Strength and mass optimisation of variable-stiffness composites in the polar parameters space

Abstract

A general theoretical and numerical framework for the strength and mass optimisation of variable-stiffness composite laminates (VSCLs) is presented in this work. The optimisation is performed in the context of the first-level problem of the multi-scale two-level optimisation strategy (MS2LOS) for VSCLs. Both the failure load maximisation problem (subject to a constraint on the mass) and the mass minimisation one (with a constraint on the VSCL strength) are solved for two benchmark structures. The effect of the presence of a constraint on the maximum tow curvature is also investigated. The solutions are searched by means of a deterministic algorithm by considering different scenarios in terms of the VSCL macroscopic behaviour: the orthotropy type and shape, the direction of the main orthotropy axis and the thickness of the laminate are tailored either globally (uniform over the structure) or locally. The polar method is used to represent the point-wise elastic response of the VSCL at the macroscopic scale. The distributions of the polar parameters and of the thickness are described through B-spline entities: their properties are exploited in computing physical and geometrical response functions of the VSCL as well as their gradient. The VSCL strength at the macroscopic scale is assessed using a laminate-level failure criterion in the space of polar parameters. Numerical results show considerable improvements with respect to both quasi-homogeneous isotropic structures and an optimised VSCL solution taken from the literature obtained by using the design approach based on lamination parameters. These results confirm the effectiveness of the proposed strategy and the great potential of VSCLs.

Appendices

Appendix 1. : On the choice of the threshold value of the laminate failure index

In (12), the LFI is defined as the integral average over the laminate thickness of the ply-level failure index (PFI) defined in (10).

If a laminate is subjected to a given loading condition, with a given load factor λ, such that the maximum value (over the surface) of the LFI is F = 1, two situations may occur:

• The most probable situation wherein some of the layers of the laminate have a PFI greater than one, while the others have a PFI lower than one.

• The unlikely situation wherein the PFI is uniform over the thickness and assumes a unit value.

In the first scenario, the failure of the structure has already been overcome, i.e. failure occurred at a load factor lower than the considered one. For such a reason, using a threshold value F_Th = 1 represents a non-conservative choice.

On the other end of the spectrum, if the maximum LFI is F = 1/N_ply, where N_ply is the number of plies composing the laminate, it may mean that:

• Likely, the PFI assumes values both greater and lower than 1/N_ply, but always lower than one.

• In the limit and unrealistic case, the PFI is equal to one in only one of the layers, while it is null in all the others.

Clearly, using a threshold value F_Th = 1/N_ply results in an over-conservative constraint. An intermediate value has to be used.

In this appendix, the effect of the use of different threshold values for the application of the LFI introduced in Section 3.2 on the optimised solution is assessed. Three possible values are considered: the two extreme values, F_Th = 1 and F_Th = 1/N_ply, and an intermediate one, F_Th = 1/2, i.e. the value used to obtain the main results of this work. The effect of this choice is shown by solving optimisation problem P1 on benchmark structure S1 for VSCLs-C2. Of course, the value of the reference load is adapted to the chosen value of F_Th (see Section 2.1). The laminate is considered composed of 16 plies, as in the work by Khani et al. (2011). The results are summarised in Table 12, which is completed by Fig. 18.

Table 12 Summary of the VSCLs-C2 solutions of problem P1 on benchmark structure S1 for different values of F_Th

Fig. 18

Benchmark structure S1 - Optimisation problem P1 - VSCLs-C2: optimal distributions of the mechanical design variables obtained with different values of F_Th

From the analysis of the results presented in Table 12, it can be observed that by decreasing the value of F_Th, even if the absolute value of the failure load ($\overline {F}_{\mathrm {x, F}}$) decreases, the possible relative gain on the failure load, represented by the load factor λ_F, increases. This means that, if the use of a more conservative (i.e. lower) value of F_Th reveals necessary, a higher λ_F of the optimised solutions will be obtained. Finally, it is noteworthy that, whichever the choice of the value F_Th used, the found load factor is greater than that obtained by Khani et al. (2011)².

A study about the choice of F_Th is part of the current research activities of the group.

Appendix 2.: Setting the p-Norm for a given maximum relative difference

Consider an indexed set of positive values (q_e ≥ 0 with $e \in \left \lbrace 1,2,\ldots , N_{e}\right \rbrace$) whose maximum is $q_{\max \limits }:= \max \limits _{e} q_{e}$. In agreement with (27),

$$ {pn}(q_{e}) := \left( \sum\limits_{e} {q_{e}^{p}}\right)^{1/p} \!= q_{\max}\left( \sum\limits_{e} {\eta_{e}^{p}}\right),^{1/p},\quad \text{with } \eta_{e}:= q_{e}/q_{\max}. $$

(A2.1)

Since 0 ≤ η_e ≤ 1 and $\max \limits _{e} \eta _{e} = 1$,

$$ 1 \le \left( \sum\limits_{e} {\eta_{e}^{p}}\right)^{1/p} \le N_{e}^{1/p} $$

(A2.2)

and, consequently,

$$ q_{\max} \le {pn}(q_{e}) \le q_{\max} N_{e}^{1/p} . $$

(A2.3)

