5.1 DISPLACEMENT FIELD

In the finite element method, as previously pointed out, the domain of the body is divided into small regions called elements. Assuming that these elements are small, one can use low-order polynomials to describe the displacement field of the element. For example, if the deformation of the body is negligible or small, only six coordinates are sufficient to define the rigid-body translation and rotation. In this special case, first-order polynomials can be used to describe exactly the rigid-body motion. For a small finite element on a deformable body, on the other hand, the deformation within the element can be in general much smaller than the overall deformation of the body, and therefore, one can still justify using a lower-order polynomial to describe the displacement of a small region on the body. After introducing the low-order polynomials to define the equations of motion of the finite elements, the body equations of motion can be obtained by assembling the equations of motion of its elements using the connectivity conditions at the finite element boundaries. By using this procedure, one can develop a powerful tool for the computer-aided analysis of structural components that have complex geometric shapes.

Separation of Variables

Finite element approximations are based on the concept of the separation of variables briefly introduced in Chapter 1 and used in the examples presented in several chapters of this book. The displacement of the finite element can be written as the product of two sets of functions: one set depends on the spatial coordinates, whereas the other set depends on time only. The separation of variables can conveniently be achieved by assuming that the position vector of arbitrary material points on the element can be written as polynomials of the spatial local coordinates of the element. The coefficients of these polynomials in the case of dynamics are the time-dependent variables. One can select points on the element, called nodes, and assign variables such as displacements, rotations, and slopes as nodal coordinates. Knowing the coordinates of these nodes in the reference configuration, one can write a set of algebraic equations using the assumed polynomial displacement field and determine the polynomial coefficients in terms of the nodal variables. In the formulation discussed in this section, all the components of the position vector are interpolated using polynomials that have the same order. For example, for a given domain defined by the spatial coordinates x₁, x₂, and x₃, the kth component of the position vector can be written as

5.1

The coefficients a_ik, i = 0, 1, 2, 3, …, in the case of dynamics, are assumed to depend only on time. In this case, the above assumed displacement field can be written as the product of functions that depend only on the spatial coordinates x = [x₁ x₂ x₃]^T and a vector of time-dependent coordinates. To see this separation of variables, the preceding equation can be written as

5.2

Denoting the space-dependent row in this equation as P_k(x) and the time-dependent vector as a_k(t), the preceding equation can be written as

5.3

where

5.4

One can select, as discussed in Chapter 1 and demonstrated by the examples presented at the end of this section, position coordinates and possibly spatial derivatives of the coordinates at selected points and write the coefficients a_k(t) in terms of these position coordinates and their derivatives. The number of the selected position coordinates and their derivatives must be equal to the number of the polynomial coefficients in order to have a number of algebraic equations, which can be solved for these coefficients. Let e be the vector of coordinates that may include position coordinates and their spatial derivatives at selected points at known local positions on the element. Substituting the values of the spatial coordinates in the preceding equation, one can write the following relationship between the selected coordinates and the coefficients of the polynomial:

5.5

where B_p is a constant square nonsingular matrix and a is the total vector of the polynomial coefficients. The preceding equation can be solved for the coefficients a in terms of the selected coordinates as . Substituting this equation into the assumed displacement field of Equation 3, one can write the displacement field in terms of the selected coordinates as

5.6

In this equation,

5.7

One, therefore, can write the position vector r using Equation 6 as

5.8

where . In this approach, S(x) is called the shape function matrix. The position vector field can then be written as the product of the space-dependent matrix S(x) and the vector of time-dependent coordinates e(t).

Using this approach of the separation of variables, and assuming that the continuum is divided into a number of n_e finite elements as shown in Figure 1, the displacement field of a finite element j can be written as

5.9

where r^j is the global position vector of an arbitrary point on the finite element j as shown in Figure 1, S^j = S^j(x^j) is a shape function matrix that depends on the element spatial coordinates defined in the element reference configuration, and e^j = e^j (t) is the vector of time-dependent nodal coordinates that define the displacements and possibly their spatial derivatives at a set of nodal points selected for the finite element.

Modes of Displacement

The selection of the number of nodal points and the number of coordinates at each node is one of the important factors that determine the accuracy of the finite element approximation. The general theory of continuum mechanics discussed in this book shows that the matrix of position vector gradients leads to nine independent modes of displacements for an infinitesimal volume at a material point. The polar decomposition theorem shows that the matrix of position vector gradients can be decomposed as the product of an orthogonal matrix and a symmetric matrix. The orthogonal matrix is function of three independent parameters that define the rotation of the infinitesimal volume, whereas the symmetric matrix is a function of six independent deformation parameters that define the strain components. Therefore, if the translation of the infinitesimal volume is considered, one has a total of 12 displacement modes for the infinitesimal volume: 3 translations, 3 rotations, and 6 deformation modes. Motivated by this simple and general continuum mechanics description of the motion of the infinitesimal volume, the vector of nodal coordinates can be selected in the three-dimensional analysis to consist of three translations and nine components of the position vector gradients. The nine components of the position vector gradients account for three rotations and six deformation modes. In this case, the vector of nodal coordinates e^j of the finite element at a node k can be written as

5.10

where r^jk is the absolute (global) position vector of the node k of the finite element j and is the vector of position gradients obtained by differentiation with respect to the spatial coordinate x_l, l = 1, 2, 3. Note that is the matrix of the position vector gradients at node k, where . Note also that the last three vector elements of the vector of nodal coordinates e^jk in Equation 10 are the columns of the matrix of position vector gradients that enters in the formulation of the strain tensors. It is important that the chosen assumed displacement field and nodal coordinates correctly describe an arbitrary displacement including rigid-body displacement. This is an essential requirement, particularly in problems in the field of MBS dynamics in which the system components may undergo finite rotations.

Nodal Coordinates

In the finite element literature, one can find a large number of finite elements that have been developed to suit varieties of applications. Some elements take the shape of trusses, some the shape of beams, some the shape of rectangle, triangle or plate, solid (prism), tetrahedral, and many other shapes. These elements employ different numbers and types of nodal coordinates. For example, solid, triangle, rectangle, and tetrahedral elements employ, for the most part, only displacement coordinates. Rotations, slopes, or gradients are not commonly used for these elements. Conventional beam, plate, and shell elements employ, in addition to the displacement coordinates, infinitesimal rotation coordinates. Arbitrary rigid body motion, however, cannot be defined as linear functions of infinitesimal rotations. Some newer beam, plate, and shell elements use finite rotations as nodal coordinates and define a rotation field that is independent from the displacement field. Because in the general theory of continuum mechanics, the rotation field is defined from the displacement field using the matrix of position vector gradients, the use of an independent finite rotation field can lead to coordinate redundancy, which can be a source of fundamental and numerical problem (Ding et al., 2014). Discussion on other types of finite elements and finite element formulations is presented in a later section.

