Joints in composites

Excellent websites for composites in general are the JEC website (www. jeccomposites.com) and the Net-Composites website (www.netcomposites.com). Two good bolting websites are due to Bolt Science (www.boltscience.com) and Nord-lock (www.nord-lock.com). Dedicated composite bolted joints websites are less common, but the EU Framework five Project BOJCAS (Bolted Joint in Composite Aircraft Structures) [19] produced a website which can be found at the following URL (http //www.smr.ch/bojcas/). The author s personal website (www.fem-ireland.ie) provides a list of many journal papers on the subject. 

McCarthy MA. BOJCAS bolted joints in composite aircraft structures. Air Space Eur 2001 3/4(3) 139-42. 

Ko HY, Kwak BM. Contact analysis of mechanically fastened joints in composite laminates by linear complementarity problem formulation. Compos Struct 1998 ... 

Camanho PP, Matthews FL. A progressive damage model for mechanically fastened joints in composite laminates. J Compos Mater 1999 33 2248-80. 

Banea MD, daSilva LFM. Adhesively bonded joints in composite materials an overview. In Proceedings of the institution of mechanical engineers. Part L. Journal of materials design and applications, 223(1) 2009. pp. 1—18. 

Gunnion AJ, Herszberg I. Parametric study of scarf joints in composite structures. Compos Stmct 2006 75(1) 364—76. 

Atas A, Mohamed GF, Soutis C. Progressive failure of bolted joints in composite laminates. Plast Rubber Compos Macromol Eng 2012 41(4—5) 209—14. 

Bemasconi A, Kharshiduzzaman Md, Anodio, LF, Bordegoni M, Re GM, Braghin F, Comolli L. J of Adhe, 2013, Development of a Monitoring System for Crack Growth in Bonded Single-Lap Joints Based on the Strain Field and Visualization by Augmented Reality. Banea MD, da Silva LFM. Proc. ImechE, Part L, 223, 1-18. Adhesively bonded joints in composite materials an overview. 

Hart-Smith L J (1974a), Advances in the analysis and design of adhesive-bonded joints in composite aerospace structures . Proceedings of the 19thNational SAMPE Symposium and Exhibition, 23-25April, Buena Park, CA Society for the Advancement of Material and Process Engineering (SAMPE), 711-137. 

Vaidya UK, Gautam ARS, Hosur M, Dutta P (2006) Experimental-numerical studies of transverse impact response of adhesively bonded lap joints in composite structures. Int J Adhes Adhes 26 184—198... 

Andre, A., Linghoff, D., 2009. Damage modelling of adhesive joint in composite reinforced metallic beams. In NSCC, September. 

The two major classes of laminate joints are bonded joints as in Figure 7-39 and bolted joints as in Figure 7-40. Often, the two classes are combined, for example, as in the bonded-bolted joint of Figure 7-41. Joints involving composite materials are often bonded because of the natural presence of resin in the composite and are often also bolted for... 

One of the ways to increase the bearing strength of a joint is to use metal inserts as in the shimmed joint of Figure 7-40. Another way is to thicken a section of the composite laminate as in the reinforced-edge joint in Figure 7-40. 

For non-interacting fluid elements, the RTD function is thus equivalent to the joint PDF of the concentrations. In composition space, the joint PDF would he on a one-dimensional sub-manifold (i.e., have a one-dimensional support) parameterized by the age a. The addition of micromixing (i.e., interactions between fluid elements) will cause the joint PDF to spread in composition space, thereby losing its one-dimensional support. 

Because the random velocity field U(x, t) appears in (1.28), p. 16, a passive scalar field in a turbulent flow will be a random field that depends strongly on the velocity field (Warhaft 2000). Thus, turbulent scalar mixing can be described by a one-point joint velocity, composition PDF /u,< (V, i/r,x, t) defined by... 