If a maximum relative difference $d_{\max \limits }$ is prescribed between the value of the p-Norm and the value $q_{\max \limits }$, one can write

$$ q_{\max} N_{e}^{1/p} \le q_{\max} \left( 1+d_{\max}\right) , $$

(A2.4)

which is satisfied if

$$ p \ge \frac{\log N_{e}}{\log \left( 1+d_{\max}\right)} . $$

(A2.5)

If p is required to be an integer, then it suffices to set

$$ p_{\min}= \Bigg\lceil\frac{\log N_{e}}{\log \left( 1+d_{\max}\right)}\Bigg\rceil . $$

(A2.6)

Appendix 3.: Failure response function gradient

According to (38) and (30),

$$ \begin{array}{@{}rcl@{}} \frac{\partial f_{\mathrm{F}}}{\partial \xi^{(i_{1}, i_{2})}}&=&\frac{\partial \bigg(pn\left( \lambda_{\mathrm{F},e}^{-1}\right)\bigg)} {\partial \xi^{(i_{1}, i_{2})}} \\&=& -\left( f_{\mathrm{F}} + 1 \right)^{1-p} \sum\limits_{e} \left[ \lambda_{\mathrm{F},e}^{-(p+1)} \frac{\partial \lambda_{\mathrm{F},e}}{\partial \xi^{(i_{1}, i_{2})}} \right] . \end{array} $$

(A3.1)

The factor of safety of the e-th element reads

$$ \lambda_{\mathrm{F},e} = \frac{\lambda_{\text{FN},e}}{\lambda_{\text{FD},e}} = \frac{-L_{e} + \sqrt{{{\varDelta}}_{e}}}{2 Q_{e}} , \text{with } {{\varDelta}}_{e} := {L_{e}^{2}} + 4 Q_{e} {F}_{\text{Th}} . $$

(A3.2)

The derivative of λ_F,e with respect to the design variable $\xi ^{(i_{1}, i_{2})}$ is

$$ \frac{\partial \lambda_{\mathrm{F},e}}{\partial \xi^{(i_{1}, i_{2})}} =\frac{1}{\lambda_{\text{FD},e}} \left( \frac{\partial\lambda_{\text{FN},e}}{\partial \xi^{(i_{1}, i_{2})}} - \lambda_{\mathrm{F},e} \frac{\partial\lambda_{\text{FD},e}}{\partial \xi^{(i_{1}, i_{2})}} \right) , $$

(A3.3)

where the derivatives of the quantities λ_FN,e and λ_FD,e read:

$$ \begin{array}{@{}rcl@{}} \frac{\partial \lambda_{Ne}}{\partial \xi^{(i_{1}, i_{2})}} =& - \frac{\partial L_{e}}{\partial \xi^{(i_{1}, i_{2})}} + \frac{L_{e} \frac{\partial L_{e}}{\partial \xi^{(i_{1}, i_{2})}} + 2 \mathrm{F}_{\text{Th}} \frac{\partial Q_{e}}{\partial \xi^{(i_{1}, i_{2})}}}{\sqrt{{{\varDelta}}_{e}}} , \\ \frac{\partial \lambda_{De}}{\partial \xi^{(i_{1}, i_{2})}} = & 2 \frac{\partial Q_{e}}{\partial \xi^{(i_{1}, i_{2})}}. \end{array} $$

(A3.4)

By injecting (A3.4) in (A3.3), one obtains

$$ \begin{aligned} \frac{\partial \lambda_{\mathrm{F},e}}{\partial \xi^{(i_{1}, i_{2})}} =c_{2,e}\frac{\partial Q_{e}}{\partial \xi^{(i_{1}, i_{2})}} +c_{1,e} \frac{\partial L_{e}}{\partial \xi^{(i_{1}, i_{2})}} , \end{aligned} $$

(A3.5)

where

$$ c_{2,e} = \frac{\mathrm{F}_{\text{Th}}-\lambda_{\mathrm{F},e}\sqrt{{\Delta}_{e}}}{Q_{e}\sqrt{{{\varDelta}}_{e}}} , \quad c_{1,e} = -\frac{\lambda_{\mathrm{F},e}}{\sqrt{{{\varDelta}}_{e}}} . $$

(A3.6)

The derivatives of the terms Q_e and L_e can be inferred from (34) and (35):

$$ \begin{array}{@{}rcl@{}} \frac{\partial Q_{e}}{\partial \xi^{(i_{1}, i_{2})}} &= & ~ \boldsymbol{\varepsilon}_{\text{Ref},e}^{\mathrm{T}} \frac{\partial}{\partial \xi^{(i_{1}, i_{2})}} \left( \frac{\mathbf{G}_{\text{lam},e}}{t_{e}} \right) \boldsymbol{\varepsilon}_{\text{Ref},e}\\ &&+ 2 \boldsymbol{\varepsilon}_{\text{Ref},e}^{\mathrm{T}} \frac{\mathbf{G}_{\text{lam},e}^{\mathrm{T}}}{t_{e}} \boldsymbol{\mathfrak{B}}_{e} \boldsymbol{\mathfrak{L}}_{e} \frac{\partial \mathbf{u}_{\text{Ref}}}{\partial \xi^{(i_{1}, i_{2})}},\\ \frac{\partial L_{e}}{\partial \xi^{(i_{1}, i_{2})}} &=& \boldsymbol{\varepsilon}_{\text{Ref},e}^{\mathrm{T}} \frac{\partial}{\partial \xi^{(i_{1}, i_{2})}} \left( \frac{\mathbf{g}_{\text{lam},e}}{t_{e}} \right) \\&&+ \frac{\mathbf{g}_{\text{lam},e}^{\mathrm{T}}}{t_{e}} \boldsymbol{\mathfrak{B}}_{e} \boldsymbol{\mathfrak{L}}_{e} \frac{\partial \mathbf{u}_{\text{Ref}}}{\partial \xi^{(i_{1}, i_{2})}}, \end{array} $$

