1 Introduction
2 Phase-field variational formulations
2.1 Energy functionals
2.1.1 Optimal profiles
2.2 Variational formulations within a quasi-static evolution
2.2.1 Evolution in terms of energy variations
2.2.2 Evolution in terms of PDEs
3 Numerical discretization strategy
3.1 Time discretization
3.2 Spatial discretization with a focus on the the phase-field problem
3.2.1 Bivariate NURBS discretization
3.3 Symmetric linear complementarity problem
4 Numerical results
4.1 Double cantilever beam
h | Elements | Control points |
---|---|---|
[mm] | (–) | (–) |
\(l_0\) | 4624 | 4900 |
\(l_0/2\) | 18,496 | 19,044 |
\(l_0/4\) | 72,900 | 73,984 |
\(l_0/6\) | 160,000 | 161,604 |
h | \(a_0\) | \(\Delta a\) | \(G_c^{Eff}\) | \(\Delta G_c^{Eff}\) | Conv. rate | Error |
---|---|---|---|---|---|---|
[mm] | [mm] | [mm] | [kN/mm] | [kN/mm] | [–] | [%] |
\(l_0\) | 0.1397 | 0.5000 | 0.0183 | 0.0083 | – | 82.9 |
\(l_0/2\) | 0.1654 | 0.5441 | 0.0145 | 0.0045 | 0.883 | 45.1 |
\(l_0/4\) | 0.1833 | 0.5704 | 0.0123 | 0.0023 | 0.968 | 23.2 |
\(l_0/6\) | 0.1922 | 0.5750 | 0.0116 | 0.0016 | 0.895 | 16.4 |
h | \(a_0\) | \(\Delta a\) | \(G_c^{Eff}\) | \(\Delta G_c^{Eff}\) | Conv. rate | Error |
---|---|---|---|---|---|---|
[mm] | [mm] | [mm] | [kN/mm] | [kN/mm] | [–] | [%] |
\(l_0\) | 0.1544 | 0.5000 | 0.0175 | 0.0075 | – | 74.6 |
\(l_0/2\) | 0.1801 | 0.5588 | 0.0129 | 0.0029 | 1.371 | 28.9 |
\(l_0/4\) | 0.1944 | 0.5778 | 0.0110 | 0.0010 | 1.536 | 10.1 |
\(l_0/6\) | 0.1987 | 0.5800 | 0.0105 | 0.0005 | 1.709 | 5.42 |
4.2 SEN tensile test
h | \(G_c^{Eff}\) | \(\Delta G_c^{Eff}\) | Conv. rate | Error | R\(_n\) | Time spent |
---|---|---|---|---|---|---|
[mm] | [kN/mm] | [kN/mm] | [-] | [%] | [kN] | [h:min] |
\(l_0/2\) | \(3.395\cdot 10^{-3}\) | \(6.947\cdot 10^{-4}\) | - | 25.7 | 0.652 | 08:45 |
\(l_0/4\) | \(3.275\cdot 10^{-3}\) | \(5.749\cdot 10^{-4}\) | 0.273 | 21.3 | 0.640 | 33:41 |
h | \(G_c^{Eff}\) | \(\Delta G_c^{Eff}\) | Conv. rate | Error | R\(_n\) | Time spent |
---|---|---|---|---|---|---|
[mm] | [kN/mm] | [kN/mm] | [-] | [%] | [kN] | [h:min] |
\(l_0/2\) | \(3.034\cdot 10^{-3}\) | \(3.341\cdot 10^{-4}\) | – | 12.4 | 0.622 | 07:45 |
\(l_0/4\) | \(2.914\cdot 10^{-3}\) | \(2.139\cdot 10^{-4}\) | 0.643 | 7.90 | 0.593 | 24:10 |
4.3 SEN shear test
h | \(\Delta a\) | \(G_c^{Eff}\) | \(\Delta G_c^{Eff}\) | Conv. rate | Error | R\(_n\) | Time spent |
---|---|---|---|---|---|---|---|
[mm] | [mm] | [kN/mm] | [kN/mm] | [-] | [%] | [kN] | [h:min] |
\(l_0/2\) | \(2.731\cdot 10^{-1}\) | \(4.043\cdot 10^{-3}\) | \(1.343\cdot 10^{-4}\) | – | 49.5 | 0.499 | 11:52 |
\(l_0/4\) | \(3.011\cdot 10^{-1}\) | \(3.666\cdot 10^{-3}\) | \(9.664\cdot 10^{-4}\) | 0.475 | 35.8 | 0.428 | 63:19 |
h | \(\Delta a\) | \(G_c^{Eff}\) | \(\Delta G_c^{Eff}\) | Conv. rate | Error | R\(_n\) | Time spent |
---|---|---|---|---|---|---|---|
[mm] | [mm] | [kN/mm] | [kN/mm] | [-] | [%] | [kN] | [h:min] |
\(l_0/2\) | \(3.142\cdot 10^{-1}\) | \(3.578\cdot 10^{-3}\) | \(8.781\cdot 10^{-4}\) | – | 32.5 | 0.428 | 16:28 |
\(l_0/4\) | \(3.571\cdot 10^{-1}\) | \(3.191\cdot 10^{-3}\) | \(4.908\cdot 10^{-4}\) | 0.839 | 18.2 | 0.406 | 74:47 |
4.4 Quarter of annulus test
h | Elements | Control points |
---|---|---|
[mm] | [–] | [–] |
\(l_0/2\) | 10,000 | 10,404 |
\(l_0/4\) | 40,000 | 40,804 |
\(l_0/6\) | 90,000 | 91,204 |
\(l_0/8\) | 160,000 | 161,604 |
h | \(G_c^{Eff}\) | \(\Delta G_c^{Eff}\) | Conv. rate | Error | Time |
---|---|---|---|---|---|
[mm] | [kN/mm] | [kN/mm] | [-] | [%] | [h: min] |
\(l_0/2\) | \(5.427\cdot 10^{-3}\) | \(2.727\cdot 10^{-3}\) | – | 100.0 | 01:39 |
\(l_0/4\) | \(3.988\cdot 10^{-3}\) | \(1.288\cdot 10^{-3}\) | 1.082 | 47.7 | 04:43 |
\(l_0/6\) | \(3.544\cdot 10^{-3}\) | \(8.440\cdot 10^{-3}\) | 1.042 | 31.3 | 13:14 |
\(l_0/8\) | \(3.283\cdot 10^{-3}\) | \(5.830\cdot 10^{-3}\) | 1.286 | 21.6 | 36:19 |
h | \(G_c^{Eff}\) | \(\Delta G_c^{Eff}\) | Conv. rate | Error | Time |
---|---|---|---|---|---|
[mm] | [kN/mm] | [kN/mm] | [-] | [%] | [h: min] |
\(l_0/2\) | \(4.735\cdot 10^{-3}\) | \(2.035\cdot 10^{-3}\) | – | 75.4 | 01:28 |
\(l_0/4\) | \(3.488\cdot 10^{-3}\) | \(7.881\cdot 10^{-3}\) | 1.368 | 29.2 | 06:32 |
\(l_0/6\) | \(3.147\cdot 10^{-3}\) | \(4.470\cdot 10^{-3}\) | 1.399 | 16.6 | 09:28 |
\(l_0/8\) | \(2.936\cdot 10^{-3}\) | \(2.360\cdot 10^{-3}\) | 2.220 | 9.00 | 29:31 |