# The Isotropic and Cubic Material Designs. Recovery of the Underlying Microstructures Appearing in the Least Compliant Continuum Bodies

^{*}

## Abstract

**:**

## 1. Introduction

_{K}(x) and eigenstate tensors

**ω**

_{K}(x) (K = 1, 2, 3 for two dimensional bodies—in short 2D) are design variables (see [49,50,54]). An extremely important feature of the non-homogeneous solutions, found by IMD method, is the emergence of subdomains where the Poisson ratio is negative (practically in the whole range from 0 to −1). The numerical results shown in the present paper for the deep beam supported at two bottom corner nodes and loaded with one vertical force applied to the upper face fully confirm this feature and clearly shows the auxetic subdomains.

_{0}, μ

_{0}) and (k

_{1}, μ

_{1}), respectively. The inhomogeneity is modeled by varying geometry of the layout of these phases within RVEs corresponding to the subsequent points x, thus making RVEs x-dependent. The homogenized fields k

^{H}(x) and μ

^{H}(x) are constructed by the strict formulae of the theory of homogenization. The modeling includes the case of the phase 0 being a void, leading to the porous material solution.

^{H}(x) as well as μ(x) and μ

^{H}(x) for subsequent points x ∈ Ω. Thus, calculated in this way the effective moduli k

^{H}(x) and μ

^{H}(x) correspond to an internal structure on the micro-scale, shown explicitly in the figures. Particularly interesting is the form and shape of the microstructures at those points where optimal material exhibits auxetic properties. The recovery of microstructures is performed by a direct combinatorial search. Alternatively, one can apply the available inverse homogenization techniques (see [55]).

## 2. Isotropic and Cubic Material Design. Least Compliant 2D Structures

^{3}be the set representing the undeformed body and Ω

^{φ}$\subset $ R

^{3}, Ω

^{φ}= φ(Ω) be the set representing the body after it is deformed, where φ: Ω→R

^{3}is the smooth enough, injective and orientation-preserving vector field defining the deformation. The sets Ω and Ω

^{φ}are called the reference and deformed configurations, respectively. Materials for which the Cauchy stress tensor

**σ**(x

^{φ}) at any point x

^{φ}= φ(x) is completely determined by the deformation gradient $\nabla $ φ(x) at the corresponding point x ∈ Ω (i.e., by Fréchet derivative of the mapping φ at x) are called elastic materials. A little more mathematically precisely: a material is elastic if there exists a mapping $C:\mathsf{\Omega}\times {M}_{+}^{3}\to {S}^{3}$, called the response function, such that $\sigma \left(\phi \left(x\right)\right)=C\left(x,\nabla \phi \left(x\right)\right)$; here ${M}_{+}^{3},\text{\hspace{0.17em}}{S}^{3}$ denote the set of all real square matrices with positive determinant and the set of all symmetric matrices of 3rd order, respectively. A material is called homogeneous in a reference configuration Ω if its response function is independent of the point x ∈ Ω, i.e., $\sigma \left(\phi \left(x\right)\right)=C\left(\nabla \phi \left(x\right)\right)$, otherwise the material is called non-homogeneous in a reference configuration Ω. The term isotropy is in the simplest way interpreted as the property of material whose response to a given load is the same in all directions. A little more mathematically precisely, an elastic material is isotropic at a point x ∈ Ω if the response function satisfies the following condition: $\forall F\in {M}_{+}^{3}\text{\hspace{0.17em}}\forall Q\in {O}_{+}^{3}\text{\hspace{1em}}C\left(x,F\text{\hspace{0.17em}}Q\right)=C\left(x,F\right)$, i.e., if the Cauchy stress tensor is unaltered when the reference configuration is subjected to an arbitrary rotation $Q\in {O}_{+}^{3}$ around the point x ∈ Ω. If the above condition is met only for rotations $Q\in G$, where $G\subset {O}_{+}^{3}$ is the subgroup of the group ${O}_{+}^{3}$, then material is said to be anisotropic at a point x ∈ Ω. An elastic material is isotropic if it is isotropic at all points x ∈ Ω.

