Finite element analysis of functionally graded hyperelastic beams under plane stress

Abstract

An alternative finite element formulation to analyze highly deformable beams composed of functionally graded (FG) hyperelastic material is presented. The 2D beam element adopted has a general order and seven degrees of freedom per node, allowing both axial and shear effects, as well as cubic variation of transverse strains. The constitutive law employed is the hyperelastic compressible neo-Hookean model for plane stress conditions. The material coefficients vary along the beam thickness according to the power law. The nonlinear system of equilibrium equations is solved numerically by the Newton–Raphson iterative technique. Full integration scheme and division in load steps are employed to obtain accuracy and stability, respectively. Four illustrative examples involving highly deformable elastic beams under plane stress and static conditions are analyzed: cantilever beam under free-end shear force, semicircular cantilever beam, column under buckling and shallow thin arch. The effects of the mesh refinement, the FG power coefficient and the transverse enrichment scheme on the beam behavior are investigated. In general, mesh refinement provides more accurate results, the power coefficient has a more significant effect on displacements and the second enrichment rate is needed to correctly predict the nonlinear variation of transverse strains across the thickness. This nonlinear variation, together with the moderate values of strains and stresses achieved, reinforces the need of adopting a nonlinear hyperelastic model. Finally, the determination of the out-of-plane strain from the plane components is solved numerically by a proposed Newton’s method.

Appendices

Appendix A: Out-of-plane normal strain

The numerical calculation of the out-of-plane strain C₃₃ is described in the present section.

From expression (29), another nonlinear relationship involving strains can be obtained:

$$\begin{aligned} K{ \ln }\left[ {\sqrt {\left( {C_{11} C_{22} - C_{12}^{2} } \right)C_{33} } } \right] = \mu \left( {1 - C_{33} } \right) \hfill \\ \quad \Rightarrow \left( {C_{11} C_{22} - C_{12}^{2} } \right)C_{33} = \left\{ {{ \exp }\left[ {\frac{\mu }{K}\left( {1 - C_{33} } \right)} \right]} \right\}^{2} . \hfill \\ \end{aligned}$$

(42)

Applying the Newton’s method to the above expression:

$$r = C_{33} \left( {C_{11} C_{22} - C_{12}^{2} } \right) - \left\{ {{ \exp }\left[ {\frac{\mu }{K}\left( {1 - C_{33} } \right)} \right]} \right\}^{2} \text{,}$$

(43)

$$\frac{{{\text{d}}r}}{{{\text{d}}C_{33} }} = \left( {C_{11} C_{22} - C_{12}^{2} } \right) + 2\frac{\mu }{K}\left\{ {{ \exp }\left[ {\frac{\mu }{K}\left( {1 - C_{33} } \right)} \right]} \right\}^{2} \text{,}$$

(44)

$$r + \frac{{{\text{d}}r}}{{{\text{d}}C_{33} }}\Delta C_{33} = 0 \Rightarrow \Delta C_{33} = - \left( {\frac{{{\text{d}}r}}{{{\text{d}}C_{33} }}} \right)^{ - 1} r\text{,}$$

(45)

The value of C₃₃ is updated until the residual r is sufficiently small. The first trial for this strain may be the unit value (i.e., the undeformed situation) or the incompressible value:

$$C_{33} = \frac{1}{{C_{11} C_{22} - C_{12}^{2} }} \Rightarrow J = 1\text{,}$$

(46)

Appendix B: Consistent tangent operator

Once all the strains are determined, the consistent tangent operator $\wp = {{\partial {\mathbf{S}}} \mathord{\left/ {\vphantom {{\partial {\mathbf{S}}} {\partial {\mathbf{C}}}}} \right. \kern-0pt} {\partial {\mathbf{C}}}}$ can be analytically obtained as follows.

According to expression (30), differentiation of the stress tensor S regarding the stretch tensor C results in:

$$\frac{{\partial {\mathbf{S}}}}{{\partial {\mathbf{C}}}} = - \mu \left( {{\mathbf{C}}^{ - 1} \otimes \frac{{\partial C_{33} }}{{\partial {\mathbf{C}}}} + C_{33} \frac{{\partial {\mathbf{C}}^{ - 1} }}{{\partial {\mathbf{C}}}}} \right)\text{,}$$

(47)

where the symbol $\otimes$ denotes the tensor product. One should remember that the partial derivatives in (47) are performed in respect to the plane strains C₁₁, C₁₂ and C₂₂.

