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:

where δlk is the Kronecker delta function and μ is the shear modulus of the mold material.

The fundamental traction ˜plk , measured at the point y on a surface with unit normal n, is:

Discretizing the surface of the mold into a total of N elements transforms Eq. 22 to:

where Γn refers to the nth surface element on the domain.

Substituting the appropriate linear shape functions into Eq. 25, the linear boundary element formulation for the mold deformation model is obtained. The equation is applied at each node on the discretized mold surface, thus giving a system of 3N linear equations, where N is the total number of nodes. Each node has eight associated quantities: three components of displacement, three components of traction, a temperature and a heat flux. The steady state thermal model supplies temperature and flux values as known quantities for each node, and of the remaining six quantities, three must be specified. Moreover, the displacement values specified at a certain number of nodes must eliminate the possibility of a rigid-body motion or rigid-body rotation to ensure a non-singular system of equations. The resulting system of equations is assembled into a integrated matrix, which is solved with an iterative solver.

2.4 Shrinkage and warpage simulation of the molded part

Internal stresses in injection-molded components are the principal cause of shrinkage and warpage. These residual stresses are mainly frozen-in thermal stresses due to inhomogeneous cooling, when surface layers stiffen sooner than the core region, as in free quenching. Based on the assumption of the linear thermo-elastic and linear thermo-viscoelastic compressible behavior of the polymeric materials, shrinkage and warpage are obtained implicitly using displacement formulations, and the governing equations can be solved numerically using a finite element method.

With the basic assumptions of injection molding [12], the components of stress and strain are given by:

The deviatoric components of stress and strain, respectively, are given by

Using a similar approach developed by Lee and Rogers [13] for predicting the residual stresses in the tempering of glass, an integral form of the viscoelastic constitutive relationships is used, and the in-plane stresses can be related to the strains by the following equation: