Ulam-Hyers stabilities of fractional functional differential equations

J. Vanterler da C. Sousa; E. Capelas de Oliveira; F. G. Rodrigues; J. Vanterler da C. Sousa; E. Capelas de Oliveira; F. G. Rodrigues

doi:10.3934/math.2020092

AIMS Mathematics

2020, Volume 5, Issue 2: 1346-1358. doi: 10.3934/math.2020092

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Research article

Ulam-Hyers stabilities of fractional functional differential equations

1.
Department of Applied Mathematics, Imecc-Unicamp, 13083-859, Campinas, SP, Brazil
2.
Departamento de Matemáticas, Universidad de la Serena, Benavente 980, La Serena, Chile

Received: 12 October 2019 Accepted: 17 January 2020 Published: 20 January 2020
MSC : 26A33, 34A08, 34K37, 34K20

From the first results on Ulam-Hyers stability, what has been noted is the exponential growth of the researchers dedicated to investigating Ulam-Hyers stability of fractional differential equation solutions whether they are functional, evolution, impulsive, among others. However, some issues and problems still need to be addressed. An intensifying problem is the small amount of work on Ulam-Hyers stability of solutions of fractional functional differential equations through more general fractional operators. In this sense, in this paper, we present a study on the Ulam-Hyers and UlamHyers-Rassias stabilities of the solution of the fractional functional differential equation using the Banach fixed point theorem.

Keywords:

Citation: J. Vanterler da C. Sousa, E. Capelas de Oliveira, F. G. Rodrigues. Ulam-Hyers stabilities of fractional functional differential equations[J]. AIMS Mathematics, 2020, 5(2): 1346-1358. doi: 10.3934/math.2020092

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Abstract

1. Introduction

Fracture mechanics is a fundamental field of study in mechanical sciences. Over the years, extensive research has focused on studying fracture processes under static loading conditions, providing valuable insights into crack initiation, propagation, and structural failure. However, in many real-world scenarios, materials are subjected to dynamic loading conditions, where cracks can propagate at high speeds and exhibit unique fracture behaviors ^{[7,39,43,45,64,73,77,80]}. Understanding the mechanics of dynamic fracture is hence crucial for enhancing the safety and reliability of structural components and advancing the design of resilient materials. The mechanics underlying dynamic fractures presents many more intricacies than underlying quasi-static fracture ^{[58,59,60,61]}. Experimental studies, however, show its practical relevance^{[6,17,20,23,25,68,79,81]} and also elucidate its loading rate-dependent behavior ^{[18,24,26,27,30,40,44,70,72,78]}. More specifically, the study of dynamic fracture finds application in engineering ^[8,18,54], automotive safety ^[33], medicine ^[69], material science ^[28], and biology ^[74]. Examples of dynamic fracture events include rapid crack propagation in structural components under time-dependent loading due to, e.g., impacts, explosions, or earthquakes ^[24], fragmentation of brittle materials under high-velocity impacts, and crack propagation in geological formations due to seismic activity. The study of dynamic fracture mechanics aims to unravel the underlying mechanisms governing the initiation, propagation, and arrest of cracks under these extreme loading conditions.

This paper presents a physico-mathematical model that addresses dynamic fracture propagation in brittle materials by applying a purely hemi-variational continuum mechanics-based strain gradient theory. This paper is based on experimental analyses conducted in ^[41], which delved into the dependence of the fracture path on the loading rate. In a theoretical paper aimed at predicting mathematically the fracture path and the related branching phenomena, the same authors utilized a microplane model ^[42] in a 3D numerical simulation. In works ^[5,22,55,71], in order to describe the experimental results on dynamic fracture propagation reported in ^[41], different approaches were used like peridynamic, microplane, and phase-field damage modeling. The recent work of Nguyen et al. ^[38] gives a reformulation, in terms of sprain and spress, of a subset of micromorphic theories discussed more than half a century ago by Germain ^[21], Mindlin ^[34], and Eringen ^[16], whose micromechanical foundations for granular systems have been elaborated in a variational approach for finite deformation ^[36], geometrically nonlinear strain gradient models ^[4], and n-th order micromorphic materials ^[37]. Following this granular micromechanical approach and utilizing hemivariational methods, a group of researchers in a series of papers ^{[10,35,47,49,50,52,67]} have clearly established the physical meaning of boundary conditions associated with strain gradient theories ^{[2,3,11,12,13,14,15,62,63,66,76]}. Moreover, they have shown that for heterogeneous systems, strain gradient theories with appropriate elastic and dissipative energies can predict a multitude of phenomena, such as evolution of chirality and size of localization/fracture zone, that are oftentimes ignored or overlooked in classical approaches ^[9,19,29,31]. The strain gradient with the inertia gradient effects approach was used in ^[46,65], the treatment being purely analytical, without application of numerical methods such as the finite element method (FEM). In the papers of Rao et al. ^[56,57], strain gradient effects in the modeling of fracture mechanics have been considered with a two-scale asymptotic analysis. In this study, we employ an isotropic model to investigate the behavior of experiments presented in ^[41]. Our objective is to tune the evolution of damage based on discerning the local strain/stress state, namely, tension/compression. Specifically, we aim at mitigating, avoiding actually, its occurrence in compression. The utilization of an isotropic model accounts for uniformity of the microstructure with respect to directions.

