### BEAM AND PLATE FORMULATIONS IN PERIDYNAMIC FRAMEWORK

Dr. Erkan Oterkus gave a lecture on beam and plate formulations in peridynamic framework during the workshop “Encounter of the third kind” on “Generalized continua and microstructures, organised by M&MoCS. The International Research Center for Mathematics & Mechanics of Complex Systems (M&MoCS) is a Research Center of the Università dell’Aquila. It was established by the Dipartimento di Ingegneria delle Strutture, delle Acque e del Terreno (DISAT) and the Dipartimento di Matematica Pura e Applicata (DMPA). Its administrative headquarters are located in L’Aquila. M&MoCS was established with the financial and logistical support of Provincia di Latina and the Fondazione Tullio Levi-Civita, with which it shares the mission of developing and disseminating scientific culture in the region. It was also founded together with the Dipartimento di Strutture of Università di Roma Tre.

### Pre-cracked thin square plate subjected to velocity boundary condition1

The pre-cracked thin square plate has dimensions 0.05 m x 0.05 m and it is subjected to velocity boundary condition along its two horizontal edges.

Peridynamic Theory (PT) is used to predict the mechanical behaviour of the plate, the crack propagation speed and the crack propagation path. PT is a new theory introduced in 2000 by Dr. Stewart Silling: it can be defined as a generalization of Classical Continuum Mechanics (CCM) whose governing equation is integral instead of partial differential as for CCM. Therefore, the formulation remains valid even when a crack occurs during the deformation process of the material. Other advantages of PT over CCM are as follow: no need for an external crack growth criteria (in PT, the damage of the material is naturally present in its formulation), no need for computationally expensive re-meshing procedures, no need of external equations for describing interfaces, capability to predict crack initiation, capability to capture complex behaviours such as crack branching, crack arrest and coalition of multiple cracks. Moreover, PT is also suitable for modeling multiscale and multiphysics problems.

In this study, two simulations have been carried out using two different values for the velocity boundary condition: 20 m/s and 50 m/s respectively. The main assumptions are as follow: homogeneous material, isotropic material, prototype micro-elastic brittle material (PMB), Poisson's ratio = 1/3 and plane stress problem. The programming language here employed is Fortran 2003. Some features of the simulation are listed here below:

Young's modulus = 192.0 GPa

Material's volumetric mass density = 8,000 Kg/m^3

Crack length = 0.01 m

Velocity boundary condition for the first simulation = 20 m/s

Velocity boundary condition for the second simulation = 50 m/s

Number of real particles along x-direction = 500

Number of real particles along y-direction = 500

Number of real particles along z-direction = 1

Number of boundary particles in the bottom boundary region = 1500

Number of boundary particles in the upper boundary region = 1500

Grid spacing (discretization parameter) = 0.0005 m

Horizon per unit of grid spacing = 3.015

time step = 1.3367 E-8 s

Number of time steps for the first simulation = 1,250.

Number of time steps for the second simulation = 1,000.

Critical stretch = 0.04472

The videos of the two simulations are showed here below:

### Finite Slab with Time-Dependent Surface Temperature

The length of the 1D slab here considered is 0.01 m and its initial temperature is 0 K. The left edge of the slab is kept at 0 K throughout the whole simulation, whilst the temperature of the right edge grows linearly with time.

Bond-based Peridynamic Theory is used to predict the time evolution of the temperature field within the slab.

The programming language here employed is Fortran 2003. Some features of the simulation are listed here below:

Specific heat capacity at constant volume of the material = 64.0 J/KgK

Thermal conductivity of the material: 233.0 W/mK

Material's volumetric mass density = 260.0 Kg/m^3

Slab length = 0.01 m

Area of the slab section = 1.0e-8 m^2

Coefficient for the linear temperature increment of the slab's right edge = 500.0

Number of real particles = 100

Number of virtual particles at the slab's left edge = 3

Number of virtual particles at the slab's right edge = 3

Grid spacing (discretization parameter) = 0.0001 m

Horizon per unit of grid spacing = 3.015

Time step = 1.0 e-6 s

Number of time steps = 50,000.

