Noether’s theorem in peridynamics

Abstract

By introducing a new nonlocal argument, the Lagrangian formulation of peridynamics is investigated. The peridynamic Euler–Lagrange equation is derived from Hamilton’s principle, and Noether’s theorem is extended into peridynamics. With the help of the peridynamic Noether’s theorem, the conservation laws relevant to energy, linear momentum, angular momentum and the Eshelby integral are determined. The results show that the peridynamic conservation laws exist only in a spatial integral form rather than in a pointwise form due to nonlocality. In bond-based peridynamics, energy conservation requires that the influence function is independent of the relative displacement field, or energy dissipation will occur. In state-based peridynamics, the angular momentum conservation causes a constraint on the constitutive relation between the force vector-state and the deformation vector-state. The Eshelby integral of peridynamics is given, which can be used to judge nucleation of defects and to calculate the energy release rates caused by damage, fracture and phase transition.

Keywords

Noether’s theorem conservation law peridynamics Euler–Lagrange equation

1. Introduction

Noether’s theorem is commonly used to show the conservation of physical quantities as a consequence of the symmetry properties of the Lagrangian action functional. In classical continuum mechanics, various conservation quantities (e.g. the Eshelby tensor or J-integral) derived from Noether’s theorem have been applied to analysis for elastic waves, fracture, damage and inhomogeneity of materials [1 –4].

With the introduction of nonlocality, Edelen [5 –7] first studied the reformulation of Noether’s theorem in the framework of nonlocal field theory. The nonlocal argument specified by Edelen [5] is too general in form to correlate with concrete physical laws. Later, Edelen [8] simplified the nonlocal argument into a linear integral operator on the field variable. Based on Eringen’s nonlocal constitutive model [9], Vukobrat and Kuzmanović [10] addressed the conservation laws in nonlocal elasticity. Lazar and Kirchner [11] discussed the energy-momentum tensor in nonlocal elasticity and nonlocal micropolar elasticity, and issued some interesting results on the interaction between dislocation and disclination. Huang [12] proposed a nonlocal form of Noether’s theorem relevant to physically based nonlocal elasticity theory [13].

Peridynamics is a new branch of continuum mechanics developed during the last two decades. It was first advanced by Silling [14] and has been applied to various topics relevant to impact, fracture and damage [14 –18]. In peridynamics, Silling [14] introduced the internal long-range body force to represent the interactions within a body, but shunned the concepts of stress and strain. An integral operator was used to formulate the constitutive relation, correlating the internal long-range body force with relative displacement. Since there is no requirement of differentiability for the displacement field in the motion equations of peridynamics, this theory is suitable for studying phenomena with discontinuities and fragmentation. Some typical examples can be found in Madenci and Oterkus [17] and Bobaru et al. [18]

Noether’s theorem is established on the basis of variational calculus. Within the framework of peridynamics, nonlocal variational calculus has been systematically investigated by Du and his collaborators [19, 20]. However, very few results are known on the form of Noether’s theorem and the relevant conservation laws in peridynamics. So the objective of the current work is to clarify this subject.

The paper is organized as follows: in section 2, we propose the Lagrange formulation of peridynamics based on a so-called nonlocal argument. Noether’s theorem is then extended into peridynamics. According to the extended Noether’s theorem, in section 3 we investigate the conservation laws in linear bond-based peridynamics. In section 4, the motion equation of state-based peridynamics is derived from the Euler–Lagrange equation given in section 2. The state-based peridynamic conservation laws are investigated. Finally, we close this paper by making some concluding remarks.

Notation: A compact notation is used, with boldface letters being vectors or tensors. The index rules and summation convention are adopted. Partial derivatives with respect to coordinates are represented as $a_{, k} = \partial a / \partial x^{k}$ . Partial derivatives with respect to time are denoted by an upper dot, i.e. $\overset{\cdot}{a} = \partial a / \partial t$ . The norm |·| is defined as $\sqrt{(\cdot) (\cdot)}$ . For example, $| r | = \sqrt{r \cdot r}$ . Other symbols will be introduced in the text where they appear for the first time.