**σ**(x) and symmetric part of the displacement gradient called linearized strain tensor

**ε**(x), i.e., $\sigma \left(x\right)=C\left(x,\epsilon \left(x\right)\right)$, where $C\left(x,\epsilon \left(x\right)\right)=C\left(x\right)\epsilon \left(x\right)$ and $C=C\left(x\right)={C}_{ij}{}_{kl}\left(x\right){e}_{i}\otimes {e}_{j}\otimes {e}_{k}\otimes {e}_{l}$, (i, j, k, l = 1, 2, 3) is the 4th rank Hooke tensor; the orthonormal vectors

**e**

_{i}, i = 1, 2, 3 span the Euclidean space of points in R

^{3}, the summation convention over small Latin indices being applied. The components C

_{ijkl}are subject to the symmetry conditions: C

_{ijkl}= C

_{klij}, C

_{ijkl}= C

_{jikl}, C

_{ijkl}= C

_{ijlk}, and to the positivity condition: C

_{ijkl}ε

_{ij}ε

_{kl}≥ cε

_{ij}ε

_{j}i.e., in Ω for some c > 0. Other properties of linear elastic materials known as isotropic-, cubic-, transversely isotropic-, tetragonal- hexagonal-, rhombic-, monoclinic-, and triclinic systems are defined mathematically by various spectral representations of elasticity tensor

**C**, commonly known as: isotropic and orientation Hooke tensors [58]. The spectral decomposition formula involves independent constitutive moduli (eigenvalues and tensor products of eigenstates—often called projectors), whose number varies between 2 and 21 for isotropic and anisotropic Hooke tensors, respectively.

_{1}of the boundary Γ of the given, two-dimensional design domain Ω $\subset $ R

^{2}. The body is fixed on the boundary Γ

_{2}, a part of Γ. The paper deals with two problems: (i) designing moduli of an isotropic body; and (ii) designing moduli of a body made of a cubic material. In the former optimum design problem the design variables are: the bulk k = k(x) and the shear moduli μ = μ(x), x ∈ Ω defining the non-homogeneous, isotropic, 4th rank Hooke tensors

**C**:

**II**stands for the 4th rank identity tensor in 2D case. The representation of the flexibility tensor, or the tensor inverse to

**C**, reads

_{0}is the referential elastic modulus, |Ω| represents the area of the design domain and $\mathrm{tr}\text{}C=2k+4\mu $. Condition (4) determines the set ℜ of admissible pairs $(k,\mu )$ of non-negative moduli corresponding to the isotropic Hooke tensor

**C**.

**t**done on the displacements $u=\left({u}_{1},{u}_{2}\right)$ caused by the same loading. More precisely, let us define the linear form

**v**=

**0**on the boundary Γ

_{2}(dot “·” denotes the scalar product

**t**·

**v**= t

_{i}·v

_{i}). The compliance of the structure is defined as the functional

**u**depends on

**C**and thus depends on (k, μ), which will be written as $u=u\left(k,\mu \right)$. Let us introduce the set

**v**and $\tau \cdot \epsilon ={\tau}_{ij}{\epsilon}_{ij}$ is the scalar product.

**Theorem**

**1.**

_{min}and μ(x) ≥ μ

_{min}, where k

_{min}> 0, μ

_{min}> 0 are given and fixed, could prevent from the emergence of the mentioned degenerated optimal materials, but then the simplicity of the IMD method would be lost and the Equation (11) cease to hold.

**Remark**

**1.**

**m**(x),

**n**(x)) and three elastic moduli (a(x), b(x), c(x)), x ∈ Ω involved in Walpole’s [58] representation of a non-homogeneous Hooke tensor

**C**of cubic symmetry in the 2D setting:

**m**=

**e**

_{1},

**n**=

**e**

_{2}, then

**m***,

**n***) at each point x ∈ Ω minimizing Compliance (8) or

**Theorem**

**2.**

**m***,

**n***) follow the trajectories of principal stresses τ* (see [53]).

**C**= (C

_{ijkl}) (cf. [47], Equation II.4).

**E**

^{I}of the isotropic tensor

**C**

^{I}is expressed in terms of the bulk and shear moduli (k, μ), or in terms of the Young’s modulus and the Poisson ratio (E, ν) as follows

**E**

^{C}representing the cubic symmetry tensor

**C**

^{C}referred to its principal directions is expressed in terms of the moduli involved in Equation (14)

**C**

^{C}by 45° gives the matrix representation as below:

**E**

^{C}will be assumed as in Equation (22), where 0 ≤ α ≤ 1.