The paper is organized as follows. In Section 2, we introduce the strain-gradient continuum model–elastic and damage components-employed to investigate numerically the propagation of dynamic fracture. In Section 3, we apply the presented model to an experimental case study that is then used for comparison with experimental data and other theoretical approaches available in the literature by solving through the weak form package of the software COMSOL Multiphysics and the weak form of its governing equations. Finally, conclusions are drawn in Section 4.

2. Formulation of the problem

2.1. Preliminary definitions

Let us consider a 2D continuum body $\mathfrak{V}\subset E^{2}$ , where the domain $E^{2}$ is the Euclidean two-dimensional space. The position of every point in the body is established through the utilization of $X$ coordinates in a frame of reference. The set of kinematic descriptors for such a model contains both the displacement field $u = u(X, t)$ and the damage field $\omega = \omega(X, t)$ . The damage state of a material point $X$ is therefore characterized, at time $t$ , by a scalar internal variable $\omega$ , which is assumed to take values within the range [0, 1]. The case $\omega = 0$ corresponds to the undamaged state, while $\omega = 1$ corresponds to complete failure. We assume the material to be not self-healing and, hence, $\omega$ is assumed to be a nondecreasing function of time. If we consider the material as a microstructured one, it becomes significant to allow the deformation energy density to vary based on the second gradient of the displacement. Thus, because of objectivity, the potential energy $U$ is assumed to be a function of the strain tensor $G$ , of its gradient $\nabla G$ , and of the damage $\omega$ , i.e., $U = U(G, \nabla G, \omega)$ . Here, $G = \frac{1}{2}\left(F^{T}F-I\right)$ , where $F = \nabla\chi$ is the deformation gradient tensor, $\chi = X+u$ is the placement function, and $F^{T}$ is the transpose of $F$ . For the small displacement approximation that is dealt with in this work, the strain tensor $G$ is assumed to be equal to $E = Sym(\nabla u)$ , the symmetric part of the displacement gradient.

2.2. The Energy functional

The energy functional $\mathcal{E}(u, \omega)$ depends on the displacement $u$ and the damage $\omega$ and is given as follows ^[53]:

$\begin{equation} \begin{split} \mathcal{E}(u, \omega) & = \int_{\mathfrak{V}}\left[U_e(G, \nabla G, \omega)+U_d(\omega)-b^{ext}\cdot u-m^{ext}:\nabla u \right]dA \\ & -\int_{\partial\mathfrak{V}}[t^{ext}\cdot u+\tau^{ext}\cdot[(\nabla u)n]]ds-\int_{\left[\partial\partial\mathfrak{V}\right]}f^{ext}\cdot u \end{split} \end{equation}$

(2.1)

where $\partial\mathfrak{B}$ are the regular parts of the boundary of $\mathfrak{B}$ , the symbol $[\partial\partial\mathfrak{B}]$ denotes the regular parts of the boundary of $\partial\mathfrak{B}$ , i.e., its wedges, and $n$ is the unit external normal. The symbols (·) and (:) indicate the scalar product between vectors or tensors, respectively. The quantities $b^{ext}$ and $m^{ext}$ are, respectively, the external body force and double force (per unit area), while the quantities $t^{ext}$ and $\tau^{ext}$ are, respectively, the external force and double force (per unit length). Finally, the quantity $f^{ext}$ is the external concentrated force which is applied on the vertices $[\partial\partial\mathfrak{B}]$ .

2.3. Variational principle

In order to get governing equations for this model, we resort to the variational principle thoroughly presented in ^{[32,48,49,51,53,75]}, where the case of one-dimensional bodies was considered. We assume that the descriptors $u(X, t)$ and $\omega(X, t)$ verify the condition

$\begin{equation} \delta\mathcal{E}(u, \omega, \dot{u}, \dot{\omega})\leq\delta\mathcal{E}(u, \omega, \nu, \beta), \qquad\forall\nu, \ \forall\beta\geq0, \end{equation}$

(2.2)

where $\nu$ and $\beta$ are compatible virtual velocities and dots represent derivation with respect to time.

