Modified couple stress-based third-order theory for nonlinear analysis of functionally graded beams

Modified couple stress-based third-order theory for nonlinear analysis of functionally graded beams

A. Arbind, J.N. Reddy^*; A. R. Srinivasa

Department of Mechanical Engineering Texas. A & M University, College Station, TX 77843-3123, USA

ABSTRACT

A microstructure-dependent nonlinear third-order beam theory which accounts for through-thickness power-law variation of a two-constituent material is developed using Hamilton's principle. The formulation is based on a modified couple stress theory, power-law variation of the material, and the von Kármán nonlinear strains. The modified couple stress theory contains a material length scale parameter that can capture the size effect in a functionally graded material beam. The influence of the material length scale parameter on linear bending is investigated. The finite element models are also developed to determine the effect of the geometric nonlinearity and microstructure-dependent constitutive relations on linear and nonlinear response.

Keywords: Modified couple stress theory, equations of motion, general third-order beam theory, von Kármán nonlinearity, functionally graded beam, nonlinear finite element model.

1 BACKGROUND

1.1 Microstructural Effects

In recent years a number of attempts have been made to bring microstructural length scales into the continuum description of beams and plates. Microstructure-dependent theories were developed for the Bernoulli-Euler beam by Park and Gao(2006 and 2008) and for the Timoshenko and Reddy-Levinson beams and Mindlin plates by Ma, Gao, and Reddy (2008, 2010 and 2011), using a modified couple stress theory proposed by Yang et al. (2002), which contains only one material length scale parameter. Recently, Reddy (2011) developed a complete formulation of functionally graded beams with microstructure-dependent constitutive relations for Bernoulli–Euler and Timoshenko beams. The formulation accounted for temperature-dependent material properties and the von Kármán nonlinearity, which may have significant contribution to the response of beam-like elements used in micro- and nano-scale devices such as biosensors and atomic force microscopes (see, e.g., Li et al. (2003) and Pei et al. (2004)). The present paper is an extension of the work of Reddy (2011) to a general third-order theory that contains the Reddy third-order beam theory as well as the Bernoulli–Euler and Timoshenko beam theories as special cases. The following review of literature provides a background for the present study.

1.2 An Overview of Beam Theories

Mathematical models used to determine the response of beams under external loads are deduced from the three-dimensional elasticity theory through a series assumption concerning the kinematics of deformation and constitutive behavior. The kinematic assumptions exploit the fact that such structures do not experience significant strains or stresses associated with the cross-sectional dimensions (e.g., transverse normal and shear strains and stresses). For example, the solution of the three-dimensional elasticity problem associated with a straight beam is reformulated as a one-dimensional problem in terms of displacements whose form is presumed on the basis of an educated guess concerning the nature of the deformation.

The theories based on assumed form of the displacement field have received the most attention, especially in the context finite element model development. In these theories, the displacement field of the beam is expanded in terms of increasing powers of the coordinate in the thickness (i.e., height) direction. The higher-order terms would have diminishing returns compared to the lower-order terms due to the smallness of thickness compared to the length. The third-order expansion of the displacement field is optimal because it gives quadratic variation of transverse strain and stress, and requires no "shear correction factor" compared to the Timoshenko theory, where the transverse strains and stresses are constant through the plate thickness (see Reddy (2002 and 2007)).

The simplest shear deformation beam theory is the Timoshenko beam theory, which is based on the displacement field

Here is the rotation of a transverse normal line

The Bernoulli–Euler hypothesis of straight lines normal to the axis of the beam before deformation remain (a) straight after deformation, (b) inextensible, and (c) rotate as rigid lines to remain perpendicular the bent axis of the beam (see Fig. 1) require

Consequently, the transverse shear strain y_xz is zero.