The results are in good agreement with the analytical solution of the problem. Two videos of the simulation are showed here below:

### Plate Under Thermal Shock with Insulated Boundaries

The length and width of the 2D plate here considered are both 10 m. The initial temperature is 0 K. At the left edge of the plate there are three vertical layers of virtual particles which are subjected to a thermal shock, which is mathematically represented as follows: T = 5.0*time*exp(-2*time).

Bond-based Peridynamic Theory is used to predict the time evolution of the temperature field within the plate for the first 6 seconds.

The programming language here employed is Fortran 2003. Some features of the simulation are listed here below:

Specific heat capacity at constant volume of the material = 1.0 J/KgK

Thermal conductivity of the material: 1.0 W/mK

Material's volumetric mass density = 1.0 Kg/m^3

Plate length = 10 m

Plate width = 10 m

Plate thickness = 1 m

Number of real particles along x-direction = 300

Number of real particles along y-direction = 300

Number of virtual particles at the plate's left edge = 900

Grid spacing (discretization parameter) = 0.0333 m

Horizon per unit of grid spacing = 3.015

Time step = 5.0 e-4 s

Number of time steps = 12,000.

Hosseini-Tehrani and Eslami (2000) have used the Boundary Element Method to solve the same problem. The results are in close agreement. Two videos of the simulation are showed here below:

### Plate Under Thermal Shock with Boundaries at T = 0 K

The length and width of the 2D plate here considered are both 10 m. The initial temperature is 0 K. At the left edge of the plate there are three vertical layers of virtual particles which are subjected to a thermal shock, which is mathematically represented as follows: T = 5.0*time*exp(-2*time). The remaining three edges of the plate are kept at T = 0 K throughout the whole simulation.

Bond-based Peridynamic Theory is used to predict the time evolution of the temperature field within the plate for the first 6 seconds.

The programming language here employed is Fortran 2003. Some features of the simulation are listed here below:

Specific heat capacity at constant volume of the material = 1.0 J/KgK

Thermal conductivity of the material: 1.0 W/mK

Material's volumetric mass density = 1.0 Kg/m^3

Plate length = 10 m

Plate width = 10 m

Plate thickness = 1 m

Number of real particles along x-direction = 300

Number of real particles along y-direction = 300

Number of virtual particles at the plate's left edge = 900

Grid spacing (discretization parameter) = 0.0333 m

Horizon per unit of grid spacing = 3.015

Time step = 5.0 e-4 s

Number of time steps = 12,000 (video no.1) and 36,000 (video no.2).

Two videos of the simulation are showed here below:

### Dissimilar Materials with a Pre-Existing Insulated Crack:

The length and width of the 2D plate here considered are both 2 cm. A crack, considered insulated, is embedded in the middle of the plate. The initial temperature is 0 K. The plate is made by two different materials: material no.1 for the upper half and material no.2 for the lower half. The only difference between the two materials is their value of thermal conductivity [W/cmK]. At both upper and lower edge of the plate there are three horizontal layers of virtual particles which are kept at constant temperature throughout the simulation: 100 K and -100 K respectively. The two vertical edges are considered to be insulated.

Bond-based Peridynamic Theory is used to predict the time evolution of the temperature field within the plate until 0.5 seconds.

The programming language here employed is Fortran 2003. Some features of the simulation are listed here below:

Specific heat capacity at constant volume of the material = 1.0 J/KgK

Thermal conductivity of material no.2: 1.14 W/cmK

Thermal conductivity of material no.1: 0.114 W/cmK

Material's volumetric mass density = 1.0 Kg/cm^3

Plate length = 2 cm

Plate width = 2 cm

Plate thickness = 0.01 cm

Crack length = 1 cm

Number of real particles along x-direction = 200

Number of real particles along y-direction = 200

Number of virtual particles at the plate's lower edge = 600

Number of virtual particles at the plate's upper edge = 600

Grid spacing (discretization parameter) = 0.01 cm

Horizon per unit of grid spacing = 3.015

Time step = 1.0 e-4 s

Number of time steps = 5,000.

The same simulation has been carried out by using ANSYS. The results are in close agreement. Two videos of the simulation are showed here below: