A Novel Peridynamic Mindlin Plate Formulation Without Limitation on Material Constants

In this study, a new peridynamic Mindlin plate formulation is introduced by utilising Euler-Lagrange equations. The classical strain energy density of a material point is converted to its corresponding peridynamic form by using Taylor’s expansions. The formulation is suitable for thick plates by considering the transverse shear deformation. Material constants do not have any limitation in the current formulation. Different types of boundary conditions are considered in numerical examples including simply supported, clamped and mixed (clamped-supported). To verify the current formulation, peridynamic solutions of the transverse displacements and rotations are compared against solutions obtained from finite element analysis.


Introduction
Solid mechanics is an important research area focusing on how the materials and structures deform if they are subjected to external loading conditions. As a result of the deformation, permanent damages can occur including plasticity and crack occurrence. Predicting crack initiation and propagation is still one of the challenging areas of solid mechanics. Moreover, with the advancement of technology, interest on nano-scale structures is increasing. Analysis of nano-scale structures can require non-classical approaches including length scale parameters. There are various non-classical approaches available in the literature, and amongst these, peridynamics [1] is the focus of this study. There has been a rapid progress in peridynamics research during the recent years . An extensive review of peridynamics is given by Javili et al. [25].
Original peridynamic formulation is suitable for analysis of three-dimensional structures. However, this formulation can be computational expensive for structures which have special shapes that can be considered as beams, plates and shells. To model such structures, standard formulation should be modified by taking into account rotational degrees of freedom. Hence, three-dimensional geometries can be represented by one-dimensional models as beams or two-dimensional models as plates and shells. Such formulations are currently available for peridynamics. For instance, a peridynamic formulation to analyse thin plates was introduced in [26]. A non-ordinary state-based Euler-Bernoulli beam formulation was developed in [27]. In another study [28], they extended this formulation to represent the Kirchhoff-Love plate theory. State-based peridynamics was used to develop peridynamic Euler beam and Kirchhoff plate formulations in [29] and [30], respectively. To analyse thick plates, it is essential to take into account transverse shear deformations. Transverse shear deformations were taken into account in [31] as a characteristic of peridynamic Mindlin plate formulation. However, this formulation is limited to Poisson's ratio of 1/3 for isotropic materials.
Hence, a new peridynamic Mindlin plate formulation is introduced in this study which does not have any limitations on material constants. The formulation is developed by using the Euler-Lagrange equation in conjunction with Taylor's expansion. To verify the formulation, several numerical examples are considered for a Mindlin plate under different boundary conditions, and peridynamic and finite element analysis results are compared with each other.

Classical Mindlin Plate Formulation
Mindlin developed a plate theory for thick plates by considering transverse shear deformation. According to Mindlin plate theory, a transverse normal to the mid-plane of the plate in the undeformed state remains straight, and there is no change in its length during deformation. The displacement components of any material point can be represented in terms of mid-plane ( xy plane) displacements and rotations as where x and y denote the mid-plane rotations about positive y-direction and negative x-direction, respectively. Moreover, w denotes the mid-plane transverse displacements. The positive set of the degrees-of-freedom is shown in Fig. 1.
The stress-strain relationships can be written for isotropic materials as: where E, G, and ν represent elastic modulus, shear modulus and Poisson's ratio, respectively. Note that the transverse normal stress, zz , is considered to be small compared with in-plane stresses. Thus, it is discarded from the stress components set for simplification. The stresses can also be written in indicial notation as: where The linear elastic strain energy density of the Mindlin plate can be expressed as Inserting Eqs. (10), (11) and (17) into Eq. (19) and rearranging indices yields For a particular material point on the mid-plane, the average strain energy density can be reasonably calculated by integrating the strain energy density function, Eq. (21), through the transverse direction and dividing by the thickness as

