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Three-dimensional printing is becoming increasingly popular because of its extensive applicability. However, printing materials remain limited. To determine the mechanical properties of polylactic acid (PLA) and copper powder-filled polylactic acid (PLA-Cu) materials subjected to static and dynamic loading, stress–strain curves were obtained under the conditions of different strain rates using a universal material testing machine and a separated Hopkinson pressure bar experimental device. Scanning electron microscopy (SEM) was used to analyze the micro-morphology of the quasi-static compression fracture and dynamic impact sections. The results revealed that the yield stress and elastic modulus of the two materials increased with increasing strain rate. When the strain rate reached a critical point of 0.033 s−1, the rate of crack propagation in the PLA samples increased, resulting in the material undergoing a change from ductile to brittle. The strength of the material subjected to dynamic loading was significantly higher than that subjected to quasi-static loading. The SEM image of the PLA-Cu material revealed that copper powder was evenly distributed throughout the 3D-printed sample and that stress initially began to concentrate at the defect site corresponding to the interface between the copper powder and PLA matrix; this resulted in comparatively lower toughness. This finding was consistent with the photographs captured via high-speed photography, which confirmed that the destruction of the specimen was accompanied by an explosive crushing process. Additionally, a Zhu–Wang–Tang constitutive model was used to fit the experimental results and establish a viscoelastic constitutive model of the material. By comparing the dynamic stress–strain curve to the theoretically predicted curve, we found that the established constitutive model could predict the mechanical properties of the PLA-Cu material with reasonable accuracy when the strain was below 7%.