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Multiscale Science and Engineering

, Volume 1, Issue 2, pp 150–160 | Cite as

Strain-Gradient Elasticity Theory for the Mechanics of Fiber Composites Subjected to Finite Plane Deformations: Comprehensive Analysis

  • Chun Il KimEmail author
Original Research
  • 143 Downloads

Abstract

A general model for the mechanics of fiber-reinforced composites is delivered in finite-plane elastostatics. Within the framework of the nonlinear strain-gradient theory, we obtain the Euler equilibrium equations and admissible boundary conditions. In particular, we present a third gradient continuum model and associated formulations. Soft composite materials are also considered by employing the Mooney-Rivlin energy potential. The presented model can be used as an alternative two-dimensional Cosserat theory of nonlinear elasticity.

Keywords

Finite plane deformations Fiber-reinforced materials Extension Flexure Third gradient continuum model 

Introduction

The mechanics of fiber-reinforced solids is a well established subject [1, 2] that has significantly advanced our understanding of continua with distinct micro structures (microstructured continua). Most of conventional approaches presume the continuous distribution of fibers within the matrix materials (see, for example, [3, 4, 5] and references therein). Such idealization furnishes the advantage of the continuum setting and the associated mathematical framework via a homogenization process. Within this prescription, the composites can be regarded as the special class of anisotropic materials where their response functions depend on the first gradient of deformation (the deformation gradient tensor), typically augmented by the constraint of bulk incompressibility or fiber inextensibility. In the latter case, the resulting equilibrium equations turn out to be so constrained that the final deformations are almost entirely determined by their kinematic relations. As a result, the model has inevitable limitations in the predictions of the mechanics of fiber composites, particularly, those arise in fibers [6, 7]. Nevertheless, continuum-based models have been broadly adopted in the relevant subject of studies (see, for example, [7, 8]) for their relative simplicity and accessibility in the modeling and analysis.

Recently, a considerable advance in the continuum theory of fiber-reinforced solids was made by incorporating the microstructural effects of fibers, bending resistance of fibers in particular [9, 10, 11]. This includes the “refinement” of the classical continuum theory by means of the higher gradient of deformation to achieve the more realistic description of microstructured continua. The concept has been successfully implemented in a wide range of subjects arising from the materials science and biological engineering (see, for example, [12, 13, 14]). In the case of fiber composites, this means the incorporation of the bending resistance of fibers (and possibly extension) into the models of deformation in which the fibers’ bending resistance is assigned to the changes in curvature (flexure) of fibers explicitly [9]. Current applications of the theory are developed in [15, 16, 17, 18] and the mathematical aspects of the subject are discussed in [19, 20, 21]. A general theory for the mechanics of an elastic solid with fibers resistant to flexure, stretch and twist is presented in [10] within the scope of the constraint Cosserat theory of elasticity. Further, authors in [22, 23, 24] proposed a second-gradient theory of elasticity for the mechanics of meshed structures and further investigated the corresponding shearing properties when they are subjected to the plane bias-extension. To this end, authors in [25, 26, 27] developed the continuum models for the mechanics of fiber-reinforced composite where the bending resistance of fibers is incorporated via the computation of the second gradient of the deformation.

In this paper, we present a comprehensive continuum model for an elastic solid reinforced with fibers resistant to extension and flexure. This includes the constitutive formulations of the composites consisting of unidirectional fibers and/or bidirectional fibers, and also covers the cases of inextensible and extensible fibers. The fibers are modeled as continuously distributed spatial rods of Kirchhoff type such that the kinematics are defined by their position and director fields [28, 29, 30]. Emphasis is placed on consistency and clarity in the derivations of the Euler equilibrium equations and the admissible boundary conditions where tensorial notation and the associated components are often interchanged without distinction. In addition, integral theorems for high order tensors with a mixed basis (current and referential) are further exploited and reformulated within the current context to promote possible applications in the relevant subjects. We show, throughout rigorous derivations, that the bending resistance of fibers is accounted for the second gradient of the deformation, whereas fibers extension remains dependent on the first gradient of deformation.

More importantly, we present a third order gradient model in which the rate of the changes in curvature (flexure) of fibers is characterized by the third gradient of the continuum deformation. The energy variation with respect to the third gradient yields a forth order tensor which maps a third arclength derivative (third order tensor) into a first order tensor in the deformed configuration. A closely related physical phenomenon of this result is the independent rotation of material particles (see, for example, [31, 32]), and therefore the model can be used as an alternative 2D Cosserat theory of nonlinear elasticity. Lastly, a complete constitutive framework for soft composite materials is formulated by employing the Mooney-Rivlin energy potential. Via the variational computation of the first and the second invariants of the deformation gradient tensor, the expressions of stresses and Euler equilibrium equations are obtained in the form of systems of Partial Differential Equations. The resulting model can be easily adopted in the analysis of fiber-reinforced soft composites experiencing large extension and possible flexure [33].

Throughout the paper, we make use of a number of well-established symbols and conventions such as \({\mathbf {A}}^{T},\,{\mathbf {A}}^{-1},\ {\mathbf {A}}^{*} \) and \(tr({\mathbf {A}}).\) These are the transpose, the inverse, the cofactor and the trace of a tensor \({\mathbf {A}}\), respectively. The tensor product of vectors is indicated by interposing the symbol \(\otimes \),and the Euclidian inner product of tensors \({\mathbf {A}}\), \({\mathbf {B}}\) is defined by \({\mathbf {A\cdot B}}=tr({\mathbf {AB}}^{T})\); the associated norm is \(\ \left| {\mathbf {A}}\right| =\sqrt{{\mathbf {A\cdot A}}}\). The symbol \(\text { ``}{{\mathbf {\cdot }}}\text {'' }\) is used to denote the usual Euclidian norm of vectors. Latin and Greek indices take values in \(\{1,2\}\) and, when repeated, are summed over their ranges. Lastly, the notation \(F_{{{\mathbf {A}}}}\) stands for the tensor-valued derivatives of a scalar-valued function \(F({\mathbf {A}})\).

Unidirectional Fiber Composites with Fibers Resistant to Flexure

The mechanical responses of a matrix material can be described via the first gradient of deformation that
$$\begin{aligned} \frac{d{\mathbf {r}}({\mathbf {X)}}}{d{\mathbf {X}}}={\mathbf {F}}=F_{iA} \mathbf {e}_{\mathbf {i}}\otimes {\mathbf {E}}_{\mathbf {A}}, \end{aligned}$$
(1)
where, \({\mathbf {F}}\) is well known deformation gradient tensor with mixed bases of \(\{{\mathbf {e}}_{i}\}\) (current) and \(\{{\mathbf {E}}_{A}\}\) (referential), respectively. \({\mathbf {F}}\) offers a useful connection of fibers’ kinematics:
$$\begin{aligned} {\mathbf {d}}=\frac{d{\mathbf {r}}({\mathbf {X)}}}{d{\mathbf {X}}} \frac{d{\mathbf {X}}(S)}{dS}={\mathbf {FD}}, \end{aligned}$$
(2)
through which the unit tangent to the fibers trajectory (\({\mathbf {D}}\)) in the reference configuration is mapped to its counterpart (\({\mathbf {d}}\)) in the deformed configuration (i.e. \(d{\mathbf {e}}_{\mathbf {i}}=F_{iA}({\mathbf {e}}_{\mathbf {i}}\otimes {\mathbf {E}}_{A})D_{B}{{\mathbf {E}}_{\mathbf {B}}}= F_{iA}D_{B}\delta _{AB}{{\mathbf {e}}_{\mathbf {i}}}=F_{iA}D_{A}{{\mathbf {e}} _{\mathbf {i}},\ }\delta _{AB}:\) Kronecker delta.). Eq. (1) can be obtained by taking the derivative of \({\mathbf {r}}(S)\) with respect to the arclength (S) along a fiber in the reference configuration, upon making the identifications \(d{\mathbf {X}}/dS={\mathbf {D}}\). Alternatively, we compute
$$\begin{aligned} {\mathbf {d}}=\frac{d{\mathbf {r}}(S)}{dS}=\frac{d{\mathbf {r}}(S(s) )}{ds}\frac{ds}{dS}=\lambda {\mathbf {l}}\text {,} \end{aligned}$$
(3)
where s is the arclength parameter in the current configuration, \(\mathbf {l }=d{\mathbf {r}}/ds\) is the unit tangent to the fibers trajectory (i.e. \( {\mathbf {l\cdot l=1}}\)) and \(\lambda =\left| {\mathbf {d}}\right| \). Eq. (3) is particularly useful in the formulation of the fibers extension. In order to incorporate fibers resistant to flexure, it is required to compute the curvature on a material point as
$$\begin{aligned} {\mathbf {g}}=\frac{d^{2}{\mathbf {r}}(S)}{dS^{2}}=\frac{d(\frac{\mathbf { r(}S)}{dS})}{dS}=\frac{\partial ({\mathbf {FD)}}}{\partial {\mathbf {X}}} \frac{d{\mathbf {X}}}{dS}=\nabla [{\mathbf {FD]D}}. \end{aligned}$$
(4)
Without loss of generality, most fibers are initially undeformed and straight. Even slightly curved fibers can be idealized as ‘fairly straight’ fibers, considering their length scales with respect to those of matrix materials. Accordingly, the gradient of the unit tangent in the reference configuration vanishes identically (i.e. \(\nabla {\mathbf {D}}=0\)), and therefore Eq. (4) reduces to
$$\begin{aligned} {\mathbf {g}}=\nabla [\mathbf {FD}]\mathbf {D}=({\varvec{\nabla }} [\mathbf {F}]\mathbf {D})\mathbf {D}=(\mathbf {GD})\mathbf {D}. \end{aligned}$$
(5)
In the above, we introduce the convention of the second gradient of deformation as
$$\begin{aligned} {\varvec{\nabla }} \mathbf {F}=\mathbf {G},\text { third order tensor}:{\mathbf {G}}=G_{iAB}( {\mathbf {e}}_{\mathbf {i}}{\otimes \mathbf {E}}_{A}\otimes {\mathbf {E}}_{B}), \end{aligned}$$
(6)
and it’s compatibility condition is given by
$$\begin{aligned} G_{iAB}=F_{iA,B}=F_{iB,A}=G_{iBA}. \end{aligned}$$
(7)
From Eq. (6), it is not difficult to show
$$\begin{aligned}(\mathbf {GD})\mathbf {D}&=[G_{iAB}({\mathbf {e}}_{\mathbf {i}}\otimes {\mathbf {E}}_{A}{\otimes \mathbf {E}}_{B})D_{C}{\mathbf {E}}_{C}]D_{D} \\ {\mathbf {E}}_{D}&=G_{iAB}D_{A}D_{B}{\mathbf {e}}_{\mathbf {i}} =\mathbf {G}(\mathbf {D}\otimes \mathbf {D}) \end{aligned}. $$
(8)
Clearly, \({\mathbf {G}}\) maps (\({\mathbf {D\otimes D}}\)) into the current configuration and yields
$$\mathbf {G}(\mathbf {D}\otimes \mathbf {D})=G_{iAB}({\mathbf {e}}_{\mathbf {i}}\otimes {\mathbf {E}}_{A}{\otimes \mathbf {E}}_{B})D_{C}D_{D}( {\mathbf {E}}_{C}{\otimes \mathbf {E}}_{D})=G_{iAB}D_{A}D_{B} {\mathbf {e}}_{i}=g_{i}{\mathbf {e}}_{i}. $$
(9)
The results in Eqs. (2)–(9) imply that the desired energy density is required to be a function of both the first and the second gradient of deformation (i.e. \({\mathbf {F}}\) and \({\mathbf {G}}\)). In particular, it is proposed by [9] that the fibers’ bending energy is entirely governed by the second gradient of deformation as
$$\begin{aligned} W({\mathbf {F}},{\mathbf {G}})=W({\mathbf {F}})+W({\mathbf {G}}),\ \ W({\mathbf {G}})\equiv \frac{1}{2}C\left( {\mathbf {F}}\right) \left| {\mathbf {g}}\right| ^{2}. \end{aligned}$$
(10)
The concept has been widely and successfully adopted in the relevant studies (see, for example, [10, 11] and [21]). In practice, Eq. (10) is typically augmented by the constraint of bulk incompressibility to deliver
$$\begin{aligned} U({\mathbf {F}},{\mathbf {G}},\, p)=W({\mathbf {F}})+W({\mathbf {G}})-p(J-1), \end{aligned}$$
(11)
where \(J=\det ({\mathbf {F}})\) and p is a Lagrange multiplier field. Further, we note here that \(C({\mathbf {F}})\) which refers to the material property of fibers, is, in general, independent of the deformation gradient. Henceforth, it is assumed that
$$\begin{aligned} C({\mathbf {F)}}=C. \end{aligned}$$
(12)

