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Fig.2 A discrete model composed of 7 cells; Fig.3 A discrete model composed of n cells; n = L/a = 3. N. Challamel et al.: Analytical length scale calibration of nonlocal continuum

Figure 2 A discrete model composed of 7 cells; Fig.3 A discrete model composed of n cells; n = L/a = 3. N. Challamel et al.: Analytical length scale calibration of nonlocal continuum

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Fig. 1 Buckling of microstructured column composed of repetitive cells; n = L/a is the number of cells. The buckling of a pinned-pinned column of length L which is modeled by some finite rigid segments and elastic rotational springs of stiffness k, is studied (see Fig. 1). The column is axially loaded by a concentrated force denoted by P. The column is composed of n repetitive cells of size denoted by a. In other words, the total length of the structure L is equal to L =n x a, the number of repetitive cells multiplied by the size of each cell. Such a model may have different engineering applications. The cell size a may be related to the interatomic distance for a physical model where the microstructure is directly related to the atomic discreteness of the matter. On the other hand, for granular applications, the microstructure can be related to some micromechanics length scale associated with the grain size or the size of aggregates in heterogeneous materials. For larger scale applications, such as repetitive frame structural design, this cell can also be related to some macroscopic length scale associated with the size of the repetitive frame scale. The free system composed of n-cells has n degrees of freedom (6), 02,...,0,). This system can also be referred as Hencky’s chain ( [33], see more recently [34]). As detailed by El Naschie [34], this problem has been already investigated in term of discrete element method, in order to approximate a continuous problem with a finite number of degrees of freedom. The purpose of the present paper is different in the sense that we are looking for the equivalent generalized continuum (or nonlocal continuum) that is able to reproduce the main features of a discrete medium, the last one is considered as the reference one. The internal elastic potential energy U is written as
Fig. 1 Buckling of microstructured column composed of repetitive cells; n = L/a is the number of cells. The buckling of a pinned-pinned column of length L which is modeled by some finite rigid segments and elastic rotational springs of stiffness k, is studied (see Fig. 1). The column is axially loaded by a concentrated force denoted by P. The column is composed of n repetitive cells of size denoted by a. In other words, the total length of the structure L is equal to L =n x a, the number of repetitive cells multiplied by the size of each cell. Such a model may have different engineering applications. The cell size a may be related to the interatomic distance for a physical model where the microstructure is directly related to the atomic discreteness of the matter. On the other hand, for granular applications, the microstructure can be related to some micromechanics length scale associated with the grain size or the size of aggregates in heterogeneous materials. For larger scale applications, such as repetitive frame structural design, this cell can also be related to some macroscopic length scale associated with the size of the repetitive frame scale. The free system composed of n-cells has n degrees of freedom (6), 02,...,0,). This system can also be referred as Hencky’s chain ( [33], see more recently [34]). As detailed by El Naschie [34], this problem has been already investigated in term of discrete element method, in order to approximate a continuous problem with a finite number of degrees of freedom. The purpose of the present paper is different in the sense that we are looking for the equivalent generalized continuum (or nonlocal continuum) that is able to reproduce the main features of a discrete medium, the last one is considered as the reference one. The internal elastic potential energy U is written as
This kinematics constraint can be expressed in an equivalent vectorial format as
This kinematics constraint can be expressed in an equivalent vectorial format as
The fundamental buckling load of the elastic system composed of 7 cells is simply computed from the singularity condition which allows one to calculate the smallest value denoted by a,, or by py, In Eq. (6), the subscript n for the calculation of the buckling load, referres to the number of cells in the discrete model. The kinematically constrained system composed of n-cells has in fact n — 1 degrees of freedom (n degrees of freedom for the free system and one additional constraint). The equilibrium is obtained from the variational equality |U — dW. = 0, leading to the stiffness matrix For instance, for n = 2 (structural system shown in Fig. 2), we calculate
