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The basis of the finite element method is that the domain of the problem (that is, the volume of the object under consideration) is divided into smaller subunits, called finite elements. In the case of isoparametric elements it is further assumed that each element has a local coordinate system, named the natural coordinates, and the coordinates and shape of the element are discretized using the same functions. The discretization process is established by interpolating the geometry in terms of the coordinates of the nodes that define the geometry of a finite element, and the shape functions: where is the number of nodes and are the natural coordinates. Similarly, the motion is described in terms of the current position of the same particles: Quantities such as displacement, velocity and virtual velocity can be discretized in a similar way.
In deriving the discretized equilibrium equations, the integrations performed over the entire volume can be written as a sum of integrations constrained to the volume of an element. For this reason, the discretized equations are defined in terms of integrations over a particular element . The discretized equilibrium equations for this particular element per node is given by where The linearization of the internal virtual work can be split into a material and an initial stress component : The constitutive component can be discretized as follows: The term in parentheses defines the constitutive component of the tangent matrix relating node to node in element : Here, the linear strain-displacement matrix relates the displacements to the small-strain tensor in Voigt Notation: Or, written out completely, The spatial constitutive matrix is constructed from the components of the fourth-order tensor using the following table; where
The initial stress component can be written as follows: For the pressure component of the external virtual work, we find where,