Equilibrium and Boundary Conditions

Common approaches to obtain equilibrium equations and boundary conditions are utilizing the virtual work statement and variational principles;
$$\begin{aligned} \dot{E}=P, \end{aligned}$$
(13)
where P is the virtual power of the applied loads and the superposed dot refers to the variational and/or Gateâux derivative (i.e. \(\overset{ \cdot }{E}=\frac{dE}{d\varepsilon };\varepsilon \ll 1\).). To proceed, we obtain from Eq. (11) that
$$\begin{aligned} E=\int \limits _{\Omega }U\left( {\mathbf {F,G}},\,p\right) dA=\int \limits _{\Omega }[W({\mathbf {F}})+W({\mathbf {G}})-p(J-1)]dA, \end{aligned}$$
(14)
which is the energy density functional for the fiber composites. Since the conservative loads are characterized by the existence of a potential L such that \(P=\dot{E}\), in the present case, the problem of determining equilibrium deformations is reduced to the problem of minimizing the potential energy \(E-L\). Therefore, from Eq. (14), the induced variation (i.e. \((\dot{*})=\partial (*)/\partial \varepsilon \)) of E can be evaluated as
$$\begin{aligned} \dot{E}=\int \limits _{\Omega }(\dot{W}({\mathbf {F}})+\dot{W}({\mathbf {G}})-p\dot{J})dA. \end{aligned}$$
(15)
In order to compute variational derivatives of the response function, we use chain rule in the form
$$\begin{aligned} \dot{W}({\mathbf {F}})=W_{{\mathbf {F}}}{\mathbf {\cdot \dot{F}},\ }\dot{W}({\mathbf {G}} )=W_{{\mathbf {G}}}{\cdot \dot{\mathbf {G}}}\text { and }\dot{J}=J_{{\mathbf {F}}} {\cdot \dot{\mathbf {F}}}, \end{aligned}$$
(16)
where the subscripts denote corresponding partial derivatives;
$$\begin{aligned} W_{{\mathbf {F}}}=\frac{\partial W}{\partial {\mathbf {F}}}= W_{F_{iA}}({{\mathbf {e}}_{i}\otimes {\mathbf {E}}_{A}}),\, W_{{\mathbf {G}}} =\frac{\partial W}{\partial {\mathbf {G}}}=W_{F_{iAB}}( {\mathbf {e}}_{\mathbf {i}}{\otimes \mathbf {E}}_{A}\otimes {\mathbf {E}}_{B}),\text { and }J_{{\mathbf {F}}}=\frac{\partial J}{ \partial {\mathbf {F}}}={\mathbf {F}}^{*}=F_{iA}^{*} ({\mathbf {\mathbf {e}} _{i}\otimes {\mathbf {E}}_{A}}), \end{aligned}$$
(17)
respectively. While the first term of the above is explicitly determined upon the selection of the types of matrix materials (e.g. Neo-Hookean, Mooney-Rivlin etc..), the expression of the second term addressing the bending stiffness of fibers is not apparent. To observe this, we substitute Eq. (10)\(_{2}\) into (16) and thereby obtain
$$\begin{aligned} C\left( \frac{1}{2}\mathbf {g}\cdot \dot{\mathbf {g}}\right) =C{\mathbf {g}\cdot \dot{\mathbf {g}}}=W_{{\mathbf {G}}} {\cdot \dot{\mathbf {G}}.} \end{aligned}$$
(18)
Since \({\dot{\mathbf {g}}=\dot{\mathbf {G}}({\mathbf {D\otimes D}})}\) for initially straight fibers [see, Eq. (5)], we evaluate from the above as
$$\begin{aligned} {\mathbf {g}\cdot \dot{\mathbf {g}}}={\mathbf {g}\cdot \dot{\mathbf {G}}}({\mathbf {D\otimes D}})=g_{i}{\mathbf {e}} _{i}\cdot [\dot{G}_{jAB}({\mathbf {e}}_{j}\otimes {\mathbf {E}}_{A}{\otimes \mathbf {E}}_{B})D_{C}D_{D} ({\mathbf {E}}_{C}\otimes {\mathbf {E}}_{D})]. \end{aligned}$$
(19)
Now, the linear transformation of the high order tensor \({\dot{\mathbf {G}}}\) in Eq. (19)\(_{3}\) yields
$$\begin{aligned} g_{i}\delta _{ij}\dot{G}_{jAB}\delta _{AC}\delta _{BD}D_{C}D_{D}= g_{i}\dot{G}_{iAB}D_{A}D_{B}={\mathbf {g}\cdot \dot{\mathbf {g}}}. \end{aligned}$$
(20)
But from Eq. (18), \(g_{i}\dot{G}_{iAB}D_{A}D_{B}\) is then satisfy
$$\begin{aligned} Cg_{i}D_{A}D_{B}\dot{G}_{iAB}=W_{{\mathbf {G}}}{\cdot \dot{\mathbf {G}}}= W_{G_{iAB}}\dot{G}_{iAB}. \end{aligned}$$
(21)
Comparing both sides of Eq. (21), we find
$$\begin{aligned} W_{G_{iAB}}=Cg_{i}D_{A}D_{B}\text { or alternatively, }W_{{\mathbf {G}}}=C (\mathbf {g}\otimes \mathbf {D}\otimes \mathbf {D}). \end{aligned}$$
(22)
Equation (22) can also be obtained by utilizing the inner product of tensors as
$$\begin{aligned} C{\mathbf {g}\cdot \dot{\mathbf {G}}}({\mathbf {D\otimes D}})& = Ctr[{\dot{\mathbf {G}}}(\mathbf {D}\otimes \mathbf {D})\otimes \mathbf {g}]\\ &=Ctr[\dot{\mathbf {G}}(\mathbf {D}\otimes \mathbf {D} \otimes \mathbf {g})]={\dot{\mathbf {G}}}\cdot \mathbf {C}(\mathbf {D}\otimes \mathbf {D}\otimes \mathbf {g})^{T} \nonumber \\& = \dot{\mathbf {G}}\cdot C(\mathbf {g}\otimes D\otimes D)= W_{\mathbf {G}}\cdot \dot{\mathbf {G}}. \end{aligned}$$
(23)
Therefore, from Eqs. (15)–(16) and (23), the energy variation becomes
$$\begin{aligned} \dot{E}=\int \limits _{\Omega }(W_{{\mathbf {F}}}{\mathbf {\cdot \dot{F}}}+C(\mathbf {g}\otimes \mathbf {D}\otimes \mathbf {D})\cdot \dot{\mathbf {G}}-pJ_{{\mathbf {F}}}\cdot \dot{\mathbf {F}})dA. \end{aligned}$$
(24)
The above energy variation form is not directly applicable, since the orders of gradient fields are not coincide each other (i.e. \({\dot{\mathbf {F}}}\): first gradient field, \({\dot{\mathbf {G}}}\): second gradient field). To reduce the high order term, we write (see also, [34, 35, 36])
$$\begin{aligned} g_{i}D_{A}D_{B}\dot{F}_{iA,B}=\left( g_{i}D_{A}D_{B}\dot{F}_{iA}\right) _{,B}-\left( g_{i}D_{A}D_{B}\right) _{,B}\dot{F}_{iA}. \end{aligned}$$
(25)
The equivalent tensorial notation of the above can be found as
$$\begin{aligned} C(\mathbf {g\otimes D\otimes D})\cdot \nabla \dot{{\mathbf {F}}}=Div[C( {\mathbf {g\otimes D\otimes D}})^{T}{\dot{\mathbf {F}}}^{T}]-Div[C({\mathbf {g\otimes D\otimes D}})]{\cdot \dot{\mathbf {F}}}, \end{aligned}$$
(26)
which can be easily seen by applying the linear transform of high order tensors (e.g. \(Div[({\mathbf {g\otimes D\otimes D}})^{T}{\dot{\mathbf {F}} }^{T}]=Div(g_{i}D_{A}D_{B}\dot{F}_{iA}{\mathbf {E}}_{B})=\left( g_{i}D_{A}D_{B}\dot{F}_{iA}\right) _{,B}\)). Thus, combining Eqs. (24 )and (26) furnishes
$$\begin{aligned} \dot{E}=\int \limits _{\Omega }[W_{{\mathbf {F}}}-Div(C( {\mathbf {g\otimes D\otimes D}})-p{\mathbf {F}}^{*}]\cdot \dot{\mathbf {F}}dA+\int \limits _{\partial \Omega }[C({\mathbf {g\otimes D\otimes D}}) ^{T}{\dot{\mathbf {F}}}^{T}]{\mathbf {\cdot N}}dS, \end{aligned}$$
(27)
where \({\mathbf {N}}\) is the rightward unit normal to the boundary curve \(\partial \Omega \) in the sense of the Green–Stoke’s theorem. Consequently, we obtain
$$\begin{aligned} \dot{E}=\int \limits _{\Omega }{\mathbf {P}\cdot \dot{\mathbf {F}}}dA+\int \limits _{\partial \Omega }[C({\mathbf {g\otimes D\otimes D}})^{T}{\dot{\mathbf {F}}}^{T}] {\mathbf {\cdot N}}dS, \end{aligned}$$
(28)
and
$$\begin{aligned} \mathbf {P}=P_{iA}(\mathbf {e}_{i}\otimes \mathbf {E}_{A})=(W_{F_{iA}} -Cg_{i,B}D_{A}D_{B}-pF_{iA}^{*})({\mathbf {e}}_{i}{\otimes \mathbf {E}}_{A}). \end{aligned}$$
(29)
