![Figure 23: Comparison of minimum principal shell forces [N/mm], after the initial pre-compression (row 1) and at the end of the analysis (row 2). DNS (column 1) vs. Multiscale (column 2).](/figures/figure-23-comparison-of-minimum-principal-shell-forces-n-mm-5vc6h8sb.webp)
Q2. What is the principle of virtual work for the 3D continuum case?
Neglecting inertia forces, and in absence of body forces and boundary traction, the Principle of Virtual Work for the 3D continuum case at the micro-scale (RVE) reads:ˆ
Q3. How many discretizations have been used for the multiscale simulations?
For the multiscale simulations three macro-scale discretizations have been used, in order to check the regularization properties of the method with respect to the mesh size.
Q4. What is the common method used to study masonry?
A popular method commonly used nowadays to study masonry, accounting for its heterogeneous micro-structure, is micro-/meso-modeling, also known as Direct Numerical Simulation (DNS), [4, 5, 6, 7, 8, 9, 10, 11].
Q5. What are the two scalar measures of a general state of stress?
In order to identify “loading”, “unloading” or “reloading” conditions of a general state of stress, two scalar measures are introduced, termed as equivalent stresses τ+ and τ−.
Q6. Why is the multiscale simulation carried out at every integration point?
It should be noted that in the present implementation, every integration point of the macro-scale has its own RVE (due to the non-linearity of the problem), and the multiscale simulation is carried out from the very beginning at every integration point.
Q7. What constraint can be applied to the micro in-plane displacement fluctuations?
Aµ ∇s ( P θ̃µ ) dA = ˆ ∂Aµ ( P θ̃µ ) ⊗s n dS = 0 (25)Eq. (24) and Eq. (25) provide the minimal kinematic constraint to be applied to the micro in-plane displacement fluctuations ũ0,µ and to the micro out-of-plane rotation fluctuations θ̃µ.
Q8. What is the generalized unknown vector in the local coordinate system of the shell?
The generalized unknown vector in the local coordinate system of the shell is defined as:û = [ u0 uz θ ]T = [ ux uy uz θx θy ]T (1) where u0 = [ ux uy]
Q9. What is the reason for the apparent plastic behavior of the walls?
As discussed by the authors of the experimental campaign, this apparent plastic behavior may be attributed to a redistribution of bending moment along diagonal cracks to horizontal bending along the vertical edges, where the bending restraint provided by the return walls has additional capacity to accept transfer of load from the diagonal bending mechanism.
Q10. What is the effect of this assumption on the RVE calculations?
This assumption reduces the range of applicability of this method, with respect to those methods using a 3D RVE, but it greatly simplifies the macro-micro scale transition (both scale share the same theory), and it reduces the computational cost of the RVE calculations.
Q11. How can the authors obtain the macroscopic generalized strain field?
The macroscopic generalized strain ε̂m , in each point xm of the macro-scale domain and at each instant t, can be obtained as the surface average of the microscopic generalized strain field ε̂µ defined at each point xµ of the micro-scale domain and at each instant t:ε̂m(xm, t) = 1Aµ ˆ
Q12. What is the main concept of the proposed computational homogenization framework?
This section gives the main concepts and basic equations of the proposed computational homogenization framework, where classical first order homogenization is extended to the case of shell theory.