2. Lagrangian formulation of peridynamics

2.1. Euler–Lagrange equation

A continuum occupies the domain $Ω$ in the three-dimensional Euclidean space. Let every particle in the continuum be referred to by the orthogonal Cartesian coordinates $x = {x^{1}, x^{2}, x^{3}}$ specifying its position in $Ω$ . The field argument $ϕ_{k} = ϕ_{k} (t, x)$ is defined on $Ω$ , and the nonlocal argument $〈 h | ϕ_{k} 〉$ of $ϕ_{k}$ is defined as

〈 h | ϕ_{k} 〉 = ϕ_{k} (t, x) \int_{Ω} h (| ξ |, | η |) d v (z) - \int_{Ω} h (| ξ |, | η |) ϕ_{k} (t, z) d v (z),

(1)

where $ξ = z - x$ , $η_{k} = ϕ_{k} (t, x) - ϕ_{k} (t, z)$ , $η = | η_{k} |$ and $h = h (| ξ |, η)$ is referred to as the influence function. Clearly, equation (1) can be abbreviated to

〈 h | ϕ_{k} 〉 = \int_{Ω} h (| ξ |, η) η_{k} d v (z) .

(2)

It is easy to verify that $〈 h | ϕ_{k} 〉$ satisfies the zero mean condition:

\int_{Ω} 〈 h | ϕ_{k} 〉 d v (x) = 0 .

(3)

Due to equation (3), the nonlocal argument determined by equation (1) is in essence different from the definition given by Edelen [5, 8]. The latter fails to satisfy the zero mean condition.

The Lagrangian of peridynamics can be taken as $L = L (ϕ_{k}, {\overset{\cdot}{ϕ}}_{k}, 〈 h | ϕ_{k} 〉) (k = 1, 2, 3)$ . So the action functional of $ϕ_{k}$ can be written as

A [ϕ_{k}] = \int_{t_{0}}^{t_{1}} \int_{Ω} L (ϕ_{k}, {\overset{\cdot}{ϕ}}_{k}, 〈 h | ϕ_{k} 〉) d v (x) d t .

(4)

Suppose $ϕ_{k} (t, x)$ , $h (| ξ |, η)$ and L are suitably smooth functions. In terms of Hamilton’s principle, we have $δ A [ϕ_{k}] = 0$ . So the fundamental lemma of variation yields the below results:

\frac{d}{d t} (\frac{\partial L}{\partial {\overset{\cdot}{ϕ}}_{k}}) - \frac{\partial L}{\partial ϕ_{k}} = 〈 g | ϕ_{k}, \frac{\partial L}{\partial 〈 h | ϕ_{k} 〉} 〉 .

(5)

Equation (5) is also referred to as the peridynamic Euler–Lagrange equation, which can be directly acquired by simplifying the relevant equations in Huang [12]. The right-side term of equation (5) is the peridynamic force density, which reads

〈 g | ϕ_{k}, \frac{\partial L}{\partial 〈 h | ϕ_{k} 〉} 〉 = \frac{\partial L}{\partial 〈 h | ϕ_{k} 〉} \int_{Ω} g (| ξ |, η) d v (z) - \int_{Ω} g (| ξ |, η) \frac{\partial L}{\partial 〈 h | ϕ_{k} 〉} d v (z),

(6)

where

g (| ξ |, η) = h (| ξ |, η) + η \frac{\partial h (| ξ |, η)}{\partial η} .

(7)

Interchanging $x$ and $y$ in the integral of equation (6) leads to

\int_{Ω} 〈 g | ϕ_{k}, \frac{\partial L}{\partial 〈 h | ϕ_{k} 〉} 〉 d v (x) = 0,

(8)

which shows that the peridynamic force density satisfies the zero mean condition automatically. In physics, the peridynamic force density represents an internal long-range body force applied on a particle by other particles within the body. Every particle is subjected to such a force. In terms of the action and reaction law, the sum of all such forces must be zero. Therefore, equation (8) is just an embodiment of the action and reaction law.