It is clear from the above that, the resulting stress field depends on both the first and the second gradient of deformation (i.e. \(W_{{\mathbf {F}}}=W_{ {\mathbf {F}}}({\mathbf {F}}),\,{\mathbf {g=g}}({\mathbf {G}})\)). In particular, the bending resistance of fibers is incorporated into the models of deformation via \(C({\mathbf {g\otimes D\otimes D}})\). The corresponding Euler equation then satisfy
$$\begin{aligned} Div({\mathbf {P}})=0\text { or }P_{iA,A}{\mathbf {e}}_{i}=0, \end{aligned}$$
(30)
which hold on \(\Omega \). By decomposing \({\mathbf {P}}\cdot \dot{\mathbf {F}}\) as in Eqs. (25) and (26) (i.e. \({\mathbf {P}}\cdot \dot{\mathbf {F}}= {\mathbf {P}}\cdot \frac{\partial {\dot{{\varvec{\chi }}}}}{\partial {\mathbf {X}}} =Div({\mathbf {P}}^{T}{\dot{{\varvec{\chi }}}})-Div({\mathbf {P}}) {\dot{{\varvec{\chi }}}}\)), Eq. (28) may be rearranged as
$$\begin{aligned} \dot{E}=\int \limits _{\partial \Omega }{\mathbf {\mathbf {P}}}^{T}{\dot{\varvec{\chi }} \cdot N}dS-\int \limits _{\Omega }Div({\mathbf {\mathbf {P}}}){\dot{\varvec{\chi }}} dA+\int \limits _{\partial \Omega }[C({\mathbf {g\otimes D\otimes D}})^{T} {\dot{\mathbf {F}}}^{T}]{\mathbf {\cdot N}}dS, \end{aligned}$$
(31)
where \(\int _{\Omega }Div({\mathbf {P}}^{T}{\dot{\varvec{\chi }}} )dA=\int _{\partial \Omega }{\mathbf {P}}^{T}{\dot{\varvec{\chi }}\cdot N} dS\). With the Euler equation satisfied \(Div({\mathbf {P}})=0\), the above reduces to
$$\begin{aligned} \dot{E}=\int \limits _{\partial \Omega }{\mathbf {P}}^{T}{\dot{\varvec{\chi }} \cdot \mathbf {N}}dS+\int \limits _{\partial \Omega }[C({\mathbf {g\otimes D\otimes D}}) ^{T}{\dot{\mathbf {F}}}^{T}]{\mathbf {\cdot N}}dS. \end{aligned}$$
(32)
Now, we make use of the normal–tangential decomposition of \({\dot{\mathbf {F}}} \) as
$$\begin{aligned} {\dot{\mathbf {F}}}=\nabla {\dot{\varvec{\chi }}}=\nabla {\dot{\varvec{\chi }} ({\mathbf {T\otimes T}})+}\nabla {\dot{\varvec{\chi }}({\mathbf {N\otimes N}})=\dot{{\varvec{\chi }}}}^{\prime } {\mathbf {\otimes T+}}\dot{{\varvec{\chi }}},_{N}{\mathbf {\otimes N,}} \end{aligned}$$
(33)
where \({\mathbf {T}}={\mathbf {X}}^{^{\prime }}(S)={\mathbf { k\times N}}\) is the unit tangent to \(\partial \Omega \), and \(\dot{\varvec{\chi }}^{\prime }\) and \({\dot{\varvec{\chi }}}_{,N}\) are the tangential and normal derivatives of \({\dot{\varvec{\chi }}}\) on \(\partial \Omega \) (i.e. \(\dot{\chi } ^{^{\prime }}{\mathbf {e}}_{i}=\dot{\chi } _{i,A}X^{\prime }{\mathbf {e}}_{i}=\dot{\chi } _{i,A}T_{A}{\mathbf {e}}_{i},\,\dot{\chi }_{,N}{\mathbf {e}}_{i}=\dot{\chi }_{i,A}\frac{\partial X_{A}}{ \partial N}{\mathbf {e}}_{i}=\dot{\chi }_{i,A}N_{A} {\mathbf {e}}_{i}\)). Accordingly, Eq. (32) can be rewritten as
$$\begin{aligned} \dot{E}=\int \limits _{\partial \Omega }{\mathbf {P}}^{T}{\dot{\varvec{\chi }} \cdot N}dS+\int \limits _{\partial \Omega }[C({\mathbf {g\otimes D\otimes D}})^{T}(\dot{\varvec{\chi }}^{\prime }{\otimes \mathbf {T}+\dot{{\varvec{\chi }}}},_{N} {\mathbf {\otimes N}})^{T}]{\mathbf {\cdot N}}dS. \end{aligned}$$
(34)
Performing tensor linear transform and using the identity; \(\mathbf {P}^{T}{\dot{\varvec{\chi }}\cdot \mathbf {N}=tr({\mathbf {N\otimes P}}^{\mathbf {T}}{\dot{\varvec{\chi }}})}=tr(({\mathbf {N\otimes }} \dot{\varvec{\chi }})\mathbf {P})={\mathbf {\mathbf {P}}}^{\mathbf {T}}{\dot{\varvec{\chi }}\cdot \mathbf {N}}\), Eq. (34) becomes
$$\begin{aligned} \dot{E}=\int \limits _{\partial \Omega }{\mathbf {P N\cdot }} \dot{\varvec{\chi }} dS+\int \limits _{\partial \Omega }[C({\mathbf {D\cdot N}})( {\mathbf {D\cdot T}}){\mathbf {g}}\cdot {\dot{\varvec{\chi }}}^{\prime }+C({\mathbf {D\cdot N}})({\mathbf {D\cdot N}}){\mathbf {g}}\cdot {\dot{\varvec{\chi }},}_{N}]dS. \end{aligned}$$
(35)
Since
$$\begin{aligned} ({\mathbf {D\cdot N}})({\mathbf {D\cdot T}}){\mathbf {g}}\cdot \dot{\varvec{\chi }}^{\prime }={[{\mathbf {(D\cdot N)(D\cdot T) g}}\cdot {\dot{\varvec{\chi }}}]}^{\prime }{-[(\mathbf {D\cdot N)(D\cdot T)}{\mathbf {g}}]}^{\prime }{\cdot \dot{\varvec{\chi }}}, \end{aligned}$$
(36)
we find
$$\begin{aligned} \dot{E}& = \int \limits _{\partial \Omega }[{\mathbf {PN}}-\{C({\mathbf {D\cdot N)(D\cdot T)g}}\}^{\prime }]\cdot \dot{{\varvec{\chi }}}dS \nonumber \\&+\int \limits _{\partial \Omega }[{C(\mathbf {D\cdot N)(D\cdot N)g}}]\cdot {\dot{\varvec{\chi }}} ,_{N}dS+\int \limits _{\partial \Omega }[C({\mathbf {D\cdot N)(D\cdot T}}){\mathbf {g}}\cdot {\dot{\varvec{\chi }}}]^{\prime }dS. \end{aligned}$$
(37)
The above equation is equivalent to
$$\begin{aligned} \dot{E}& = \int \limits _{\partial \Omega }\left[ \mathbf {\mathbf {P}N\mathbf {-}}\frac{d}{dS }\{C{\mathbf {(D\cdot N)( D\cdot T)g}}\}\right] \cdot \dot{{\varvec{\chi }}}dS \nonumber \\&+\int \limits _{\partial \Omega }[C{\mathbf {(D\cdot N)(D\cdot N)g}}]\cdot \dot{{\varvec{\chi }}} ,_{N}dS+\sum \left\| C({\mathbf {D\cdot N)( D\cdot T}})({\mathbf {g\cdot }}\dot{{\varvec{\chi }}})\right\| , \end{aligned}$$
(38)
where the double bar symbol refers to the jump across the discontinuities on the boundary \(\partial \Omega \) (i.e. \(\left\| *\right\| =(*)^{+}-(*)^{-}\)) and the sum refers to the collection of all discontinuities. In addition, the principle of virtual work \(\dot{E}=P\) states that the admissible mechanical powers are necessary to have the following form
$$\begin{aligned} P=\int \limits _{\partial \Omega _{t}}{\mathbf {t\cdot }} \dot{{\varvec{\chi }}}dS+ \int \limits _{\partial \Omega }{\mathbf {m\cdot }}\dot{\varvec{\chi }} ,_{N}dS+\sum \left\| {\mathbf {f}}\cdot \dot{{\varvec{\chi }}}\right\| . \end{aligned}$$
(39)
Consequently, by comparing Eqs. (38) and (39), we observe that
$$\begin{aligned} {\mathbf {t}}& = {\mathbf {PN-}}\frac{d}{ds}\left[ C{\mathbf {g}}({\mathbf {D\cdot T}})( {\mathbf {D\cdot N}})\right] {\mathbf {,}} \nonumber \\ {\mathbf {m}} &= C{\mathbf {g}}({\mathbf {D\cdot N}}){\mathbf {\mathbf {\mathbf {(}}D\cdot N)}}, \nonumber \\ {\mathbf {f}} &= C{\mathbf {g}}({\mathbf {D\cdot T}})({\mathbf {D\cdot N}}). \end{aligned}$$
(40)
For example, if the fibers’ directions are either normal or tangential to the boundary (i.e. \(({\mathbf {D\cdot T}})({\mathbf {D\cdot N}})=0\) ), Eq. (40) further reduces to
$$\begin{aligned} t_{i}{\mathbf {e}}_{i} & = P_{iA}N_{A}{\mathbf {e}}_{i}, \nonumber \\ m_{i}{\mathbf {e}}_{i} & = Cg_{i}D_{A}N_{A}D_{B}N_{B}{\mathbf {e}} _{i}, \nonumber \\ f_{i}{\mathbf {e}}_{i} & = 0, \end{aligned}$$
(41)
where
$$\begin{aligned} P_{iA}({\mathbf {e}}_{i}\otimes {\mathbf {E}}_{A})=(W_{F_{iA}}- Cg_{i,B}D_{A}D_{B}-pF_{iA}^{*})(\mathbf {e} _{i}{{\mathbf {\otimes E}}}_{A}),\ g_{i} {\mathbf {e}}_{i}=F_{iA,B}D_{A}D_{B} {\mathbf {e}}_{i}. \end{aligned}$$

