A thermodynamic framework has been developed for a class of amorphous polymers used in fused deposition modeling (FDM), in order to predict the residual stresses and the accompanying distortion of the geometry of the printed part (warping). When a polymeric melt is cooled, the inhomogeneous distribution of temperature causes spatially varying volumetric shrinkage resulting in the generation of residual stresses. Shrinkage is incorporated into the framework by introducing an isotropic volumetric expansion/contraction in the kinematics of the body. We show that the parameter for shrinkage also appears in the systematically derived rate-type constitutive relation for the stress. The solidification of the melt around the glass transition temperature is emulated by drastically increasing the viscosity of the melt. In order to illustrate the usefulness and efficacy of the constitutive relation that has been developed, we consider four ribbons of polymeric melt stacked on top of each other such as those extruded using a flat nozzle: each layer laid instantaneously and allowed to cool for one second before another layer is laid on it. Each layer cools, shrinks and warps until a new layer is laid, at which time the heat from the newly laid layer flows into the previous laid layer and heats up the bottom layers. The residual stresses of the existing and newly laid layers readjust to satisfy equilibrium. Such mechanical and thermal interactions amongst layers result in a complex distribution of residual stresses. The plane strain approximation predicts nearly equibiaxial tensile stress conditions in the core region of the solidified part, implying that a preexisting crack in that region is likely to propagate and cause failure of the part during service. The free-end of the interface between the first and the second layer is subjected to the largest magnitude of combined shear and tension in the plane with a propensity for delamination.

已经开发出一种热力学框架，用于熔合沉积建模（FDM）中使用的一类无定形聚合物，以预测残余应力以及印刷零件的几何形状的伴随变形（翘曲）。当聚合物熔体冷却时，温度的不均匀分布会导致空间变化的体积收缩，从而导致残余应力的产生。通过在人体运动学中引入各向同性的体积膨胀/收缩，将收缩纳入框架。我们表明，收缩率的参数也出现在系统得出的应力率型本构关系中。通过急剧增加熔体的粘度来模拟熔体在玻璃化转变温度附近的固化。为了说明已建立的本构关系的有用性和有效性，我们考虑了四层相互叠置的聚合物熔体带，例如使用扁平喷嘴挤出的带：每层均瞬时铺放并冷却一秒钟在其上放置另一层之前。每层冷却，收缩和翘曲直到铺设新层，这时新铺设的层产生的热量流入先前的铺设层并加热底层。现有层和新铺设层的残余应力会重新调整以满足平衡。层之间的这种机械和热相互作用导致残余应力的复杂分布。平面应变近似可预测凝固零件核心区域的几乎双轴拉伸应力条件，暗示该区域中先前存在的裂纹很可能会传播并导致零件在维修期间失效。第一层和第二层之间的界面的自由端在平面中承受了最大的组合剪切和拉伸强度，具有分层的倾向。