An Extension of the Strain Transfer Principle for Fiber Reinforced Materials

Fiber optical strain sensors are used to measure the strain at a particular sensor position inside the fiber. In order to deduce the strain in the surrounding matrix material, one can employ the strain transfer principle. Its application is based on the assumption that the presence of the fiber does not impede the deformation of the matrix material in fiber direction. In fact, the strain transfer principle implies that the strain in fiber direction inside the fiber carries over verbatim to the strain inside the matrix material. For a comparatively soft matrix material, however, this underlying assumption may not be valid. To overcome this drawback, we propose to superimpose the matrix material with a one-dimensional model of the fiber, which takes into account its elastic properties. The finite element solution of this model yields a more accurate prediction of the strain inside the fiber in fiber direction at low computational costs.

The rst attempt to analytically model the stress transfer from a uniaxially loaded matrix material to an embedded ber was made by Cox, , leading to the emergence of the research eld referred to as shear-lag theory today (see for instance Nayfeh, ; McCartney, ; Nairn, ), with various applications to ber optical sensors described, e. g., in Li, Zhou, Liang, et al., ; Li, Zhou, Ren, et al., ; Zhou et al., . Here, the uniaxial stress in ber direction is related to the shear stress at the ber matrix interface, which is recognized as the dominating mechanism of stress transfer from the matrix to the ber material.
In order to deduce the full strain state inside the material surrounding the ber instead of only two stress components, one can alternatively employ the strain transfer principle (STP) described in Lekhnitskii, ; Kollár, Van Steenkiste, . The STP postulates a linear relationship between the strain tensor inside the sensor and the strain tensor of the far eld of the surrounding matrix material (i. e., as though there was no ber present). This linear relationship can be expressed analytically in the form of the strain transfer tensor and it is valid for orthotropic matrix as well as orthotropic ber materials; see Kollár, Van Steenkiste, . An extension of this model to coated bers and temperature di erences between the matrix and the ber is also available in Van Steenkiste, Kollár, . The predictions of the analytical STP have been con rmed by various experimental works, for instance Luyckx et al., ; Voet et al., ; Lammens et al., . The STP yields particularly good results when the material properties of the matrix and the ber are similar, or when the ber material is softer than the matrix. In these cases the ber does not restrain the deformations of the matrix material in ber direction under a certain load. The strain in ber direction in the matrix material is transferred verbatim to the strain inside the ber in ber direction and vice versa. In case the matrix material is softer than the ber, however, the ber itself may restrain deformations of the entire part/matrix material, and the strain in ber direction no longer carries over verbatim from the matrix material. The magnitude of this e ect also depends on the ber diameter, part dimensions as well as load conditions under consideration.
To overcome this drawback one has to take into consideration the entire geometry of the part, the embedded ber as well as the load conditions. However, since the ber diameter is usually small compared to the part dimensions, a fully resolved nite element (FE) model is often impractical. Instead, we propose an extension of the STP. Our method combines the practical bene ts of the STP with the improved accuracy of a fully resolved nite element model. In particular, we can continue to simulate the deformation of the matrix material in the absence of the ber.
We refer to our proposed approach as the extended STP. We apply the classical STP to deduce all strain components except the strain in ber direction from a FE model of the matrix material without the ber. By contrast, the strain in ber direction is derived from the solution of a modi ed FE model. The latter is obtained by superimposing the elastic properties of the bulk matrix material with a one-dimensional model of the ber. This does not require the ber to be resolved in the computational mesh.
The paper is structured as follows. Section states the linear elasticity problem for a matrix material part with an embedded ber. Section introduces the strain transfer principle and recalls the results from the existing analytical theory. Section presents our extension to this theory. In Section we derive the variational form of our elasticity model, and Section presents detailed numerical results of our extended STP in comparison with the original STP and with fully resolved nite element computations.
Nomenclature: We denote by : the double contraction of a rank-tensor with a matrix , i. e., ( : ) = ℓ ℓ ℓ . We also use : for the double contraction of two matrices, i. e., : = . Moreover, I denotes the identity on rank-tensors, i. e. I : = holds for any matrix of appropriate dimensions. ⊗ denotes the outer product between matrices and , i. e. ( ⊗ ) ℓ = ℓ . Finally · denotes the usual dot product between vectors and , i. e., · = .
L E E F Let Ω ⊂ R denote the domain ( = ) occupied by the part under consideration. Furthermore, let C : Ω → (Sym( )) denote the sti ness tensor eld on Ω, i. e. for every material point ∈ Ω, C( ) is a linear map between symmetric × strain matrices and symmetric × stress matrices. The function : Ω → R denotes a force density eld, e. g., due to gravity. Let ad denote the set of all admissible displacements of the part which satisfy the given boundary conditions.
The elastic deformation energy of a displacement eld ∈ ad on the domain without embedded ber is given by where ( ) denotes the symmetric displacement gradient The equilibrium solution, which solves the minimization problem will be denoted by nf , where the subscript denotes the absence of the ber. We refer the reader to Braess, for an account of the mathematical theory.
Let : [ , ] → Ω denote the arc-length parameterization of a curve which models the center of the ber. Consequently, denotes the total length of the ber. Let us assume that each point of the ber ( ) is tied to the corresponding point in the domain Ω, i. e., we do not consider slip between ber and matrix material. Given a deformation eld ∈ ad on the domain, the energy of a one-dimensional ber generally consists of three parts, modelling the stretching energy stretch ( ), the bending energy bend ( ) and the twisting energy twist ( ), respectively. Following Spencer, ; Spencer, Soldatos, , the stretching energy is given by where denotes the e ective elastic modulus of the ber material de ned below, is the cross-sectional area and ( ) is the strain in ber direction. Similarly, the bending energy is given by where denotes the cross-sectional moment of inertia and denotes the curvature of (i. e., the derivative of the bending angle). For the twisting energy we have where denotes the shear modulus, is the torsion constant for the section and ( ) denotes the derivative of the torsion angle of . We consider only homogeneous bers for which , , , and are constant. However, the curvature may vary along the ber.
Usually, the radius of curvature of the ber, / ( ), is much larger than the radius of the ber itself. Similarly, the length over which the ber twists by a full turn / ( ) is usually much larger than . Furthermore, we have ∝ , ∝ and ∝ , from where we conclude ( ) and ( ) . As a consequence, the bending and twisting energy terms can be neglected compared to the stretching term for most applications involving only small deformations of the domain Ω.
Therefore, we neglect the bending and twisting energies and consider The equilibrium solution, which solves the minimization problem will be denoted by f , where the subscript denotes the presence of the ber in the deformation energy.
We come back to the de nition of the e ective stretching Young's modulus of the ber material. In order to compensate for the existing matrix material in the volume occupied by the ber, is de ned as where is the stretching Young's modulus of the ber and is the stretching Young's modulus of the matrix in the local ber direction = ( ).
is calculated from the matrix compliance as For the case ≤ which indicates soft ber material, this results in f = nf . Otherwise, f and nf are di erent.
The ber cross-section is assumed to be of circular shape, such that = holds, and the strain in ber direction ( ) is given by These relations allow us to evaluate and minimize the total deformation energy in ( . ). Notice that this does not require to resolve the ber in the computational mesh.

