Accurate and Efficient Thermal Stress Analyses of Functionally Graded Solids Using Incompatible Graded Finite Elements

Abstract

Functionally graded materials have found a wide usage in high temperature applications. The smooth transition from one material to another, in graded materials, may reduce thermal stresses, residual stresses and stress concentration factors as well as utilize properties of both materials. To perform accurate and efficient finite element analysis for heat transfer and transient thermal stress analyses in two-dimensional functionally graded materials, incompatible graded finite elements are developed and verified. User-defined subroutines in ABAQUS are developed to address the gradation of material properties within an element. An emphasis is made on an incompatible graded finite element (QM6) which is accurate and efficient compared to linear four-node (Q4) and quadratic eight-node (Q8) elements. With the help of posteriori error estimation, a critical comparison is made among three types of solid elements. Modified 6-node (QM6) incompatible graded elements provide better accuracy than Q4 elements and take less computational time than Q8 elements, thereby showing QM6 as an optimal element for engineering analysis.

1. Introduction

Finite element methods (FEM) have been used to reduce the number of physical prototypes and experiments in developing new products. Different physical phenomena such as wave propagation, thermal transport, structural behavior or fluid mechanics, are normally described by partial different equations. To solve these differential equations, different numerical techniques have been developed, one of them being finite element analysis. In any engineering problem, by determining the value of some basic unknowns, the behavior of the entire structure can be predicted and these basic unknowns are known as field variables [1]. These unknown field variables are infinite but finite element converts them into finite number by dividing the solution region into smaller parts called elements and expressing the field variables as approximating or interpolating functions within each of the elements. Thus, the accuracy of the entire system depends upon the selection of element interpolation function [2]. Due to this factor, different finite elements having different shapes and interpolation functions have been developed. In our paper, we have used two-dimensional elements that is 4-noded, 6-noded and 8-noded quadrilateral elements. Here, we have carried out a comparative study between these three elements to have an in depth information on their level of accuracy and computational time required.

Graded materials are a multi-component composite characterized by a macroscopic compositional gradient from one component to other. This composite, however, can fully utilize the properties of each component as it contains significant proportion of the composite materials in pure form. For example, the toughness of a metal can be mated with the refractoriness of a ceramic, without any compromise in the toughness of the metal side or the refractoriness of the ceramic side. Owing to this property, a significant number of research [3, 4, 5, 6] has been carried out on these materials showing their use as a structural component in transportation, biomedical engineering, electronic engineering, aerospace and many more, especially in the high temperature environment. In various high temperature environment structural components are subjected to sudden temperature fluctuations which results into thermal induced stresses or motions [7]. It is very much required to reduce these stresses to prevent any catastrophic malfunctions and [8, 9] shows the effectiveness of graded materials, with optimized composition, in doing so. Birman and Byrd [10] reviewed various developments in FGMs along with their theory and applications, various heat transfer issues, stress, stability and dynamic analysis, testing, manufacturing and design, and fractures. They showed that heat transfer and thermal stress analysis of FGM is analytically complicated and closed form solutions are possible only for few cases. Thus, numerical solution can be used to solve such thermal problems as well as mechanical problems more accurately. Chung and Chang [11] analyzed elastic, rectangular and simply supported graded plate with linear temperature change in thickness direction. They had only varying coefficient of thermal expansion following power, sigmoid or exponential variation (three cases) where they verified the analytical solution by numerical analysis. Erdogan and Wu [12] used numerical method to understand the distribution of thermal stresses, determine stress intensity factor for embedded and surface cracks and then presented the results for crack/contact problem in a FGM layer under tension in the interior region and compression near the surface. Similarly, Jin and Paulino [13] studied thermally nonhomogeneous materials under transient thermal loading conditions using Laplace Transform and an asymptotic analysis. Hein, Storm and Kuna [14], on the other hand, showed the effectiveness of layered or graded materials by carrying out numerical analysis using both FDM and FEM, where they showed the preference of FEM with regards to universal usability because of its easier applicability on arbitrary geometries compared to other methods. Similarly, Fuchiyama and Noda [15] also developed computer programs to analyze transient heat transfer and transient thermal stress of FGM using finite element method. Hence, we can thereby observe that finite element modelling is a very efficient method of solving thermal and mechanical problems in a FGM. However, during numerical modelling of graded material the main issue is to address the varying material property. Kim and Paulino [16] addressed this issue by evaluating the material properties directly at the gauss points, interpolating them from the nodal points using the same shape function (isoparametric elements). They used these isoparametric elements to solve the boundary value problems involving continuously non-homogeneous isotropic and orthotropic materials and then compared the performance of graded materials with that of homogeneous materials showing the effectiveness of graded finite elements while determining the local stress. Martἰnez-Paῆeda and Gallego [17] used this concept to carry out numerical analysis of quasi-static fracture in FGMs. They used ABAQUS [18] for numerical modelling and USDFLD subroutine [18, 19] to address the material gradation. Burlayenko et.al. [20] Also used isoparametric elements and showed the detailed finite element formulation of a coupled thermo-mechanical problem in an FGM plate. They also used various subroutines to capture the property gradation instead of using the conventional homogeneous elements in a layered model. Similarly, [21, 22] both use the ABAQUS software in which subroutines are used to address the material variation. Using this concept, we have carried out thermo-mechanical analysis of two dimensional functionally graded plate using ABAQUS and the subroutines.

Often we determine our results through simulation or experimentation but it is very much necessary to validate these results in order to establish its credibility. Robert G. Sargent [23] have discussed various validation techniques for simulation models elaborating their significance. For transient response, comparing one set of results with the other is always a subjective task. Various studies [24, 25] has been carried out for determining the error or correlation between transient data. However, each of these error measures had different issues. Russell [26, 27] has then developed a new set of error measure, addressing these issues, mainly biasing problem. We, thereby used this Russell Error measure to establish a correlation between the results obtained from simulation, using Q4, QM6 and Q8 elements, and analytical results or results obtained from literature.

2. Isoparametric Formulation for Graded Elements

Finite element formulation can be based on principle of virtual displacement [1]. Let us assume that the system at its equilibrium configuration is subjected to virtual displacement relative to that position. Then the equilibrium condition can be stated in the form:

$$\int {\left\{\text{δε}\right\}}^{\text{T}}\left\{\text{σ}\right\}\text{dV}=\int {\left\{\text{δu}\right\}}^{\text{T}}\left\{\text{F}\right\}\text{dV}+{\left\{\text{δu}\right\}}^{\text{T}}\left\{\text{ф}\right\}\text{dS}$$ (1)

Let displacement {u} be interpolated over an element, then,

$$\left\{\text{u}\right\}=[\text{N}]\left\{\text{d}\right\}\text{where}\left\{\text{u}\right\}={\lfloor \text{u\hspace{0.17em}v\hspace{0.17em}w}\rfloor }^{\text{T}}$$ (2)

and {d} lists the nodal displacement degree of freedom of an element.