(A3.7)

By injecting (A3.7) in (A3.5), one obtains

$$ \begin{array}{@{}rcl@{}} \frac{\partial \lambda_{\mathrm{F},e}}{\partial \xi^{(i_{1}, i_{2})}} &= & \boldsymbol{\varepsilon}_{\text{Ref},e}^{\mathrm{T}} \left( c_{2,e} \frac{\partial\left( {\mathbf{G}_{\text{lam},e}}/{t_{e}} \right)} {\partial \xi^{(i_{1}, i_{2})}} \boldsymbol{\varepsilon}_{\text{Ref},e} + c_{1,e} \frac{\partial\left( {\mathbf{g}_{\text{lam},e}}/{t_{e}} \right)} {\partial \xi^{(i_{1}, i_{2})}} \right)\\ && + \boldsymbol{\psi}^{\mathrm{T}}_{e} \frac{\partial \mathbf{u}_{\text{Ref}}}{\partial \xi^{(i_{1}, i_{2})}} , \end{array} $$

(A3.8)

where ψ_e is defined as

$$ \boldsymbol{\psi}_{e} := \boldsymbol{\mathfrak{L}}_{e}^{\mathrm{T}} \boldsymbol{\mathfrak{B}}_{e}^{\mathrm{T}} \left( 2 c_{2,e} \frac{\mathbf{G}_{\text{lam},e}}{t_{e}} \boldsymbol{\varepsilon}_{\text{Ref},e} + c_{1,e} \frac{\mathbf{g}_{\text{lam},e}}{t_{e}} . \right) $$

(A3.9)

By injecting (A3.8) in (A3.1) and by taking into account for the B-spline blending functions local support property, one obtains

$$ \begin{array}{@{}rcl@{}} \frac{\partial f_{\mathrm{F}}}{\partial \xi^{(i_{1}, i_{2})}} &= -&\left( f_{\mathrm{F}}+1 \right)^{1-p} \left\lbrace \sum\limits_{e \in S_{i_{1}i_{2}}} \left[ \lambda_{\mathrm{F},e}^{-(p+1)}\frac{\partial\xi_{e}}{\partial\xi^{(i_{1}, i_{2})}} \right. \right. \\ && \cdot \left. \left. \boldsymbol{\varepsilon}_{\text{Ref},e}^{\mathrm{T}} \left( c_{2,e} \frac{\partial\left( {\mathbf{G}_{\text{lam},e}}/{t_{e}} \right)} {\partial \xi_{e}} \boldsymbol{\varepsilon}_{\text{Ref},e} + c_{1,e} \frac{\partial\left( {\mathbf{g}_{\text{lam},e}}/{t_{e}} \right)} {\partial \xi_{e}} \right) \right] \right. \\ && + \left. \boldsymbol{\psi}^{\mathrm{T}} \frac{\partial \mathbf{u}_{\text{Ref}}}{\partial \xi^{(i_{1},i_{2})}} \right\rbrace, \end{array} $$

(A3.10)

where

$$ \boldsymbol{\psi} := {\sum}_{e} \left[ \lambda_{\mathrm{F},e}^{-(p+1)} \boldsymbol{\psi}_{e} \right]. $$

(A3.11)

Consider the upper part of (26), with λ = 1:

$$ \mathbf{K^{*}_{\text{FF}}} \mathbf{u}^{*}_{\mathrm{Ref,F}} + \mathbf{K^{*}_{\text{FD}}} \mathbf{u}^{*}_{\mathrm{Ref,D}} = \mathbf{f}^{*}_{\mathrm{Ref,F}} . $$

(A3.12)

Inasmuch as external forces and known DOFs do not depend upon the design variables of the problem at hand, i.e.

$$ \frac{\partial \mathbf{f}^{*}_{\mathrm{Ref,F}}}{\partial \xi^{(i_{1},i_{2})}} = \mathbf{0} \quad\!\text{and}\quad\! \frac{\partial \mathbf{u}^{*}_{\mathrm{Ref,D}}}{\partial \xi^{(i_{1},i_{2})}} = \mathbf{0} , $$

(A3.13)

the derivative of (A3.12) reads

$$ \mathbf{K^{*}_{\text{FF}}} \frac{\partial \mathbf{u}^{*}_{\mathrm{Ref,F}}}{\partial \xi^{(i_{1},i_{2})}} + \frac{\partial \mathbf{K^{*}_{\text{FF}}}}{\partial \xi^{(i_{1},i_{2})}} \mathbf{u}^{*}_{\mathrm{Ref,F}} + \frac{\partial \mathbf{K^{*}_{\text{FD}}}}{\partial \xi^{(i_{1},i_{2})}} \mathbf{u}^{*}_{\mathrm{Ref,D}} = \mathbf{0} . $$

(A3.14)

