11. Quasistatics

11.1. Quasistatic Assumption

As discussed previously in the context of a Linear Elastic IBVP, the quasistatic approximation is appropriate when inertial forces are negligible compared to the internal and applied forces in a system. The question of what is negligible generally relies on intuition, and numerical experimentation is one way to gain this intuition.

Omission of the inertial term in the discrete equations of motion, (10.38), yields a quasistatic problem of the form

(11.1)\[\mathbf{F}^{\mathrm{int}}(\mathbf{d}(t)) = \mathbf{F}^{\mathrm{ext}}\]

subject to only one initial condition of the form

(11.2)\[\mathbf{d}(0) = \mathbf{d}_0 .\]

Note that the time variable, \(t\) may correspond to real time (e.g., if rate-dependent material response is considered) but need not have physical meaning for rate independent behavior. For example, it is common for \(t\) to be taken as a generic parameterization for the applied loading on the system as discussed below.

11.2. Internal Force Vector

The quantity \(\mathbf{F}^{\mathrm{int}} \left(\mathbf{d}(t)\right)\) is known as the internal force vector and consists of that set of forces that are variationally consistent with the internal stresses in the body undergoing analysis. The generic expression for an element in this vector is

(11.3)\[ F^{\mathrm{int}}_P = \int_{\varphi^h_t (\Omega)} \left( \sum_{j=1}^3 N_{A,j} \left(\varphi^{-1}_t (\mathbf{x}) \right) T^h_{ij} \right) \mathrm{d}v.\]

This vector-valued operator is generally a nonlinear function of the unknown solution vector \(\mathbf{d}(t)\) due to the possible material nonlinearity and/or geometric nonlinearity inherent in the definition of the Cauchy stress \(T^h_{ij}\) in (11.3). As implied by our notation, we assume the solution vector \(\mathbf{d}\) to be smoothly parameterized by \(t\) which may represent time or some other loading parameter.

11.3. External Force Vector

The external load vector \(\mathbf{F}^{\mathrm{ext}}(t)\) must equilibrate the internal force vector, as is clear from (11.1). As presented in the previous chapter, the expression of an element \(F^{\mathrm{ext}}_P\) of \(\mathbf{F}^{\mathrm{ext}}(t)\) is

(11.4)\[ F^{\mathrm{ext}}_{P} = \int_{\varphi^h_t(\Omega)} N_A \left( \varphi^{-1}_t (\mathbf{x}) \right) f_i (t) \mathrm{d}v + \int_{\varphi^h_t(\Gamma_{\sigma})} N_A \left( \varphi^{-1}_t (\mathbf{x}) \right) \cdot \bar{t}_i (t) \mathrm{d}a ,\]

where the explicit dependence of \(f_i\) and \(\bar{t}_i\) upon \(t\) has been indicated and where \(P=ID(i,a)\) as given in (10.27). In other words, we assume that the prescribed external force loadings \(f_i\) and prescribed surface tractions \(\bar{t}_i\) are given functions of \(t\).

(11.4) implies no dependence of either \(\bar{t}_i\) or \(f_i\) upon \(\varphi_t(\mathbf{x})\) (and thus \(\mathbf{d}\)). Provided no such dependence exists, the external force is completely parameterized by \(t\), and the sole dependence of the equilibrium equations on \(\mathbf{d}\) occurs through \(\mathbf{F}^{\mathrm{int}}\). However , it is important to realize that some important loading cases are precluded by this assumption. Perhaps the most important being the case of pressure loading, where the direction of applied traction is opposite to the surface normal, which in large deformation problems depends upon \(\varphi_t(\mathbf{x})\). Such a load is sometimes called a follower force and will, in general, contribute additional nonlinearities. Such nonlinearities are handled notationally, simply by recognizing that the traction \(\bar{t}_i\) now depends on \(\varphi_t(\mathbf{x})\), i.e.,

(11.5)\[ F^{\mathrm{ext}}_{P} = \int_{\varphi^h_t(\Omega)} N_A \left( \varphi^{-1}_t (\mathbf{x}) \right) f_i \mathrm{d}v + \int_{\varphi^h_t(\Gamma_{\sigma})} N_A \left( \varphi^{-1}_t (\mathbf{x}) \right) \cdot \bar{t}_i (t,\varphi_t(\mathbf{x})) \mathrm{d}a .\]

11.4. Incremental Load Approach

We may now summarize the global solution strategy applied to quasistatic nonlinear solid mechanics applications. We assume that we are interested in the solution \(\mathbf{d}(t)\) over some time interval of interest for \(t\):

(11.6)\[ t \in [0,\mathrm{T}]\]

We subdivide this interval of interest into a set of sub-intervals via

(11.7)\[ [0,\mathrm{T}] = \bigcup_{n=0}^{N-1} \left[ t_n , t_{n+1} \right],\]

where \(n\) is an index on the time steps or intervals, and \(N\) is the total number of such increments. We assume that \(t_0=0\) and that \(t_N=\mathrm{T}\), but we do not, in general, assume that all time intervals \(\left[ t_n, t_{n+1} \right]\) have the same width.

With this notation, the incremental load approach attempts to solve the following problem successively in each time interval \(\left[ t_n, t_{n+1} \right]\):

Given the solution \(\mathbf{d}_n\) corresponding to time level \(t_n\), find \(\mathbf{d}_{n+1}\) corresponding to \(t_{n+1}\) satisfying:

(11.8)\[\mathbf{F}^{\mathrm{int}} \left( \mathbf{d}_{n+1} \right) = \mathbf{F}^{\mathrm{ext}} \left( \mathbf{d}_{n+1} \right) .\]

where we have included an assumed dependence of the external loading on deformation \(\varphi_t(\mathbf{x})\).

This governing equation is also often expressed by introducing the concept of a residual vector \(\mathbf{r}(\mathbf{d}_{n+1})\):

(11.9)\[\mathbf{r} \left( \mathbf{d}_{n+1} \right) = \mathbf{F}^{\mathrm{ext}} \left( \mathbf{d}_{n+1} \right) - \mathbf{F}^{\mathrm{int}} \left( \mathbf{d}_{n+1} \right) .\]

Solution of (11.8), therefore, amounts to finding the root of the equation

(11.10)\[\mathbf{r} \left( \mathbf{d}_{n+1} \right) = 0 .\]

The importance of stating equilibrium in this manner will be made much clearer in the Chapter discussing nonlinear equation solving, (chapter Section 13). For the moment, the physical meaning of this approach is depicted graphically in Fig. 11.1. Starting with an initial equilibrium state \(t_n\), so that \(\mathbf{r}(\mathbf{d}_n)=0\), we introduce an increment in the prescribed load and attempt to find that displacement increment, \(\mathbf{d}_{n+1}-\mathbf{d}_n\), that will restore equilibrium (i.e., result in satisfaction of (11.10)). This will require a nonlinear equation solving technique for determination of \(\mathbf{d}_{n+1}\), a topic that will be discussed further in Section 13.

Fig. 11.1 Simple illustration of the incremental load approach to quasistatics problems