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Minimization of Shear Energy in Two Dimensional Continua with Two Orthogonal Families of Inextensible Fibers: The Case of Standard Bias Extension Test

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Abstract

In this paper we consider Pipkin-type bi-dimensional continua with two orthogonal families of inextensible fibers. We generalize the representation formula due to Rivlin (J. Ration. Mech. Anal. 4(6):951–974, 1955) valid for planar placement fields. They are the sum of two vector functions each of which depends on one real variable only, are piece-wise \(\mathscr{C}^{2}\) and may exhibit jumps of their first gradients on some inextensible fibers. Subsequently we consider a deformation energy depending only on the shear deformation relative to the inextensible families of fibers. In the suitably introduced space of configurations representing considered constrained kinematics, we formulate the relative energy minimization problem for the standard bias extension test problem, i.e., elongation of specimens which (i) have the shape of a rectangle with one side exactly three times longer than the other; (ii) are subject to a relative displacement of shorter sides in the direction parallel to the longer one. By exploiting the material and geometric symmetries, we reduce the aforementioned minimization problem to the determination of a piece-wise real function defined in a real interval. A delicate calculation of the energy first variation produces a necessary stationarity condition: it consists of an integral equation which is to be satisfied by the unknown function. The crucial points of this deduction are represented by (i) the reduction of two-dimensional integrals to one-dimensional integrals by the Fubini Theorem and (ii) the determination of the set of admissible kinematical functions on the basis of imposed boundary conditions, which implies a further integral constraint condition on the unknown function. Therefore the energy minimization problem requires the introduction of a global Lagrange multiplier. The established integral equations are solved numerically with a scheme based on a contraction type iterative process. Finally, the equilibrium shapes of the specimen undergoing large deformations, as determined by the presented model, are shown and briefly discussed.

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Notes

  1. One could also apply the previous proposition to the function \(\mathbf{r}^{ (3,3 )} (\xi_{1},\xi _{2} )-\mathbf{u}_{0}\).

  2. Note that this development is not possible in the neighborhood of the reference configuration.

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Acknowledgements

The authors must thank the Università di Roma La Sapienza and the MEMOCS Research Centre for their support. The fruitful scientific discussions with P. Seppecher, C. Boutin, A. Luongo, U. Andreaus, M. Cuomo, A. Cazzani, L. Lozzi and E. Turco greatly influenced the investigations which were presented in this paper.

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Correspondence to A. Madeo.

Appendix

Appendix

In this appendix we supply some more details about the derivation of the Euler-Lagrange conditions used in the paper.

1.1 6.1 Deformation Energy Functional

The functional whose first variation has to be calculated is:

$$\begin{aligned} \mathscr{A}_{W} \bigl(\mu (\cdot ) \bigr) =&\int _{B}W\, dm+\varLambda C (\mu_{,\xi} ) \\ =&\int _{B} \biggl[\frac{1}{2} \bigl( \bigl( \mu_{,\xi} (\xi_{1} ) \bigr)^{2} \bigl(1- \bigl( \mu_{,\xi} (\xi_{2} ) \bigr)^{2} \bigr)+ \bigl( \mu_{,\xi} (\xi_{2} ) \bigr)^{2} \bigl(1- \bigl( \mu_{,\xi} (\xi_{1} ) \bigr)^{2} \bigr) \\ &{}+2\mu_{,\xi} (\xi_{1} )\mu_{,\xi} (\xi _{2} )\sqrt{ \bigl(1- \bigl(\mu_{,\xi} (\xi_{1} ) \bigr)^{2} \bigr) \bigl(1- \bigl(\mu_{,\xi} (\xi_{2} ) \bigr)^{2} \bigr)} \bigr) \biggr]\, dm \\ &{}+ \biggl(\mu (H-1 )+\int_{1}^{3}\sqrt{1- (\mu_{,\xi } )^{2}}d\eta-3-u_{0} \biggr)\varLambda. \end{aligned}$$
(57)

Remark

The reader should note that this functional maps a real function \(\mu\) defined in the interval \([1,3 ]\) into a real number. However the function \(\mu\) is calculated in some occurrences in the variable \(\xi_{1}\) and in some other occurrences in the variable \(\xi_{2}\) when forming the integrand function appearing inside the square brackets of the previous equation. This integrand is indeed a function of the two variables \((\xi_{1},\xi_{2} )\) and its domain of integration is the two-dimensional domain \(B\).

