Specimen geometry and specimen size dependence of the \({\mathcal {R}}\)-curve and the size effect law from a cohesive model point of view


An analytic model has been developed for a Compact Tension specimen subjected to a controlled displacement and corresponding load within a cohesive model framework. The model is able to capture the material response while the Fracture Process Zone is being developed, obtaining the evolution of multiple variables such as the crack opening and the cohesive stresses, for an arbitrary Cohesive Law shape. The crack growth prediction based on the \({\mathcal {R}}\)-curve and the nominal strength prediction based on Bažant’s Size Effect Law have been implemented using the output variables available from the proposed analytic model. The minimum specimen size has been found in order to properly apply \({\mathcal {R}}\)-curve based methods. The study has concluded that only the cohesive model is able to properly capture the changes of the Specimen Geometry and Specimen Size, as unlike in other theories, no Linear Elastic Fracture Mechanics assumptions are made.

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This work has been partially funded by the Spanish Government through the Ministerio de Economía y Competitividad, under contracts MAT2013-46749-R (subprogram MAT) and MAT2015-69491-C3-1-R.

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Correspondence to Adrián Ortega.

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Daniel Trias: Serra Húnter Fellow.


Appendix 1: Stress intensity factors equations

The Stress Intensity Factor \(\bar{K}^P\) due the load P is defined (Tada et al. 2000)

$$\begin{aligned}&\bar{K}^P=\dfrac{2+\bar{a}}{\left( 1-\bar{a}\right) ^{3/2}}F_1 \end{aligned}$$
$$\begin{aligned}&F_1=0.886+4.64\bar{a}-13.32\bar{a}^2+14.72\bar{a}^3-5.6\bar{a}^4\nonumber \\ \end{aligned}$$

The Stress Intensity Factor \(K_i^Q\) due a point load Q at a distance of \(\bar{a}_i\) measured from the load line, positioned at the crack surface, is defined (Newman et al. 2010)

$$\begin{aligned}&K_i^Q=\dfrac{Q}{hW^{1/2}}\bar{K}_i^Q\text {;}\nonumber \\&\bar{K}_i^Q=\left( \dfrac{2}{\pi \left( \bar{a}-\bar{a}_i\right) }\right) ^{1/2}F_2 \end{aligned}$$
$$\begin{aligned}&F_2=\left( 1+A_1\varDelta +A_2\varDelta ^2\right) \nonumber \\&\quad \left[ 1-1.05\left( 1-\bar{a}\right) ^9\left( \varDelta /\varDelta _0\right) ^3\right] /\left( 1-\varDelta \right) ^{3/2} \end{aligned}$$
$$\begin{aligned}&\varDelta =\dfrac{\bar{a}-\bar{a}_i}{1-\bar{a}_i} \text {;} \nonumber \\&\varDelta _0=0.8\bar{a}+0.2 \end{aligned}$$
$$\begin{aligned}&A_1=3.6+12.5\left( 1-\bar{a}\right) ^8 \text {;} \nonumber \\&A_2=5.1-15.32\bar{a}+16.58\bar{a}^2-5.97\bar{a}^3 \end{aligned}$$

The non-dimensional stress intensity factor \(\bar{K}_i^\sigma \) caused by a constant cohesive stress of normalized width \(\varDelta \bar{a}\) and centered at \(a_i\) (Mall and Newman 1985):

$$\begin{aligned}&\bar{K}_i^{\sigma }=\dfrac{1}{\left( 1-\bar{a}\right) ^{3/2}\left( 8\pi \right) ^{1/2}}\left[ 2B\left( 1+A_1+A_2\right) \sqrt{B^2+\left( 1-\bar{a}\right) B} \right. \nonumber \\&\quad +\,\left. \left( 1-\bar{a}\right) \left( 5+A_1-3A_2\right) \sqrt{B^2+\left( 1-\bar{a}\right) B} \right. \nonumber \\&\quad \left. +\,\left( 1-\bar{a}\right) ^2\left( 3-A_1+3A_2\right) \ln \left( \sqrt{B}+\sqrt{B+1-\bar{a}}\right) \right] \Big |^{B=\bar{a}-\bar{a}_i-\varDelta \bar{a}/2}_{B=\bar{a}-\bar{a}_i+\varDelta \bar{a}/2}\nonumber \\ \end{aligned}$$

The crack opening at a distance \(\bar{a}_i\) caused by the load P is obtained

$$\begin{aligned} \hat{\omega }_i^P=\int _{\bar{a}_i}^{\bar{a}}2\bar{K}^P\bar{K}^Q_id\bar{a} \end{aligned}$$