The Timoshenko beam theory is based on the first two assumptions of the Bernoulli–Euler hypothesis, and the normality of the assumption is not invoked, making the rotation θ_x to be independent of ∂w/∂x. As a result, the transverse shear strain γ_xz is nonzero but independent of z. This leads to the introduction of a shear correction factor in the evaluation of the transverse shear force. The finite element models of the theory require only C⁰-continuity, that is, the variables of the theory (u, w, θ_x) be continuous between elements; however, they can exhibit spurious transverse shear strains even in pure bending, known as the shear locking, as the beam length-to-height ratio, L/h, becomes very large. The spurious transverse shear stiffness stems from an interpolation inconsistency that prevents the condition of Eq. (2) from being satisfied as the beam becomes thin, that is, L/h → ∞. The shear locking phenomenon can be alleviated by using a reduced integration in the evaluation of transverse shear stiffness terms in the element stiffness matrix or by using higher-order approximations of the variables (u, w, θ_x). Although the reduced integration solution is the most economical alternative, the process allows some elements to exhibit spurious displacement modes, i.e., deformation modes that result in zero strain at the numerical integration points.

Second-order and higher-order beam theories relax the Bernoulli-Euler hypothesis further by allowing the straight lines normal to the beam axis before deformation to become curves after deformation. However, most published theories still assume inextensibility of these lines. Second-order plate theories are not popular because of the fact that they too require shear correction factors and while not improving over the Timoshenko beam theory. The third-order theories provide a slight increase in accuracy relative to the Timoshenko beam theory, at the expense of an increase in computational effort, but do not require a shear correction factor. From the finite element model development standpoint, the third-order beam theory of Reddy (see Reddy (1984, 1987, 1990, 2002 and 2011)), which is based on the displacement field

requires C¹-continuity of the transverse displacement component but less sensitive to shear locking. In the Reddy beam theory, the transverse shear stresses are zero at the top and bottom of the beam.

In this study, we consider a general third-order theory based on the following displacement field

Finite element model of this theory requires only C⁰-continuity of all field variables (u, w, θ_x, ϕ_x, ψ_x, θ_z, ϕ_z), and no shear correction factors are needed.

1.3 Functionally Graded Materials

Elimination of stress concentrations in structural elements is mitigated by suitable design of materials, and functionally gradient materials (FGMs) are a class of materials that are designed to eliminate stress concentrations (see Hasselman and Youngblood (1978), Yamanouchi et al. (1990) and Koizumi (1993)). These materials were proposed as thermal barrier materials for applications in space planes, space structures, nuclear reactors, turbine rotors, flywheels, and gears, to name only a few. One reason for increased interest in FGMs is that it may be possible to create certain types of FGM structures capable of adapting to operating conditions. A most common FGM is one in which two materials are mixed to achieve a composition that provides certain functionality. For example, for thermal-barrier structures, two-constituent FGMs are made of a mixture of ceramic and metals. The ceramic constituent of the material provides the high temperature resistance due to its low thermal conductivity. The ductile metal constituent, on the other hand, prevents fracture due to high temperature gradient in a very short period of time. The gradation in properties of the material reduces thermal stresses, residual stresses, and stress concentrations. A number of other investigations dealing with thermal stresses and deformations in beams, plates, and cylinders have been published in the literature (see, for example, Noda and Tsuji (1991), Obta, Noda, and Tsuji (1991), Reddy and Chin (1998), Praveen and Reddy (1998), Praveen, Chin, and Reddy (1999), Reddy (2000), and Aliaga and Reddy (2004), among others). The work of Praveen and Reddy (1998), Reddy (2000) and Aliaga and Reddy (2004) considered von Kármán nonlinearity, and Aliaga and Reddy (2004) used the third-order plate theory of Reddy (1984, 1987, 1990 and 2011).

1.4 Present Study

The objective of the present study is to develop a general third-order beam theory that accounts for through-thickness power-law variation of a two-constituent material with temperature dependent material properties, modified couple stress theory, and the von Kármán nonlinear strains. In particular, we extend the modified couple stress theory of Yang et al. (2002) to the case of functionally graded beams (Reddy (2011)) using the third-order kinematics with the von Kármán nonlinearity and derive the equations of motion using Hamilton's principle (see Reddy (2002 and 2013)). Also, a nonlinear finite element model is developed for the general third-order theory of beams. To date no such study is reported in the literature. Since most nanoscale devices involve beam-like elements that may be functionally graded and undergo moderately large rotations, the newly developed third-order beam theory can be used to capture the size effects in functionally graded microbeams. Moreover, the bending-extensional coupling is captured through the von Kármán nonlinear strains.

