Peridynamics is a conception of continuum mechanics that is aggressive against deformations with discontinuities, abnormally fractures.
Wednesday, February 8, 2012
Purpose of peridynamics
The peridynamic access is based on basic equations, in adverse with the classical access of continuum mechanics, which is based on fractional cogwheel equations. Since fractional derivatives do not abide on able surfaces and added singularities, the classical equations of continuum mechanics cannot be activated anon if such appearance are present in a deformation. The basic equations of the peridynamic access can be activated directly, because they do not crave fractional derivatives.
The adeptness to administer the aforementioned equations anon at all credibility in a algebraic archetypal of a deforming anatomy helps the peridynamic access abstain the charge for the appropriate techniques of breach mechanics. For example, in peridynamics, there is no charge for a abstracted able advance law based on a accent acuteness factor.
The adeptness to administer the aforementioned equations anon at all credibility in a algebraic archetypal of a deforming anatomy helps the peridynamic access abstain the charge for the appropriate techniques of breach mechanics. For example, in peridynamics, there is no charge for a abstracted able advance law based on a accent acuteness factor.
Definition and basic terminology
The basal blueprint of peridynamics is the afterward blueprint of motion:
\rho(x)\ddot u(x,t)=\int_R f(u(x',t)-u(x,t),x'-x,x)dV_{x'} + b(x,t)
where x is a point in a physique R, t is time, u is the displacement agent field, and ρ is the accumulation physique in the undeformed body. x' is a copy capricious of integration.
The agent admired action f is the force physique that x' exerts on x. This force physique depends on the about displacement and about position vectors amid x' and x. The ambit of f are force per aggregate squared. The action f is alleged the "pairwise force function" and contains all the basal (material-dependent) properties. It describes how the centralized armament depend on the deformation.
The alternation amid any x and x' is alleged a "bond." The concrete apparatus in this alternation charge not be specified. It is usually affected that f vanishes whenever x' is alfresco a adjacency of x (in the undeformed configuration) alleged the horizon.
Peridynamics-horizon.jpg
The appellation "peridynamic," an adjective, was proposed in the year 2000 and comes from the prefix peri, which agency all around, near, or surrounding; and the basis dyna, which agency force or power. The appellation "peridynamics," a noun, is a beneath anatomy of the byword peridynamic archetypal of solid mechanics.
\rho(x)\ddot u(x,t)=\int_R f(u(x',t)-u(x,t),x'-x,x)dV_{x'} + b(x,t)
where x is a point in a physique R, t is time, u is the displacement agent field, and ρ is the accumulation physique in the undeformed body. x' is a copy capricious of integration.
The agent admired action f is the force physique that x' exerts on x. This force physique depends on the about displacement and about position vectors amid x' and x. The ambit of f are force per aggregate squared. The action f is alleged the "pairwise force function" and contains all the basal (material-dependent) properties. It describes how the centralized armament depend on the deformation.
The alternation amid any x and x' is alleged a "bond." The concrete apparatus in this alternation charge not be specified. It is usually affected that f vanishes whenever x' is alfresco a adjacency of x (in the undeformed configuration) alleged the horizon.
Peridynamics-horizon.jpg
The appellation "peridynamic," an adjective, was proposed in the year 2000 and comes from the prefix peri, which agency all around, near, or surrounding; and the basis dyna, which agency force or power. The appellation "peridynamics," a noun, is a beneath anatomy of the byword peridynamic archetypal of solid mechanics.
Pairwise force functions
Using the abbreviated characters u = u(x,t) and u' = u(x',t) Newton's third law places the afterward brake on f:
\displaystyle f(u-u', x-x', x') = -f(u'-u, x'-x, x)
for any x,x',u,u'. This blueprint states that the force body agent that x exerts on x' equals bare the force body agent that x' exerts on x. Balance of angular drive requires that f be alongside to the agent abutting the askew position of x to the askew position of x':
\displaystyle ((x'+u')-(x+u))\times f(u'-u, x'-x, x)=0.
A pairwise force action is authentic by a blueprint of | f | against band addendum e, authentic by
\displaystyle e=|(x'+u')-(x+u)|-|x'-x|.
A schematic of a pairwise force action for the band abutting two archetypal credibility is apparent in the afterward figure:
\displaystyle f(u-u', x-x', x') = -f(u'-u, x'-x, x)
for any x,x',u,u'. This blueprint states that the force body agent that x exerts on x' equals bare the force body agent that x' exerts on x. Balance of angular drive requires that f be alongside to the agent abutting the askew position of x to the askew position of x':
\displaystyle ((x'+u')-(x+u))\times f(u'-u, x'-x, x)=0.
A pairwise force action is authentic by a blueprint of | f | against band addendum e, authentic by
\displaystyle e=|(x'+u')-(x+u)|-|x'-x|.
A schematic of a pairwise force action for the band abutting two archetypal credibility is apparent in the afterward figure:
Damage
Damage is congenital in the pairwise force action by acceptance bonds to breach if their addendum exceeds some assigned value. After a band breaks, it no best sustains any force, and the endpoints are finer torn from anniversary other. If a band breaks, the force it was accustomed is redistributed to added bonds that accept not yet broken. This added amount makes it added acceptable that these added bonds will break. The action of band accident and amount redistribution, arch to added breakage, is how cracks abound in the peridynamic model.
Peridynamic states
The approach declared aloft assumes that anniversary peridynamic band responds apart of all the others. This is an oversimplification for a lot of abstracts and leads to restrictions on the types of abstracts that can be modeled. In particular, this acceptance implies that any isotropic beeline adaptable solid is belted to a Poisson arrangement of 1/4.
To abode this abridgement of generality, the abstraction of "peridynamic states" was introduced. This allows the force body in anniversary band to depend on the stretches in all the bonds affiliated to its endpoints, in accession to its own stretch. For example, the force in a band could depend on the net aggregate changes at the endpoints. The aftereffect of this aggregate change, about to the aftereffect of the band stretch, determines the Poisson ratio. With peridynamic states, any actual that can be modeled aural the accepted approach of continuum mechanics can be modeled as a peridynamic material, while application the advantages of the peridynamic approach for fracture.
To abode this abridgement of generality, the abstraction of "peridynamic states" was introduced. This allows the force body in anniversary band to depend on the stretches in all the bonds affiliated to its endpoints, in accession to its own stretch. For example, the force in a band could depend on the net aggregate changes at the endpoints. The aftereffect of this aggregate change, about to the aftereffect of the band stretch, determines the Poisson ratio. With peridynamic states, any actual that can be modeled aural the accepted approach of continuum mechanics can be modeled as a peridynamic material, while application the advantages of the peridynamic approach for fracture.
Subscribe to:
Comments (Atom)