2.2. Noether’s theorem

Consider the infinitesimal transformations of the Lie group

t \mapsto \bar{t} = t + ϵ^{0} T,

(9)

x^{k} \mapsto {\bar{x}}^{k} = x^{k} + ϵ^{j} X_{j}^{k},

(10)

ϕ_{k} (t, x) \mapsto {\bar{ϕ}}_{k} (\bar{t}, \bar{x}) = ϕ_{k} (t, x) + ϵ^{α} U_{k α},

(11)

where $ϵ^{0}$ and $ϵ^{j}$ are infinitesimal parameters independent of the space-time coordinates. T (sometimes also denoted as $X_{0}^{0}$ ), $X_{j}^{k}$ and $U_{k α} α = 0, 1, 2, 3$ are the generators of the Lie group. The action functional $A [ϕ_{k}]$ is said to be symmetric if it is form-invariant with respect to the infinitesimal transformations in equations (9)–(11), i.e.

\int_{\bar{Ω}} L [{\bar{ϕ}}_{k} (\bar{t}, \bar{x}), 〈 h | {\bar{ϕ}}_{k} 〉] d v (\bar{x}) = \int_{Ω} L [ϕ_{k} (t, x), 〈 h | ϕ_{k} 〉] d v (x) .

(12)

It should be noted that $〈 h | {\bar{ϕ}}_{k} 〉$ in the left-hand term of equation (12) is an integral defined on $\bar{Ω}$ , but the integral domain of $〈 h | ϕ_{k} 〉$ in the right-hand term is $Ω$ . The Noether theorem asserts that equation (12) holds if and only if

\frac{d}{d t} \int_{Ω} (\frac{\partial L}{\partial {\overset{\cdot}{ϕ}}_{k}} {\overset{\cdot}{ϕ}}_{k} - L) T d v (x) = 0

(13)

and

\int_{Ω} {\frac{d}{d t} [\frac{\partial L}{\partial {\overset{\cdot}{ϕ}}_{k}} (U_{jk} - ϕ_{k, i} X_{j}^{i})] + \partial_{k} [(L - \int_{Ω} h (| ξ |, η) \frac{\partial L}{\partial 〈 h | ϕ_{i} 〉} η_{i} d v (y)) X_{j}^{k}]} d v (x) = 0 .

(14)

Equations (13) and (14) are general forms of the conservation laws in peridynamics. They are global and not pointwise equations due to the occurrence of the nonlocal argument in the Lagrangian. As a consequence of the invariance of the Lagrangian action functional under the infinitesimal transformations of the Lie group, equations (13) and (14) can be also acquired by simplifying the equation below

\int_{t_{1}}^{t_{2}} \int_{Ω} \partial_{γ} [(L δ_{β}^{γ} - \frac{\partial L}{\partial ϕ_{k, γ}} ϕ_{k, β}) X_{α}^{β} + \frac{\partial L}{\partial ϕ_{k, γ}} U_{k α}] d v (x) d t = \int_{t_{1}}^{t_{2}} \int_{Ω} \partial_{k} [X_{α}^{k} \int_{Ω} h (| ξ |, η) \frac{\partial L}{\partial 〈 h | ϕ_{i} 〉} η_{i} d v (y)] d v (x) d t,

(15)

where the Latin indices have the range 1, 2, 3; while the Greek indices run from 0 to 3. Thus, we have ${x^{α}} = {x^{0}, x^{k}} = {t, x^{k}}$ . Equation (15) is a revised version of equation (45) in Huang [12].

3. Conservation laws in bond-based peridynamics

Consider a peridynamic body free of external body forces. We set $ϕ^{i} = y^{i} (x^{k}, t) = x^{i} + u^{i} (x^{k}, t)$ , where $u^{i} (x^{k}, t)$ is the contravariant component of the displacement field. It is easy to see that ${\overset{\cdot}{ϕ}}^{i} = {\overset{\cdot}{y}}^{i} = {\overset{\cdot}{u}}^{i}$ . The Lagrangian function of peridynamics is assumed to take the following form:

L = \frac{1}{2} ρ {\overset{\cdot}{y}}_{i} {\overset{\cdot}{y}}^{i} - \frac{1}{2} 〈 h | y_{i} 〉 y^{i},

(16)

where $ρ$ denotes the mass density. The term $〈 h | y_{i} 〉 y_{i}$ represents the work done by the internal long-range force.