Equation (2) can also be written as:

$${\text{ u}}^{\text{e}}=\sum _{\text{i}=1}^{\text{m}}{\text{N}}_{\text{i}}{\text{u}}_{\text{i}}^{\text{e}}$$ (3)

Which is the displacements for an isoparametric finite element.

Here, N_i are shape functions

• u_i is the nodal displacement corresponding to node i

• m is the number of nodal points in the elements.

Now, for a Q4 element, the shape functions is given by:

$${\text{N}}_{\text{i}}=\frac{\left(1+{\text{ξξ}}_{\text{i}}\right)\left(1+{\text{ηη}}_{\text{i}}\right)}{4},\text{ i}=1,\dots \mathrm{..},4$$ (4)

Where (ξ,η) denote intrinsic coordinates in the interval [-1,1] and (ξ_i, η_i) denote the local coordinates of node i.

The limitation of Q4 element is that it cannot exhibit pure bending. When bent, it displays shear strain as well as the expected bending strain. This strain needs strain energy, thus in case of pure bending deformation, the bending moment required to produce it will be more than the actual value, which is also known as shear locking behavior. Let us elaborate the case of shear locking.

The strain energy U is given by:

$$\text{U}=\frac{1}{2}\int {\left\{\text{ε}\right\}}^{\text{T}}\left[\text{E}\right]\left\{\text{ε}\right\}\text{ dVwhere},\text{ for 2D case},\left\{\text{ε}\right\}={\lfloor {\text{ε}}_{\text{x}}{\text{ ε}}_{\text{y}}{\text{ γ}}_{\text{xy}}\rfloor }^{\text{T}}$$ (5)

where, [E] is taken according to Plane stress or Plane strain condition.

• dV= t dx dy

For the actual block of material, subjected to pure bending, we have strains as:

$${\text{ε}}_{\text{x}}=-\frac{{\text{θ}}_{\text{b}}\text{y}}{2\text{a}}{\text{ ε}}_{\text{y}}=\text{ν}\frac{{\text{θ}}_{\text{b}}\text{y}}{2\text{a}}{\text{ γ}}_{\text{xy}}=0$$ (6)

Now, for strain energy in theQ4 element, let us subject the element to pure bending. Then, the top and bottom side remains unchanged whereas there is horizontal displacement is of θ_elb/2. So the element strains become,

$${\text{ε}}_{\text{x}}=-\frac{{\text{θ}}_{\text{el}}\text{y}}{2\text{a}}{\text{ ε}}_{\text{y}}=0{\text{ γ}}_{\text{xy}}=-\frac{{\text{θ}}_{\text{el}}\text{x}}{2\text{a}}$$ (7)

The main concern in equation (7) is the nonzero shear strain γ_XY, which should be zero in bending and thereby creates the spurious shear strain. Equation (5) and (6) yield strain energy U_b in the actual block whereas equations (5) and (7) yield strain energy U_el in the element. We know that work done by moment is equal to strain energy stored, hence,

$$\frac{{\text{M}}_{\text{b}}{\text{θ}}_{\text{b}}}{2}={\text{U}}_{\text{b}}\frac{{\text{M}}_{\text{el}}{\text{θ}}_{\text{el}}}{2}={\text{U}}_{\text{el}}$$ (8)

We can observe that θ is directly proportional to strain energy U and inversely proportional to bending moment M. So for θ_el = θ_b, M_el is greater than M_b and U_el is greater than U_b. Now, if M_el = M_b, then the ratio of rotation is:

$$\frac{{\text{θ}}_{\text{el}}}{{\text{θ}}_{\text{b}}}=\frac{1-{\text{ν}}^2}{1+\frac{1-\text{ν}}{2}{\left(\frac{\text{a}}{\text{b}}\right)}^2}$$ (9)

The ratio of (θ_el/θ_b)² approaches zero as aspect ratio a/b increases without limit and this is the condition for shear locking. Due to shear locking bending is excluded from the element behavior and this results into high strain energy.

To avoid shear locking in Q4, incompatible generalized degree of freedoms (a_i in Equation (10) below) can be added to represent pure bending which leads to incompatible 6-node quadrilateral element. For Q6 element, the displacement field is given by:

$$\begin{gathered} {\text{u}}^{\text{e}}=\sum _{\text{i}=1}^4{\text{N}}_{\text{i}}{\text{u}}_{\text{i}}+\left(1-{\text{ξ}}^2\right){\text{a}}_1+\left(1-{\text{η}}^2\right){\text{a}}_2 \\ {\text{v}}^{\text{e}}=\sum _{\text{i}=1}^4{\text{N}}_{\text{i}}{\text{v}}_{\text{i}}+\left(1-{\text{ξ}}^2\right){\text{a}}_3+\left(1-{\text{η}}^2\right){\text{a}}_4 \end{gathered}$$

In this case, N_i are the same shape functions as for Q4 elements.

From the above equations (10) we can see that the displacement field is augmented by modes that describe the state of constant curvature. Now, let us observe the modes associated with degrees of freedom a₂ and a_3$.$ The quadratic x² and y² introduced allow the elements to curve in between the nodes and there will be model bending with x or y axis as the neutral axis. Now, in pure bending the shear stress in the element is:

$${\text{γ}}_{\text{xy}}=\sum _{\text{i}=1}^4\frac{\partial {\text{N}}_{\text{i}}}{\partial \text{y}}{\text{u}}_{\text{i}}+\sum _{\text{i}=1}^4\frac{\partial {\text{N}}_{\text{i}}}{\partial \text{x}}{\text{v}}_{\text{i}}-\frac{2\text{y}}{{\text{b}}^2}{\text{a}}_2-\frac{2\text{x}}{{\text{a}}^2}{\text{a}}_3$$ (11)

Here, the negative terms balance out the positive terms, thereby minimizing the shear strain. As shown in fig 3, when incompatible elements Q6 are loaded, a gap appears between elements. Although no gaps appear in actual physical condition, one may question the effectiveness of these elements. However, these models provide a satisfactory results as in a refined mesh, each element is in a state of constant strain. The sides of undeformed elements remain straight, thereby producing a deformation in a state of constant strain. Convergence, in the case of Q6 maybe from above as the coarser mesh may be overly flexible, as opposed to the case of Q4 elements where convergence is from below.

