Computational Modeling of Fiber Composites with Thick Fibers as Homogeneous Structures with Use of Couple Stress Theory

Abstract

Unidirectional fiber-reinforced elastomers are investigated. A respective finite strain model is formulated within the couple stress theory, and a specific new form of strain energy density is implemented for the three-dimensional finite element analysis. The homogeneous anisotropic model is based on kinematics and constitutive equations proposed by Spencer and Soldatos (International Journal of Non-Linear Mechanics 42:355–368, 2007) and includes additional material parameter regulating bending stiffness of the material regardless of its tensile stiffness. The procedure of determination of the additional material parameter is offered for the case of simple beam under small strains. Numerical simulations of four-point bending test are presented to demonstrate advantage of the new model.

Appendix

The material parameter k ₂ can be determined under condition that the tensile force in the direction of fiber remains the same in both the homogenous and heterogeneous models under uniaxial tension ${P}^{\mathrm{het}}={P}^{\hom }$.

Consider $\mathbf{A}=\left(\begin{array}{c}\hfill \mathbf{1}\hfill \\ {}\hfill \mathbf{0}\hfill \\ {}\hfill \mathbf{0}\hfill \end{array}\right)$ and uniaxial tension in the direction X ₁. We consider small tensile strains in the fiber direction; therefore linearized constitutive equations might be used. ${\sigma}_{ij}=2{F}_{ik}{F}_{jl}\frac{\partial W}{\partial {C}_{ikl}}$ is general constitutive relation for incompressible hyperelastic material (hydrostatic pressure is absent). The strain energy density is given by $W=\frac{\mu }{2}\left({I}_1-3\right)+{k}_2{\left({I}_4-1\right)}^2$. The right Cauchy–Green deformation tensor is $C=\left(\begin{array}{ccc}\hfill {\lambda}_1^2\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill {\lambda}_2^2\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill {\lambda}_3^2\hfill \end{array}\right),$ and incompressibility condition holds ${\lambda}_1{\lambda}_2{\lambda}_3=1$, where λ ₁, λ ₂, λ ₃ are principal stretch ratios.

Consequently, we can write that

$$ {\sigma}_{11}^{\hom }=\mu \left(2{\lambda}^2+\frac{1}{\lambda}\right)+4{k}_2\left({\lambda}^4-{\lambda}^2\right) $$

(74)

where $\uplambda ={\uplambda}_1$ is the stretch ratio in the direction of fiber.

For the small strains, it takes form

$$ {\sigma}_{11}^{\hom }=\left(3\mu +8{k}_2\right)\cdot {\varepsilon}_{11}={E_1}^{\hom}\cdot {\varepsilon}_{11}. $$

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The tensile force acting on the section will be

$$ {P}^{\hom }={\displaystyle {\int}_S{\sigma}_{11}^{\hom }}\mathrm{d}S=S\cdot {\sigma}_{11}^{\hom }=S\cdot \left(3\upmu +8{\mathrm{k}}_2\right)\cdot {\varepsilon}_{11}. $$

(76)

As to the heterogeneous model, the tensile force in linearized case is as follows:

$$ {P}^{\mathrm{het}}={\displaystyle {\int}_S{\sigma}_{11}^{\mathrm{het}}\mathrm{d}S}=S\cdot \left(\left(1-{\psi}_{\mathrm{f}}\right)\cdot {E}_{\mathrm{m}}+{\psi}_{\mathrm{f}}\cdot {E}_{\mathrm{f}}\right)\cdot {\varepsilon}_{11}, $$

(77)

where ψ _f is the fiber volume fraction and E _m and E _f are matrix and fiber moduli.

Consequently, we have condition

$$ \left(3\mu +8{k}_2\right)\cdot {\varepsilon}_{11}=\left(\left(1-{\psi}_{\mathrm{f}}\right)\cdot {E}_{\mathrm{m}}+{\psi}_{\mathrm{f}}\cdot {E}_{\mathrm{f}}\right)\cdot {\varepsilon}_{11}, $$

(78)

and so it follows that

$$ {k}_2=\frac{1}{8}\left(\left(1-{\psi}_f\right)\cdot {E}_m+{\psi}_f\cdot {E}_f-3\mu \right). $$

(79)