Consider now the arbitrary auxiliary vector υ, which can be reordered such that $\left .\boldsymbol \upsilon ^{*}\right .^{\mathrm {T}} := \left \lbrace \left .\boldsymbol \upsilon ^{*}_{\mathrm {F}}\right .^{\mathrm {T}}, \left .\boldsymbol \upsilon _{\mathrm {D}}^{*}\right .^{\mathrm {T}} \right \rbrace$. If the components of such vector corresponding to the known DOFs of (26) are set null, i.e. $\boldsymbol \upsilon ^{*}_{\mathrm {D}}=\mathbf {0}$, the following relation holds:

$$ \begin{aligned} \boldsymbol\psi^{\mathrm{T}} \frac{\partial \mathbf{u}_{\text{Ref}}}{\partial \xi^{(i_{1},i_{2})}} \equiv & \left\lbrace \begin{array}{c} \boldsymbol\psi^{*}_{\mathrm{F}} \\ \boldsymbol\psi^{*}_{\mathrm{D}} \end{array} \right\rbrace^{\mathrm{T}} \frac{\partial}{\partial \xi^{(i_{1},i_{2})}}\left\lbrace \begin{array}{c} \mathbf{u}^{*}_{\mathrm{Ref,F}} \\ \mathbf{u}^{*}_{\mathrm{Ref,D}} \end{array} \right\rbrace \\ = & ~ \left.\boldsymbol\psi^{*}_{\mathrm{F}}\right.^{\mathrm{T}} \frac{\partial \mathbf{u}^{*}_{\mathrm{Ref,F}}}{\partial \xi^{(i_{1},i_{2})}} + \left.\boldsymbol\psi^{*}_{\mathrm{D}}\right.^{\mathrm{T}} \underbrace{\frac{\partial \mathbf{u}^{*}_{\mathrm{Ref,D}}}{\partial \xi^{(i_{1},i_{2})}}}_{= \mathbf{0}} \\ & + \left.\boldsymbol\upsilon^{*}_{\mathrm{F}}\right.^{\mathrm{T}} \underbrace{\left( \mathbf{K^{*}_{\text{FF}}} \frac{\partial \mathbf{u}^{*}_{\mathrm{Ref,F}}}{\partial \xi^{(i_{1},i_{2})}} + \frac{\partial \mathbf{K^{*}_{\text{FF}}}}{\partial \xi^{(i_{1},i_{2})}} \mathbf{u}^{*}_{\mathrm{Ref,F}} + \frac{\partial \mathbf{K^{*}_{\text{FD}}}}{\partial \xi^{(i_{1},i_{2})}} \mathbf{u}^{*}_{\mathrm{Ref,D}} \right)}_{= \mathbf{0}} \\ & + \underbrace{\left.\boldsymbol\upsilon^{*}_{\mathrm{D}}\right.^{\mathrm{T}}}_{= \mathbf{0}} \left( \frac{\partial \mathbf{K^{*}_{\text{DF}}}}{\partial \xi^{(i_{1},i_{2})}} \mathbf{u}^{*}_{\mathrm{Ref,F}} + \frac{\partial \mathbf{K^{*}_{\text{DD}}}}{\partial \xi^{(i_{1},i_{2})}} \mathbf{u}^{*}_{\mathrm{Ref,D}} \right) , \end{aligned} $$

(A3.15)

where the last null term is added for the sole purpose of obtaining a more compact expression. Equation (A3.15) simplifies to

$$ \boldsymbol\psi^{\mathrm{T}} \frac{\partial \mathbf{u}_{\text{Ref}}}{\partial \xi^{(i_{1},i_{2})}} \equiv \left( \left.\boldsymbol\psi^{*}_{\mathrm{F}}\right.^{\mathrm{T}} + \left.\boldsymbol\upsilon^{*}_{\mathrm{F}}\right.^{\mathrm{T}} \mathbf{K^{*}_{\text{FF}}}\right) \frac{\partial \mathbf{u}^{*}_{\mathrm{Ref,F}}}{\partial \xi^{(i_{1},i_{2})}} + \boldsymbol\upsilon^{\mathrm{T}}\frac{\partial \mathbf{K} }{\partial \xi^{(i_{1},i_{2})}} \mathbf{u}_{\text{Ref}} . $$

(A3.16)

Consider the term $\frac {\partial \mathbf {K} }{\partial \xi ^{(i_{1},i_{2})}} \mathbf {u}_{\text {Ref}}$ in (A3.16). The stiffness matrix of the structure is defined as

$$ \mathbf{K} := {\sum}_{e} \boldsymbol{\mathfrak{L}}_{e}^{\mathrm{T}} \mathbf{K}_{e} \boldsymbol{\mathfrak{L}}_{e} . $$

(A3.17)

K_e is the stiffness matrix of the element and, for a shell element, it is defined as:

$$ \mathbf{K}_{e} := {\int}_{A_{e}} \boldsymbol{\mathfrak{B}}_{e}^{\mathrm{T}} \mathbf{K}_{\text{lam},e} \boldsymbol{\mathfrak{B}}_{e} dS , $$