2 CONSTITUTIVE MODELS

2.1 Material Variation through the Thickness

Consider a straight beam of rectangular cross section, with width b and total height (or thickness) h. The x coordinates is taken along the length of the beam and it passes through the geometric centroid of the cross section, and the z-axis is taken transverse to the beam length such that the beam cross section is in the yz plane. We assume that the material of the beam is isotropic but varies from one kind of material at the bottom, z = -h/2 to another material on the top, z = h/2. A typical material property of the FGM through the beam thickness is assumed to be represented by a power-law (see Praveen and Reddy (1998) and Reddy (2000))

Where, P_c and P_m are the values of a typical material property, such as the modulus, density, and conductivity, n of a ceramic material and metal, respectively; denotes the volume fraction exponent, called power-law index. When n = 0, we obtain the single-material plate (with property P_c). When FGMs are used in high-temperature environment, the material properties are temperature-dependent and they can be expressed as (see Reddy (2011))

where c₀ is a constant appearing in the cubic fit of the material property with temperature; and c_-1, c₁, c₂, and c₃ coefficients of T^-1, T, T², and T³, obtained after factoring out c₀ from the cubic curve fit of the property. The modulus of elasticity, conductivity, and the coefficient of thermal expansion are considered to vary according to Eqs. (6) and (7).

The present study is concerned with the development of the equations of motion associated with a general third-order beam theory which accounts for microstructural length scale, two-constituent material variation through beam height, and the von Kármán nonlinearity. Solutions of bending, vibration, and buckling are presented for the linear case to study various parametric effects on the response. Following this introduction, the governing equations of the general third-order FGM beams with the microstructure-dependent length scale parameter are presented. Existing beam theories are derived as special cases of the theory presented herein. The equations derived are then specialized to the linear case to develop the bending, buckling, and vibration frequencies.

3 THE GENERAL THIRD-ORDER BEAM THEORY

3.1 Modified Couple Stress Theory

According to the modified couple stress theory, the strain energy of an elastic beam can be expressed as

where σ_ij are the components of the symmetric part of the Cauchy stress tensor, m_ij denote the components of the deviatoric part of the symmetric couple stress tensor, and χ_ij are the components of the symmetric curvature tensor

where ω_i are the components of the rotation vector

In the following sections we replace (x₁, x₂, x₃) with (x, y, z), χ₁₂ - χ_xy, χ₂₃ = χ_yz, and ω₂ = ω_y.

3.2 Equations of Equilibrium

We begin with the following displacement field for a straight beam bent by forces in the xy plane (i.e., bending about the y-axis):

where are midplane displacements along the x and z directions, respectively, and and have the meaning

For a general third order beam theory, we take m = 3 and p = 2 in Eq. (11). Then the displacement field would be same as (5) with and . The nonzero von Kármán nonlinear strains (i.e., simplified Green-Lagrange strain tensor components) associated with the displacement field (11) are

and rotation and curvature components are

Where = max (m - 1, p) and

Next, Hamilton's Principle is used to derive the equations of motion, incorporating modified couple stress terms and through-thickness variation of the material (see Reddy (2002)):

Where δK is the virtual kinetic energy, δU is the virtual strain energy, and δV is the virtual work done by external forces. The virtual kinetic energy expression is

where

The expression for the virtual strain energy is

Various stress resultants used in Eq. (19) are defined as

and virtual work done by external forces is

where

Here and denote the distributed axial load, transverse load, and body couple about the y axis (all measured per unit volume of the beam), respectively. Substituting the expressions δU, δV, and δK into Hamilton's principle (16), performing integration-by-parts with respect to as well as x to relieve the generalized displacements and of any differentiations, and using the fundamental lemma of calculus variations, we obtain the following equations of motion:

In Eqs.(23) and(24), i varies from 0 to m and 0 to p respectively. There are a total of (m + p + 2) equations of motion.The natural boundary conditions involve specifying the following generalized forces (when the corresponding generalized displacements are not specified):

3.3 Generalized Force-Displacement Relations

For the general third-order beam theory, 2-D plane stress state is assumed to write the relations between symmetric part of stress and strain for isotropic and linear elastic material as

where ℓ is a material length scale parameter, which is the square root of the ratio of the modulus of curvature to the modulus of shear, and it is a physical a property measuring the effect of couple stress (see Mindlin (1963)).

The stress resultants can be expressed as

where

3.4 Virtual work statements: weak forms

The weak forms of equations of motion (23)–(24) for static case are (see Reddy (2004))

where the generalized force are

In terms of displacement the weak forms can be given as

3.5 Finite element model

Let's approximate the degrees of freedom for static case as following

where (x) and (x) are the interpolation functions for and respectively. In case of conventional beam (i.e. without microstructure dependance) the interpolation function can be taken as Lagrange interpolation polynomials, whereas in case of microstructure dependent beam (x) are Lagrange polynomials and (x) are Hermite interpolation polynomials as the first derivative of are also primary variable in the virtual work statement (35). and are the nodal values associated with and respectively (see Reddy (2004)). Substituting and in the virtual work statement (34) and (35), we get the finite element equation as

where N = (m + p + 2) is the total no. of equations of motion and

The components of the elemental stiffness matrix K in (37) are

The components of force vector are

The nonlinear equations (37) are solved using Newton's iterative method (see Reddy (2004)), which involves the computation of the coefficients of the element tangent stiffness matrix T^e, which has similar structure as the stiffness matrix in Eq. (37). By applying Newton's iterative method, the algebraic equation for (s + 1)th iteration can be given as

where T^e is the tangent matrix and its components are

Here R^α is the residual vector

Hence, the components of tangent matrix are

4 SPECIALIZATIONS TO OTHER BEAM THEORIES

4.1 Reddy Third-Order Beam Theory

The Reddy third-order beam theory is based on the displacement field

which is a special case of the general beam theory with m = 3, p = 0, and

The nonzero strains are

where

and

The equations of motion of the Reddy third-order beam theory take the form

where

The natural boundary conditions involve specifying the following generalized forces:

4.2 The Timoshenko (First-Order) Beam Theory

The Timoshenko first-order beam theory is based on the displacement field (see Reddy (2011))

which is again a special case of the general third-order beam theory with m = 1, p = 0, and

The nonzero strains are

where

The shear stress resultant in the Timoshenko beam theory is defined as

and K is the shear correction factor. The equations of motion of the Timoshenko beam theory are

The natural boundary conditions involve specifying the following generalized forces:

4.3 The Bernoulli–Euler Beam Theory

The Bernoulli–Euler beam theory is based on the displacement field

This displacement field is a special case of the general third order beam theory with m = 1, p = 0, and

The equations of motion can be obtained from the Reddy third-order beam theory by some changes. We have

The natural boundary conditions are

In all of the special cases considered in this section, the transverse normal strain is zero, ε_zz = 0; this requires us to use the one-dimensional stress-strain relations σ_xx = Eε_xx and σ_xz = Gγ_xz. That is, we set ν = 0 in the definition of , , and of Eq. (29).

5 ANALYTICAL SOLUTION

Analytical solution for a simply supported beam is obtained for a general third-order beam theory neglecting geometric nonlinearity. The equations of equilibrium for linear static case are

Static linear equations of motion in terms of displacement

Let the solution for the degrees of freedom be of the form

and the applied transverse force is given as

Substituting and in the above equation, we get

For the above equations to be true, each coefficient of sine and cosine term should be equal to zero. Hence, for the r^th coefficient we have

where

and

For the analytical solution, homogeneous and functionally graded simply supported beams with (n = 1) of following geometric and material parameter are considered:

Results for nondimensional central transverse deflection ( = wEI/q₀L⁴) are tabulated for uniform (q = q₀) and sinusoidal load (q = q₀ sin(x/L)) for q₀ = 1 N/m for both homogeneous and FGM beam and also, the general third-order beam theory (TOBT)(considering m = 3 in horizontal displacement, u₁ and p = 2 in transverse displacement, u₃) results are compared with Euler–Bernoulli beam (EBT) and Timoshenko beam (TBT)(see Reddy (2011)) in Table 1 for the load acting as body force and also as the normal traction force on the top of the beam. In Table 2, the non dimensional central transverse deflection for uniform load (acting as body force) for FGM beam are tabulated considering m = 3 and p = 1,2,3 for the expression of horizontal displacement, u₁ and transverse displacement, u₃ [see Eq. (11)] respectively for different microstructural length scale factor. By comparing the nondimensional central transverse deflection for (p = 1,2,3), we can see that the term corresponding to p = 3 in the displacement field is not contributing significantly. So, further in the present study, for all the numerical examples m = 3 and p = 2 are considered for the horizontal and transverse displacement field.

6 FINITE ELEMENT SOLUTION

6.1 Linear solution

For numerical example, again the same beam specified as Eq. (80) is considered for the simply supported boundary condition. The linear finite element and analytical solutions are compared in Table 3 for microstructure length scale factor (ℓ = 0,0.6h,h) where, h is the height if the beam. In case of conventional beam (i.e., ℓ = 0), thirty quadratic Lagrange elements are used in half beam (exploiting the symmetry). For microstructure dependent beam, are approximated as linear Lagrange interpolation function whereas, Hermite cubic interpolation functions are used for . Forty-five such elements are used for the half domain to get the solution. The boundary condition can be given as:

6.2 Nonlinear solution

Nonlinear solutions are presented in this section for pinned–pinned and clamped–clamped boundary conditions for the aforementioned beam. The deflection under uniformly distributed transverse load acting on the top surface of the beam as traction, q₀ = 1 N/m for homogeneous and functionally graded beams with power-law index n• • and • • •• for various microstructure length scale factors are presented in Figures 2 and 3 for pinned–pinned and clamped–clamped boundary conditions, respectively. Again, for nonlinear solution also, thirty quadratic Lagrange elements are considered for conventional beam (i.e., • • •) and, forty-five elements are considered for microstructure dependent beam. The error tolerance for Newton's method (see Reddy (2004)) is taken as 10^-4. The linear solutions are also shown in the same figures with dotted lines for comparison. Further, to see the effect of nonlinearity the plots for maximum deflection verses transverse load applied are presented in Figures 4, 5, and ⁶ for both homogeneous and functionally graded beam considering pinned–pinned and clamped–clamped boundary conditions. Also, in the figures, nonlinear solution for Euler–Bernoulli and Timoshenko beam (see Reddy (2011)) are plotted, for comparison, using the same loads.

6.3 Summary

In the present study, the governing equations for a general third order theory for functionally graded beam considering the modified couple stress theory and von Kármán nonlinearity are developed using Hamilton's principle. The nonlinear finite element model is also developed using weak formulation of the equations. Analytical solution for the linear case is presented for the pinned-pinned boundary condition, and linear finite element solution is compared with the analytical solution obtained. Nonlinear finite element solutions for different boundary conditions for FGM beams with different power-law index n and different microstructure length-scale factor ℓ are presented. These results are the first of its kind, as there are no solutions available in the literature.

Acknowledgements The research reported herein was carried out under National Science Foundation research grants CMMI-1000790. The second author also acknowledges support NSF grant CMMI-1030836.

Received in 02 Apr 2013

In revised form 30 Apr 2013

• Aliaga JW and Reddy JN (2004), Nonlinear thermoelastic analysis of functioally graded plates using the third order shear deformation theory, Int. J. Comp. Eng. Sci. 05, 753.
• Hasselman DPH and Youngblood GE (1978), Enhanced thermal stress resistance of structural ceramics with thermal conductivity gradient. Journal of the American Ceramic Society 61(1, 2), 4953.
• Koizumi M (1993), The Concept of FGM. Ceramic Transactions, Functionally Gradient Materials34, 310.
• Li X, Bhushan B, Takashima K, Baek CW, and Kim YK (2003), Mechanical characterization of micro/nanoscale structures for MEMS/NEMS applications using nanoindentation techniques. Ultramicroscopy 97: 481494
• Ma HM, Gao XL, and Reddy JN (2008), A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. J MechPhys Solids 56: 33793391
• Ma HM, Gao XL, and Reddy JN (2010), A non classical Reddy-Levinson beam model based on a modified couple stress theory. Int J Multiscale Computational Engineering 8(2): 167180
• Ma HM, Gao XL, and Reddy JN (2011), A non-classical Mindlin plate model based on a modified couple stress theory. ActaMechanica, DOI 10.1007/s00707-011-0480-4
• Mindlin RD (1963), Influence of couple-stresses on stress concentrations. Experimental Mechanics. 3: 1-7.
• Noda N (1991), Thermal Stresses in Materials with Temperature-Dependent Properties, Applied Mechanics Reviews, 383397.
• Noda N and Tsuji T (1991), Steady thermal stresses in a plate of functionally gradient material with temperature-dependent properties. Transactions of Japan Society of Mechanical Engineers Series A, 57: 625631.
• Obata Y, Noda N and Tsuji T. (1992), Steady thermal stresses in a functionally gradient material plate. Transactions of Japan Society of Mechanical Engineers 58: 16891695.
• Park SK and Gao XL (2008), Variational formulation of a modified couple stress theory and its application to a simple shear problem. Z angew Math Phys 59: 904917
• Park SK and Gao XL (2006), Bernoulli-Euler beam model based on a modified couple stress theory. J MicromechMicroeng 16: 23552359.
• Pei J, Tian F, and Thundat T (2004), Glucose biosensor based on the microcantilever. Analytical Chem 76: 292297
• Praveen GN and Reddy JN (1998), Nonlinear transient thermoelastic analysis of functionally graded ceramic-metal plates. Journal of Solids and Structures 35(33): 44574476.
• Praveen GN, Chin CD, and Reddy JN (1999), Thermoelastic analysis of functionally graded ceramic-metal cylinder. Journal of Engineering Mechanics, ASCE 125(11): 12591266.
• Reddy JN (1984), A refined nonlinear theory of plates with transverse shear deformation. Int J Solids Struct 20: 881896.
• Reddy JN (1984), A simple higher-order theory for laminated composite plates. J ApplMech51: 745752.
• Reddy JN (1987), A small strain and moderate rotation theory of laminated anisotropic plates. J ApplMech 54: 623626.
• Reddy JN (1990), A general non-linear third-order theory of plates with moderate thickness. Int J Non-Linear Mech25(6): 677686.
• Reddy JN and Chin CD (1998), Thermomechanical analysis of functionally graded cylinders and plates. J. Thermal Stresses 26(1): 593626.
• Reddy JN (2000), Analysis of functionally graded plates. International Journal for Numerical Methods in Engineering, 663684.
• Reddy JN (2002), Energy Principles and Variational Methods in Applied Mechanics. 2nd ed. John Wiley & Sons, NY
• Reddy JN (2004), An Introduction to Nonlinear Finite Element Analysis, Oxford University Press, Oxford, UK.
• Reddy JN (2007), Theory and Analysis of Elastic Plates and Shells. 2nd ed. CRC Press, Boca Raton, FL
• Reddy JN (2011), A general nonlinear third-order theory of functionally graded plates. International Journal of Aerospace and Lightweight Structures 1(1): 121.
• Reddy JN (2011), Microstructure-dependent couple stress theories of functionally graded beams. Journal of Mechanics and Physics of Solids. 59: 2382-2399.
• Reddy JN (2013), Introduction to Continuum Mechanics. 2nd ed., Cambridge University Press, New York.
• Yamanouchi M, Koizumi M, Hirai T, and Shiota I (eds) (1990), Proceedings of the First International Symposium on Functionally Gradient Materials, Japan.
• Yang F, Chong ACM, Lam DCC, and Tong P (2002), Couple stress based strain gradient theory for elasticity. Int. J. Solids Struct. 39: 27312743

Publication Dates

• Publication in this collection
  13 Dec 2013
• Date of issue
  May 2014

History

• Received
  02 Apr 2013
• Accepted
  30 Apr 2013