2.4. Reduction from the 3D Mindlin elastic strain gradient energy to the 2D explicit form

A general form of the deformation energy density of an elastic 3D isotropic strain gradient elastic material is provided in ^[34]. In our case, we have

$\begin{equation} U_{e}(G, \nabla G, \omega) = \mathbb{H}(-TrG+\gamma)(U^I_e+U^{II}_e)+\mathbb{H}(TrG-\gamma)\left[(1-r\omega)U^I_e+(1-m\omega)U^{II}_e\right] \end{equation}$

(2.3)

where $\mathbb{H}$ is the Heaviside function, $TrG$ is the trace of $G$ , $r$ affects on the 1st gradient residual inside the fracture, and $m$ affects on microstructure due to damage. Parameters $r$ and $m$ should be as close as possible to the value 1. $\gamma$ is a constitutive parameter representing the threshold of the trace of $G$ controlling the bi-modular behavior of damage, which stipulates that damage evolution occurs only when the trace of the strain tensor $G$ is positive and exceeds a predefined threshold. By implementing this condition, we ensure that damage progression is assessed solely in regions experiencing tensile stresses. The first gradient part $U^I_e$ of the deformation energy density above reads as

$\begin{equation} U^I_{e}(G, \nabla G) = \frac{\lambda}{2}G_{ii}G_{jj}+\mu G_{ij}G_{ij} \end{equation}$

(2.4)

and the strain gradient part $U^{II}_e$ reads as

$\begin{equation} U^{II}_{e}(G, \nabla G, \omega) = 2c_{3}G_{ii, j}G_{jh, h}+\frac{c_{4}}{2}G_{ii, k}G_{ll, k}+2c_{5}G_{ij, i}G_{jm, m}+c_{6}G_{ij, k}G_{ij, k}+2c_{7}G_{ij, k}G_{ki, j} \end{equation}$

(2.5)

where $\lambda$ , $\mu$ , $c_{3}$ , $c_{4}$ , $c_{5}$ , $c_{6}$ , $c_{7}$ $\in\mathbb{R}$ are independent parameters. Our model accounts for an isotropic material, because the key focus is on the examination of damage evolution exclusively under tension while avoiding its manifestation in compression, aimed at capturing the behavior of cementitious materials. The quantities $\lambda$ and $\mu$ in Eq (2.4) are referred to as Lamé parameters and it is a classical result in mechanics that they can be easily related to the Young modulus $Y$ and to the Poisson's ratio $\nu$ . For the 2D case, we have

$\begin{equation} \lambda = \frac{Y\nu}{(1+\nu)(1-2\nu)}, \qquad\mu = \frac{Y}{2(1+\nu)}. \end{equation}$

(2.6)

The coefficients $c_{3}$ – $c_{7}$ can instead be identified in terms of Young modulus and Lamé parameters, and an internal characteristic length–also referred to as the intergranular distance–within the granular micro-mechanics-based framework presented in ^[4] is as follows:

$\begin{equation} \begin{cases} & c_{3} = YL^{2}\frac{\nu}{112(1+\nu)(1-2\nu)} = \frac{L^{2}}{112}\lambda, \\ & c_{4} = YL^{2}\frac{\nu}{112(1+\nu)(1-2\nu)} = \frac{L^{2}}{112}\lambda, \\ & c_{5} = YL^{2}\frac{7-8\nu}{2240(1+\nu)(1-2\nu)} = \frac{L^{2}}{1120}(7\mu+3\lambda), \\ & c_{6} = YL^{2}\frac{7-18\nu}{1120(1+\nu)(1-2\nu)} = \frac{L^{2}}{1120}(7\mu-4\lambda), \\ & c_{7} = YL^{2}\frac{7-8\nu}{2240(1+\nu)(1-2\nu)} = \frac{L^{2}}{1120}(7\mu+3\lambda). \end{cases} \end{equation}$

(2.7)

We now introduce the dissipated energy $U_d$ as

$\begin{equation} U_d(\omega) = \frac{1}{2}k_{\omega}\omega^{2} \end{equation}$

(2.8)

where $k_{\omega}$ represents the resistance to damage. The kinetic energy $U_{k}$ is instead assumed to having the following expression:

$\begin{equation} U_{k} = \frac{1}{2}\rho\big\Vert\frac{\partial u}{\partial t}\big\Vert^2 = \frac{1}{2}\rho \left( \frac{\partial^2 u_1}{\partial t^2}+ \frac{\partial^2 u_2}{\partial t^2}\right). \end{equation}$

(2.9)

where $\rho$ is material density per unit area and $u_{1}$ and $u_{2}$ are the displacement components in the horizontal and vertical directions, respectively. It is easy to prove ^[1] that having a nonzero kinetic energy in the form above is equivalent to have $b^{ext} = -\rho\ddot{u}$ .

