Contact forces (Cauchy's hypothesis)
force exerted by $\Omega_t \setminus U_t$ on $U_t$ $ = \displaystyle \int_{\partial U_t} s(t,y,\nu(y)) \, \mathrm{d} \mathcal{H}^{n-1}(y)$
Cauchy's theorem
$s(t,y,\nu) = T(t,y) \nu$
Some remarks on the peridynamics model
Gianluca Orlando
Joint works with:
G. M. Coclite (Politecnico di Bari), S. Dipierro (UWA)
F. Maddalena (Politecnico di Bari), E. Valdinoci (UWA)
Third Workshop on Mathematical Challenges in Ecology and Biology
November 19-21, 2025
Classical mechanics
Classical mechanics
Balance of linear momentum (no body forces)
In spatial coordinates: $\hspace{0.2cm} \rho \dot v(t,y) - \mathrm{div}_y T(t,y) = 0$, $\hspace{0.2cm} t > 0$, $\hspace{0.2cm} y \in \Omega_t$
In material coordinates: $\hspace{0.2cm} \rho_0 \partial_{tt} y(t,x) - \mathrm{div}_x S(t,x) = 0$, $\hspace{0.2cm} t > 0$, $\hspace{0.2cm} x \in \Omega$
Constitutive assumptions
Hyperelasticity: $\hspace{0.2cm} \displaystyle S = \frac{\partial W}{\partial F}(F)$, $\hspace{0.2cm} F = \nabla y^\top$, $\hspace{0.2cm} \displaystyle \int_\Omega W(\nabla y) \, \mathrm{d} x$
Prototypical model: $\hspace{0.2cm} \partial_{tt} u(t,x) - \Delta u(t,x) = 0 \hspace{0.2cm}$ (linearization $y = x + \varepsilon u$)
Caveats
Pros:
- Works very well
Cons:
- No nucleation of discontinuities (cracks)
- No long-range forces
- What if $\nabla u$ is undefined?
Peri (= "near") + dynamics (= "force")
S. Silling, Journal of the Mechanics and Physics of Solids 48, 175-209 (2000)
Long-range forces
\[ \mathrm{div}_x S(t,x) \leadsto \int_{B_{\delta}(x)} F(t, x, x', y(t,x),y(t,x')) \, \mathrm{d} x' \]Peridynamic equation of motion
\[ \rho_0 \partial_{tt} y(t,x) - \int_{B_{\delta}(x)} F(t, x, x', y(t,x),y(t,x')) \, \mathrm{d} x' = 0 \]Peri (= "near") + dynamics (= "force")
Constitutive assumptions
$\delta > 0$, $\hspace{0.2cm} \alpha \in (0,1)$
\[ \begin{split} \hspace{-0.5cm} \int_{B_{\delta}(x)} F(t, x, x', y(t,x),y(t,x')) \, \mathrm{d} x' & = \int_{B_{\delta}(x)} f(x'-x, u(t,x')-u(t,x)) \, \mathrm{d} x' \\ & = \mathrm{p.v.} \! \int_{B_{\delta}(x)} \! \chi_\delta(x'-x)\frac{u(t,x') - u(t,x)}{|x' - x|^{d + 2 \alpha}} \, \mathrm{d} x' \end{split} \]
\[ \begin{split} \hspace{-0.5cm} \int_{B_{\delta}(x)} F(t, x, x', y(t,x),y(t,x')) \, \mathrm{d} x' & = \int_{B_{\delta}(x)} f(x'-x, u(t,x')-u(t,x)) \, \mathrm{d} x' \\ & = \mathrm{p.v.} \! \int_{B_{\delta}(x)} \! \frac{u(t,x') - u(t,x)}{|x' - x|^{1 + 2 \alpha}} \, \mathrm{d} x' \hphantom{\chi_\delta(x'-x)} \end{split} \]
Prototypical model: $\hspace{0.2cm} \displaystyle \partial_{tt} u(t,x) - \mathrm{p.v.} \int_{B_{\delta}(x)} \frac{u(t,x') - u(t,x)}{|x' - x|^{1 + 2 \alpha}} \, \mathrm{d} x' = 0$
Remarks
- $f(x'-x,u'-u) = - f(x-x',u-u')$
- $\displaystyle f(x'-x, u'-u) = \partial_{u'-u} \Big( \frac{1}{2} \frac{|u'-u|^2}{|x'-x|^{1 + 2 \alpha}} \Big)$
Objectives
Prototypical classical waves
\[ \partial_{tt} u(t,x) - \partial_{xx} u(t,x) = 0 \]Prototypical peridynamic waves
\[ \partial_{tt} u(t,x) - \mathrm{p.v.} \int_{B_{\delta}(x)} \frac{u(x') - u(x)}{|x' - x|^{1 + 2 \alpha}} \, \mathrm{d} x' = 0 \]Questions
- Are they related?
