Attila Kocsis

Noël Challamel

György Károlyi

Discrete and nonlocal models of Engesser and Haringx elastica

Keywords: Lattice, Discrete elastica, Shear effect, Nonlocal beam mechanics, Timoshenko beam elements, Scale effect, Buckling, Bifurcation, Post-buckling, Finite Difference Methods 2

In this paper a generalized discrete elastica including both bending and shear elastic interactions is developed and its possible link with nonlocal beam continua is revealed. This lattice system can be viewed as the generalization of the Hencky bar-chain model, which can be retrieved in the case of infinite shear stiffness. The shear contribution in the discrete elastica is introduced by following the approach of Engesser (normal and shear forces are aligned with and perpendicular to the link axis, respectively) and that of Haringx (shear force is parallel to end section of links), both supported by physical arguments. The nonlinear analysis of the shearable-bendable discrete elastica under axial load is accomplished. Buckling and post-buckling of the lattice systems are analysed in a geometrically exact framework. The buckling loads of both the discrete Engesser and Haringx elastica are analytically calculated, and the post-buckling behavior is numerically studied for large displacement. Nonlocal Timoshenko-type beam models, including both bending and shear stiffness, are then built from the continualization of the discrete systems. Analytical solutions for the fundamental buckling loads of the nonlocal Engesser and Haringx elastica models are given, and their first post-buckling paths are numerically computed and compared to those of the discrete Engesser and Haringx elastica. It is shown that the nonlocal Timoshenko-type beam models efficiently capture the scale effects associated with the shearable-bendable discrete elastica.

Introduction

In this paper the possible link between discrete beam systems and their continuum analogies for structural mechanics applications is investigated. One related question is whether we can build nonlocal beam models from discrete systems composed of rigid elements linked by concentrated or distributed interaction.

A pioneer model for relating discrete beam mechanics with its continuum analogue is the so-called Hencky

INTRODUCTION

2 bar-chain model, which is composed of rigid elements connected by concentrated rotational springs (Hencky [START_REF] Hencky | Über die angenäherte Lösung von Stabilitätsproblemen im Raummittels der elastischen Gelenkkette[END_REF]). Hencky [START_REF] Hencky | Über die angenäherte Lösung von Stabilitätsproblemen im Raummittels der elastischen Gelenkkette[END_REF] considered this lattice system for estimating the buckling load of the Euler-Bernoulli beam (Euler [START_REF] Euler | Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici lattissimo sensu accepti[END_REF]). Hencky [START_REF] Hencky | Über die angenäherte Lösung von Stabilitätsproblemen im Raummittels der elastischen Gelenkkette[END_REF] presented some solutions for small number of links, namely n = 2, 3 and 4, and showed the efficiency of this computation method for approximating the buckling load of the continuous

Euler-Bernoulli beam. The exact buckling load valid for any number of links was derived by Wang [START_REF] Wang ; Salvadori | Numerical computation of buckling loads by finite differences[END_REF][START_REF] Wang | Applied Elasticity[END_REF] by solving a linear, discrete boundary value problem. In fact, as pointed out by Silverman [START_REF] Silverman ; Salvadori | Numerical computation of buckling loads by finite differences[END_REF], the difference equations of Hencky bar-chain model are the central finite difference formulation of the continuous Euler-Bernoulli beam equations. This strong connection between lattice mechanics and finite difference method was also discussed by Maugin [START_REF] Maugin | Nonlinear waves in elastic crystals[END_REF] and Challamel et al [START_REF] Challamel | Revisiting finite difference and finite element methods applied to structural mechanics within enriched continua[END_REF]. It is worth mentioning that alternative lattice models with distributed microstructure can be related to the finite element method, as recently studied by Kocsis and Challamel [START_REF] Kocsis | On the post-buckling of distributed microstructured system: the Finite Element elastica[END_REF]. Going back to Hencky's model and due to the interest of the mechanics community in understanding the discretization properties of nonlinear systems, the post-buckling behavior of Hencky bar-chain model was reconsidered in the 80s independently by El Naschie et al [START_REF] El Naschie | A simple discrete element method for the initial postbuckling of elastic structures[END_REF] and Gáspár and Domokos [START_REF] Gáspár | Global investigation of discrete models of the Euler buckling problem[END_REF]. Gáspár and Domokos [START_REF] Gáspár | Global investigation of discrete models of the Euler buckling problem[END_REF], Domokos [START_REF] Domokos | Qualitative convergence in the discrete approximation of the euler problem[END_REF], Domokos and Holmes [START_REF] Domokos | Euler's problem, Euler's method, and the standard map; or, the discrete charm of buckling[END_REF] Károlyi and Domokos [START_REF] Károlyi | Symbolic dynamics of infinite depth: finding global invariants for BVPs[END_REF], and Kapsza et al [START_REF] Kapsza | Regular and random patterns in complex bifurcation diagrams[END_REF] revealed how rich and complex equilibrium states the Hencky bar-chain model can possess, contrary to the Bernoulli-Euler beam. This complex behavior is because of the discrete system is spatially chaotic as pointed out by Domokos and Holmes [START_REF] Domokos | Euler's problem, Euler's method, and the standard map; or, the discrete charm of buckling[END_REF]. In fact, the discrete Hencky bar-chain model shows spatial chaotic behavior not only if axially compressed and simply supported, but also under different (even non-conservative) loadings and boundary conditions (see Kocsis and Károlyi [START_REF] Kocsis | Buckling under nonconservative load: conservative spatial chaos[END_REF][START_REF] Kocsis | Conservative spatial chaos of buckled elastic linkages[END_REF] and Kocsis [START_REF] Kocsis | Spatial chaos in discrete mechanical systems: elastic linkages and elastic web of links[END_REF]). Discrete microstructured systems with both shear and bending effects were considered for instance by Ostoja-Starzewski [START_REF] Ostoja-Starzewski | Lattice models in micromechanics[END_REF], Zhang et al [START_REF] Zhang | Eringens small length scale coefficient for buckling of nonlocal Timoshenko beam based on microstructured beam model[END_REF] and Duan et al [START_REF] Duan | Development of analytical vibration solutions for microstructured beam model to calibrate length scale coefficient in nonlocal Timoshenko beams[END_REF] in different physical systems.

Ostoja-Starzewski presented results for dumbbell particles (vertical rigid bars) of X-braced girder geometry, whereas Zhang et al [START_REF] Zhang | Eringens small length scale coefficient for buckling of nonlocal Timoshenko beam based on microstructured beam model[END_REF] and Duan et al [START_REF] Duan | Development of analytical vibration solutions for microstructured beam model to calibrate length scale coefficient in nonlocal Timoshenko beams[END_REF] used generalized Hencky-type systems with bending-shear interaction for buckling (Zhang et al [START_REF] Zhang | Eringens small length scale coefficient for buckling of nonlocal Timoshenko beam based on microstructured beam model[END_REF]) and vibrations behavior (Duan et al [START_REF] Duan | Development of analytical vibration solutions for microstructured beam model to calibrate length scale coefficient in nonlocal Timoshenko beams[END_REF], respectively). Recently, Dell'Isola et al [START_REF] Dell'isola | Large deformations of planar extensible beams and pantographic lattices: Heuristic homogenisation, experimental and numerical examples of equilibrium[END_REF] has investigated the behavior of a pantographic sheet or pantographic lattice composed of extensionable springs with concentrated rotational stiffnesses. This two-dimensional lattice can be viewed as a possible two-dimensional generalization of Hencky's system with an extensibility condition for each cell Lattice beam models have a wide range of applications ranging from stability in construction engineering [START_REF] Hegedűs | Structural stability in engineering practice[END_REF], through computer graphics and image analysis [START_REF] Bruckstein | Discrete elastica[END_REF] to nanostructures [START_REF] Wang | Application of nonlocal continuum mechanics to static analysis of micro-and nano-structures[END_REF] and biomechanics [START_REF] Coleman | Theory of sequence-dependent DNA elasticity[END_REF]. As an example for biomechanical application, a 3D discrete model was developed by Coleman et al [START_REF] Coleman | Theory of sequence-dependent DNA elasticity[END_REF] for studying sequence-dependent DNA elasticity, while the 2D counterpart of this model, the discrete planar Cosserat rod was developed by Kocsis [START_REF] Kocsis | Buckling analysis of the discrete planar cosserat rod[END_REF].

The discrete elastica problem has been recently revisited from a nonlocal point of view by Challamel et al [START_REF] Challamel | Discrete and non-local elastica[END_REF], who showed the efficiency of nonlocal mechanics to capture the scale effects of the discrete elastica. The nonlocal beam model considered by Challamel et al [START_REF] Challamel | Discrete and non-local elastica[END_REF] was built by continualization of the governing difference equations of the discrete elastica, leading to an equivalent stress gradient model (as used by Eringen [START_REF] Eringen | On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves[END_REF] for axial lattice dynamics). They [START_REF] Challamel | Discrete and non-local elastica[END_REF] also showed that the nonlocal terms of the nonlocal continualized beam model strongly depend on the lattice spacing of Hencky bar-chain model.