In the ANCF discussed in this chapter, no infinitesimal or finite rotations are used as nodal coordinates. In this formulation, only absolute position vectors consistent with the general continuum mechanics are interpolated, and the position vector gradient field is obtained by differentiation of the position vector field with respect to the spatial coordinates of the finite element. As will be shown in this chapter, in addition to having a description consistent with the general continuum mechanics theory, the formulation discussed in this chapter has many unique features that make it suited for the analysis of large deformation and large rotation of very flexible structures.

Inertia Forces

In order to formulate the inertia forces of the finite element, one must obtain an expression for the acceleration vector. Differentiating the global position vector of Equation 9 with respect to time, the absolute velocity vector v^j of an arbitrary material point on the element j can be written as

5.15

Differentiating this equation with respect to time, the acceleration vector a^j can be written in the case of the ANCF as

5.16

The virtual work of the inertia forces of the finite element can then be defined as

5.17

where ρ^j and V^j are, respectively, the mass density and volume of the finite element. It is important to point out that because of the principle of conservation of mass, ρ^jdV^j = constant, the mass density ρ^j, and the volume V^j in the reference configuration can be used. For the simplicity of the notation in this chapter, we use ρ^j instead of to denote the density in the reference configuration because in most of the developments presented in this chapter and the following one, the inertia can be formulated using the reference configuration. A specific mention will be made if ρ^j is the mass density associated with the current configuration, as it is the case when the equations that govern the fluid motion are discussed.

The virtual change in the position vector of the material point can be written as

5.18

Using the preceding two equations with the expression for the acceleration of Equation 16 and keeping in mind that the time-dependent nodal coordinates do not depend on the spatial coordinates and can be factored out of the integration sign, one obtains the following equation for the virtual work of the inertia forces:

5.19

This equation can be written as

5.20

where M^j is the symmetric mass matrix of the finite element j defined as

5.21

This mass matrix is constant in both two- and three-dimensional cases. This is a unique feature of the ANCF because other known three-dimensional finite element formulations that give complete information about the rotation at the nodes and use infinitesimal or finite rotation parameters as nodal coordinates do not lead to a constant mass matrix in the case of three-dimensional analysis. This important property of the formulation simplifies the governing equations significantly since it leads to zero centrifugal and Coriolis forces when the body experiences an arbitrary large deformation and finite rotation.

By using the preceding equations, the virtual work of the inertia forces can be written as

5.22

where is the vector of the inertia forces that takes the following simple form:

5.23

One can show that, for many ANCF finite elements, the mass matrix remains the same under an orthogonal coordinate transformation.

Elastic Forces

An expression for the virtual work of the stresses of the continuum was obtained in Chapter 3 in terms of the Green–Lagrange strain tensor and the second Piola–Kirchhoff stress tensor. For the finite element j, the virtual work of the stresses can be written as

5.24

In this equation, is the second Piola–Kirchhof stress tensor and ϵ^j is the Green–Lagrange strain tensor at an arbitrary material point on the finite element j. The stress and strain tensors used in Equation 24 are defined in the reference configuration. The virtual strain can be expressed in terms of the virtual changes of the position vector gradients as

5.25

The second Piola–Kirchhoff stresses are related to the Green–Lagrange strains using the constitutive equations

5.26

where E^j is the fourth-order tensor of elastic coefficients. Substituting the preceding two equations into the expression of the virtual work of the stresses, using the definition of the matrix of position vector gradients, and using the expression of the gradients in terms of the finite element nodal coordinates, one can show that the virtual work of the stresses of the finite element j can be written as

5.27

In this equation, is the vector of the elastic forces associated with the nodal coordinates of the finite element j. This vector, which is the result of the deformation of the continuum, takes a more complex form as compared to the simple expression of the inertia forces obtained previously in this section. Because the integrals for the stress forces are, in general, highly nonlinear functions in the nodal and spatial coordinates, numerical integration methods are often used for the evaluation of the nonlinear generalized stress forces.

Curved Geometry

ANCF finite elements can be used to conveniently describe complex shapes. These shapes can represent curved geometry in the undeformed reference configurations. In these cases, the position vector gradients in the undeformed reference configuration are not orthogonal unit vectors because of the continuum initial geometry. Examples of these applications, which were presented in Chapter 2 and include tires, belt drives, and tank cars, are shown in Figure 4. As discussed in Chapter 2, in these cases, the strains evaluated using the matrix of position vector gradients in the initial configuration must be identically equal to zero. In order to explain how the strains are formulated in these cases of initial curved geometry, the three different configurations shown in Figure 5 were considered in Chapter 2. Because complex geometries can always be obtained by changing the shape of simple objects, the first configuration in Figure 5 depicts the simple geometry, which can represent straight metal sheets described conveniently using ANCF finite elements. In this simple geometry configuration, the location of the material points is defined using the position vector of the ANCF element or an assembly of ANCF finite elements. The second configuration is the reference configuration, which defines the initial undeformed geometry. In this undeformed reference configuration, the position of the material points is defined by the vector . One can write , where , and the superscript that indicates the finite element number is dropped for simplicity. The third configuration shown in Figure 5 defines the current deformed configuration. In this configuration, the position of the material points is defined by the vector . As discussed in Chapter 2, one can write , with . It follows that , where .

The volume of the curved structure is related to the volume of the straight structure (Figure 5) using the relationship , where is the determinant of the matrix of position vector gradients . Therefore, integration with respect to the domain can be converted to integration with respect to the straight domain by using the transformation . This allows for using the original dimensions of the simpler geometry to carry out the integrations associated with the initially curved configuration. Note that the matrix is constant.

The matrix of position vector gradients is the matrix, which is used to determine the Lagrangian strain tensor as . This matrix can be defined using the ANCF description as

5.40

where, as previously defined, . Therefore, the Lagrangian strain tensor can be written as

5.41

Note that the relationship between the volume in the current deformed configuration and the volume in the curved reference configuration can be written as , where is the determinant of the matrix of position gradients . It follows that . Using the relationship , one has .

