Due to the extension of the weak form of equilibrium from Eqs. (
15) to (
17), special care must be taken regarding
\(\varvec{\varepsilon }_c\). If
\(\varvec{\varepsilon }_c\) is assumed to be an arbitrary function, then it would accumulate all of the strains, which would result in
\(\varvec{\varepsilon }_p={\mathbf {0}}\) and
\(\varvec{\sigma }_p={\mathbf {0}}\). As a result, the beam node can move independently of the solid surface. To determine how to reduce the solution space of
\(\varvec{\varepsilon }_c\), it is crucial to examine the relation of the kinematic assumption; see Eqs. (
8) to (
13). The assumption of an inextensible and plane cross-section is included with an additional deviation of
\(\varvec{\omega }_C\) from
\(\Gamma _S\). The face
\(\Gamma _B\) can only perform a rigid translation and rotation according to Eq. (
3). To achieve the correct deformation state inside the element, the correction strains
\(\varvec{\varepsilon }_C\) must remove the constraint of an inextensible and plane cross-section on
\(\Gamma _B\). To construct these strains, the same approach is used as that in [
16,
17]. Following this approach with some slight modifications, the continuum strains are represented using the following equation:
$$\begin{aligned} \varvec{\varepsilon }+\varvec{\varepsilon }_{c}={\mathbf {E}}_{1}\,\varvec{\varepsilon }_{B}+\tilde{\varvec{\varepsilon }}+{\mathbf {E}}_{2}\,{\varvec{\varepsilon }}_{c\omega }. \end{aligned}$$
(19)
In Eq. (
19), the term
\(\varvec{\varepsilon }_{c}={\mathbf {E}}_{2}\,{\varvec{\varepsilon }}_{c\omega }\) is used to remove the constraint of an inextensible and plane cross-section on the beam side
\(\Gamma _B\). Like the expression for the classical beam strains,
\({\mathbf {E}}_{2}\,{\varvec{\varepsilon }}_{c\omega }\) is composed of functions over the cross-section arranged in
\({\mathbf {E}}_2\), which are scaled by scalar values in
\(\varvec{\varepsilon }_{c\omega }\) that represent higher-order beam strains. The functions in
\({\mathbf {E}}_2\) are constructed using the general deformation functions
\(\varvec{\omega }_1\) and
\(\varvec{\omega }_2\), where
\(\varvec{\omega }_1\) represents cross-sectional warping in the local
\({\mathbf {A}}_1\)-direction and
\(\varvec{\omega }_2\) represents the contraction of the cross-section in the local
\({\mathbf {A}}_2\)-direction:
$$\begin{aligned} {\mathbf {E}}_{2}=\left[ \begin{array}{ccc} \varvec{\omega }_{{1}} &{} 0 &{} 0 \\ 0 &{} \varvec{\omega }_{{2},{2}} &{} 0 \\ 0 &{} 0 &{} \varvec{\omega }_{{1},{2}}. \end{array}\right] . \end{aligned}$$
(20)
In the definition of
\({\mathbf {E}}_2\) in Eq. (
20), it is clear that the strains
\(\varvec{\varepsilon }_c\) are introduced as independent strains and are not defined through additional displacements on the beam surface, despite the fact that
\(\varvec{\omega }_1\) and
\(\varvec{\omega }_2\) can be interpreted as superposed displacements on
\(\Gamma _B\). For
\(\varvec{\varepsilon }_c\) to not accumulate all of the strains, the general displacement functions
\(\varvec{\omega }_1\) and
\(\varvec{\omega }_2\) must fulfill the following orthogonality conditions:
$$\begin{aligned} \int _{h}\varvec{\omega }_{{1}}^{T}\,\text {d}\Gamma _{B}&={\mathbf {0}},\nonumber \\ \int _{h}\xi _{2}\,\varvec{\omega }_{{1}}^{T}\,\text {d}\Gamma _{B}&={\mathbf {0}},\nonumber \\ \int _{h}\varvec{\omega }_{{2}}^{T}\,\text {d}\Gamma _{B}&={\mathbf {0}}. \end{aligned}$$
(21)
The constraints in Eq. (
21) result from the interpretation of
\(\varvec{\omega }_1\) and
\(\varvec{\omega }_2\) as superposed displacements on
\(\Gamma _B\) and compared with Eq. (
9). Within the reformulated 2D solid displacements [Eq. (
9)], the rigid body translation and rotation are described using the beam kinematics, and therefore they must be removed from
\(\varvec{\omega }_1\) and
\(\varvec{\omega }_2\). As demonstrated in [
16,
17], the representation of
\(\varvec{\varepsilon }_c={\mathbf {E}}_{2}\,{\varvec{\varepsilon }}_{c\omega }\) with the constraints in Eq. (
21) allow the enriching of classical beam strains to represent a complete continuum strain state.