A nonlinear two-dimensional (2D) continuum with a latent internal structure is introduced as a coarse model of a plane network of beams which, in turn, is assumed as a model of a pantographic structure made up by two families of equispaced beams, superimposed and connected by pivots. The deformation measures of the beams of the network and that of the 2D body are introduced and the former are expressed in terms of the latter by making some kinematical assumptions. The expressions for the strain and kinetic energy densities of the network are then introduced and given in terms of the kinematic quantities of the 2D continuum. To account for the modelling abilities of the 2D continuum in the linear range, the eigenmode and eigenfrequencies of a given specimen are determined. The buckling and post-buckling behaviour of the same specimen, subjected to two different loading conditions are analysed as tests in the nonlinear range. The problems have been solved numerically by means of the COMSOL Multiphysics finite element software.

1. Introduction

The type of pantographic structure considered in this paper can be described as follows. Consider a hyperelastic beam of uniform cross-section orthogonal to a straight axis. Take a set of these beams so that the cross sections have the same orientation, and the axes are parallel and lie on a plane at a constant pitch. Then duplicate the set and rotate the new plane by π/2. Superpose the two planes and connect the beams where they cross each other by small cylinders that we will call ‘pivots’. Finally, cut a rectangle whose sides intersect the beam axes at π/4 (figure 1).

Figure 1. Sample of a pantographic sheet made of polyamide by three-dimensional printing. (Online version in colour.)

Such structures, which we will refer to as ‘pantographic’, have been largely analysed in a number of preceding papers [1–12] where only specimens subject to planar motions have been studied.

The aim of this paper is to take into account also large, out-of-plane, motions.

This point can be addressed in a number of ways. One resides in assuming that each element of the structure is a three-dimensional nonlinear hyperelastic Cauchy body [13]. Besides, one can assume that the beams are modelled as one-dimensional continua (Euler, Timoshenko or more sophisticated beams [14]). In both cases, the solution even of simple problems requires a numerical approach that can lead to very cumbersome calculations, giving results that, due to their intrinsic complexity, may prove to be scarcely reliable. This is why it is necessary look for alternative approaches. The authors maintain that a way to overcome such difficulties can be found by modelling the pantographic structure as a two-dimensional (2D) continuum endowed with a suitable internal structure.

As will be seen in the following, although the proposed model will not be able to describe the mechanical behaviour of the pantograph in a detailed way, it gives a fair account of many relevant phenomena both qualitatively and quantitatively.

The procedure adopted takes into account some results presented in [15,16] and can be arranged in the following steps.

At first, the pantographic sheet is viewed as a beam network. This means that each beam is modelled as a nonlinear hyperelastic body whose motion is described by the placement of a line (the beam axis) and a rotation field that describes the placement of the beam cross sections (assumed to behave as 2D rigid bodies). In addition, the assumption that the pivots can undergo only torsional deformation, is made. The distance between two adjacent pivots is assumed as the ‘pitch’ of the pantographic structure and is considered as a vanishing quantity.

The continuum chosen for describing the coarse behaviour of the pantographic sheet is a 2D (initially flat) body endowed with an internal structure characterized by two orthogonal tensor fields whose values at a point where two beams meet give account of their cross-section rotations.

The equivalence between the 2D model and the beam network is obtained by requiring that the strain energy density be the same for some selected classes of regular motions.

The assumption that the pivots can only undergo torsional deformation is shown to result in some internal constraints for the 2D model that allow both the rotation fields to be expressed terms of the transplacement of the 2D surface. The internal microstructure then becomes ‘latent’ (see [17]) with the consequence that the 2D continuum results are of the second gradient type [18–28]. Using the reduced kinematics, an expression of the kinetic energy of the 2D body is derived from the one of the network by following an equivalence procedure analogous to that used for the strain energy described before. If compared with a classical plate model, the proposed 2D continuum is certainly more complex. This is due to the fact that it is aimed to describe, at least in an approximate way, the mechanical behaviour of the beams that make up the pantographic sheet. To do that, the introduction of a microstructure richer than that used for a standard plate is compulsory. Of course, the larger the number of the parameters introduced in the microstructure, the greater is the accuracy in the description of the behaviour of the beams. In this sense, the proposed model can be certainly enriched in order to improve its simulation ability. Obviously, this is a cost that deserves to be carefully evaluated. It is worth adding that the proposed continuous model can be properly employed not only to describe the synthetic metamaterial we have in mind but also to describe biological phenomena characterized by the presence of a lattice microstructure of fibres as in fibre-reinforced materials [29,30], or trabecular bones [31,32].

