Problems for Chapter 3

Constitutive Models: Relations between Stress and Strain

3.5.  Hyperelasticity

3.5.1.      Derive the stress-strain relations for an incompressible, Neo-Hookean material subjected to

3.5.1.1.            Uniaxial tension

3.5.1.2.            Equibiaxial tension

3.5.1.3.            Pure shear

Derive expressions for the Cauchy stress, the Nominal stress, and the Material stress tensors (the solutions for nominal stress are listed in the table in Section 3.5.6).  You should use the following procedure: (i) assume that the specimen experiences the length changes listed in 3.5.6; (ii) use the formulas in Section 3.5.5 to compute the Cauchy stress, leaving the hydrostatic part of the stress p as an unknown; (iii) Determine the hydrostatic stress from the boundary conditions (e.g. for uniaxial tensile parallel to ${e}_{1}$ you know ${\sigma }_{22}={\sigma }_{33}=0$; for equibiaxial tension or pure shear in the ${e}_{1},{e}_{2}$ plane you know that ${\sigma }_{33}=0$ )

3.5.2.      Repeat problem 3.5.1 for an incompressible Mooney-Rivlin material.

3.5.3.      Repeat problem 3.5.1 for an incompressible Arruda-Boyce material

3.5.4.      Repeat problem 3.5.1 for an incompressible Ogden material.

3.5.5.      Using the results listed in the table in Section 3.5.6, and the material properties listed in Section 3.5.7, plot graphs showing the nominal stress as a function of stretch ratio $\lambda$ for each of (a) a Neo-Hookean material; (b) a Mooney-Rivlin material; (c) the Arruda-Boyce material and (c) the Ogden material when subjected to uniaxial tension, biaxial tension, and pure shear (for the latter case, plot the largest tensile stress ${S}_{1}$ ).

3.5.6.      A foam specimen is idealized as an Ogden-Storakers foam with strain energy density

$\stackrel{˜}{U}=\frac{2\mu }{{\alpha }_{}^{2}}\left({\lambda }_{1}^{\alpha }+{\lambda }_{2}^{\alpha }+{\lambda }_{3}^{\alpha }-3+\frac{1}{\beta }\left({J}^{-\alpha \beta }-1\right)\right)$

where $\mu ,\alpha$ and $\beta$ are material properties.  Calculate:

3.5.6.1.            The Cauchy stress in a specimen subjected to a pure volume change with principal stretches ${\lambda }_{1}={\lambda }_{2}={\lambda }_{3}=\lambda$

3.5.6.2.            The Cauchy stress in a specimen subjected to volume preserving uniaxial extension

3.5.6.3.            The Cauchy stress in a specimen subjected to uniaxial tension, as a function of the tensile stretch ratio ${\lambda }_{1}$. (To solve this problem you will need to assume that the solid is subjected to principal stretches ${\lambda }_{1}$ parallel to ${e}_{1}$, and stretches ${\lambda }_{2}$ parallel to ${e}_{2}$ and ${e}_{3}$.  You will need to determine ${\lambda }_{2}$ from the condition that ${\sigma }_{22}={\sigma }_{33}=0$ in a uniaxial tensile test.

3.5.7.      Suppose that a hyperelastic solid is characterized by a strain energy density $\overline{U}\left({\overline{I}}_{1},{\overline{I}}_{2},J\right)$ where

${\overline{I}}_{1}=\frac{{B}_{kk}}{{J}^{2/3}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\overline{I}}_{2}=\frac{1}{2}\left({\overline{I}}_{1}^{2}-\frac{{B}_{ik}{B}_{ki}}{{J}^{4/3}}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}J=\sqrt{\mathrm{det}B}$

are invariants of the Left Cauchy-Green deformation tensor ${B}_{ij}={F}_{ik}{F}_{jk}$Suppose that the solid is subjected to an infinitesimal strain, so that B can be approximated as ${B}_{ij}\approx {\delta }_{ij}+{\epsilon }_{ij}$, where ${\epsilon }_{ij}$ is a symmetric infinitesimal strain tensor.   Linearize the constitutive equations for ${\epsilon }_{ij}<<1$, and show that the relationship between Cauchy stress ${\sigma }_{ij}$ and infinitesimal strain ${\epsilon }_{ij}$ is equivalent to the isotropic linear elastic constitutive equation.   Give formulas for the bulk modulus and shear modulus for the equivalent solid in terms of the derivatives of $\overline{U}$.

3.5.8.      The constitutive law for a hyperelastic solid is derived from a strain energy potential $U\left({I}_{1},{I}_{2},{I}_{3}\right)$, where

${I}_{1}=\text{trace}\left(B\right)={B}_{kk}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{I}_{2}=\frac{1}{2}\left({I}_{1}^{2}-B\cdot \cdot B\right)=\frac{1}{2}\left({I}_{1}^{2}-{B}_{ik}{B}_{ki}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{I}_{3}=\mathrm{det}B={J}^{2}$

are the invariants of the Left Cauchy-Green deformation tensor $B=F\cdot {F}^{T}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{B}_{ij}={F}_{ik}{F}_{jk}$.

3.5.8.1.            Calculate the Cauchy stress induced in the solid when it is subjected to a rigid rotation, followed by an arbitrary homogeneous deformation.   Hence, demonstrate that the constitutive law is isotropic.

3.5.8.2.            Apply the simple check described in Section 3.1 to test whether the constitutive law is objective.

3.5.9.      The strain energy density of a hyperelastic solid is sometimes specified as a function of the right Cauchy-Green deformation tensor ${C}_{ij}={F}_{ki}{F}_{kj}$, instead of  ${B}_{ij}$ as described in Section 3.5.  (This procedure must be used if the material is anisotropic, for example)

3.5.9.1.            Suppose that the strain energy density has the general form $W\left({C}_{ij}\right)$.  Derive formulas for the Material stress, Nominal stress and Cauchy stress in the solid as functions of ${F}_{ij}$, ${C}_{ij}$ and $\partial W/\partial {C}_{ij}$

3.5.9.2.            Apply the simple check described in Section 3.1 to demonstrate that the resulting stress-strain relation is objective.

3.5.9.3.            Calculate the Cauchy stress induced in the solid when it is subjected to a rigid rotation, followed by an arbitrary homogeneous deformation.   Hence, demonstrate that the constitutive law is not, in general, isotropic.

3.5.9.4.            Suppose that the constitutive law is simplified further by writing the strain energy density as a function of the invariants of C, i.e.  $W\left({C}_{ij}\right)=U\left({I}_{1},{I}_{2},{I}_{3}\right)$, where

${I}_{1}={C}_{kk}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{I}_{2}=\frac{1}{2}\left({I}_{1}^{2}-{C}_{ik}{C}_{ki}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{I}_{3}=\mathrm{det}C={J}^{2}$

Derive expressions relating the Cauchy stress components to $\partial U/\partial {I}_{j}$.

3.5.9.5.            Demonstrate that the simplified constitutive law described in 3.5.9.4 characterizes an isotropic solid.