May 2021 Volume 3

FORGING RESEARCH

Eq. (19) cannot be uniquely determined because of the undefined value of the effective stress when the effective strain rate approaches zero. This difficulty is overcome by approximating the constitutive relationship [9]:

5.2.2 Explicit finite element formulation Explicit finite element formulation is based on the dynamic equilibrium at time t:

(22)

(20)

where R is the external force vector and F is the nodal point force vector. By requiring the nodal mass matrix to be a diagonal or lumped mass matrix, the main advantage of this method is that solving the nodal acceleration vector ü does not involve factorization of a coefficient matrix.Then the calculation can be performed on the element level with relatively small memory and disk space requirements. Using this approach, a system having a very large number of degrees of freedom can be solved very effectively. The above equation can then be expressed as:

which results in:

(21)

By employing the variational approach and applying the Taylor series truncated after the first derivative, the global tangent stiffness matrix is obtained, which – in metal forming modeling – is significantly affected by a number of constraints including material incompressibility, contact (sliding, sticky and impenetrable), boundary conditions and friction. The stiffness matrix is solved by using the Gaussian quadrature with full or selective reduced integration to correct the stiffening behavior resulting from the full integration and element locking. The implicit finite element methods are most often applied by using rigid-plastic flow models to analyze the bulk forming processes (upsetting, heading, forging, extrusion, etc.) or by using elasto-plastic flow models, because of their ability to easily calculate the elastic recovery (springback) after the force is released and to analyze the sheet metal forming processes such as stamping, deep drawing, etc. When they are used for solving these relatively simple bulk or sheet metal forming models, especially in the 2D space for axisymmetric, plane-stress or plane-strain cases, the implicit finite element methods are highly effective and efficient because of their smaller stiffness matrix, allowable larger time steps, more mature iterative solving algorithms and fewer convergence issues. The accuracy is also satisfactory. When the implicit finite element methods are introduced to the more complex 3D analyses, there are some inherent issues that make the simulations difficult. The stiffness matrix is considerably larger. Solving the hugely populated system of linear equations, even with advanced solution methods, can be very laborious and costly. The contact regime is comprised of surfaces instead of the lines used in the 2D cases. Non-matching element sizes and geometric curvatures between the master (tooling) and slave (workpiece) surfaces can limit the time step size, leading to a lengthy simulation and high computational cost. Convergence is not guaranteed either, so building a model that can run smoothly becomes a time-consuming and challenging effort. All these issues acted as a catalyst in the development and applications of explicit finite element formulation, which is discussed next.

(23)

To solve the nodal velocities and displacements through time, the central difference method is used. This method assumes that the nodal accelerations are constant within a time increment. This assumption is valid for the method to achieve accurate results because the time increment is very tiny. The velocities and displacements are determined from the following equations:

(24)

(25)

The name central difference comes from the fact that the central point of a time increment is used in the calculation. Once the velocities and displacements are determined, the strain rate, strain, stresses, and nodal forces can be calculated through the time increment. After this, the simulation goes forward to the next time increment until the entire time span is covered. Because these variables can be calculated explicitly, this method is referred to as an explicit integration method . In the practical use of the explicit time integration, the most important factor is the proper selection of the time increment . For stability, the time increment has to be smaller than a critical time increment; otherwise, the round-off errors in the simulation can grow without bound (a blow-up). Theoretically, the critical time increment for a linear system is determined by the highest frequency in the system:

(26)

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FIA MAGAZINE | MAY 2021