The scalar fields appearing in Figs. 3.7 to 3.9 were taken from the same DNS database as the velocities shown in Figs. 2.1 to 2.3. The one-point joint velocity, composition PDF found from any of these examples will be nearly Gaussian, even though the temporal and/or spatial variations are distinctly different in each case.11 Due to the mean scalar... 

Thus, in Chapter 6, the transport equations for /++ x, t) and the one-point joint velocity, composition PDF /u+V, + x. / ) are derived and discussed in detail. Nevertheless, the computational effort required to solve the PDF transport equations is often considered to be too large for practical applications. Therefore, in Chapter 5, we will look at alternative closures that attempt to replace /++ x, t) in (3.24) by a simplified expression that can be evaluated based on one-point scalar statistics that are easier to compute. 

Alternatively, an LES joint velocity, composition PDF can be defined where both (j> andU are random variables Aj 0 U 4 U >4 x, t). In either case, the sample space fields U and0 are assumed to be known. 

The turbulence models discussed in this chapter attempt to model the flow using low-order moments of the velocity and scalar fields. An alternative approach is to model the one-point joint velocity, composition PDF directly. For reacting flows, this offers the significant advantage of avoiding a closure for the chemical source term. However, the numerical methods needed to solve for the PDF are very different than those used in standard CFD codes. We will thus hold off the discussion of transported PDF methods until Chapters 6 and 7 after discussing closures for the chemical source term in Chapter 5 that can be used with RANS and LES models. 

As we saw in Chapter 1, the one-point joint velocity, composition PDF contains random variables representing the three velocity components and all chemical species at a particular spatial location. The restriction to a one-point description implies the following. 

The joint velocity, composition PDF is defined in terms of the probability of observing the event where the velocity and composition random fields at point x and time t fall in the differential neighborhood of the fixed values V and ip ... 

The joint velocity, composition PDF transport equation can be derived starting from the transport equations6 for U and 0 given in Chapter l 7... 

We start by considering an arbitrary measurable10 one-point11 scalar function of the random fields U and 0 Q U, 0). Note that, based on this definition, Q is also a random field parameterized by x and t. For each realization of a turbulent flow, Q will be different, and we can define its expected value using the probability distribution for the ensemble of realizations.12 Nevertheless, the expected value of the convected derivative of Q can be expressed in terms of partial derivatives of the one-point joint velocity, composition PDF 13... 

In summary, due to the linear nature of the derivative operator, it is possible to express the expected value of a convected derivative of Q in terms of temporal and spatial derivatives of the one-point joint velocity, composition PDF. Two-point information about the random fields U and

expected value and derivative operators commute, and does not appear in the final expression (i.e., (6.9)). 

Note that A, and , will, in general, depend on multi-point information from the random fields U and 0. For example, they will depend on the velocity/scalar gradients and the velocity/scalar Laplacians. Since these quantities are not contained in the one-point formulation for U(x, t) and 0(x, f), we will lump them all into an unknown random vector Z(x, f).16 Denoting the one-point joint PDF of U, 0, and Z by /u,,z(V, ip, z x, t), we can express it in terms of an unknown conditional joint PDF and the known joint velocity, composition PDF ... 

We have seen that the joint velocity, composition PDF treats both the velocity and the compositions as random variables. However, as noted in Section 6.1, it is possible to carry out transported PDF simulations using only the composition PDF. By definition, x, t) can be found from /u,< >(V, 0 x, t) using (6.3). The same definition can be used with the transported PDF equation derived in Section 6.2 to find a transport equation for / (0 x, r). 

We shall see that a conditional acceleration model in the form of (6.48) is equivalent to a stochastic Lagrangian model for the velocity fluctuations whose characteristic correlation time is proportional to e/k. As discussed below, this implies that the scalar flux (u,

joint velocity, composition PDF level, and thus that a consistent scalar-flux transport equation can be derived from the PDF transport equation. 

As noted earlier, the extension of the conditional acceleration model in (6.48) to the joint velocity, composition PDF is trivial ... 

---