- Does peridynamics allow for discontinuity nucleation?
Convergence of operators
Lemma
For $u \in C^\infty_c(\mathbb{R})$, we have that
\[ K_\delta[u](x) = \frac{1}{\delta^{2(1-\alpha)}} \mathrm{p.v.} \int_{B_{\delta}(x)} \frac{u(x') - u(x)}{|x' - x|^{1 + 2 \alpha}} \, \mathrm{d} x' \hspace{0.2cm} \stackrel{\delta \to 0}{\longrightarrow} \hspace{0.2cm} \gamma_\alpha^2 \partial_{xx} u(x) \]Back-of-the-envelope computation
For $u$ smooth, we have that
\[ \begin{split} & \mathrm{p.v.} \int_{B_{\delta}(x)} \frac{u(x') - u(x)}{|x' - x|^{1 + 2 \alpha}} \, \mathrm{d} x' = \mathrm{p.v.} \int_{\{ |z| \leq \delta \}} \frac{u(x+z) - u(x)}{|z|^{1 + 2 \alpha}} \, \mathrm{d} z \\ & \quad = \mathrm{p.v.} \int_{\{ |z| \leq \delta \}} \frac{\partial_{x} u(x) z + \frac{1}{2} \partial_{xx} u(x) z^2 + O(|z|^3)}{|z|^{1 + 2 \alpha}} \, \mathrm{d} z \\ & \quad = \delta^{2-2\alpha} \int_{\{ |w| \leq 1 \}} \frac{\frac{1}{2} \partial_{xx} u(x) w^2}{|w|^{1 + 2 \alpha}} \, \mathrm{d} w + \delta^{3-2\alpha} \int_{\{ |w| \leq 1 \}} \frac{O(1)}{|w|^{1 + 2 \alpha - 3}} \, \mathrm{d} w \\ & \quad = \delta^{2-2\alpha} \frac{1}{2} \partial_{xx}u(x) \int_{\{ |w| \leq 1 \}} |w|^{1-2\alpha} \, \mathrm{d} w + \delta^{3-2\alpha} O(1) \\ & \quad = \delta^{2-2\alpha} \partial_{xx}u(x) \frac{1}{2(1-\alpha)} + \delta^{3-2\alpha} O(1) \\ & \quad = \delta^{2(1-\alpha)} \gamma_\alpha^2 \partial_{xx}u(x) + \delta^{3-2\alpha} O(1) \end{split} \]hence
\[ K_\delta[u](x) = \frac{1}{\delta^{2(1-\alpha)}} \mathrm{p.v.} \int_{B_{\delta}(x)} \frac{u(x') - u(x)}{|x' - x|^{1 + 2 \alpha}} \, \mathrm{d} x' = \gamma_\alpha^2 \partial_{xx}u(x) + \delta O(1) \]Solutions to peridynamic equation
G. M. Coclite, S. Dipierro, F. Maddalena, E. Valdinoci, Nonlinearity 32, 1 (2018)
In the linear case, \[ \begin{cases} \partial_{tt} u(t,x) - K_\delta[u](t,x) = 0 & (t,x) \in (0,+\infty) \times \mathbb{T}^1 \\ u(0,x) = u_0(x) \, , \quad \partial_t u(0,x) = v_0(x) & x \in \mathbb{T}^1 \end{cases} \] reads, in Fourier coefficients, \[ \begin{cases} \partial_{tt} \hat u(t,k) + \omega_\delta^2(k) \hat u(t,k) = 0 & (t,k) \in (0,+\infty) \times \mathbb{Z} \\ \hat u(0,k) = \hat u_0(k) \, , \quad \partial_t \hat u(0,k) = \hat v_0(k) & k \in \mathbb{Z} \end{cases} \] Solutions in Fourier space are given by \[ \hat u(t,k) = \hat u_0(k) \cos(\omega_\delta(k) t) + \frac{\sin(\omega_\delta(k) t)}{\omega_\delta(k)} \hat v_0(k) \]Back-of-the-envelope computation
Peridynamic waves
Energy and solution space
Lemma
The quantity
\[ \frac{1}{2} \int_{\mathbb{T}^1} |\partial_t u_\delta(t,x)|^2 \, \mathrm{d} x + \frac{1}{2} \int_{\mathbb{T}^1} \int_{B_{\delta}(x)} \frac{|u_\delta(t,x') - u_\delta(t,x)|^2}{|x' - x|^{1 + 2 \alpha}} \, \mathrm{d} x' \, \mathrm{d} x \]is conserved along solutions.