From equation (42), the following scalar function can be defined:

$$fC_{33} = \left\{ {{ \exp }\left[ {\frac{\mu }{K}\left( {1 - C_{33} } \right)} \right]} \right\}^{2} \Rightarrow f = C_{11} C_{22} - C_{12}^{2} .$$

(48)

A variation in any one of the plane strains causes a variation in the function f, which in turn causes a variation in the out-of-plane strain C₃₃. Thus, the derivatives of C₃₃ regarding the plane strains can be determined as follows:

$$\frac{{\partial C_{33} }}{{\partial {\mathbf{C}}}} = \frac{{\partial C_{33} }}{\partial f}\frac{\partial f}{{\partial {\mathbf{C}}}} = \left[ {\frac{\partial f}{{\partial C_{33} }}} \right]^{ - 1} \frac{\partial f}{{\partial {\mathbf{C}}}} = {{\frac{\partial f}{{\partial {\mathbf{C}}}}} \mathord{\left/ {\vphantom {{\frac{\partial f}{{\partial {\mathbf{C}}}}} {\frac{\partial f}{{\partial C_{33} }}}}} \right. \kern-0pt} {\frac{\partial f}{{\partial C_{33} }}}}\text{,}$$

(49)

$$\frac{\partial f}{{\partial {\mathbf{C}}}} = \left[ {\begin{array}{*{20}c} {C_{22} } & { - 2C_{12} } \\ { - 2C_{12} } & {C_{11} } \\ \end{array} } \right].$$

(50)

Considering only the plane strains and, thus, a stretch tensor C represented by a square matrix 2 × 2, the inverse of C is:

$${\mathbf{C}}^{ - 1} = \frac{1}{{J^{2} }}\left[ {\begin{array}{*{20}c} {C_{22} } & { - 2C_{12} } \\ { - 2C_{12} } & {C_{11} } \\ \end{array} } \right] = \frac{1}{{J^{2} }}{\mathbf{C}}_{\text{aux}} \text{,}$$

(51)

where ${\mathbf{C}}_{\text{aux}}$ is an auxiliary 2 × 2 matrix. Consequently, the derivative of ${\mathbf{C}}^{ - 1}$ in respect to C is:

$$\frac{{\partial {\mathbf{C}}^{ - 1} }}{{\partial {\mathbf{C}}}} = \frac{\partial }{{\partial {\mathbf{C}}}}\left( {\frac{1}{{J^{2} }}{\mathbf{C}}_{\text{aux}} } \right) = {\mathbf{C}}_{\text{aux}} \otimes \frac{{\partial \left( {J^{ - 2} } \right)}}{{\partial {\mathbf{C}}}} + \frac{1}{{J^{2} }}\frac{{\partial {\mathbf{C}}_{\text{aux}} }}{{\partial {\mathbf{C}}}}\text{,}$$

(52)

$$\frac{{\partial \left( {J^{ - 2} } \right)}}{{\partial {\mathbf{C}}}} = - \frac{1}{{\left( {J^{2} } \right)^{2} }}\frac{{\partial \left( {J^{2} } \right)}}{{\partial {\mathbf{C}}}} = - \frac{1}{{\left( {J^{2} } \right)^{2} }}\left[ {\frac{\partial f}{{\partial {\mathbf{C}}}}C_{33} + f\frac{{\partial C_{33} }}{{\partial {\mathbf{C}}}}} \right]\text{,}$$

(53)

$$\frac{{\partial \left( {C_{\text{aux}} } \right)_{11} }}{{\partial C_{22} }} = \frac{{\partial \left( {C_{\text{aux}} } \right)_{22} }}{{\partial C_{11} }} = 1,\frac{{\partial \left( {C_{\text{aux}} } \right)_{12} }}{{\partial C_{12} }} = \frac{{\partial \left( {C_{\text{aux}} } \right)_{21} }}{{\partial C_{12} }} = - 1.$$

(54)