Let us consider a thick rectangular plate(Figure 1)occupying the domain *Ω* = {0 ≤ *x*_{1} ≤ *ℓ*_{1}, 0 ≤ *x*_{2} ≤ *h*, 0 ≤ *x*_{3} ≤ *ℓ*_{3}} in the Cartesian Coordinate system *Ox*_{1}*x*_{2} *x*_{3}, which contains a band crack located in the region Ω′ = {(*ℓ*_{1} /2 − *ℓ*_{0}/2) ≤ *x*_{1} ≤ (*ℓ*_{1} / 2 + *ℓ*_{0} / 2), *x*_{2} = *h*_{c} ± 0, 0 ≤ *x*_{3} ≤ *ℓ*_{3}} . Assume that on the crack’s edge planes, the opening uniformly distributed normal forces with intensity p act.

Figure 1 Geometry of a rectangular plate with a band crack.

The field equations and boundary conditions for the case under consideration are:

$$\begin{array}{}{\displaystyle \frac{\mathrm{\partial}{\sigma}_{ij}}{\mathrm{\partial}{x}_{i}}=0\phantom{\rule{thinmathspace}{0ex}},{\sigma}_{ii}={A}_{ij}{\epsilon}_{jj}\phantom{\rule{thinmathspace}{0ex}},{A}_{ij}={A}_{jj},\phantom{\rule{thinmathspace}{0ex}}{\sigma}_{ij}=2{\mu}_{ij}{\epsilon}_{ij},\phantom{\rule{thinmathspace}{0ex}}\text{at}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}i\ne j\phantom{\rule{thinmathspace}{0ex}},{\epsilon}_{ij}=\frac{1}{2}(\frac{\mathrm{\partial}{u}_{i}}{\mathrm{\partial}{x}_{j}}+\frac{\mathrm{\partial}{u}_{j}}{\mathrm{\partial}{x}_{i}}),}\\ \\ {\displaystyle \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{u}_{2}{|}_{{x}_{1}=0}={u}_{2}{|}_{{x}_{1}={\ell}_{1}}=0\phantom{\rule{thinmathspace}{0ex}},{u}_{2}{|}_{{x}_{3}=0}={u}_{2}{|}_{{x}_{3}={\ell}_{3}}=0\phantom{\rule{thinmathspace}{0ex}},{\sigma}_{1k}{|}_{{x}_{1}=0}={\sigma}_{1k}{|}_{{x}_{1}={\ell}_{1}}=}\\ \\ {\displaystyle =0\phantom{\rule{thinmathspace}{0ex}},{\sigma}_{3k}{|}_{{x}_{3}=0}={\sigma}_{3k}{|}_{{x}_{3}={\ell}_{3}}=0\phantom{\rule{thinmathspace}{0ex}},}\\ \\ {\displaystyle \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\sigma}_{2i}{|}_{{x}_{2}=0}=0\phantom{\rule{thinmathspace}{0ex}},{\sigma}_{2i}{|}_{{x}_{2}=h}=0\phantom{\rule{thinmathspace}{0ex}},{\sigma}_{2i}{|}_{{\mathrm{\Omega}}^{\prime}}=-p{\delta}_{i}^{2}\phantom{\rule{thinmathspace}{0ex}},\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}i,j=1,2,3;\phantom{\rule{thinmathspace}{0ex}}k=1,3.}\end{array}$$(1)

To solve the foregoing problem (1), we employ the 3D FEM and for this purpose we introduce the following functional:

$$\begin{array}{}{\displaystyle \mathit{\Pi}=\frac{1}{2}\underset{\mathit{\Omega}}{\int}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\underset{-}{\int}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\underset{\mathit{\Omega}{\phantom{\rule{thinmathspace}{0ex}}}^{\prime}}{\int}({\sigma}_{11}{\epsilon}_{11}+2{\sigma}_{12}{\epsilon}_{12}+2{\sigma}_{13}{\epsilon}_{13}+2{\sigma}_{23}{\epsilon}_{23}+{\sigma}_{22}{\epsilon}_{22}+{\sigma}_{33}{\epsilon}_{33})\phantom{\rule{negativethinmathspace}{0ex}}d\mathit{\Omega}}\\ {\displaystyle -\underset{0}{\overset{{\ell}_{3}}{\int}}\underset{0}{\overset{{\ell}_{1}}{\int}}p\phantom{\rule{thinmathspace}{0ex}}{u}_{2}{|}_{{x}_{2}=h}d{x}_{1}d{x}_{3}-\underset{0}{\overset{{\ell}_{3}}{\int}}\underset{0}{\overset{{\ell}_{1}}{\int}}p\phantom{\rule{thinmathspace}{0ex}}{u}_{2}{|}_{{x}_{2}=0}d{x}_{1}d{x}_{3}}\end{array}$$(2)

Using the Ritz technique [13], *δΠ* = 0, we obtain the equilibrium equation and boundary conditions in Eq. (1). In this way, the validity of the functional (2) for the FEM modelling of the considered problem is proven.

In these cases, the solution domain *Ω* is divided into a finite number of finite elements, which are selected as standard rectangular prisms (bricks) with eight nodes. Selection of the number of degrees of freedom (NDOF) is determined from the requirements that the boundary conditions should be satisfied with very high accuracy and that the numerical results obtained for various NDOFs should converge.

After consideration of the FEM modelling, the ERR (denoted by *γ*) is calculated by using the expression:

$$\begin{array}{}{\displaystyle \gamma \approx \frac{1}{2}\frac{U({S}_{c}+\mathrm{\Delta}{S}_{c}(s/{\ell}_{1}))-U({S}_{c})}{\mathrm{\Delta}{S}_{c}(s/{\ell}_{1})}.}\end{array}$$(3)

In Eq. (3), *U* is the strain energy, *S*_{c} shows the area of the crack’s edge surface, Δ*S*_{c} is the increment of the crack’s edge area, and *s* / *ℓ*_{1} = (0.5*ℓ*_{3} − *x*_{3})/ *ℓ*_{1}. In the solution procedure, the values of Δ*S*_{c} are selected regarding convergence of the numerical results obtained for *γ*.

Ethical approval: The conducted research is not related to either human or animals use.

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