The continualization method used in Challamel et al [START_REF] Challamel | Discrete and non-local elastica[END_REF] implemented stress gradient operators, but strain gradient operators can also be used for other continuous approximations of the discrete problem (Challamel et al [START_REF] Challamel | Higher-order gradient elasticity models applied to geometrically nonlinear discrete systems[END_REF]). In other words, nonlocal Euler-Bernoulli beam model can be built from Hencky bar-chain model by using a continualization method, both in the linear and in the nonlinear range.

The present paper generalizes these results by including the effect of shear deformation. The influence of

The buckling of Hencky bar-chain model extended by some shear effects has been studied by Zhang et al [START_REF] Zhang | Eringens small length scale coefficient for buckling of nonlocal Timoshenko beam based on microstructured beam model[END_REF], who also continualized this discrete structural model to obtain a nonlocal Timoshenko beam model.

However, Zhang et al [START_REF] Zhang | Eringens small length scale coefficient for buckling of nonlocal Timoshenko beam based on microstructured beam model[END_REF] only considered the linearized buckling problem and obtained an Engesser-type formula for the buckling load. In this paper we introduce two generalized Hencky bar-chain models, both of which possess bending and shear interactions. One model turns out to be a lattice system corresponding to the Engesser elastica, while the other model is a lattice system corresponding to the Haringx elastica.

Geometrically exact nonlocal Engesser and Haringx elastica are then asymptotically obtained from a continualization process applied to the discrete Haringx and Engesser elastica. Nonlocal Engesser and Haringx elastica are also obtained by using Eringen's nonlocal elasticity [START_REF] Eringen | On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves[END_REF] for small displacements in order to verify and comprehend the continualized nonlocal models. It is shown that the continualized nonlocal models can be obtained from Eringen's nonlocal elasticity if nonlocality affects only the bending, but not the shear terms of the constitutive law. Analytical solutions for the buckling loads of the introduced discrete and nonlocal models are derived and numerical solutions for the first post-buckling paths are given for the discrete and nonlocal models for large displacements. Numerically obtained bifurcation diagrams of the discrete systems are also plotted.

The generalized Hencky bar-chain model

The model of the generalized Hencky bar-chain model, a shearable-bendable elastic linkage, is shown in Figure 1 (a). There are N links connected by frictionless hinges in the x, y-plane. Initially, each link is a rectangular object, its longer side length, also called the length of the link, is a 0 parallel to the x-axis. In the unloaded state the linkage is straight, its total length is L = N • a 0 . Upon loading each link can undergo rigid-body-like rotation, and can be distorted due to shearing. The rigid-body-like rotation of link i is denoted by θ i . Both the link axis and the end sections of the link rotate by this angle while the length of the link remains a 0 . Then the link can be distorted such that the link axis is rotated by an additional (shear) angle γ i , while the end sections of the link keep their angle θ i , and the length of the link becomes a i . The shear force that develops during distortion will be defined later, since it depends on the applied shear strain theory. The neighboring end sections of adjacent links are connected by linear rotational springs of stiffness ρ. These are called bending springs. The moment in the ith bending spring is given by the material law:

M i = ρ (θ i+1 -θ i ) . (1)
Hinge 0, at the starting end of the linkage, is fixed, while hinge N is supported by a roller which constraints translation along the y axis. Hence, the supports are at the same height, y 0 = y N , leading to the kinematic boundary condition:

y N -y 0 = 0. ( 2
)
The structure is loaded by a horizontal, compressive (axial) force P at hinge N . There are zero moments at the ends of the linkage, defining the force boundary conditions M 0 = M N = 0, which, along with the discrete constitutive law Eq. ( 1), yield: We will implement two types of shear strain theories for the links. First we follow Timoshenko's finite shear strain theory [START_REF] Timoshenko | Theory of elastic stability[END_REF], assuming that the shear deformation preserves the initial length of the link axis.

θ 0 = θ 1 , θ N +1 = θ N . (3)
Hence, after the links are distorted, their length remain constant, a i = a 0 . This approach is discussed in Section 3.

Then we apply Love's finite shear strain theory [START_REF] Love | A treatise on the mathematical theory of elasticity[END_REF], assuming that the shear deformation preserves the initial area of the links. In this case, the length of the ith link axis becomes a i = a 0 / cos γ i after the shear deformation. This approach is detailed in Section 4.

Discrete and nonlocal Engesser elastica

The deformed link i, following Timoshenko's finite shear strain theory, is shown in Figure 2. The length of the link is preserved during the shear deformation, a i = a 0 , while the area of the link decreases.

If the linkage is a discrete model of a homogeneous, prismatic beam of bending stiffness EI and shear stiffness κGA, then the stiffness of the bending springs is ρ = EI/a 0 and the stiffness of the shearable links is k = κGAa 0 . The total potential energy of the structure is the sum of internal and external potentials:

Π tot = 1 2 ρ N -1 i=1 (θ i+1 -θ i ) 2 + 1 2 k N i=1 γ 2 i + P a 0 N i=1 cos (θ i + γ i ) . (4)
The dimensionless total potential energy is obtained by dividing Eq. ( 4) by ρ:

Π = 1 2 N -1 i=1 (θ i+1 -θ i ) 2 + α 2N 2 N i=1 γ 2 i + β N 2 N i=1 cos (θ i + γ i ) (5)
Here the following stiffness and load parameters are introduced:

α = κGAL 2 EI , β = P L 2 EI . (6)
In an equilibrium configuration the total potential energy is stationary [START_REF] Gantmacher | Lectures in Analytical Mechanics[END_REF]. This condition yields the 2N equilibrium equations of the structure:

∂Π ∂θ i = -(θ i+1 -2θ i + θ i-1 ) - β N 2 sin (θ i + γ i ) = 0, (7)
∂Π ∂γ i = α N 2 γ i - β N 2 sin (θ i + γ i ) = 0, i = 1, . . . , N. (8)
If Eqs. ( 7) are summed, and the static boundary conditions Eq. ( 3) are taken into account, then the kinematic boundary condition Eq. ( 2) is satisfied for β = 0.

Note the physical meaning of Eq. ( 7): It is the equilibrium of moments of link i. Now let us decompose the force P . The axial component, P cos(θ i + γ i ), has no apparent effect since the link is inextensible.

The component perpendicular to the link axis, P sin(θ i + γ i ), shears the link, and cause the shear strain γ i . This is meant by Eq. ( 8) in a non-dimensional form. Therefore, the introduced model inherits a discrete

Engesser-type approach: the normal and shear forces are aligned with and perpendicular to the link axis, respectively [START_REF] Bažant | Stability of Structures -elastic, inelastic, fracture, and damage theories[END_REF].

By introducing

Ψ i = θ i + γ i , (9)
the angle of the link axis, the equilibrium equation system Eqs. ( 7)-( 8) can be written as:

Ψ i+1 -2Ψ i + Ψ i-1 - β α (sin Ψ i+1 -2 sin Ψ i + sin Ψ i-1 ) + β N 2 sin Ψ i = 0, i = 1, . . . , N (10)
with boundary conditions Ψ 0 = Ψ 1 and Ψ N +1 = Ψ N . Note that Eq. ( 10) is identical to the difference equation system of Hencky bar-chain model as α → ∞, i.e. as the links become unshearable.

Buckling load of the discrete Engesser elastica

The trivial solution for Eq. ( 10), for fixed stiffness parameter α, is: Ψ i = 0, and β is arbitrary. This solution defines the trivial equilibrium states of the structure. A Hessian, formed by the partial derivatives of Eq. [START_REF] Gáspár | Global investigation of discrete models of the Euler buckling problem[END_REF] with respect to Ψ i , around the trivial state is the following:

H = -C 1 - β α + β N 2 I. ( 11
)
Here I is the N -by-N identity matrix, while C is the N -by-N modified continuant matrix defined as

C i,i = 2, except for C 1,1 = C N,N = 1, C i,i+1 = C i,i-1 = -1,
λ i = 4 sin 2 iπ 2N , i = 0, . . . , N -1 (12)
(see page 232 of [START_REF] Rózsa | Linear Algebra and its Applications[END_REF] for details). There is a trivial solution of i = 0 and β = 0, which implies P = 0 or EI → ∞. The other solutions give the buckling loads of the structure:

β cr,i = 4N 2 sin 2 iπ 2N 1 + 4N 2 sin 2 iπ 2N α , i = 1, . . . , N -1. (13)
It is important to observe that all of these N -1 buckling loads are compressive, thus the structure can buckle only under compression. The fundamental buckling load of the discrete Engesser elastica is obtained from Eq. ( 13) with i = 1 as:

β dE = 4N 2 sin 2 π 2N 1 + 4N 2 sin 2 π 2N α = β H 1 + β H α . (14)
Here

β H = 4N 2 sin 2 π 2N (15)
is the fundamental buckling load of Hencky's system [START_REF] Wang | Applied Elasticity[END_REF][START_REF] Domokos | Euler's problem, Euler's method, and the standard map; or, the discrete charm of buckling[END_REF][START_REF] Challamel | Discrete systems behave as nonlocal structural elements: Bending, buckling and vibration analysis[END_REF], to which Eq. ( 14) tends in the unshearable limit, i.e. as κGA → ∞, α → ∞. In view of this, the formula that has been developed for the fundamental buckling load of our shearable, bendable discrete model, Eq. ( 14), follows the Föppl-Papkovich summation pattern [START_REF] Tarnai | Structural Stability in Engineering Practice[END_REF].