Example: Neo-Hookean Type Materials

The energy density function of the Neo-Hookean material is given by
$$\begin{aligned} W({\mathbf {F}})=\frac{\mu }{2}(I_{1}-3)-\mu \log I_{3}+\frac{\lambda }{2}(\log I_{3})^{2};\,\mu ,\,\lambda >0. \end{aligned}$$
(42)
By setting \(I_{3}=1\), the incompressible model can be obtained from the above as
$$\begin{aligned} W({\mathbf {F}})=\frac{\mu }{2}(I_{1}-3)=\frac{\mu }{2}({\mathbf {F\cdot F-}}3). \end{aligned}$$
(43)
Taking derivative of the above with respect to \({\mathbf {F}}\) and subsequently substituting it into Eq. (29), we obtain
$$\begin{aligned} {\mathbf {P}}=P_{iA}(\mathbf {e}_{i}\otimes \mathbf {E}_{A})=(\mu F_{iA} -Cg_{i,B}D_{A}D_{B}-pF_{iA}^{*})(\mathbf {e}_{i}\otimes \mathbf {E}_{A}). \end{aligned}$$
(44)
Therefore, the Euler equilibrium Eq. (30) yields
$$\begin{aligned} P_{iA,A}{\mathbf {e}}_{i}=(\mu F_{iA,A}-p_{,A}F_{iA}^{*}-Cg_{i,AB}D_{A}D_{B}){\mathbf {e}}_{i}=0,\ \ \because F_{iA,A}^{*}=0\text { ( Piola's identity).} \end{aligned}$$
(45)
In the case of an incompressible fiber-reinforced material which consists of a single family of fibers (i.e. \({\mathbf {D}}={\mathbf {E}}_{1},\,D_{1}=1,\,D_{2}=0\) ) and is subjected to the plane deformations, Eq. (45) further reduces to
$$\begin{aligned} (\mu \chi _{i,AA}-C\chi _{i,1111}-p_{,A}\varepsilon _{ij}\varepsilon _{AB}\chi _{j,B}){\mathbf {e}}_{i}& = 0\text { and} \nonumber \\ \chi _{1,1}\chi _{2,2}-\chi _{1,2}\chi _{2,1}& = 1;\text {incompressibility condition }(\det {\mathbf {F}}\,=\,1). \end{aligned}$$
(46)
In the above, \(g_{i}=F_{i1,1}\), \(F_{iA}^{*}=\varepsilon _{ij}\varepsilon _{AB}F_{jB}\) and \(\varepsilon _{ij}\) is the 2-D permutation; \(\varepsilon _{12}=-\varepsilon _{21}=1,\varepsilon _{11}=-\varepsilon _{22}=0\). The solutions of Eq. (46) (i.e. \(\chi _{1},\)\(\chi _{2}\) and p) can be accommodated via the commercial packages (e.g. Matlab, COMSOL etc...). Clearly, the resulting equations (i.e. Eqs. (44)–(46)) are compatible to those in [25], yet they are addressed here in the more general and comprehensive manner.