Inserting equation (16) into equation (5) yields

ρ {\overset{\cdot\cdot}{y}}_{i} + 〈 κ | y_{i} 〉 = 0

(17)

where $κ$ is represented by

κ = h (| ξ |, η) + \frac{1}{2} η \frac{\partial h (| ξ |, η)}{\partial η} .

(18)

Clearly, if $κ = κ (| ξ |, η)$ is known while $h (| ξ |, η)$ is unknown, then solving equation (18) leads to

h (| ξ |, η) = 2 η^{- 2} \int_{0}^{η} η κ (| ξ |, η) d η .

(19)

Substituting equation (16) into equations (13) and (14), we have

\frac{d}{d t} \int_{Ω} (\frac{1}{2} ρ {\overset{\cdot}{y}}_{k} {\overset{\cdot}{y}}^{k} + \frac{1}{2} 〈 h | y_{k} 〉 y^{k}) T d v (x) = 0

(20)

\int_{Ω} {\frac{d}{d t} [ρ {\overset{\cdot}{y}}^{k} (U_{jk} - y_{k, i} X_{j}^{i})] + \partial_{k} [(\frac{1}{2} ρ {\overset{\cdot}{y}}_{i} {\overset{\cdot}{y}}^{i} - 〈 h | y_{i} 〉 y^{i}) X_{j}^{k}]} d v (x) = 0 .

(21)

In classical elasticity, Fletcher proved the completeness of conservation laws under the infinitesimal transformations below [2]:

\begin{matrix} t \mapsto \bar{t} = t + ϵ (ν t + c_{0}), x^{i} \mapsto {\bar{x}}^{i} = x^{i} + ϵ (ν x^{i} + ϵ_{jk}^{i} x^{j} b^{k} + c^{i}), \\ y_{i} \mapsto {\bar{y}}_{i} = y_{i} + ϵ (- ν y_{i} + ϵ_{ijk} y^{j} b^{k} + ϵ_{ijk} x^{j} c^{k} + d_{i}), \end{matrix}

(22)

where $ϵ$ is an infinitesimal parameter, and $ν, a_{i}, b_{i}, c_{α}$ and $d_{i}$ are arbitrary real constants. In terms of equation (22), we turn now to investigating the concrete forms of the conservation laws under four typical transformations.

3.1. Case 1: $\bar{t} = t + ϵ^{0}, {\bar{x}}^{k} = x^{k}, {\bar{y}}^{k} = y^{k}$

The transformations above are equivalent to taking $T = 1, X_{j}^{i} = 0$ and $U_{k α} = 0$ . Under this case, equation (21) becomes an identity, while equation (20) reduces to

\frac{d}{d t} \int_{Ω} (\frac{1}{2} ρ {\overset{\cdot}{y}}_{k} {\overset{\cdot}{y}}^{k} + \frac{1}{2} 〈 h | y_{k} 〉 y^{k}) d v (x) = 0 .

(23)

Equation (23) corresponds to the conservation of energy, which shows that the total energy on $Ω$ , including the sum of the kinetic energy and the internal long-range interaction potential energy, is a constant.

Multiplying ${\overset{\cdot}{y}}^{i}$ on both sides of equation (17) and integrating, we have

\frac{d}{d t} \int_{Ω} \frac{1}{2} ρ {\overset{\cdot}{y}}_{k} {\overset{\cdot}{y}}^{k} d v (x) + \int_{Ω} 〈 κ | y_{k} 〉 {\overset{\cdot}{y}}^{k} d v (x) = 0 .

(24)

It is easy to verify that

\int_{Ω} 〈 κ | y_{k} 〉 {\overset{\cdot}{y}}^{k} d v (x) = \frac{d}{d t} \int_{Ω} \frac{1}{2} 〈 κ | y_{k} 〉 y^{k} d v (x) - \int_{Ω} \frac{1}{2} 〈 \overset{\cdot}{κ} | y_{k} 〉 y^{k} d v (x) .