The unbendable case implies EI → ∞. Then the buckling load equals the shear stiffness of the structures, which corresponds to the elastic web of links [START_REF] Kocsis | Spatial chaos in discrete mechanical systems: elastic linkages and elastic web of links[END_REF][START_REF] Kocsis | Spatially chaotic bifurcations of an elastic web of links[END_REF], a discrete model of Timoshenko's unbendable, inextensible, but shearable column [START_REF] Timoshenko | Theory of elastic stability[END_REF].

Note that Eq. ( 15) has already been given by Zhang et al [START_REF] Zhang | Eringens small length scale coefficient for buckling of nonlocal Timoshenko beam based on microstructured beam model[END_REF], corresponding to a different micromechanical model of the nonlocal Timoshenko beam.

Asymptotic limit: the local Engesser elastica

In the continuum limit Eq. ( 10) yields the local Engesser elastica:

d 2 dξ 2 Ψ - β α sin Ψ + β sin Ψ = 0, (16)
with ξ = s/L being the non-dimensional arc-length parameter, d 2 / dξ 2 denotes the second derivative with respect to ξ, and Ψ(ξ) = θ(ξ) + γ(ξ). It can be also written as

d 2 Ψ dξ 2 1 - β α cos Ψ + β 1 + 1 α dΨ dξ 2 sin Ψ = 0, (17)
which is similar to the so-called Timoshenko elastica equation given by Atanackovic [START_REF] Atanackovic | Stability theory of elastic rods, Series on Stability[END_REF] (see Eq. (3.3.57) on page 131).

The solution of the linearized version of Eq. ( 16) is

Ψ(ξ) = A sin β 1 -β/α ξ + B cos β 1 -β/α ξ . (18)
Satisfying pinned-pinned boundary conditions, Ψ (0) = Ψ (1) = 0, yields infinitely many buckling load parameters:

β lE,r = r 2 π 2 1 + r 2 π 2 α , r = 1, 2, . . . . (19)
The fundamental buckling load of the local Engesser elastica is Eq. ( 19) at r = 1:

β lE = π 2 1 + π 2 α , (20)
or, in a dimensional form,

P lE = P E 1 + P E κGA (21)
with P E = π 2 EI/L 2 being Euler's formula. Note that Eq. ( 21) is the critical force proposed by Engesser [START_REF] Bažant | Stability of Structures -elastic, inelastic, fracture, and damage theories[END_REF][START_REF] Engesser | Die Knickfestigkeit gerader Stäbe[END_REF] for the buckling load of a shearable and flexural column with pinned ends. Hence, Eq. ( 14) is a discrete version of the Engesser formula.

Geometrically exact continualized nonlocal Engesser elastica

Let us introduce the rotation field Ψ(ξ) of non-dimensional coordinate ξ = x/L and write its Taylor expansion [START_REF] Greenberg | Advanced Engineering Mathematics[END_REF]:

Ψ ξ + 1 N = ∞ k=0 d k dξ k Ψ(ξ) k! 1 N k = e 1 N d dξ Ψ(ξ). (22)
Then the following identity can be written:

Ψ i+1 -2Ψ i + Ψ i-1 = Ψ ξ + 1 N -2Ψ(ξ) + Ψ ξ - 1 N = 4 sinh 2 1 2N d dξ Ψ(ξ) ≈ 1 N 2 • d 2 dξ 2 1 -1 12N 2 d 2 dξ 2 Ψ(ξ) (23)
using a Padé approximant of order [START_REF] Euler | Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici lattissimo sensu accepti[END_REF][START_REF] Euler | Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici lattissimo sensu accepti[END_REF]. Similarly,

sin Ψ i+1 -2 sin Ψ i + sin Ψ i-1 = 4 sinh 2 1 2N d dξ (sin Ψ(ξ)) ≈ 1 N 2 • d 2 dξ 2 1 -1 12N 2 d 2 dξ 2 sin Ψ(ξ). (24)
Then the geometrically exact differential equation of the continualized nonlocal Engesser elastica yields:

d 2 Ψ dξ 2 + β sin Ψ -β 1 α + 1 12N 2 d 2 dξ 2 (sin Ψ) = 0. ( 25
)
This equation shows that the nonlocal effect is added to the shear effect in the nonlocal elastica. The symmetrical roles played by the shear effect and the nonlocal parameter in the shear buckling equations have been already outlined in [START_REF] Challamel | Higher-order shear beam theories and enriched continuum[END_REF]. Now analytical solution for the geometrically exact continualized nonlocal Engesser elastica is derived.

We follow the work of Lembo [START_REF] Lembo | On nonlinear deformations of nonlocal elastic rods[END_REF], who developed analytical solutions for the nonlocal Euler-Bernoulli elastica for large displacements using elliptic integrals. The methodology is the same for the nonlocal Engesser elastica investigated in this paper. Let us introduce

dy * dξ = sin Ψ (26)
and rewrite Eq. ( 25) as:

d 2 Ψ dξ 2 + β dy * dξ -β 1 α + 1 12N 2 d 3 y * dξ 3 = 0. ( 27
)
Integrating the above equation once one obtains

dΨ dξ + βy * -β 1 α + 1 12N 2 d 2 y * dξ 2 = C 1 (28)
with C 1 being an arbitrary constant. Multiplying Eq. ( 28) with ( 26) yields

dΨ dξ sin Ψ + βy * dy * dξ -β 1 α + 1 12N 2 d 2 y * dξ 2 dy * dξ = C 1 dy * dξ , (29)
which can be integrated as

-cos Ψ + β 2 (y * ) 2 - β 2 1 α + 1 12N 2 dy * dξ 2 = C 0 + C 1 y * (30)
with C 0 being another arbitrary constant.

Note that C 1 = 0 due to the boundary condition Eq. ( 2). Hence, the following formula can be obtained:

β 2 (y * ) 2 - β 2 1 α + 1 12N 2 dy * dξ 2 = C 0 + 1 - dy * dξ 2 , (31)
which shows that the shear effect and the nonlocal small length term play similar roles.

Differentiating Eq. ( 31) yields

β y * -ε 2 d 2 y * dξ 2 + d 2 y * dξ 2 1 -dy * dξ 2 = 0 with ε 2 = 1 α + 1 12N 2 (32)
It is possible to reformulate Eq. ( 32), by using the complementary spatial coordinates

dx * dξ 2 + dy * dξ 2 = 1 → dx * dξ d 2 x * dξ 2 + dy * dξ d 2 y * dξ 2 = 0, (33)
as

d 2 x * dξ 2 1 + βε 2 dx * dξ = -βy * dy * dξ . ( 34
)
This equation has been obtained by Lembo [47] for the nonlocal shear inextensible column, i.e. for α → ∞ and with 2 c /L 2 = 1/(12N 2 ) for the nonlocal correspondence. An integration of Eq. ( 34) alternatively gives:

(y * ) 2 = 2 β c - dx * dξ 1 + βε 2 2 dx * dξ . ( 35
)
where c is an integration constant. It is finally possible to rewrite the differential equation in terms of x * :

d 2 x * dξ 2 2 = 2β 1 -dx * dξ 2 1 + βε 2 dx * dξ 2 c - dx * dξ 1 + βε 2 2 dx * dξ . ( 36
)
The solution of this equation can be expressed in term of Weierstrass's elliptic functions (see Lembo [START_REF] Lembo | On nonlinear deformations of nonlocal elastic rods[END_REF]).

The correspondence with the solution of Lembo [START_REF] Lembo | On nonlinear deformations of nonlocal elastic rods[END_REF] is obtained from the shear and nonlocal factor:

ε 2 = 1 α + ˆ c 2 with ˆ c 2 = 2 c L 2 = 1 12N 2 . ( 37
)
Lembo [START_REF] Lembo | On nonlinear deformations of nonlocal elastic rods[END_REF] considered the nonlocal Euler-Bernoulli beam asymptotically obtained from α → ∞.