To test the model in the linear range the natural frequencies of a specimen have been evaluated. As a sample of nonlinear analysis, the buckling and post-buckling behaviour of the same specimen subjected to a uniform axial displacement along a part of the boundary has been analysed. Besides, the case of a tangential uniform displacement along the same side has been studied. We remark that the nonlinear analysis can be extended also to dynamic problems, and in particular to the wave propagation. In fact, nonlinearity associated with gradient elasticity can allow to capture, as it is well known, very interesting phenomena (see e.g. [33]). However, this kind of analysis is beyond the scope of this paper. All the numerical analyses have been made by means of the COMSOL Multiphysics^® finite-element software.

2. Planar beams’ network

As a first step, the pantographic structure described in the Introduction is modelled as a planar network of beams. This means that, in the reference configuration, the straight axes of the two families of beams are assumed to lie in the same plane. They are denoted by $^{1 j} C$ with j={1,…,b}, and $^{2 i} C$ with i={1,…,a}, where the first left superscript (whose values are 1, 2) denotes the family and second the element of that family.

Now we make the following assumptions on the constraints to the network.

Assumption 2.1.

Where the beams’ axes cross each other (joints), the points belonging to the two curves must share the same position in any configuration.

Assumption 2.2.

Each beam is shear undeformable, that is, the cross sections remain orthogonal to the tangent vector at each point of the axis in any configuration.

Assumption 2.3.

At the joints the cross sections of the two beams are rigidly connected apart from the rotation about the line in which they intersect; in view of assumption 2.2, they remain orthogonal to the tangent plane of the network.

On the reference network plane we define a Cartesian orthogonal coordinate system (X₁,X₂) whose associated base of unit vectors is (D₁,D₂), which are chosen to be directed as $^{1 j} C$ and $^{2 i} C$ , respectively. The unit vector normal to the network plane is D₃=D₁×D₂. We assume that the spacing between the beams of the network is the same for both the families and will be denoted by ℓ, so that:

— the coordinates of a point $^{1 j} X \in^{1 j} C$ are (X₁,jℓ);
— the coordinates of a point $^{2 i} X \in^{2 i} C$ are (iℓ,X₂).

In addition, we put

— ^αE as the unit tangent vector directed along $^{α h} C$ , where α=(1,2) stands for the specific family and h=(j,i) identifies the beam in the family;
— N=¹E×²E;
— ^αM=N×^αE,

where ^αM and N are assumed to be directed along the principal inertial axes of the beam cross sections. Besides, we assume that ¹E, ¹M coincide with D₁,D₂, respectively; ²E, ²M coincide with D₂,D₁, respectively, while N = D₃.

A deformed configuration of the beam axes ^αh𝒸 is described by the transplacements

\begin{aligned} ^{α h} χ :^{α h} C & \to^{α h} 𝒸 \\ and^{α h} X & \mapsto^{α h} x \end{aligned}}

2.1

and, in view of the assumption 2.1, the following conditions must be satisfied: denoting by ^ijX the joint that in the reference configuration occupies the position with coordinates (iℓ,jℓ),

^{1 j} χ (^{i j} X) =^{2 i} χ (^{i j} X) =^{i j} x .

2.2

Now, we define by

^{α h} t = \frac{\partial^{α h} χ}{\partial X_{α}},

2.3

the tangent vector to the curve ^αh𝒸 and by

^{α h} u =^{α h} χ (^{α h} X) -^{α h} X,

2.4

the displacement of the beam axes.