S T P
The STP states that there exists a linear relationship between the strain at the center of the ber and the strain inside the matrix material. In other words, there exists ∈ (Sym( )) such that depends only on the geometry of the ber described by its radius and the material parameters of the matrix and ber and possibly the ber orientation if any of the materials are anisotropic. Equation ( . ) is exact only if nf and f are identical outside of the ber. In general, this is approximately ful lled if the ber has only little in uence on the overall deformation f . This is for instance the case if , i. e., the ber is a relatively soft inclusion in the matrix material.
Analytical representation There exists an analytical representation of the strain transfer principle by Kollár, Van Steenkiste, , which considers a homogeneous, uncoated orthotropic ber with elliptic cross-section embedded into a homogeneous orthotropic ber reinforced composite with coinciding ber directions. Under the assumptions of perfect bonding between ber and matrix material, small deformations, and a uniform stress distribution in the ber, the displacement and stress continuity conditions at the ber matrix interface are evaluated using expressions from Lekhnitskii, to describe the linearly elastic e ect of the elliptic inclusion in the matrix material. Residual strains in the ber are neglected and the ber is assumed to have in nite length. This leads to a : relation between ( ( f )) and ( ( nf )) , for ( f ) and ( nf ) in Voigt notation, i. e., We assume here that the rst axis of the coordinate system is aligned with the ber direction. We additionally assume constant temperatures, i. e. Δ = . The detailed derivation in Kollár, Van Steenkiste, then results in a sparse strain transfer matrix with the non-zero entries expressed by the relations Here Θ( f ) is the angular displacement of the sensor, which we ignore here. Moreover, we denote the entries of the sti ness tensor C in Voigt notation by , ≤ , ≤ and, similarly, the entries of the compliance tensor S = C − in Voigt notation by , ≤ , ≤ . The superscript · denotes material parameters of the ber and no superscript denotes matrix material parameters. Furthermore, = = denotes the lengths of the semi-axes of the ber's cross-section. The matrices in ( . b) are given by where denotes the zero vector ∈ R × . The matrix is similar to but with ber material parameters instead of . Finally, by Lekhnitskii, , the parameters , and are related to the entries of S via