Unidirectional Fiber Composites with Extensible Fibers

In order to incorporate fibers resistant to extension, it is required to formulate the fiber stretch as
$$\begin{aligned} \lambda ^{2}={\mathbf {FD\cdot FD}} \end{aligned}$$
(47)
Accordingly, the expression of the fiber strain is obtained by
$$\begin{aligned} \varepsilon =\frac{1}{2}(\lambda ^{2}-1)=\frac{1}{2}\left[ {\mathbf {FD\cdot FD-}}1\right] . \end{aligned}$$
(48)
Equation (48) indicates that the mechanical response of the fiber stretch is the function of the first gradient of deformation (i.e. \(\varepsilon =\varepsilon ({\mathbf {F}})\)) and therefore, in the case of extensible fibers, \({\mathbf {F}}\) dependent energy function can be identified as
$$\begin{aligned} W({\mathbf {F,}}\,\varepsilon ({\mathbf {F}}))=W({\mathbf {F}})+\frac{1}{2} E\varepsilon ^{2}=W({\mathbf {F}})+\frac{1}{2}E\left[ \frac{1}{2}({\mathbf {FD\cdot FD-}} 1)\right] , \end{aligned}$$
(49)
where E is the elastic modulus of fibers in extension. Substituting the above into Eq. (11) furnishes
$$\begin{aligned} U({\mathbf {F}},{\mathbf {G,\ }}p) &=W({\mathbf {F}})+\frac{E}{4}({\mathbf {FD\cdot FD-}} 1)^{2}\\ & \quad +\,\frac{1}{2}C{\mathbf {G(D\otimes D)\cdot G(D\otimes D)}}-p(J-1). \end{aligned}$$
(50)
Here, we make use of the relations established in Eqs. (5) and (8 ). Thus, the virtual work statement now yields
$$\begin{aligned} \dot{E}=\int \limits _{\Omega }\dot{U}\left( {\mathbf {F}},\ {\mathbf {G,\ }}p\right) dA=P. \end{aligned}$$
(51)
To get the required expression, we compute the induced variation as
$$\begin{aligned} \dot{U}\left( {\mathbf {F}},\ {\mathbf {G,\ }}p\right) =\dot{W}({\mathbf {F}})+\frac{E }{4}({\mathbf {FD\cdot FD-}}1\dot{)}^{2}+\frac{1}{2}C[\mathbf {G(D\otimes D)\cdot G(D\otimes D)\dot{]}}^{2}-[p(J-1)\dot{]}, \end{aligned}$$
(52)
and thereby obtain
$$\begin{aligned} \dot{U}\left( {\mathbf {F}},\ {\mathbf {G,\ }}p\right) =W_{{\mathbf {F}}}\mathbf { \cdot \dot{F}+}\frac{E}{2}({\mathbf {FD\cdot FD-}}1){\dot{\mathbf {F}}D\cdot FD+}C[ {\mathbf {G(D\otimes D)\cdot \dot{G}}({\mathbf {D\otimes D}})-}pJ_{{\mathbf {F}}}\cdot \dot{\mathbf {F}}. \end{aligned}$$
(53)
In view of Eqs. (17) and (19)–(22) and using the following identity,
$$\begin{aligned} {\mathbf {FD\cdot }}\dot{\mathbf {F}}{\mathbf {D}}=tr({\mathbf {FD}}\otimes \dot{\mathbf {F}}{\mathbf {D}} )=tr(({\mathbf {FD\otimes D}})\dot{{\mathbf {F}}}^{T})={\mathbf {FD\otimes D\cdot }\dot{\mathbf {F}},} \end{aligned}$$
(54)
Eq. (53) reduces to
$$\begin{aligned} \dot{U}\left( {\mathbf {F}},\ {\mathbf {G,\ }}p\right) =\left( W_{{\mathbf {F}}}+ \frac{E}{2}({\mathbf {FD\cdot FD-}}1)({\mathbf {FD\otimes D)-}}pJ_{{\mathbf {F}}}\right) {\cdot \dot{\mathbf {F}}+}[C({\mathbf {G(D\otimes D))\otimes (D\otimes D)]\cdot \dot{G}}}. \end{aligned}$$
(55)
By applying the similar procedures as in Eqs. (25)–(27), we find
$$\begin{aligned} \dot{E}=\int \limits _{\Omega }[W_{{\mathbf {F}}}-Div\{C( {\mathbf {g\otimes }}D\otimes D)\}+\frac{E}{2}({\mathbf {FD\cdot FD-}}1)( {\mathbf {FD\otimes D}})-p{\mathbf {F}}^{*}]{\cdot \dot{\mathbf {F}}} dA+\int \limits _{\partial \Omega }[C({\mathbf {g\otimes D\otimes D}})^{T} {\dot{\mathbf {F}}}^{T}]{\mathbf {\cdot N}}dS, \end{aligned}$$
(56)
where \({\mathbf {g}}=G({\mathbf {D\otimes D}})\). Hence, the Piola stress \(\mathbf {P}\) in the case of extensible fibers can be expressed as
$$\begin{aligned} {\mathbf {P}}=W_{{\mathbf {F}}}-Div(C({\mathbf {g\otimes D\otimes D}}))+\frac{E}{2}({\mathbf {FD\cdot FD-}}1)(\mathbf { FD\otimes D)}-p{\mathbf {F}}^{*}, \end{aligned}$$
(57)
or equivalently
$$\begin{aligned} P_{iA} ({\mathbf {e}}_{i}{\mathbf {\otimes E}}_{A}) & = \left(W_{F_{iA}}- Cg_{i,B}D_{A}D_{B}\right.\\ & \left.\quad +\,\frac{E}{2}(F_{jC}D_{C}F_{jD}D_{D}- 1)F_{iB}D_{A}D_{B}-pF_{iA}^{*} \right)(\mathbf {e}_{i} {\mathbf {\otimes E}}_{A} ). \end{aligned}$$
(58)
Finally, the corresponding Euler equation is necessary to satisfy
$$\begin{aligned} P_{iA,A}{\mathbf {e}}_{i} & = \left[ ( W_{F_{iA}})_{,A}+\frac{E}{2} (F_{iB,A}F_{jC}F_{jD}+F_{iB}F_{jC,A}F_{jD}+F_{iB}F_{jC}F_{jD,A})D_{A}D_{B}D_{C}D_{D}\right. \\ & \left.\quad -\frac{E}{2}F_{iB,A}D_{A}D_{B}-p_{,A}F_{iA}^{*}-Cg_{i,AB}D_{A}D_{B}\right] {\mathbf {e}}_{i} =0. \end{aligned}$$
(59)
For the Neo-Hookean type material (see, Eq. (42) ) reinforced with a single family of fibers (i.e. \({\mathbf {D}}={\mathbf {E}}_{1},\,D_{1}=1,\,D_{2}=0\) ), the above becomes
$$\begin{aligned}[\mu \chi _{i,AA}-p_{,A}\varepsilon _{ij}\varepsilon _{AB}\chi _{j,B}-C\chi _{i,1111}-\frac{1}{2}E\chi _{i,11}+\frac{1}{2}E(\chi _{i,11}\chi _{j,1}\chi _{j,1}+\chi _{i,1}\chi _{j,11}\chi _{j,1}+\chi _{i,1}\chi _{j,1}\chi _{j,11})]{\mathbf {e}}_{i}=0. \end{aligned}$$
(60)
It is also noted here that Eqs. (58) and (60) reduce to Eqs. (29) and (46) in the limit of vanishing E, respectively. Since the boundary integral in Eq. (56) is the same as that in the inextensible fibers (Eq. (27)), the resulting boundary conditions remain intact, except the expression of \({\mathbf {P}}\), where the explicit form is given in Eq. (58). Numerical schemes (e.g Finite Element Analysis) can be employed to obtain the solution of Eq. (60).