Using Eq. ( 26), Eq. ( 28) and Eq. ( 31) can be equivalently written as:

dΨ dξ = 1 α + 1 12N 2 sin 2 Ψ + 2 β (C 0 + cos Ψ) 1 α + 1 12N 2 cos Ψ -1 β ( 38
)
which reduces to the local Euler-Bernoulli elastica as α → ∞ and N → ∞:

dΨ dξ = -2β(C 0 + cos Ψ). (39)
This equation can be solved using elliptic integrals [START_REF] Euler | Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici lattissimo sensu accepti[END_REF], [START_REF] Timoshenko | Theory of elastic stability[END_REF].

In order to derive the buckling load of the continualized nonlocal Engesser elastica, Eq. ( 25) can be linearized as

1 -β 1 α + 1 12N 2 d 2 Ψ dξ 2 + βΨ = 0. ( 40
)
This equation is solved by

Ψ(ξ) = A sin β 1 -β 1 α + 1 12N 2 ξ + B cos β 1 -β 1 α + 1 12N 2 ξ . (41)
Satisfying pinned-pinned local boundary conditions, Ψ (0) = Ψ (1) = 0, yields infinitely many buckling load parameters of the continulized nonlocal Engesser elastica:

β nlE,r = r 2 π 2 1+ r 2 π 2 12N 2 1 + r 2 π 2 α 1+ r 2 π 2 12N 2 , r = 1, 2, . . . . (42)
The fundamental buckling load of the continualized nonlocal Engesser elastica is Eq. ( 42) at r = 1:

β nlE = π 2 1+ π 2 12N 2 1 + π 2 α 1+ π 2 12N 2 = β nle 1 + β nle α . ( 43
)
Here

β nle = π 2 / 1 + π 2 /(12N 2
) is the fundamental buckling load parameter of the nonlocal elastica [START_REF] Challamel | Discrete and non-local elastica[END_REF]. In terms of dimensional quantities, the fundamental buckling load is:

P nlE = P nle 1 + P nle κGA (44)
where

P nle = EIπ 2 L 2 c , with L c = L 2 + 2 c π 2 , c = a 0 2 √ 3 (45)
is the fundamental (dimensional) buckling load of the non-local elastica [START_REF] Challamel | Discrete and non-local elastica[END_REF]. It is important to outline that the nonlocal characteristic length, c , of the nonlocal model is proportional to the length of a link, a 0 , of the discrete model. Hence, the lattice size of the system is captured by the nonlocal model.

Note that Eq. ( 44) is an Engesser-type formula [START_REF] Bažant | Stability of Structures -elastic, inelastic, fracture, and damage theories[END_REF]. One only needs to replace Euler's buckling load (i.e. the buckling load of the local Bernoulli-Euler beam) with the buckling load of the nonlocal elastica (which is the nonlocal Bernoulli-Euler beam) in Engesser's famous formula to get the fundamental buckling load of the nonlocal Engesser elastica.

Eringen nonlocal Engesser elastica for small displacements

Wang et al [START_REF] Wang | Buckling analysis of micro-and nanorods/tubes based on nonlocal Timoshenko beam theory[END_REF] investigated the buckling of nonlocal Engesser-type columns where the nonlocal term only affects the bending part of the constitutive law. Reddy [START_REF] Reddy | Nonlocal theories for bending, buckling and vibration of beams[END_REF] investigated the buckling of nonlocal Engesser-type columns with both the bending part and the shear part being affected by the nonlocal terms.

Zhang et al [START_REF] Zhang | Eringens small length scale coefficient for buckling of nonlocal Timoshenko beam based on microstructured beam model[END_REF] presented a unified formulation of both nonlocal models and showed that a generalized Hencky model including some discrete shear stiffness behaves as a nonlocal Timoshenko column where the bending part of the constitutive law is affected by the small length terms (supporting the model of Wang et al [START_REF] Wang | Buckling analysis of micro-and nanorods/tubes based on nonlocal Timoshenko beam theory[END_REF] from a micromechanical point of view). Now we give the differential equations of the axially loaded nonlocal Engesser elastica for small displacements and its buckling loads, and compare it to the buckling loads of the continualized model presented in this paper.

Let us start from the principle of virtual work for establishing the governing equations of the Eringen nonlocal Engesser column under axial compression, for small displacements:

L 0 M δψ + V δ(w -ψ) -P w δw dx = 0. ( 46
)
Here M (x) and V (x) are the internal bending moment and shear force, respectively. The translation perpendicular to the line of action of the compressive force P is denoted by w(x), and the rotation of the cross-section is ψ(x). The shear deformation is γ(x) = w (x)ψ(x). The equilibrium equations can be written as:

M + V = 0, V = P w → M = -P w , V = P w . ( 47
)
The column is incompressible, the nonlocal (stress gradient) material law is:

M -2 1 M = EIψ , V -2 2 V = κGA(w -ψ). (48)
Note that the nonlocal characteristic lengths, 1 and 2 , are different for bending and shearing.

Substituting Eq. ( 47) in (48) yields the internal forces in terms of the kinematic variables,

M = EIψ -P 2 1 w , V = κGA(w -ψ) + P 2 2 w , (49)
which can be substituted back in Eq. ( 48), yielding

EIψ -P 2 1 w + κGA(w -ψ) + P 2 2 w = 0, κGA(w -ψ ) + P 2 2 w (4) -P w = 0. (50)
It is equivalent to the results of Zhang et al [START_REF] Zhang | Eringens small length scale coefficient for buckling of nonlocal Timoshenko beam based on microstructured beam model[END_REF].

It is possible to express the above coupled differential equations as a sixth-order differential equation for

ŵ(ξ) = w(x)| x=ξL /L.
In a non-dimensional form it is:

β α ˆ 2 2 ŵ(6) + ŵ(4) 1 -β 1 α + ˆ 2 1 + β ŵ = 0. (51)
Here α and β are defined by Eq. ( 6), while ˆ 1 = 1 /L, and

ˆ 2 = 2 /L.
If it is assumed that nonlocality only affects bending, i.e. 2 = 0, as proposed by Wang et al [START_REF] Wang | Buckling analysis of micro-and nanorods/tubes based on nonlocal Timoshenko beam theory[END_REF], then Eq. ( 51) reduces to a fourth-order differential equation which is solved by

ŵ(ξ) = A cos    β 1 -β 1 α + ˆ 2 1 ξ    + B sin    β 1 -β 1 α + ˆ 2 1 ξ    . ( 52
)
For simply supported columns the associated local boundary conditions are ŵ (0) = ŵ (1) = 0, and the fundamental buckling load is the same as Eq. ( 43) if

ˆ 1 = ˆ c = 1/(2 √ 3N ).
It coincides with the results of Wang et al [START_REF] Wang | Buckling analysis of micro-and nanorods/tubes based on nonlocal Timoshenko beam theory[END_REF] and Zhang et al [START_REF] Zhang | Eringens small length scale coefficient for buckling of nonlocal Timoshenko beam based on microstructured beam model[END_REF]. Note that the fourth-order differential equation (Eq. ( 51) with 2 = 0) 1 for different values of N and α. The buckling load of Hencky bar-chain model is also given for completeness. Note that the convergence to the local solution as N increases is faster for linkages with lower shear stiffness.

coincides

Numerical solutions of the discrete and nonlocal Engesser elastica

In this section first numerical solutions are given for the first post-buckling path of the discrete and nonlocal Engesser elastica for some smaller values of N , and then bifurcation diagrams of the discrete Engesser elastica of a few links are plotted.

Figure 3 shows the first post-buckling path of the discrete Engesser elastica and that of the geometrically exact continualized nonlocal Engesser elastica for different values of N and α. The numerical solutions are obtained by using the simplex algorithm [START_REF] Domokos | A global, direct algorithm for path-following and active static control of elastic bar structures[END_REF] as follows.

For the discrete Engesser elastica the difference equation system Eq. ( 10) was solved. For fixed number of links N there are N equations to solve, given by Eq. ( 10), with the boundary conditions Ψ 0 = Ψ 1 and Ψ N +1 = Ψ N . The deviation of these differences from zero are the errors that are minimized by the algorithm. The unknowns of the equation system are the angles Ψ 1 , Ψ 2 , • • • , Ψ N and the load parameter β (the stiffness parameter α is fixed). Hence, equilibrium paths in the Ψ 1...Nβ space are obtained as solutions of the simplex algorithm [START_REF] Domokos | A global, direct algorithm for path-following and active static control of elastic bar structures[END_REF]. The first post-buckling path, starting from the first bifurcation point of the trivial equilibrium branch, was followed and the projection of the path to the Ψ 1β plane is shown in Figure 3 in black color.