Considering the partition (17) of the domain B and defining \(\underline{\Delta}=\Delta_{11}\cup\Delta_{21}\cup\Delta_{22}\cup\Delta_{21}\), it is possible to rewrite the functional as follows:

$$\begin{aligned} \mathscr{A}_{W} =&\int_{\Delta_{01}\cup\Delta_{32}}\frac{1}{2} \bigl(1- \bigl(\mu_{,\xi} (\xi_{2} ) \bigr)^{2} \bigr)dm+\int_{\Delta _{10}\cup\Delta_{23}}\frac{1}{2} \bigl(1- \bigl( \mu_{,\xi} (\xi _{1} ) \bigr)^{2} \bigr)dm \\ &{}+\int_{\underline{\Delta}}\frac{1}{2} \bigl[ \bigl( \mu_{,\xi} (\xi_{1} ) \bigr)^{2} \bigl(1- \bigl( \mu_{,\xi} (\xi_{2} ) \bigr)^{2} \bigr)+ \bigl( \mu_{,\xi} (\xi_{2} ) \bigr)^{2} \bigl(1- \bigl( \mu_{,\xi} (\xi_{1} ) \bigr)^{2} \bigr) \\ &{}+2\mu_{,\xi} (\xi_{1} )\mu_{,\xi} ( \xi_{2} )\sqrt{ \bigl(1- \bigl(\mu_{,\xi} (\xi_{1} ) \bigr)^{2} \bigr) \bigl(1- \bigl(\mu_{,\xi} (\xi_{2} ) \bigr)^{2} \bigr)} \bigr]\, dm \\ &{}+ \biggl(\int_{1}^{3} \Bigl( \mu_{,\xi} (\xi_{1} )+\sqrt {1- \bigl(\mu_{,\xi} ( \xi_{1} ) \bigr)^{2}} \Bigr)d\xi _{1}-2-u_{0} \biggr)\varLambda, \end{aligned}$$
(58)

which, by using the assumed symmetry of the solution, becomes

$$\begin{aligned} \mathscr{A}_{W} =&\int_{\Delta_{01}\cup\Delta_{32}}\frac{1}{2} \bigl(1- \bigl(\mu_{,\xi} (\xi_{2} ) \bigr)^{2} \bigr)dm+\int_{\Delta _{10}\cup\Delta_{23}}\frac{1}{2} \bigl(1- \bigl( \mu_{,\xi} (\xi _{1} ) \bigr)^{2} \bigr)dm \\ &{}+\int_{\underline{\Delta}} \bigl[ \bigl(\mu_{,\xi} (\xi _{1} ) \bigr)^{2} \bigl(1- \bigl(\mu_{,\xi} ( \xi_{2} ) \bigr)^{2} \bigr)+\mu_{,\xi} ( \xi_{1} )\mu_{,\xi} (\xi_{2} ) \\ &{}\times\sqrt{ \bigl(1- \bigl( \mu_{,\xi} (\xi_{1} ) \bigr)^{2} \bigr) \bigl(1- \bigl(\mu_{,\xi} (\xi_{2} ) \bigr)^{2} \bigr)} \bigr] \, dm \\ &{}+ \biggl(\int_{1}^{3} \bigl( \mu_{,\xi} (\xi_{1} )+\sqrt {1- \bigl(\mu_{,\xi} ( \xi_{1} ) \bigr)^{2}} \bigr)d\xi _{1}-2-u_{0} \biggr)\varLambda. \end{aligned}$$
(59)

1.2 6.2 Calculation of the First Variation of the Deformation Energy Functional

The first variation of the deformation energy functional can be expressed as:

$$ \delta\mathscr{A}_{W}=\int_{B}\delta W\, dm+ \varLambda\delta C (\mu _{,\xi} )+C (\mu_{,\xi} )\delta\varLambda. $$
(60)

