Q2. What is the effect of a phase-field variable on the functional of a s?
After reaching the peak load the functional of the smeared crack decreases to asymptotically reach the discrete value without the introduction of a constraint that prevents the phase-field variable from decreasing locally.
Q3. What is the reason why the -convergence proofs are invalidated?
A monotonically increasing order parameter seems to prevent the construction of a proper functional, and would therefore invalidate a basic assumption of the theory on which the convergence proofs are based.
Q4. How is the bar of length measured?
The bar of length L is discretised such that regions of large gradients of d and u are meshed using the smallest element size h while the mesh is coarsened towards both ends of the bar.
Q5. What is the key issue in the phase-field approach to brittle fracture?
A crucial issue in the phase-field approach to brittle fracture is the requirement that the functional Πℓ, which describes the distributed crack surface, approaches the functional Π for the discrete crack for ℓ→ 0.
Q6. What is the basic idea of phase-field models for brittle fracture?
Phasefield models have now been applied to a variety of fracture problems, including dynamic fracture (Borden et al., 2012; Hofacker and Miehe, 2013), cohesive fracture (Verhoosel and de Borst, 2013), and finite deformations (Hesch et al., 2017).
Q7. What is the effect of a finite length in numerical computations?
When a correction is made for the fact that specimens have a finite length, and the appropriate boundary conditions are imposed, the numerical solutions tend to replicate the analytical solution for specimens which areshort relative to the value of the internal length scale.
Q8. What is the mesh size for the smallest length scale?
It is noted that the mesh size has been taken proportional to the internal length scale for this set of computations: h/ℓ = 1/32, which leads to a mesh more than ten times as fine as the previous set of calculations for the smallest length scale ℓ/L = 0.05.
Q9. What is the simplest equation to describe the discontinuity of a one-dimensional problem?
In a one-dimensional setting the exponentialfunction d(x) = e− |x| 2ℓ (1)is used to approximate the discontinuous function of Figure 1(a), with ℓ being the internal length scale parameter.
Q10. What is the error in the smeared approximation of the sharp crack?
For the error in the smeared approximation of the sharp crack, the following measure was taken:E = |Γℓ − Γ|Γ , (23)where Γℓ is evaluated numerically at max (d) = dmax with dmax = 0.99 and assuming that Γ = A/2 is the theoretical final crack surface.
Q11. What is the energy functional for brittle fracture?
As a starting point the authors consider the energy functional for brittle fracture in a Griffith sense (Francfort and Marigo, 1998):Π =∫Ωψe(ǫ) dV +∫ΓGc dA , (7)where the elastic energy density ψe is a function of the infinitesimal strain tensor ǫ: ψe = ψe(ǫ).
Q12. What is the error in the graph?
The error defined in Equation (23) exhibits a minimum, when plotted as a function of the internal length scale ℓ, see also the curve marked in Figure 3 by ’+’ symbols, which was obtained for 600 elements over the bar.
Q13. How does the error in the equations of Equation 26-7 differ from the numerical ones?
When using a much finer mesh (h/ℓ = 1/32), the curve marked with squares is obtained, which does not exhibit an increase in the error E for small values of the internal length scale ℓ, and levels off, albeit at a very small non-zero value.
Q14. What is the function that describes the discrete crack surface?
A natural requirement is that the functional that describes the smeared crack surface, converges to the original functional that describes the discrete crack surface.
Q15. What are the main advantages of phase-field models?
In this context, phase-field models have become increasingly popular for simulating a host of physical phenomena which exhibit sharp interfaces.