The rotation of the cross sections is given by the orthogonal tensor fields

^{α h} R : (^{α} E,^{α} M, N) \mapsto (^{α h} e,^{α h} m,^{α h} n),

2.5

where

^{α h} e = \frac{^{α h} t}{∥^{α h} t ∥}

2.6

is the unit tangent vector along the deformed axis ^αh𝒸, and Span{^αhn,^αhm} define the new orientation of the cross sections [34–38].

In addition, in view of the assumption 2.2, the following conditions must be satisfied:

\begin{aligned} ^{1 j} n (^{i j} X) =^{2 i} n (^{i j} X) =:^{i j} n, \end{aligned}

2.7

\begin{aligned} ^{i j} n = \frac{^{1 j} e \times^{2 i} e}{∥^{1 j} e \times^{2 i} e ∥} \end{aligned}

2.8

\begin{aligned} and^{α h} m =^{i j} n \times^{α h} e . \end{aligned}

2.9

The local deformation measures for the beam ^αh() are assumed to be

^{α h} ε = ∥^{α h} t ∥ - 1

2.10

and the components of the skew tensor ^αhW(X_α)=^αhR^T(X_α) ^αhR′(X_α),¹ which can be made explicit as follows:

^{α h} W =^{α h} κ_{1}^{α} M \land N +^{α h} κ_{2} N \land^{α} E +^{α h} κ_{3}^{α} E \land^{α} M .

3. A two-dimensional continuum with (latent) internal microstructure

Let us consider now a 2D continuum embedded in a three-dimensional space as a surface at each point of which two three-dimensional rigid bodies are attached.

We assume that, in the reference configuration, the surface is a rectangle $R$ and the orientation of each one of the two rigid bodies are the same at each point of the 2D continuum (figure 2).

Figure 2. Scheme of the 2D continuum with local microstructure in the reference configuration and in the present one. (Online version in colour.)

In the plane of the rectangle, a Cartesian coordinate system is introduced with the associated orthonormal base (D₁,D₂,D₃=D₁×D₂). In this way, a point $X \in R$ will have the coordinates (X₁,X₂).

A deformed configuration can be identified by a transplacement

\begin{aligned} χ : R & \to S \\ and X & \mapsto x \end{aligned}}

3.1

and two orthogonal tensor fields

\begin{aligned} ^{1} R : (D_{1}, D_{2}, D_{3}) & \mapsto (^{1} a_{1},^{1} a_{2},^{1} a_{3}) \\ and^{2} R : (D_{1}, D_{2}, D_{3}) & \mapsto (^{2} a_{1},^{2} a_{2},^{2} a_{3}), \end{aligned}}

3.2

where (¹a₁,¹a₂,¹a₃) and (²a₁,²a₂,²a₃) give the orientation of the microstructure in the deformed configuration as an effect of the rotations ¹R and ²R. We remark that the proposed continuum differs from a typical micropolar (or Cosserat-type) continuum as, e.g. proposed in [39,40], because of the presence of two distinct rotations.

Finally, we will call

u (X) = x - X = u_{1} (X) D_{1} + u_{2} (X) D_{2} + u_{3} (X) D_{3},

3.3

the displacement field defined on

R

Now, we define by

^{α} t = \frac{\partial χ}{\partial X_{α}} α = 1, 2,

3.4

the vectors tangent to the deformed coordinate lines, while

^{α} e = \frac{^{α} t}{∥^{α} t ∥}

3.5

are the corresponding unitary tangent vectors. In addition, we define by

n = \frac{^{1} e \times^{2} e}{∥^{1} e \times^{2} e ∥}

3.6

the unit normal to the deformed surface.