Bidirectional and extensible fiber composites

Based on the frame work developed in the previous sections, we now consider fiber-composites reinforced with bidirectional and extensible fibers. For this purpose, we assign two parametric trajectories and their arch length derivatives for each fiber family as
$$\begin{aligned} \frac{d{\mathbf {r}}(S)}{dS}={\mathbf {FL}}=\lambda {\mathbf {l,\ }}\lambda = \frac{ds}{dS},\ \frac{d{\mathbf {r}}(S)}{dS}={\mathbf {FM}}=\mu \mathbf {m, }\text { and }\mu =\frac{du}{dU}\text {,} \end{aligned}$$
(61)
where \({\mathbf {L}}\) and \({\mathbf {M}}\) are the unit tangent to the fiber trajectory in the reference configuration, and \({\mathbf {l}}\) and \({\mathbf {m}}\) are their counterparts in the deformed configuration, respectively. Without loss of generality, it is assumed that bidirectional fibers are initially orthonormal:
$$\begin{aligned} {\mathbf {L\cdot M}}={\mathbf {0.}} \end{aligned}$$
(62)
Combining Eqs. (61)–(62) furnishes a useful fiber decomposition of the deformation gradient that
$$\begin{aligned} {\mathbf {F}}=\lambda {\mathbf {l\otimes L+}}\mu {\mathbf {m\otimes M.}} \end{aligned}$$
(63)
Thus, we have, for example, \({\mathbf {L}}=L_{A}{\mathbf {E}}_{A}\) and \({\mathbf {l}}= l_{i}{\mathbf {e}}_{i}\) to yield
$$\begin{aligned} \lambda l_{i}{\mathbf {e}}_{i}=F_{iA}({\mathbf {e}}_{i}{\mathbf {\otimes E}}_{A})L_{B} {\mathbf {E}}_{B}=F_{iA}L_{A}{\mathbf {e}}_{i}, \end{aligned}$$
(64)
where \(\{{\mathbf {E}}_{A}\},\ \{{\mathbf {e}}_{i}\}\) are the orthonormal bases in the reference and deformed configurations. Also, the expressions for the geodesic curvatures of the parametric curves can be derived from Eqs. (5)–(9) that
$$\begin{aligned} {\mathbf {g}}_{1}=\frac{d^{2}{\mathbf {r}}(S)}{dS^{2}}=\mathbf {G(L\otimes L)},\text { and }{\mathbf {g}}_{2}=\frac{d^{2}{\mathbf {r}}(U)}{dU^{2}}= {\mathbf {G(M\otimes M)}}. \end{aligned}$$
(65)
The forgoing developments suggest that the second gradient (\({\mathbf {G}}\)) dependent energy function requires
$$\begin{aligned} W({\mathbf {G}})\equiv \frac{1}{2}C_{1}\left( {\mathbf {F}}\right) \left| {\mathbf {g}}_{1}\right| ^{2}+\frac{1}{2}C_{2}\left( {\mathbf {F}}\right) \left| {\mathbf {g}}_{2}\right| ^{2}=\frac{1}{2}C_{i}\left( {\mathbf {F}} \right) \left| {\mathbf {g}}_{i}\right| ^{2}. \end{aligned}$$
(66)
Further, in view of Eqs. (47)–(48), the expression of \(W(\mathbf { F})\), accounting for the stretches of the bidirectional fibers, is obtained by
$$\begin{aligned} W({\mathbf {F,}}\,\varepsilon _{i}({\mathbf {F}}))=W({\mathbf {F}})+\frac{1}{2} E_{i}\varepsilon _{i}^{2}=W({\mathbf {F}})+\frac{1}{2}E_{1}\left[ \frac{1}{2}(\mathbf { FL\cdot FL-}1)\right] +\frac{1}{2}E_{2}\left[ \frac{1}{2}({\mathbf {FM\cdot FM-}}1)\right] . \end{aligned}$$
(67)
In the above, \(E_{i}\) are \(C_{i}\) the fibers mechanical responses against the extension and flexure, respectively. By substituting Eqs. (66)–(67) into Eq. (11), we find
$$\begin{aligned} U({\mathbf {F}},{\mathbf {G,\ }}p)& = W({\mathbf {F}})+\frac{1}{2}E_{i}\varepsilon _{i}^{2}+\frac{1}{2}C_{i}\left( {\mathbf {F}}\right) \left| {\mathbf {g}} _{i}\right| ^{2} \nonumber \\& = W({\mathbf {F}})+\frac{E_{1}}{4}({\mathbf {FL\cdot FL-}}1)^{2}+\frac{E_{2}}{4}( {\mathbf {FM\cdot FM-}}1)^{2} \nonumber \\&{\mathbf {+}}\frac{C_{1}}{2}({\mathbf {G(L\otimes L))\cdot G(L\otimes L)+}}\frac{ C_{2}}{2}({\mathbf {G(M\otimes M))\cdot G(M\otimes M)}}-p(J-1), \end{aligned}$$
(68)
which is the explicit form of the energy density function for an incompressible elastic solid reinforced with the bidirectional and extensible fibers. Now, the variational derivatives of Eq. (68) yields
$$\begin{aligned} \dot{U}\left( {\mathbf {F}},\ {\mathbf {G,\ }}p\right)& = \left( W_{{\mathbf {F}}}{\mathbf {+}} \frac{1}{2}E_{1}({\mathbf {FL\cdot FL-}}1)({\mathbf {FL\otimes L)\mathbf {+}}}\frac{ 1}{2}E_{2}{\mathbf {(\mathbf {FM\cdot FM-}}1)({\mathbf {FM\otimes M)}}-}pJ_{\mathbf { F}}\right) {\mathbf {\cdot \dot{F}}} \nonumber \\&{\mathbf {+}}[C_{1}({\mathbf {G(L\otimes L))\otimes (L\otimes L)+}}C_{2}(\mathbf { G(M\otimes M))\otimes (M\otimes M)]\cdot \dot{G}.} \end{aligned}$$
(69)
Therefore, from the virtual work statement, we obtain
$$\begin{aligned} \overset{\cdot }{E}& = \int \limits _{\Omega }\dot{U}\left( {\mathbf {F}},\ \mathbf {G,\ }p\right) dA \nonumber \\& = \int \limits _{\Omega }\left[ \left\{ W_{{\mathbf {F}}}{\mathbf {+}}\frac{E_{1}}{2}(\mathbf {FL\cdot FL-}1)({\mathbf {FL\otimes L)\mathbf {+}}}\frac{E_{2}}{2}\mathbf {(\mathbf { FM\cdot FM-}1)({\mathbf {FM\otimes M)}}-}pJ_{{\mathbf {F}}}\right\} {\mathbf {\cdot \dot{F}}} \right. \nonumber \\&\left. +\{C_{1}({\mathbf {g}}_{1}{\mathbf {\otimes L\otimes L)+}}C_{2}(\mathbf { g}_{2}{\mathbf {\otimes M\otimes M)\}}\cdot \dot{\mathbf {G}}}\right] dA, \end{aligned}$$
(70)
where \({\mathbf {g}}_{1}={\mathbf {G(L\otimes L)}}\) and \({\mathbf {g}}_{2}=\mathbf { G(M\otimes M)}\), respectively (see, Eq. 65). Applying the same reduction process as in Eqs. (26)–(28), the above may be recast as
$$\begin{aligned} \dot{E}=\int \limits _{\Omega }{\mathbf {P\cdot \dot{F}}}dA+\int \limits _{\partial \Omega }[C_{1}({\mathbf {g}}_{1}{\mathbf {\otimes L \otimes L}})^{T} {\dot{\mathbf {F}}}^{T}+C_{2}({\mathbf {g}}_{2}{\mathbf {\otimes M \otimes M}})^{T}{\dot{\mathbf {F}}}^{T}]{\mathbf {\cdot N}}dS, \end{aligned}$$
(71)
and
$$\begin{aligned} {\mathbf {P}}& = W_{{\mathbf {F}}}-Div\{C_{1}({\mathbf {g}}_{1} {\mathbf {\otimes L \otimes L}})+C_{2}( \mathbf {g}_{2}{\mathbf {\otimes M \otimes M}})\} \nonumber \\&{\mathbf {+}}\frac{E_{1}}{2}({\mathbf {FL\cdot FL-}}1)({\mathbf {FL\otimes L}})+ \frac{E_{2}}{2}({\mathbf {FM\cdot FM-}}1)({\mathbf {FM\otimes M}})-p{\mathbf {F}} ^{*}). \end{aligned}$$
(72)
Consequently, the corresponding Euler equation is found as
$$\begin{aligned} P_{iA,A}{\mathbf {e}}_{i}& = \left[ ( W_{F_{iA}})_{,A}-pF_{iA}^{*}+\frac{E_{1}}{2} (F_{iB,A}F_{jC}F_{jD}+F_{iB}F_{jC,A}F_{jD}+F_{iB}F_{jC}F_{jD,A})L_{A}L_{B}L_{C}L_{D} \right. \nonumber \\&+\frac{E_{2}}{2} (F_{iB,A}F_{jC}F_{jD}+F_{iB}F_{jC,A}F_{jD}+F_{iB}F_{jC}F_{jD,A})M_{A}M_{B}M_{C}M_{D} \nonumber \\&\left. -\frac{E_{1}}{2}F_{iB,A}L_{A}L_{B}-\frac{E_{2}}{2} F_{iB,A}M_{A}M_{B}-C_{1}(g_{1})_{i,AB}L_{A}L_{B}-C_{2}(g_{2})_{i,AB}M_{A}M_{B}\right] {\mathbf {e}}_{i}=0, \end{aligned}$$
(73)
which hold on \(\Omega \). In cases where the Neo-Hookean type material (see, Eq. 42) is reinforced with initially orthonormal set of fibers,
$$\begin{aligned} {\mathbf {L}}={\mathbf {E}}_{1},\,L_{1}=1,\,L_{2}=0,\ {\mathbf {M}}={\mathbf {E}} _{2},\,M_{1}=0,\,M_{2}=1, \end{aligned}$$
(74)
Eq. (73) further reduces to
$$\begin{aligned}&\left[ \mu \chi _{i,AA}-p_{,A}\varepsilon _{ij}\varepsilon _{AB}\chi _{j,B}-C_{1}\chi _{i,1111}-\frac{1}{2}E_{1}\chi _{i,11}+\frac{1}{2} E_{1}(\chi _{i,11}\chi _{j,1}\chi _{j,1}+\chi _{i,1}\chi _{j,11}\chi _{j,1}+\chi _{i,1}\chi _{j,1}\chi _{j,11})\right. \nonumber \\&\left. \quad -C_{2}\chi _{i,2222}-\frac{1}{2}E_{2}\chi _{i,22}+\frac{1}{2}E_{2}(\chi _{i,22}\chi _{j,2}\chi _{j,2}+\chi _{i,2}\chi _{j,22}\chi _{j,2}+\chi _{i,2}\chi _{j,2}\chi _{j,22})\right] {\mathbf {e}}_{i}=0. \end{aligned}$$
(75)
We also note here that Eqs. (46) and (60) are the special cases of the above when \(E_{1}=E_{2}=C_{2}=0\) (unidirectional fiber) and \( E_{2}=C_{2}=0\) (unidirectional and extensible fiber), respectively.

Boundary conditions: Bidirectional fibers

It is clear from the second of Eqs. (28) and (71) that the boundary conditions are not compatible due to the introduction of an additional fiber family. To see this, we utilize the identity of \({\mathbf {P}}\cdot \dot{\mathbf {F}}=Div({\mathbf {P}}^{T}{\dot{\varvec{\chi }}})-Div( {\mathbf {P}}){\dot{\varvec{\chi }}}\) (see, Eq. (28)) and subsequently obtain from Eq. (28) that
$$\begin{aligned} \dot{E}=\int \limits _{\partial \Omega }{\mathbf {P}}^{T}{\dot{\varvec{\chi }} \cdot N}dS-\int \limits _{\Omega }Div({\mathbf {P}}){\dot{\varvec{\chi }}} dA+\int \limits _{\partial \Omega }[C_{1}({\mathbf {g}}_{1}{\mathbf {\otimes L \otimes L}})^{T}{\dot{\mathbf {F}}}^{T}+C_{2}({\mathbf {g}}_{2} {\mathbf {\otimes M \otimes M}})^{T}{\dot{\mathbf {F}}}^{T}]\mathbf {\cdot N }dS, \end{aligned}$$
(76)
where \(\int _{\Omega }Div({\mathbf {P}}^{T}{\dot{\varvec{\chi }}} )dA=\int _{\partial \Omega }{\mathbf {P}}^{T}{\dot{\varvec{\chi }}\cdot N} dS.\ \)Since the Euler equation remains valid (i.e. \(Div({\mathbf {P}})= 0\)), the above reduces to
$$\begin{aligned} \dot{E}=\int \limits _{\partial \Omega }{\mathbf {P}}^{T}{\dot{\varvec{\chi }} \cdot N}dS+\int \limits _{\partial \Omega }[C_{1}({\mathbf {g}}_{1}{\mathbf {\otimes L \otimes L}})^{T}{\dot{\mathbf {F}}}^{T}+C_{2}({\mathbf {g}}_{2}{\mathbf {\otimes M \otimes M}})^{T}{\dot{\mathbf {F}}}^{T}]\mathbf { \cdot N}dS. \end{aligned}$$
(77)
Applying the normal–tangential decomposition of \({\dot{\mathbf {F}}}\) as in Eq. (34), we arrive
$$\begin{aligned} \dot{E}=\int \limits _{\partial \Omega }{\mathbf {P}}^{T}{\dot{\varvec{\chi }} \cdot N}dS+\int \limits _{\partial \Omega }[(C_{1}({\mathbf {g}}_{1}{\mathbf {\otimes L \otimes L}})^{T}+C_{2}(\mathbf {g}_{2}{\mathbf {\otimes M \otimes M}})^{T})({\dot{\varvec{\chi }}}^{\prime } {\mathbf {\otimes T+\dot{\varvec{\chi }}},}_{N}{\mathbf {\otimes N}})^{T}]\mathbf { \cdot N}dS. \end{aligned}$$
(78)
Now, the linear transform of the above yields
$$\begin{aligned} \dot{E}& = \int \limits _{\partial \Omega }{\mathbf {P}N\cdot \dot{\varvec{\chi }}} dS+\int \limits _{\partial \Omega }[C_{1}({\mathbf {L \cdot N}} )({\mathbf {L\cdot T}})\mathbf {g}_{1}{{\cdot \dot{\varvec{\chi }}}^{\prime }+}C_{1} {\mathbf {(L \cdot N)(L \cdot N)}}\mathbf {g}_{1}{\cdot {\dot{\varvec{\chi }},}}_{N}]dS \nonumber \\&+\int \limits _{\partial \Omega }[C_{2}({\mathbf {M \cdot N}})({\mathbf {M\cdot T}}){\mathbf {g}}_{2}{\cdot \dot{\varvec{\chi }}}^{\prime }+C_{2} ({\mathbf {M \cdot N}})({\mathbf {M\cdot N}}){\mathbf {g}}_{2}\cdot {\dot{\varvec{\chi }},}_{N}]dS. \end{aligned}$$
(79)
In addition, it is not difficult to show the identity,
$$\begin{aligned} ({\mathbf {L\cdot N}})({\mathbf {L\cdot T}})\mathbf {g}\cdot \dot{{\varvec{\chi }}}^{\prime }=[({\mathbf {L\cdot N}})({\mathbf {L\cdot T}}) {\mathbf {g}}\cdot {\dot{\varvec{\chi }}}]^{\prime }-({\mathbf {L \cdot N}})({\mathbf {L\cdot T}}){\mathbf {g}}]^{\prime }{\cdot \dot{\varvec{\chi }},} \end{aligned}$$
and similarly for \({\mathbf {M}}\). Therefore, we obtain
$$\begin{aligned} \dot{E}& = \int \limits _{\partial \Omega }\left[ {\mathbf {PN}}-\frac{d}{dS}\{C_{1}({\mathbf {L\cdot N}})({\mathbf {L\cdot T}}){\mathbf {g}}_{1}\}-\frac{d}{dS}\{C_{2}({\mathbf {M \cdot N}})({\mathbf {M\cdot T}}){\mathbf {g}}_{2}\}\right] \cdot \dot{{\varvec{\chi }}}dS \nonumber \\&+\int \limits _{\partial \Omega }[C_{1}({\mathbf {L\cdot N}})({\mathbf {L\cdot N}}){\mathbf {g}}_{1}+C_{2}({\mathbf {M\cdot N}})({\mathbf {M\cdot N}}){\mathbf {g}}_{2}]\cdot \dot{{\varvec{\chi }}},_{N}dS \nonumber \\&+\sum \left\| [C_{1}({\mathbf {L\cdot N}})({\mathbf {L \cdot T}}){\mathbf {g}}_{1}+C_{2}({\mathbf {M \cdot N}})({\mathbf {M\cdot T}}){\mathbf {g}}_{2}]\cdot \dot{\varvec{\chi }}) \right\| . \end{aligned}$$
(80)
Consequently, in view of the admissible mechanical power, Eq. (38), the expressions of the boundary conditions can be identified as
$$\begin{aligned} {\mathbf {t}}& = {\mathbf {PN-}}\frac{d}{ds}\left[ C_{1}{\mathbf {g}}_{1}(\mathbf { L\cdot T})({\mathbf {L\cdot N}})+C_{2}{\mathbf {g}}_{2}({\mathbf {M\cdot T}})(\mathbf { M\cdot N})\right] , \nonumber \\ {\mathbf {m}}& = C_{1}{\mathbf {g}}_{1}({\mathbf {L\cdot N}})({\mathbf {L\cdot N}})+C_{2}{\mathbf {g}}_{2}({\mathbf {M\cdot N}})({\mathbf {M\cdot N}}), \nonumber \\ {\mathbf {f}}& = C_{1}{\mathbf {g}}_{1}({\mathbf {L\cdot N}})({\mathbf {L}}\cdot {\mathbf {T}}){\mathbf {g}}_{1}+C_{2}{\mathbf {g}}_{2}({\mathbf {M\cdot N}})({\mathbf {M\cdot T}}). \end{aligned}$$
(81)
In particular, if the fiber’s directions are either normal or tangential to the boundary (i.e. \(({\mathbf {L\cdot T}})({\mathbf {L\cdot N}})=0\) and \((\mathbf { M\cdot T})({\mathbf {M\cdot N}})=0\)), Eq. (81) further reduces to
$$\begin{aligned} t_{i}{\mathbf {e}}_{\mathbf {i}}& = P_{iA}N_{A}{\mathbf {e}}_{\mathbf {i}}, \nonumber \\ m_{i}{\mathbf {e}}_{\mathbf {i}}& = (C_{1}(g_{1})_{i}L_{A}N_{A}L_{B}N_{B}+C_{2}(g_{2})_{i}M_{A} N_{A}M_{B}N_{B}){\mathbf {e}}_{\mathbf {i}}, \nonumber \\ f_{i}{\mathbf {e}}_{\mathbf {i}}& = 0, \end{aligned}$$
(82)
where
$$\begin{aligned} P_{iA}& = \mu F_{iA}+\frac{E_{1}}{2}(F_{jC}F_{jD}L_{C}L_{D}{\mathbf {-}} 1)F_{iB}L_{A}L_{B} \nonumber \\&+\frac{E_{2}}{2}(F_{jC}F_{jD}M_{C}M_{D}{\mathbf {-}} 1)F_{iB}M_{A}M_{B}-pF_{iA}^{*}-C_{1}(g_{1})_{i,B}L_{A}L_{B}-C_{2}(g_{2})_{i,B}M_{A}M_{B}, \end{aligned}$$
(83)
in the case of the Neo-Hookean type material. The boundary conditions in Eq. (82) together with the Eq. (46)\(_{2}\) and (75) constitute the systems of non-linear Partial Differential Equations (PEDs), where the solution can be accommodated via the Finite element analysis (FEA). The corresponding numberical procedures are reserved for the sake of conciseness and coherence, but can be found in [26, 27]. One of the important features of the present model is that it predicts the smooth transitions of shear stress fields as opposed to the classical first-order theory where a significant discontinuity is present (see, Fig. 1 (a)). The result also agrees with the study presented in the work of Boisse et. al. [37] (see, also, Fig. 1 b). Figure 2 illustrates the deformation profiles predicted by the presented model which clearly indicates that the second-order theory is capable of capturing the fibers resistant to flexure.
Fig. 1

Shear strain contour: 1st gradient (left) VS 2nd gradient (right)

Fig. 2

Deformed configurations with respect to \(C_{1}/\mu \)

Further considerations

For the analysis of the more sophisticated materials such as living tissues, bones and composites with interconnected arrays of fibers, it is of necessary to develop higher order gradient models [31]. This can be achieved, within the framework of the proposed model, by introducing higher gradient dependent energy potentials into the models of deformation. For example, Eq. (10) can be augmented to integrate the third gradient of deformation that
$$\begin{aligned} W({\mathbf {F}},{\mathbf {G,H}})=W({\mathbf {F}})+W({\mathbf {G}})+W({\mathbf {H}}) ,\,W({\mathbf {H}})\equiv \frac{1}{2}A\left( {\mathbf {F}}\right) \left| {\varvec{\alpha }} \right| ^{2}. \end{aligned}$$
(84)
In the above, \({\mathbf {H}}\) denotes the third gradient of deformation;
$$\begin{aligned} \nabla {\mathbf {G=H=}}H_{iABC}({\mathbf {e}}_{\mathbf {i}}{\mathbf {\otimes E }}_{A}{\mathbf {\otimes E}}_{B}{\mathbf {\otimes E}}_{C}), \end{aligned}$$
(85)
and \(A\left( {\mathbf {F}}\right) \) is the material constant associated with \( {\mathbf {H,}}\) which can be assumed constant (i.e. \(A\left( {\mathbf {F}}\right) =A\)) similarly as in Eq. (12). To observe the connection between \( {\mathbf {H}}\) and \({\varvec{\alpha }}\) (third order arclength derivative), we evaluate
$$\begin{aligned} {{\varvec{\alpha }}}=\frac{d^{3}{\mathbf {r}}(S)}{dS^{3}}=\frac{d(\mathbf { G(D\otimes D)})}{dS}=\nabla [\mathbf {G(D\otimes D)]D=H(D\otimes D\otimes D),} \end{aligned}$$
(86)
where, from Eqs. (4) and (5),\(\frac{d^{2}{\mathbf {r}}(S)}{ dS^{2}}={\mathbf {G(D\otimes D)}}\) and \(\nabla [\mathbf {G(D\otimes D)]=\nabla {\mathbf {G(D\otimes D)}}}\) for initially straight fibers. Further, combining Eqs. (85) and (86) furnishes
$$\begin{aligned} {\varvec{\alpha }}=\alpha _{i}{\mathbf {e}}_{i}=H_{iABC}({\mathbf {e}}_{\mathbf {i}} {\mathbf {\mathbf {\otimes E}}}_{A}{\mathbf {\otimes E}}_{B}{\mathbf {\otimes E}}_{C}) {\mathbf {(D\otimes D\otimes D)}}=H_{iABC}D_{A}D_{B}D_{C}{\mathbf {e}}_{i}, \end{aligned}$$
and therefore we find
$$\begin{aligned} \alpha _{i}{\mathbf {e}}_{i}=\chi _{i,ABC}D_{A}D_{B}D_{C}{\mathbf {e}}_{i}. \end{aligned}$$
(87)
Eq. (87) implies that the third gradient of deformation (\({\mathbf {H)}}\) is related to the rate of changes in curvature and/or flexure of fibers. The immediate physical relevance of the results is the micromechanical responses of materials such as micropolar rotations (see, for example, [31] and references therein). Using chain rule, we compute the variational derivatives of \(W({\mathbf {H}})\) with respect to the third gradient of deformation as
$$\begin{aligned} \dot{W}({\mathbf {H}})=W_{{\mathbf {H}}}\cdot {\dot{\mathbf {H}}}=W_{H_{iABC}}( {\mathbf {e}}_{i}{\mathbf {\otimes E}}_{A}{\mathbf {\otimes E}}_{B}{\mathbf {\otimes E}} _{C})\dot{H}_{jDEF}({\mathbf {e}}_{j}{\mathbf {\otimes E}}_{D}{\mathbf {\otimes E}} _{E}{\mathbf {\otimes E}}_{F})=W_{H_{iABC}}\dot{H}_{iABC}. \end{aligned}$$
(88)
Thus, from the second of Eq. (84) and Eq. (88), it is straightforward to show
$$\begin{aligned} \dot{W}({\mathbf {H}})=(\frac{1}{2}A{\varvec{\alpha }}\cdot {\varvec{\alpha }}\dot{)}=A{\varvec{\alpha }}\cdot {\dot{\varvec{\alpha }}=W_{{\mathbf {H}}}\cdot {\dot{\mathbf {H}}},} \end{aligned}$$
(89)
where \(\dot{A}=0\) for constant A (i.e. \(A\left( {\mathbf {F}}\right) =A\)). Now, we substitute Eq. (86) into Eq. (89) and thereby obtain
$$\begin{aligned} W_{H_{iABC}}\dot{H}_{iABC}=A\alpha _{i}D_{A}D_{B}D_{C}\dot{H}_{iABC}, \end{aligned}$$
(90)
where \({\dot{{\varvec{\alpha }}}=\dot{H}({\mathbf {D\otimes D\otimes D}})}\) for initially straight fibers (i.e. \({\dot{\mathbf {D}}}=0\)). Consequently, the explicit expression of \(W_{{\mathbf {H}}}\) can be found as
$$\begin{aligned} W_{H_{iABC}}=A\alpha _{i}D_{A}D_{B}D_{C}, \end{aligned}$$
(91)
or equivalently
$$\begin{aligned} W_{H_{iABC}}({\mathbf {e}}_{i}{\mathbf {\otimes E}}_{A}{\mathbf {\otimes E}}_{B} {\mathbf {\otimes E}}_{C})=A\alpha _{i}D_{A}D_{B}D_{C}({\mathbf {e}}_{i}\mathbf { \otimes E}_{A}{\mathbf {\otimes E}}_{B}{\mathbf {\otimes E}}_{C})\Leftrightarrow W_{{\mathbf {H}}}=A{\varvec{\alpha \otimes D\otimes D\otimes D}}. \end{aligned}$$
(92)
The obtained expressions in Eqs. (88)–(92) are then used to determine the Euler equilibrium equations via virtual work statement \(\dot{E} =P\). For example, in the case of unidirectional and inextensible fiber composites, we obtain
$$\begin{aligned} \dot{E}=\int \limits _{\Omega }(W_{{\mathbf {F}}}{\mathbf {\cdot \dot{F}}}+W_{{\mathbf {G}}} {\mathbf {\cdot \dot{G}}}+W_{{\mathbf {H}}}{\mathbf {\cdot \dot{H}}}-p\dot{J})dA, \end{aligned}$$
(93)
where \(W_{{\mathbf {G}}}=C{\mathbf {(\mathbf {g\otimes }}D\otimes D)}\) and \(W_{ {\mathbf {H}}}=A({\varvec{\alpha }} {\mathbf {\otimes D\otimes D\otimes D}})\). We are currently working on the evaluation of Eq. (93) and the associated boundary conditions. Our intention is to report elsewhere.