Peridynamic Mindlin Plate Formulation
Peridynamics (PD) is a continuum mechanics formulation in which material points can interact with each other in a non-local manner. The range of non-local interactions is defined as "horizon," H . The equation of motion peridynamics can be expressed in discrete form as for the material point k where N is the number of material points inside the horizon, is density, is the displacement, t is time, V is volume and is the body load vector. The interaction force vector, (k)(j) , between material points k and j can be defined as (see Fig. 2) The PD equations of motion can be obtained from Euler-Lagrange's equation as: where L is the Lagrangian which is the difference between the total kinetic energy, T , and total potential energy, U , that can be expressed as and where the generalised displacement vector, , and generalised body force density vector, , can be defined as and Here, b and b z correspond to moment and transverse force, respectively. Inserting Eqs. (1), (2) and (3) into (26) and discretizing the function gives the kinetic energy of the system as Fig. 2 PD force density factor and horizon [32] Assuming the system is holonomic, the first term of Euler-Lagrange equation can be obtained by substituting Eq. (30) into (25): In PD theory, the strain energy density has a non-local form. Therefore, the strain energy of a certain material point k can be expressed as where (k) represents the displacement vector of the material point k , and (i k ) (i = 1, 2, 3, ⋯) represents the displacement vector of the i th material point inside the horizon of the material point k.
The total potential energy given in Eq. (27)  As explained in Appendix, the strain energy densities of the material point k and j can be expressed in peridynamic form by transforming all differential terms in Eq. (22) as with n 1 = cos ,n 2 = sin and is the orientation of peridynamic bond (interaction). Inserting Eqs. (36) and (37)  where c b and c s represent PD material parameters related with bending and transverse shear deformations, respectively, which are defined as (37)

Boundary Conditions
As shown in Fig. 3, boundary conditions are imposed by creating a fictitious domain, R c , which is located outside of the real domain, R . The thickness of this fictitious layer can be specified as twice of the horizon size if ≠ 1∕3 , or the size of the horizon if = 1∕3.

Clamped Boundary Condition
The clamped boundary condition can be imposed by specifying zero transverse displacement and zero rotation for the material points at the clamped boundary as In this study, these conditions can be obtained via mirror image of the transverse displacements of the material points adjacent to the clamped end and anti-symmetric image of rotation fields as shown in Fig. 4 as

Simply Supported Boundary Condition
The simply supported boundary condition can be obtained by imposing zero transverse displacement and curvature for the material point adjacent to the constrained boundary. The constraint condition related with transverse displacement can be achieved by imposing antisymmetrical transverse displacement fields in the fictitious region as shown in Fig. 5 which can be expressed as The curvature condition can be defined as

Mindlin Plate Subjected to Simply Supported Boundary Conditions
As the first numerical example, a simply supported Mindlin plate is considered as shown in

Mindlin Plate Subjected to Clamped Boundary Conditions
In the second case, the same problem is considered as in the previous example case except the boundary conditions being clamped instead of simply supported as shown in Fig. 10.
As depicted in Figs. 11 and 12, the PD solution of the transverse displacement,w , and rotations, L , along the central x− and y − axes are compared with the FE method results, and there is a very good agreement between the PD and the FE method results.

Mindlin Plate Subjected to Mixed (Clamped-Simply Supported) Boundary Conditions
In this final numerical case, as opposed to first and second numerical cases, Mindlin plate is subjected to mixed (clamped-simply supported) boundary conditions as shown in

Conclusions
In this study, a new Mindlin plate formulation was developed. In the current formulation, there is no limitation on material constants as in bond-based peridynamics. Euler-Lagrange equation was utilised in conjunction with Taylor's expansion to determine the equations of motion. Three different numerical examples were considered for a Mindlin plate subjected to different boundary conditions. Peridynamic results were compared against results obtained from finite element analysis, and a good agreement was obtained between the two approaches verifying the newly developed formulation.

Appendix
As explained above, the strain energy density function of Mindlin plate can be expressed as: Location, x(m) In order to obtain the strain energy density function in PD form, it is necessary to transform each differential terms into their equivalent nonlocal form. This can be achieved by utilizing Taylor expansion as: As shown in Fig. 18    Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/.