Peridynamics

For any ${\displaystyle {\bf {x}}}$ and ${\displaystyle {\bf {x'}}}$ belonging to the neighborhood ${\displaystyle B_{\delta }({\bf {x}})}$, the following relationship holds: ${\displaystyle {\bf {f}}(-\eta ,-\xi )=-{\bf {f}}(\eta ,\xi )}$ . This expression reflects the principle of action and reaction, commonly known as Newton's Third Law. It guarantees the conservation of linear momentum in a system composed of mutually interacting particles.[1]

For any ${\displaystyle {\bf {x}}}$ and ${\displaystyle {\bf {{x}'}}}$ belonging to the neighborhood ${\displaystyle B_{\delta }({\bf {x}})}$, the following condition holds: ${\displaystyle (\xi +\eta )\times {\bf {f}}(\xi ,\eta )=0}$ . This condition arises from considering the relative deformed ray-vector connecting ${\displaystyle {\bf {x}}}$ and ${\displaystyle {\bf {{x}'}}}$ as ${\displaystyle \xi +\eta }$ . The condition is satisfied if and only if the pairwise force density vector has the same direction as the relative deformed ray-vector. In other words, ${\displaystyle {\bf {f}}(\xi ,\eta )=f(\xi ,\eta )(\xi +\eta )}$ for all ${\displaystyle \xi }$ and ${\displaystyle \eta }$ , where ${\displaystyle f(\xi ,\eta )}$ is a scalar-valued function.[1]

An hyperelastic material is a material with constitutive relation such that:[1]

${\displaystyle \int _{\Gamma }{\bf {f}}({\bf {\xi }},{\bf {\eta }})\cdot d{\bf {\eta }}=0\,,\quad \forall {\text{ closed curve }}\Gamma ,\ \ \ \ \forall {\bf {\xi }}\neq {\bf {{0},}}}$

or, equivalently, by Stokes' theorem

${\displaystyle \nabla _{\bf {\eta }}\times {\bf {f}}({\bf {\xi }},{\bf {\eta }})={\bf {{0}\,}}}$ ,${\displaystyle \forall \,{\bf {\xi }},\,{\bf {\eta }}}$

and, thus,

${\displaystyle {\bf {f}}({\bf {\xi }},{\bf {\eta }})=\nabla _{\bf {\eta }}\Phi ({\bf {\xi }},\,{\bf {\eta }})\,\forall {\bf {\xi }},\,{\bf {\eta }}\,.}$

In the equation above ${\displaystyle \Phi ({\bf {\xi }},{\bf {\eta }})}$ is the scalar valued potential function in ${\displaystyle C^{2}(\mathbb {R} ^{n}\setminus {\bf {{\{0\}}\times \mathbb {R} ^{n})}}}$.[1] Due to the necessity of satisfying angular momentum conservation, the condition below on the scalar valued function ${\displaystyle f({\bf {\xi }},{\bf {\eta }})}$ follows[1]

${\displaystyle {\frac {\partial f({\bf {\xi }},{\bf {\eta }})}{\partial {\bf {\eta }}}}=g({\bf {\xi }},{\bf {\eta }})({\bf {\xi }}+{\bf {\eta }}).}$

where ${\displaystyle g({\bf {\xi }},{\bf {\eta }})}$ is a scalar valued function. Integrating both sides of the equation, the following condition on ${\displaystyle g({\bf {\xi }},{\bf {\eta }})}$ is obtained[1]

${\displaystyle {\bf {f}}({\bf {\xi }},{\bf {\eta }})=h(|{\bf {\xi }}+{\bf {\eta }}|,{\bf {\xi }})({\bf {\xi }}+{\bf {\eta }})}$,   

for ${\displaystyle h(|{\bf {\xi }}+{\bf {\eta }}|,{\bf {\xi }})}$ a scalar valued function. The elastic nature of ${\displaystyle {\bf {f}}}$ is evident: the interaction force depends only on the initial relative position between points ${\displaystyle {\bf {x}}}$ and ${\displaystyle {\bf {x}}'}$ and the modulus of their relative position, ${\displaystyle |{\bf {\xi }}+{\bf {\eta }}|}$, in the deformed configuration ${\displaystyle \Omega _{t}}$ at time ${\displaystyle t}$. Applying the isotropy hypothesis, the dependence on vector ${\displaystyle {\bf {\xi }}}$ can be substituted with a dependence on its modulus ${\displaystyle |{\bf {\xi }}|}$,[1]

${\displaystyle {\bf {f}}({\bf {\xi }},{\bf {\eta }})=h(|{\bf {\xi }}+{\bf {\eta }}|,|{\bf {\xi }}|)({\bf {\xi }}+{\bf {\eta }}).}$

Bond forces can, thus, be considered as modeling a spring net that connects each point  ${\displaystyle {\bf {x}}\in \Omega _{0}}$ pairwise with ${\displaystyle {\bf {x}}'\in B_{\delta }({\bf {x}})\cap \Omega _{0}}$.

