【正文】 nd 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 threedimensional 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 ponents of displacement, three ponents 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 rigidbody motion or rigidbody rotation to ensure a nonsingular system of equations. The resulting system of equations is assembled into a integrated matrix, which is solved with an iterative solver. Shrinkage and warpage simulation of the molded part Internal stresses in injectionmolded ponents are the principal cause of shrinkage and warpage. These residual stresses are mainly frozenin 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 thermoelastic and linear thermoviscoelastic pressible 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 ponents of stress and strain are given by: The deviatoric ponents 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 inplane 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 quasinumerical 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, nhcuj7d3 is the displacements, and {re} is the element load vector which can be evaluated by: The use of a full threedimensional FEM analysis can achieve accurate warpage results, however, it is cumbersome when the shape of the part is very plicated. In this paper, a twodimensional FEM method, based on shell theory, was used because most injectionmolded parts have a sheetlike 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 threenode shell element is a bination 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 planestretching deformation of the CST and platebending deformation of the DKT, and correspondingly, the element stiffness matrix to describe warpage can also be divided into the stretchingstiffness matrix and bendingstiffness matrix. Fig. 3a–c. Deformation deposition of shell element in the local coordinate system. a Inplane stretching element b Platebending element c Shell element 3 Experimental validation To assess the usefulness of the proposed model and developed program, verification is important. The distortions obtained from the simulation model are pared to the ones from SL injection molding experiments whose data is presented in the literature [8]. A mon injection molded part with the dimensions of 36366 mm is considered in the experiment, as shown in Fig. 4. The thickness dimensions of the thin walls and rib are both mm。 and polypropylene was used as the injection material. The injection machine was a production level ARGURY Hydronica 320210750 with the following process parameters: a melt temperature of 250 ?C。 an ambient temperature of 30 ?C。 an injection pressure of MPa。 an injection time of 3 s。 and a cooling time of 48 s. The SL material used, Dupont SOMOST