## 3. Optimum Design of a Deep Beam by the IMD Method

_{x}= 4L and height L

_{y}= L, L being a referential length (equal to 1 m in calculations), simply supported at bottom corners and subjected to the in-plane vertical force T = −T

**e**

_{2}centered on the upper edge (see Figure 1). The right support has the possibility of free horizontal displacement and the loading T is represented by the traction

**t**= (0, t) modeled by a smoothing weight function, i.e., $T={\displaystyle \underset{0}{\overset{{L}_{x}}{\int}}t\left(s\right)\text{\hspace{0.17em}}ds}\cong 1.0\left[N\right]$, $t\left(s\right)={t}_{\mathrm{max}}\mathrm{exp}\left(-{\left(\left(s-{L}_{x}/2\right)/w\right)}^{2}\right)$, ${t}_{\mathrm{max}}\cong 3.76\text{\hspace{0.17em}}\left[N/m\right]$, $w=0.15$ (see Figure 1). The finite element mesh used is composed of E

_{x}× E

_{y}= 100 × 25 = 2500 bilinear finite elements C2D4. The value of the referential Young’s modulus E

_{0}is assumed equal to 1.0 N/m

^{2}.

^{−7}to ~4.0. Inside the inner subdomain bounded by the strips mentioned above, the optimal modulus k* reaches close to zero values, but this observation does not apply to the optimal modulus μ*, which reaches also small but positive values, hence there a degenerated material emerges with the extreme negative value of Poisson’s ratio ν* equal to −1. It is also worth noting that below the applied load, a characteristic “bubble” subarea emerges in which the optimal Poisson’s ratio ν* reaches a maximum positive value equal to 1 (red and violet color in Figure 2g,h, respectively). This is due to the fact that the optimal bulk modulus k* is positive, and the optimal shear modulus μ* is equal (or close) to zero, so according to the second formula in Equation (13), the degenerated material once more emerges with extreme positive value of Poisson’s ratio ν* equal to 1. At the same time, in the subdomains on the left and right side of the “bubble”, an auxetic material with a Poisson’s ratio value of about −0.3 emerges (green color). Under these subdomains the Poisson ratio decreases rapidly to negative values: from −0.7 (blue color) to −1 (violet color) (see Figure 3). This phenomenon is known; it is called the indentation behavior of auxetic materials, which improves indentation resistance when compared to conventional materials (see [5,6]). In the example considered, the reason of this phenomenon is that the local increasing of the indentation resistance contributes to the decreasing of the compliance, i.e., decreasing of the work done by the loading

**T**on the displacement

**u**. Note that the region ν* = 0 degenerates to a contour dividing the subdomains corresponding to ν* < 0 and ν* > 0 (see Figure 3c). The distribution of the optimal moduli k*, μ* shows the appearance of the auxetic material (k* < μ*), cf. Figure 4.

## 4. Optimum Design of a Deep Beam by the CMD Method

_{1212}= 0, which means that the underlying microstructures should have zero stiffness due to shear. The optimal cubic material forming the stiffest deep beam of Figure 1 satisfies the mentioned properties. The layout of the optimal moduli a*, c* is similar (see Figure 5), but not identical with the layout of k, μ of the isotropic design of Figure 2. However, in the cubic design, one can note the subdomains of negative Poisson ratio (see Figure 6). Distributions of optimal pairs of points (a*/2, c*/2) are shown in Figure 7. Since the moduli (a*/2, c*/2) are counterparts of the moduli (k*, μ*) it is worth showing the results of the IMD and CMD methods in one figure (see Figure 8). We note that CMD results lie in a broader domain. To explain it let us note that the optimum design method IMD involves two unknown fields, the method CMD involves three unknown fields, while the same compliance functional is minimized. Consequently, the IMD method imposes stronger constraints thus delivering the results lying in a narrower domain than the CMD results.

## 5. Effective Properties of Periodic Composites—Numerical Homogenization

_{e}, e being a small parameter. Upon rescaling we work with the basic cell Y of a rectangular shape parameterized by the (y

_{1}, y

_{2}) Cartesian system. The elastic moduli of the basic cell are still denoted by C

_{ijkl}, but they are viewed as functions of argument y = (y

_{1}, y

_{2}); with y being a point of Y. The basic cell problem of the homogenization theory reads:

**v**defined on Y.

_{ijkl}suffer jumps within Y. Having solved Problem (20) one can define the homogenized moduli by

**C**

^{H}thus constructed is invariant with respect to translations and rotations of the periodicity cell Y.

_{k}of Y to make it possible to satisfy the periodicity conditions involved in Equation (24). The simplest FE algorithms start from the approximation of the test fields

**v**=

**Nq**, leading to the strain approximation of the form $\epsilon \left(v\right)=Bq$,

**N**being the matrix of the shape functions, and

**B**being defined according to the definition of strain. The periodicity conditions are fulfilled by identifying the nodes at opposite edges, by virtue of the mesh being properly introduced. For the k-th element we define the matrices

_{k}being the FE stiffness matrix of Y and H = Σ H

_{k}being the matrix comprising the self-equilibrated pseudo load vectors. The components of these pseudo load vectors are determined by the layout of the elastic moduli within Y; they vanish if the layout of the elastic properties is homogeneous within Y.