If ${\displaystyle |{\bf {\eta }}|\ll 1}$, the peridynamic kernel can be linearised around ${\displaystyle {\bf {\eta }}={\bf {0}}}$:[1]

${\displaystyle {\bf {f}}({\bf {\xi }},{\bf {\eta }})\approx {\bf {f}}({\bf {\xi }},{\bf {{0})+\left.{\frac {\partial {\bf {f}}({\bf {\xi }},{\bf {\eta }})}{\partial {\bf {\eta }}}}\right|_{{\bf {\eta }}={\bf {0}}}{\bf {\eta }};}}}$

then, a second-order micro-modulus tensor can be defined as

${\displaystyle {\bf {C}}({\bf {\xi }})=\left.{\frac {\partial {\bf {f}}({\bf {\xi }},{\bf {\eta }})}{\partial {\bf {\eta }}}}\right|_{{\bf {\eta }}={\bf {0}}}={\bf {\xi }}\otimes \left.{\frac {\partial f({\bf {\xi }},{\bf {\eta }})}{\partial {\bf {\eta }}}}\right|_{{\bf {\eta }}={\bf {0}}}+f_{0}I}$

where ${\displaystyle f_{0}:=f({\bf {\xi }},{\bf {0}})}$ and ${\displaystyle I}$ is the identity tensor. Following application of linear momentum balance, elasticity and isotropy condition, the micro-modulus tensor can be expressed in this form[1]

${\displaystyle {\bf {C}}({\bf {\xi }})=\lambda (|{\bf {\xi }}|){\bf {\xi }}\otimes {\bf {\xi }}+f_{0}I.}$

Therefore for a linearised hyperelastic material, its peridynamic kernel holds the following structure[1]

${\displaystyle {\bf {f}}({\bf {\xi }},{\bf {\eta }})\approx {\bf {f}}({\bf {\xi }},{\bf {0}})+\left(\lambda (|{\bf {\xi }}|){\bf {\xi }}\otimes {\bf {\xi }}+f_{0}I\right){\bf {\eta }}.}$

The peridynamic kernel is a versatile function that characterizes the constitutive behavior of materials within the framework of peridynamic theory. One commonly employed formulation of the kernel is used to describe a class of materials known as prototype micro-elastic brittle (PMB) materials. In the case of isotropic PMB materials, the pairwise force is assumed to be linearly proportional to the finite stretch[7] experienced by the material, defined as

${\displaystyle s:=(|{\bf {\xi }}+{\bf {\eta }}|-|{\bf {\xi }}|)/|{\bf {\xi }}|}$,

so that

${\displaystyle \mathbf {f} ({\bf {\eta }},{\bf {\xi }})=f(|{\bf {\xi }}+{\bf {\eta }}|,|{\bf {\xi }}|){\bf {{n},}}}$

where

${\displaystyle {\bf {{n}:=({\bf {\xi }}+{\bf {\eta }})/|{\bf {\xi }}+{\bf {\eta }}|}}}$

and where the scalar function ${\displaystyle f}$ is defined as follow[7]

${\displaystyle f=cs\mu (s,t)=c\;{\frac {|{\bf {\xi }}+{\bf {\eta }}|-|{\bf {\xi }}|}{|{\bf {\xi }}|}}\mu (s,t),}$ with

${\displaystyle \mu (s,t)=\left\{{\begin{array}{ll}1\,,&{\text{ if }}s\left(t^{\prime },{\bf {\xi }}\right)<s_{0}\,,\\0\,,&{\text{ otherwise, }}\end{array}}\ \ \ \ {\text{ for all }}0\leq t^{\prime }\leq t\right.;}$

The constant ${\displaystyle c}$ is referred to as the micro-modulus constant, and the function ${\displaystyle \mu (s,t)}$ serves to indicate whether, at a given time ${\displaystyle t'\leq t}$, the bond stretch ${\displaystyle s}$ associated with the pair ${\displaystyle ({\bf {x,\,x'}})}$ has surpassed the critical value ${\displaystyle s_{0}}$. If the critical value is exceeded, the bond is considered broken, and a pairwise force of zero is assigned for all ${\displaystyle t\geq t'}$.[1]

After a comparison between the strain energy density value obtained under isotropic extension respectively employing peridynamics and classical continuum theory framework, the physical coherent value of micro-modulus ${\displaystyle c}$ can be found[7]

${\displaystyle c={\frac {18k}{\pi \delta ^{4}}},}$

where ${\displaystyle k}$ is the material bulk modulus.