For the nonlocal model, introducing the auxiliary variable κ, the nonlinear, second-order differential equation ( 25) is written in two nonlinear, first order differential equations:

Ψ = κ, κ = β sin Ψ 1 + κ 2 1 α + 1 12N 2 β cos Ψ 1 α + 1 12N 2 -1 . ( 53
)
Using local boundary conditions, κ(0) = κ(1) = 0, the equation system can be solved with the shooting method as follows. Satisfying the close-end boundary condition κ(0) = 0, the far-end boundary condition κ(1) can be numerically computed as for an initial value problem (IVP), if some value for Ψ(0) is set.

For solving the differential equation system [START_REF] Domokos | Buckling of a cord under tension[END_REF] as an IVP, a predictor corrector method was used. The differential equation system was discretized and the non-dimensional spatial variable ξ was divided in 1000 equal parts. Starting with κ 0 = κ(0) = 0 and Ψ 0 = Ψ(0), the values of κ i = κ(i/1000) and Ψ i = Ψ(i/1000) were predicted by the (explicit) Euler method and corrected by the Adams-Moulton method [START_REF] Hairer | Solving Ordinary Differential Equations I: Nonstiff Problems[END_REF]. This scheme was embedded in the path-following simplex algorithm [START_REF] Domokos | A global, direct algorithm for path-following and active static control of elastic bar structures[END_REF]. The two unknowns of the algorithm are the angle Ψ(0) and the load parameter β (the stiffness parameter α is fixed). The error function of the algorithm is the value of κ(1). This scheme hence also yields equilibrium paths as solution. The first post-buckling path was followed starting from the first bifurcation point of the trivial equilibrium branch.

The result is shown in Figure 3 in gray color.

The numerically computed fundamental buckling loads correspond to the analytical results summarized in Table 1. The first post buckling paths of the discrete and nonlocal models correlate well up to Ψ 1 < π/4.

Beyond that some deviation can be observed which decreases as N increases.

The simplex algorithm is also capable to give global solutions for nonlinear boundary value problems without iterations [START_REF] Gáspár | A parallel algorithm for the global computation of elastic bar structures[END_REF]. This is done by scanning the representation space spanned by the unknowns (variables and parameter) of the problem for solutions. The drawback of the algorithm is that the required computational capacity increases exponentially with the number of unknowns. Therefore, this algorithm is applied for the discrete Engesser elastica of only a few links. The equations to solve and the unknowns are the same as those for the path-following process: the equations are Eq. ( 10) and the unknowns are The second-order equilibrium paths bifurcate from the trivial branch. They branch off the trivial equilibrium path at the buckling loads obtained analytically, given by Eq. ( 13) and in Table 1. Besides, there are many other equilibrium paths either bifurcating from the second-order branches (as pitchfork bifurcations), or appearing as separated branches (through saddle-node bifurcations).

The appearance of a multitude of higher order branches with the increase of beta is seminiscent of the behavior of similar spatially chaotic systems [START_REF] Domokos | Euler's problem, Euler's method, and the standard map; or, the discrete charm of buckling[END_REF][START_REF] Károlyi | Symbolic dynamics of infinite depth: finding global invariants for BVPs[END_REF][START_REF] Kapsza | Regular and random patterns in complex bifurcation diagrams[END_REF][START_REF] Kocsis | Buckling under nonconservative load: conservative spatial chaos[END_REF][START_REF] Kocsis | Conservative spatial chaos of buckled elastic linkages[END_REF][START_REF] Kocsis | Spatial chaos in discrete mechanical systems: elastic linkages and elastic web of links[END_REF].

Discrete and nonlocal Haringx elastica

A deformed link of the discrete model (Figure 1 (a)), following Love's finite shear strain theory [START_REF] Love | A treatise on the mathematical theory of elasticity[END_REF],

is shown in Figure 7. The area of the link is preserved during the shear deformation, while its side length increases as a i = a 0 / cos γ i .

If the linkage is a discrete model of a homogeneous, prismatic beam of bending stiffness EI and shear stiffness κGA, then the total potential energy of the structure is:

Π tot = 1 2 ρ N -1 i=1 (θ i+1 -θ i ) 2 + 1 2 k N i=1 tan 2 γ i + P a 0 N i=1 cos (θ i + γ i ) cos γ i ( 54
)
with the same bending and shear stiffness values as in Eq. ( 4). The dimensionless potential energy is:

Π = 1 2 N -1 i=1 (θ i+1 -θ i ) 2 + α 2N 2 N i=1 tan 2 γ i + β N 2 N i=1 cos (θ i + γ i ) cos γ i . ( 55
)
Here the stiffness and load parameters, α and β, are the same as those in Eq. ( 6).

The equilibrium equations of the structure are:

∂Π ∂θ i = -(θ i+1 -2θ i + θ i-1 ) - β N 2 sin (θ i + γ i ) cos γ i = 0, (56)
∂Π ∂γ i = α tan γ i -β sin θ i N 2 cos 2 γ i = 0, i = 1, . . . , N. (57)
Summing Eqs. [START_REF] Hodges | On tension buckling in shear-flexible composite beams[END_REF] and taking into account the static boundary conditions Eq. ( 3) yield the kinematic boundary condition Eq. ( 2) for non-zero load.

Similarly to the previous model, Eq. ( 56) is the equilibrium of moments of the ith link (see Figure 1 (b)). However, now P sin θ i is the shear force on link i, which is aligned with the end sections and not perpendicular to the link axis. Therefore, this discrete model follows a Haringx-type approach [START_REF] Haringx | On the Buckling and Lateral Rigidity of Helical Springs[END_REF][START_REF] Bažant | Stability of Structures -elastic, inelastic, fracture, and damage theories[END_REF].

The difference equation system, Eqs. ( 56)-( 57), can be reformulated as a difference equation for the rotations θ i , uncoupled from the shear deformation γ i :

θ i+1 -2θ i + θ i-1 + β N 2 1 + β α cos θ i sin θ i = 0, i = 1, . . . , N. (58)
To obtain the above equation system tan γ i = β α sin θ i is expressed from Eq. ( 57) and substituted in Eq. ( 56)

along with using the trigonometric identity sin(θ i + γ i ) = sin θ i cos γ i + cos θ i sin γ i .

Note that Eq. ( 58) is the same as the difference equation system of Hencky bar-chain model in the unshearable limit, i.e. as α → ∞

Buckling load of the discrete Haringx elastica

The trivial equilibrium state of the model for fixed stiffness parameter α is defined by: θ i = γ i = 0, and β is arbitrary. Around this state the Hessian given by the partial derivatives of Eq. ( 58) with respect to θ i is:

H = -C + β N 2 1 + β α I. (59)
Here I and C are the same as in Section 3.1.

If the determinant of H is zero, det -C + β/N 2 (1 + β/α) I = 0, i.e. if β/N 2 (1 + β/α
) is equal to an eigenvalue of the modified continuant matrix C (given by Eq. ( 12)), then the trivial equilibrium state is critical. It yields the buckling loads of the structure in the following Haringx-type formula:

β cr,2i-1,2i = α 2 -1 ± 1 + 16 N 2 α sin 2 iπ 2N , i = 0, . . . , N -1. (60)
Eq. ( 60) at i = 0, with subtraction, yields a buckling load that corresponds to a pure shear buckling mode under tension. (Note that positive load means compression.) This is the smallest tensile buckling load, hence it is the fundamental tensile buckling load parameter of the discrete Haringx elastica:

β tens dH = -α. (61)
Hence, pure shear buckling appears in our bendable, shearable model if the shear strain follows the theory proposed by Love. See, for example, the discrete model of a cord [START_REF] Domokos | Buckling of a cord under tension[END_REF], the discrete planar Cosserat rod [START_REF] Kocsis | Buckling analysis of the discrete planar cosserat rod[END_REF],

or the discrete model of DNA [START_REF] Kocsis | DNA stretching modeled at the base pair level: Overtwisting and shear instability in elastic linkages[END_REF], for micromechanical models that can buckle under tension in a shear mode, or the works of Kelly [START_REF] Kelly | Tension buckling in multilayer elastomeric bearings[END_REF], Hodges et al [START_REF] Hodges | On tension buckling in shear-flexible composite beams[END_REF] and Aristizabal-Ochoa [START_REF] Aristizabal-Ochoa | Tension and compression stability and second order analyses of threedimensional multicolumn systems: effects of shear deformations[END_REF].

For each nonzero i, Eq. ( 60) corresponds to a mixed buckling mode, with one compressive and one tensile buckling load.

For the unshearable case, κGA → ∞, i.e. α → ∞, this model is also identical to the Hencky chain, and yields the fundamental buckling load β H (see Eq. ( 15)). Meanwhile, for the unbendable case, EI → ∞, the structure yields the discrete model of a cord [START_REF] Domokos | Buckling of a cord under tension[END_REF], and undergoes pure shear buckling under tension at the critical load Eq. ( 61).