As mentioned in Chapter 2, the procedure described in this section to model the initial curvature, which is the same as the one used in the literature for modeling the initially curved configuration of belt drives and rubber chains (Dufva et al., 2007; Maqueda et al., 2010), will lead to zero strains for an arbitrary initially curved geometry. Using the ANCF finite elements discussed later in this chapter, the constant matrix of position vector gradients can be determined in a straightforward manner. In the ANCF description, the assumed displacement field can be written as , where is the global position vector, is the vector of the element spatial coordinates, is time, is the element shape function matrix, and is the vector of the element nodal coordinates that include absolute position and gradient coordinates. In the ANCF description, the vector of nodal coordinates can be written as , where is the vector of nodal coordinates in the reference configuration and is the vector of nodal displacements. Using this partitioning, the assumed displacement field can be written as . Using the general continuum mechanics description , where is the absolute position vector of an arbitrary point in the reference configuration and is the displacement vector, one can write and . By appropriate choice of the elements of the vector , initially curved structures can be defined in a straightforward manner using ANCF finite elements.

As previously mentioned, integration with respect to the domain can be converted to integration with respect to the straight element domain by using the transformation , where is the determinant of the matrix of position vector gradients . In the ANCF description, the matrix is constant, while . Table 1 explains how the basic continuum mechanics description is implemented using ANCF finite elements.

Gaussian Quadrature

In the Gaussian quadrature formulas, the integral is evaluated by approximating the function f(x) by a polynomial P_n(x) defined at unequally spaced base points. This function approximation can lead to an error δ(x), as previously mentioned. The locations of the base points are determined by developing a set of algebraic equations that make the integral of the error function equal to zero. The solution of these algebraic equations, which are obtained using the properties of the orthogonal polynomials, defines the base points. One can then write the integral I in the following form:

5.47

The domain of integration can be changed from x ∈ [a, b] to ξ ∈ [−1, 1] by using the substitution

5.48

The steps used to determine the base points are first to employ the Lagrangian interpolating polynomials to approximate P_n(x) and δ(x). The Lagrangian interpolating polynomials do not require equally spaced base points. The resulting polynomials are then expressed in terms of the orthogonal Legendre polynomials. The Legendre polynomial orthogonality conditions are used to define a set of algebraic equations that make the integral of the error function equal to zero. This set of algebraic equations defines the base points for different orders of the polynomial P_n(x). The integral can then be written in terms of the function evaluated at these base points multiplied by weight factors or weight coefficients. This procedure for determining the base points is described in detail by Carnahan et al. (1969). The results are weight factors, which depend on the order of the polynomial or the number of base points selected to approximate f(x). These weight factors are presented in tables in mathematics handbooks or in textbooks on the subject of numerical analysis. In general, if the integral of the error function becomes zero and the domain of integration is changed to [−1, 1], the integral I can be written in terms of the function at the base points and the weight coefficients as

5.49

where w_i, i = 1, 2, …, m, are the weight factors. These weight factors are called Gauss–Legendre coefficients. The weight factors are selected in order to achieve the greatest accuracy. Symmetrically located base points have the same weight coefficients. Note that if g(ξ) is approximated by one quadrature point, there is only one base point ξ₁ = 0. In this case, the integral is given by . If g(ξ) is an arbitrary linear function (n = 1), this integration result obtained using one quadrature point is exact. In this case, w₁ represents the length of the domain of integration, that is, w₁ = 2, and g(ξ₁) is the height used to determine the area under the curve. In this case, the function g(ξ) is evaluated at ξ₁ = 0, which represents the center of the interval. If g(ξ), on the other hand, represents a higher-order function, the one-point quadrature integration is an approximation. In general, a polynomial of degree n requires m = (n + 1)/2 quadrature base points for exact integration, where in this simple rule n is assumed an odd number.

Generalization

The procedure used for integrating a function that depends on one variable can be generalized to the case in which the function depends on two or three variables, as it is the case in some finite element assumed displacement fields. In the two-dimensional case, consider the function g(ξ, η). It is assumed that the domains of integration are changed as discussed before such that ξ ∈ [−1, 1], and η ∈ [−1, 1]. The integral of the function g(ξ, η) can then be written as

5.50

Performing the second integration with respect to η, one obtains

5.51

Following a similar procedure, one can show that in the case of a function that depends on three variables, ξ, η, and ζ, one has the following Gauss quadrature formula:

5.52

If the original function is expressed in terms of the coordinates x₁, x₂, and x₃, that is, f = f(x₁, x_2, x₃), the preceding formula requires using the relationship dx₁dx₂dx₃ = Jdξdηdζ, where J is the determinant of the Jacobian of the coordinate transformation.

Using the Gauss quadrature formulas presented in this section, a systematic procedure for the numerical evaluation of the nonlinear stress elastic forces of the finite elements can be developed. The number of quadrature points used in the numerical integration defines the accuracy of the integration, and therefore, this number must be carefully selected in order to avoid increasing the computational cost or obtaining inaccurate results. There is no advantage gained from using a number of quadrature points larger than the number that gives exact evaluation of the integral (full integration). In some applications, on the other hand, there is an advantage in selecting a number of points that does not yield exact evaluation of the integrals. This is the case of reduced integration, which is commonly adopted in the finite element computational algorithms and will be discussed in a later section.

General Continuum Mechanics Approach and Classical Theories

It is important to point out that in all the finite elements that are discussed in this chapter, classical theories can be used by introducing a local element frame. Therefore, for beam elements, one can still use Euler–Bernoulli and Timoshenko beam theories; and, for plate and shell elements, one can still use Kirchhoff and Mindlin plate theories. This can always be accomplished by using the element local frame, which serves the only purpose of measuring the deformation (Shabana, 2013), and such a frame does not enter into the formulation of the inertia forces. Therefore, it is important to distinguish between this local element frame and the corotational frame used in the finite element literature. If the general continuum mechanics approach is used instead of the classical theories, one obtains more general formulations that relax the assumptions of Euler–Bernoulli, Timoshenko, Kirchhoff, and Mindlin theories. In these general formulations, the element cross section is allowed to deform.

Gradient Vectors

Some of the elements presented in this chapter employ a complete set of gradient vectors as nodal coordinates. These elements allow, in a straightforward manner, for the use of a general continuum mechanics approach to formulate the elastic forces. The use of these elements also allows for using more general constitutive relationships. Elements that do not employ a complete set of parameters required to evaluate all the gradient vectors are called in this book, gradient deficient. The use of a general continuum mechanics approach with these elements is not as straightforward as compared to elements that have a complete set of parameters (spatial coordinates). The latter elements are called fully parameterized elements.

Locking Problems

ANCF finite elements were introduced to deal with very flexible components. These finite elements perform well in the case of very flexible bodies, and efficient solutions for large deformations of very flexible bodies can be obtained because a nonincremental solution procedure can be used with these ANCF elements. As the element stiffness decreases, ANCF elements become more efficient. Some researchers, however, used ANCF finite elements in the analysis of thin and stiff structures. In this case, some elements exhibit locking problems when the general continuum mechanics approach is used to formulate the elastic forces. The general continuum mechanics approach leads to what is called ANCF-coupled deformation modes (Hussein et al., 2007). These modes, which couple the deformation of the cross section and other deformations such as bending, can have high frequencies and can be a source of numerical problems (Schwab and Meijaard, 2005). Several techniques were proposed in the literature to solve the locking problems and improve the element performance.