Fig 3:

Incompatibility between adjacent Q6 elements

These elements pass the patch test thereby guaranteeing convergence towards exact results with mesh refinement. Q6 elements are converted to QM6 by selective integration, which uses different integration rules to treat different parts of the stiffness matrix. In this paper, we have developed QM6 graded elements.

Besides Q4 and QM6 elements, Q8 elements are also used. The shape functions for Q8 are given by:

For corner nodes (i=1, 2, 3, 4)

$${\text{N}}_{\text{i}}=\frac{1}{4}\left(1+{\text{ξξ}}_{\text{i}}\right)\left(1+{\text{ηη}}_{\text{i}}\right)\left({\text{ξξ}}_{\text{i}}+{\text{ηη}}_{\text{i}}-1\right)$$ (12)

For mid side nodes,

$$\text{If\hspace{0.17em}}{\xi }_{\text{i}}=0\text{\hspace{0.17em}then},{\text{N}}_{\text{i}}=\frac{1}{2}\left(1-{\text{ξ}}^2\right)\left(1+{\text{ηη}}_{\text{i}}\right)\left(\text{i.e. for\hspace{0.17em} }5,7\right)$$ (13)

$$\text{If\hspace{0.17em}}{\text{η}}_{\text{i}}=0\text{\hspace{0.17em}then},{\text{N}}_{\text{i}}=\frac{1}{2}\left(1-{\text{η}}^2\right)\left(1+\xi {\xi }_{\text{i}}\right)\left(\text{i.e. for\hspace{0.17em} }6,8\right)$$ (14)

The shape functions formulated in above cases are used to determine the strain and then stresses. The strain is obtained by:

$${\text{ε}}^{\text{e}}={\text{B}}^{\text{e}}{\text{u}}^{\text{e}}$$ (15)

Where, u^e is the nodal displacement vector, and B^e is the strain-displacement matrix of shape function derivatives i.e. [B^e] = [∂][N] for compatible elements (Q4 and Q8), and for QM6 element [B] is given by:

$$\left[\text{B}\right]=\left[{\text{B}}_{\text{d}} {\text{B}}_{\text{a}}\right]$$ (16)

where [Bd] operates on nodal degree of freedom and [Ba] operates on node-less degree of freedom. Hence, [Bd] is identical to [B] of a Q4 element. To obtain [B_a], [B] is given by:

$$\left[\begin{array}{c}{N}_{1,\xi }\\ N_{1,\eta }\\ N_{1,\xi }\\ N_{1,\eta }\end{array}\begin{array}{c}0\\ 0\\ 0\\ 0\end{array}\begin{array}{c}{N}_{2,\xi }\\ N_{2,\eta }\\ N_{2,\xi }\\ N_{2,\eta }\end{array}\begin{array}{c}0\\ 0\\ 0\\ 0\end{array}\begin{array}{c}{N}_{3,\xi }\\ N_{3,\eta }\\ N_{3,\xi }\\ N_{3,\eta }\end{array}\begin{array}{c}0\\ 0\\ 0\\ 0\end{array}\begin{array}{c}{N}_{4,\xi }\\ N_{4,\eta }\\ N_{4,\xi }\\ N_{4,\eta }\end{array}\begin{array}{c}0\\ 0\\ 0\\ 0\end{array}\begin{array}{c}-2\xi \\ 0\\ 0\\ 0\end{array}\begin{array}{c}0\\ -2\xi \\ 0\\ 0\end{array}\begin{array}{c}0\\ 0\\ -2\eta \\ 0\end{array}\begin{array}{c}0\\ 0\\ 0\\ -2\eta \end{array}\right]$$ (17)

where, N_1,η is derived from the shape function of quadrilateral Q4 and is equal to $\frac{-(1-\mathrm{ }\mathrm{\xi })\mathrm{ }}{4}$. Other terms is derived similarly. The last 4 columns in matrix given by equation (17) are multiplied by a_i to get strains. However, QM6 elements formulated in this way fails to represent constant stress or constant strain states unless they are rectangular. The strain energy of elements is given as:

$$\text{U}=\frac{1}{2}({\int }_{\text{V}}{\sigma }^{\text{T}}\text{ε\hspace{0.17em}}\text{dV})=\frac{1}{2}({\int }_{\text{V}}{\sigma }^{\text{T}}\text{B\hspace{0.17em}}\text{dV})\tilde{\text{u}}+\frac{1}{2}({\int }_{\text{V}}{\sigma }^{\text{T}}{\text{B}}_{\text{a}}\text{dV})\text{a}$$ (18)

Q4 element fulfills both compatibility and completeness requirement irrespective of the shape of the element. The incompatible (QM6) element will also fulfill completeness if the strain energy associated with the incompatible modes vanish for all constant strain/states. Let a vector of constants [σ₀] represent any state of uniform stress. We desire that degree of freedom remain zero when a typical element displays an arbitrary constant stress state [σ₀]. This requires that load terms associated with ai be zero.

$$\frac{1}{2}({\int }_{\text{V}}{\sigma }_0^{\text{T}}{\text{B}}_{\text{a}}\text{dV})\text{a}=\frac{1}{2}{\sigma }_0^{\text{T}}({\int }_{\text{v}}{\text{B}}_{\text{a}}\text{dV})\text{a=0}$$ (19)

Thus, QM6 element would satisfy the requirement:

$${\int }_{\text{V}}{\text{B}}_{\text{a}}\text{dv\hspace{0.17em}=0}$$ (20)

Strain displacement matrix can be modified to satisfy the completeness requirement as follows:

$${\text{B}}_{\text{a}}^{\text{m}}={\text{B}}_{\text{a}}-\frac{1}{\text{V}}{\int }_{\text{V}}{\text{B}}_{\text{a}}\text{dv}$$ (21)

Drawback associated with incompatible elements is that there is lack of a bound-on displacement which is a less important factor than the accuracy of parent elements.

Now, stress is obtained as:

$${\text{σ}}^{\text{e}}={\text{D}}^{\text{e}}\left(\text{x}\right){\text{ε}}^{\text{e}}$$ (22)

Where D^e(x) = D^e(x, y) is the constitutive matrix, which is the function of position for nonhomogeneous materials.

