Classical continuum theory can be applied to several challenging problems, but its governing equation has a limitation that it cannot be applied to any discontinuity such as a crack, since the partial derivatives with respect to space are not defined at a crack. The formulations therefore do not apply as soon as the discontinuity occurs in the body. Classical theory involves partial differential equations with respect to space and these spatial derivatives are not defined at the discontinuities. Thus, the main limitation of classical theory is that it is not applicable due to its governing equations involving partial derivatives with respect to space when there is a discontinuity such as a crack. a) Local (b) Peridynamics Figure 1.1: Comparison between local and non-local theory.
The stress at the crack tips tends to be infinite in classical theory, which leads to the concept of Linear Elastic Fracture Mechanics (LEFM) which assumes pre-existing cracks in the material where the crack initiation and crack path are treated separately using parameters external as critical energy. release rate. The critical rate of energy release is not part of the governing equations of classical theory. The solutions of classical theory are independent of crack size, but experiments showed that materials with small cracks have more fracture resistance.
Also classical theory predicts no dispersion which is not true as experiments show otherwise for the propagation of shorter wavelength elastic plane waves in elastic solids. However, classical theory is not capable of distinguishing different length scales, but can capture failure processes using the finite element method (FEM). In fracture mechanics the main concern is the introduction of pre-existing cracks rather than new cracks. When dealing with pre-existing crack growth, FEM using traditional elements have the limitation of recombination after each incremental growth of a crack.
Basic Definitions
- Equation of Motion
- Relative Position
- Relative Displacement
- Conservation laws
- Bond stretch
The law of linear and rotational quantity must be followed by the force function f, i.e. The above expressions mean that the relative position is parallel to the force between the two particles. The general form of the binding force for the basic theory can be written as
As in the classical theory, where there is a strain, here is a scalar extension of the bond0s0b which is defined as. The value of the spring constant c is found by equating the energy density within the horizon in peridynamic theory to the strain energy under isotropic stretching from continuum mechanics for the same deformation. The critical extent for bond failure is therefore related to the gas energy release rate.
This is achieved by equating work done to break all bonds in a unit area to energy release rate Silling and Askari (2005) described that introducing failure at the interaction level leads to local damage at a point given by. So we can get the definition of the local damage at a time as the ratio of the amount of broken interactions to the total amount of interactions.
Horizon size
Advantages and Limitations
LITERATURE REVIEW
A peridynamic formulation for a unidirectional fiber-reinforced composite lamina based on homogenization and mapping between the micro-level peridynamic bond elastic and fracture parameters and the macro-level composite parameters was developed by Wenke et al. The model is then used to analyze the fracture in the splitting mode (mode II) under a dynamic load of 0◦ of the lamina. Appropriate scaling factors are used in the model to have an elastic strain energy for a fixed non-local interaction distance (peridynamic horizon).
The convergence behavior of peridynamic solutions in terms of the size of the non-local region compared to the classical (local) mechanics model is also studied. 17] for plane stress and plane strain was studied and validated using a two-dimensional rectangular plate with a circular hole in the center under constant tensile stress. The problem of crack propagation in thin orthotropic flat plates under bending loads was studied by Tastan et al.
Furthermore, it appears that the numerically calculated crack patterns reasonably follow the orthotropic properties of the models. Michael et al.[19] developed a plate model as a two-dimensional approximation of the three-dimensional bond-based theory of peridynamics via an asymptotic analysis.
APPLICATION TO EULER BERNOULLI BEAM
Formulation
- Equation of Motion
- Validation with classical theory
Micropotentials are also functions of the transverse degrees of freedom of material points. The PD equation of motion at a material point xk is found using the principle of virtual work, which is. The total kinetic energy of the system is the result of bending and transverse shear deformation, the total potential energy is obtained by summing the micropotentials.
Using dimensional analysis we find that the term P∞ i=1. uj −uk)Vi represents the expletive. Using classical continuum mechanics, the strain energy density for material points can be represented as The validity of the equation needs to be checked and checked, so we restrict the horizon nsize to zero to get the classical continuum theory, that is, δ → 0.