The local deformation measures of the 2D continuum are assumed to be

^{α} ε = ∥^{α} t ∥ - 1,^{1} e \cdot^{2} e = \sin γ

3.7

and

\begin{aligned} ^{1} W & :=^{1} R^{T} \frac{\partial^{1} R}{\partial X_{1}} =^{1} κ_{1} D_{2} \land D_{3} +^{1} κ_{2} D_{3} \land D_{1} +^{1} κ_{3} D_{1} \land D_{2} \\ and^{2} W & :=^{2} R^{T} \frac{\partial^{2} R}{\partial X_{2}} =^{2} κ_{1} D_{2} \land D_{3} +^{2} κ_{2} D_{3} \land D_{1} +^{2} κ_{3} D_{1} \land D_{2} . \end{aligned}}

3.8

Note that ^αε and γ represent the ‘stretch’ of the surface, while the six components of ¹W and ²W are ‘curvatures’ evaluated along the coordinate lines.

Now, we assume that the following constraints hold at each point $X \in R$ :

Constraint (1) ¹a₁=¹e and ²a₁=²e.
Constraint (2) ¹a₃=²a₃=n.
Constraint (3) ¹a₂=n×¹e and ²a₂=n×²e.

Owing to these constraints, the rotations ¹R,²R, and consequently the orientation of the two rigid bodies that describe the local microstructure at X, are completely determined by the present shape of the coordinate lines passing through the point X. This means that the local structure becomes latent in the sense of [17].

It is also worth noting that, due to the constraints introduced, γ results to be the amplitude of the relative rotation between the two rigid bodies that describe the microstructure at X.

The strain measures, written in terms of the displacement components, result to be

\begin{aligned} ^{1} κ_{1} & = - \frac{n \cdot g_{1} - \sin γ n \cdot c_{1}}{∥^{1} e \times^{2} e ∥}^{1} κ_{2} = n \cdot c_{1}^{1} κ_{3} = -^{1} m \cdot c_{1} \\ and^{2} κ_{1} & = \frac{n \cdot g_{2} - \sin γ n \cdot c_{2}}{∥^{1} e \times^{2} e ∥}^{2} κ_{2} = n \cdot c_{2}^{2} κ_{3} = -^{2} m \cdot c_{2}, \end{aligned}}

3.9

where ^αm=n×^αe. The expressions for c₁,c₂,g₁,g₂ are given in the appendix.

Note that

^{α} κ_{1} = \frac{\partial n}{\partial X_{α}} \cdot^{α} m,^{α} κ_{2} = \frac{\partial^{α} e}{\partial X_{α}} \cdot n and^{α} κ_{3} = \frac{\partial^{α} m}{\partial X_{α}} \cdot^{α} e,

3.10

as the frames that describe the cross sections can be interpreted as Darboux frames. In view of the assumed constraints, ^ακ₁, ^ακ₂, ^ακ₃ are the geodesic torsion, the normal curvature and the geodesic curvature multiplied by ∥^αt∥, respectively, as X_α is not a unitary speed parametrization.

Expression (3.9) can be obtained from expressions (3.8) following the procedures sketched in the appendix.

4. The Planar beams’ network as a two-dimensional continuum

In this section, the 2D model introduced in §3 is used in order to give a coarse description of the mechanical behaviour of the network described in §2.

We make the following assumptions:

Assumption 4.1.

The network’s joint ^ijX whose coordinates in the reference position are (iℓ,jℓ) is identified with the point X of the continuum that share the same reference coordinates [41].

Assumption 4.2.

The axes of the two beams of the network that lie along the coordinate lines through ^ijX are identified with the corresponding coordinate lines through the point X in the 2D continuum; the sections attached to the beam axes along X₁,X₂ are identified with the rigid bodies of the 2D continuum denoted by the subscripts 1,2, respectively.

Using the definitions introduced in the §§2 and 3, the local deformation measures of the 2D continuum are assumed to be

^{α} ε =^{α h} ε,^{1} e \cdot^{2} e =^{1 j} e \cdot^{2 i} e = \sin γ,

4.1

where ^αt are the vectors defined in (3.4), and

^{1} W =^{1 h} W (X_{1}) and^{2} W =^{2 h} W (X_{2}),

4.2

that is, in components

^{α} κ_{1} =^{α h} κ_{1},^{α} κ_{2} =^{α h} κ_{2} and^{α} κ_{3} =^{α h} κ_{3} .