Mooney-Rivlin Type Materials

In cases of soft materials-based composites, such as carbon rubber-fiber composites and polymer composites, a different type of energy functions can be considered. The Mooney-Rivlin model may be suitable for the above mentioned cases (see, for example, [33]) where the energy density function is given by
$$\begin{aligned} W({\mathbf {F}})=\frac{\mu }{2}(I_{1}-3)+\frac{\lambda }{2}(I_{2}-3);\,\mu ,\,\lambda >0. \end{aligned}$$
(94)
The variational computation of Eq. (94) then yields
$$\begin{aligned} \dot{W}({\mathbf {F}})=W_{{\mathbf {F}}}{\mathbf {\cdot \dot{F}}}=\left[ \frac{\mu }{2} (I_{1})_{{\mathbf {F}}}{\mathbf {+}}\frac{\lambda }{2}(I_{2})_{{\mathbf {F}}}\right] \mathbf { \cdot \dot{F}}=[\mu {\mathbf {F+}}\lambda {\mathbf {F}}\{({\mathbf {F\cdot F)I-F}}^{T} {\mathbf {F}}\}]{\mathbf {\cdot \dot{F}},} \end{aligned}$$
(95)
where \({\mathbf {I}}\) is the identity tensor. For desired applications, the above energy variation form can be employed instead of that from the Neo-Hookean model. For example, in the case of unidirectional and inextensible fibers, we combine Eqs. (24) and (95), and thereby obtain
$$\begin{aligned} \dot{E}=\int \limits _{\Omega }({\mathbf {[}}\mu {\mathbf {F+}}\lambda {\mathbf {F}}\{( {\mathbf {F\cdot F)I-F}}^{T}{\mathbf {F}}\}]{\mathbf {\cdot \dot{F}}}+C\mathbf { {\mathbf {(g\otimes }}D\otimes D)\cdot \dot{G}}-pJ_{{\mathbf {F}}}\mathbf {\cdot \dot{F}})dA. \end{aligned}$$
(96)
Applying Green–Stoke’s theorem, Eq. (96) becomes
$$\begin{aligned} \dot{E}=\int \limits _{\Omega }[\mu {\mathbf {F+}}\lambda {\mathbf {F}}\{({\mathbf {F\cdot F}}){\mathbf {I-F}}^{T}{\mathbf {F}}\}-Div(C{\mathbf { g\otimes D\otimes D}})-p{\mathbf {F}}^{*}]{\mathbf {\cdot \dot{F}}} dA+\int \limits _{\partial \Omega }[C({\mathbf {g\otimes D\otimes D}})^{T} {\dot{\mathbf {F}}}^{T}]{\mathbf {\cdot N}}dS. \end{aligned}$$
(97)
Therefore we find
$$\begin{aligned} {\mathbf {P}}=P_{iA}({\mathbf {e}}_{i}{\mathbf {\otimes E}}_{A})=[\mu F_{iA}+ \lambda F_{iB}(F_{jC}F_{jC}\delta _{AB}-F_{jA}F_{jB})- Cg_{i,B}D_{A}D_{B}-pF_{iA}^{*}]({\mathbf {e}}_{i} {\mathbf {\otimes E}}_{A}), \end{aligned}$$
(98)
which may be served as the expression of the Piola stress for soft composite materials. Also, the corresponding Euler equilibrium equation can be derived as
$$\begin{aligned} P_{iA,A}{\mathbf {e}}_{i}=[\mu F_{iA,A}+\lambda \{F_{iB}(F_{jC}F_{jC}\delta _{AB}- F_{jA}F_{jB})\}_{,A}-Cg_{i,AB}D_{A}D_{B}-p_{,A}F_{iA}^{*}] \mathbf {e}_{i}=0, \end{aligned}$$
(99)
which hold on \(\Omega \). In the case of an elastic medium reinforced with a single family of fibers (i.e. \({\mathbf {D}}={\mathbf {E}}_{1},\,D_{1}=1,\,D_{2}=0\) ), Eq. (99) furnishes
$$\begin{aligned}{}[\mu \chi _{i,AA}-p_{,A}\varepsilon _{ij}\varepsilon _{AB}\chi _{j,B} +\lambda \{\chi _{i,B}(\chi _{j,C}\chi _{j,C}\delta _{AB}- \chi _{j,A}\chi _{j,B})\}_{,A}-C\chi _{i,1111}]\mathbf {e}_{i}=0. \end{aligned}$$
(100)
The admissible boundary conditions and numerical schemes for high-order coupled differential equations remain to be established in order to fully determine the solution of the above systems of PDEs. The implementation of the proposed model is currently in progress and results will be presented soon.

Conclusion

We present a complete second order gradient theory describing the mechanics of an elastic solids reinforced with fibers resistant to flexure and extension. This includes the formulations of the constitutive relations for the general class of fiber composites consisted with unidirectional and bidirectional families of fibers. The bending resistance of fibers is computed via the arclength derivatives along the directions of fibers’ parametric segments. We then demonstrate the derivation of the Euler equilibrium equations in which the constraint of bulk incompressibility is supplemented. A complete analysis of the admissible boundary conditions is also presented for the sake of clarity. It is found that the expressions of the admissible boundary tractions are heavily dependent on the types of fibers. However, the overall structures of the boundary forces are, in principal, remain intact regardless of the fibers’ characteristics.

In particular, we consider a high order gradient model and formulate it’s energy variations. It turns out that the variational derivative with respect to the third gradient of deformation yields a fourth order tensor which maps a third order arclength derivative (third order tensor in nature) into a first order tensor in the reference configuration. The results may be further exploited in the general formulation of the third gradient elasticity theory for the applications of microstructured continua, for example, living tissues and composites with the interconnected arrays of fibers. Lastly, the energy potential of the Mooney-Rivlin type is examined through which the complete expressions of the associated stress fields and the Euler equilibrium equation are obtained. Potential application of the proposed model is the analysis of the mechanical behavior of soft composite materials such as carbon rubber-fiber composites and polymer composites. We also note here that the implementations of the aforementioned models are currently underway and the results will be available soon.

Notes

Acknowledgements

This work was supported by the Natural Sciences and Engineering Research Council of Canada via Grant #RGPIN 04742 and the University of Alberta through a start-up grant. The author would like to thank Dr. David Steigmann for discussions concerning the underlying theory.

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Copyright information

© Korean Multi-Scale Mechanics (KMSM) 2019

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of AlbertaEdmontonCanada

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