(25)

Substituting equation (25) into equation (24) yields

\frac{d}{d t} \int_{Ω} (\frac{1}{2} ρ {\overset{\cdot}{y}}_{k} {\overset{\cdot}{y}}^{k} + \frac{1}{2} 〈 κ | y_{k} 〉 y^{k}) d v (x) - \int_{Ω} \frac{1}{2} 〈 \overset{\cdot}{κ} | y_{k} 〉 y^{k} d v (x) = 0 .

(26)

As is well known, the energy of elastic motion is of conservation. Therefore, equation (26) should be the same as equation (23) in form. This is true if and only if the influence function is independent of the relative displacement. Under this case, we have $h (| ξ |) = κ (| ξ |)$ . In contrast, if the influence function depends on the relative displacement, it is necessary to cause the dissipation of energy.

3.2. Case 2: $\bar{t} = t, {\bar{x}}^{k} = x^{k}, {\bar{y}}^{k} = y^{k} + ϵ^{k}, ϵ^{0} = ϵ^{1} = ϵ^{2} = ϵ^{3}$

The transformations above represent the rigid body translations. Under these transformations, equation (20) becomes an identity, while equation (21) reduces to

\frac{d}{d t} \int_{Ω} ρ {\overset{\cdot}{y}}_{k} d v (x) = 0,

(27)

which is the integral representation of the conservation law of linear momentum, and it can be acquired directly from the integral of equation (17) over the spatial domain $Ω$ .

3.3. Case 3: $\bar{t} = t, {\bar{x}}^{k} = x^{k}, {\bar{y}}^{k} = y^{k} + ϵ_{kij} y^{i} ϵ^{j}, ϵ^{0} = ϵ^{1} = ϵ^{2} = ϵ^{3}$

The transformations above characterize the rigid body rotations. According to these transformations, equations (20) and (21) reduce to

\frac{d}{d t} \int_{Ω} ρ ϵ_{kij} y^{i} {\overset{\cdot}{y}}^{j} d v (x) = 0 .

(28)

Therefore, the conservation of total angular momentum on $Ω$ is associated with the invariance of the action functional under rigid body rotation.

By interchanging $x$ and $z$ , it is easy to verify that

\int_{Ω} ϵ_{kij} y^{i} (x) 〈 κ | y^{j} 〉 d v (x) = 0 .

(29)

Therefore, equation (28) can be also derived from equation (17).

3.4. Case 4: $\bar{t} = t, {\bar{x}}^{k} = x^{k} + ϵ^{k}, {\bar{u}}^{k} = u^{k}, ϵ^{0} = ϵ^{1} = ϵ^{2} = ϵ^{3}$

The transformations above correspond to the coordinate translations that are identical to setting $T = 0, X_{j}^{i} = δ_{j}^{i}, U_{k α} = 0$ . Thus, equation (20) reduces to an identity and equation (21) leads to

\int_{Ω} [- \frac{d}{d t} (ρ {\overset{\cdot}{y}}^{k} y_{k, j}) + \partial_{j} (\frac{1}{2} ρ {\overset{\cdot}{y}}_{i} {\overset{\cdot}{y}}^{i} - 〈 h | y_{i} 〉 y^{i})] d v (x) = 0 .

(30)

Under the static-equilibrium state, equation (30) further reduces to

\int_{\partial Ω} 〈 h | y_{i} 〉 y^{i} n_{j} d v (x) = 0 .

(31)

The integral on the right-hand side of equation (31) is the so-called Eshelby integral. For the two-dimensional case, it is also called the J-integral. In classical elasticity, the Eshelby integral is independent of the integral path. However, the same conclusion is no longer available in peridynamics. This is because equation (31) holds only on $Ω$ as a whole. For any $v \subset Ω$ , it ceases to be valid.