The fundamental buckling load of the elastic system composed of 7 cells is simply computed from the singularity condition which allows one to calculate the smallest value denoted by a,, or by py, In Eq. (6), the subscript n for the calculation of the buckling load, referres to the number of cells in the discrete model. The kinematically constrained system composed of n-cells has in fact n — 1 degrees of freedom (n degrees of freedom for the free system and one additional constraint). The equilibrium is obtained from the variational equality |U — dW. = 0, leading to the stiffness matrix For instance, for n = 2 (structural system shown in Fig. 2), we calculate
Table 1 Calculation of the buckling load with respect to the number of cells. 3 Buckling model — Use of Lagrange multiplier Following the reasoning of Challamel et al. [35], the buckling problem of this elastic system with the kinematics constrain Eq. (5), can be presented in the simple form
Table 1 Calculation of the buckling load with respect to the number of cells. 3 Buckling model — Use of Lagrange multiplier Following the reasoning of Challamel et al. [35], the buckling problem of this elastic system with the kinematics constrain Eq. (5), can be presented in the simple form
This mathematical formulation, based on the calculation of a, from Eq. (14), is strictly equivalent to the one given by Eq. (8). For instance, for n = 2, we compute with the method based on the Lagrange multiplier, the buckling load as possesses the tridiagonal matrix property, and which can be expressed in a dimensionless format as
This mathematical formulation, based on the calculation of a, from Eq. (14), is strictly equivalent to the one given by Eq. (8). For instance, for n = 2, we compute with the method based on the Lagrange multiplier, the buckling load as possesses the tridiagonal matrix property, and which can be expressed in a dimensionless format as
It is also possible to expand g,, (3) directly from Eq. (19) and then obtain the contribution of d,, (3) a where Lc ° is given in Eq. (15) and dy, (3) is the additional term in the calculation of the determinant of the square matrix defined in Eq. (20) for all terms except the last one, which means that d,, ((3) can be simply expressed as The insertion of Eq. (23) into Eq. (21) leads to the relationship between g,, (3) and det x ° as By developing now according to the last row of Eq. (20), the following relationship is obtained a:
It is also possible to expand g,, (3) directly from Eq. (19) and then obtain the contribution of d,, (3) a where Lc ° is given in Eq. (15) and dy, (3) is the additional term in the calculation of the determinant of the square matrix defined in Eq. (20) for all terms except the last one, which means that d,, ((3) can be simply expressed as The insertion of Eq. (23) into Eq. (21) leads to the relationship between g,, (3) and det x ° as By developing now according to the last row of Eq. (20), the following relationship is obtained a:
We introduce now the dimensionless determinant It is also convenient to introduce a slightly different matrix from A’ ° which only differs by the last (n,n ) term of rq ° as
We introduce now the dimensionless determinant It is also convenient to introduce a slightly different matrix from A’ ° which only differs by the last (n,n ) term of rq ° as
Figure 4 shows that the normalized buckling load p is increasing with the number of cells n, i.e. the size of the repetitive structure. This softening effect with small scale structure is typically found with nonlocal elasticity model such as Eringen’s nonlocal model (see for instance Elishakoff et al. [14]). Following this softening behaviour, it is chosen in this part, to describe the microstructured beam by an equivalent continuum based on nonlocal elasticity of Eringen’s type. Fig. 5 Comparison between the exact discretized problem and the nonlocal Eringen’s solution for eg = fe =>. Fig.4 Evolution of the normalized buckling load with respect to the number n of cells. The buckling equations of Euler-Bernoulli column generalized to Eringen’s nonlocal elasticity are recalled for the hinge- inge case (see also [38] for this problem). Eringen’s nonlocal law is used for the equivalent continuum [1] 5 Nonlocal equivalent continuum
Figure 4 shows that the normalized buckling load p is increasing with the number of cells n, i.e. the size of the repetitive structure. This softening effect with small scale structure is typically found with nonlocal elasticity model such as Eringen’s nonlocal model (see for instance Elishakoff et al. [14]). Following this softening behaviour, it is chosen in this part, to describe the microstructured beam by an equivalent continuum based on nonlocal elasticity of Eringen’s type. Fig. 5 Comparison between the exact discretized problem and the nonlocal Eringen’s solution for eg = fe =>. Fig.4 Evolution of the normalized buckling load with respect to the number n of cells. The buckling equations of Euler-Bernoulli column generalized to Eringen’s nonlocal elasticity are recalled for the hinge- inge case (see also [38] for this problem). Eringen’s nonlocal law is used for the equivalent continuum [1] 5 Nonlocal equivalent continuum
Related topics:
EngineeringMathematicsNonlocal ElasticityEigenvalues and EigenvectorsScale Effects
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