Peridynamics - AbsoluteAstronomy.com

Peridynamics is a formulation of continuum mechanics

Continuum mechanics

Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and the mechanical behavior of materials modelled as a continuous mass rather than as discrete particles...

that is oriented toward deformations with discontinuities, especially fracture

Fracture

A fracture is the separation of an object or material into two, or more, pieces under the action of stress.The word fracture is often applied to bones of living creatures , or to crystals or crystalline materials, such as gemstones or metal...

Purpose of peridynamics

The peridynamic theory is based on integral equation

Integral equation

In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. There is a close connection between differential and integral equations, and some problems may be formulated either way...

s, in contrast with the classical theory of continuum mechanics, which is based on partial differential equation

Partial differential equation

In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...

s. Since partial derivative

Partial derivative

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant...

s do not exist on crack surfaces and other singularities

Mathematical singularity

In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...

, the classical equations of continuum mechanics

Continuum mechanics

Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and the mechanical behavior of materials modelled as a continuous mass rather than as discrete particles...

cannot be applied directly when such features are present in a deformation. The integral equations of the peridynamic theory can be applied directly, because they do not require partial derivatives.

The ability to apply the same equations directly at all points in a mathematical model of a deforming structure helps the peridynamic approach avoid the need for the special techniques of fracture mechanics

Fracture mechanics

Fracture mechanics is the field of mechanics concerned with the study of the propagation of cracks in materials. It uses methods of analytical solid mechanics to calculate the driving force on a crack and those of experimental solid mechanics to characterize the material's resistance to fracture.In...

. For example, in peridynamics, there is no need for a separate crack growth law based on a stress intensity factor

Stress Intensity Factor

The stress intensity factor, K, is used in fracture mechanics to predict the stress state near the tip of a crack caused by a remote load or residual stresses. It is a theoretical construct usually applied to a homogeneous, linear elastic material and is useful for providing a failure criterion...

Definition and basic terminology

The basic equation of peridynamics is the following equation of motion

Equation of motion

Equations of motion are equations that describe the behavior of a system in terms of its motion as a function of time...

where

is a point in a body

is time,

is the displacement vector

Displacement (vector)

A displacement is the shortest distance from the initial to the final position of a point P. Thus, it is the length of an imaginary straight path, typically distinct from the path actually travelled by P...

field, and

is the mass density in the undeformed body.
x' is a dummy variable of integration.

The vector valued function

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

is the force density that

exerts on

. This force density depends on the relative displacement and relative position vectors between

and

. The dimension

Dimension

In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...

s of

are force per volume squared. The function

is called the "pairwise force function" and contains all the constitutive

Constitutive equation

In physics, a constitutive equation is a relation between two physical quantities that is specific to a material or substance, and approximates the response of that material to external forces...

(material-dependent) properties. It describes how the internal forces depend on the deformation.

The interaction between any

and

is called a "bond." The physical mechanism in this interaction need not be specified.
It is usually assumed that

vanishes whenever

is outside a neighborhood of

(in the undeformed configuration) called the horizon.

The term "peridynamic," an adjective, was proposed in the year 2000 and comes from the prefix peri, which means all around, near, or surrounding; and the root dyna, which means force or power. The term "peridynamics," a noun, is a shortened form of the phrase peridynamic model of solid mechanics.

Pairwise force functions

Using the abbreviated notation

and

Newton's third law places the following restriction on

for any

. This equation states that
the force density vector that

exerts on

equals minus the force density vector that

exerts on

. Balance of angular momentum

Angular momentum

In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...

requires that

be parallel to the vector connecting the deformed position of

to the deformed position of

A pairwise force function is specified by a graph of

versus bond elongation

, defined by

A schematic of a pairwise force function for the bond connecting two typical points is shown in the following figure:

Damage

Damage is incorporated in the pairwise force function by allowing bonds to break when their elongation exceeds some prescribed value. After a bond breaks, it no longer sustains any force, and the endpoints are effectively disconnected from each other. When a bond breaks, the force it was carrying is redistributed to other bonds that have not yet broken. This increased load makes it more likely that these other bonds will break. The process of bond breakage and load redistribution, leading to further breakage, is how cracks grow in the peridynamic model.

Peridynamic states

The theory described above assumes that each peridynamic bond responds independently of all the others. This is an oversimplification for most materials and leads to restrictions on the types of materials that can be modeled. In particular, this assumption implies that any isotropic linear elastic solid is restricted to a Poisson ratio of 1/4.

To address this lack of generality, the idea of "peridynamic states" was introduced. This allows the force density in each bond to depend on the stretches in all the bonds connected to its endpoints, in addition to its own stretch. For example, the force in a bond could depend on the net volume changes at the endpoints. The effect of this volume change, relative to the effect of the bond stretch, determines the Poisson ratio. With peridynamic states, any material that can be modeled within the standard theory of continuum mechanics

Continuum mechanics

Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and the mechanical behavior of materials modelled as a continuous mass rather than as discrete particles...

can be modeled as a peridynamic material, while retaining the advantages of the peridynamic theory for fracture.

External links

The source of this article is wikipedia, the free encyclopedia. The text of this article is licensed under the GFDL.

Purpose of peridynamics

Definition and basic terminology

Pairwise force functions

Damage

Peridynamic states

Further reading

External links