Using the Taylor series, transverse displacements are expressed in terms of the displacements of their principal material points and without considering higher-order terms. Vj+ ˆbk (3.21) The summation sign includes all material points in the horizon of the main material point to the left and right of the material point. Vj+ ˆbk (3.23) With some algebraic calculations, we obtain the final equation of motion for peridynamics as
We can calculate peridynamic material parameterized by comparing the curvature of the material point in peridynamics with curvature in classical theory under a simple loading condition with constant curvature υ, i.e. this shows that we can recover classical theory by reducing the horizon size in peridynamics to zero.
APPLICATION TO TIMOSHENKO BEAM
Formulation
- Equations of Motion
- Validation with classical theory
The PD equation of motion at a material point xk is found using the principle of virtual work, which is. Let ˜αkj(κkj) and ˆαkj(ϑkj) be the micropotentials between material points resulting from bending and transverse shear deformation. Putting these values of the peridynamic forces and the corresponding shear angles and curvature into the EOM, we get
Using the Taylor series, we can express the out-of-plane rotation and the transverse shear strain at the material point j as . Let Vj = A∆ξjk, where ∆ξjki represents the spacing between two consecutive material points, and replace the summation by integration as ∆ξjk approaches zero.
APPLICATION TO MICROPOLAR BEAM
- Geometry
- General Equation of motion
- Constitutive relations
- Equation of motion
Neglecting the microrotation and setting micropolar constant η as zero, we obtain the EOM for Timoshenko beam by adjusting the constants as . Further neglecting the shear deformation and adjusting the constants, we obtain EOM for Euler Bernoulli beam as.
PROBLEM STATEMENT
Problem
- Solution
Two fictitious volumes are created on the left and right hand side of the beam with length equal to horizon size. Boundary condition is applied by expanding the deformed shape of the beam in the fictitious volumes as shown. Similarly, it is also done for other support. The displacements of the material points in the fictitious volume are specified as.
The reduced displacement matrix contains displacements of material points that are only in the beam. By putting stiffness, transformation, reduced displacement and body force matrix into the following equation, we will get the unknown displacements.
RESULTS AND DISCUSSIONS
Euler Bernoulli Beam
- Simply Supported Beam
The maximum deflection is at the center of the beam and is analytically equal to 6.25 mm.
Location (m)
Cantilever Beam
The maximum deflection is at the free end of the beam and is analytically equal to 100 mm. The maximum deflection is at the free end of the beam and is analytically equal to 100 mm.
Clamped Clamped Beam
The maximum deflection is at the center of the beam and is analytically equal to 1.6 mm.
Timoshenko Beam
- Simply Supported Beam
- Cantilever Beam
- Clamped Clamped Beam
These are the plots between the transverse displacement and the rotation of the material points and their coordinates. The maximum displacement is in the center and is equal to 6.25 mm. The maximum rotation is at the support and is equal to 1.5×10−6 and is equal to zero at the midpoint of the beam analytically. For rotation the graph is somewhat non-uniform, the graph for NE=800 overlaps with the analytical one at the beginning, then from X=0.2m to X=0.7m, NE=200 overlaps with the analytical one, then at the end NE=400 overlaps with the analytical one .
For transverse displacement, from NE=100, they start to overlap and it fully complies with the analytical solution. The maximum displacement is at the free end and is equal to 100 mm. The maximum rotation is at the free end and is equal to 0.15 rad. For transverse displacement, from NE=100, the graph starts to overlap and it fully obeys the analytical solution, only for NE=50 the graph appears to have less displacement values.
Also for rotation the graph starts to overlap from NE=100 and fully satisfies the analytical solution, only for NE=50 the graph seems to have fewer rotation values. The maximum displacement is in the center and is equal to 1.6 mm. The maximum rotation is at a distance of 0.25 m from each support and is equal to 4.5×10−3 and is equal to zero at each support and at the center of the beam. For the transverse displacement, up to a distance of 0.5 m, all graphs appear to overlap, but after 0.5 m the graph with NE=50 starts to diverge. For NE =50 the maximum deflection value is X=0.55 m.
For rotation, all graphs appear to overlap, but show a higher value of maximum rotation.
SCOPE OF PRESENT WORK
A still relevant contribution of Gabrio Piola to continuum mechanics: the creation of peridynamics, non-local and higher gradient continuum mechanics.