4. Conservation laws in state-based peridynamics

4.1. Motion equation

Let $ϕ^{i} = y^{i} (x^{k}, t) = x^{i} + u^{i} (x^{k}, t)$ . The deformation vector-state is defined as

Y^{i} = Y^{i} (x^{k}) < ξ^{k} > = y^{i} (z^{k}, t) - y^{i} (x^{k}, t) = ξ^{i} + u^{i} (z^{k}, t) - u^{i} (x^{k}, t) = - η^{i}

(32)

where $ξ^{k} = z^{k} - x^{k}$ . Assume the influence function $h = h (| ξ |, η)$ takes the form below

h (| ξ |, η) = {\begin{matrix} 1, ξ \in H . \\ 0, otherwise \end{matrix}

(33)

where H is a spherical neighborhood of radius $δ$ centered at $x \in Ω$ in $R^{3}$ . Thus, equations (2) and (5) can be rewritten as

[Y^{k} (x^{j}) < ξ^{j} >] : = - 〈 h | y^{k} 〉 = \int_{H} Y^{k} d v (z),

(34)

\frac{d}{d t} (\frac{\partial L}{\partial {\overset{\cdot}{y}}^{k}}) - \frac{\partial L}{\partial y^{k}} = - \int_{H} (\frac{\partial L}{\partial [Y^{k} (x^{j}) < ξ^{j} >]} - \frac{\partial L}{\partial [Y^{k} (z^{j}) < - ξ^{j} >]}) d v (z),

(35)

where $[Y^{k} (z^{j}) < - ξ^{j} >]$ is acquired by interchanging $x$ and $z$ in equation (34). By the equality below (see the Appendix)

\frac{\partial L}{\partial [Y^{k} (x^{j}) < ξ^{j} >]} = \frac{\partial L}{\partial Y^{k} (x^{j}) < ξ^{j} >},

(36)

equation (35) leads to

\frac{d}{d t} (\frac{\partial L}{\partial {\overset{\cdot}{y}}^{k}}) - \frac{\partial L}{\partial y^{k}} = - \int_{H} (\frac{\partial L}{\partial Y^{k} (x^{j}) < ξ^{j} >} - \frac{\partial L}{\partial Y^{k} (z^{j}) < - ξ^{j} >}) d v (z) .

(37)

The Lagrangian of peridynamics is written as

L = \frac{1}{2} ρ {\overset{\cdot}{y}}^{k} {\overset{\cdot}{y}}_{k} - ψ ([Y^{k} (x^{j}) < ξ^{j} >]) .

(38)

Substituting equation (38) into equation (37) yields

ρ {\overset{\cdot\cdot}{y}}_{k} = \int_{H} (\frac{\partial ψ}{\partial Y^{k} (x^{j}) < ξ^{j} >} - \frac{\partial ψ}{\partial Y^{k} (z^{j}) < - ξ^{j} >}) d v (z),

(39)

which represents the motion equation of peridynamics. This equation was first given in Silling et al. [15]. Later, it was derived from the variational method by Bobaru et al. [18].

4.2. Conservation laws

By equations (32)–(34), (36) and (38), equations (13) and (14) lead to

\frac{d}{d t} \int_{Ω} (\frac{1}{2} ρ {\overset{\cdot}{y}}_{k} {\overset{\cdot}{y}}^{k} + ψ) T d v (x) = 0

(40)

\int_{Ω} {\frac{d}{d t} [ρ {\overset{\cdot}{y}}^{k} (U_{jk} - y_{k, i} X_{j}^{i})] + \partial_{k} [(\frac{1}{2} ρ {\overset{\cdot}{y}}_{i} {\overset{\cdot}{y}}^{i} - ψ + \int_{H} \frac{\partial ψ}{\partial η^{i}} η^{i} d v (z)) X_{j}^{k}]} d v (x) = 0 .