Now, from equation (1) we get,

$${\text{k}}^{\text{e}}{\text{u}}^{\text{e}}={\text{F}}^{\text{e}}$$ (23)

F^e is the load vector and the element stiffness matrix is given by:

$${\text{k}}^{\text{e}}=\underset{{\text{V}}_{\text{e}}}{\iiint }{\text{B}}^{{\text{e}}^{\text{T}}}{\text{D}}^{\text{e}}\left(\text{x}\right){\text{B}}^{\text{e}}{\text{dV}}_{\text{e}}$$ (24)

In which V_e is the domain of element (e). This element level stiffness matrix can be used for the entire system.

3. Thermo-mechanical Analysis of Graded Materials

The isoparametric formulation allows quadrilateral and hexahedral elements to have non-rectangular shapes [1]. The shape functions in this case, are used to interpolate both displacement field as well as element geometry. Also, in graded materials, the same shape functions are used to interpolate the material properties at nodal points to each of the Gaussian points. We can now obtain the interpolated Young’s modulus E=E(x) and Poisson’s ratio ν=ν(x), similar to equation (3), as:

$$\text{E}=\sum _{\text{i}=1}^{\text{m}}{\text{N}}_{\text{i}}{\text{E}}_{\text{i}},\text{ν}=\sum _{\text{i}=1}^{\text{m}}{\text{N}}_{\text{i}}{\text{ν}}_{\text{i}}$$ (25)

As the material properties in our case are represented in terms of volume fraction (V) of a material phase, p, we represent it too in terms of the generalized isoparametric formulation by interpolation function as:

$${\text{V}}^{\text{p}}=\sum _{\text{i}=1}^{\text{m}}{\text{N}}_{\text{i}}{\text{V}}_{\text{i}}^{\text{p}}$$ (26)

Where, ${\mathrm{V}}_{\mathrm{i}}^{\mathrm{p}}$ (i=1,2,…m) are the values of V^p at the nodal points.

The finite element analyses are generally based on the approximation of displacements. All the material points within the inhomogeneous isotropic body in the 2-D domain are assumed to be X. As the stress in the materials are developed due to temperature variation, we need to first determine temperature field and then the corresponding stresses [14]. We know the transient heat conduction equation as:

$$\text{ρ}\left(\overrightarrow{\text{x}}\right)\text{c}\left(\overrightarrow{\text{x}}\right)\frac{\partial \text{T}\left(\overrightarrow{\text{x}},\text{t}\right)}{\partial \text{t}}=\nabla .\left(\text{k}\left(\overrightarrow{\text{x}}\right)\nabla \text{T}\left(\overrightarrow{\text{x}},\text{t}\right)\right)$$ (27)

Where, ρ($\overrightarrow{\mathrm{x}}$) is the material density

• $\mathrm{c}\left(\overrightarrow{\mathrm{x}}\right)$ is the specific heat

• k$\left(\overrightarrow{\mathrm{x}}\right)$ is the thermal conductivity

• $\mathrm{T}\left(\overrightarrow{\mathrm{x}},\mathrm{t}\right)$ is the temperature

Equation (27) is converted into 1-Dimensional equation as the boundary conditions on each surface x=0 and x=w, where w is the width of the plate, are uniform. So the equation is written in the form:

$$\text{ρ}\left(\overrightarrow{\text{x}}\right)\text{c}\left(\overrightarrow{\text{x}}\right)\frac{\partial \text{T}\left(\overrightarrow{\text{x}},\text{t}\right)}{\partial \text{t}}=\frac{\partial }{\partial \text{x}}\left(\text{k}\left(\overrightarrow{\text{x}}\right)\frac{\partial \text{T}\left(\overrightarrow{\text{x}},\text{t}\right)}{\partial \text{x}}\right)$$ (28)

Equation (28) can be solved by using the following initial condition:

$$\text{T}\left(\text{x},0\right)={\text{T}}_0,0\le \text{x}\le \text{b}$$ (29)

And two boundary conditions. We can apply different conditions on both boundaries (x₀ ϵ {0, w}). For instance NEUMANN condition i.e.:

$${\frac{\partial \text{T}\left(\text{x},\text{t}\right)}{\partial \text{x}}|}_{{\text{x=x}}_{\text{0}}}=\text{f}\left(\text{t}\right),\text{ where f}\left(\text{t}\right)\text{is arbitary}$$ (30)

Or DIRICHLET condition:

$$\text{T}\left({\text{x}}_0,\text{t}\right)=\overline{\text{T}\left(\text{t}\right)}$$ (31)

Or convective heat transfer from environment into the material i.e.

$${\text{h}\left({\text{T}}_{\text{e}}\left(\text{t}\right)-\text{T}\left({\text{x}}_0,\text{t}\right)\right)=-\text{k}\left({\text{x}}_0\right)\frac{\partial \text{T}\left(\text{x},\text{t}\right)}{\partial \text{x}}|}_{\text{x}={\text{x}}_0}$$ (32)

Where, h is the convective heat transfer from environment.

Now, as stated before, thermal strains are the determined based on these temperature loads. As we are dealing with a 2-D model and from the compatibility equation, we have,

$${\text{2ε}}_{\text{xy,yx}}{\text{=ε}}_{\text{xx,yy}}{\text{+ε}}_{\text{yy,xx}}$$ (33)

As we are dealing with linear materials,

$${\text{ε}}_{\text{xx}}=\frac{1-\text{ν}{\left(\text{x}\right)}^2}{\text{E}\left(\text{x}\right)}\left({\text{σ}}_{\text{xx}}-\frac{\text{ν}\left(\text{x}\right)}{1-\text{ν}\left(\text{x}\right)}{\text{σ}}_{\text{yy}}\right)+\left(1+\text{ν}\left(\text{x}\right)\right)\text{α}\left(\text{x}\right)\left(\text{T}\left(\text{x}\right){\text{-T}}_0\right)$$ (34)

$${\text{ε}}_{\text{yy}}=\frac{1-\text{ν}{\left(\text{x}\right)}^2}{\text{E}\left(\text{x}\right)}\left({\text{σ}}_{\text{yy}}-\frac{\text{ν}\left(\text{x}\right)}{1-\text{ν}\left(\text{x}\right)}{\text{σ}}_{\text{xx}}\right)+\left(1+\text{ν}\left(\text{x}\right)\right)\text{α}\left(\text{x}\right)\left(\text{T}\left(\text{x}\right)-{\text{T}}_0\right)$$ (35)

$${\text{ε}}_{\text{xy}}=\frac{1+\text{ν}\left(\text{x}\right)}{\text{E}\left(\text{x}\right)}{\text{τ}}_{\text{xy}}$$ (36)