The crack opening at a distance \(\bar{a}_i\) caused by a constant cohesive stress of normalized length \(\varDelta \bar{a}\) and centred at \(\bar{a}_j\) is obtained

$$\begin{aligned} \hat{\omega }_{ij}=\int _{\bar{a}_i}^{\bar{a}}2\bar{K}^{\sigma }_j\bar{K}^Q_id\bar{a} \end{aligned}$$

Appendix 2: Elastic CT compliance

The dimensionless elastic compliance for an isotropic material is defined (Tada et al. 2000)

$$\begin{aligned} \bar{C}= & {} \left( \dfrac{1+\bar{a}}{1-\bar{a}}\right) ^2\left( 2.1630+12.219\bar{a}-20.065\bar{a}^2\right. \nonumber \\&\left. -\,0.9925\bar{a}^3+20.609\bar{a}^4-9.9314\bar{a}^5\right) \end{aligned}$$

Appendix 3: Cohesive solution extended for orthotropic materials

The differential equation that defines the stress state of an orthotropic material with the principal directions aligned normal to the crack growth direction depends on the roots of the polynomial (Ortega et al. 2014):

$$\begin{aligned} \lambda p^4+2\rho \sqrt{\lambda }p^2+1=0 \end{aligned}$$

Defining the principal directions as in Fig. 2, in the plane stress case, \(\lambda \) and \(\rho \) are expressed as:

$$\begin{aligned} \begin{array}{ll} \lambda =\dfrac{E_{22}}{E_{11}} \text {, }&\rho = \dfrac{\sqrt{\lambda }}{2G_{12}} \left( E_{11}-2\nu _{12}G_{12}\right) \end{array} \end{aligned}$$

where \(E_{11}\) and \(E_{22}\) are the elastic moduli, \(G_{12}\) is the shear modulus, and \(\nu _{12}\) is the Poisson’s ratio. In the plane strain case, \(\lambda \) and \(\rho \) are obtained by replacing \(E_{11}\), \(E_{22}\) and \(\nu _{12}\) in Eq. (31) by:

$$\begin{aligned} E'_{11}= & {} \dfrac{E_{11}}{1-\nu _{13}\nu _{31}} \text {, } E'_{22}=\dfrac{E_{22}}{1-\nu _{23}\nu _{32}} \text {, } \nonumber \\ \nu '_{12}= & {} \dfrac{\nu _{12}+\nu _{13}\nu _{32}}{1-\nu _{13}\nu _{31}} \end{aligned}$$

To ensure the positive definiteness of the strain energy, it must be ensured that:

$$\begin{aligned} \begin{array}{lll} \lambda>0&\text { and }&\rho >-1 \end{array} \end{aligned}$$

The anisotropy of the material is easily described by the parameters \(\lambda \) and \(\rho \). For an isotropic material, the parameters take the values \(\lambda =\rho =1\). However, for a cubic material, it only needs to be ensured that \(\lambda =1\) and that \(\rho \ne 1\).

In order to solve the cohesive model for an orthotropic material, the SIF, and therefore, the other variables defined in Sect. 2 need to be expressed as a function of the geometry, \(\lambda \) and \(\rho \).

$$\begin{aligned}&\bar{K}^P\left( \bar{a},\lambda ,\rho \right) \text {;}\quad \bar{K}_i^Q\left( \bar{a},\lambda ,\rho \right) \text {;}\quad \bar{K}_i^{\sigma }\left( \bar{a},\lambda ,\rho \right) \end{aligned}$$
$$\begin{aligned}&\beta _i \left( \bar{a},\lambda ,\rho \right) \text {;}\quad \hat{\omega }^P_i \left( \bar{a},\lambda ,\rho \right) \text {;}\quad \hat{\omega }_{ij}^\sigma \left( \bar{a},\lambda ,\rho \right) \end{aligned}$$

The equation \(\bar{K}^P\left( \bar{a},\lambda ,\rho \right) \) is found in Ortega et al. (2014), as for the rest of the Eqs. 34 and 35, they can be obtained using the finite elements or equivalent method.

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Ortega, A., Maimí, P., González, E.V. et al. Specimen geometry and specimen size dependence of the \({\mathcal {R}}\)-curve and the size effect law from a cohesive model point of view. Int J Fract 205, 239–254 (2017). https://doi.org/10.1007/s10704-017-0195-1

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  • R-curves
  • Cohesive zone modelling
  • J-integral
  • Bridging
  • Crack growth