## 6. Recovery of the Underlying Microstructures by the Combinatorial Homogenization-Based Process

_{0}(x), μ

_{0}(x) and k

_{1}(x), μ

_{1}(x) we state the problem of recovery the two-phase microstructure within Y(x) such that the homogenized tensor

**C**

^{H}is isotropic of moduli k

^{H}(x), μ

^{H}(x), coinciding with the moduli k*(x), μ*(x). Let us note that the moduli k

_{0}(x), μ

_{0}(x) and k

_{1}(x), μ

_{1}(x) must be appropriately chosen once and then should serve for each x. One of the choices is k

_{0}(x) = 0, μ

_{0}(x) = 0, which means that the problem reduces to finding the layout of the isotropic material of moduli k

_{1}(x), μ

_{1}(x) within Y(x) to achieve given moduli k*(x), μ*(x) by the homogenization method.

_{0}(x), μ

_{0}(x) and k

_{1}(x), μ

_{1}(x) given a priori. The unknown is the layout of these phases within the cell Y(x) such that the resulting homogenized tensor

**C**

^{H}is of cubic symmetry and is characterized by the moduli k

^{H}(x), μ

^{H}(x), α

^{H}(x) coinciding with k*(x), μ*(x), α*(x). The final computational results will refer to the case of the material M0 being a void.

^{I}cells as hexagons constructed by two subsequent rotations of a basic material domain by the angle 120° (see Figure 9). The periodicity assumptions are imposed by identifying the degrees of freedom at the opposite sides of the hexagon. Irrespective of the properties of the rotated part, the resulting homogenized tensor is characterized by all conditions of isotropy. This method has been proposed in [64].

^{C}.

^{H}, c

^{H}are almost equal to a*, c* (found by the CMD method), while the additional condition of b* being zero will be omitted. Nevertheless some microstructures appear to have very small b*, which will be discussed in Section 7.2.

## 7. Analysis of the Ranges of the Effective Moduli of the Isotropic and Cubic-Symmetry Composites. Selecting Microstructures of Extremal Properties

_{0}, μ

_{0}) = (k

_{1}, μ

_{1}·10

^{−9}). The isotropic material phase M1 is characterized by (k

_{1}, μ

_{1}) = (0.6667, 0.4). Then the Young’s modulus and the Poisson ratio are (E, ν) = (1.0, 0.25); the moduli are scaled by the reference modulus E

_{0}= 11.25.

^{I}and Y

^{C}are divided into sixteen eight-node elements (see Figure 9 and Figure 10). To each element, either the material phase M1 (black) or the void phase M0 (white) is assigned. For each possible non-degenerated layouts of the phases, the effective moduli (k

^{H}, μ

^{H}) or (k

^{H}, μ

^{H}, α

^{H}) are computed using Equation (20). The volume fraction of the material is defined as ρ = (area of material elements or of the phase M1)/|Y|.

#### 7.1. The Isotropic Composites of Negative Poisson’s Ratio

^{H}, μ

^{H}) constitute a cloud of points (see Figure 12). The dashed red line k = μ separates the composites of the positive and negative Poisson ratio. All results lie within the Cherkaev–Gibiansky bounds (see [66]) being tighter than the Hashin–Shtrikman bounds.

^{H}. The microstructures of ν

^{H}close to −1 are characterized by the modulus k

^{H}close to zero (cf. Figure 13).

#### 7.2. The Cubic Composites of Extremal Properties

^{H}, μ

^{H}, α

^{H}) constitute a dense cloud (see Figure 14a). Upon projecting them on the plane α

^{H}= 1, we obtain a plane cloud of results (see Figure 14b). The plane k = μ separates the composites of the positive and negative Poisson ratio. All results lie within the Cherkaev–Gibiansky bounds [66], referring to the isotropic composites.

^{H}is close to 1 and other microstructures for which ν

^{H}is close to −1. Some of them are characterized by a very small modulus b

^{H}, but this is not a rule (cf. Figure 15).

## 8. Recovery of the Optimal Microstructure Corresponding to the IMD and CMD Designs

^{H}, μ

^{H}) characterizing the given layout of the phases and the tessellated patterns. The positions of the pairs of the central nodes of the RVEs are shown within the design domain, see Figure 17.