E S T P
We now consider the case > , which is relevant for ber Bragg grating applications. A typical case is that of a glass ber embedded in either an isotropic or ber reinforced plastic. For the sake of simplicity, we ignore here the soft protective coating of the ber. We propose the following extension of the STP (in tensor notation) Equation ( . ) lets us recover the strain tensor ( f ) using the strain eld ( nf ) computed from the solution nf of ( . ) (without a ber), and using the strain in ber direction ( f ) computed from the solution f of ( . ) (taking the sti ening due to the ber into account). Note that since we consider a one-dimensional ber, the full ( f ) tensor is not directly computable from f in a meaningful way, but only the component ( f ) in ber direction is available. We remark that ( . ) is equivalent to the original STP, but with the strain in ber direction corrected.

V F N D
Problem ( . ) will be solved using the method of nite elements. The displacement ∈ ad minimizing the energy in ( . ) is characterized by the variational formulation where ad = { ∈ (Ω) | = on Γ } and Γ is the boundary of Ω with imposed Dirichlet boundary conditions = for the displacement. The corresponding test space is given by = { ∈ (Ω) | = on Γ }. For the nite element discetization, the domain Ω ⊂ R is approximated by a tetrahedral mesh consisting of a set of tetrahedrons T (Ω) such that the ber is approximated by a set of edges E ( ) of the mesh. The set of admissible displacements ad is approximated by functions ad,ℎ which are piecewise linear on each tetrahedron, globally continuous and satisfy the Dirichlet boundary conditions. Similarly, the set of test functions is approximated by functions ℎ which are piecewise linear on each tetrahedron, globally continuous and are zero on the boundary Γ . Due to the linearity, ( ) and ( ) are constant on each tetrahedron. In this discrete setting, the variational form ( . ) becomes For an edge ∈ E ( ) and its incident vertices and , the strain in ber direction can be calculated as where ( ) and ( ) denote the nodal displacements in and .

N D
We test the proposed extension of the STP on an example of intermediate complexity. The geometry is a mm × mm × mm plate with a mm diameter bore located at ( , ) = ( mm, mm), as shown in Fig. . a. As matrix material we use a ber reinforced plastic, for which we compute the e ective sti ness tensor by homogenization, described below in Section . . The sensor ber is made of glass. All FE computations were performed using D /FE CS . ; see Alnaes et al., ; Logg, Mardal, Wells, . The strain transfer described in ( . ) is also computed numerically. Further details are given in the following sections.

. D M G
The geometry shown in Fig. . a was created using F CAD . Notice that this geometry also contains a ber of diameter mm and length = mm along the path shown in Fig. .  --cbna page of ( , , ) = (− mm, mm, mm) and ends at ( , , ) = ( mm, mm, mm). The sole purpose of resolving the relatively thick ber in the mesh is to create a reference nite element solution to compare to the results obtained by the STP. The ber cross-section itself was split into quadrants such that after meshing there will be an edge following the center of the ber. For meshing, the geometry was exported from F CAD as a STEP le. The STEP le was then loaded into G using the O CASCADE plugin. The boundaries of the di erent regions were associated using the "Coherence" function in G and di erent subdomains, boundaries and the ber center line were labeled. The characteristic length used for meshing was calculated from curvature, such that the mesh is more re ned near the ber. The resulting mesh at the boundary and inside of the domain can be seen in Fig. . c and Fig. . d, respectively. The mesh contains nodes and tetrahedral elements. The G mesh including subdomains, boundaries and paths was then loaded and converted to the XDMF format using .
As was mentioned above, the mesh with the three-dimensional ber resolved is used for the purpose of computing reference solutions. However, the same mesh was also used when computing the strains for the embedded one-dimensional ber using the STP. In this case, the material inside of the meshed ber was set equal to the matrix material. We followed this procedure to avoid the in uence of di erent meshes on the solutions. In practice, there would be no need to re ne the mesh close to the ber, nor to resolve the ber in the mesh.

. H M M
For the demonstration, we used glass ber reinforced polypropylene as matrix material. For glass we use a Young's modulus of = GPa and a Poisson ratio of = . . For polypropylene we use = .
GPa and = . . The bers are assumed to be parallel and of in nite length in -direction with a diameter which results in a ber volume fraction of %. The e ective sti ness matrix was computed using from Ospald, , which employs a Lippmann-Schwinger approach (see Moulinec, Suquet, ) with a conjugate gradient solver on a staggered grid described in Kabel, Böhlke, Schneider, ; Schneider, Ospald, Kabel, b. Using a laminate mixing rule at the interfaces, see Schneider, Ospald, Kabel, a, only a resolution of × × voxels is required for the representative volume element to achieve a su cient accuracy. The method also allows the computation of e ective material properties for other ber distributions, e. g., for injection molded parts.
Using these settings, the homogenized matrix material sti ness tensor in Voigt notation reads .
which enters the computation of the strain transfer matrix in the following section. For our FE https://github.com/nsch oe/meshio calculations, we assume the reinforcement bers to be oriented in the -direction of our plate. In this instance, one has to swap the with the -axis of C to obtain the correct material law.

. C S T M
For the embedded ber, we assume the same properties of glass as above, i. e. = GPa and a Poisson ratio of = . . The strain transfer matrix between strain tensors in Voigt notation representing the respective strain transfer tensor in ( . ) can then be computed analytically as demonstrated in Section . Alternatively, we can evaluate it numerically. We chose the latter approach using Ospald, in a similar fashion as for the homogenization. As representative volume element for the latter, we chose a × box with a disc of diameter . placed at the center representing the ber, while the remaining domain represents the matrix material. Due to use of periodic boundary conditions for the displacements, the diameter of the disc has to be su ciently small and the resolution su ciently large (in our case × voxels). For the identi cation of the strain transfer matrix, six linearly independent load cases with prescribed strain ( ) are required. The prescribed strain represents the far eld or matrix strain at the ber position. The computed strain eld ( ) for prescribed strain ( ) is evaluated at the center of the domain to obtain the strain inside the ber ( ) = ( ) ( . , . ). The transfer matrix is then given by the relation where the strains in each column are given in Mandel notation (the notation is only important to interpret the numerical values below). In our instance, the ber is oriented in -direction, i. e. parallel to the reinforcement bers of the matrix material. In order to compute the strain transfer matrix for instances where the sensor ber has an angle to the reinforcement bers we keep the sensor ber oriented in -direction but rotate the matrix material around the -axis. The rotated C (in full tensor notation) is then given by where the sum is carried out over all free indices (using Einstein summation) and = represents the rotation matrix for rotation by the angle around the -axis Similarly as for C of our plate, one has to swap the -with the -axis of to obtain the strain transfer matrix in the correct coordinate system for our example. Furthermore, if the ber orientation (in the --plane) has an angle ≠ to the -axis one has to rotate around the -axis by angle , i. e. ℓ = ℓ , where again the sum is carried out over all free indices (Einstein summation) and = represents the rotation matrix for a rotation by the angle around the -axis = cos − sin sin cos .
Note that also has to be blown up to a full -tensor and then converted back to a matrix in Mandel notation.
In the rst horizontal section of the path of the sensor ber in our example (see Fig. .

b) we have
= and the computed (and properly rotated) strain transfer matrix is given by In the middle of the arc section at an angle of = , the strain transfer matrix is given by . and nally in the vertical section ( = ) we have = . − Note that in all sections the strain in ber direction is always transferred verbatim from the matrix material, as recognized, e. g., from the unit diagonal entry in the rst and the last ber sections.