(A3.18)

where, A_e is the area of e-th element. By taking into account for (A3.17) and (A3.18) and for the local support property, the term $\frac {\partial \mathbf {K} }{\partial \xi ^{(i_{1},i_{2})}} \mathbf {u}_{\text {Ref}}$ in (A3.16) can be expressed as:

$$ \begin{aligned} \tilde{\boldsymbol\psi}_{i_{1}i_{2}} := & \frac{\partial \mathbf{K}}{\partial \xi^{(i_{1},i_{2})}} \mathbf{u}_{\text{Ref}} \\ = & \sum\limits_{e \in S_{i_{1}i_{2}}} \left[ \frac{\partial \xi_{e}}{\partial \xi^{(i_{1},i_{2})}} {\int}_{A_{e}} \boldsymbol{\mathfrak{L}}_{e}^{\mathrm{T}} \boldsymbol{\mathfrak{B}}_{e}^{\mathrm{T}} \frac{\partial \mathbf{K}_{\text{lam},e}}{\partial \xi_{e}} \boldsymbol{\mathfrak{B}}_{e} \boldsymbol{\mathfrak{L}}_{e} dS \right] \mathbf{u}_{\text{Ref}} . \end{aligned} $$

(A3.19)

The arbitrary vector $\boldsymbol \upsilon ^{*}_{\mathrm {F}}$ can be chosen in such a way that the term multiplying $\frac {\partial \mathbf {u}_{\mathrm {Ref,F}}}{\partial \xi ^{(i_{1},i_{2})}}$ vanishes, i.e.

$$ \mathbf{K}^{*}_{\text{FF}} \boldsymbol\upsilon^{*}_{\mathrm{F}} = - \boldsymbol\psi^{*}_{\mathrm{F}} . $$

(A3.20)

By using (A3.19) and the expression of $\boldsymbol \upsilon ^{*}_{\mathrm {F}}$ obtained by solving the auxiliary system of (A3.20), (A3.16) simplifies to

$$ \boldsymbol\psi^{\mathrm{T}} \frac{\partial \mathbf{u}_{\text{Ref}}}{\partial \xi^{(i_{1},i_{2})}} = \boldsymbol\upsilon^{\mathrm{T}} \tilde{\boldsymbol\psi}_{i_{1}i_{2}} . $$

(A3.21)

Therefore, the final expression of the generic component of the gradient of f_F is

$$ \begin{aligned} \frac{\partial f_{\mathrm{F}}}{\partial \xi^{(i_{1}, i_{2})}} = -\left( f_{\mathrm{F}}+1 \right)^{1-p} \left\lbrace l_{i_{1}i_{2}} + \boldsymbol\upsilon^{\mathrm{T}} \tilde{\boldsymbol\psi}_{i_{1}i_{2}} \right\rbrace , \end{aligned} $$

(A3.22)

with

$$ \begin{array}{@{}rcl@{}} l_{i_{1}i_{2}} &:= & {\sum}_{e \in S_{i_{1}i_{2}}} \left[\lambda_{\mathrm{F},e}^{-(p+1)} \frac{\partial\xi_{e}}{\partial\xi^{(i_{1}, i_{2})}} \right. \\ && \left. \cdot \ \boldsymbol{\varepsilon}_{\text{Ref},e}^{\mathrm{T}} \left( c_{2,e} \frac{\partial \left( {\mathbf{G}_{\text{lam},e}}/{t_{e}} \right)} {\partial \xi_{e}} \boldsymbol{\varepsilon}_{\text{Ref},e} + c_{1,e} \frac{\partial\left( {\mathbf{g}_{\text{lam},e}}/{t_{e}} \right)} {\partial \xi_{e}} \right) \right] .\\ \end{array} $$

(A3.23)

Remark A3.1

The analytic expression of the derivatives $\frac {\partial \mathbf {K}_{\text {lam},e} }{\partial \xi _{e}}$, $\frac {\partial }{\partial \xi _{e}} \left (\frac {\mathbf {g}_{\text {lam},e}}{t_{e}} \right )$ and $\frac {\partial }{\partial \xi _{e}} \left (\frac {\mathbf {G}_{\text {lam},e}}{t_{e}} \right )$ appearing in the above formulae is provided in Appendix D.

Remark A3.2

The solution of the auxiliary system of (A3.20), needed to compute the gradient of f_F, can be obtained through a FE analysis wherein all DOFs corresponding to the known DOFs of (26) are set null (i.e. $\boldsymbol \upsilon ^{*}_{\mathrm {D}}=\mathbf {0}$) and whose generalised external nodal forces ψ are computed using (A3.11). Of course, the components of ψ corresponding to the known DOFs of (26), i.e. ψ_D, are discarded by the FE solver because applied on constrained DOFs.

Appendix 4.: Analytical expression of the laminate stiffness and strength matrices and vector and their gradient

Remark A4.1

The hypotheses of point-wise fully orthotropy and quasi-homogeneity of the considered VSCLs (corresponding to the conditions of (7)) hold in the following. All quantities in this section are referred to a generic element e of the structure model, but the index e is omitted for the sake of readability.