Where E(x) is the Young’s Modulus of Elasticity, $\mathrm{\nu }\left(\mathrm{x}\right)$ is the Poisson’s ratio and α(x) coefficient of thermal expansion. As the temperature loads are applied through x=0 to x=w, so no strains were developed along the y-direction and also stress along x-direction was zero due to free boundaries and extensibility. Substituting these values in equation (29), integrating twice, and rearranging, we get,

$${\text{σ}}_{\text{yy}}\text{=-}\frac{\text{α}\left(\text{x}\right)\text{E}\left(\text{x}\right)}{\text{1-ν}\left(\text{x}\right)}\left(\text{T}\left(\text{x}\right){\text{-T}}_{\text{0}}\right)\text{+}\frac{\text{E}\left(\text{x}\right)}{\text{1-ν}{\left(\text{x}\right)}^{\text{2}}}\text{Mx+}\frac{\text{E}\left(\text{x}\right)}{\text{1-ν}{\left(\text{x}\right)}^{\text{2}}}\text{N}$$ (37)

Where M and N are integration constants which are determined using the equilibrium conditions:

$${\int }_0^{\text{W}}{\text{σ}}_{\text{yy}}\text{dx}=0\text{ and }{\int }_0^{\text{W}}{\text{σ}}_{\text{yy}}\text{xdx}=0$$ (38)

4. Finite Element Modeling Procedure

The basic property of graded materials is the continuously changing material properties, along a certain direction. This can be incorporated in the numerical models by creating number of layers in the direction of variation [15]. Each layer is treated as a homogeneous material with its own material properties which leads to a stepwise change in properties along the direction of material gradient. This method, however, requires a lot of work as one needs to calculate material property of each layer manually and then input it. Also, for better results one must use finer meshes. Owing to these drawbacks, it is better to incorporate the material gradation in the element level and this can be achieved by using graded finite elements. We used ABAQUS as the modeling software and utilized the user-subroutine feature in it to develop a graded element using the constitutive law of material properties. Besides using compatible graded 4-noded and 8-noded quadrilateral elements, we also developed modified 6-noded quadrilateral elements and compared the results obtained from all three elements for accuracy and efficiency. The varying mechanical properties within the elements, are addressed using UMAT subroutine, UMATHT is used for varying thermal properties and USDFLD is used to capture the variation in density. To understand how these subroutines were developed, you look into the pseudo codes prepared for the materials in Appendix1.

5. Error Calculation

There are numerous error measures that have been developed in order to quantify the error in between two sets of transient data. One may bias their results against one set of data assuming it to be absolutely correct, which may not be true, thereby resulting in an erroneous evaluation. Other issues that may arise is that the physical interpretation of the results and the basis for the error factor may not be well understood. Also some error measures may not easily point to the issue resulting to the high errors or the reason behind it. Counteracting among all these points, we can observe that [26, 27] Russell Error Measurement is one of the suitable technique. This error measure is not biased towards either of the transient response and hence is suitable for statistical evaluation of multiple point systems. Also, as shown in the paper, it takes in account all the issues stated above, thereby resulting in an unbiased error measurement technique.

5.1. Russell Error

Russell error [26,27] is a mathematical tool that provides a robust means for determining the phasing, difference in magnitude, and overall error (comprehensive error) between two functions, or set of data.

5.1.1. Relative Magnitude Error:

Let $\overrightarrow{\mathrm{p}}\mathrm{ }\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{ }\overrightarrow{\mathrm{m}}$ be the vectors of data which are being considered. There magnitudes are given as s₁ and s₂, respectively. The difference between the relative magnitudes is:

$$\text{rme}=\frac{{\text{s}}_1^2-{\text{s}}_2^2}{{\text{s}}_1{\text{s}}_2}$$ (39)

This equation is unbiased towards either response, which implies, the absolute value of rme is the same for $\overrightarrow{\mathrm{p}}\mathrm{ }\mathrm{v}\mathrm{s}\mathrm{ }\overrightarrow{\mathrm{m}}$ as well as $\overrightarrow{\mathrm{m}}$ vs $\overrightarrow{\mathrm{p}}$. For ease in computing, we can write relative magnitude error as:

$$\text{rme}=\frac{{\sum }_{\text{i}=1}^{\text{N}}{\text{p}}_{\text{i}}^2-{\sum }_{\text{i}=1}^{\text{N}}{\text{m}}_{\text{i}}^2}{\sqrt{{\sum }_{\text{i=1}}^{\text{N}}{\text{p}}_{\text{i}}^{\text{2}}{\sum }_{\text{i}=1}^{\text{N}}{\text{m}}_{\text{i}}^2}}$$ (40)

where, N is the number of data points that are measured.

The obtained relative error is unbounded, but as we need to combine phase error, which is bounded, to magnitude error in order to obtain comprehensive error (which is not dominated by either phase or magnitude), it is desirable to have the measure of magnitude error on the same relative scale as phase error. Maintaining the signed unbiased nature, the magnitude error factor is defined as:

$${\text{M}}_{\text{R}}=\text{sign }\left(\text{rme}\right){\text{log}}_{10}\left(\text{1}+\left|\text{rme}\right|\right)$$ (41)

5.1.2. Phase Error:

Phase error is determined on the basis of phase correlation, which is the normal correlation computed on set of data that fluctuates according to time. By equating the phase correlation directly to the phase shift between two trigonometric functions, we obtain the phase error as:

$${\text{P}}_{\text{R}}=\frac{1}{\text{π}}{\text{cos}}^{-1}\left(\frac{{\sum }_{\text{i}=1}^{\text{N}}{\text{p}}_{\text{i}}{\text{m}}_{\text{i}}}{\sqrt{{\sum }_{\text{i}=1}^{\text{N}}{\text{p}}_{\text{i}}^{\text{2}}{\sum }_{\text{i}=1}^{\text{N}}{\text{m}}_{\text{i}}^{\text{2}}}}\right)$$ (42)

5.1.3. Comprehensive Error:

The magnitude and phase error can be combined into a single comprehensive error C_R:

$${\text{C}}_{\text{R}}=\sqrt{\frac{\text{π}}{4}\left({\text{M}}_{\text{R}}^2-{\text{P}}_{\text{R}}^{\text{2}}\right)}$$

Here, the factor п/4 is used in the comprehensive error as this factor keeps comprehensive error in the similar scale as magnitude and phase errors as well as can be forgiving when one of the error measures is small or can be more restrictive if both the phase and magnitude error are approaching the acceptance criteria.