4.3

Note that, in view of (4.1)₂, the relative rotation of the two rigid bodies attached at a point of the 2D body is identified with the relative rotation of the tangents to the two beams of the network crossing at that point which, in turn, describes the torsion of a pivot.

Owing to the constraints introduced, the shear deformation in the beams and pivots has been disregarded. This assumption seems to be fair, generally speaking. However, when one requires a metamaterial in which the sliding between the two families of beams should be allowed, at least the shear deformation of the pivots must be taken into account.

5. Energy and dynamical equations

Let us consider now a representative elementary volume (REV) which is a square of edge length ℓ centred at a network’s joint ^ijX.

The strain energy density of the 2D continuum is assumed to be

π = \frac{1}{2} [K_{e} (^{1} ε^{2} +^{2} ε^{2}) + K_{s} γ^{2} + K_{t} (^{1} κ_{1}^{2} +^{2} κ_{1}^{2}) + K_{n} (^{1} κ_{2}^{2} +^{2} κ_{2}^{2}) + K_{g} (^{1} κ_{3}^{2} +^{2} κ_{3}^{2})],

5.1

where

K_{e} = \frac{Y A}{ℓ}, K_{s} = \frac{G J_{p}}{h_{p} ℓ^{2}}, K_{t} = \frac{G J_{t}}{ℓ}, K_{n} = \frac{Y J_{f 1}}{ℓ} and K_{g} = \frac{Y J_{f 2}}{ℓ}

5.2

G being the shear modulus, J_p the torsional inertia of the pivot and J_t,J_f1,J_f2 the torsional and flexural inertia of the beams’ cross sections (see [42–44] for an identification method suitable for the polyamide).

Note that K_e, K_t, K_n and K_g are the stiffnesses related to the elongation, twisting, normal and geodesic bending of the beams embedded in the REV, while K_s is the shear stiffness between beams belonging to the two different families.

The angular velocity of the beam cross sections is described by the skew tensor field

^{α} V := {^{α} R}^{T}^{α} \dot{R} =^{α} ω_{1} D_{2} \land D_{3} +^{α} ω_{2} D_{3} \land D_{1} +^{α} ω_{3} D_{1} \land D_{2},

where the dot denotes the time derivative.

The kinetic energy density is

τ = \frac{1}{2} ϱ_{s} ({\dot{u}}_{1}^{2} + {\dot{u}}_{2}^{2} + {\dot{u}}_{3}^{2}) + \frac{1}{2} [J_{1} ({^{1} ω_{1}}^{2} + {^{2} ω_{1}}^{2}) + J_{2} ({^{1} ω_{2}}^{2} + {^{2} ω_{2}}^{2}) + J_{3} ({^{1} ω_{3}}^{2} + {^{2} ω_{3}}^{2})],

5.3

where

J_{1} = \frac{ϱ J_{t}}{ℓ}, J_{2} = \frac{ϱ J_{f 1}}{ℓ} and J_{3} = \frac{ϱ J_{f 2}}{ℓ}

5.4

ϱ_s being the apparent mass density per unit area and ϱ the apparent mass density.

Note that J₂ and J₃ are the moments of inertia of the beam cross section with respect to the principal inertia axes, that is ^αM, N (see §2) while J₁ is the polar moment of inertia evaluated with respect to the centre of the section.

The virtual work equation runs as follows:

δ W_{(e)} + δ W_{(i)} + δ W_{(s)} + δ W_{(l)} = 0,

5.5

where the internal elastic work done on a given subdomain

B

R

δ W_{(e)} = - \int_{B} δ π d S

5.6

and the inertial work can be expressed by

δ W_{(i)} = \int_{B} δ τ d S,

5.7

or, after an integration by parts, in an explicit form as reported in the appendix.

The external forces acting on $B$ are the surface forces, exerted by systems located outside $R$ which give the virtual work

δ W_{(s)} = \int_{B} F_{i} δ u_{i} d S

5.8

and the contact forces acting on

B

by the parts of

R

located outside

B

which yield the virtual work

δ W_{(l)} = \int_{\partial B} f_{i} δ u_{i} d l .

