figure 2.7(b) displays the response of the shear stressτ. It is clearly seen that the shear stress response is affected oppositely of the case of normal stress.
2.2 Studies on Mixed Mode Interface Fracture Interface Characterization
very flexible, and hence unsuitable for testing. In the present study, the stiffness has been measured for the specific set-up and the maximum testing angle αwas found to be 30 degrees.
A test program to obtain constitutive parameters for the mixed mode model was per-formed. A total number of nine specimens, three for each of the mixed mode angles 00, 150, and 300, were tested. A two step inverse analysis to couple the experiments in the δn−δt space, was established. First step is to translate each experiment into a bilin-ear stress-crack opening relationship (for Mode I) and a bilinbilin-ear stress-crack tangential opening (for Mode II). This is shown for an experiment on a test withα=300, in Figure 2.9.
0 0.2 0.4 0.6 0.8 1
0 0.5 1 1.5 2 2.5 3
Crack deformation δn,δt [mm]
Stress σ,τ [MPa]
Experimental results Mode II Bilinear approximation Mode II Experimental results Mode I Bilinear approximation Mode I
Mixed Mode Angle ψ=300
Figure 2.9Example on experimental data from a test withα= 300and the approximation of two bilinear curves in Mode I and II.
As displayed in Figure 2.9, an experiment on a specimen with an inclination angle of 300 produces a larger Mode I response compared to the Mode II response. The translation of the experimental data into a bilinear shape makes later numerical interpretation simple.
The final step in the inverse analysis is to couple the bilinear curves obtained in theδn−δt
andσ−τ space. The kink points in the bilinear curves are coupled according to Equation (2.4). When applying a bilinear shape, a total number of 4 kink points need to be coupled.
Two stress kink points need to be coupled in theσ−τ space, and two crack deformation kink points need to be coupled in theδn−δtspace. Coupling of the stress kink points are shown in Figure 2.10(a)-(b), and coupling of the deformation kink points are shown in Figure 2.11(a)-(b).
As observed in the figures, coupling of the experimental data is possible and acceptable.
However, a large amount of scatter is present for the kink points in the deformation measurements, Figure 2.11(a), though the scale of the deformations has to be considered when comparing to the deformations in Figure 2.11(b).
It should be noted that only reliable results are obtained from experiments on low mixed
Department of Civil Engineering - Technical University of Denmark 21
Interface Characterization 2.2 Studies on Mixed Mode Interface Fracture
mode angles, and via extrapolation of these, it is possible to derive the pure Mode II curve. The final results, which can be given as input in the constitutive model are: (i) a pure Mode I and II curve, and (ii) exponentsm andn for use in the failure criterion (Equation (2.4)). The values of the pure Mode I and II curves are shown in Table 2.1.
Mode I σmax1 [MPa] σmax2 [MPa] δmax2n [mm] δnmax3 [mm] Gf [n/mm]
n-direction 3.0 0.4 0.02 0.5 0.12
Mode II τmax1[MPa] τmax2[MPa] δmax2t [mm] δtmax3 [mm] Gf [n/mm]
t-direction 3.5 0.5 0.02 0.77 0.23
Table 2.1Pure Mode I and II parameters (Figure 2.5), obtained in the inverse analysis.
The results in Table 2.1 are found for the exponents m = n = 2. Optimization of the exponents can not be justified on the small amount of data available and has not been utilized in the present study. The two curves are found as two bilinear curves wherea1
anda2are the slopes of the two line segments andb2 is the cross point of the second line segment and the normalized stress axis (y-axis).
Since data has solely been collected for low mixed mode angles, extrapolated data for high mixed mode angles is less reliable. The ideal case, and a major improvement of the data collected, is to carry out tests using a biaxial testing machine capable of changing the ratio of shear and normal deformation. Then by testing different mixed mode angles a full set of stress deformation curves in a mixed mode angle range from 00to 900 could be collected.
0 1 2 3 4
0 1 2 3 4
σ [MPa]
τ [MPa]
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
σ [MPa]
τ [MPa]
Figure 2.10(a) Plot of coupling stress kink points, (σmax1, τmax1) (b) Plot of coupling stress kink points, (σmax2, τmax2).
