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Figure 10.3 Indentation mode of collapse for a threepoint bend configuration f
Consider first the case of threepoint bending. Then the collapse load F on the upper indenter can be derived by a simple upper bound calculation. Two segments of the upper face, of wavelength X, are rotated through a small angle B. The resulting collapse load is given by
where Mp = bt2/4 is the full plastic moment of the facesheet section. Minimization of this upper bound solution for F with respect to the free parameter X gives an indentation load FI of
We note in passing that the same expression for Fj and X as given by equations (10.13) and (10.14) are obtained by a lower bound calculation, by f y considering equilibrium of the face sheet and yield of the face sheet and core. We assume that the bending moment in the face sheets attains a maximum local value of —Mp at the edge of the indenter and also a value of Mp at a distance X from the edge of the indenter. It is further assumed that the foam core yields with a compressive yield strength oy and exerts this level of stress on the face sheet, as shown in Figure 10.3. Then, force equilibrium on the segment of a face sheet of length (2X + a) gives
and moment equilibrium provides
Relations (10.15) and (10.16) can be rearranged to the form of (10.13) and (10.14), demonstrating that the lower and upper bound solutions coincide. We conclude that, for a rigidperfectly plastic material response, these bounds give exact values for the collapse load and for the span length X between plastic hinges. The presence of two indenters on the top face of the sandwich beam in fourpoint bending results in a collapse load twice that for threepoint bending, but with the same wavelength X.
Core shear
When a sandwich panel is subjected to a transverse shear force the shear force is carried mainly by the core, and plastic collapse by core shear can result. Two competing collapse mechanisms can be identified, as shown in Figure 10.4 for the case of a beam in threepoint bending. Mode A comprises plastic hinge formation at midspan of the sandwich panel, with shear yielding of the core. Mode B consists of plastic hinge formation both at midspan and at the outer supports.
Consider first collapse mode A (see Figure 10.4). A simple work balance gives the collapse load FA, assuming that the face sheets on the right half of the sandwich panel rotate through an angle B, and that those on the left half rotate through an angle —B. Consequently, the foam core shears by an angle B. On equating the external work done FIB/2 to the internal work dissipated within the core of length (I + 2H) and at the two plastic hinges in the face sheets, we obtain
where r'y is the shear yield strength for the foam core. Typically, the shear strength of a foam is about twothirds of the uniaxial strength, tC = 2oC/3.
We note from equation (10.17) that FA increases linearly with the length of overhang, H, beyond the outer supports.
Second, consider collapse mode B. As sketched in Figure 10.4, this collapse mechanism involves the formation of plastic hinges in the face sheets at both midspan and at the outer supports. The core undergoes simple shear over the length, L, between the outer supports, with no deformation beyond the outer supports. A work calculation gives for the plastic collapse load FB,
4bt2 f c
Since the two calculations given above are upper bounds for the actual collapse load, the lower is taken as the best estimate for the actual collapse load. It is instructive to compare the collapse loads as a function of overhang length H, as sketched in Figure 10.5. Collapse mode A is activated for small lengths of overhang, whereas collapse mode B has the lower collapse load and is activated for large overhangs. The transition length of overhang, Ht, is determined by equating (10.17) and (10.18), giving
1 t2 ofy
Mode B
Mode B
Overhang, h
Figure 10.5 The competition between collapse modes A and B for core
In order to gage the practical significance of the overhang, let us take some representative values for a typical sandwich panel comprising aluminum skins and a metallic foam core, with c/l = 0.1, t/c = 0.1, r^/af = 0.005. Then, the transition overhang length, Ht, is given by Ht = 0.1l: that is, an overhang of length 10% that of the sandwich panel span l is sufficient to switch the collapse mode from mode A to mode B. Furthermore, the enhancement in collapse load due to plastic bending of the face sheets above the load required to shear the core is about 20% for a small overhang, H — Ht, and is about 40% for H > Ht. In much of the current literature on sandwich panels, a gross approximation is made by neglecting the contribution of the face sheets to the collapse load.
Parallel expressions can be derived for the collapse of a sandwich beam in fourpoint bending by core shear. The collapse load for mode A becomes
The transition length of overhang at which the expected collapse mode switches from mode A to mode B is given by the same expression (10.19) as for a beam in threepoint bending.