5.9

We remark that because of the nature of the equation (5.1), which involves the second gradient of the independent displacement field u, the proposed 2D continuum is also able to sustain double forces and corner forces. In what follows, for the sake of simplicity, we impose only geometric boundary conditions.

6. Numerical examples

Let us consider a rectangular specimen made up by two families of beams that in the reference configuration are orthogonal (figure 1) and directed along D₁ and D₂, respectively. The edges of the specimen are directed along B₁ and B₂ obtained by a counterclockwise rotation of π/4 of D₁ and D₂. Their lengths are 21 cm and 7 cm along B₁ and B₂, respectively.

The beams have both rectangular cross sections of area A and dimensions 1×1.6 mm, while the pivots are cylinders 1 mm long and with circular cross sections of diameter 0.9 mm. The pitch between two adjacent beams is $ℓ = 7 / \sqrt{2} mm$ . The beams and the pivots are made of polyamide. The material is assumed to be linear hyperelastic and isotropic with Young’s modulus Y =1600 MPa and Poisson’s coefficient ν=0.3.

The specimen is modelled by the 2D body described in §4 that in the reference configuration is a rectangle with the same dimensions of the specimen, i.e. 7 and 21 cm.

As a first example, we consider the case in which the specimen is restrained along the shortest sides. One of them is fixed (u=0, ¹R=I and ²R=I, where I is the identity tensor), while the other can slide in the plane of the figure in the direction orthogonal to it (u=−λB₁ with λ≥0, ¹R=I and ²R=I).

When λ reaches the value of 5.6 cm, an out-of-plane buckling is detected (figures 3 and 4a). The corresponding critical load is found to be approximately 5.8 N. The map of the strain energy density in that configuration is given in figure 4b while the contributions of the deformation terms are shown in figure 5.

Figure 3. Compressive force versus out-of-plane displacement at the central point of the sample. (Online version in colour.)

Figure 4. Specimen subjected to edge displacement along B₁. (a) Buckled shape and (b) energy density map. (Online version in colour.)

Figure 5. Energy density contributions. Extension (a), geodesic bending (b), normal bending (c), twisting (d) and shear (e) energy densities. (Online version in colour.)

Furthermore, the case in which the boundary conditions on the moving side are changed in u=−λB₂ with λ≥0, ¹R=I and ²R=I, which corresponds to a tangential uniform displacement, is examined. The first two critical values of λ are found to be approximately 2.2 and 4.7 cm (figure 6a), and the corresponding critical loads are approximately 10 and 23.5 N, respectively (figure 6b). Note that the coordinates of point P_b referred to in figure 6 are $((23 / 30 L) / \sqrt{(} 2), (l + 23 / 30 L) / \sqrt{(} 2))$ ; P_b is a point that shows an out-of-plane displacement very close to the maximum value. The first two bifurcation modes are reported in figure 7a,b, respectively. The map of the contributions to the strain energy density in the first configuration, with out-of-plane antisymmetric peaks, is given in figure 8.

Figure 6. Specimen subjected to edge displacement along B₂. (a) Out-of-plane displacement of point P_b versus imposed displacement and (b) shear force versus out-of-plane displacement of point P_b. (Online version in colour.)

Figure 7. Shear test: Buckled shapes of the first two bifurcation modes. Colours indicate values of the out-of-plane displacement. (a) First buckling mode and (b) second buckling mode. (Online version in colour.)

Figure 8. Shear test: Energy density contributions. Extension (a), geodesic bending (b), normal bending (c), twisting (d) and shear (e) energy densities. (Online version in colour.)

To test the dynamical behaviour of the specimen, the vibrations around two different configurations have been examined. They are obtained by putting λ=0 and λ=49 mm in the first example. We will refer to them as reference and prestretched, respectively. Note that the value λ=49 mm is 70% of the side length. The first twelve vibration modes have been determined for both the cases. To increase the mass of the beams, their sections have been given the following dimensions: 2.5×1.6 mm.