We have

$$\begin{aligned} &\int_{B}\delta W\, dm+\varLambda\delta C ( \mu_{,\xi } ) \\ &\quad = -\int_{\Delta_{01}\cup\Delta_{32}}\mu_{,\xi} (\xi _{2} )\delta\mu_{,\xi} (\xi_{2} )dm-\int _{\Delta_{10}\cup \Delta_{23}}\mu_{,\xi} (\xi_{1} )\delta \mu_{,\xi} (\xi _{1} )dm \\ &\qquad{} +\int_{\underline{\Delta}} \biggl(2\mu_{,\xi} ( \xi_{1} ) \bigl(1- \bigl(\mu_{,\xi} (\xi_{2} ) \bigr)^{2} \bigr)+\frac {1-2 (\mu_{,\xi} (\xi_{1} ) )^{2}}{\sqrt{1- (\mu _{,\xi} (\xi_{1} ) )^{2}}}\mu_{,\xi} ( \xi_{2} )\sqrt{1- \bigl(\mu_{,\xi} (\xi_{2} ) \bigr)^{2}} \\ &\qquad {}+ \biggl(1-\frac{\mu_{,\xi} (\xi_{1} )}{\sqrt{1- (\mu _{,\xi} (\xi_{1} ) )^{2}}} \biggr)\varLambda \biggr)\delta\mu _{,\xi} (\xi_{1} ) \\ &\qquad {}+\int_{\underline{\Delta}} \biggl(-2\mu_{,\xi} ( \xi_{2} ) \bigl( \bigl(\mu_{,\xi} (\xi_{1} ) \bigr)^{2} \bigr)+\frac {1-2 (\mu_{,\xi} (\xi_{2} ) )^{2}}{\sqrt{1- (\mu _{,\xi} (\xi_{2} ) )^{2}}}\mu_{,\xi} ( \xi_{1} ) \\ &\qquad {}\times\sqrt{1- \bigl(\mu_{,\xi} (\xi_{1} ) \bigr)^{2}} \biggr)\delta\mu _{,\xi} (\xi_{2} )dm. \end{aligned}$$
(61)

Due to the symmetry condition, we have that

$$\begin{aligned} &\int_{\underline{\Delta}} \biggl(-2\mu_{,\xi} (\xi_{2} ) \bigl( \bigl(\mu_{,\xi} (\xi_{1} ) \bigr)^{2} \bigr)+ \frac{1-2 (\mu_{,\xi} (\xi_{2} ) )^{2}}{\sqrt{1- (\mu_{,\xi } (\xi_{2} ) )^{2}}}\mu_{,\xi} (\xi_{1} )\sqrt {1- \bigl( \mu_{,\xi} (\xi_{1} ) \bigr)^{2}} \biggr)\delta \mu_{,\xi } (\xi_{2} )dm \\ &\quad =\int_{\underline{\Delta}} \biggl(-2\mu_{,\xi} (\xi_{1} ) \bigl( \bigl(\mu_{,\xi} (\xi_{2} ) \bigr)^{2} \bigr)+ \frac{1-2 (\mu_{,\xi} (\xi_{1} ) )^{2}}{\sqrt{1- (\mu_{,\xi } (\xi_{1} ) )^{2}}}\mu_{,\xi} (\xi_{2} )\sqrt {1- \bigl( \mu_{,\xi} (\xi_{2} ) \bigr)^{2}} \biggr)\delta \mu_{,\xi } (\xi_{1} )dm, \end{aligned}$$
(62)

and

$$ \int_{\Delta_{01}\cup\Delta_{32}}\mu_{,\xi} (\xi_{2} )\delta\mu _{,\xi} (\xi_{2} )dm=\int_{\Delta_{10}\cup\Delta_{23}} \mu_{,\xi } (\xi_{1} )\delta\mu_{,\xi} ( \xi_{1} )dm. $$
(63)