Using (6), (5) can be rewritten as

$$ \mathbf{K}_{\text{lam}} = \left[\! \begin{array}{ccc} t\mathbf{A^{*}} & \mathbf{0} & \mathbf{0}\\ ~ & \frac{t^{3}}{12}\mathbf{A^{*}} & \mathbf{0}\\ \text{sym} & ~ & t\mathbf{H^{*}} \end{array} \!\right] . $$

(A4.1)

Matrices A^∗ and H^∗ can be computed as a function of their PPs, thanks to (2) and (1). The PPs of matrix A^∗ can be obtained by means of (17) as

$$ \begin{aligned} &T_{0}^{\mathrm{A}^{*}}\!\equiv T_{0}^{\mathrm{Q}_{\text{in}}} , T_{1}^{\mathrm{A}^{*}}\!\equiv T_{1}^{\mathrm{Q}_{\text{in}}} , R_{0}^{\mathrm{A}^{*}}\!=R_{0}^{\mathrm{Q}_{\text{in}}} \lvert\rho_{0K}\rvert , R_{1}^{\mathrm{A}^{*}}\!=R_{1}^{\mathrm{Q}_{\text{in}}}\rho_{1} , \\&{{{\varPhi}}}_{0}^{\mathrm{A}^{*}}\!={{{\varPhi}}}_{1}^{\mathrm{A}^{*}}\!+ \frac{\pi}{4} K^{\mathrm{A}^{*}} , {{{\varPhi}}}_{1}^{\mathrm{A}^{*}}\!=\frac{\pi}{2}\phi_{1} , \end{aligned} $$

(A4.2)

where

$$ K^{\mathrm{A}^{*}} = \begin{cases} 0 & \quad \text{if}~ \rho_{0K} \ge 0 , \\ 1 & \quad \text{if}~ \rho_{0K} < 0 , \end{cases} $$

(A4.3)

and can be related to the problem design variables through (20). The PPs of matrix H^∗ depend upon those of A^∗ and of the constitutive tow as follows (Montemurro 2015a; 2015b):

$$ T^{\mathrm{H}^{*}} \! = T^{\mathrm{Q}_{\text{in}}} , R^{\mathrm{H}^{*}} \! = \frac{\mathrm{R}^{\mathrm{Q}_{\text{out}}}}{\mathrm{R}_{1}^{\mathrm{Q}_{\text{in}}}} R_{1}^{\mathrm{A}^{*}} , {{\varPhi}}^{\mathrm{H}^{*}} \! = {{\varPhi}}^{\mathrm{Q}_{\text{out}}} + {{{\varPhi}}}_{1}^{\mathrm{Q}_{\text{in}}} - {{{\varPhi}}}_{1}^{\mathrm{A}^{*}} . $$

(A4.4)

Regarding the strength properties of the laminate, by employing (14), (13) and the main results obtained by Catapano and Montemurro (2019), one can write:

$$ \frac{\mathbf{G}_{\text{lam}}}{t} = \left[\! \begin{array}{ccc} \mathbf{G}_{\mathrm{A}}^{*} & \mathbf{0} & \mathbf{0}\\ ~ & \frac{t^{2}}{12}\mathbf{G}_{\mathrm{A}}^{*} & \mathbf{0}\\ \text{sym} & ~ & \mathbf{G}_{\mathrm{H}}^{*} \end{array} \!\right] , \frac{\mathbf{g}_{\text{lam}}}{t} = \left\lbrace\! \begin{array}{c} \mathbf{g}_{\mathrm{A}}^{*} \\ \mathbf{0} \\ \mathbf{0} \end{array} \!\right\rbrace . $$

(A4.5)

Of course, also matrices $\mathbf {G}_{\mathrm {A}}^{*}$ and $\mathbf {G}_{\mathrm {H}}^{*}$ and vector $\mathbf {G}_{\mathrm {A}}^{*}$ can be computed as a function of their PPs. Such PPs depend on those of matrix A^∗ and of the constitutive tow as follows (Catapano and Montemurro 2019):

$$ \begin{aligned} &T_{0}^{\mathrm{G}_{\mathrm{A}}^{*}} \! \equiv T_{0}^{\mathrm{G}_{\text{in}}} , T_{1}^{\mathrm{G}_{\mathrm{A}}^{*}} \! \equiv T_{1}^{\mathrm{G}_{\text{in}}} , \\&R_{0}^{\mathrm{G}_{\mathrm{A}}^{*}} \! = \frac{R_{0}^{\mathrm{G}_{\text{in}}}}{R_{0}^{\mathrm{Q}_{\text{in}}}} R_{0}^{\mathrm{A}^{*}} , R_{1}^{\mathrm{G}_{\mathrm{A}}^{*}} \! = \frac{R_{1}^{\mathrm{G}_{\text{in}}}}{R_{1}^{\mathrm{Q}_{\text{in}}}} R_{1}^{\mathrm{A}^{*}} , \\&{{{\varPhi}}}_{0}^{\mathrm{G}_{\mathrm{A}}^{*}} \! = {{{\varPhi}}}_{0}^{\mathrm{G}_{\text{in}}} -{{{\varPhi}}}_{0}^{\mathrm{Q}_{\text{in}}} +{{{\varPhi}}}_{0}^{\mathrm{A}^{*}} , \\&{{{\varPhi}}}_{1}^{\mathrm{G}_{\mathrm{A}}^{*}} \! = {{{\varPhi}}}_{1}^{\mathrm{G}_{\text{in}}} -{{{\varPhi}}}_{1}^{\mathrm{Q}_{\text{in}}} +{{{\varPhi}}}_{1}^{\mathrm{A}^{*}} , \end{aligned} $$