Following the same approach[10] the micro-modulus constant ${\displaystyle c}$ can be extended to ${\displaystyle c({\bf {\xi }},\delta )}$, where ${\displaystyle c}$ is now a micro-modulus function. This function provides a more detailed description of how the intensity of pairwise forces is distributed over the peridynamic horizon ${\displaystyle B_{\delta }({\bf {x}})}$. Intuitively, the intensity of forces decreases as the distance between ${\displaystyle {\bf {x}}}$ and ${\displaystyle {\bf {x}}'\in B_{\delta }({\bf {x}})}$ increases, but the specific manner in which this decrease occurs can vary.

The micro-modulus function is expressed as[11]

${\displaystyle c({\bf {\xi }},\delta ):=c({\bf {{0},\delta )k({\bf {\xi }},\delta )\,,}}}$

where the constant ${\displaystyle c({\bf {{0},\delta )}}}$ is obtained by comparing peridynamic strain density with the classical mechanical theories;[12] ${\displaystyle k({\bf {\xi }},\delta )}$ is a function defined on ${\displaystyle \Omega _{0}}$ with the following properties (given the restrictions of momentum conservation and isotropy)[11]

${\displaystyle \left\{{\begin{array}{l}k({\bf {\xi }},\delta )=k(-{\bf {\xi }},\delta )\,,\\\lim _{{\bf {\xi }}\rightarrow {\bf {0}}}k({\bf {\xi }},\delta )=\max _{{\bf {\xi }}\ \in \mathbb {R} ^{n}}\{k({\bf {\xi }},\delta )\}\,,\\\lim _{{\bf {\xi }}\rightarrow \delta }k({\bf {\xi }},\delta )=0\,,\\\int _{\mathbb {R} ^{n}}\lim _{\delta \rightarrow 0}k({\bf {\xi }},\delta )d{\bf {x}}=\int _{\mathbb {R} ^{n}}\Delta ({\bf {\xi }})d{\bf {x}}=1\,,\end{array}}\right.}$

where ${\displaystyle \Delta ({\bf {\xi }})}$ is the Dirac Delta function.

Pictorial representation of some widely used micro-modulus function ${\displaystyle c({\bf {\xi }},\delta )=c({\bf {{0},\delta )k({\bf {\xi }},\delta )}}}$.

The simplest expression for the micro-modulus function is[11]

${\displaystyle c({\bf {{0},\delta )k({\bf {\xi }},\delta )=c{\bf {{1}_{B_{\delta }({\bf {x}}')}}}}}}$,

where ${\displaystyle {\bf {{1}_{A}}}}$: ${\displaystyle X\rightarrow \mathbb {R} }$ is the indicator function of the subset ${\displaystyle A\subset X}$, defined as

${\displaystyle \mathbf {1} _{A}(x):={\begin{cases}1,&x\in A\,,\\0,&x\notin A\,,\end{cases}}\;\;;}$

It is characterized by ${\displaystyle k({\bf {\xi }},\delta )}$ to a be a linear function[13]

${\displaystyle k({\bf {\xi }},\delta )=\left(1-{\frac {|{\bf {\xi }}|}{\delta }}\right){\bf {{1}_{B_{\delta }({\bf {x}}')}.}}}$

If one wants to reflects the fact that most common discrete physical systems are characterized by a Maxwell-Boltzmann distribution, in order to include this behavior in peridynamics, the following expression for ${\displaystyle k({\bf {\xi }},\delta )}$ can be utilized[14]

${\displaystyle k({\bf {\xi }},\delta )=e^{-(|{\bf {\xi }}|/\delta )^{2}}{\bf {{1}_{B_{\delta }({\bf {x}}')};}}}$

In the literature one can find also the following expression for the ${\displaystyle k({\bf {\xi }},\delta )}$function[11]

${\displaystyle k({\bf {\xi }},\delta )=\left(1-\left({\frac {\xi }{\delta }}\right)^{2}\right)^{2}{\bf {{1}_{B_{\delta }({\bf {x}}')}.}}}$

Overall, there exists a wide ranges of expression for the micro-modulus and in general, for the peridynamic kernel, depending on the specific material property to be modeled. The above list is, thus, not exhaustive.[11]