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Arrhenius-type temperature sensitivity and exponential pressure dependence, η0(T, P) can be represented with reasonable accuracy as follows:
Equations 11 and 12 constitute a five-constant (n, τ∗, B, Tb, β) representation for viscosity. The shear rate for viscosity calculation is obtained by:
Based on the above, we can infer the following filling pressure equation from the governing Eqs. 1–4:
where S is calculated by S = b0/(b−z)2 η dz. Applying the Galerkin method, the pressure finite-element equation is deduced as:
where l_ traverses all elements, including node N, and where I and j represent the local node number in element l_ corresponding to the node number N and N_ in the whole, respectively. The D(l_) ij is calculated as follows:
where A(l_) represents triangular finite elements, and L(l_) i is the pressure trial function in finite elements.
Temperature field. To determine the temperature profile across the gap, each triangular finite element at the surface is further pided into NZ layers for the finite-difference grid.
The left item of the energy equation (Eq. 4) can be expressed as:
where TN, j,t represents the temperature of the j layer of node N at time t. The heat conduction item is calculated by:
where l traverses all elements, including node N, and i and j represent the local node number in element l corresponding to the node number N and N_ in the whole, respectively.
The heat convection item is calculated by:
For viscous heat, it follows that:
Substituting Eqs. 17–20 into the energy equation (Eq. 4), the temperature equation becomes:
2.3 Structural analysis of the mold
The purpose of structural analysis is to predict the deformation occurring in the photopolymer mold due to the thermal and mechanical loads of the filling process. This model is based on a three-dimensional thermoelastic boundary element method (BEM). The BEM is ideally suited for this application because only the deformation of the mold surfaces is of interest. Moreover, the BEM has an advantage over other techniques in that computing effort is not wasted on calculating deformation within the mold.
The stresses resulting from the process loads are well within the elastic range of the mold material. Therefore, the mold deformation model is based on a thermoelastic formulation. The thermal and mechanical properties of the mold are assumed to be isotropic and temperature independent.
Although the process is cyclic, time-averaged values of temperature and heat flux are used for calculating the mold deformation. Typically, transient temperature variations within a mold have been restricted to regions local to the cavity surface and the nozzle tip [8]. The transients decay sharply with distance from the cavity surface and generally little variation is observed beyond distances as small as 2.5 mm. This suggests that the contribution from the transients to the deformation at the mold block interface is small, and therefore it is reasonable to neglect the transient effects. The steady state temperature field satisfies Laplace’s equation 2T = 0 and the time-averaged boundary conditions. The boundary conditions on the mold surfaces are described in detail by Tang et al. [9]. As for the mechanical boundary conditions, the cavity surface is subjected to the melt pressure, the surfaces of the mold connected to the worktable are fixed in space, and other external surfaces are assumed to be stress free.
The derivation of the thermoelastic boundary integral formulation is well known [10]. It is given by:
where uk, pk and T are the displacement, traction and temperature,α, ν represent the thermal expansion coefficient and Poisson’s ratio of the material, and r = |y−x|. clk(x) is the surface coefficient which depends on the local geometry at x, the orientation of the coordinate frame and Poisson’s ratio for the domain [11]. The fundamental displacement ˜ulk at a point y in the xk direction, in a three-dimensional infinite isotropic elastic domain, results from a unit load concentrated at a point x acting in the xl direction and is of the form: