On Energy Release Rate for Propagation of a Straight Crack in a Cosserat Elastic Body

Abstract

In this paper, we extend some results involving the energy release rate in the case of the propagation of a straight crack in an elastic solid. These results, approached by Gurtin and Yatomi in classical elasticity, are generalized in order to cover a Cosserat-type elastic body. We also investigate the effects of the microinertia and the couple stresses on the energy release rate.

1. Introduction

It is unanimously accepted that the non-classical theories of elasticity have as their main goal to eliminate experimental aspects that cannot be described by elasticity based on the classical theory. If a body’s intimate structure is considered for behavior, classical elasticity theory does not offer a good description, for example, in polymers, human and animal bones, and graphite. This is even more pronounced in materials with specific porous or granular internal structures. For more information, we recommend [1,2].

The first researcher who approached microstructure theories in a large number of studies was Eringen, who is considered a true pioneer in this field. We suggest [3,4]. After the way was opened by Eringen, many other researchers tackled a large number of aspects of microstructure, such as [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]. In [7], Ciarletta investigated the bending of microstretch elastic material plates with no thermal effects using the main ideas of Eringen. In [5], Iesan and Pompei find a exact solution of the Boussinesq–Somigliana–Galerkin type for a boundary-value problem in the same context of an elastic material with microstretch structure. In [12,14], the authors present some aspects regarding the propagation of waves through bodies with different microstructures. One of the first and most important microstructures is the Cosserat type. The Cosserat continuum, proposed by the French brothers Cosserat, introduces the rotational kinematic degree of freedom in addition to the translation components of the Cauchy model. A large number of influential scientists have granted great importance to this type of structure. A long time ago, the study of crack propagation was raised, at first, obviously, in the case of elastic or thermoelastic media. The first studies dedicated to this matter include Eringen [22], Freund [23], Gurtin [24], Gurtin and Yatomi [25], and so on. For instance, Freund [18] proved that energy flux depends on the elastic field near the tip. Further, Gurtin [24] discussed some thermodynamically cohesive models of mechanical fractures.

Gurtin and Yatomi [25] provided some expressions for the energy release rates in the presence of a propagating crack for a body that is hyperelastic and is subject to finite deformation.

The authors of [26] address the physical meaning of path-independent integrals for elastodynamically propagating cracks, the relation of these integrals to the energy release rates for propagating cracks, and the relation between these integrals and time-dependent stress-intensity factors in general mixed-mode dynamic crack propagation. Recently, some studies have presented new results regarding the energy release rate for crack propagation in bodies with microstructures. For examples, Zhang and Wang [27] theoretically investigated magneto–thermo–electro–elastic crack branching in these materials. A mathematical model was presented by Tian and Rajapakse [28] to find the parameters of fracture in the case of a finite impermeable crack in a magneto–electro–elastic plane actuated by magneto–electro–mechanical loading.

A study of a semi-infinite crack that is constantly propagated in an elastic material was conduected by Gourgiotis and Piccolroaz [29]. The body is subjected to some plane-strain shear loadings.

Our present paper tackles the problem of the propagation of a straight crack in a Cosserat elastic material.

2. Preliminaries

An open region D of the Euclidean three-dimensional space $R^3$ is considered, and an elastic Cosserat material fills this domain. As usual, the closure of D is denoted by $\overline{D}$, and its boundary $\partial D$ is assumed to be a regular surface so as to allow application of the divergence theorem. We adopt the Cartesian vector and tensor notation and use two types of variables: the points of D have the spatial variables $x_m$, and we use the temporal variable t, with $t\in [0,t_0]$. When the meaning is unambiguous, we do not specify the dependence of a function on its spatial and/or temporal variables. We use the known rule of summation in the case of repeated indices.

Regarding the differentiation of functions, we use the following two rules: (1) a superposed dot on a function denotes the partial differentiation of the function with respect to t; and (2) the partial differentiation with respect to spatial variables $x_m$ will be designated with the respective subscript preceded by a comma: $f,m$.

Notations for the mechanical quantities:

-

$v_m$ the components of the displacement vector field;

-

${\varphi }_m$ the components of the microrotation vector;

-

$e_{mn}$ the components of the strain tensor;

-

${ϵ}_{mn}$ the components of the couple strain tensor;

-

${\epsilon }_{mnk}$ the permutation symbol (the Ricci’s tensor);

-

${\tau }_{mn}$ the components of the stress tensor;

-

${\sigma }_{mn}$ the components of the couple stress tensor;

-

$\varrho$ the reference mass density;

-

$J_{mn}$ the components of the microinertia;

-

$f_m$ the components of the body force;

-

$g_m$ the components of the couple body force;

-

$n_k$ are the director cosines of the unit normal, outward to the boundary $\partial D$;

-

${\tau }_m$ the components of the surface traction vector;

-

${\sigma }_m$ the components of the couple surface traction vector;

-

$\mathcal{E}$ the dynamic energy release rate;

The deformation of a Cosserat elastic body will be described using the following internal constitutive variables, as functions depending on the time $t\in [0,t_0]$ and on point position $x\in D$:

-

$v_m(x,t)$ the component elements of the vector of displacement;

-

${\varphi }_m(x,t)$ the component elements of the tensor of Cosserat displacement.

The strain tensors have the component elements ${ϵ}_{mn}$ and $e_{mn}$, and these are introduced with help of the components of displacement by means of the next kinematic relations:

$$\begin{array}{c}\hfill {ϵ}_{mn}=v_{n,m}+{\epsilon }_{nmk}{\varphi }_k,\\ \hfill e_{mn}={\varphi }_{n,m}.\end{array}$$

(1)

Because our subsequent considerations are made only in the context of a linear theory, it is normal to consider that the density of the internal energy E is a quadratic form regarding all its constitutive variables. Thus, by assuming that our body in its reference state is free from stress and has zero intrinsic body couples, E has the following expression:

$$\begin{array}{c}\hfill E=\frac{1}{2}{C}_{klmn}{ϵ}_{kl}{ϵ}_{mn}+B_{klmn}{ϵ}_{kl}{e}_{mn}+\frac{1}{2}{A}_{klmn}{e}_{kl}{e}_{mn}.\end{array}$$

(2)

Taking into account the expression of the density of internal energy using Clausius–Duhem inequality, we can obtain the constitutive relations that give the expressions for the stress tensors ${\tau }_{mn}$ and ${\sigma }_{mn}$ in terms of the strain tensors:

$$\begin{array}{c}\hfill {\tau }_{mn}=\frac{\partial E}{\partial {ϵ}_{mn}}=C_{klmn}{ϵ}_{kl}+B_{klmn}{e}_{kl},\\ \hfill {\sigma }_{mn}=\frac{\partial E}{\partial e_{mn}}=B_{klmn}{ϵ}_{kl}+A_{klmn}{e}_{kl},\end{array}$$

(3)

obtained in the case of a homogenous Cosserat elastic body.

Above tensors $C_{klmn}$, $B_{klmn}$, and $A_{klmn}$ are elastic constants that characterize the material properties and obey these relations of symmetry:

$$\begin{array}{c}\hfill C_{mnkl}=C_{klmn},A_{klmn}=A_{mnkl}.\end{array}$$

(4)

By using the procedure of Eringen [4], we can deduce the equations of motion. If it is assumed that mass forces and the couples of forces are missing, then the motion equations have the following form:

$$\begin{array}{c}\hfill \varrho {\ddot{v}}_m={\tau }_{nm,n},\\ \hfill J_{mn}{\ddot{\varphi }}_n={\sigma }_{nm,n}+{\epsilon }_{mnk}{\tau }_{nk},\end{array}$$

(5)

in which $\varrho$ is the density of the body in its reference state, and $J_{mn}$ are the components of the tensor of inertia, with $J_{mn}=J_{nm}$.

Further, ${\epsilon }_{mnk}$ is the permutation symbol (the Ricci’s tensor).

We assume that Equations (1), (3), and (5) take place for all $(x,t)\in B\times [0,t_0]$.

Further, we suppose that our body D in moment t has a straight crack, denoted by $C_t$, and the tip of this crack at moment t is denoted by ${\mathbf{y}}_t$. Assume that ${\mathbf{y}}_t$ is a function of class $C^2$ regarding the time variable t, and its velocity is ${\mathbf{w}}_t=d{\mathbf{y}}_t/dt$.

Let us denote by $D_0$ the domain covered by ${\mathbf{y}}_t$ for all $t\in [0,t_0]$.

In what follows, we consider certain applications of the form $\phi (x,t)$, depending on $x\in D\backslash C_t$ and on $t\in [0,t_0]$. According to a suggestion given by Gurtin and Yatomi in [20], we introduce the next definition.

Definition 1.

We say that a function f is called a $C^m$ fracture field for a positive integer m if the following two conditions are met:

(1) f admits derivatives of any order i, $i\le m$, except for the points in the crack;

(2) all derivatives $f^{\left(i\right)}$, $i\le n$, are continuous applications except for the points in the crack, and functions $f^{\left(i\right)}$ are continuous at points approaching the crack from both sides except in points at the tip of the crack.

Let us specify some theoretical issues that we use below.

(i) The notation $f\in L^p\left(D\right)$ means that we are in the case $f(x,t)\in L^p\left(D\right)$ for any $t\in [0,t_0]$;

(ii) Let us introduce the family $D_{\beta }$ of smooth surfaces $D_{\beta }$ of D, having the property: $area\left(D\backslash D_{\beta }\right)\to 0$ as $\beta \to 0$. If $h\in L^p\left(D\right)$, we have

$$\begin{array}{c}\hfill \underset{\beta \to 0}{lim}{\int }_{D_{\beta }}{\left|h(x,t)\right|}^{p}dV={\int }_D{\left|h(x,t)\right|}^{p}dV.\end{array}$$

(6)

(iii) If the limit from (6) takes place uniformly with regards to $t\in [0,t_0]$, we consider that $f\in L^p\left(D\right)$ belongs uniformly in time.

(iv) Consider that $\mathbf{r}$ is the vector of the position from the tip. If f is a function that depends on t in the form $f\left({\mathbf{y}}_t+\mathbf{r},t\right)$, then the derivative of f with respect to t, denoted by $f^{\prime }$, is computed as follows:

$$\begin{array}{c}\hfill f^{\prime }=\frac{\partial f}{\partial t}+c_i\frac{\partial f}{\partial x_i}=\dot{f}+d_{c}f,\end{array}$$

(7)

holding $\mathbf{r}$ fixed. Here, we use the notation

$$d_{c}f=c_i\frac{\partial f}{\partial x_i}.$$

(v) In the case of three scalar functions u, v, and w with properties $u,w\in C^1$, and $\dot{u}=v+d_{c}w$, we use the rule:

$${\int }_{D}udV={\int }_{D}vdV+{\int }_{\partial D}w\mathbf{c}.\mathbf{n}dA,$$

in which $\mathbf{n}$ is a normal versor oriented outward from $\partial D$.

(vi) If $\mathbf{u}$ is a $C^1$ fracture field so that $div\mathbf{u}\in L^1\left(D\right)$, then the divergence theorem gives:

$${\int }_{\partial D}\mathbf{u}.\mathbf{n}dA={\int }_{D}div\mathbf{u}dV,$$

called the “flow-divergence formula”.

Statements (v) and (vi) above are easy to prove. However, for more details, we recommend [20].

3. Basic Results

We assume that the displacement vector of components $v_m$ and the couple displacement vectors of components ${\varphi }_m$ are $C^3$ fracture fields in the sense of the above definition. Further, we suppose that the tensors of stress ${\tau }_{mn}$ and ${\sigma }_{mn}$ are $C^1$ fracture fields.

We can now define the surface traction vector of components ${\tau }_l$ and the couple surface traction vector of components ${\sigma }_l$ with the help of the next equations:

$$\begin{array}{c}\hfill {\tau }_l={\tau }_{kl}{n}_k,{\sigma }_l={\sigma }_{kl}{n}_k,\end{array}$$

(8)

where $n_k$ are the director cosines of the unit normal outward to the border $\partial D$.

By definition, the application $\mathcal{E}$ defined in any t of the interval $[0,t_0]$ is called a rate of the dynamic energy release if $\mathcal{E}$ has the following expression:

$$\begin{array}{c}\hfill \mathcal{E}={\int }_{\partial D}\left({\tau }_m{\dot{v}}_m+{\sigma }_m{\dot{\varphi }}_m\right)dA-\frac{d}{dt}{\int }_D\left[\frac{1}{2}\left(\varrho {\dot{v}}_m{\dot{v}}_m+J_{mn}{\dot{\varphi }}_m{\dot{\varphi }}_n\right)\right.\\ \hfill \left.+\frac{1}{2}{C}_{klmn}{ϵ}_{kl}{ϵ}_{mn}+B_{klmn}{ϵ}_{kl}{e}_{mn}+\frac{1}{2}{A}_{klmn}{e}_{kl}{e}_{mn}\right]dV.\end{array}$$

(9)

We will consider S a surface that includes its inside at time $t_1$ at the tip of the crack, where $t_1\in [0,t_0]$ and so $\partial S$ and ${\mathbf{y}}_t\in S$ intersects the crack $C_t$ only once.

It can be shown that the rate of dynamic energy release does not depend on the choice of surface S.

Now, we prove a property of the crack tip.

Proposition 1.

The rate of dynamic energy release $\mathcal{E}$ is intrinsic to the tip of the crack.

Proof. 

First of all, observing the expression of the internal density, we can write:

$$\begin{array}{c}\hfill \dot{E}={\tau }_{mn}{\dot{ϵ}}_{mn}+{\sigma }_{mn}{\dot{e}}_{mn}.\end{array}$$

(10)

Then, considering the theorem of divergence and Equation (8), we can write:

$$\begin{array}{c}\hfill {\int }_{\partial (D\backslash S)}\left({\tau }_m{\dot{v}}_m+{\sigma }_m{\dot{\varphi }}_m\right)dA={\int }_{D\backslash S}{\left({\tau }_{mn}{\dot{v}}_n+{\sigma }_{mn}{\dot{\varphi }}_n\right)}_{,m}dV.\end{array}$$

(11)

Further, if we consider the motion Equations (5) and (10), we deduce:

$$\begin{array}{c}\hfill {\int }_{\partial (D\backslash S)}\left({\tau }_m{\dot{v}}_m+{\sigma }_m{\dot{\varphi }}_m\right)dA\\ \hfill ={\int }_{D\backslash S}\left(\varrho {\ddot{v}}_m{\dot{v}}_m+J_{mn}{\ddot{\varphi }}_m{\dot{\varphi }}_n+{\tau }_{mn}{\dot{ϵ}}_{mn}+{\sigma }_{mn}{\dot{e}}_{mn}\right)dV\\ \hfill =\frac{d}{dt}{\int }_{D\backslash S}\frac{1}{2}\left(\varrho {\dot{v}}_m{\dot{v}}_m+J_{mn}{\dot{\varphi }}_m{\dot{\varphi }}_n+C_{klmn}{ϵ}_{kl}{ϵ}_{mn}+2B_{klmn}{ϵ}_{kl}{e}_{mn}+A_{klmn}{e}_{kl}{e}_{mn}\right)dV.\end{array}$$

(12)

After these calculations, dynamic energy release rate receives the form:

$$\begin{array}{c}\hfill \mathcal{E}={\int }_{\partial D}\left({\tau }_m{\dot{v}}_m+{\sigma }_m{\dot{\varphi }}_m\right)dA-\frac{d}{dt}{\int }_D\left[\frac{1}{2}\left(\varrho {\dot{v}}_m{\dot{v}}_m+J_{mn}{\dot{\varphi }}_m{\dot{\varphi }}_n\right)\right.\\ \hfill \left.+\frac{1}{2}{C}_{klmn}{ϵ}_{kl}{ϵ}_{mn}+B_{klmn}{ϵ}_{kl}{e}_{mn}+\frac{1}{2}{A}_{klmn}{e}_{kl}{e}_{mn}\right]dV,\end{array}$$

which is even (9), and thus we conclude the proof of this proposition. □

In our next result, we obtain another expression for the rate of dynamic energy release.

Theorem 1.

For any $t\in [0,t_0]$, the rate of dynamic energy release can be written in the following form:

$$\begin{array}{c}\hfill \mathcal{E}\left(t\right)=\underset{S\to {\mathbf{y}}_t}{lim}{\int }_S\left\{\left[\frac{1}{2}\left(\varrho d_c{v}_m{d}_c{v}_m+J_{mn}{d}_c{\varphi }_m{d}_c{\varphi }_n\right)\right.\right.\\ \hfill \left.+\frac{1}{2}{C}_{klmn}{ϵ}_{kl}{ϵ}_{mn}+B_{klmn}{ϵ}_{kl}{e}_{mn}+\frac{1}{2}{A}_{klmn}{e}_{kl}{e}_{mn}\right]c_s\\ \hfill \left.-{\tau }_{sm}{d}_c{v}_m-{\sigma }_{sm}{d}_c{\varphi }_m\right\}{n}_{s}dA,\end{array}$$

(13)

where S is a surface that includes the inside of the tip of the crack. Further, $c_s$ represent the elements of the speed of the tip ${\mathbf{y}}_t$, and $d_c$ appears in (7).

Proof. 

If we consider Rule (7), we obtain

$$\begin{array}{c}\hfill {\int }_{\partial D}\left({\tau }_m{\dot{v}}_m+{\sigma }_m{\dot{\varphi }}_m\right)dA={\int }_{\partial D}\left({\tau }_m{v}_m^{\prime }+{\sigma }_m{\varphi }_m^{\prime }\right)dA-{\int }_{\partial D}\left({\tau }_m{d}_c{v}_m+{\sigma }_m{d}_c{\varphi }_m\right)dA.\end{array}$$

(14)

If we take into account the kinematic Equation (1), we are led to

$$\begin{array}{c}\hfill {\left({\tau }_{mn}{u}_m^{\prime }+{\sigma }_{mn}{\varphi }_m^{\prime }\right)}_{,n}=\varrho {\ddot{v}}_m{v}_m^{\prime }+J_{mn}{\ddot{\varphi }}_m{\varphi }_n^{\prime }+{\tau }_{mn}{ϵ}_{mn}^{\prime }+{\sigma }_{mn}{e}_{mn}^{\prime },\end{array}$$

(15)

where the following notations are used:

$${ϵ}_{mn}^{\prime }=v_{n,m}^{\prime }+{\epsilon }_{nmk}{\varphi }_k^{\prime },e_{mn}^{\prime }={\varphi }_{n,m}^{\prime }.$$

Applying the second derivative, according to the rule in (7), we deduce:

$$\begin{array}{c}\hfill {\ddot{v}}_m=v_m^{″}-2d_c{v}_m^{\prime }+d_c^2{v}_m-d_c{v}_m,d_c^2{v}_m=v_{m,kl}{c}_l{c}_k,\\ \hfill {\ddot{\varphi }}_m={\varphi }_m^{″}-2d_c{\varphi }_m^{\prime }+d_c^2{\varphi }_m-d_c{\varphi }_m,d_c^2{\varphi }_{mm}={\varphi }_{m,kl}{c}_l{c}_k.\end{array}$$

(16)

We obtain a relation similar to the one in (11) using (7) again:

$$\begin{array}{c}\hfill {\int }_{\partial D}\left({\tau }_m{v}_m^{\prime }+{\sigma }_m{\varphi }_m^{\prime }\right)dA\\ \hfill ={\int }_D\left(\varrho {\ddot{v}}_m{v}_m^{\prime }+J_{mn}{\ddot{\varphi }}_m{\varphi }_n^{\prime }+{\tau }_{mn}{ϵ}_{mn}^{\prime }+{\sigma }_{mn}{e}_{mn}^{\prime }\right)dV,\end{array}$$

(17)

and by using (16), we also deduce that

$$\begin{array}{c}\hfill \frac{1}{2}\frac{d}{dt}\left(\varrho {\dot{v}}_m{\dot{v}}_m+J_{mn}{\dot{\varphi }}_m{\dot{\varphi }}_n\right)=\varrho {\ddot{v}}_m{v}_m^{\prime }+J_{mn}{\ddot{\varphi }}_m{\varphi }_n^{\prime }\\ \hfill -\varrho \left(v_m^{″}-2d_c{v}_m^{\prime }-d_c{v}_m\right)d_c{v}_m-J_{mn}\left({\varphi }_m^{″}-2d_c{\varphi }_m^{\prime }-d_c{\varphi }_m\right)d_c{\varphi }_n\\ \hfill -\frac{1}{2}{d}_c\left(\varrho d_c{v}_m{d}_c{v}_m+J_{mn}{d}_c{\varphi }_m{d}_c{\varphi }_n\right).\end{array}$$

(18)

Now, if we take into account Equation (2) and use the relation (7), the following equality is obtained

$$\begin{array}{c}\hfill \dot{E}=E^{\prime }-d_{c}E,\end{array}$$

(19)

so that from previous equations we deduce:

$$\begin{array}{c}\hfill \frac{d}{dt}{\int }_D\left[\frac{1}{2}\left(\varrho {\dot{v}}_m{\dot{v}}_m+J_{mn}{\dot{\varphi }}_m{\dot{\varphi }}_n\right)\right.\\ \hfill \left.+\frac{1}{2}{C}_{klmn}{ϵ}_{mn}{ϵ}_{mn}+B_{ijmn}{ϵ}_{mn}{e}_{mn}+\frac{1}{2}{A}_{klmn}{ϵ}_{mn}{e}_{mn}\right]dV\\ \hfill =\frac{d}{dt}{\int }_D\left(E+\varrho {\dot{v}}_m{\dot{v}}_m+J_{mn}{\dot{\varphi }}_m{\dot{\varphi }}_n\right)dV\\ \hfill ={\int }_D\left[E^{\prime }+\frac{1}{2}\frac{d}{dt}\left(\varrho {\dot{v}}_m{\dot{v}}_m+J_{mn}{\dot{\varphi }}_m{\dot{\varphi }}_n\right)+\frac{1}{2}\left(\varrho d_c{v}_m{d}_c{v}_m+J_{mn}{d}_c{\varphi }_m{d}_c{\varphi }_n\right)\right]dV\\ \hfill -{\int }_{\partial D}\left[E+\frac{1}{2}\left(\varrho d_c{v}_m{d}_c{v}_m+J_{mn}{d}_c{\varphi }_m{d}_c{\varphi }_n\right)\right]c_s{n}_{s}dA,\end{array}$$

(20)

where $c_s$ represent the elements of the speed of the tip, and $n=\left(n_s\right)$ is the outward normal to the border $\partial D$.

By taking into account Equations (11), (12), (17) and (20), the following relation is obtained:

$$\begin{array}{c}\hfill \mathcal{E}\left(t\right)={\int }_{\partial D}\left\{\left[\frac{1}{2}\left(\varrho d_c{v}_m{d}_c{v}_m+J_{mn}{d}_c{\varphi }_m{d}_c{\varphi }_n\right)\right.\right.\\ \hfill \left.+\frac{1}{2}{C}_{klmn}{ϵ}_{kl}{ϵ}_{mn}+B_{klmn}{ϵ}_{kl}{e}_{mn}+\frac{1}{2}{A}_{klmn}{e}_{kl}{e}_{mn}\right]c_s\\ \hfill \left.-{\tau }_{mk}{d}_c{v}_m-{\sigma }_{mk}{d}_c{\varphi }_m\right\}{n}_{k}dA\\ \hfill -{\int }_D\left[E^{\prime }+\frac{1}{2}\left(\varrho {\dot{v}}_m{\dot{v}}_m+J_{mn}{\dot{\varphi }}_m{\dot{\varphi }}_n\right)-{\tau }_{mn}{ϵ}_{mn}^{\prime }+{\sigma }_{mn}{e}_{mn}^{\prime }-\varrho {\ddot{v}}_m{v}_m^{\prime }-J_{mn}{\ddot{\varphi }}_m{\varphi }_n^{\prime }\right]dV.\end{array}$$

(21)

Finally, we take into account that surface S includes the tip of its crack in its inside at time t. Based on Proposition 1, we know that the rate of the dynamic energy release $\mathcal{E}$ is intrinsic relative to the tip of the crack, as such, $\mathcal{E}$ can be written in the form:

$$\begin{array}{c}\hfill \mathcal{E}\left(t\right)={\int }_S\left\{\left[\frac{1}{2}\left(\varrho d_c{v}_m{d}_c{v}_m+J_{mn}{d}_c{\varphi }_m{d}_c{\varphi }_n\right)\right.\right.\\ \hfill \left.+\frac{1}{2}{C}_{klmn}{ϵ}_{kl}{ϵ}_{mn}+B_{klmn}{ϵ}_{kl}{e}_{mn}+\frac{1}{2}{A}_{klmn}{e}_{kl}{e}_{mn}\right]c_s\\ \hfill \left.-{\tau }_{mk}{d}_c{v}_m-{\sigma }_{mk}{d}_c{\varphi }_m\right\}{n}_{k}dA\\ \hfill -{\int }_S\left(E^{\prime }+\frac{1}{2}\left(\varrho {\dot{v}}_m{\dot{v}}_m+J_{mn}{\dot{\varphi }}_m{\dot{\varphi }}_n\right)-{\tau }_{mn}{ϵ}_{mn}^{\prime }-{\sigma }_{mn}{e}_{mn}^{\prime }-\varrho {\ddot{v}}_m{v}_m^{\prime }-J_{mn}{\ddot{\varphi }}_m{\varphi }_n^{\prime }\right)dV.\end{array}$$

In this last relation, we take into account that the area of S is tending to zero, and so we obtain (13), which concludes the demonstration of Theorem 1. □

In Theorem 2 that follows, we obtain another expression of the rate of dynamic energy release at an arbitrary fixed time $t\in [0,t_0]$.

Theorem 2.

For a fixed time $t_1\in [0,t_0]$, the rate of dynamic energy release can be written in the form:

$$\begin{array}{c}\hfill \mathcal{E}\left(t_1\right)=\left(\frac{d}{dt}\right){\int }_D{\int }_{t_1}^t\left[\left({\tau }_{mn}\left(t\right)-{\tau }_{mn}\left(s\right)\right){\dot{ϵ}}_{mn}\left(s\right)+\left({\sigma }_{mn}\left(t\right)-{\sigma }_{mn}\left(s\right)\right){\dot{e}}_{ij}\left(s\right)\right.\\ \hfill \left.+\varrho \left({\ddot{v}}_m\left(t\right)-{\ddot{v}}_m\left(s\right)\right){\dot{v}}_m\left(s\right)+J_{mn}\left({\ddot{\varphi }}_m\left(t\right)-{\ddot{\varphi }}_m\left(s\right)\right){\dot{\varphi }}_m\left(s\right)\right]dsdV.\end{array}$$

(22)

Proof. 

We start with the observation that:

$$\begin{array}{c}\hfill \mathcal{E}=\frac{d}{dt}\left({\int }_{t_1}^t{\int }_{\partial D}\left({\tau }_m{\dot{v}}_m+{\sigma }_m{\dot{\varphi }}_m\right)dAds\right)\\ \hfill -\frac{d}{dt}\left({\int }_D\left[E+\frac{1}{2}\left(\varrho {\dot{v}}_m{\dot{v}}_m+J_{mn}{\dot{\varphi }}_m{\dot{\varphi }}_n\right)\right]dV\right).\end{array}$$

(23)

Then, we have

$$\begin{array}{c}\hfill {\int }_{t_1}^t{\int }_{\partial D}\left({\tau }_m{\dot{v}}_m+{\sigma }_m{\dot{\varphi }}_m\right)dAds={\int }_{t_1}^t{\int }_{\partial D}\left[{\tau }_m\left(t\right){\dot{v}}_m\left(s\right)+{\sigma }_m\left(t\right){\dot{\varphi }}_m\left(s\right)\right]dAds\\ \hfill +{\int }_{t_1}^t{\int }_{\partial D}\left\{\left[{\tau }_m\left(s\right)-{\tau }_m\left(t\right)\right]{\dot{v}}_m\left(s\right)+\left[{\sigma }_m\left(s\right)-{\sigma }_m\left(t\right)\right]{\dot{\varphi }}_m\left(s\right)\right\}dAds.\end{array}$$

Now, it is easy to verify that we have:

$$\begin{array}{c}\hfill \frac{d}{dt}\left({\int }_{t_1}^t{\int }_{\partial D}\left\{\left[{\tau }_m\left(s\right)-{\tau }_m\left(t\right)\right]{\dot{v}}_m\left(s\right)+\left[{\sigma }_m\left(s\right)-{\sigma }_m\left(t\right)\right]{\dot{\varphi }}_m\left(s\right)\right\}dAds\right)\\ \hfill =-{\int }_{t_1}^t{\int }_{\partial D}\left[{\tau }_m\left(t\right){\dot{v}}_m\left(s\right)+{\sigma }_m\left(t\right){\dot{\varphi }}_m\left(s\right)\right]dAds.\end{array}$$

According to these last two equations, we deduce that:

$$\begin{array}{c}\hfill {\left(\frac{d}{dt}\right)}_{t_1}\left({\int }_{t_1}^t{\int }_{\partial D}\left({\tau }_m{\dot{v}}_m+{\sigma }_m{\dot{\varphi }}_m\right)dAds\right)\\ \hfill ={\left(\frac{d}{dt}\right)}_{t_1}\left({\int }_{\partial D}\left\{\left[v_m\left(t\right)-v_m\left(t_1\right)\right]{\tau }_m\left(t\right)+\left[{\varphi }_m\left(t\right)-{\varphi }_m\left(t_1\right)\right]{\sigma }_m\left(t\right)\right\}dA\right).\end{array}$$

Now, we consider the motion Equation (5), the kinematic relations (1) and the tractions on the surface from (8), so that we obtain the next identity:

$$\begin{array}{c}\hfill {\left(\frac{d}{dt}\right)}_{t_1}\left({\int }_{\partial D}\left\{\left[v_m\left(t\right)-v_m\left(t_1\right)\right]{\tau }_m\left(t\right)+\left[{\varphi }_m\left(t\right)-{\varphi }_m\left(t_1\right)\right]{\sigma }_m\left(t\right)\right\}dA\right)\\ \hfill ={\left(\frac{d}{dt}\right)}_{t_1}\left({\int }_D\left\{\varrho \left[v_m\left(t\right)-v_m\left(t_1\right)\right]{\ddot{v}}_m\left(t\right)+J_{mn}\left[{\varphi }_m\left(t\right)-{\varphi }_m\left(t_1\right)\right]{\ddot{\varphi }}_n\left(t\right)\right\}dV\right)\\ \hfill +{\left(\frac{d}{dt}\right)}_{t_1}\left({\int }_D\left\{\left[{ϵ}_{mn}\left(t\right)-{ϵ}_{mn}\left(t_1\right)\right]{\tau }_{mn}\left(t\right)+\left[e_{mn}\left(t\right)-e_{mn}\left(t_1\right)\right]{\sigma }_{mn}\left(t\right)\right\}dV\right).\end{array}$$

(24)

According to the Equation (24), one can write:

$$\begin{array}{c}\hfill \mathcal{E}\left(t_1\right)={\left(\frac{d}{dt}\right)}_{t_1}\left({\int }_D\left\{\varrho \left[v_m\left(t\right)-v_m\left(t_1\right)\right]{\ddot{v}}_m\left(t\right)+J_{mn}\left[{\varphi }_m\left(t\right)-{\varphi }_m\left(t_1\right)\right]{\ddot{\varphi }}_n\left(t\right)\right\}dV\right)\\ \hfill +{\left(\frac{d}{dt}\right)}_{t_1}\left({\int }_D\left\{\left[{ϵ}_{mn}\left(t\right)-{ϵ}_{mn}\left(t_1\right)\right]{\tau }_{mn}\left(t\right)+\left[e_{mn}\left(t\right)-e_{mn}\left(t_1\right)\right]{\sigma }_{mn}\left(t\right)\right\}dV\right)\\ \hfill -{\left(\frac{d}{dt}\right)}_{t_1}\left({\int }_D\left[E+\frac{1}{2}\left(\varrho {\dot{v}}_m{\dot{v}}_m+J_{mn}{\dot{\varphi }}_m{\dot{\varphi }}_n\right)\right]dV\right).\end{array}$$

(25)

Further, it is not difficult to verify that:

$$\begin{array}{c}\hfill {\int }_{t_1}^t\left[{\tau }_{mn}\left(t\right){\dot{ϵ}}_{mn}\left(s\right)+{\sigma }_{mn}\left(t\right){\dot{e}}_{mn}\left(s\right)\right]ds\\ \hfill =\left[{ϵ}_{mn}\left(t\right)-{ϵ}_{mn}\left(t_1\right)\right]{\tau }_{mn}\left(t\right)+\left[e_{mn}\left(t\right)-e_{mn}\left(t_1\right)\right]{\sigma }_{mn}\left(t\right).\end{array}$$

(26)

Now, by integrating over D for both sides of Equation (26), we deduce:

$$\begin{array}{c}\hfill {\int }_D\left\{{\int }_{t_1}^t\left[{\tau }_{mn}\left(t\right){\dot{ϵ}}_{mn}\left(s\right)+{\sigma }_{mn}\left(t\right){\dot{e}}_{mn}\left(s\right)\right]ds\right\}dV\\ \hfill ={\int }_D\left\{\left[{ϵ}_{mn}\left(t\right)-{ϵ}_{mn}\left(t_1\right)\right]{\tau }_{mn}\left(t\right)+\left[e_{mn}\left(t\right)-e_{mn}\left(t_1\right)\right]{\sigma }_{mn}\left(t\right)\right\}dV.\end{array}$$

(27)

By using some similar calculus, we can obtain an analogous relation:

$$\begin{array}{c}\hfill {\int }_D{\int }_{t_1}^t\left[\varrho {\ddot{v}}_m\left(t\right){\dot{v}}_m\left(s\right)+J_{mn}{\ddot{\varphi }}_m\left(t\right){\dot{\varphi }}_n\left(s\right)\right]dsdV\\ \hfill ={\int }_D\left\{\varrho \left[v_m\left(t\right)-v_m\left(t_1\right)\right]{\ddot{v}}_m\left(t\right)+J_{mn}\left[{\varphi }_m\left(t\right)-{\varphi }_m\left(t_1\right)\right]{\ddot{\varphi }}_n\left(t\right)\right\}dV.\end{array}$$

(28)

Now, we consider Equation (10) in order to obtain:

$$\begin{array}{c}\hfill \frac{d}{dt}{\int }_D\left[E+\frac{1}{2}\left(\varrho {\dot{v}}_m{\dot{v}}_m+J_{mn}{\dot{\varphi }}_m{\dot{\varphi }}_n\right)\right]dV\\ \hfill =\frac{d}{dt}{\int }_D{\int }_{t_1}^t\left[{\tau }_{mn}\left(t\right){\dot{ϵ}}_{mn}\left(s\right)+t_{mn}\left(t\right){\dot{e}}_{mn}\left(s\right)\right.\\ \hfill \left.+\varrho {\ddot{v}}_m\left(t\right){\dot{v}}_m\left(s\right)+J_{mn}{\ddot{\varphi }}_m\left(t\right){\dot{\varphi }}_n\left(s\right)\right]dsdV.\end{array}$$

(29)

Ultimately, we consider the relations (27)–(29) so that from (25), we arrive at identity (22), and the proof of the theorem ends. □

Our results are placed within a linear theory of Cosserat elastic media. After formulating the basic equations and conditions in this context, we approached the rate of dynamic energy release due to the propagation of a straight crack in this type of material. We insisted on highlighting the effect of microinertia and the couples stresses on this rate, these being contributions of the Cosserat structure. Now, we emphasize the fact that compared to classical elasticity:

a. There was only one balance equation, namely ${\left(5\right)}_1$; now there is another one, namely ${\left(5\right)}_2$, in which microinertia appears.

b. There was only one stress tensor; now there is another one that appears in both balance equations.

c. There was only one strain tensor; now there is another one that appears in both constitutive equations.

d. There was only one constitutive equation; now there is another one.

It is not difficult to observe that these contributions are not so obvious as to cause spectacular and meaningful changes in the propagation of a straight crack. In our basic results, concretized in estimations (13) and (22), we deduced certain alternative formulations for the rate of dynamic energy release. These refer to some boundary concrete integrals calculated on a surface that includes the crack in its inside, and this surface shrinks to the tip of the crack.