^{H}, μ

^{H}, α

^{H}) is set up, disclosing that the modulus α

^{H}usually deviates from α* = 0. The results show that at many points the optimal Poisson’s ratio assumes negative values and can reach the lower bound −1.

## 9. Final Remarks

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**The plate supported at two bottom corner nodes and loaded with one vertical force applied to the upper face (

**left**); and modeled by a smoothing weight function (

**right**).

**Figure 2.**First six figures: Volume-render (

**left**); and contour (

**right**) views of the optimal and relative (i.e., divided by E

_{0}) layouts of (

**a**,

**b**) bulk k*; (

**c**,

**d**) shear μ* and (

**e**,

**f**) Young’s E* moduli, respectively. The last two figures: (

**g**) Volume-render and (

**h**) contours views of optimal distribution of the optimal Poisson’s ratio ν*.

**Figure 3.**Two first figures: (

**a**) The height field view and (

**b**) the scatter plot view of the distribution of the optimal Poisson ratio ν*; the negative values are shown in blue color and the positive values are shown in red. The third figure (

**c**) shows the domains where the optimal Poisson ratio ν* assumes positive and negative values.

**Figure 5.**(

**a**) Volume-render and (

**b**) contours views of the optimal and relative (i.e., divided by E

_{0}) layouts of a* and c* moduli, respectively.

**Figure 6.**(

**a**) Volume-render and (

**b**) contour views of optimal distribution of the optimal Poisson’s ratio ν*.

**Figure 8.**Distribution of optimal pairs of points (k*, μ*) (blue triangles) and (a*/2, c*/2) (red diamonds) for IMD and CMD method, respectively.

**Figure 9.**Basic structure and the cell of periodicity Y

^{I}exhibiting homogenized exact isotropic properties of a periodic body.

**Figure 10.**Basic structure and cells of periodicity Y

^{C}exhibiting homogenized exact cubic symmetry properties of a periodic body.

**Figure 11.**Microstructure with cubic symmetry properties of

**C**proposed in [65]. Material 2 is isotropic, white is void. Courtesy of the authors. (Reproduced with permission from Engineering Transaction; published by Institute of Fundamental Technology Research Polish Academy of Sciences, Warsaw, National Engineering School of Metz, and Poznan University of Technology, 2017.)

**Figure 12.**The sets of points of coordinates (k

^{H}, μ

^{H}) corresponding to the isotropic periodic composites of the two-phase cells (see Figure 9) for all possible black and white non-degenerated layouts for the given mesh. The axes are scaled differently, thus changing the Cherkaev–Gibiansky bounds into squares.

**Figure 13.**Selected auxetic 1st rank isotropic microstructures and the effective moduli. Cells of periodicity, and the tessellated patterns.

**Figure 14.**(

**a**) The sets of points of coordinates (k

^{H}, μ

^{H}, α

^{H}) corresponding to the periodic composites of cubic symmetry (see the two-phase cell shown in Figure 10). The plane k = μ is marked by a dashed red line (note that the plane k = μ is visible as a line) Black points on the plane α = 0 refer to microstructures from Figure 11; (

**b**) The set of points (k

^{H}, μ

^{H}, α

^{H}) projected onto α

^{H}= 1 plane.

**Figure 15.**Selected auxetic 1st rank cubic microstructures and the effective moduli. Cells of periodicity, and images upon their tessellated patterns.

**Figure 17.**Optimal isotropic microstructures approximating the IMD results with the error (Equation (28)) less than 0.001.

**Figure 18.**Optimal cubic microstructures approximating the CMD results with the error (Equation (27)) less than 0.001.

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**MDPI and ACS Style**

Czarnecki, S.; Łukasiak, T.; Lewiński, T.
The Isotropic and Cubic Material Designs. Recovery of the Underlying Microstructures Appearing in the Least Compliant Continuum Bodies. *Materials* **2017**, *10*, 1137.
https://doi.org/10.3390/ma10101137

**AMA Style**

Czarnecki S, Łukasiak T, Lewiński T.
The Isotropic and Cubic Material Designs. Recovery of the Underlying Microstructures Appearing in the Least Compliant Continuum Bodies. *Materials*. 2017; 10(10):1137.
https://doi.org/10.3390/ma10101137

**Chicago/Turabian Style**

Czarnecki, Sławomir, Tomasz Łukasiak, and Tomasz Lewiński.
2017. "The Isotropic and Cubic Material Designs. Recovery of the Underlying Microstructures Appearing in the Least Compliant Continuum Bodies" *Materials* 10, no. 10: 1137.
https://doi.org/10.3390/ma10101137