# On the Finite Element Implementation of Functionally Graded Materials

## Abstract

**:**

## 1. Introduction

## 2. Numerical Formulation

#### 2.1. Gauss Integration Point-Based Variation

#### 2.2. Nodal-Based Variation via Temperature Dependence

## 3. Results

#### 3.1. Uniform Displacement Perpendicular to the Material Gradient Direction

#### 3.2. Uniform Traction Perpendicular to the Material Gradient Direction

#### 3.3. Uniform Traction Parallel to the Material Gradient Direction

## 4. Conclusions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Sketch outlining the (

**a**) gradual variation of Young’s modulus E along the y-coordinate, as captured by (

**b**) homogeneous elements and (

**c**) graded elements.

**Figure 2.**Sketch outlining the gradual variation of Young’s modulus E and its associated interpolation by means of temperature-based generalized isoparametric graded element for an equivalent interpolation of thermal and mechanical strains.

**Figure 4.**Boundary value problems under consideration: (

**a**) functionally graded plate with spatially varying Young’s modulus subjected to (

**b**) uniform displacement perpendicular to the material gradient direction, (

**c**) uniform traction perpendicular to the material gradient direction, and (

**d**) uniform traction in the direction parallel to material gradation—consistent units.

**Figure 5.**Uniform displacement perpendicular to the material gradient for different kinds of elements: (

**a**) Q4; (

**b**) Q4R; (

**c**) Q8; and (

**d**) Q8R—consistent units.

**Figure 6.**Uniform displacement perpendicular to the material gradient. Error analysis for the Q8 element. Consistent units.

**Figure 7.**Uniform displacement perpendicular to the material gradient, (

**a**) tensile stress for one Q8 element in the x-direction; (

**b**) Young’s modulus interpolation through different schemes; and (

**c**) mesh-sensitivity error analysis—consistent units.

**Figure 8.**Uniform traction perpendicular to the material gradient for different kinds of elements: (

**a**) Q4; (

**b**) Q4R; (

**c**) Q8; and (

**d**) Q8R—consistent units.

**Figure 9.**Uniform traction perpendicular to the material gradient. Error analysis for the Q8 element—consistent units.

**Figure 10.**Uniform traction perpendicular to the material gradient, tensile (

**a**) stress and (

**b**) strain for one Q8 element in the x-direction—consistent units.

**Figure 11.**Uniform traction parallel to the material gradient for different kinds of elements: (

**a**) Q4 and (

**b**) Q8—consistent units.

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**MDPI and ACS Style**

Martínez-Pañeda, E.
On the Finite Element Implementation of Functionally Graded Materials. *Materials* **2019**, *12*, 287.
https://doi.org/10.3390/ma12020287

**AMA Style**

Martínez-Pañeda E.
On the Finite Element Implementation of Functionally Graded Materials. *Materials*. 2019; 12(2):287.
https://doi.org/10.3390/ma12020287

**Chicago/Turabian Style**

Martínez-Pañeda, Emilio.
2019. "On the Finite Element Implementation of Functionally Graded Materials" *Materials* 12, no. 2: 287.
https://doi.org/10.3390/ma12020287