Novel weak form quadrature elements for non-classical higher order beam and plate theories
Based on Lagrange and Hermite interpolation two novel versions of weak form quadrature element are proposed for a non-classical Euler-Bernoulli beam theory. By extending these concept two new plate elements are formulated using Lagrange-Lagrange and mixed Lagrange-Hermite interpolations for a non-classical Kirchhoff plate theory. The non-classical theories are governed by sixth order partial differential equation and have deflection, slope and curvature as degrees of freedom. A novel and generalize way is proposed herein to implement these degrees of freedom in a simple and efficient manner. A new procedure to compute the modified weighting coefficient matrices for beam and plate elements is presented. The proposed elements have displacement as the only degree of freedom in the element domain and displacement, slope and curvature at the boundaries. The Gauss-Lobatto-Legender quadrature points are considered as element nodes and also used for numerical integration of the element matrices. The framework for computing the stiffness matrices at the integration points is analogous to the conventional finite element method. Numerical examples on free vibration analysis of gradient beams and plates are presented to demonstrate the efficiency and accuracy of the proposed elements.
Keywords: Quadrature element, gradient elasticity theory, weighting coefficients, non-classical dofs, frequencies, mixed interpolation
Department of Aerospace Engineering, Indian Institute of Science
Bengaluru 560012, India
In recent decades the research in the field of computational solid and fluid mechanics focused on developing cost effective and highly accurate numerical schemes. Subsequently, many numerical schemes were proposed and applied to various engineering problems. The early research emphasized on the development of finite element and finite difference methods[1, 2, 3], these methodologies had limitations related to the computational cost. Alternatively, differential quadrature method (DQM) was proposed by Bellman  which employed less number of grid points. Later, many enriched versions of differential quadrature method were developed, for example, differential quadrature method [5, 6, 7, 8, 9, 10], harmonic differential quadrature method[11, 12], strong form differential quadrature element method (DQEM) [13, 14, 15, 16, 17, 18, 19], and weak form quadrature element method [20, 21, 22, 23]. The main theme in these improved DQ versions was to develop versatile models to account for complex loading, discontinuous geometries and generalized boundary conditions.
Lately, much research inclination is seen towards the strong and weak form DQ methods due their versatality[13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. The strong form differential quadrature method which is built on governing equations, require explicit expressions for interpolation functions and their derivatives, and yield unsymmetric matrices. In contrast, the weak form quadrature method is fomulated using variation principles, and the weighting coefficients are computed explicitly at integration points using the DQ rule, leading to symmetric matrices. The aforementioned literature forcussed on developing DQ schemes for classcial beam and plate theories which are governed by fourth order partial differential equations. The DQ solution for the sixth and eighth order differential equations using GDQR technique is due to Wu et al. [24, 25]. In their research, they employed strong form of governing equation in conjunction with Hermite interpolation function to compute the weighting coefficients and demonstrated the capability for structural and fluid mechanics problems. Recently, Wang et al.  proposed a strong form differential quadrature element based on Hermite interpolation to solve a sixth order partial differential equation associated with a non-local Euler-Bernoulli beam. The capability of the element was demonstrated through free vibration analysis. In this article the main focus is to propose a weak form quadrature beam and plate element for non-classical higher order theories, which are characterized by sixth order partial differential equations. As per the authors knowledge no such work is reported in the literature till date.
The non-classical higher order theories unlike classical continuum theories are governed by sixth order partial differential equations[27, 28, 29, 30, 31, 32]. These non-classical continuum theories are modified versions of classical continuum theories incorporating higher order gradient terms in the constitutive relations. The higher order terms consists of stress and strain gradients accompanied with intrinsic length which accounts for micro and nano scale effects[27, 28, 29, 30, 31, 32]. These scale dependent non-classical theories are efficient in capturing the micro and nano scale behaviours of structural systems[29, 30, 31]. One such class of non-classical gradient elasticity theory is the simplified theory by Mindlin et al. , with one gradient elastic modulus and two classical lame constant for structural applications [32, 33, 34]. This simlified theory was applied earlier to study the static, dynamic and buckling behaviour of gradient elastic beams [35, 36, 37] and plates [38, 39, 40] by developing analytical solutions. Pegios et al.  developed a finite element model for static and stability analysis of gradient beams. The numerical solution of 2-D and 3-D gradient elastic structural problems using finite element and boundary element methods can be found in .
