The paper presents an extension of the particle-in-cell (PIC) method for solving engineering problems involving history-dependent materials, such as penetration, impact, and large rotations of solid bodies. The proposed method combines the advantages of both Eulerian and Lagrangian schemes to avoid mesh distortion and ensure accurate tracking of material points through the complete deformation history. The key features include: 1. **Fixed Eulerian Grid**: A fixed computational grid is used to determine spatial gradients, which helps in avoiding mesh distortion and ensuring numerical stability. 2. **Material Points**: Material points are interpreted as Lagrangian frames, allowing the convection term in the acceleration associated with Eulerian formulations to be omitted. 3. **Mapping Between Material Points and Grid**: Variables are mapped between material points and the grid, enabling the use of both Eulerian and Lagrangian schemes while avoiding mesh tangling. 4. **History-Dependent Variables**: The constitutive equations are applied at material points, allowing for the tracking of history-dependent variables like plastic strain and strain-hardening parameters. 5. **Numerical Algorithm**: The algorithm involves explicit time integration to update velocities and positions of material points, followed by mapping these values back to the grid nodes to handle convection. The paper includes several numerical examples to demonstrate the robustness of the proposed method, including large rotation tests, vibration of an elastic cylinder, impact of elastic and inelastic bodies, and bouncing bar. These examples show that the method can accurately reproduce elastic behavior and handle impact problems without special algorithms for interfaces, achieving no slip conditions. The method also handles "mixed cells" naturally, making it suitable for problems with different constitutive equations within the same element. The authors conclude that the method combines the benefits of both Eulerian and Lagrangian schemes, avoiding mesh distortion and ensuring accurate tracking of material points. The method is efficient for impact problems and has potential applications in various engineering contexts, including solid-solid and solid-fluid interfaces.The paper presents an extension of the particle-in-cell (PIC) method for solving engineering problems involving history-dependent materials, such as penetration, impact, and large rotations of solid bodies. The proposed method combines the advantages of both Eulerian and Lagrangian schemes to avoid mesh distortion and ensure accurate tracking of material points through the complete deformation history. The key features include: 1. **Fixed Eulerian Grid**: A fixed computational grid is used to determine spatial gradients, which helps in avoiding mesh distortion and ensuring numerical stability. 2. **Material Points**: Material points are interpreted as Lagrangian frames, allowing the convection term in the acceleration associated with Eulerian formulations to be omitted. 3. **Mapping Between Material Points and Grid**: Variables are mapped between material points and the grid, enabling the use of both Eulerian and Lagrangian schemes while avoiding mesh tangling. 4. **History-Dependent Variables**: The constitutive equations are applied at material points, allowing for the tracking of history-dependent variables like plastic strain and strain-hardening parameters. 5. **Numerical Algorithm**: The algorithm involves explicit time integration to update velocities and positions of material points, followed by mapping these values back to the grid nodes to handle convection. The paper includes several numerical examples to demonstrate the robustness of the proposed method, including large rotation tests, vibration of an elastic cylinder, impact of elastic and inelastic bodies, and bouncing bar. These examples show that the method can accurately reproduce elastic behavior and handle impact problems without special algorithms for interfaces, achieving no slip conditions. The method also handles "mixed cells" naturally, making it suitable for problems with different constitutive equations within the same element. The authors conclude that the method combines the benefits of both Eulerian and Lagrangian schemes, avoiding mesh distortion and ensuring accurate tracking of material points. The method is efficient for impact problems and has potential applications in various engineering contexts, including solid-solid and solid-fluid interfaces.