10.3 Collapse mechanism maps for sandwich panels shear
and that for mode B is
It is assumed that the operative collapse mechanism for a sandwich beam is the one associated with the lowest collapse load. This can be shown graphically
F = F/biof are plotted, with c/i and t/c as axes. The map is drawn for the f selected values ocY/oJY = 0.005 and a/i = 0.1. (b) The effect of the size of the indenter a/i upon the relative dominance of the collapse mechanisms for threepoint bending by plotting a nondimensional measure of the collapse load F = F/biof on a diagram with the nondimensional axes c/i and t/c, for selected values of a/i and ocy/of. An example is given in Figure 10.6(a) for the case of threepoint bending, with a/i = 0.1 and oy/of = 0.005. It is assumed that the overhang H exceeds the transition value Ht so that core shear is by mode B, as depicted in Figure 10.4. The regimes of dominance for each collapse mechanism are marked: for example, it is clear that failure is by face yield for thin facesheets (small t/c) and long beams (small c/i). The boundaries of the indentation regime are sensitive to the value taken for a/i: with diminishing a/i the magnitude of the indentation load drops and this regime enlarges on the map, as illustrated in Figure 10.6(b). It is striking that the boundary between the core shear and the indentation regimes has a large curvature, with the indentation mechanism operating at a large values of t/c as well as at small values. This is a consequence of plastic hinge formation within the face sheets in the core shear collapse modes: the collapse load for core shear increases quadratically with increasing t/c due to the contribution from face sheet bending, as seen by examining the first term on the righthand side of relations (10.17) and (10.18). The contours of collapse load in Figure 10.6(b) show that the load increases along the leading diagonal of the map, with increasing c/l and t/c.
A similar map can be constructed for fourpoint bending; this is illustrated in Figure 10.7, for the same values a/l = 0.1 and oy/of = 0.005 as for threepoint bending, but with the added parameter s/l = 0.5. A comparison with the map of Figure 10.6(a) reveals that the domain of face yield shrinks slightly for fourpoint bending, and indentation almost disappears as a viable mechanism. Core shear dominates the map for the values of parameters selected.
Now, some words of caution. The collapse mechanisms described neglect elastic deformation and assume perfectly plastic behavior. Alternative failure modes are expected when the face sheets are made of a monolithic ceramic or composite layers, and behave in an elasticbrittle manner. Then, collapse is dictated by fracture of the face sheets, as analysed by Shuaeib and Soden (1997) and Soden (1996). The above treatment has been limited to the case of flatbottomed indenters. An alternative practical case is the loading of sandwich beams by rollers, of radius R. This case is more complex because the contact size increases with increasing load. The failure modes of core shear, face yield and indentation have been observed for sandwich beams in a fourpoint bend with rollerindenters by Harte (1999), and good agreement between the measured strengths of the beams and theoretical predictions are observed, upon making use of the formulae given above but with a = 0.
10.4 Case study: the threepoint bending of a sandwich panel
A set of threepoint bend experiments on foamcored sandwich panels has been performed by BartSmith (1999), using a foam core of Alporas aluminum alloy (relative density p/ps = 0.08) and 6061T6 aluminum alloy face sheets. The sandwich beams were loaded by flatbottom indenters of size a/l = 0.1; the pertinent geometric and material parameters are summarized in Table 10.1 for two specimen designs, 1 and 2. These designs are based on the collapse load predictions, such that Design 1 sits within the core shear regime, whereas Design 2 lies just within the indentation regime, as marked in Figure 10.6(a).
Table 10.1 Geometric and material parameters for Designs 1 and 2
Parameter Design 1 Design 2
Table 10.1 Geometric and material parameters for Designs 1 and 2
Parameter Design 1 Design 2
t, mm 
79 
40 
t, mm 
0.8 
0.8 
c, mm 
10 
10 
b, mm 
20 
21 
H, mm 
18 
21 
a, mm 
7.9 
4.0 
Ef, GPa 
70 
70 
af, MPa 
263 
263 
Ec, GPa 
0.236 
0.263 
ay, MPa 
1.5 
1.5 
The measured load F versus displacement S curves are shown in Figure 10.8; it was found that the Designs 1 and 2 failed by core shear and by indentation, respectively. For comparison purposes, the predicted elastic stiffness and collapse load have been added to each plot by employing relations (10.1), (10.13) and (10.18). Additionally, the loaddeflection responses were calculated by a finite element procedure, using the constitutive model of
Design 1: Core shear
"5 400
Design 1: Core shear
"5 400
(Eq. 10.1) 
H» 
Displacement (mm) Design 2: Indentation Displacement (mm) Design 2: Indentation 1000 200 0 1000 200 0

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