In this paper, we propose for the first time, two novel versions of weak form quadrature beam elements to solve a sixth order partial differential equation encountered in higher order non-classical elasticity theories. The two versions of quadrature beam element are based on Lagrange and continuous Hermite interpolations, respectively. Further, we extend this concept and develop two new types of quadrature plate elements for gradient elastic plate theories. The first element employs Lagrange interpolation in and direction and second element is based on Lagrange-Hermite mixed interpolation with Lagrange interpolation in and Hermite in direction. These elements are formulated with the aid of variation principles, differential quadrature rule and Gauss Lobatto Legendre (GLL) quadrature rule. Here, the GLL points are used as element nodes and also to perform numerical integration to evaluate the stiffness and consistent mass matrices. The proposed elements have displacement, slope and curvature as the degrees of freedom at the element boundaries and only displacement in the domain. A new way to incorporate the non-classical boundary conditions associated with the gradient elastic beam and plate theory is proposed and implemented. The novelty in the proposed scheme is the way the classical and non-classical boundary conditions are represented accurately and with ease. It should be noted that the higher order degrees of freedom at the boundaries are built into the formulation only to enforce the boundary conditions.
The paper is organized as follows, first the theoretical basis of gradient elasticity theory required to formulate the quadrature elements is presented. Next, the quadrature elements based on Lagrange and Hermite interpolations functions for an Euler-Bernoulli gradient beam are formulated. Later, the formulation for the quadrature plate elements are given. Finally, numerical results on free vibration analysis of gradient beams and plates are presented to demonstrate the capability of the proposed elements followed by conclusions.
1 Strain gradient elasticity theory
In this study, we consider Mindlin’s  simplified strain gradient micro-elasticity theory with two classical and one non-classical material constants. The two classical material constants are Lame constants and the non-classical one is related to intrinsic bulk length . The theoretical basis of gradient elastic theory required to formulate the quadrature beam and plate elements are presented in this section.
1.1 Gradient elastic beam theory
where , are Lam constants. is the Laplacian operator and I is the unit tensor. , denotes Cauchy and higher order stress respectively, and () are the classical strain and its trace which are expressed in terms of displacement vector w as
From the above equations the constitutive relations for an Euler-Bernoulli gradient beam can be defined as
For the above state of stress and strain the strain energy in terms of displacements for a beam defined over a domain can be written as 
The kinetic energy is given as
where , , and are the Young’s modulus, area, moment of inertia, and density, respectively. is transverse displacement and over dot indicates differentiation with respect to time.
Using the The Hamilton’s principle:
we get the following weak form expression for elastic stiffness matrix ‘K’ and consistent mass matrix ‘m’ as
The governing equation of motion for a gradient elastic Euler-Bernoulli beam is obtained as
The above sixth order equation of motion yields three independent variables related to deflection , slope and curvature and six boundary conditions in total, as given below
Classical boundary conditions :
Non-classical boundary conditions :
where , and are shear force, bending moment and higher order moment, respectively.
1.2 Gradient elastic plate theory
The strain-displacement relations for a Kirchhoff’s plate theory are defined as 
where and are the classical and gradient elastic strain energy given by
The kinetic energy is given by
Using the The Hamilton’s principle:
we obtain the following expression for elastic stiffness and mass matrix for a gradient elastic plate
where , are classical and non-classical elastic stiffness matrix defined as
The equation of motion for a gradient elastic Kirchhoff plate considering the inertial effect is obtained as:
the associated boundary conditions for the plate with origin at and domain defined over (), (), are listed below.
Classical boundary conditions :
Non-classical boundary conditions :
where and are the length and width of the plate. , are the shear force, , are the bending moment and , are the higher order moment. The different boundary conditions employed in the present study for a gradient elastic Kirchhoff plate are
Simply supported on all edges SSSS:
Free on all edges FFFF:
Simply supported and free on adjacent edges SSFF:
for the SSFF plate at and , condition is enforced. The above boundary conditions are described by a notation, for example, consider a SSFF plate, the first and second letter correspond to and edges, similarly, the third and fourth letter correspond to the edges and , respectively. Further, the letter S, C and F correspond to simply supported, clamped and free edges of the plate.
2 Quadrature element for a gradient elastic Euler-Bernoulli beam
Two novel quadrature elements for a gradient Euler-Bernoulli beam are presented in this section. First, the quadrature element based on Lagrangian interpolation is formulated. Later, the quadrature element based on continuous Hermite interpolation is developed. The procedure to modify the DQ rule to implement the classical and non-classical boundary conditions are explained. A typical N-node quadrature element for an Euler-Bernoulli gradient beam is shown in the Figure 1.