It is generally suggested that if all the three types of error are each less than 0.2 the correlation between the data can be accepted. This is the same criteria that we have adopted in this paper.

6. Numerical Examples

6.1. A graded plate subjected to tension

Before carrying out transient thermal analysis, we can show the advantages of QM6 graded elements over graded Q4 elements in a mechanical problem. Consider a functionally graded square plate with same width and length (i.e. L=9) subjected to tensile load applied parallel to the material gradation on the right edge. Fig 4 shows an isotropic square plate with Young’s modulus varying in the horizontal direction and with zero Poisson’s ratio assumed which is simply supported on the left edge. The tensile load is applied by constant stress distribution along the right edge. The magnitude of 1 unit force was applied.

Fig. 4:

A graded plate under tension in the direction of gradation with either exponential or linear variation of Young’s modulus E=E(x) where E₁ = E(x=0) and E₂ = E(x=L).

The exponential variation of the Young’s modulus along the y-direction is given by

$$\text{E}\left(\text{x}\right)={\text{E}}_1{\text{e}}^{\text{γx}},\text{ γ}=\frac{1}{\text{L}}\text{ln}\left(\frac{{\text{E}}_2}{{\text{E}}_1}\right)$$

and the linear variation is given by

$$\text{E}\left(\text{x}\right)={\text{E}}_1+\text{γx },\text{ γ}=\frac{{\text{E}}_2-{\text{E}}_1}{\text{L}}$$

with E₁ = 1.0 and E₂ = 8.0.

These material property variations are coded using Abaqus user subroutine UMAT. The plate is discretized into 9 by 9 meshes, using Q4 and QM6 elements. The nodal stresses along x=0 are plotted for comparison in each of the elements.

Figures 5(a) and 5(b) compare an element-level nodal stress σ_xx versus x in a plate with exponentially and linearly graded moduli, respectively, subjected to tension load parallel to the gradation with the exact solution. Element-level stresses can demonstrate element quality. Q4 graded elements with 2x2 Gaussian quadrature yield piecewise linear element-level stresses, while stress results using QM6 and Q8 elements match with the exact solution. A typical FEA will take averages stresses for the node shared with adjacent elements. But right and left corner nodes at x=0 and x=9 are not shared and the averaged stresses at these nodes are still way off from the exact solution. The maximum relative error is about 10% for exponential variation, and 28% for linear variation. Stresses along domain boundaries and edges need to be accurately calculated because higher stresses typically arise on these boundaries than internal domains. Note that incompatible QM6 graded elements are capable of representing accurate stress solutions for graded materials with arbitrary material gradation, and more efficient than Q8 elements. For non-zero Poisson’s ratio (e.g. v=0.3), no analytical solution is available, but QM6 elements are in good comparison with reference solutions obtained by Q8 elements (see Figures 6(a) and 6(b)). Due to this evident advantage, we explore the characteristics of QM6 graded elements in ensuring transient thermal stress analysis for graded media.

Fig. 5.

Element-field stress distribution in a plate with zero Poisson’s ratio under tension applied parallel to a) exponential material gradation b) linear material gradation

Fig. 6.

Element-field stress distribution in a plate with 0.3 Poisson’s ratio under tension applied parallel to a) exponential material gradation b) linear material gradation

6.2. A graded plate subject to thermal shock

For transient thermal analysis, consider an FGM plate (with p=1) consisting of titanium carbide (TiC) and the silicon carbide (SiC) [14] (see Fig 7). It is subjected to thermal shock on the edge with pure TiC at x=0. The entire system is initially at T=0⁰C and then the TiC edge is suddenly heated to 900⁰C.

Fig 7:

Schematic diagram of the FGM plate

Such a severe thermal shock results in development of thermal stresses in the plate. The material properties are assumed to vary continuously along the thickness direction based on power-law function. The volume fraction of the constituents are assumed to obey:

$$\text{V}\left(\text{x}\right)={\left(\frac{\text{x}}{\text{w}}\right)}^{\text{p}}$$ (44)

where, p is the material parameter and w is the thickness of the plate. The Young’s Modulus and Poisson’s ratio are assumed to be constant and there is only variation of the thermal properties which are expressed by means of the rule of mixtures of a two-phase materials as follows:

$$\text{ρ}\left(\text{x}\right)=\text{V}\left(\text{x}\right){\text{ρ}}_{\text{2}}+\left(1-\text{V}\left(\text{x}\right)\right){\text{ρ}}_1,$$ (45)

$$\text{c}\left(\text{x}\right)=\text{V}\left(\text{x}\right){\text{c}}_2+\left(1-\text{V}\left(\text{x}\right)\right){\text{c}}_1$$ (46)

$$\text{k}\left(\text{x}\right)={\text{k}}_{\text{1}}\left\{1+\frac{3\left({\text{k}}_2-{\text{k}}_1\right)\text{V}\left(\text{x}\right)}{3{\text{k}}_1+\left({\text{k}}_2-{\text{k}}_1\right)\left(1-\text{V}\left(\text{x}\right)\right)}\right\}$$ (47)

$$\text{α}\left(\text{x}\right)=\text{V}\left(\text{x}\right){\text{α}}_2+\left(1-\text{V}\left(\text{x}\right)\right){\text{α}}_1$$ (48)

Figure 8 shows the temperature developed at the non-dimensional time τ= 0.001, 0.01, 0.1. We obtain similar temperature profile for all the three types of elements used which is plotted in the figure. We can see at τ= 0.001, temperature remains at the initial value (T=0) for maximum portion of the strip and rises rapidly near the thermally shocked edge, x=0. For τ= 0.01, the temperature starts dropping around the middle portion of the strip and at τ= 0.1, temperature decay is within the full width of the strip.

Fig 8:

Temperature distribution within the plate

Figure 9 show thermal stresses developed at the non-dimensional time τ= 0.001, 0.01, 0.1. At each time, we can observe that the thermal stress reaches the peak value at X=0. The stress profile for τ= 0.001 is sharper than the other two as the temperature suddenly rises near the edge at this time. For τ= 0.1, there is minimum stress developed as there is linear distribution of temperature within the full width.

Fig 9:

Stress distribution within the plate at the non-dimensional time τ= 0.001, 0.01, 0.1

These results are estimated by error measures as shown in Figure 10. The obtained data are compared against the theoretical data in the literature [14]. Looking at the Russell Error obtained, all the errors from the three element types are within the error limit of 0.2. However, the error for Q4 elements are larger than that for QM6 and Q8. For instance, at t=0.001 sec, the error produced by Q4 elements is 100.7% greater than Q8, and error produced by QM6 elements is 36.41% greater than that produced by Q8. This shows that Q8 is the most accurate, and then QM6 and finally Q4 is the least accurate element, in this analysis.