(41)

Corresponding to the four transformations in sections 3.1–3.4, equations (40) and (41) reduce to

\frac{d}{d t} \int_{Ω} (\frac{1}{2} ρ {\overset{\cdot}{y}}_{k} {\overset{\cdot}{y}}^{k} + ψ) d v (x) = 0,

(42)

\frac{d}{d t} \int_{Ω} ρ {\overset{\cdot}{y}}_{k} d v (x) = 0,

(43)

\frac{d}{d t} \int_{Ω} ρ ϵ_{kij} y^{i} {\overset{\cdot}{y}}^{j} d v (x) = 0,

(44)

\int_{Ω} {\frac{d}{d t} (ρ {\overset{\cdot}{y}}^{k} y_{k, j}) - \partial_{j} [\frac{1}{2} ρ {\overset{\cdot}{y}}_{i} {\overset{\cdot}{y}}^{i} - ψ + \int_{H} \frac{\partial ψ}{\partial η^{i}} η^{i} d v (z)]} d v (x) = 0 .

(45)

They represent the conservation laws of energy, linear momentum, angular momentum and the Eshelby integral, respectively. By equations (39) and (44), it is easy to find that the conservation of angular momentum holds if and only if

\int_{Ω} \int_{H} ϵ^{ijk} Y_{j} (x^{s}) < ξ^{s} > \frac{\partial ψ}{\partial Y^{k} (x^{s}) < ξ^{s} >} d v (z) d v (x) = 0 .

(46)

Clearly, equation (46) is a constraint on the constitutive relation between the force vector-state and the deformation vector-state caused by the conservation of angular momentum. Compared with Proposition 8.1 in Silling et al. [15], equation (46) is a relaxed constraint condition, because equation (46) is necessarily valid if Proposition 8.1 holds, but the contrary statement is not always true.

Under the static-equilibrium state, equation (45) reduces to

\int_{\partial Ω} [ψ - \int_{H} \frac{\partial ψ}{\partial η^{i}} η^{i} d v (z)] n_{j} d v (x) = 0,

(47)

where the integrand is referred to as the peridynamic Eshelby integrand, which reads

E_{p} = ψ - \int_{H} \frac{\partial ψ}{\partial η^{i}} η^{i} d v (z) .

(48)

It should be emphasized that equations (31) and (47) hold only when no interfaces caused by damage, fracture and phase transition occur within the body. If there are voids, cracks and phase transitions in the body, equations (31) and (47) will be no longer be equal to zero. Therefore, the Eshelby integrals can be used to calculate the energy release rates caused by damage, fracture and phase transition.

In classical elasticity, the Eshelby tensor is represented as

E_{j}^{i} = W δ_{j}^{i} - \frac{\partial W}{\partial (\partial_{k} u_{i})} \partial_{k} u_{j} .

(49)

Comparing equations (48) and (49), we easily find that there are one-to-one correspondence relations between various kinds of mathematical quantities of the peridynamic Eshelby integrand and the Eshelby tensor, for example, the peridynamic potential energy $ψ$ to the elastic strain energy W, the relative displacement vector-state $η^{i}$ to the gradient of displacement $\partial_{k} u_{j}$ , and the inner product of the vector-state [15] to the inner product of the tensor. These correspondences imply that equation (48) is perhaps degraded into equation (49) under some limit conditions, but this awaits verification.

5. Summary

We have defined a new nonlocal argument. Based on this argument, the Lagrangian formulation of peridynamics is systematically established. The peridynamic Euler–Lagrange equation is derived from Hamilton’s principle, and Noether’s theorem is extended into peridynamics.

We use the peridynamic Noether’s theorem to deduce the bond-based and state-based conservation laws relevant to energy, linear momentum, angular momentum and the Eshelby integral. The results show that the peridynamic conservation laws exist only in an integral form, rather than in a pointwise form.

In bond-based peridynamics, the conservation of linear and angular momentum is satisfied automatically, while energy conservation requires that the influence function is independent of the relative displacement field, or energy dissipation will occur.

In state-based peridynamics, the conservation of angular momentum causes a constraint on the constitutive relation between the force vector-state and the deformation vector-state. Compared with Proposition 8.1 in Silling et al. [15], it is a weaker constraint.

The peridynamic Eshelby integral and its conservation law have been determined. They can be used to judge nucleation of defects and to calculate the energy release rates caused by damage, fracture and phase transition.