It can be noticed from the Figure 1, each interior node has only displacement as degrees of freedom and the boundary has 3 degrees of freedom , , . The new displacement vector now includes the slope and curvature as additional degrees of freedom at the element boundaries given by: . The procedure to incorporate these extra boundary degrees of freedom in to the formulation will be presented next for Lagrange and continuous Hermite interpolation based quadrature elements.
2.1 Lagrange interpolation based quadrature beam element
The displacement for a N-node quadrature beam is assumed as:
The first order derivative of the above interpolation function can be written as,
The conventional higher order weighting coefficients are computed as
Here, and are weighting coefficients for second and third order derivatives, respectively.
The sixth order partial differential equation given in Equation (9) renders slope and curvature as extra degrees of freedom at the element boundaries. To account for these extra boundary degrees of freedom in the formulation, the derivatives of conventional weighting function , , and are modified as follows:
First order derivative matrix:
Second order derivative matrix:
Third order derivative matrix:
Here and are the coordinate and weights of GLL quadrature. is the Dirac-delta function.
2.2 Hermite interpolation based quadrature beam element
For the case of quadrature element based on continuous Hermite interpolation the displacement for a N-node gradient beam element is assumed as
The th order derivative of with respect to is obtained from Equation (39) as
here and are the coordinate and weights of GLL quadrature. The consistent mass matrix remains the same as given by Equation (38).
Combining the stiffness and mass matrix, the system of equations after applying the boundary conditions can be expressed as
where the vector contains the boundary related non-zero slope and curvature dofs. Similarly, the vector includes all the non-zero displacement dofs of the beam. In the present analysis the boundary force vector is assumed to be zero, . Now, expressing the dofs in terms of , the system of equations reduces to
Here, is the modified stiffness matrix associated with dofs. The above system of equations leads to an Eigenvalue problem and its solutions renders frequencies and corresponding mode shapes.
3 Quadrature element for gradient elastic Kirchhoff plate
In this section, we formulate two novel quadrature elements for non-classical gradient Kirchhoff plate. First, the quadrature element based on Lagrange interpolation in and direction is presented. Next, the quadrature element based on Lagrange-Hermite mixed interpolation, with Lagrangian interpolation is direction and Hermite interpolation assumed in direction is formulated. The GLL points in and directions are used as element nodes. Similar to the beam elements discussed in the section 2, the plate element also has displacement as the only degrees of freedom in the domain and at the edges it has 3 degrees of freedom , or , or depending upon the edge. At the corners the element has five degrees of freedom, , , , and . The new displacement vector now includes the slope and curvature as additional degrees of freedom at the element boundaries given by: , where . A quadrature element for a gradient Kirchhoff plate with grid is shown in the Figure 2.
Here, are the number of grid points in and directions, respectively. It can be seen from the Figure 2, the plate element has three degrees of freedom on each edge, five degrees of freedom at the corners and only displacement in the domain. The slope and curvature dofs related to each edge of the plate are highlighted by the boxes. The transformation used for the plate is and with .
3.1 Lagrange interpolation based quadrature element for gradient elastic plates
The displacement for a node quadrature plate element is assumed as
where is the nodal displacement vector for the plate and , are the Lagrange interpolation functions in and directions, respectively. The slope and curvature degrees of freedom at the element boundaries are accounted while computing the weighting coefficients of higher order derivatives as discussed in section 2.1. Substituting the above Equation (48) in Equation (20) we get the stiffness matrix for a gradient elastic quadrature plate element as
where ( and are abscissas and weights of GLL quadrature rule. and are the classical and non-classical strain matrices at location for gradient elastic plate. and are the constitutive matrices corresponding to classical and gradient elastic plate. The classical and non-classical strain matrices are defined as
The classical and non-classical constitutive matrices are given as
The diagonal mass matrix is given by
3.2 Mixed interpolation based quadrature element for gradient elastic plates
The quadrature element presented here is based on mixed Lagrange-Hermite interpolation, with Lagrangian interpolation is assumed in direction and Hermite interpolation in direction. The displacement for a node mixed interpolation quadrature plate element is assumed as
where is the nodal displacement vector of the plate and and are the Lagrange and Hermite interpolation functions in and directions, respectively. The formulations based on mixed interpolation methods have advantage in excluding the mixed derivative dofs at the free corners of the plate. The modified weighting coefficient matrices derived in section 2.1, using Lagrange interpolations and those given in section 2.2, for Hermite interpolations are used in forming the element matrices. Substituting the above Equation (56) in Equation (20), we get the stiffness matrix for gradient elastic quadrature plate element based on mixed interpolation as