3. Numerical results

The weak form package of the commercial software COMSOL Multiphysics has been employed to perform the numerical simulation presented in this section, based on the modeling presented so far.

3.1. Description of the numerical experiments

In the current section, we present numerical simulations to show the capabilities of the presented model to describe initiation and growth of damage localization zones. Following ^[41], we consider a 2D square specimen with a notch (pre-crack). The followig type of numerical experiment is considered: the left edge of the notch (the blue line) is fixed, i.e., cannot move, along the direction $\hat{e_{x}}$ , while the right edge of the notch (the red line) is subjected to an applied velocity in the $\hat{e_{x}}$ direction (from left to right). Such an applied velocity will be also referred to as crack-opening velocity.

Figure 1 illustrates this loading condition. In the original paper ^[41], there are twelve cases of different applied velocities, the loading-rates being in the range from 0.045 m/s to 4.298 m/s. In this section, only four cases of tested velocities are considered for the study, which are Case 1: 0.491 m/s; Case 2: 1.375 m/s; Case 3: 3.268 m/s, and Case 4: 4.298 m/s. In the current study, the applied velocity is assumed to depend upon time–the data of the actual loading history used in the experiments was not reported in the original paper, as follows:

$\begin{equation} v = \begin{cases} \frac{t}{t_{0}}v_{0} & t\leq t_{0}\\ v_{0} & t > t_{0} \end{cases} \end{equation}$

(3.1)

with $t_{0}$ = 100 $\mu s$ and $t$ = 500 $\mu s$ . The prescribed displacement velocity as a function of time for $v_{0}$ = 3.268 m/s is reported in Figure 2.

The constitutive parameters used in the numerical simulations are reported in , where $t_{f}$ is the time horizon of the simulations and $\triangle t$ the time step. It should be noted that since simulations were performed for the 2D case, to take into account the (uniform) thickness of the specimen–25 mm–some constitutive parameters were adapted by multiplying them by this value.

Figure 1. Schematics of analyzed domains and considered boundary conditions.

$Y$	$\nu$	$L$	$\rho$	$k_{\omega}$	$t_{f}$	$\triangle t$	$\gamma$	$r$	$m$
15 $GPa$	0.18	0.01118 $m$	2400 $kg/m^{3}$	5 $J/m^{2}$	500 $\mu s$	0.5 $\mu s$	0.0012	0.99999	0.99999

DOF	N. of elements	Max. element size, $m$	Reaction force, $kN$	Reaction integration over time
656	110	0.1	77.35	17.6204
684	115	0.08	76.87	17.5195
752	128	0.066	77.15	17.5876
796	136	0.04	76.16	17.4080
1108	195	0.026	76.17	17.4330
1614	292	0.02	76.01	17.3816
2768	514	0.0134	75.43	17.2840
3526	661	0.0106	75.52	17.3038
5200	987	0.008	75.32	17.2693
14500	2818	0.004	75.50	17.3041
51046	10060	0.002	75.17	17.2411
186276	36991	0.001	74.80	17.1749

Applied $v_0$ (m/s)	Exp. results ^[41]	Numerical results obtained with various approaches by
Applied $v_0$ (m/s)	Exp. results ^[41]	Ozbolt ^[42]	This paper	Hai ^[22]	Wu's IH-PD ^[71]	Wu's FH-PD ^[71]	Bui ^[5]	Qinami ^[55]
0.491	4.05	4.12	2.88	-	-	-	-	-
1.375	4.64	3.54	3.68	3.75	$\thickapprox$ 4.3	$\thickapprox$ 5.5	$\thickapprox$ 3.3	$\thickapprox$ 4.7
3.268	4.59	4.76	4.99	4.70	$\thickapprox$ 4.6	$\thickapprox$ 6.1	$\thickapprox$ 3.3	$\thickapprox$ 6.1
4.298	5.66	5.15	5.00	-	$\thickapprox$ 5.6	$\thickapprox$ 6.1	-	$\thickapprox$ 6.5

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AIMS Mathematics

Ulam-Hyers stabilities of fractional functional differential equations

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Abstract

1. Introduction

2. Formulation of the problem

2.1. Preliminary definitions

2.2. The Energy functional

2.3. Variational principle

2.4. Reduction from the 3D Mindlin elastic strain gradient energy to the 2D explicit form

3. Numerical results

3.1. Description of the numerical experiments

3.2. Convergence analysis

3.3. Numerical results with varying loading rate.

4. Conclusions

Author contributions

Use of AI tools declaration

Acknowledgments

Conflict of interest

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