(A4.6)

$$ \begin{aligned} &T^{\mathrm{G}_{\mathrm{H}}^{*}} \! \equiv T^{\mathrm{G}_{\text{out}}} , R^{\mathrm{G}_{\mathrm{H}}^{*}} \! = \frac{R^{\mathrm{G}_{\text{out}}}}{R_{1}^{\mathrm{Q}_{\text{in}}}} R_{1}^{\mathrm{A}^{*}} , {{\varPhi}}^{\mathrm{G}_{\mathrm{H}}^{*}} \! = {{\varPhi}}^{\mathrm{G}_{\text{out}}} +{{{\varPhi}}}_{1}^{\mathrm{Q}_{\text{in}}} -{{{\varPhi}}}_{1}^{\mathrm{A}^{*}} , \end{aligned} $$

(A4.7)

$$ \begin{aligned} &T^{\mathrm{g}_{\mathrm{A}}^{*}} \! \equiv T^{\mathrm{g}_{\text{in}}} , R^{\mathrm{g}_{\mathrm{A}}^{*}} \! = \frac{R^{\mathrm{g}_{\text{in}}}}{R_{1}^{\mathrm{Q}_{\text{in}}}} R_{1}^{\mathrm{A}^{*}} , {{\varPhi}}^{\mathrm{g}_{\mathrm{A}}^{*}} \! = {{\varPhi}}^{\mathrm{g}_{\text{in}}} -{{{\varPhi}}}_{1}^{\mathrm{Q}_{\text{in}}} +{{{\varPhi}}}_{1}^{\mathrm{A}^{*}} . \end{aligned} $$

(A4.8)

The derivatives of K_lam, G_lam/t and g_lam/t with respect to the value of the generic dimensionless variable ξ are:

$$ \frac{\partial\mathbf{K}_{\text{lam}}}{\partial\xi}= \begin{cases} \left[\!\begin{array}{ccc} t\frac{\partial\mathbf{A^{*}}}{\partial\xi} & \mathbf{0} & \mathbf{0}\\ ~ & \frac{t^{3}}{12}\frac{\partial\mathbf{A^{*}}}{\partial\xi} & \mathbf{0}\\ \text{sym} & ~ & t\frac{\partial\mathbf{H^{*}}}{\partial\xi} \end{array} \!\right] & \quad \text{if}~ \xi=\rho_{0K}, \rho_{1}, \phi_{1} , \\ t_{\text{RSol}}\left[\!\begin{array}{ccc} \mathbf{A^{*}} & \mathbf{0} & \mathbf{0}\\ ~ & \frac{t^{2}}{4}\mathbf{A^{*}} & \mathbf{0}\\ \text{sym} & ~ & \mathbf{H^{*}} \end{array} \!\right] & \quad \text{if}~ \xi=\tau , \end{cases} $$

(A4.9)

$$ \frac{\partial}{\partial\xi} \left( \frac{\mathbf{G}_{\text{lam}}}{t}\right) = \begin{cases} \left[\!\begin{array}{ccc} \frac{\partial\mathbf{G}_{\mathrm{A}}^{*}}{\partial\xi} & \mathbf{0} & \mathbf{0}\\ ~ & \frac{t^{2}}{12}\frac{\partial\mathbf{G}_{\mathrm{A}}^{*}}{\partial\xi} & \mathbf{0}\\ \text{sym} & ~ & \frac{\partial\mathbf{G}_{\mathrm{H}}^{*}}{\partial\xi} \end{array} \!\right] & \quad \text{if}~ \xi=\rho_{0K}, \rho_{1}, \phi_{1} , \\ t_{\text{RSol}}\left[\!\begin{array}{ccc} \mathbf{0} & \mathbf{0} & \mathbf{0}\\ ~ & \frac{t}{6}\mathbf{G}_{\mathrm{A}}^{*} & \mathbf{0}\\ \text{sym} & ~ & \mathbf{0} \end{array} \!\right] & \quad \text{if}~ \xi=\tau , \end{cases} $$

(A4.10)

$$ \frac{\partial}{\partial\xi} \left( \frac{\mathbf{g}_{\text{lam}}}{t}\right) = \begin{cases} \left\lbrace\begin{array}{c} \frac{\partial\mathbf{g}_{\mathrm{A}}^{*}}{\partial\xi} \\ \mathbf{0} \\ \mathbf{0} \end{array} \!\right\rbrace & \quad \text{if}~ \xi=\rho_{0K}, \rho_{1}, \phi_{1} , \\ \left\lbrace\begin{array}{c} \mathbf{0} \end{array} \!\right\rbrace & \quad \text{if}~ \xi=\tau . \end{cases} $$

(A4.11)

The derivatives of A^∗ and $\mathbf {G}_{\mathrm {A}}^{*}$ are of type:

$$ \begin{aligned} &\frac{\partial\mathbf{X}}{\partial\rho_{0K}}= a_{\mathrm{X}} \left[\!\begin{array}{rrr} c_{4}^{\mathrm{X}} & -c_{4}^{\mathrm{X}} & s_{4}^{\mathrm{X}}\\ ~ & c_{4}^{\mathrm{X}} & -s_{4}^{\mathrm{X}} \\ \text{sym} & ~ & -c_{4}^{\mathrm{X}} \end{array} \!\right] , \end{aligned} $$

(A4.12)

$$ \begin{aligned} &\frac{\partial\mathbf{X}}{\partial\rho_{1}}= b_{\mathrm{X}} \left[\!\begin{array}{rrr} 4c_{2}^{\mathrm{X}} & 0 & +2s_{2}^{\mathrm{X}}\\ ~ & -4c_{2}^{\mathrm{X}} & +2s_{2}^{\mathrm{X}} \\ \text{sym} & ~ & 0 \end{array} \!\right] , \end{aligned} $$

