Finite Plane Elastostatics Research Articles

Mechanics of hyperelastic composites reinforced with nonlinear elastic fibrous materials in finite plane elastostatics

A model for the mechanics of a hyperelastic material reinforced with unidirectional and bidirectional fibers is presented in finite plane elastostatics. This includes the refinement/development of a series of continuum-based prediction models to accommodate the nonlinear responses of both the matrix material and the reinforcing fibers. The kinematics of the embedded fibers, including the torsional kinematics between two adjoining fibers, are formulated via the first and second gradient of continuum deformations. Within the framework of variational principles and a virtual work statement, the Euler equation and the admissible boundary conditions are derived. To this end, a set of inhouse experiments are performed for the purpose of cross-examination and model implementation. The obtained models successfully predict the strain-stiffening responses of the elastomer – polyester fiber composites together with other key design considerations such as, deformation profiles, shear strain distributions, and the deformed configurations of a local unit fiber mesh. The Euler – Almansi strain integrate model is also proposed through which the strain-softening behaviors of a certain type of polyurethane fiber composites are predicted, yet further implementation of the obtained Euler – Almansi model remains to be determined due to the paucity of available data. The practical utility of the proposed models may be expected in the design and analysis of hyperelastic composites exhibiting strain-stiffening/softening responses by providing instant estimations of the resultant properties of intended composites.

A model for hyperelastic materials reinforced with fibers resistance to extension and flexure

A model for the mechanics of a soft hyperelastic material reinforced with long fibers is presented in finite plane elastostatics. The strain energy potential of the composite is refined by the Mooney Rivlin model to accommodate the hyperelastic behaviors of the matrix material. Within the framework of the strain gradient theory, the kinematics of the fibers is formulated and subsequently integrated into the model of continuum deformation. A rigorous derivation of the Euler equation and the associated boundary conditions are presented by virtue of variational principles and a virtual work statement. In particular, the obtained model successfully predicts the ‘J-shaped’ stress-strain responses, deformation profiles and shear strain angles of the elastomer - polyester fiber composites subjected to uniaxial extension. The proposed model may also serve as an alternative of the Holzapfel model in the design and analysis of biomimetic composites with rapid strain-stiffening behaviors by providing direct estimations of the stress-strain responses of intended composites.

Mechanics of third-gradient continua reinforced with fibers resistant to flexure in finite plane elastostatics

A third-gradient continuum model is developed for the deformation analysis of an elastic solid, reinforced with fibers resistant to flexure. This is framed in the second strain gradient elasticity theory within which the kinematics of fibers are formulated, and subsequently integrated into the models of deformations. By means of variational principles and iterated integrations by parts, the Euler equilibrium equation is obtained which, together with the constraints of bulk incompressibility, compose the system of the coupled nonlinear partial differential equations. In particular, a rigorous derivation of the admissible boundary conditions arising in the third gradient of virtual displacement is presented from which the expressions of the triple forces are derived. The resulting triple forces are, in turn, coupled with the Piola-type triple stress and are necessary to determine a unique deformation map. The proposed model predicts smooth and dilatational shear angle distributions, as opposed to those obtained from the first- and second-gradient theory where the resulting shear zones are either non-dilatational or non-smooth.

Uniformity of stresses inside a parabolic inhomogeneity in finite plane elastostatics

Using complex variable techniques, we establish the uniformity of the internal Piola stresses within a parabolic inhomogeneity embedded in a matrix from a particular class of compressible hyperelastic materials of harmonic type subjected to uniform remote Piola stresses. In addition, two Piola stress components inside the inhomogeneity simply coincide with their remotely prescribed counterparts in the matrix, while the two other internal Piola stress components can be determined via the solution of a quadratic equation, and are found to be independent of the single geometric parameter characterizing the inhomogeneity–matrix system. We derive a complete solution to the corresponding boundary value problem describing the mechanical behavior of the composite.

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

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.

Gradient elasticity theory for fiber composites with fibers resistant to extension and flexure

A model for the mechanics of an elastic solid, reinforced with bidirectional fibers is presented in finite plane elastostatics. The fiber’s resistance to stretch and flexure are accounted for with variational computations of first and second gradient of deformations, respectively. Within the framework of strain-gradient elasticity, the Euler equation and necessary boundary conditions are formulated. A rigorous derivation of the corresponding linear theory is also developed from which, a complete analytical solution is obtained for small deformations superposed on large. In particular, we assimilate plane bias extension test results indicating that the proposed model successfully predicts smooth transitions of shear strain fields unlike those depicted by the first gradient theory where significant discontinuities are present. The proposed model can serve as an alternative 2D Cosserat theory of non-linear elasticity.