2.2 Studies on Mixed Mode Interface Fracture Interface Characterization
0 0.01 0.02 0.03 0.04 0.05 0
0.01 0.02 0.03 0.04 0.05
δn [mm]
δt [mm]
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
δn [mm]
δt [mm]
Figure 2.11(a) Plot of coupling crack deformation kink points, (δnmax2, δtmax2) (b) Plot of coupling crack deformation kink points, (δnmax3, δtmax3).
Department of Civil Engineering - Technical University of Denmark 23
Interface Characterization 2.2 Studies on Mixed Mode Interface Fracture
Chapter 3
Composite Elements
An important issue in the cement-based overlay system for stiffening orthotropic steel bridge decks, is cracking of the overlay. The composite strength of the system is closely related to cracking of the overlay, and subsequently cracking of the steel-concrete interface (debonding). Penetration of a vertical crack in the overlay, might for a certain crack width, initiate large stresses at the interface. Increased interfacial stresses will consequently lead to debonding. Using numerical tools can help identifying important parameters, which influence the performance, with regards to cracking of a steel deck stiffened with a cement-based overlay. Through testing and numerical simulations it might be possible to identify the significance of overlay and interface cracking in relation to the composite behavior.
The major outcome ofPaper V is a number of numerical parametric studies on different constitutive parameters of the overlay and interface with regards to the composite behavior between a cement-based overlay and steel plate. BothPaper III andPaper IV contains numerical and experimental studies of overlay fracture. In the present chapter, a short overview is given on numerical and experimental studies on small composite elements.
The reader is referred to the appended papers for a review in greater details.
3.1 Numerical Studies on Composite Beams
The composite behavior between the overlay and a the steel plate can be analyzed using a simple three point bending test. Consider a composite beam exposed to negative bending as shown in Figure 3.1. The composite beam is viewed as the very top part of the bridge deck, turned up-side down for convenience. Cracking and debonding can be analyzed as discrete processes, vertical cracking in the overlay and horizontal cracking at the interface, respectively.
Loading of the composite beam will at some point cause cracking of the overlay as the overlay reaches its tensile strength. As the vertical crack propagates through the overlay, its crack front will at some stage be opposed by the steel plate. The opposition of the steel plate will lead to an increase of normal stress in the plane perpendicular to the vertical crack tip, i.e. in the plane of the steel concrete interface. The increase in horizontal stresses is likely to introduce cracking of the steel-concrete interface. This situation can be analyzed using finite elements. Consider a close up look of the part where the overlay
25
Composite Elements 3.1 Numerical Studies on Composite Beams
Figure 3.1Three point bending set-up: simulating a negative bending moment in a bridge deck.
crack initiates as sketched in Figure 3.2(a). Two situations are analyzed: (i) an overlay Crack Mouth Opening Displacement (CMOD) of 0 mm, and (ii) an overlay crack opening of 0.03 mm. The shear and normal stress can be plotted along the interface to analyze the problem, cf. Figure 3.2(b).
(a)
0 0.5 1 1.5 2
−2 0 2
X−coordinate/h
c [mm/mm]
Stress [MPa]
σ, CMOD=0.03mm σ, CMOD=0.00mm
τ, CMOD=0.00mm τ, CMOD=0.03mm
(b)
Figure 3.2Stress distribution along the interface for a CMOD value of zero and 0.03 mm. (a) Interfacial forces and configuration. (b) Stress distribution along the interface versus the x-coordinate normalized with the concrete heighthc. Dashed lines represent shear stressτ and solid line represent the normal stress σ.
The first situation corresponds to a sound overlay with no cracking, whereas the second situation corresponds to the initiation of a small overlay crack. The interfacial stresses (normal, σ, and shear, τ) can be plotted in a stress vs. x-coordinate diagram (x = 0 is the location of the vertical overlay crack). It is observed from Figure 3.2(b), that the interfacial stresses change dramatically for an increase of crack opening from 0 to 0.03 mm. The normal stresses change from compression to tension, which in many cases are