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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:
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.