The fundamental (compressive) buckling load parameter of the discrete Haringx elastica is Eq. ( 60) at i = 1 with addition:

β comp dH = α 2 1 + 4 β H α -1 . ( 62
)

Asymptotic limit: the local Haringx elastica

The local Haringx elastica can be asymptotically obtained from Eq. ( 58):

d 2 θ dξ 2 + β 1 + β α cos θ sin θ = 0 ( 63
)
with ξ = s/L being the non-dimensional arc-length parameter.

The governing equations of the local Haringx elastica are given in the literature by Koiter [START_REF] Koiter | Elastic stability of solids and structures[END_REF], Goto et al [START_REF] Goto | Elliptic integral solutions of plane elastica with axial and shear deformations[END_REF], Atanackovic [START_REF] Atanackovic | Stability theory of elastic rods, Series on Stability[END_REF], Huang and Kardomateas [START_REF] Huang | Buckling and initial post-buckling behavior of sandwich beams including transverse shear[END_REF], Attard [START_REF] Attard | Finite strain-beam theory[END_REF] and Humer [START_REF] Humer | Exact solutions for the buckling and post-buckling of shear-deformable beams[END_REF], as:

EI d 2 θ ds 2 + P sin θ + P 2 κGA sin θ cos θ = 0, (64)
with θ(s) = θ(ξ)| ξ=s/L . It can also be obtained from Reissner's geometrically exact approach [START_REF] Reissner | On one-dimensional finite-strain beam theory[END_REF]. Eq. ( 64)

is the dimensional version of Eq. ( 63), and hence validates that the lattice system introduced in Section 4 is a micromechanical model of the Haringx elastica.

The linearized version of Eq. ( 63) is solved by

θ(ξ) = A sin β (1 + β/α)ξ + B cos β (1 + β/α)ξ . ( 65
)
With pinned-pinned boundary conditions, θ (0) = θ (1) = 0, the above solution yields for the buckling load:

β (1 + β/α) = r 2 π 2 , r = 0, 1, 2, . . . . (66)
For r = 0 critical loads corresponding to mixed (shear-bending) modes are obtained:

β lH,r = α 2 -1 ± 1 + 4 r 2 π 2 α , r = 1, 2, . . . . (67)
For r = 0 there is a nontrivial solution, the pure shear buckling load of the local Haringx elastica, which is the fundamental tensile buckling load parameter:

β tens lH = -α. (68)
Hence, similarly to the discrete model, the (local) continuum can also buckle under tension in a pure shear mode (the axis of the buckled rod remains straight). The fundamental (compressive) buckling load parameter can be obtained from Eq. ( 67) at r = 1:

β comp lH = α 2 1 + 4 π 2 α -1 . ( 69
)
In a dimensional form the fundamental tensile and compressive buckling loads of the local Haringx elastica are:

P tens lH = -κGA, (70)
P comp lH = κGA 2 1 + 4 P E κGA -1 , (71)
with P E = π 2 EI/L 2 being Euler's force. Note that Eq. ( 71) is the Haringx formula, the critical force proposed by Haringx [START_REF] Bažant | Stability of Structures -elastic, inelastic, fracture, and damage theories[END_REF][START_REF] Haringx | On the Buckling and Lateral Rigidity of Helical Springs[END_REF] originally for buckling of springs.

Geometrically exact continualized nonlocal Haringx elastica

The governing difference equation, Eq. ( 58), can be continualized based on Eq. ( 23):

d 2 θ dξ 2 + β 1 - 1 12N 2 d 2 dξ 2 • 1 + β α cos θ sin θ = 0, (72)
which can be reformulated as:

1 - β 12N 2 cos θ + β α cos 2θ d 2 θ dξ 2 + β 12N 2 sin θ + 2β α sin 2θ dθ dξ 2 + β sin θ + β 2α sin 2θ = 0. (73)
No analytical solution for the geometrically exact continualized nonlocal Haringx elastica has been found.

Next the buckling load of the continualized nonlocal Haringx elastica is developed.

The linearization of Eq. ( 72) leads to:

1 - β (1 + β/α) 12N 2 d 2 θ dξ 2 + β (1 + β/α) θ = 0. ( 74
)
The solution of this differential equation is:

θ(ξ) = A sin β (1 + β/α) 1 -β(1+β/α) 12N 2 ξ + B cos β (1 + β/α) 1 -β(1+β/α) 12N 2 ξ . ( 75
)
Using pinned-pinned, local boundary conditions, θ (0) = θ (1) = 0, the critical load parameters of the continualized nonlocal Haringx elastica can be obtained:

β nlH,r = α 2   1 ± 1 + 4 r 2 π 2 1 + r 2 π 2 12N 2 α    , r = 0, 1, 2 . . . . (76)
For r = 0 a nontrivial solution exists, that is the fundamental tensile buckling load parameter of the nonlocal Haringx elastica:

β tens nlH = -α. ( 77
)
This is the same pure shear buckling load that was derived for both the discrete and the local continuum

Haringx elastica. This buckling load hence is the same irrespectively to the modelling scale.

Eq. ( 76) at r = 1 yields the fundamental compressive buckling load parameter of the nonlocal Haringx elastica:

β comp nlH = α 2    1 + 4 π 2 1 + π 2 12N 2 α -1    = α 2 1 + 4 β nle α -1 , (78)
with β nle being the fundamental buckling load parameter of the nonlocal elastica [START_REF] Challamel | Discrete and non-local elastica[END_REF]. Note that there are infinitely many tensile and compressive buckling loads of the nonlocal Haringx elastica, while the discrete model has only a finite number of buckling loads.

In a dimensional form the fundamental buckling loads of the Haringx elastica are:

P tens nlH = -κGA, (79)
P comp nlH = κGA 2 1 + 4 P nle κGA -1 . (80)
Here P nle is the fundamental buckling load of the nonlocal elastica, given already by Eq. [START_REF] Greenberg | Advanced Engineering Mathematics[END_REF].

It is worth comparing the results obtained for the local and nonlocal continuum models. The only difference is that in Eq. ( 71) the buckling load of the (local) Bernoulli-Euler beam is used in the Haringxtype formula, while in Eq. ( 80) the buckling load of the nonlocal Bernoulli-Euler beam appears.

Eringen nonlocal Haringx elastica for small displacements

Buckling of nonlocal Haringx elastica has not been presented in the literature to the best of our knowledge. Let us start from the principle of virtual work for establishing the governing equations of the axially compressed Eringen nonlocal Haringx elastica for small displacements (see for example [START_REF] Challamel | Higher-order shear beam theories and enriched continuum[END_REF]):

L 0 M δψ + V δ(w -ψ) -P w δw + P (w -ψ)δ(w -ψ) dx = 0. (81)
Here M (x) and V (x) are the internal bending moment and shear force, respectively. The equilibrium equations can be written as:

M + V + P (w -ψ) = 0, V = P ψ → M = -P w , V = P ψ . ( 82
)
The column is inextensible, the nonlocal (stress gradient) material law is given by Eq. ( 48). Casting Eq. ( 82) in ( 48) yields the internal forces in terms of the kinematic variables,

M = EIψ -P 2 1 w , V = κGA(w -ψ) + P 2 2 ψ , (83)
which can be substituted back in Eq. ( 48), yielding the coupled system of differential equations,

EIψ -P 2 1 w + κGA(w -ψ) + P 2 2 ψ + P (w -ψ) = 0, κGA(w -ψ ) + P 2 2 ψ -P ψ = 0. (84)
This differential equation system can be written as a sixth-order, non-dimensional differential equation:

β 2 α ˆ 2 1 ˆ 2 2 ŵ(6) + 1 -β 1 + β α 2 1 - β 2 α 2 2 ŵ(4) + β 1 + β α ŵ = 0. (85)
Here ŵ(ξ) = w(x)| x=ξL /L, ˆ 1 = 1 /L, and ˆ 2 = 2 /L, and Eq. ( 6) gives α and β.

If it is assumed that nonlocality only affects bending, i.e. 2 = 0, then Eq. ( 85) reduces to a fourth-order differential equation solved by:

ŵ(ξ) = A sin β (1 + β/α) 1 -β (1 + β/α) 2 1 ξ + B cos β (1 + β/α) 1 -β (1 + β/α) 2 1 ξ . (86)
For simply supported columns the associated boundary conditions are ŵ (0) = ŵ (1) = 0, and the same critical loads can be derived for the Eringen nonlocal model as for the continualized nonlocal model, Eq. (76), with

ˆ 1 = ˆ c = 1/(2 √ 3N ).
We can conclude that the generalized Hencky bar-chain model, which incorporates shear deformation following Love's finite strain theory, i.e. the discrete Haringx elastica, can be associated with a stress gradient elasticity model where the nonlocality affects only the material law for bending, but not for shearing. The fundamental (compressive) buckling loads of the discrete model, its continualized counterpart (the continualized nonlocal Haringx elastica), and the Eringen nonlocal Haringx elastica are all given by Haringx-type formulae. All the models can buckle under tension in a pure shear mode, and also in mixed buckling modes.