Surface Green’s functions in finite plane elastostatics of harmonic materials

The closed-form representations of surface Green’s functions corresponding to the action of a concentrated force applied at the boundary of a region occupied by a particular class of compressible hyperelastic materials of harmonic type, has been derived. In our analysis, we consider both a bounded region in the form of a circular disk and an unbounded region with either an elliptical hole or a parabolic boundary.

A concentrated couple inside or outside a circular inclusion in finite plane elastostatics

We consider a circular elastic inclusion embedded in an infinite matrix each from a particular class of compressible hyperelastic materials of harmonic type. A concentrated couple is applied either inside the circular inclusion or in the matrix. Closed-form solutions of the corresponding boundary value problems are obtained using complex variable methods, in particular the principle of analytic continuation. Our analysis reveals several interesting conclusions including: the sum σ11+σ22 of the normal stresses inside the inclusion remains constant when the concentrated couple is located in the surrounding matrix; the sum of the normal stresses is zero everywhere in the matrix when the concentrated couple is located inside the inclusion.

Mechanics of an elastic solid reinforced with bidirectional fiber in finite plane elastostatics: complete analysis

A continuum-based model is presented for the mechanics of bidirectional composites subjected to finite plane deformations. This is framed in the development of a constitutive relation within which the constraint of material incompressibility is augmented. The elastic resistance of the fibers is accounted for directly via the computation of variational derivatives along the lengths of bidirectional fibers. The equilibrium equation and necessary boundary conditions are derived by virtue of the principles of virtual work statement. A rigorous derivation of the corresponding linear theory is developed and used to obtain a complete analytical solution for small deformations superposed on large. The proposed model can serve as an alternative 2D Cosserat theory of nonlinear elasticity.

Mechanics of fiber composites with fibers resistant to extension and flexure

A model of elastic solids reinforced with fibers resistant to extension and bending is formulated in finite-plane elastostatics. The linear theory of the proposed model is also derived through which a complete analytical solution is obtained. The presented model can serve as an alternative two-dimensional Cosserat theory of non-linear elasticity.

A circular inclusion with inhomogeneous non-slip imperfect interface in harmonic materials.

In this work, a rigorous study is presented for the problem associated with a circular inclusion embedded in an infinite matrix in finite plane elastostatics where both the inclusion and matrix are comprised a harmonic material. The inclusion/matrix boundary is treated as a circumferentially inhomogeneous imperfect interface that is described by a linear spring-type imperfect interface model where in the tangential direction, the interface parameter is infinite in magnitude and in the normal direction, the interface parameter is finite in magnitude (the so-called non-slip interface condition). Through the repeated use of the technique of analytic continuation, the boundary value problem for four analytic functions is reduced to solve a single first-order linear ordinary differential equation with variable coefficients for a single analytic function defined within the inclusion. The unknown coefficients of said function are then found via various analyticity requirements. The method is illustrated, using a specific example of a particular class of inhomogeneous non-slip imperfect interface. The results from these calculations are then contrasted with the results from the homogeneous imperfect interface. These comparisons indicate that the circumferential variation of interface damage has a pronounced effect on the average boundary stress.

Neutrality of an elliptical inhomogeneity in finite plane elastostatics of harmonic materials

We examine the neutrality of an elliptical inhomogeneity embedded in a particular class of compressible hyperelastic materials of harmonic type when a uniform Piola stress field is prescribed in the surrounding matrix. The present method is based on a spring-type imperfect interface model and on the proper choice of imperfect interface function, which realizes the same degree of imperfection in both normal and tangential directions. The analysis indicates that, in general, the neutral shape and the single imperfect interface function are both dependent on the magnitude of the prescribed stress field.

A crack with surface elasticity in finite plane elastostatics

A crack with surface elasticity in finite plane elastostatics

A nanosized circular inhomogeneity in finite plane elastostatics

We consider the deformation of a nanocomposite by incorporating the effect of surface elasticity on the finite plane deformations of a circular inhomogeneity embedded in a particular class of compressible hyperelastic materials of harmonic type subjected to uniform remote Piola stresses. We incorporate the surface mechanics by using a version of the continuum-based surface/interface model of Gurtin and Murdoch. A complete solution is derived by reducing the original boundary value problem to two coupled first-order differential equations which can be solved analytically. The solution clearly demonstrates that the stress fields in the composite are size dependent and that the stress field inside a circular inhomogeneity is, in general, non-uniform. Two special cases which admit an internal uniform, yet size-dependent stress field, are discussed.

A circular inclusion with imperfect interface in finite plane elastostatics

We consider a circular elastic inclusion embedded in a particular class of harmonic materials subjected to remote uniform stresses. The imperfect interface can be rate dependent as well as rate independent. First, we study the situation in which both rate-depending slip and diffusional relaxation are present on the sharp inclusion-matrix imperfect interface. It is found that in general, the internal Piola stresses within the inclusion are spatially non-uniform and decay with two relaxation times. Interestingly, the average mean Piola stress within the circular inclusion is time independent. Some extreme cases for the imperfect interface are discussed in detail. Particularly, we find a simple condition leading to internal uniform Piola stresses that decay only with a single relaxation time. Second, we investigate a rate-independent spring-type imperfect interface on which normal and shear tractions are proportional to the corresponding displacement jumps. It is found that in general, the internal Piola stresses are intrinsically non-uniform. A special kind of the spring-type interface leading to internal uniform Piola stresses is also found.

