# Infinitesimal Deformation Linear Elasticity Static Analysis

## Infinitesimal Deformation Linear Elastic Static Analysis

In this section, the elastic static analysis is formulated on the basis of the infinitesimal deformation theory, which assumes linear elasticity as a stress-strain relationship.

### Basic equations

The equilibrium equation, mechanical boundary conditions, and geometric boundary conditions (basic boundary conditions) of solid mechanics are given by the following equations (see Fig. 2.1.1):

where $\sigma$, $\overline{t}$ and $S_t$ denote stress, surface force, and body force, respectively. $S_t$ and $S_u$ represent the geometric and mechanical boundaries, respectively.

Fig. 2.1.1 Boundary value problem in solid mechanics (infinitesimal deformation problem)

The strain-displacement relation in infinitesimal deformation problems is given by the following equation:

Furthermore, the stress-strain relationship (constitutive equation) in linear elastic bodies is given by the following equation:

where, $C$ is a fourth-order elasticity tensor.

### Principle of Virtual Work

The principle of the virtual work related to the infinitesimal deformation linear elasticity problem, which is equivalent to the basic equation Eq.$\eqref{eq:2.1.1}$, Eq.$\eqref{eq:2.1.2}$ and Eq.$\eqref{eq:2.1.3}$, is expressed as:

Moreover, considering the constitutive equation Eq.$\eqref{eq:2.1.5}$, Eq.i$\eqref{eq:2.1.6}$, is expressed as follows:

In Eq.$\eqref{eq:2.1.8}$, $\varepsilon$ is the strain tensor and $C$ is the forth-order enasticity tensor. In this case, if the strain tensor $\sigma$ and $\varepsilon$ are represented by vector formats $\hat{\sigma}$ and $\hat{\varepsilon}$, respectively, the consitutive equation Eq.$\eqref{eq:2.1.5}$ is expressed as follows

where $D$ is an elastic matrix.

Considering that the $\hat{\sigma}$, $\hat{\varepsilon}$ and Eq.$\eqref{eq:2.1.9}$ are expressed in vector format, Eq.$\eqref{eq:2.1.8}$ is expressed as follows:

Eq.$\eqref{eq:2.1.10}$ and Eq.$\eqref{eq:2.1.7}$ are the principles of the virtual work discretized in this development code.

### Formulation

If the principle of virtual work, Eq.$\eqref{eq:2.1.10}$, is discretized for each finite element, the following equation is obtained:

Using the displacement of the nodes that compose each element, the displacement field is interpolated as follows:

The strain at this moment, using Eq.$\eqref{eq:2.1.4}$, is given as follows:

When Eq.$\eqref{eq:2.1.12}$ and Eq.$\eqref{eq:2.1.13}$ are substituted into Eq.$\eqref{eq:2.1.11}$, the following equation is obtained:

Eq.$\eqref{eq:2.1.14}$ can be summarized as

In this case, the components of the matrix and vector defined by Eq.$\eqref{eq:2.1.16}$ and Eq.$\eqref{eq:2.1.17}$ can be calculated for each finite and overlapped element:

if Eq.$\eqref{eq:2.1.15}$ is true for an arbitary virtual displacement $\delta U$, tha following equation is obtained:

Meanwhile, the displacement boundary conditioni Eq.$\eqref{eq:2.1.3}$ is expressed as follows:

By solving Eq.$\eqref{eq:2.1.18}$ based on the constraint condition Eq.$\eqref{eq:2.1.19}$, it is possible to define the node displacement $U$.