Remark
We have that
\[ \frac{1}{2} \int_{\mathbb{T}^1} \int_{B_{\delta}(x)} \frac{|u(x') - u(x)|^2}{|x' - x|^{1 + 2 \alpha}} \, \mathrm{d} x' \, \mathrm{d} x \sim [u]_{H^\alpha(\mathbb{T}^1)}^2 \]Solutions to peridynamic equation
Theorem
Let $u_0 \in H^\alpha(\mathbb{T}^1)$, $v_0 \in L^2(\mathbb{T}^1)$. There exists a unique distributional solution $u_\delta \in L^1_{\mathrm{loc}}((0,+\infty); H^\alpha(\mathbb{T}^1))$ to
\[ \begin{cases} \partial_{tt} u_\delta(t,x) - K_\delta[u_\delta](t,x) = 0 & (t,x) \in (0,+\infty) \times \mathbb{T}^1 \\ u_\delta(0,x) = u_0(x) \, , \quad \partial_t u_\delta(0,x) = v_0(x) & x \in \mathbb{T}^1 \end{cases} \] Moreover, it is given by \[ u_\delta(t,x) = \sum_{k \in \mathbb{Z}} \Big( \hat u_0(k) \cos(\omega_\delta(k) t) + \frac{\sin(\omega_\delta(k) t)}{\omega_\delta(k)} \hat v_0(k) \Big) e^{2 \pi i k x} \]and satisfies $u_\delta \in C([0,+\infty); H^\alpha(\mathbb{T}^1)) \cap C^1([0,+\infty); L^2(\mathbb{T}^1))$.
Solutions to peridynamic equation
Theorem
Let $s \in \mathbb{R}$. Let $u_0 \in $$\, H^s$$(\mathbb{T}^1)$, $v_0 \in $$\, H^{s-\alpha}$$(\mathbb{T}^1)$. There exists a unique distributional solution $u_\delta \in L^1_{\mathrm{loc}}((0,+\infty); $$\, H^s$$(\mathbb{T}^1))$ to
\[ \begin{cases} \partial_{tt} u_\delta(t,x) - K_\delta[u_\delta](t,x) = 0 & (t,x) \in (0,+\infty) \times \mathbb{T}^1 \\ u_\delta(0,x) = u_0(x) \, , \quad \partial_t u_\delta(0,x) = v_0(x) & x \in \mathbb{T}^1 \end{cases} \] Moreover, it is given by \[ u_\delta(t,x) = \sum_{k \in \mathbb{Z}} \Big( \hat u_0(k) \cos(\omega_\delta(k) t) + \frac{\sin(\omega_\delta(k) t)}{\omega_\delta(k)} \hat v_0(k) \Big) e^{2 \pi i k x} \]and satisfies $u_\delta \in C([0,+\infty); $$\, H^s$$(\mathbb{T}^1)) \cap C^1([0,+\infty); $$\, H^{s-\alpha}$$(\mathbb{T}^1))$.
Vanishing horizon
Theorem [Coclite, Dipierro, Maddalena, O., Valdinoci]
Let $s \in \mathbb{R}$. Let $u_0 \in$$\, H^s$$(\mathbb{T}^1)$, $v_0 \in $$\, H^{s-\alpha}$$(\mathbb{T}^1)$.
Let $u_\delta \in C([0,+\infty); $$\, H^s$$(\mathbb{T}^1)) \cap C^1([0,+\infty); $$\, H^{s-\alpha}$$(\mathbb{T}^1))$ solve
\[ \begin{cases} \partial_{tt} u_\delta(t,x) - K_\delta[u_\delta](t,x) = 0 & (t,x) \in (0,+\infty) \times \mathbb{T}^1 \\ u_\delta(0,x) = u_0(x) \, , \quad \partial_t u_\delta(0,x) = v_0(x) & x \in \mathbb{T}^1 \end{cases} \]Let $u \in \in C([0,+\infty); $$\, H^s$$(\mathbb{T}^1)) \cap C^1([0,+\infty); $$\, H^{s-1}$$(\mathbb{T}^1))$ solve
\[ \begin{cases} \partial_{tt} u(t,x) - \gamma_\alpha^2 \Delta u(t,x) = 0 & (t,x) \in (0,+\infty) \times \mathbb{T}^1 \\ u(0,x) = u_0(x) \, , \quad \partial_t u(0,x) = v_0(x) & x \in \mathbb{T}^1 \end{cases} \]Then, locally in time,
\[ \| u_\delta - u \|_{L^\infty_t H^s_x} + \|\partial_t u_\delta - \partial_t u \|_{L^\infty_t H^{s-1}_x} \to 0 \quad \text{as } \delta \to 0 \]Dispersion relation
G. M. Coclite, S. Dipierro, G. Fanizza, E. Valdinoci, Nonlinearity 35, 11 (2022)