Footnotes

Appendix

Equation (36) is proven as follows. By the chain rule, we have

(50)

\frac{\partial L}{\partial Y^{k}} = \frac{\partial L}{\partial [Y^{j}]} \frac{\partial [Y^{j}]}{\partial Y^{k}} .

It should be noted that both L and $[Y^{k}]$ are functionals of the vector-state $Y_{k}$ . So $\partial L / \partial Y^{k}$ and $\partial [Y^{j}] / \partial Y^{k}$ represent two Frechet derivatives. Comparing equation (36) with equation (50), we see that equation (36) holds if only $\partial [Y^{j}] / \partial Y^{k} = δ_{k}^{j}$ , where $δ_{k}^{j}$ is the second-order unit tensor. Next, we will show that this is true. In terms of equation (34), it is easy to verify that

(51)

d [Y^{j}] = \int_{H} d Y^{j} d v (z) .

On the other hand, $d [Y^{j}]$ can be written as (referring to equations (23) and (11) in Silling et al. [15])

(52)

d [Y^{j}] = \int_{H} \frac{\partial [Y^{j}]}{\partial Y^{k}} d Y^{k} d v (z) .

Equation (52) minus equation (51) yields

(53)

\int_{H} (\frac{\partial [Y^{j}]}{\partial Y^{k}} - δ_{k}^{j}) d Y^{k} d v (z) = 0 .

Equation (53) holds for any $d [Y^{k}]$ if and only if

(54)

\frac{\partial [Y^{j}]}{\partial Y^{k}} = δ_{k}^{j} .

Substituting equation (54) into equation (50) leads to equation (36). Thus, the proof is completed.

Acknowledgements

The author is very grateful to the reviewers for their comments, suggestions and help!

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The support of the National Nature Science Foundation of China (grant number 11672129) and the Research Fund of State Key Laboratory of Mechanics and Control of Mechanical Structures (Nanjing University of Aeronautics and Astronautics) (MCMS-I-0218G01) is gratefully acknowledged.

ORCID iD

Zaixing Huang

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Noether’s theorem in peridynamics

Abstract

Keywords

1. Introduction

2. Lagrangian formulation of peridynamics

2.1. Euler–Lagrange equation

2.2. Noether’s theorem

3. Conservation laws in bond-based peridynamics

3.1. Case 1: t ¯ = t + ϵ 0 , x ¯ k = x k , y ¯ k = y k

3.2. Case 2: t ¯ = t , x ¯ k = x k , y ¯ k = y k + ϵ k , ϵ 0 = ϵ 1 = ϵ 2 = ϵ 3

3.3. Case 3: t ¯ = t , x ¯ k = x k , y ¯ k = y k + ϵ kij y i ϵ j , ϵ 0 = ϵ 1 = ϵ 2 = ϵ 3

3.4. Case 4: t ¯ = t , x ¯ k = x k + ϵ k , u ¯ k = u k , ϵ 0 = ϵ 1 = ϵ 2 = ϵ 3

4. Conservation laws in state-based peridynamics

4.1. Motion equation

4.2. Conservation laws

5. Summary

Footnotes

Appendix

Acknowledgements

Funding

ORCID iD

References

3.1. Case 1: $\bar{t} = t + ϵ^{0}, {\bar{x}}^{k} = x^{k}, {\bar{y}}^{k} = y^{k}$

3.2. Case 2: $\bar{t} = t, {\bar{x}}^{k} = x^{k}, {\bar{y}}^{k} = y^{k} + ϵ^{k}, ϵ^{0} = ϵ^{1} = ϵ^{2} = ϵ^{3}$

3.3. Case 3: $\bar{t} = t, {\bar{x}}^{k} = x^{k}, {\bar{y}}^{k} = y^{k} + ϵ_{kij} y^{i} ϵ^{j}, ϵ^{0} = ϵ^{1} = ϵ^{2} = ϵ^{3}$

3.4. Case 4: $\bar{t} = t, {\bar{x}}^{k} = x^{k} + ϵ^{k}, {\bar{u}}^{k} = u^{k}, ϵ^{0} = ϵ^{1} = ϵ^{2} = ϵ^{3}$