. S L E P
The discretization and solution of ( . ), its counterpart coming from ( . ), and the reference solution are performed using D /FE CS . ; see Alnaes et al., ; Logg, Mardal, Wells, . Meshes, subdomains and the ber path are loaded from the XDMF les generated as described in Section . . The plate is clamped on the lower boundary ( = mm, see Fig. . a) and a xed displacement of ( , , ) ᵀ mm is enforced on the upper ( = mm) boundary. The volume force is set to zero. For the solution of the arising linear systems we use the conjugate gradient method together with an AMG preconditioner from the D PETS backend. Three solutions are obtained: the displacement eld of the reference solution (with three-dimensionally resolved ber); the solution nf with the ber neglected by setting the material inside the ber subdomain to the matrix material; and the solution f from the superimposed one-dimensional ber model ( . ), where the material inside the ber subdomain is also set to the matrix material but the ber's sti ness enters through the stretching energy term.
. E In Figs. . and . we plot all components of the corresponding strain tensors along the path of the ber. The strain for the reference solution is denoted by , the strain for nf is given by and the strain for f is obtained from our extended STP given in ( . ). For comparison, the strain obtained by the original STP ( . ) is denoted by . In addition to a matrix ber volume fraction of % (right plots of Figs. . and . ), we also performed the same simulations with a ber volume fraction of % (left plots of Figs. . and . ) representing a very soft and isotropic matrix material.
Ideally, the STP solutions (red triangles) and (blue diamonds) should be identical to the reference solution (black stars). For the -and -components we can see that our extended version of the STP agrees very well with the reference solution, whereas the original STP has major deviations for the -component until after the bend of the sensor ber at around = mm as well as for the -component beginning with the bend at around = mm. This observation is independent of the matrix ber volume fraction. For the -component the extended STP and original STP are identical, since the sensor ber direction is always orthogonal to the -direction. Also they both disagree with the reference solution by a signi cant amount for the % matrix ber volume fraction case. Theand -components are one order of magnitude smaller than the other components and in theory they should be zero in view of the symmetry of the problem in -direction. Finally, for the -component the original and extended STP agree everywhere except for the bend around the hole. Here again the extended STP outperforms the original STP in the case of a % matrix ber volume fraction. The % case is indecisive.

C O
In this paper, we proposed an improvement of the original strain transfer principle, which recovers the strain components inside a ber embedded in a matrix material from simulations which do not require the ber geometry to be resolved. The matrix material itself can be isotropic or ber reinforced. The proposed modi cation to the classical STP consists of an additional term in the elastic energy of the total part, which takes into account the additional stretching energy using a simple one-dimensional ber model. This modi cation is particularly relevant for ber materials which are sti er than the matrix, as it is often the case for ber optical strain sensors made of glass. Our evaluation shows that the extended STP improves the classical STP in regions in which the presence of the sensor ber restricts the displacement of the surrounding material in ber direction.
It is a limitation that the one-dimensional ber model ( . ) does not incorporate lateral strains, which results in moderate deviations in the -component. More deviations are expected due to the neglection of bending and twisting terms. While these have only a minor contribution to the overall energy of the ber in the chosen example, they may become more relevant in other setups.
The inclusions of bending and twisting energies as well as the consideration of lateral strains are left to future research. We expect that these terms will be quite challenging to model, discretize and implement. Furthermore, as glass ber sensors are usually coated with a protective layer made of a soft material, a model for coated bers should be considered. For this case there already exists an analytical STP proposed in the literature; see for instance Van Steenkiste, Kollár, . Additionally, the integration of ideas from the shear-lag theory mentioned in the introduction might prove bene cial to address the usually vast di erences in the material properties between ber and coating materials.