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Incremental Analysis Framework

Need for Incremental Analysis

As described above, in the analysis of small-deformation problems, finite element analysis can be performed by using the principle of virtual work, which is equivalent to the fundamental equations such as the equilibrium equations, and discretizing this equation with finite elements. The same basic principle of virtual work is also used in the analysis of finite-deformation problems involving large deformation of structures.

However, in finite-deformation problems, even if material linearity is assumed, the equation of the principle of virtual work becomes nonlinear with respect to displacement. To solve nonlinear equations, iterative calculations are generally performed using an iterative method.

In these iterative calculations, an incremental analysis method is used in which the calculation is performed piecewise for small load increments and the increments are accumulated until the final deformed state is reached. When a small-deformation problem is assumed, no particular distinction is made between the configurations before and after deformation when defining strain and stress. That is, under the small-deformation assumption, it does not matter whether the configuration used to describe the fundamental equations is the configuration before or after deformation.

When incremental analysis is performed for a finite-deformation problem, the formulation differs depending on whether the initial configuration or the configuration at the start of the increment is taken as the reference configuration. The former is called the total Lagrange method, and the latter the updated Lagrange method. For details, refer to the references at the end of the chapter and related material.

Starting Equation: Principle of Virtual Work at Time \(t'\)

Consider an incremental analysis in which the state up to time \(t\) is known and the state at time \(t' = t + \Delta t\) is unknown (see Figure 2.2.1). The equilibrium equations, boundary conditions, and principle of virtual work at time \(t'\) (expressed in the current configuration using the Cauchy stress \(^{t'}\sigma\) and the linear part of the Almansi strain \(^{t'} A_{(L)}\)) are given in Principle of Virtual Work.

Concept of incremental analysis

Figure 2.2.1 Concept of incremental analysis

However, because this starting equation is written in the (unknown) configuration at time \(t'\), the actual solution procedure selects either the initial configuration \(V\) at time \(0\) or the current configuration \(^{t'} v\) at time \(t\) as the reference configuration, rewrites the equation in incremental form, and then solves it:

  • Total Lagrange method — Uses the initial configuration \(V\) as the reference. The second PK stress \(\boldsymbol{S}\) and Green-Lagrange strain \(\boldsymbol{E}\) are used.
  • Updated Lagrange method — Uses the current configuration \(^{t}v\) at the start of the increment as the reference. The Cauchy stress \(\boldsymbol{\sigma}\) and the linear part of the Almansi strain \(\boldsymbol{A}_{(L)}\) are used.

For the spatial discretization of both formulations (B matrix and internal force vector), see Discretization of Internal Virtual Work; for linearization and construction of the tangent stiffness, see Tangent Stiffness Matrix.