Figures 9–12 show the modal shapes associated with each one of the natural frequencies. The modal shapes are normalized to unit modal mass. The strain (figures 9 and 11) and kinetic (figures 10 and 12) energies are also reported by means of a superimposed colour map. Tables 1 and 2 summarize the eigenfrequencies for both cases analysed, emphasizing the effect of the section inertia, which at low frequencies can be neglected even if there are some small effects when the twisting energy involved is significant.

Figure 9. (a,b) Mode shapes for the reference configuration. Strain energy. (Online version in colour.)

Figure 10. (a,b) Mode shapes for the reference configuration. Kinetic energy. (Online version in colour.)

Figure 11. (a,b) Mode shapes for the prestretched configuration. Strain energy. (Online version in colour.)

Figure 12. (a,b) Mode shapes for the prestretched configuration. Kinetic energy. (Online version in colour.)

Table 1.Reference configuration: eigenfrequencies, rad s⁻¹.
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mode #	with cross-section inertia	without cross-section inertia	difference %
1	593.93	594.10	0.02
2	1393.3	1393.4	0.01
3	1522.2	1525.1	0.19
4	1615.0	1615.1	0.01
5	1636.6	1637.8	0.07
6	3192.0	3199.6	0.24
7	3207.4	3212.6	0.16
8	3596.6	3597.4	0.02
9	3630.9	3631.7	0.02
10	5142.0	5158.3	0.32
11	5298.2	5313.0	0.28
12	6046.7	6091.0	0.73

Table 2.Prestretched configuration: eigenfrequencies, rad s⁻¹.
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mode #	with cross-section inertia	without cross-section inertia	difference %
1	720.22	720.30	0.01
2	1597.3	1611.4	0.88
3	1671.0	1671.0	0.00
4	1774.5	1775.2	0.04
5	3044.3	3044.6	0.01
6	3163.5	3166.2	0.09
7	3899.7	3924.7	0.64
8	4189.7	4190.4	0.02
9	4744.6	4745.7	0.02
10	4944.8	4952.0	0.15
11	5889.7	6029.4	2.37
12	6818.8	6821.1	0.03

The cases examined have been solved numerically by using the ‘Weak Form PDE’ feature of COMSOL Multiphysics. This means that equation (5.5) has been explicitly coded, while the finite-element discretization has been made using the Argyris elements characterized by 21 d.f. already implemented in COMSOL Multiphysics. The choice of Argyris elements has been done in order to guarantee C¹ continuity. Indeed, such elements can be used to properly approximate functions in the Sobolev space H², as those involved in our problem characterized by the presence of second spatial derivatives of the transplacement. To solve the problem in a more efficient way, ad hoc codes can be implemented by using an isogeometric formulation [45–47] or discrete Hencky-type systems based on the geometry of the microstructure as done in [48–50]. It is worth mentioning also the particular adaptation of standard finite-element model for second gradient materials (see e.g. [51,52]).

7. Conclusion

Pantographic structures made up by two families of equispaced beams, superimposed and connected by pivots, have been modelled at first as a plane network of beams, and then as a 2D continuum with a suitable internal structure. The internal structure has been made latent by imposing some internal constraints. A procedure aimed to express the kinematic variables of the beam network in terms of the ones of the 2D body that we have considered as coarse descriptors has been described. Given an expression of the strain and kinetic energy densities for the beams, the corresponding energies of the network have been derived. By introducing the coarse descriptors in the preceding expressions, the energies of the 2D body have been constructed. The model obtained has been used for the analysis of the mechanical behaviour of a given specimen. The COMSOL Multiphysics finite-element software has been used in order to determine the eigenmode and eigenfrequencies of the specimen. For two different loading conditions, the nonlinear behaviour of the specimen has been determined, and the uprising of bifurcation detected and examined. It is worth mentioning that some laboratory tests on the specimen studied in this paper have been planned and will be shortly performed. They are intended to be a benchmark for the results obtained numerically.