That is a feature which does not exist for the Engesser-type discrete and nonlocal models.

The fundamental buckling loads of the discrete Haringx elastica is compared to that of the nonlocal Haringx elastica in Table 2 for different values of N and α. The buckling load of Hencky bar-chain model is also included to show the buckling load in the limit when the links become unshearable. The convergence to the local solution as N increases is faster for linkages with lower shear stiffness, similarly to the Engessertype models. The buckling loads of the Engesser-type model are lower than those of the Haringx-type model for the same shear stiffness parameter (see Table 1 vs. Table 2).

Numerical solutions of the discrete and nonlocal Haringx elastica

In this section some numerical solutions for the discrete and nonlocal Haringx elastica are given.

Figure 8 shows the first post-buckling path of the discrete Haringx elastica (with black line) and that of the geometrically exact continualized nonlocal Haringx elastica (with gray line) for different values of N and α.

For the nonlocal model the nonlinear, second-order differential equation (72) was written in the following two, nonlinear, first order differential equations:

θ = ω, (87)
ω = β sin θ + β 2α sin 2θ + ω 2 12N 2 sin θ + 2β α sin 2θ β 12N 2 cos θ + β α cos 2θ -1 . ( 88
)
Using local boundary conditions, ω (0) = ω (1) = 0, the equation system can be solved with the shooting method. A same procedure was followed as in Section 3.5.

For the results of the discrete Haringx elastica the difference equation system Eq. ( 58) was solved by the simplex algorithm [START_REF] Domokos | A global, direct algorithm for path-following and active static control of elastic bar structures[END_REF][START_REF] Gáspár | A parallel algorithm for the global computation of elastic bar structures[END_REF], c.f. Section 3.5. For fixed number of links N , there are N equations to solve given by Eq. ( 58 60), and also many other higher order paths either bifurcating from second-order branches (as pitchfork bifurcations), or appearing as separated branches (through saddle-node bifurcations). However, in this case bifurcations from the trivial equilibrium path occur also under tensile loads, and the secondary (post buckling) equilibrium paths in the negative load parameter regime can well be seen.

Conclusions

In this paper nonlocal Engesser and Haringx elastica models, including both bending and shear interactions, have been built from lattice physical systems, also called generalized Hencky bar-chain models. It has been shown that the nonlocality of these nonlocal beams was related to the lattice spacing of the generalized Hencky-type models. This connection between a bendable-shearable discrete elastica and nonlocal Timoshenko beam was already investigated by Zhang et al [START_REF] Zhang | Eringens small length scale coefficient for buckling of nonlocal Timoshenko beam based on microstructured beam model[END_REF] for a linearized Engesser-type model. In this paper, we have introduced a different lattice system and generalized the results for geometrically exact

Engesser and Haringx-type beams. Closed form solutions for the buckling loads of the lattice and nonlocal systems have been developed, and the geometrically exact nonlinear behaviour of both (Engesser and

Haringx) bendable-shearable lattice systems have been also studied far beyond the buckling loads. Analytical solution for the axially compressed, pinned-pinned, geometrically exact Engesser elastica has been given using elliptic integrals. Numerical solutions for large displacements have been computed, bifurcation diagrams of the lattice systems have been plotted (showing higher-order equilibrium paths of the discrete models), and the first post-buckling paths of the discrete and nonlocal models have been compared. It was revealed that the Haringx-type models (the lattice, the nonlocal and the local, too) can buckle under tension in a pure shear mode and also in mixed flexural-shear modes. This phenomenon has an interesting aspect in a biomechanical field, namely in DNA modelling, since the molecule may undergo a shear instability under tension as was studied by Kocsis and Swigon [START_REF] Kocsis | DNA stretching modeled at the base pair level: Overtwisting and shear instability in elastic linkages[END_REF].

In case of rigid shear interaction (infinite shear stiffness), the introduced discrete elastica systems are identical to the Hencky bar-chain model (see Hencky [START_REF] Hencky | Über die angenäherte Lösung von Stabilitätsproblemen im Raummittels der elastischen Gelenkkette[END_REF] for the linearized buckling problem, or El Naschie et al [START_REF] El Naschie | A simple discrete element method for the initial postbuckling of elastic structures[END_REF] or Gáspár and Domokos [START_REF] Gáspár | Global investigation of discrete models of the Euler buckling problem[END_REF] for the post-buckling behavior). This so-called discrete elastica has been recently revisited by Challamel et al [START_REF] Challamel | Discrete and non-local elastica[END_REF] with a nonlocal Euler-Bernoulli beam model, for the primary post-bifurcation branches. It has been shown that such kind of nonlocality belongs to the family of stress gradient Eringen's model [START_REF] Eringen | On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves[END_REF], associated with a softening effect of the small length scale parameter. As shown in the paper, a similar nonlocality also controls the behavior of nonlocal Engesser and Haringxtype Timoshenko beams. The introduced model could be generalized to include extensibility, and a parallel study between a generalized Hencky chain with axial deformations and the extensible-elastica theory (see [START_REF] Magnusson | Behaviour of the extensible elastica solution[END_REF]) could also be made. 15)). The local continuum is the case of N → ∞, given also by Eq. [START_REF] Duan | Development of analytical vibration solutions for microstructured beam model to calibrate length scale coefficient in nonlocal Timoshenko beams[END_REF]. 15)). The local continuum is the case of N → ∞, given also by Eq. (69).

Magnusson et al

Tables 602 N = 2 N = 3 N = 4 N = 5 N → ∞ discrete nonlocal discrete
N = 2 N = 3 N = 4 N = 5 N → ∞ discrete nonlocal discrete nonlocal discrete nonlocal discrete nonlocal local α = 1 2

Figure Captions

Figure 1 (

 1 Figure 1 (b) shows a schematic free body diagram of the displaced link i. If hinges 0 and N do not coincide, then the reaction force in the roller, and the vertical component of the reaction force in the fixed hinge are zero.

Ψ 1 ,

 1 Ψ 2 , . . . , Ψ N and β (the stiffness parameter α is fixed). Besides, the domain of the unknowns is bounded asΨ 1 ∈ [0, π], Ψ 2...N ∈ [-2π,2π], and β ∈ [0, 150]. Hence, only solutions found within this domain are part of the bifurcation diagram. By taking into account the symmetries of the solutions for Eq. (10), namely β → (-β) k and Ψ i → -Ψ i + kπ with k = 1, 2, 3, . . ., the bifurcation diagram is derived and plotted also for negative load parameter values, β ∈ [-150, 150]. In order to smoothen the obtained equilibrium paths a Newton-Raphson iteration was applied on the results of the scanning algorithm. The solutions, i.e. bifurcation diagrams of the discrete Engesser elastica, are shown in Figure 4, 5 and 6 for link numbers N = 2, 3 and 4, respectively.

  ) with boundary conditions θ 0 = θ 1 and θ N +1 = θ N . The unknowns of the equation system are the angles θ 1 , θ 2 , • • • , θ N and the load parameter β (the stiffness parameter α is fixed). The first (compressive) post-buckling path, starting from the first bifurcation point of the trivial equilibrium branch, was followed and shown in Figure8in black color. This is compared to the first (compressive) post-buckling path of the nonlocal Haringx elastica, shown in Figure8in gray color. Note that in these figures the angle Ψ is used and not θ. For the nonlocal model Ψ 1 = Ψ(0) is the angle of the axis tangent at ξ = 0.For the discrete model bifurcation diagrams were also constructed by using the simplex algorithm, and scanning the following domain of unknowns:θ 1 ∈ [0, π], θ 2...N ∈ [-2π,2π], and β ∈ [0, 150]. Incorporating the symmetries of the solutions for Eq. (58), i.e. β → (-β) k and Ψ i → -Ψ i + kπ with k = 1, 2, 3, . . ., the bifurcation diagram is derived and plotted also for negative load parameter values, β ∈ [-150, 150]. In order to smoothen the obtained equilibrium paths a Newton-Raphson iteration was applied on the results of the scanning algorithm. The resulting bifurcation diagrams of the discrete Haringx elastica of link number N = 2, 3 and 4 are shown in Figure 9, 10 and 11, respectively. Similarly to the bifurcation diagrams of the discrete Engesser elastica (Figures 4, 5 and 6), there are second-order equilibrium paths bifurcating from the trivial branch, corresponding to Eq. (

603Figure 1 .

