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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:
Where G1 is the relaxation shear modulus of the material. The dilatational stresses can be related to the strain as follows:
Where K is the relaxation bulk modulus of the material, and the definition of α and Θ is:
If α(t) = α0, applying Eq. 27 to Eq. 29 results in:
Similarly, applying Eq. 31 to Eq. 28 and eliminating strain εxx(z, t) results in:
Employing a Laplace transform to Eq. 32, the auxiliary modulus R(ξ) is given by:
Using the above constitutive equation (Eq. 33) and simplified forms of the stresses and strains in the mold, the formulation of the residual stress of the injection molded part during the cooling stage is obtain by:
Equation 34 can be solved through the application of trapezoidal quadrature. Due to the rapid initial change in the material time, a quasi-numerical procedure is employed for evaluating the integral item. The auxiliary modulus is evaluated numerically by the trapezoidal rule.
For warpage analysis, nodal displacements and curvatures for shell elements are expressed as:
where [k] is the element stiffness matrix, [Be] is the derivative operator matrix, {d} is the displacements, and {re} is the element load vector which can be evaluated by:
The use of a full three-dimensional FEM analysis can achieve accurate warpage results, however, it is cumbersome when the shape of the part is very complicated. In this paper, a two-dimensional FEM method, based on shell theory, was used because most injection-molded parts have a sheet-like geometry in which the thickness is much smaller than the other dimensions of the part. Therefore, the part can be regarded as an assembly of flat elements to predict warpage. Each three-node shell element is a combination of a constant strain triangular element (CST) and a discrete Kirchhoff triangular element (DKT), as shown in Fig. 3. Thus, the warpage can be separated into plane-stretching deformation of the CST and plate-bending deformation of the DKT, and correspondingly, the element stiffness matrix to describe warpage can also be pided into the stretching-stiffness matrix and bending-stiffness matrix.