In order to better understand the behavior of the finite elements introduced in the following sections, an understanding of the geometry is necessary. Some basic results from the theories of curves and surfaces, which are covered in the subject of differential geometry, can help the reader better understand and solve the problems encountered when ANCF finite elements are used.

Theory of Curves

The centerline of a beam element represents a space curve. A curve can be uniquely defined in terms of one parameter. That is, the Cartesian coordinates that define the curve can be determined once this parameter is specified. Let α be the parameter that defines the curve over the interval a ≤ α ≤ b. The curve can then be represented by the following parametric form:

5.53

The tangent vector to the curve at α is given by

5.54

If at a given point α, |dr(α)/dα| = 0, the point is called a singular point. The parameter α can be selected to be the arc length s. If the arc length is used as a parameter, the tangent vector r_s is a unit vector. That is,

5.55

In order to mathematically prove this important result, let r be the vector that defines the position of the points on a space curve. One can write the following equation:

5.56

If Δα is assumed small, Δr defines the tangent vector. In this case, if s is assumed to be the arc length of the space curve that defines the centerline of the element, then one has

5.57

This equation shows that in the limit when Δs approaches zero, one has

5.58

That is, the tangent vector obtained by differentiation with respect to the arc length is indeed a unit vector.

If a curve is parameterized by its arc length, the derivative of the unit tangent vector defines the curvature vector. That is, the curvature vector is defined as

5.59

The magnitude of the curvature vector at a given s is called the curvature and given as

5.60

Because the tangent vector r_s(s) is a unit vector, the curvature κ(s) measures the rate of change of orientation of the tangent vector, that is, it measures the amount of bending of the curve. The preceding two equations show that linear displacement fields lead to zero curvature and, therefore, such fields are not appropriate for describing the displacement of components subjected to bending. Although one can approximate a curve by a large number of straight segments, in the finite element implementation the use of linear field will require the use of a very large number of finite elements. This increases the dimensions of the problem and can lead to a very inefficient solution procedure.

Because the tangent and curvature vectors are orthogonal, a unit vector along the curvature vector defines the unit normal to the curve n given as

5.61

The unit tangent and normal vectors form a plane called the osculating plane. The radius of curvature of the curve at s is defined as R = 1/κ(s). A vector normal to the osculating plane, called the binomial vector at s, is given by

5.62

The three orthogonal unit vectors r_s, n, and b form a coordinate system called the Frenet frame.

Differentiating Equation 62 with respect to s and keeping in mind that the vectors r_ss(s) and n(s) are parallel, one obtains

5.63

This equation shows that b_s(s) is normal to r_s. Furthermore, because b(s) is a unit vector, b_s(s) and b(s) are two orthogonal vectors, and b_s(s) is parallel to n. Therefore, b_s(s) can be written in the following form:

5.64

where τ is called the torsion. The curvature and torsion uniquely define the space curve.

Theory of Surfaces

Whereas a general space curve can be defined in terms of one parameter, a surface can be completely described in terms of two parameters s₁ and s₂. In general, a surface can be described in the following parametric form (Goetz, 1970; Kreyszig, 1991):

5.65

It is required that the mapping in this equation is one to one, and the Jacobian matrix

5.66

has a rank equal to two. This condition is satisfied if (∂r/∂s₁) × (∂r/∂s₂) ≠ 0, which implies that the two columns of the Jacobian matrix in the preceding equation are linearly independent. The two vectors and represent the two tangent vectors at the point of intersection of the coordinate lines s₁ and s₂. The unit vector normal to the surface at this point can then be defined as

5.67

As in the case of curves, the surface can be defined uniquely using local geometric quantities called the first and second fundamental forms. The first fundamental form of a surface is defined as follows:

5.68

This equation shows that the first fundamental form I can be used as a measure of distance or length. Using the fact that , the first fundamental form of the preceding equation can be written as

5.69

where

5.70

These coefficients are called the coefficients of the first fundamental form. One can show that distances, angles, and areas on the surface can be expressed in terms of the first fundamental form (Goetz, 1970; Kreyszig, 1991).

The second fundamental form of a surface is defined as

5.71

where n is the unit normal and the coefficients of the second fundamental form are defined as

5.72

Because and are perpendicular to n for all values of the parameters s₁ and s₂, one has the following identities:

5.73

Using these identities, the coefficients of the second fundamental form can be written in an alternate form as follows:

5.74

where . Using the preceding equation, and the fact that

5.75

one can show that the second fundamental form can be written in the following alternate form:

5.76

This equation can be used to measure the rate of change of orientation of the tangent plane. The coefficients of the second fundamental form can be used to determine the nature of the surface in the neighborhood of an arbitrary point P. If , the surface is called elliptic. If , the surface is called hyperbolic. If , the surface is called parabolic. If L_II = M_II = N_II = 0, the surface is called planar. Figure 6 shows examples of such surface geometry

Surface Curvature

One can always define a curve on a surface if the parameters s₁ and s₂ are expressed in terms of one parameter α. Let c = c(s₁(α), s₂(α)) be a regular curve defined on the surface r = r (s₁, s₂). The normal curvature vector to the curve c at point P denoted by K_n is defined as the projection of the curvature vector c_ss of the curve onto the normal n to the surface at point P and is given by

5.77

In this equation, s is the arc length of the curve. Note that the curve c can be defined as the intersection of a plane that contains the tangent to c and the normal vector. The norm of the normal curvature vector defined in the preceding equation is called the normal curvature and is defined as

5.78

Recall that the curvature vector of the curve at a point P on the surface r is given by

5.79

where r_s is the tangent vector to the curve at P and s is the curve arc length. In deriving the preceding equation, one utilized the fact that |dr/dα| = |r_s|(ds/dα) = (ds/dα), which is the consequence of the fact that |r_s| = 1. Because r_s is orthogonal to n, one has , which leads to

5.80

Substituting Equation 79 into Equation 78 and using Equation 80, one obtains

5.81

Because the first fundamental form I is positive, the sign of k_n depends on the sign of the second fundamental form II. Using the preceding equation, one can show that k_n = 0 in all directions for a planar point. For an elliptic point, k_n ≠ 0 and has the same sign as the ratio ds₁/ds₂. In the case of a hyperbolic point, k_n can be positive, negative, or zero, depending on the sign and value of ds₁/ds₂. For a parabolic point, k_n maintains the same sign and it is zero if the second fundamental form II is equal to zero.

The directions that define the maximum or minimum values of the normal curvature can be obtained by differentiating Equation 81 with respect to the parameters s₁ and s₂, and setting the results equal to zero. That is,

5.82

Substituting Equation 81 into these equations, one obtains

5.83

This equation has a nontrivial solution if and only if the determinant of the coefficient matrix is equal to zero, that is,

5.84

The solution of this quadratic equation defines two roots k₁ and k₂. These two roots, which are called the principal curvatures, can be substituted into Equation 83 to determine the principal directions. The mean curvature K_m and the Gaussian curvature k_G at a point P on the surface are defined in terms of the principal curvatures as

5.85

These surface definitions as well as the analysis of curve and surface geometry presented in this section are important to understand the behavior of beams, plates, and shells. Some of the obtained geometric results shed light on the order of approximation that must be used when employing the finite element method to solve beam, plate, and shell problems. For example, as previously discussed, the curvature is obtained from the second derivative of the position vector. Therefore, finite elements that employ linear or bilinear approximation cannot be effectively used in bending problems because the curvature will always be equal to zero. When these linear and bilinear elements are employed, one must use a very fine mesh in order to be able to represent a space curve or a shell by straight lines or flat sections, respectively. This approach tends to be very inefficient and, therefore, the use of structural finite elements that are based on higher-order interpolations is recommended. Some of these ANCF elements are discussed in the following sections.

Kinematics of the Element

In order to understand the kinematics of the two-dimensional Euler–Bernoulli finite element discussed in this section, some differential geometry results are required. For simplicity, the superscript that indicates the element number is dropped in the following discussion.

The displacement field of the Euler–Bernoulli beam element is function of one spatial coordinate x₁ only. For a given, deformed shape of the element, the element centerline defines a space curve. The unit tangent to this space curve is defined by the vector

5.90

In this equation, s is the arc length. Because r_s is a unit vector, that is, , it follows by differentiating this equation that . This implies that the derivative of the unit tangent with respect to the arc length defines the curvature vector r_ss, which is perpendicular to r_s. The magnitude of the curvature vector κ, called the curvature, measures the rate of change of the tangent vector r_s along the arc length. Therefore, the curvature, as previously defined in this chapter, is

5.91

This curvature can be expressed in terms of derivatives with respect to the element spatial coordinate x₁(Goetz, 1970; Dmitrochenko and Pogorelov, 2003; Gerstmayr and Shabana, 2006). To this end, recall that . Because has the same direction as the unit tangent r_s, one can write and . It follows that

5.92

Because r_s is a unit vector perpendicular to r_ss, the curvature can be written upon utilizing the preceding equation as

5.93

This definition of the curvature does not imply any linearization or simplifications and can be used to define the bending strain in the large-deformation analysis of the Euler–Bernoulli beam element described in this section.

The discussion on the geometry presented in this section shows that if linear interpolation instead of the cubic interpolation is used for this element, the curvature will be zero everywhere inside the element, as previously pointed out. That is, one cannot bend this element. Therefore, a finite element mesh that employs linear interpolation will require a very large number of elements to achieve convergence in beam-bending problems. If quadratic interpolation is used, one obtains, at most, constant curvature. Elements that employ quadratic interpolations lead to zero shear forces as can be demonstrated using simple equilibrium considerations. It is, therefore, recommended to use cubic interpolation to represent beam bending.

Formulation of the Element Elastic Forces

In the case of the Euler–Bernoulli beam element, one can define one gradient vector only because the element assumed displacement field is expressed in terms of one spatial coordinate x₁. That is, this element is gradient deficient. Therefore, for this element, the only nonzero strain component is the axial strain, and the shear strain is assumed to be zero. When one or more gradient vectors are missing, the formulation of the elastic forces using the general continuum mechanics approach is not straightforward. Elements that are not gradient deficient have two gradient vectors in the planar analysis and three gradient vectors in the spatial analysis.

Because one spatial coordinate only is used for the Euler–Bernoulli beam element, cannot be determined using the element assumed displacement field. In this case, the normal to the centerline of the element remains normal, and as a consequence, the shear deformation is assumed to be equal to zero and the cross section of the element is assumed to remain rigid and perpendicular to the element centerline. For this shear nondeformable element, only the strain component ϵ₁₁ can have nonzero value, and it measures only the extensional strain. The bending strain can be defined using the curvature. In the case of two-dimensional elements, which are not gradient deficient, the component ϵ₁₁ is a function of the spatial coordinate x₂, and such a component contributes to the bending strain of the finite element, as will be demonstrated in later sections. In this case of shear deformable elements, the use of the curvature definition to define the bending strain energy is not necessary.

The elastic forces of the two-dimensional Euler–Bernoulli beam element can be obtained by using the virtual work or the strain energy. The virtual work of the elastic forces can be written as

5.94

In this equation, l is the length of the element, E is the modulus of elasticity, A is the cross-sectional area, and I is the second moment of area. It is assumed in the preceding equation that the curvature and strain are defined in terms of the reference spatial coordinate x₁. Therefore, one can use undeformed geometry data in the integrations of the preceding equation.

The strain energy for the Euler–Bernoulli beam element can be written as

5.95

The first integral in the preceding equation represents the strain energy due to the extension, whereas the second integral is the strain energy due to bending. It is important to note that because this element is gradient deficient, one must resort to the curvature definition in order to account for the bending deformation. The curvature definition requires the evaluation of the second derivatives, which is one of the disadvantages of this element. If the element has a complete set of gradients, as it is the case of the shear deformable beam element discussed in the following section, one can use the Green–Lagrange strain tensor to evaluate the elastic forces. This tensor is a function of only first derivatives of the absolute position vector.

The expression of the total strain energy presented in the preceding equation can be used in the large-deformation analysis because no assumptions are made regarding the amount of axial and bending deformations. The strain and curvature can be expressed in terms of the element nodal coordinates. The vector of the elastic forces can be determined using the virtual work as previously described or by using the strain energy as Q_s = −(∂U/∂e)^T. Different elastic force models can be developed for the Euler–Bernoulli beam element presented in this section, as discussed in the literature (Berzeri and Shabana, 2000).

Special Case

As previously mentioned, the expression of the strain energy presented in this section imposes no restrictions on the amount of the axial and bending deformations of the element. The resulting elastic forces are nonlinear functions of the element nodal coordinates. These nonlinear forces include terms that couple the axial and bending deformations. This coupling has a significant effect on the dynamics of rotating beams. The ANCF element automatically accounts for this coupling when nonlinear strain displacement relationships as the ones employed in this section are used (Berzeri and Shabana, 2002).

In many structural applications, Euler–Bernoulli beam theory has been used for small-deformation analysis. In these structural applications, the rigid-body motion is eliminated. In this special case, an assumption is made that s = x₁. Using this assumption, one has . It follows that the curvature in this special case can be written as

5.96

Recall that the curvature vector can be written as

5.97

For structural systems in which the rigid-body motion is eliminated, r₁ = u is the axial displacement, and r₂ = v is the bending displacement. In this case, the strain ϵ₁₁ is approximated as . Furthermore, the effect of the derivatives of on the curvature is neglected. The curvature can then be approximated as . Using these assumptions, the strain energy that does not account for the coupling between the axial and bending displacements can be written as

5.98

This expression for the strain energy can only be used in the small-deformation analysis of structural systems, and the use of such an expression in the case of rotating beams can lead to significant errors as the result of the neglect of the effect of the coupling between bending and axial deformations.

Formulation of the Elastic Forces

Because the shear deformable beam element used in this section is not gradient deficient, one can use the general continuum mechanics approach to formulate the element elastic forces. For simplicity, superscript j that indicates the element number will be again dropped in the discussion presented in the remainder of this section. The matrix of position vector gradients of the element can be written in this case as

5.102

The Green–Lagrange strain tensor can then be evaluated to define two normal strain components ϵ₁₁ and ϵ₂₂, and one shear strain component ϵ₁₂. The general procedure described previously in this chapter can be employed to define the elastic forces by using the constitutive equations that relate the second Piola–Kirchhoff stress tensor to the Green–Lagrange strain tensor.

The interpolating polynomials used to develop the two-dimensional shear deformable element shape function matrix were introduced in Chapter 1. These polynomials were defined as

5.103

where a_i and b_i, i = 0, 1, …, 5, are the polynomial coefficients. Note that using this representation, which is cubic in x₁ and linear in x₂, one can write the vector r as

5.104

where r₀ = r(x₂ = 0) defines the global position of the material points on the centerline of the finite element and defines the location of the points on the cross section with respect to the centerline. Note that if remains a unit vector perpendicular to the tangent to the centerline, one obtains the Euler–Bernoulli beam model. If, on the other hand, remains a unit vector that does not remain perpendicular to the tangent to the centerline, one obtains a model similar to the Timoshenko beam model. Using the preceding equation, the strain components can be written as (Sugiyama et al., 2006)

5.105

In this equation, is the tangent to the centerline of the element, and describes the rate of change of the gradient vector with respect to the spatial coordinate x₁. Because a linear interpolation with respect to x₂ is used, one can show for the finite element described in this section that

5.106

where is the gradient vector defined at node k, k = 1, 2. One can verify that in the case of a rigid-body motion is identically equal to zero. In general, this vector is constant everywhere inside the element, that is, this vector does not depend on the spatial coordinate x₁. The fact that the gradient vector can vary only linearly with respect to x₁ regardless of the load applied can introduce excessive stiffness, leading to the locking problem. This problem becomes more serious when thin and stiff structures are modeled. The use of different orders of spatial coordinate interpolations leads to different orders of interpolations for deformations and strain components. When some strain components are restricted to take certain values as the result of a low-order interpolation, the element tends to have unreasonable high stiffness. Different approaches for solving this problem can be used and these approaches, which are well documented in the finite element literature, are briefly discussed in a later section.

Patch Test

The patch tests are simple tests, which are used to check the convergence of the finite element formulation as well as the computer implementation. These tests can also be used to evaluate the element performance and stability by checking whether or not the element satisfies basic equilibrium requirements. A successful patch test for an element is an indication that if the element is used to model a structure, a refinement of the finite element mesh will produce solutions that converge to the exact solutions. To perform the patch test, one considers a finite element mesh that consists of a small number of elements with at least one node inside the patch. The patch can be subjected to forces or prescribed nodal displacements to develop problems with known exact solutions. For example, boundary nodes can be constrained just enough to eliminate the rigid-body motion of the structure. One can then apply, at the free boundary nodes, loads that correspond to the state of a constant stress. The computed stresses inside the elements are compared with the exact solution to check whether or not the two solutions agree to within the numerical errors. The patch test is repeated to cover all cases of constant stresses relevant to this element. If all the computed stress results agree with the exact solution, the element passes the patch test, and a fine mesh of a structure using this element will produce a solution that converges to the exact one.

One can also examine the strains and displacements to make sure that the computed values are correct. One must also check that in the case of prescribed displacements of the boundary nodes and in the absence of the body forces, the stress and strains must be constant in order for the element to satisfy the partial differential equations of equilibrium. If these conditions are not satisfied, then it is likely that the element assumed displacement field is not correct and/or there is a problem with the computer implementation. It is also recommended to perform an eigenvalue analysis to check that the element has the correct number of rigid-body modes for a given support condition. The eigenvalue analysis should not produce a zero eigenvalue associated with a deformation mode; otherwise, the element will exhibit unstable behavior.

Locking Problem

With regard to the finite element performance, one of the important issues that have been discussed in the literature is the locking problem. Some finite elements exhibit in some applications overly stiff behavior due to two main reasons. First, the order of the polynomial interpolation used for the element is low such that some important modes of deformations cannot be effectively captured. For example, if a linear interpolation is used for a finite element, the curvature which is necessary to describe bending will be zero everywhere inside the element. The use of such low-order finite elements for bending is therefore not recommended because a very large number of elements will be required to solve a simple bending problem. The use of such a fine finite element mesh can be very inefficient in solving beam and plate problems. The second reason for the poor performance of an element is the existence of high-frequency modes that have no significant effect on the solution but lead to a deterioration of the element performance. Such modes can be consistently eliminated using approximation methods based on coordinate reduction as described in the following chapter, or by using analytical methods by introducing kinematic algebraic constraints to prevent the motion in the direction of such stiff modes. The algebraic constraint equations can be used to systematically eliminate these stiff modes from the formulation or can be used to introduce constraint forces that can be expressed in terms of Lagrange multipliers; a subject that has been extensively covered in the MBS dynamics literature (Roberson and Schwertassek, 1988; Shabana, 2013).

As the result of low-order interpolations and the existence of high-frequency modes of deformations, the element performance deteriorates and serious numerical problems can be encountered. There are several types of locking, including volumetric, membrane, and shear locking. For example, most structural materials are nearly incompressible. Changes of the dilatation can be accompanied by large values of stresses that absorb a significant part of the energy and make the element very stiff or lock. Theoretically, an ideal solution to deal with incompressible or nearly incompressible materials is to impose the incompressibility condition as a constraint by assuming that the determinant of the matrix of position vector gradients remains equal to one and does not change. This condition arises from the relationship between the volumes in the current and reference configurations, dv = JdV, where J is the determinant of the matrix of position vector gradients, and dv and dV are, respectively, the volumes in the current and reference configurations. As previously mentioned, imposing the incompressibility condition can be accomplished using two approaches: in the first, a Lagrange multiplier technique is used, whereas in the second, a penalty method is used. The penalty method is easier to use because it is equivalent to adding a force to the equations of motion to guarantee that the incompressibility condition is satisfied. On the other hand, when the Lagrange multiplier technique is used, one must augment the equations of motion with algebraic equations that describe the constraint conditions. This leads to a system of differential and algebraic equations that must be solved simultaneously, making the numerical procedure much more complex as compared to using the penalty method.

The use of the penalty method is equivalent to changing the strain energy of the system by adding another term that enforces the incompressibility condition. Another method used in the finite element literature to solve the locking problems is to use multifield variational principles, which are also called mixed or hybrid principles. In these principles, the stress and strain components that lead to overly stiff behavior are interpolated independently of the displacements. The independent interpolation allows for using higher order for those components that are the source of the locking behavior. Examples of these mixed principles are the Hellinger–Reissner principle and the Hu–Washizu principle: the first is stress based, whereas the second is strain based. The use of these principles allows using different fields for stresses and strains to avoid the locking problem and improve the element behavior in some problems such as beam and plate bending. For this reason, the resulting elements are also called assumed strain or assumed stress elements depending on which variables are interpolated. The drawback of using the mixed principles is that the elements can exhibit instabilities in other fields and, therefore, it is important to check the accuracy of the solution obtained for other field variables.

Shear locking, which is also a source of numerical problems in beam and plate problems, is the result of excessive shear stresses. For thin elements, the cross section is expected to remain perpendicular to the element centerline or midsurface of the element. This is the basic assumption used in Euler–Bernoulli beam theory. Elements that are based on this theory do not allow for shear deformation, and therefore, such elements do not, in general, suffer from the shear locking problem. Examples of these elements are the two-dimensional Euler–Bernoulli beam element and the three-dimensional cable element discussed previously in this chapter. These elements, as demonstrated in the literature, are efficient in thin-beam applications. Shear deformable elements, on the other hand, can suffer from locking problems if they are used in thin structure applications. When these elements are used, the cross section does not remain perpendicular to the element centerline, leading to shear forces. For thin structures, the resulting shear stresses can be very high leading to numerical problems. This problem can be circumvented by using the elastic line or midsurface approaches, the mixed variational principles, or reduced integration methods.

Similarly, some shell-element formulations produce coupling between membrane and bending deformations. In these formulations, a bending of a plate leads to membrane (extension) displacements. This kinematic coupling can lead to the problem of membrane locking, which in turn leads to numerical difficulties when thin shell structures are analyzed. In some applications such as papers and cloths, bending does not produce extension. For these applications, it is recommended to use the thin-plate element formulation, which is based on the elastic midsurface approach. In the formulation of the elastic forces of this element, it is assumed that the bending and extension are not coupled. If, on the other hand, the higher-order fully parameterized plate element is used in thin structural shell applications, it is recommended to use the elastic midsurface approach, the mixed variational principles, and reduced integration methods.

Reduced Integration

One must be careful when speaking of the order of the interpolation and the performance of an element. Although low-order interpolation may necessitate the use of a larger number of elements in order to be able to capture a certain deformation mode, higher-order terms in a polynomial introduce more complex shapes that are associated with high-frequency modes of oscillations. These high-frequency modes can also lead to a deterioration of the element performance. Elimination of these modes can enhance the element performance in some applications. One method, which is recommended in the finite element literature, is to use reduced integration instead of full integration. In the reduced integration, fewer quadrature points are used in the numerical integration of the elastic forces. This is equivalent to using lower-order polynomials to approximate the integrands that appear in the elastic force expressions. Lower-order polynomials have simpler shapes, which are associated with lower modes of oscillations. Elimination of the complex shapes is equivalent to eliminating high frequencies and is equivalent to lowering the element stiffness. This can significantly enhance the element performance. Underintegration, which can be used effectively to eliminate shear locking in some applications, leads to additional computational advantage because it reduces the number of calculations by using fewer quadrature points. Reduced integration, however, should not be used if it leads to mesh instabilities or wrong solutions. It is, therefore, important that the integration rule used is tested in order to make sure that an accurate solution is obtained.

Another form of reduced integration is the selective reduced integration, which can be used to enhance the element performance in some problems. In this integration method, some terms that are the source of locking can be selected and underintegrated, whereas full integration is used for other terms that appear in the expression of the elastic forces. For example, the terms that define the contribution to the elastic forces from the volumetric strains can be underintegrated, whereas terms associated with the deviatoric strains can have a higher order of integration. This method of selective reduced integration can be effective in dealing with volumetric locking in some applications.

Reduced integration if not carefully performed can lead to instabilities. For example, if the deformations at all the selected quadrature points happen to be zero, one obtains zero strain energy for a nonrigid-body mode. In this case, the stiffness matrix is singular and the element exhibits unstable behavior. These types of modes are called in the finite element literature hourglass modes, zero energy modes, or spurious singular modes. These types of modes can be detected using an eigenvalue analysis. In this case, the number of modes of the finite element that have zero eigenvalue is higher than the number of rigid-body modes of the element because the improper selection of the quadrature points leads to a zero eigenvalue associated with a deformation mode.

Isoparametric Finite Elements

Some elements such as the two-dimensional rectangular and triangular elements and the three-dimensional solid and tetrahedral elements (Zienkiewicz, 1977) employ only position coordinates. These elements can correctly describe rigid-body motion and they are of the isoparametric type because the same shape function can be used to describe the displacement and geometry of the element. Nonetheless, the nodal coordinates do not include rotation variables, which make these elements unsuitable for beam, plate, and shell applications and also unsuitable for many MBS applications where joint constraints between bodies are often formulated in terms of rotation coordinates. Because the continuity of the rotation field at the nodal points is not guaranteed when these isoparametric elements are used, imposing MBS connectivity conditions that allow relative motion is not straightforward. Furthermore, the limitations of these conventional isoparametric elements in the analysis of bending, as the result of the low order of interpolation, are well known and have been discussed in the literature.

Use of Infinitesimal Rotation Coordinates

Another type of finite elements includes elements that employ infinitesimal rotations in addition to translational coordinates as nodal variables. Examples of these conventional finite elements are beam, plate, and shell elements. These elements, which were widely used in many structural applications, cannot correctly describe large rigid-body rotation; they can describe only infinitesimal rigid-body rotation. Because of the use of the infinitesimal rotations as nodal coordinates, one can show that the kinematic equations of these elements employ linearization (Shabana, 1996a). For this reason, these elements have been used in structural dynamics applications in the framework of an incremental solution procedure. It is known, however, that the incremental solutions based on linearized equations eventually diverge from the correct solution of the nonlinear problem. Furthermore, most MBS algorithms that are designed to solve large rigid-body rotation problems are based on nonincremental solution procedure. In order to be able to use these finite elements in MBS algorithms, the finite element FFR formulation was proposed in the early 1980s (Shabana and Wehage, 1981; Shabana, 1982). This approach, which is discussed in detail in the following chapter, leads to correct description of the rigid-body motion of the finite elements that employ infinitesimal rotations as nodal coordinates. The FFR formulation has been primarily used for small-deformation problems because the elements are assumed to undergo small displacements with respect to the floating frame, which may experience an arbitrary rigid-body motion including finite rotations. The FFR formulation remains an effective and efficient tool for modeling the small deformation of flexible bodies because it allows reducing systematically the number of deformation degrees of freedom. This formulation, which will be discussed in more detail in the following chapter, is implemented in most general purpose flexible MBS computer programs.

Perhaps, it is also important to point out that, in MBS applications, one cannot use infinitesimal or finite rotations as nodal coordinates in the interpolation of the rigid-body displacement field. Recall, as shown in Chapter 1, that the rigid-body kinematic equations are defined in terms of trigonometric functions and not in terms of angles. Trigonometric functions can be approximated by angles only when these angles are infinitesimal.

Use of Finite Rotation Coordinates

Another element formulation, which was introduced in the mid-1980s, is based on using two independent fields with finite rotation coordinates as nodal coordinates (Simo and Vu-Quoc, 1986). The resulting elements are capable of correctly describing arbitrary rigid-body displacements. In this formulation, the two independent fields are introduced for the position vector and the finite rotations of the cross section of the finite element. That is, the position and rotations are obtained from independent interpolations. As demonstrated in this book, using the polar decomposition theorem, the rotation field can be defined using the matrix of position vector gradients. That is, the position vector field is sufficient to determine the rotations of infinitesimal volumes. The use of independent displacement and rotation fields, therefore, can lead to a problem of coordinate redundancy and inconsistency in the definition of the rotation variables (Ding et al., 2014). Formulations that suffer from this problem of coordinate redundancy can lead to numerical problems in the analysis of large rotations, particularly large rigid-body rotations. For instance, some of these formulations do not automatically satisfy the principle of work and energy, and special measures must be taken in the numerical integration routines in order to satisfy this principle. On the other hand, because the ANCF position vector field is used to determine the gradients that define the rotation field, such a problem is not encountered, and ANCF finite elements automatically satisfy the principle of work and energy without the need for special measures as demonstrated in the literature (Campanelli et al., 2000).

ANCF Finite Elements

The following characteristics define the ANCF solution framework:

1. 1. The problem must be a dynamics problem that requires addressing the formulation of the inertia forces, which are expressed in terms of a constant inertia matrix and, therefore, the Coriolis and centrifugal forces are identically zero.
2. 2. The elements must be of higher order in order to provide the option for imposing continuity on gradient vectors. The vector of nodal coordinates of ANCF finite elements can consist of position and gradient coordinates and the gradients must be interpreted correctly during the entire solution procedure. Conventional finite elements do not ensure continuity of the gradient or rotation fields. As it is known continuity of the gradients ensures continuity of the rotations. The converse, however, is not, in general, true.
3. 3. The finite elements must correctly describe an arbitrary rigid body motion including arbitrary finite rotations using the assumed displacement field , where is the global position vector of an arbitrary point on the element, is the element shape function matrix, is the vector of nodal coordinates, is the vector of the element spatial coordinates, and is time.

When using ANCF finite elements, the equations of motion can be solved nonincrementally and, therefore, there is no need for the use of the corotational approach, which was introduced to remedy the problems associated with finite elements that employ infinitesimal rotations. In the FE literature, there has been frequent reference to gradients and slopes. Nonetheless, all the above conditions are not often met as the result of improper treatment of the gradients, the use of incremental procedures, and/or the focus on static procedures without addressing the formulation of the element inertia when the element experiences arbitrary finite rotations (Betsch and Stein, 1995; Milner, 1981).

Constrained Motion

As discussed in this chapter, ANCF finite elements can be used as the basis for a nonincremental solution procedure that allows for developing new computer models with significant details. One example of these models is the tracked vehicle shown in Figure 10. The links of the track chains of this vehicle are modeled as flexible bodies connected by pin (revolute) joints that allow relative rotation as well as relative deformations between the chain links. This detailed vehicle model was successfully developed using ANCF finite elements and computational MBS algorithms (Hamed et al, 2015; Wallin et al., 2013). Using ANCF finite elements, linear connectivity conditions, that define the pin (revolute) joints, were developed, allowing eliminating the dependent variables at a preprocessing stage (Garcia-Vallejo et al, 2003; Hamed et al., 2015). The two tracks of the vehicle can be treated as one ANCF finite element mesh since ANCF finite elements allow for arbitrarily large displacements. The track chains mass matrix remains constant and the centrifugal and Coriolis forces remain equal to zero. The development of such a heavily constrained model, in which the algebraic connectivity conditions are eliminated at a preprocessing stage, using ANCF finite elements is straightforward.

ANCF Reference Node

Another important concept that has been used effectively in modeling complex system is the ANCF reference node (Shabana, 2015a,b). The ANCF reference node is a node that is not associated with a particular ANCF finite element. This node can be used for assembling rigid and structural components. Such a node can also be used to describe a rigid body by imposing six rigidity constraints that eliminate the deformation modes of the reference node. The reference node can be used to develop new tire assembly models and can be used also to develop an MBS submodel at a preprocessing stage as shown in Figure 11 (Shabana, 2015a,b).

Deformation Modes

As discussed in the literature, and also in this book, fully parameterized ANCF finite elements may exhibit locking behavior, particularly in the case of thin and stiff structures. Nonetheless, the use of such fully parameterized ANCF finite elements is necessary in order to be able to capture correct physics in many applications. For example, excessive axial and bending forces lead to change in the cross-sectional dimensions of beam-like structure. Formulations that assume rigid cross section, from the outset, cannot be used to capture this physics behavior. In the finite element literature, many solutions were proposed to solve the locking problem. Some of the locking-solution techniques can be applied in the case of fully parameterized ANCF finite elements, allowing for developing new models that correctly capture the physics of the problem under consideration.