\[ \omega_\delta(\xi) = \Big( \frac{1}{\delta^{2(1-\alpha)}}\int_{\{|y| \leq \delta\}} \frac{1 - \cos(2 \pi \xi y)}{|y|^{1+2\alpha}} \, \mathrm{d} y \Big)^\frac{1}{2} \sim \min\{ \gamma_\alpha |\xi|, \lambda_\alpha |\xi|^\alpha \} \]$\alpha = 0.2$
Lemma: $\hspace{0.2cm}\big| \omega_\delta(\xi) - 2 \pi \gamma_\alpha |\xi| \big| \leq C \delta |\xi|^2$
Computation
We have that
\[ \begin{split} & \big| \omega_\delta^2(\xi) - 4 \pi^2 \gamma_\alpha^2 |\xi|^2 \big| = \Big|\frac{1}{\delta^{2(1-\alpha)}} \int_{\{|y| \leq \delta\}} \frac{1-\cos(2\pi \xi y)}{|y|^{1+2\alpha}} \, \mathrm{d} y - \frac{4 \pi^2}{2(1-\alpha)} |\xi|^2\Big| \\ & = \Big|\frac{1}{\delta^{2(1-\alpha)}} \int_{\{|y| \leq \delta\}} \frac{2 \pi^2 |\xi|^2 |y|^2 + O(|\xi|^4|y|^4)}{|y|^{1+2\alpha}} \, \mathrm{d} y - 2 \pi^2|\xi|^2 \int_{\{|w|\leq 1\}} |w|^{1-2\alpha} \, \mathrm{d} w \Big| \\ & = \Big|2 \pi^2 |\xi|^2 \int_{\{|w| \leq 1\}} \frac{|w|^2 + \delta^2 O(|\xi|^2|w|^4)}{|w|^{1+2\alpha}} \, \mathrm{d} w - 2 \pi^2|\xi|^2 \int_{\{|w|\leq 1\}} |w|^{1-2\alpha} \, \mathrm{d} w \Big| \\ & \lesssim \delta^2 |\xi|^4 \int_{\{|w| \leq 1\}} |w|^{3-2\alpha} \, \mathrm{d} w \lesssim C \delta^2 |\xi|^4 \end{split} \]Nucleation of a discontinuity?
In dimension $d=1$
\[ H^\alpha(\mathbb{T}^1) \not\subset C(\mathbb{T}^1) \quad \text{for } \alpha < \frac{1}{2} \, . \]In principle, solutions may become discontinuous.
Question:
Does there exist a solution with a continuous initial datum which evolves into a discontinuous one?
Classical waves stay continuous
A travelling discontinuity
Numerical example of singularity dissolution in peridynamics
Analytical example of singularity dissolution in peridynamics
Theorem [Coclite, Dipierro, Maddalena, O., Valdinoci]
Let $\alpha \in (0,\frac{1}{2})$.
Consider the initial data
\[ u_0(x) = \frac{1}{2} - x \, , \quad v_0(x) = 0 \quad x \in [0,1] \,. \]Let $u \in C([0,+\infty); H^\alpha(\mathbb{T}^1)) \cap C^1([0,+\infty); L^2(\mathbb{T}^1))$ be the unique solution to the peridynamic problem.
Then $u(t,\cdot)$ is continuous for all $t > 0$.
Proof
Key fact
For all $t > 0$, the trigonometric series
\[ \sum_{k=1}^{+\infty} \frac{1}{k} e^{i 2 \pi k x + i \omega_\delta(k) t} \]converges uniformly in $x \in \mathbb{T}^1$.
Proof in the spirit of Van der Corput estimates.
A. Zygmund, Trigonometric series. Vol. I, II. Cambridge University Press (1959)
Main steps
Step 1: Summation by parts
Set $\displaystyle s_k = \sum_{h=1}^{k} e^{i 2 \pi h x + i h^\alpha t}$ and sum by parts
\[ \sum_{k=1}^{N} \frac{1}{k} e^{i 2 \pi k x + i k^\alpha t} \approx \frac{1}{N} s_N - \sum_{k=1}^{N} \frac{1}{k^2} s_k \hphantom{ \hspace{0.1cm} \approx N^{-\alpha/2} + \sum_{k=1}^{N} \frac{1}{k^{1+\alpha/2}}} \]
\[ \sum_{k=1}^{N} \frac{1}{k} e^{i 2 \pi k x + i k^\alpha t} \approx \frac{1}{N} s_N - \sum_{k=1}^{N} \frac{1}{k^2} s_k \approx N^{-\alpha/2} + \sum_{k=1}^{N} \frac{1}{k^{1+\alpha/2}} \]