Analysis of local singular fields near the corner of a quarter-plane with mixed boundary conditions in finite plane elastostatics

We consider a quarter-plane of compressible hyperelastic material of harmonic-type undergoing finite plane deformations. The plane is subjected to mixed (free–fixed) boundary conditions. In contrast to the analogous case from classical linear elasticity, we find that the deformation field is smooth in the vicinity of the vertex and is actually bounded at the vertex itself. In particular, the normal displacement remains positive eliminating the possibility of material interpenetration. Finally, explicit expressions for Cauchy and Piola stress distributions are obtained in the vicinity of the vertex.

Three-phase elliptical inclusions with internal uniform hydrostatic stresses in finite plane elastostatics

We consider the internal stress field of a three-phase elliptical inclusion bonded to an infinite matrix through an interphase layer when the matrix is subjected to remote uniform stresses. The elastic materials comprising all the three phases belong to a particular class of harmonic materials, and the formed interfaces are two confocal ellipses. A condition leading to internal uniform hydrostatic stresses is derived. This condition relates the two remote principal stresses with the geometric parameters (the thickness of the interphase layer and the aspect ratio of the elliptical inclusion) of the three-phase elliptical inclusion. When this condition is met, the hoop stress in the interphase layer along the entire interphase/inclusion interface is also uniform. Five special situations of practical importance are discussed in considerable detail to demonstrate the unique phenomena inherent in harmonic materials. Our discussions indicate that when this condition is met, it is permissible for the two remote principal stresses to have opposite signs and that for given geometric and material parameters, the remote loading ratio is no longer constant and multiple external loading states exist leading to internal uniform hydrostatic stresses. It is found that this condition can be written into a hyperbola for the two remote principal stresses when the interphase layer is extremely compliant or relatively stiff or when the inclusion is almost rigid. When the magnitudes of the remote stresses are sufficiently large, this condition becomes a very simple one relating the remote loading ratio with the geometric parameters of the composite. Interestingly, it is clearly observed from the simple condition that for given geometric parameters of the three-phase elliptical inclusion, there exist two different values of the remote loading ratio, both of which lead to an internal uniform hydrostatic stress state.

On Partially Debonded Circular Inclusions in Finite Plane Elastostatics of Harmonic Materials

We investigate a partially debonded circular elastic inclusion embedded in a particular class of harmonic materials by using the complex variable method under finite plane-strain deformations. A complete (or full-field) solution is derived. It is observed that the stresses in general exhibit oscillatory singularities near the two tips of the arc shaped interface crack. Particularly the traditional inverse square root singularity for stresses is observed when the asymptotic behavior of the harmonic materials obeys a constitutive restriction proposed by Knowles and Sternberg (1975, “On the Singularity Induced by Certain Mixed Boundary Conditions in Linearized and Nonlinear Elastostatics,” Int. J. Solids Struct., 11, pp. 1173–1201). It is also found that the number of admissible states under finite plane deformations for given external loads can be two, one, or even zero.

Eshelby's conjecture in finite plane elastostatics

We consider an inhomogeneity–matrix system from a particular class of compressible hyper- elastic materials of harmonic type undergoing finiteplane deformations. We discuss the analogue of Eshelby's conjecture for this class of materials. Specifically, we examine whether the stress distribution inside an inhomogeneity is uniform if and only if the inhomogeneity is elliptic when the system is subjected to uniform remote stress.

Designing an inhomogeneity with uniform interior stress in finite plane elastostatics

We consider an inhomogeneity-matrix system from a particular class of compressible hyperelastic materials of harmonic-type undergoing finite plane deformations. We discuss the idea that by adjusting the remote (Piola) stress we can design the shape of the inhomogeneity in such a way that the interior (Piola) stress distribution remains uniform. In fact, using complex variable techniques, we show that a uniform (Piola) stress distribution can be achieved inside the inhomogeneity when the system is subjected to linear remote loading, in which case, the inhomogeneity is necessarily hypotrochoidal with three cusps. We obtain the complete solution of the corresponding inhomogeneity-matrix system in this case. In addition, we consider the more general case when the system is subjected to nonuniform remote loading characterized by stress functions in the form of nth degree polynomials in the complex variable z describing the matrix. In this case, by finding the complete solution of the system, we show that we can again obtain uniform stress inside a hypotrochoidal inhomogeneity with n + 1 cusps. Finally, we illustrate our results with an example.