Plastics Engineering - May 2014 - (Page 41)

and mass-conservation equation: together with Hooke's law: (4) where: (7) and boundary conditions: = shear stress, (8) = pressure, in Equations 5-8, σ is stress, ε is strain, n is a normal vector for polymer-core interface, and C is the mold material compliance. the supports for the cores are implemented as springs; there an additional nodal force reaction freaction is proportional to the deflection of the node u: = shear rate, T = temperature, t = time, v = melt velocity, (9) ␣ = thermal diffusivity, = shear heating, = compression heating, = polymer density, and Sρ is an additional polymer velocity source term for the nodes on the mold core/polymer melt interface that describes movement of the mold core: where u is the deformation of the interface and n is the direction of the normal to this surface. to solve fluid dynamics Equations 1-4, we use Hele-shaw approximation using triangular elements.10 For chunky moldings, we solve Equations 1-4 in true 3-D formulation using 4-node tetrahedral elements.11 the melt pressure p creates additional surface forces f acting on the boundary of the core: (5) these forces would cause deformation of the core u that we can find using momentum equations inside the core: in Equation 9, K is the spring elasticity and u0 is the equilibrium position of the node. if the support fixes deflection component ui, we select the corresponding component of the spring elasticity tensor Kii to be a large value Kfixed (much higher than the elasticity of the core itself); if the support left deflection component ui to be free, we choose the elasticity Kii to be zero. For elastic support, appropriate values for spring elasticity K allow to accurately take into account movement of the support. the off-diagonal terms of the tensor K are used to describe tilted support when movement in some directions is constrained and in other directions is free. Usually the equilibrium positions of constrained nodes u0 are set to zero; non-zero equilibrium positions can be used to simulate prescribed displacements on the node constraint or pre-stressed support (e.g. clamping force). For one-sided constraints, spring elasticity and the equilibrium position become dependent upon deflection of the constrained node u. For each component of the deflection, we consider three types of the constraints: * (A) movement in positive direction is limited up to some value u+, while movement in negative direction is free; * (B) movement in positive direction is free, while movement in negative direction is limited down to some value u-; and * (c) movement in positive direction is limited up to some value u+, while movement in negative direction is limited down to another value u-. For the one-sided constraints of type (A) we can select effective spring elasticity K and equilibrium position u0 as: (10) (6) www.plasticsengineering.org | www.4spe.org | MAY 2014 | PlAstics EnginEEring | 41 http://www.plasticsengineering.org http://www.4spe.org

Plastics Engineering - May 2014

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