Fig 10:

Comparison of comprehensive (Russell) error for all three elements. It is generally suggested that if all the three types of error are each less than 0.2 the correlation between the data can be accepted.

Figures 11 and 12 respectively depict averaged and non-averaged (element-level) stress contours for Q4 and QM6 or Q8 elements. Major difference is not noticeable in Figure 11. But in Figure 12, some differences in element-level stresses are seen in Q4 elements.

Fig 11:

Contour of averaged stress a) Q4 elements; b) QM6 or Q8 elements

Fig 12:

Contour of non-averaged stress a) Q4 elements; b) QM6 or Q8 elements

To establish the efficiency with regards to computational time, the model with finer mesh of 50,000 elements and time increment of 0.0001 sec is run in a 10 core processor computer, for each type of element. Using Q4 graded elements, the analysis time was 23 minutes and 21 seconds, and QM6 graded elements required 29 minutes and 26 seconds whereas Q8 graded elements required 39 minutes and 34 seconds.

6.3. A graded plate with transient thermal loads

Here we analyze a plate that consists of titanium alloy Ti-6Al-4V and zirconium dioxide ZrO₂ ceramic [20]. The plate has all of its properties, thermal as well as mechanical, varying along the thickness direction. Using the volume fraction function in equation (44), the volume fraction of metal is:

$${\text{V}}_{\text{m}}=\text{V}\left(\text{x}\right)$$ (49)

where m denotes metal. Then, the volume fraction of ceramic is expressed as:

$${\text{V}}_{\text{c}}=1-{\text{V}}_{\text{m}}$$ (50)

Hence, material properties are given by:

$$\text{E}\left(\text{x}\right)={\text{E}}_{\text{c}}\left\{\frac{{\text{E}}_{\text{c}}+\left({\text{E}}_{\text{m}}-{\text{E}}_{\text{c}}\right){\text{V}}_{\text{m}}^{\text{2/3}}}{{\text{E}}_{\text{c}}+\left({\text{E}}_{\text{m}}-{\text{E}}_{\text{c}}\right)\left({\text{V}}_{\text{m}}^{\frac{\text{2}}{\text{3}}}-{\text{V}}_{\text{m}}\right)}\right\}$$ (51)

$$\text{ν}\left(\text{x}\right)={\text{ν}}_{\text{m}}{\text{V}}_{\text{m}}+{\text{ν}}_{\text{c}}{\text{V}}_{\text{c}}$$ (52)

$$\text{ρ}\left(\text{x}\right)={\text{ρ}}_{\text{m}}{\text{V}}_{\text{m}}+{\text{ρ}}_{\text{c}}{\text{V}}_{\text{c}}$$ (53)

$$\text{α}\left(\text{x}\right)=\frac{\frac{{\text{α}}_{\text{m}}{\text{V}}_{\text{m}}{\text{E}}_{\text{m}}}{1-{\text{ν}}_{\text{m}}}+\frac{{\text{α}}_{\text{c}}{\text{V}}_{\text{c}}{\text{E}}_{\text{c}}}{1-{\text{ν}}_{\text{c}}}}{\frac{{\text{V}}_{\text{m}}{\text{E}}_{\text{m}}}{1-{\text{ν}}_{\text{m}}}+\frac{{\text{V}}_{\text{c}}{\text{E}}_{\text{c}}}{1-{\text{ν}}_{\text{c}}}}$$ (54)

$$\text{k}\left(\text{x}\right)={\text{k}}_{\text{c}}\left\{\frac{1+3\left({\text{k}}_{\text{m}}-{\text{k}}_{\text{c}}\right){\text{V}}_{\text{m}}}{3{\text{k}}_{\text{c}}+\left({\text{k}}_{\text{m}}-{\text{k}}_{\text{c}}\right){\text{V}}_{\text{c}}}\right\}$$ (55)

$$\text{c}\left(\text{x}\right)=\frac{{\text{c}}_{\text{m}}{\text{ρ}}_{\text{m}}{\text{V}}_{\text{m}}+{\text{c}}_{\text{c}}{\text{ρ}}_{\text{c}}{\text{V}}_{\text{c}}}{{\text{ρ}}_{\text{m}}{\text{V}}_{\text{m}}+{\text{ρ}}_{\text{c}}{\text{V}}_{\text{c}}}$$ (56)

The properties of ceramic and metal incorporated in FGM are as follows:

In order to include the variation of thermal, mechanical as well as density one needs to use UMAT, UMATHT, USDFLD together in ABAQUS. A 10 by 100, symmetric graded plate (see Figure 13) is modelled. The gradation varies from ceramic to metal. The ceramic side is heated from 300K to 1300K, while the metal side is maintained at 300K. A coupled temperature displacement analysis is carried out using CPE4T (Q4), CPEMT (QM6) and CPE8T (Q8). When a steady heat is obtained in the plate, the ceramic side is again cooled to 300K using a film convection coefficient of h=2000 W/mm² K. During the entire heating and cooling process, the obtained stresses are monitored and a comparative graph is plotted.

Fig 13:

Schematic diagram of the FGM plate

Figures 14(a) and 14(b) show the temperature profile developed at time t= 0.00032, 0.032, 0.32, 3.2, 32. A similar temperature profile is obtained for all three type of elements. In figure 14(a), when t= 0.00032, temperature remains at the initial value of 300K for maximum portion of the plate and rises rapidly near the thermally shocked ceramic edge. With increase in time, temperature distribution becomes more uniform within the plate and at t= 32, the temperature is distributed throughout the plate, reaching a steady state. Now, after a steady state is obtained at t=32, in heating, the ceramic surface is quickly cooled to the initial temperature of 300K, as shown in the cooling profile of figure 14(b). As seen in the graph, at t=0.00032, the temperature near the ceramic edge quickly drops down from 1300K to 300K, thereby creating sharp temperature change. With increase in time, the temperature decay is not that abrupt and at t=32sec, the entire plate is at 300K, thereby reaching a steady state.

Fig 14:

Temperature distribution in the FGM plate under a) heating and b) cooling

Figures 15(a) and 15(b) show thermal stresses developed within the plate at different times for both heating and cooling. Although there is a material property distribution within the plate, the maximum stress occurs at the ceramic side, and not at the inner layer of the graded plate for all time increments. Maximum stresses occurs at t=0.00032, for both heating and cooling, as there is abrupt temperature change at this time increment.