(A4.13)

$$ \begin{aligned} \frac{\partial\mathbf{X}}{\partial\phi_{1}}= & 2 \pi \rho_{0K} a_{\mathrm{X}} \left[\!\begin{array}{ccc} -s_{4}^{\mathrm{X}} & +s_{4}^{\mathrm{X}} & +c_{4}^{\mathrm{X}}\\ ~ & -s_{4}^{\mathrm{X}} & -c_{4}^{\mathrm{X}} \\ \text{sym} & ~ & +s_{4}^{\mathrm{X}} \end{array} \!\right] \\& + 2 \pi \rho_{1} b_{\mathrm{X}} \left[\!\begin{array}{ccc} -2s_{2}^{\mathrm{X}} & 0 & +c_{2}^{\mathrm{X}}\\ ~ & +2s_{2}^{\mathrm{X}} & +c_{2}^{\mathrm{X}} \\ \text{sym} & ~ & 0 \end{array} \!\right] , \end{aligned} $$

(A4.14)

where

$$ c_{4}^{\mathrm{X}}/s_{4}^{\mathrm{X}} = \cos/\sin\left( 4 \beta_{\mathrm{X}}\right) , c_{2}^{\mathrm{X}}/s_{2}^{\mathrm{X}} = \cos/\sin\left( 2 \beta_{\mathrm{X}}\right) $$

(A4.15)

and X, a_X, b_X and β_X assume the values provided in Table 13.

Table 13 Possible sets of X, a_X, b_X and β_X

The derivatives of H^∗ and $\mathbf {G}_{\mathrm {H}}^{*}$ are of type:

$$ \frac{\partial\mathbf{Y}}{\partial\rho_{0K}}= \left[ \mathbf{0} \right] , $$

(A4.16)

$$ \begin{aligned} &\frac{\partial\mathbf{Y}}{\partial\rho_{1}}= b_{\mathrm{Y}} \left[\!\begin{array}{cc} c_{2}^{\mathrm{Y}} & s_{2}^{\mathrm{Y}}\\ s_{2}^{\mathrm{Y}} & -c_{2}^{\mathrm{Y}} \end{array} \!\right] , \end{aligned} $$

(A4.17)

$$ \begin{aligned} \frac{\partial\mathbf{X}}{\partial\phi_{1}}= \pi \rho_{1} b_{\mathrm{Y}} \left[\!\begin{array}{rr} s_{2}^{\mathrm{Y}} & -c_{2}^{\mathrm{Y}}\\ -c_{2}^{\mathrm{Y}} & -s_{2}^{\mathrm{Y}} \end{array} \!\right] , \end{aligned} $$

(A4.18)

where

$$ c_{2}^{\mathrm{Y}}/s_{2}^{\mathrm{Y}} = \cos/\sin\left( 2 \beta_{\mathrm{Y}}\right) $$

(A4.19)

and Y, b_Y and β_Y assume the values provided in Table 14.

Table 14 Possible sets of Y, b_Y and β_Y

Finally, the derivatives of $\mathbf {G}_{\mathrm {A}}^{*}$ are:

$$ \frac{\partial\mathbf{g}_{\mathrm{A}}^{*}}{\partial\rho_{0K}}= \left\lbrace \mathbf{0} \right\rbrace , $$

(A4.20)

$$ \frac{\partial\mathbf{g}_{\mathrm{A}}^{*}}{\partial\rho_{1}}= R^{\mathrm{g}_{\text{in}}}\left\lbrace \begin{array}{ccc} {c_{2}^{g}} & -{c_{2}^{g}} & {s_{2}^{g}} \end{array} \right\rbrace^{\mathrm{T}} , $$

(A4.21)

$$ \frac{\partial\mathbf{g}_{\mathrm{A}}^{*}}{\partial\phi_{1}}= \pi R^{\mathrm{g}_{\text{in}}} \rho_{1} \left\lbrace \begin{array}{rrr} -{s_{2}^{g}} & {s_{2}^{g}} & {c_{2}^{g}} \end{array} \right\rbrace^{\mathrm{T}} , $$

(A4.22)

where

$$ c_{2}^{\mathrm{g}}/s_{2}^{\mathrm{g}} = \cos/\sin \bigg(2 \left( {{\varPhi}}^{\mathrm{g}_{\text{in}}} -{{{\varPhi}}}_{1}^{\mathrm{Q}_{\text{in}}} +({\pi}/{2}) \phi_{1} \right)\bigg) . $$

(A4.23)

Remark A4.2

According to (20), the partial derivatives of a generic function f(ρ_0K,ρ₁) with respect to the design variables α₀ and α₁ read:

$$ \begin{aligned} \frac{\partial f}{\partial \alpha_{0}} = & \frac{\partial f}{\partial \rho_{\mathrm{0K}}} \left( 2{\rho_{1}^{2}} - 2 \right) , \\ \frac{\partial f}{\partial \alpha_{1}} = & \frac{\partial f}{\partial \rho_{\mathrm{0K}}} \left( 2 \frac{\rho_{0K}-1}{{\rho_{1}^{2}}-1} \rho_{1} \right) + \frac{\partial f}{\partial \rho_{\mathrm{1}}} . \end{aligned} $$

(A4.24)