resid-ual stress
5.3.1 Analysis of the experimental procedure for prestressing
It is instructive to quantify the expected residual stress present in a specimen prior to a three-point bend experiment. Here we present a simple analytical description of the chosen experimental prestressing procedure. In this study, we aim to bring a sandwich panel into a state of equillibrium with a residual stress field present, comprising a tensile stress in the face sheets and a compressive stress in the core. This is illustrated in Figure 5.5 and achieved via the following steps
(i) The foam and both face sheets are unbonded and unstressed (σc= σf = 0).
(ii) A tensile stress, σf0is applied to the face sheets, where σf0< σf y. The foam core remains unstressed and unbonded (σc= 0).
(iii) The unstressed core is bonded between the face sheets. The face sheets remain at σf0for a period of ∆t whilst the adhesive cures.
(iv) The load is released on the bonded sandwich structure. In the process of reaching a state of equilibrium, the stress in the face sheet reduces to σf R via a change in strain ∆ε which results in a compressive stress of σcR in the core.
∆ε describes the loss in tensile load in the face sheet due to the mismatch in Young’s modulus of both core and face sheet materials. This loss is a consequence of the chosen experimental methodolgy in this study. ∆ε may be expressed as
∆ε = ∆σf
Ef = ∆σc
Ec (5.23)
Chapter 5 - The influence of residual stress on the elastic limit and collapse of sandwich beams
(a)
f0bt (ii) Prestress face sheets to f0
f0bt
(iii) Apply adhesive to face sheet, insert and bond core.
(iv) Release loading. Residual stress state remains in face sheet and core.
f0bt
(i) Undeformed face sheet configuration
FIGURE 5.5: (a) Schematic of experimental prestress procedure. (b) Stress and strain state in face sheet and core during and after the experimental prestress procedure. (c)
Strain state in the face sheet as a function of time.
Chapter 5 - The influence of residual stress on the elastic limit and collapse of sandwich beams
where Ef and Ecare the Young’s modulus of the face sheet and core respectively. ∆ε is the change in strain during unloading between state (iii) and (iv). As a result, the stress in the foam and the face sheets at the final state (iv), may be given as follows
σf R= σf0+ ∆σf (5.24)
σcR= 0 + ∆σc (5.25)
Equilibrium of the foam and face sheets in the final state B dictates that
2tσf R= cσcR (5.26)
Equating Equations 5.23 - 5.26 enables the prediction of the change in strain of the sandwich beam between state (iii) and (iv) as a function of the initial load in the face sheet and material Young’s modulus, face sheet thickness, t, and core thickness, c
∆ε = −
2t c
σf0
Ef
2t c +EEc
f
(5.27)
In addition to this, the final stress induced in the face sheet may also be given as follows
σf R= σf0 2EEf
c
t c+ 1
(5.28)
If the core was prestressed in compression whilst the face sheets were simultaneously prestressed in tension and subsequently bonded then there would be no loss in residual stress. In practice it is difficult to apply a compressive load to the core due to its length;
column buckling is likely to occur. A more practical, but not ideal, method has been described in general above and the detailed experimental procedure will be described in Section 5.5. This methodology is inspired by the prestressed steel reinforced concrete used in civil engineering applications.
Chapter 5 - The influence of residual stress on the elastic limit and collapse of sandwich beams
5.3.2 Limitations on material selection due to experimental method-ology
The general conclusion from the analysis in Section 5.3.1 is that the chosen experimen-tal prestressing method does not produce the same level of residual stress state as that of the ideal case. This places limits on material selection for face sheet and core materials.
If we assume that the desire is to place the sandwich beam in the maximum possible residual stress state, then we assume that we load a material to its elastic limit (εf). We now take Equation 5.29 and normalise
ˆεL= ∆ε
εf = − 1 1 +2tc EEc
f
(5.29)
This equation gives a design space for a sandwich beam in terms of the constituent geometry and stiffness properties with the goal of minimising ˆεL. The relationship be-tween ˆεLand t/c for a variety of Young’s modulus mismatches ( ¯E= Ef/Ec) is shown in Figure 5.6. It is clear that as the mismatch in Young’s modulus increases, the allowable t/c space in which a small loss in residual stress occurs becomes increasingly limited.
Consider the GFRP and H200 PVC foam from Chapter 4. The modulus mismatch is approximately ¯E= 200 and thus severely limits the maximum residual stress in the face sheet to 0 < ˆεL< 0.3 for values of t/c < 1. Thus we find that the loss in tension ∆ε is large for a GFRP face sheet prestressed to approximately its failure load. The loss in tension is in fact approximately equal to the pretension applied for a significantly large range of t/c resulting in no induced residual stress in the sandwich structure. It must be noted that geometry plays a role, but in this case the stiffness of the material is so high that the loss in tension is a dominant factor. If, in contrast, we choose a PC face sheet and H200 foam core, ( ¯E ≈ 15), we find that, for most geometries, the efficiency is higher than the GFRP and H200 sandwich panel, and it is indeed possible to achieve much lower losses, with the loss in residual stress exceeding ˆεL > 0.5. We conclude that, when using the present methodology of applying a residual stress to a sandwich
Chapter 5 - The influence of residual stress on the elastic limit and collapse of
FIGURE 5.6: Loss of residual stress during the experimental prestress procedure for a variety of Ef/Ec
panel, a system comprising PC face sheets and H200 foam core is more practical than the GFRP/PVC foam combination.