Data accessibility

This article has no additional data.

Author contributions

All authors contributed equally to this paper.

Competing interests

The authors declare that they have no competing interests.

Funding statement

No funding is declared concerning the work on the article.

Appendix A

The expressions for the variables c₁,c₂,g₁,g₂ appearing in the deformation measures (3.9) are

\begin{aligned} c_{1} & = \frac{(\nabla \nabla χ) D_{1} \otimes D_{1}}{∥ \nabla χ D_{1} ∥} c_{2} = \frac{(\nabla \nabla χ) D_{2} \otimes D_{2}}{∥ \nabla χ D_{2} ∥} \\ g_{1} & = \frac{(\nabla \nabla χ) D_{2} \otimes D_{1}}{∥ \nabla χ D_{2} ∥} g_{2} = \frac{(\nabla \nabla χ) D_{1} \otimes D_{2}}{∥ \nabla χ D_{1} ∥} . \end{aligned}

The curvatures (3.9) can be obtained from equations (3.8) following the procedure sketched below:

\begin{aligned} ^{α} R : (D_{1}, D_{2}, D_{3}) & \mapsto (^{α} a_{1},^{α} a_{2},^{α} a_{3}) \\ {(^{α} R D_{i})}^{'} = {^{α} R}^{'} D_{i} = {^{α} a_{i}}^{'} \\ {^{α} a_{i}}^{'} \cdot^{α} a_{j} = {^{α} R}^{'} D_{i} \cdot^{α} R D_{j} = (^{α} W D_{i}) \cdot D_{j} = ε_{s i j}^{α} κ_{s} . \end{aligned}

A 1

The explicit expression for the inertial virtual work of equation (5.7) is

\begin{aligned} δ W_{(i)} & = \int_{B} - ϱ_{s} ({\ddot{u}}_{1} δ u_{1} + {\ddot{u}}_{2} δ u_{2} + {\ddot{u}}_{3} δ u_{3}) \\ + \sum_{α} {- J_{2} [(n \cdot^{α} \ddot{e}) n + (\dot{n} \cdot^{α} \dot{e}) n + (n \cdot^{α} \dot{e}) \dot{n}] + J_{3} (^{α} e \cdot^{α} \dot{m})^{α} \dot{m}} \cdot δ^{α} e \\ + \sum_{α} {- J_{3} [(^{α} e \cdot^{α} \ddot{m})^{α} e + (^{α} \dot{e} \cdot^{α} \dot{m})^{α} e + (^{α} e \cdot^{α} \dot{m})^{α} \dot{e}] + J_{1} (^{α} m \cdot \dot{n}) \dot{n}} \cdot δ^{α} m \\ + \sum_{α} {- J_{1} [(^{α} m \cdot \ddot{n})^{α} m + (^{α} \dot{m} \cdot \dot{n})^{α} m + (^{α} m \cdot \dot{n})^{α} \dot{m}] + J_{2} (n \cdot^{α} \dot{e})^{α} \dot{e}} \cdot δ n d S . \end{aligned}

A 2

Footnotes

1 The prime symbol denotes differentiation of a function with respect to its own argument.

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This Issue

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences cover image

November 2017

Volume 473Issue 2207

Article Information

DOI:https://doi.org/10.1098/rspa.2017.0636
Published by:Royal Society
Online ISSN:1471-2946

History:

Manuscript received07/09/2017
Manuscript accepted16/10/2017
Published online15/11/2017
Published in print30/11/2017

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Continuum modelling of pantographic sheets for out-of-plane bifurcation and vibrational analysis

Abstract

1. Introduction

2. Planar beams’ network

Assumption 2.1.

Assumption 2.2.

Assumption 2.3.

3. A two-dimensional continuum with (latent) internal microstructure

4. The Planar beams’ network as a two-dimensional continuum

Assumption 4.1.

Assumption 4.2.

5. Energy and dynamical equations

6. Numerical examples

7. Conclusion

Data accessibility

Author contributions

Competing interests

Funding statement

Appendix A

Footnotes

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