So, replacing (62) and (63) in (61), we obtain:

$$\begin{aligned} &\int_{B}\delta W\, dm+\varLambda\delta C ( \mu_{,\xi } ) \\ &\quad =-2\int_{\Delta_{10}\cup\Delta_{23}}\mu_{,\xi} (\xi _{1} )\delta\mu_{,\xi} (\xi_{1} )dm +\int _{\underline {\Delta}} \biggl(2\mu_{,\xi} (\xi_{1} )-4 \mu_{,\xi} (\xi _{1} ) \bigl(\mu_{,\xi} ( \xi_{2} ) \bigr)^{2} \\ &\qquad {}+2\frac{1-2 (\mu_{,\xi} (\xi_{1} ) )^{2}}{\sqrt{1- (\mu_{,\xi} (\xi_{1} ) )^{2}}}\mu _{,\xi} (\xi_{2} )\sqrt{1- \bigl(\mu_{,\xi} (\xi_{2} ) \bigr)^{2}} + \biggl(1- \frac{\mu_{,\xi} (\xi_{1} )}{\sqrt {1- (\mu_{,\xi} (\xi_{1} ) )^{2}}} \biggr)\varLambda \biggr)\delta\mu_{,\xi} ( \xi_{1} ). \end{aligned}$$
(64)

Now, applying the Fubini Theorem we find:

$$\begin{aligned} &\int_{B}\delta W\, dm+ \varLambda\delta C (\mu_{,\xi} ) \\ &\quad =\int_{1}^{2} \biggl[4\mu (\xi_{1} ) \biggl(-1+\int_{1}^{1+\xi_{1}} \bigl(1- \bigl(\mu_{,\xi} (\xi_{2} ) \bigr)^{2} \bigr)d\xi_{2} \biggr) \\ &\qquad{}+2\frac{1-2 (\mu_{,\xi} (\xi_{1} ) )^{2}}{\sqrt{1- (\mu_{,\xi} (\xi_{1} ) )^{2}}}\int_{1}^{1+\xi_{1}} \mu_{,\xi} (\xi_{2} )\sqrt{1- \bigl(\mu_{,\xi } ( \xi_{2} ) \bigr)^{2}}d\xi_{2} \\ &\qquad{}+ \biggl(1-\frac{\mu_{,\xi} (\xi_{1} )}{\sqrt{1- (\mu _{,\xi} (\xi_{1} ) )^{2}}} \biggr)\varLambda \biggr]\delta\mu _{,\xi} (\xi_{1} )\, d\xi_{1} \\ &\qquad {}+\int _{2}^{3} \biggl[4\mu_{,\xi } ( \xi_{1} ) \biggl(-1+\int_{\xi_{1}-1}^{3} \bigl(1- \bigl(\mu _{,\xi} (\xi_{2} ) \bigr)^{2} \bigr)d\xi_{2} \biggr) \\ & \qquad{}+2\frac{1-2 (\mu_{,\xi} (\xi_{1} ) )^{2}}{\sqrt{1- (\mu_{,\xi} (\xi_{1} ) )^{2}}}\int_{\xi_{1}-1}^{3} \mu_{,\xi} (\xi_{2} )\sqrt{1- \bigl(\mu_{,\xi } ( \xi_{2} ) \bigr)^{2}}d\xi_{2} \\ &\qquad {}+ \biggl(1- \frac{\mu_{,\xi} (\xi_{1} )}{\sqrt{1- (\mu_{,\xi} (\xi_{1} ) )^{2}}} \biggr)\varLambda \biggr]\delta\mu_{,\xi} ( \xi_{1} )\, d\xi _{1}. \end{aligned}$$
(65)

Renaming the variables by means of the equalities \(\xi_{2}=\eta\) and \(\xi_{1}=\xi\), we are lead to define two functions \(F_{i}\) in the following way:

$$\begin{aligned} F_{1} (\mu_{,\xi},\xi,\varLambda ) =&\frac{1-2 (\mu_{,\xi } (\xi ) )^{2}}{2\sqrt{1- (\mu_{,\xi} (\xi ) )^{2}}}\ \underbrace{\int_{1}^{1+\xi}\mu_{,\xi} (\eta ) \sqrt{1- \bigl(\mu_{,\xi} (\eta ) \bigr)^{2}}d \eta}_{a_{1}(\xi )}-\mu (\xi ) \\ &{}\times \underbrace{ \biggl(1-\int_{1}^{1+\xi} \bigl(1- \bigl(\mu_{,\xi} (\eta ) \bigr)^{2} \bigr)d\eta \biggr)}_{d_{1}(\xi)} + \biggl(1-\frac{\mu_{,\xi} (\xi )}{\sqrt{1- (\mu _{,\xi} (\xi ) )^{2}}} \biggr)\varLambda, \\ F_{2} (\mu_{,\xi},\xi,\varLambda ) =&\frac{1-2 (\mu_{,\xi } (\xi ) )^{2}}{2\sqrt{1- (\mu_{,\xi} (\xi ) )^{2}}}\ \underbrace{\int_{\xi-1}^{3}\mu_{,\xi} (\eta ) \sqrt{1- (\mu_{,\xi}(\eta )^{2}}d\eta}_{a_{2}(\xi)}- \mu_{,\xi } (\xi ) \\ &{}\times \underbrace{ \biggl(1-\int_{\xi-1}^{3} \bigl(1- \bigl(\mu _{,\xi} (\eta ) \bigr)^{2} \bigr)d\eta \biggr)}_{d_{2}(\xi )} + \biggl(1-\frac{\mu_{,\xi} (\xi )}{\sqrt{1- (\mu _{,\xi} (\xi ) )^{2}}} \biggr)\varLambda. \end{aligned}$$
(66)

Therefore the first variation of the deformation energy can be represented as follows:

$$ \delta\mathscr{A}_{W}=4\int_{1}^{2}F_{1}( \mu_{,\xi},\xi,\varLambda)\delta \mu_{,\xi}d\xi+4\int _{2}^{3}F_{2}(\mu_{,\xi},\xi, \varLambda)\delta\mu_{,\xi }d\xi+C(\mu_{,\xi})\delta\varLambda. $$
(67)

1.3 6.3 Integration by Parts and the Final Expression for the Euler-Lagrange Stationary Condition

Applying an integration by parts, we find that it is possible to rewrite the variation of the deformation energy functional as follows:

$$\begin{aligned} \delta\mathscr{A}_{W} =&4\int_{1}^{2}F_{1}( \mu_{,\xi},\xi,\varLambda )\delta\mu_{,\xi}d\xi+4\int _{2}^{3}F_{2}(\mu_{,\xi},\xi, \varLambda)\delta \mu_{,\xi}d\xi+C(\mu_{,\xi})\delta\varLambda \\ =&4 \bigl[F_{1}(\mu_{,\xi},\xi,\varLambda)\delta\mu \bigr]_{1}^{2}+4 \bigl[F_{2}(\mu_{,\xi}, \xi,\varLambda)\delta\mu \bigr]_{2}^{3}-4\int _{1}^{2}\frac{\partial F_{1}(\mu_{,\xi},\xi,\varLambda)}{\partial\xi}\delta \mu\, d\xi \\ &{}-4\int _{2}^{3}\frac{\partial F_{2}(\mu_{,\xi},\xi,\varLambda )}{\partial\xi}\delta\mu\, d\xi+C( \mu_{,\xi})\delta\varLambda. \end{aligned}$$
(68)

Because of the arbitrariness of the variation \(\delta\mu\), we get that:

$$ \textstyle\begin{cases} {\displaystyle\frac{\partial F_{i}(\mu_{,\xi},\xi,\varLambda)}{\partial\xi }}=0 & \forall\xi\in I_{i}\quad \implies\quad F_{i}(\mu_{,\xi},\xi,\varLambda)=K_{i}\\ F_{2}(\mu_{,\xi},3,\varLambda)=0 & \implies\quad K_{2}=0\\ F_{1}(\mu_{,\xi},2,\varLambda)=F_{2}(\mu_{,\xi},2,\varLambda) & \implies\quad K_{1}=K_{2}=0\\ C(\mu_{,\xi})=0. \end{cases} $$
(69)

Therefore, the stationarity condition is given by the system of equations:

$$ \textstyle\begin{cases} F_{i}(\mu_{,\xi},\xi,\varLambda)=0 & \forall\xi\in I_{i}\\ C(\mu_{,\xi})=0 \end{cases} $$
(70)

with \(I_{1}= [1,2 ]\), \(I_{2}= [2,3 ]\).

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dell’Isola, F., d’Agostino, M.V., Madeo, A. et al. Minimization of Shear Energy in Two Dimensional Continua with Two Orthogonal Families of Inextensible Fibers: The Case of Standard Bias Extension Test. J Elast 122, 131–155 (2016). https://doi.org/10.1007/s10659-015-9536-3

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