 1 Figure 1. (a) Model of the generalized Hencky bar-chain. The angle of the axis of link i is Ψ i = θ i + γ i , and the angle of the end sections of link i is θ i . (b) Free body diagram of the ith link, assuming that the reaction force in the roller is zero. The moments in the bending springs are M i and M i-1 , respectively.

Figure 2 .

 2 Figure 2. Shear deformation of link i according to Timoshenko's finite shear strain theory. The length of the link axis is preserved, a i = a 0 .

Figure 3 .

 3 Figure3. First post-buckling equilibrium path of the discrete Engesser elastica (black), which is a solution of Eq. (10), and that of the geometrically exact continualized nonlocal Engesser elastica (gray), which is a solution of[START_REF] Coleman | Theory of sequence-dependent DNA elasticity[END_REF], for different values of N and α. For the discrete model Ψ 1 is the angle of the first link axis, while for the nonlocal model Ψ 1 = Ψ(0) is the angle of the tangent to the beam axis at ξ = 0.

Figure 4 .

 4 Figure 4. Bifurcation diagram of the discrete Engesser elastica of N = 2 links, for different values of the stiffness parameter α. Numerical solutions of Eq. (10) within the domain of β ∈ [0, 150], Ψ 1 ∈ [0, π] and Ψ 2 ∈ [-2π, 2π]. Solutions for negative load parameter β are derived using symmetry properties.

Figure 5 .

 5 Figure 5. Bifurcation diagram of the discrete Engesser elastica of N = 3 links, for different values of the stiffness parameter α. Numerical solutions of Eq. (10) within the domain of β ∈ [0, 150], Ψ 1 ∈ [0, π], Ψ 2...3 ∈ [-2π, 2π]. Solutions for negative load parameter β are derived using symmetry properties.

Figure 6 .

 6 Figure 6. Bifurcation diagram of the discrete Engesser elastica of N = 4 links, for different values of the stiffness parameter α. Numerical solutions of Eq. (10) within the domain of β ∈ [0, 150], Ψ 1 ∈ [0, π], Ψ 2...4 ∈ [-2π, 2π]. Solutions for negative load parameter β are derived using symmetry properties.

Figure 7 .

 7 Figure 7. Shear deformation of link i according to Love's finite shear strain theory. The length of the link axis becomes a i = a 0 / cos γ i while the area of the link is preserved.

Figure 8 .

 8 Figure 8. First compressive post-buckling equilibrium path of the discrete Haringx elastica (black), as a solution of Eq. (58), and that of the continualized nonlocal Haringx elastica (gray), which is a solution of Eq. (72) for different values of N and α. For the discrete model Ψ 1 is the angle of the first link axis, but for the nonlocal model Ψ 1 = Ψ(0) is the angle of the beam axis tangent at ξ = 0.

Figure 9 .

 9 Figure 9. Bifurcation diagram of the discrete Haringx elastica of N = 2 links, for different values of the stiffness parameter α. Numerical solutions of Eq. (58) in the domain of β ∈ [0, 150], θ 1 ∈ [0, π] and θ 2 ∈ [-2π, 2π]. Solutions for negative load parameter β are derived using symmetry properties.

Figure 10 .

 10 Figure 10. Bifurcation diagram of the discrete Haringx elastica of N = 3 links, for different values of the stiffness parameter α. Numerical solutions of Eq. (58) in the domain of β ∈ [0, 150], θ 1 ∈ [0, π], θ 2...3 ∈ [-2π, 2π]. Solutions for negative load parameter β are derived using symmetry properties.

Figure 11 .

 11 Figure 11. Bifurcation diagram of the discrete Haringx elastica of N = 4 links, for different values of the stiffness parameter α. Numerical solutions of Eq. (58) in the domain of β ∈ [0, 150], θ 1 ∈ [0, π], θ 2...4 ∈ [-2π, 2π]. Solutions for negative load parameter β are derived using symmetry properties.

  Figures 604

Figure 1 :Figure 2 :Figure 3 :

 123 Figure 1: (a) Model of the generalized Hencky bar-chain. The angle of the axis of link i is Ψi = θi + γi, and the angle of the end sections of link i is θi. (b) Free body diagram of the ith link, assuming that the reaction force in the roller is zero. The moments in the bending springs are Mi and Mi-1, respectively.

Figure 4 :

 4 Figure 4: Bifurcation diagram of the discrete Engesser elastica of N = 2 links, for different values of the stiffness parameter α. Numerical solutions of Eq. (10) within the domain of β ∈ [0, 150], Ψ1 ∈ [0, π] and Ψ2 ∈ [-2π, 2π]. Solutions for negative load parameter β are derived using symmetry properties.

Figure 5 :

 5 Figure 5: Bifurcation diagram of the discrete Engesser elastica of N = 3 links, for different values of the stiffness parameter α. Numerical solutions of Eq. (10) within the domain of β ∈ [0, 150], Ψ1 ∈ [0, π], Ψ2...3 ∈ [-2π, 2π]. Solutions for negative load parameter β are derived using symmetry properties.

Figure 6 :

 6 Figure 6: Bifurcation diagram of the discrete Engesser elastica of N = 4 links, for different values of the stiffness parameter α. Numerical solutions of Eq. (10) within the domain of β ∈ [0, 150], Ψ1 ∈ [0, π], Ψ2...4 ∈ [-2π, 2π]. Solutions for negative load parameter β are derived using symmetry properties.

Figure 7 :Figure 8 :

 78 Figure 7: Shear deformation of link i according to Love's finite shear strain theory. The length of the link axis becomes ai = a0/ cos γi while the area of the link is preserved.

Figure 9 :

 9 Figure 9: Bifurcation diagram of the discrete Haringx elastica of N = 2 links, for different values of the stiffness parameter α. Numerical solutions of Eq. (58) in the domain of β ∈ [0, 150], θ1 ∈ [0, π] and θ2 ∈ [-2π, 2π]. Solutions for negative load parameter β are derived using symmetry properties.

Figure 10 :

 10 Figure 10: Bifurcation diagram of the discrete Haringx elastica of N = 3 links, for different values of the stiffness parameter α. Numerical solutions of Eq. (58) in the domain of β ∈ [0, 150], θ1 ∈ [0, π], θ2...3 ∈ [-2π, 2π]. Solutions for negative load parameter β are derived using symmetry properties.

Figure 11 :

 11 Figure 11: Bifurcation diagram of the discrete Haringx elastica of N = 4 links, for different values of the stiffness parameter α. Numerical solutions of Eq. (58) in the domain of β ∈ [0, 150], θ1 ∈ [0, π], θ2...4 ∈ [-2π, 2π]. Solutions for negative load parameter β are derived using symmetry properties.

  with the second-order derivative of the linearized version of the continualized Engesser elastica Eq. (25), obtained from the discrete Engesser model for specific length scales. Now we can state that the introduced micromechanical model, the discrete Engesser elastica can be associated with a stress gradient elasticity model where the nonlocality affects only the material law for bending, but not for shearing. The fundamental buckling loads of the discrete model, its continualized

counterpart (the continualized nonlocal Engesser elastica), and the Eringen nonlocal Engesser elastica are all given by Engesser-type formulae. The fundamental buckling load of the discrete Engesser elastica is compared to that of the nonlocal Engesser elastica in Table

Table 1 :

 1 Buckling loads of the discrete and nonlocal Engesser elastica, according to Eqs. (14) and[START_REF] Atanackovic | Stability theory of elastic rods, Series on Stability[END_REF], for different values of N and α. Hencky bar-chain model is recovered as α → ∞ (see Eq. (

	nonlocal discrete nonlocal discrete nonlocal	local

Table 2 :

 2

		.37228 2.40454 2.54138 2.54847 2.60203 2.60436 2.63036 2.63133	2.68113
	α = 10	5.24695 5.33748 5.72381 5.74392 5.89614 5.90278 5.97686 5.97964	6.12187
	α = 100	7.44563 7.60760 8.30952 8.34654 8.62814 8.64049 8.77853 8.78371	9.05049
	α = 1000	7.93700 8.12041 8.92043 8.96286 9.28635 9.30057 9.45967 9.46565	9.77407
	α = 10000 7.99361 8.17966 8.99191 9.03503 9.36381 9.37827 9.54005 9.54614	9.85988
	α → ∞	8	9	9.37258	9.54915	π 2

Fundamental (compressive) buckling loads of the discrete and nonlocal Haringx elastica, given by Eqs. (

62

) and (78), for different values of N and α. Hencky bar-chain model is recovered as α → ∞ (see Eq. (

DISCRETE AND NONLOCAL ENGESSER ELASTICA

DISCRETE AND NONLOCAL HARINGX ELASTICA

CONCLUSIONS

Acknowledgement

The work of A. Kocsis was supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. Gy. Károlyi thanks the support from OTKA grant no. 100894.