Fig 15:

Stress Distribution in the FGM plate under a) heating b) cooling

Figure 16 shows the error obtained, while using Q4, QM6 and Q8 graded elements when compared with the results in [14]. The bar graph clearly depicits the accuracy of Q8 elements over QM6 and Q4. The point of major interest, however, is how effective QM6 elements are over Q4 elements as they clearly produce less error than the latter.

Fig 16:

Comparison of comprehensive (Russell) error for all three elements under a) heating and b) Cooling. It is generally suggested that if all the three types of error are each less than 0.2 the correlation between the data can be accepted.

Figure 17 shows σ_yy obtained using Q4 element at t=32 sec under heating and cooling, respectively. Regular response is seen along y direction. Higher stresses are seen in at or near the left edge.

Fig 17:

Contour of Stress along y-direction for Q4 element at t=32 sec under a) heating and b) cooling

To establish the efficiency with regards to computational time, a model with finer mesh of 100,000 elements and fixed time increment of 100 steps, for each type of element is run. Using Q4 graded elements, the analysis time was 1 hour and 37 minutes, QM6 graded elements required 1 hour and 54 minutes whereas Q8 graded elements required 3 hours, and 12 minutes.

7. Conclusions

This paper addresses incompatible graded finite element that is developed to analyze graded materials subjected to transient thermal analysis. The accuracy and efficient of QM6 graded finite element are compared with Q4 and Q8 elements. The obtained results can be verified by calculating error against the theoretical values determined by using formulas or through literature. The gradation of the material properties have been included in the finite element model through ABAQUS using the user-defined subroutines UMAT, UMATHT and USDFLD. These subroutines assures that the thermo-mechanical variation of material properties are included in the finite element model at element level. Thermal analysis is carried out using coupled temperature-displacement plane strain quadrilateral finite element that is CPE4T for Q4, CPE6T for QM6 and CPE8T for Q8. From the comparative error diagram we can see that QM6 provides more accurate results than Q4 and the processing time for QM6 elements is less than that of Q8. Hence, by comparing both the obtained stresses and computational time required one can come to the conclusion that QM6 elements are the best element to use, especially when carrying out large analysis as it not only saves computational time but also produces the results which are as compatible as Q8. Q4 graded elements are too stiff, thereby resisting bending by developing both normal and shear stresses, and to resolve this issue, QM6 graded elements are developed which in turns produce better results by modelling bending conditions.

Fig 1:

Four node plane element in physical space and then mapped into ξη space

Fig 2:

Displacement modes in graded Q6 elements

Table 1:

Material properties of TiC and SiC

Materials	Young’s Modulus (GPa)	Poisson’s Ratio	Coefficient of thermal expansion (10−6 K−1)	Thermal conductivity (Wm−1K)	Mass Density (gcm−3)	Specific heat (Jg−1K−1)
TiC	400	0.2	7.0	20	4.9	0.7
SiC	400	0.2	4.0	60	3.2	1

Table 2:

Thermo-mechanical properties

Property	Constituents
	Ceramic	Metal
	ZiO2	Ti - 6Al - 4V
Young’s Modulus E, [GPa]	117	66.2
Poisson’s ratio ν	0.333	0.32
Mass Density, ρ, [kg/m3]	5600	4420
Coefficient of thermal expansion α, 10−6 [1/K]	7.11	10.3
Thermal Conductivity k, [W/mK]	2.036	18.1
Specific Heat c [J/kgK]	615.6	808.3

Appendix 1

In this case all mechanical as well as thermal properties are varying along the x-direction.

1. First subroutine UMAT is called

   1. The mechanical properties such as Young’s Modulus of Elasticity, Poisson’s ratio and Coefficient of thermal expansion are read from the .inp file, which are defined by the user.
   2. V_m And V_c are determined, as in equation (33) and (39), then they are used to determine the material properties at each Gaussian point using equations (40), (41) and (43).
   3. The lame constants λ(x) and μ(x) are determined, which then used to determine β=α (3λ+2μ) at each pointwise functions of location.
   4. Matrix of material constants are stored in DDSDDE as

$$\left[\text{DDSDDE}\right]=\frac{\text{E}}{\left(1+\text{ν}\right)\left(1-2\text{ν}\right)}\left[\begin{array}{cccc}\text{1-ν}& \text{ν}& \text{ν}& \text{0}\\ \text{ν}& \text{1-ν}& \text{ν}& \text{0}\\ \text{ν}& \text{ν}& \text{1-ν}& \text{0}\\ \text{0}& \text{0}& \text{0}& \frac{\text{1–2ν}}{\text{2}}\end{array}\right]$$

This is a case for plane strain analysis, as done in this paper.

1. Now, the matrix of thermal coefficients are defined by:

$$\mathbf{\text{β}}=\frac{\text{β}\left(1+\text{ν}\right)}{\left(2-\text{ν}\right)}\left[\begin{array}{c}1\\ 1\\ 1\\ 0\end{array}\right]$$

1. The stresses in the plate is then determined using the increment of in-plane stresses which takes the following matrix form:

$$\mathrm{\Delta }\text{σ}=\text{DDSDDE}\mathrm{\Delta }\text{ε}-\mathbf{\text{β}}\text{Δθ}$$

Where, $\text{Δ}\mathrm{\epsilon }$ is the increment of in-plane strains

1. After ending UMAT, UMATHT subroutine is called

   1. The thermal properties such as conductivity and specific heat as well as densities are read from the .inp file, which are defined by the user.
   2. V_m and V_c are determined, as in equation (33) and (39), then they are used to determine the material properties at each Gaussian point using equations (44) and (45).
   3. Internal thermal energy per unit mass is determined as:

U=U (Ө), where $\frac{\partial \mathrm{U}}{\partial \mathrm{Ө}}=\mathrm{c}$(x)

c(x) is the specific heat of the material at the given Gaussian point

1. The flux is then determined as:

flux (i) =−k$\frac{\partial \mathrm{Ө}}{\partial \mathrm{x}}$

1. Call subroutine USDFLD at the end

   1. The densities of the materials are read from the .inp file, defined by the user.
   2. V_m and V_c are determined, as in equation (33) and (39), then they are used to determine the density as per equation (42).
   3. The obtained density is stored as field variable.

The combination of these three subroutines ensures that the variation of all the material properties in a graded material is properly considered and thereby evaluates accurate properties at each of the Gauss points.