Extension of a shell-like RVE model to represent transverse shear effects in textile reinforced composites

2.1 Macroscale mindlin-reissner kinematics

The operators $$(\ast )_{,\alpha }$$ and (*)_{, 3} are defined as the partial derivatives of (*) with respect to $$\xi ^{\alpha }$$ and ξ³, respectively. In the current configuration, the director $$\underline{d}$$ is rotated with respect to the original director $${\underline{\widetilde{d}}}$$ due to two factors: transverse shear $$\varphi _\lambda$$ and rotation of the reference plane by the angle $$\varphi _k$$. The transverse shear deformation causes a rotation of the director along the thickness of the material, while the rotation of the reference plane is associated with the curvature of the structure. The transport of a material line element from the undeformed to the deformed configuration is described by the following operator:

$$$$\begin{align} \mathrm{d}{\underline{r}} =\underline{\underline{F}} \cdot \mathrm{d}{\underline{\widetilde{r}}} \quad \text{with} \quad \underline{\underline{F}} = {\underline{g}_{\alpha }}\otimes \underline{\widetilde{g}}^{\alpha } + \xi ^{3}\,{\underline{d}_{,\alpha }}\otimes \underline{\widetilde{g}}^{\alpha }+{\underline{d}}\otimes {\underline{\widetilde{d}}} \end{align}$$$$(2)

The macroscale deformation gradient tensor $$\underline{\underline{F}}$$ of the three-dimensional continuum is approximated using the first-order Mindlin-Reissner theory with $$\underline{\underline{\bm {F}}}={\underline{g}_{\alpha }}\otimes \underline{\widetilde{g}}{}^{\alpha }$$ as the in-plane deformation gradient tensor of the reference surface, which corresponds to the membrane deformation tensor in shell theories. Moreover, $$\underline{\underline{\bm {K}}}=\underline{\widetilde{d}}_{,\alpha }\otimes \underline{\widetilde{g}}{}^{\alpha }$$ is the curvature tensor, and $$\underline{d}$$ represents the in-plane transverse shear. Additionally, in accordance with classical shell theories, an infinitesimal rotation description is employed for the in-plane deformation of the director, effectively restricting the main of curvature to a small value of $$H=\tfrac{1}{2}{\underline{\underline{\bm {K}}}}\cdot {\underline{\underline{\bm {F}}}^{-1}}\ll 1$$. The shell strains are obtained based on the assumed kinematics stated in assumption (2) and can be defined as follows [7]

$$$$\begin{equation} \def\eqcellsep{&}\begin{array}{lll}\underline{\underline{\bm {h}}} = \dfrac{1}{2}\ln ((\underline{\underline{\bm {F}}}\cdot \underline{\underline{\bm {F}}}^T)) \quad , & \underline{\underline{\bm {\kappa }}} = \underline{\underline{\bm {K}}}\cdot \underline{\underline{\bm {F}}}^{-1}-\underline{\underline{\bm {F}}}^{-T}\cdot \underline{\underline{\bm {K}}}{}_{0}\cdot \underline{\underline{\bm {F}}}^{-1}\quad , & {\underline{\gamma }} = {\left({\underline{g}_{\alpha }}\cdot {\underline{d}}-\underline{\widetilde{g}}_{\alpha }\cdot \underline{\widetilde{d}}\,\right)}\,\underline{g}^{\alpha }, \end{array} \end{equation}$$$$(3)

where $${\underline{\underline{\bm {h}}}}$$ is a Hencky-tensor of the membrane deformation, $${\underline{\underline{\bm {\kappa }}}}$$ is denotes a symmetric relative curvature tensor, $${\underline{\gamma }}$$ represents the transverse shear strains, and $$\ln ((\cdots ))$$ denotes the tensorial natural logarithmic function. The membrane forces are quantified as $${{\it {N}}{}_{\alpha \beta }}$$ in [N mm] units, the bending moments are symbolized as $${{\it {M}}{}_{\alpha \beta }}$$ in [N mm²] units, and the shear forces are indicated as $${{\it {S}}{}_{\alpha }}$$ in [N mm] units, with their expressions as follows

$$$$\begin{equation} {{\it {N}}{}_{\alpha \beta }} = \int _{h} {{\it {\sigma }}{}_{\alpha \beta }}\, \mathrm{d}\xi ^{3} \quad ,\quad {{\it {M}}{}_{\alpha \beta }} = \int _{h} \xi ^{3}\,{{\it {\sigma }}{}_{\alpha \beta }}\, \mathrm{d}\xi ^{3} \quad ,\quad {{\it {S}}{}_{\alpha }} = \int _{h} {{\it {\sigma }}{}_{3\alpha }}\, \mathrm{d}\xi ^{3} \,, \end{equation}$$$$(4)

Here, h represents the thickness of the shell. To define the nonlinear mechanical behavior of a shell section in terms of generalized section quantities using the ABAQUS subroutine UGENS, the force array [N_i] and strain array [E_i] are used. In these arrays, the direct membrane terms are stored first, followed by the shear membrane term, the direct and shear bending terms, and then the transverse shear components, as defined in Equation (5). The number of entries in these arrays depends on the specific element type, as only the active components are stored.

$$$$\begin{align} \def\eqcellsep{&}\begin{array}{ll} {[\bm{N}_i]} &= {\left[N_{11},N_{22},\dfrac{1}{2}(N_{12}+N_{21}),M_{11},M_{22},\dfrac{1}{2}(M_{12}+M_{21}),S_{1},S_{2}\right]}\\[3pt] {[\bm{E}_i]} &={\left[h_{11},h_{22},2\,h_{12},\kappa _{11},\kappa _{22},2\,\kappa _{12},\gamma _{1},\gamma _{2}\right]}\\[3pt] \end{array} \end{align}$$$$(5)

The linearized constitutive relations that align with the macroscopic framework can be expressed as follows

(6)

where the macroscopic constitutive tangents of a shell element are captured by an 8 × 8 stiffness matrix [$$\bm{K}_{ij}$$]. This matrix can be decomposed into four distinct parts. The first part [A] is the so-called membrane stiffness, the second part [D] represents the plate stiffness and the mixed derivatives [B] correspond to coupling stiffnesses, accounting for the coupling between in-plane and out-of-plane deformations. The last part [Q] represent the transverse shear stiffness.

2.2 Microscale projection: Second-order homogenization

The displacements of the cross-section are linearly approximated, which leads to deviations when the cross-section undergoes large rotations.

2.3 Microscale projection: Rotation-based homogenization

It is important to emphasize that only surfaces with a Gaussian curvature $$ \mathrm{det}[{\it {K}_{\alpha \beta }}] = 0$$ are considered as so-called developable surfaces. This means that the curved surface can be flattened without distortion. Please note that the sphere is the only closed surface with a constant Gaussian curvature. In general, undesired surface effects increase with increasing Gaussian curvature. Therefore, either small RVEs of highly curved surfaces or large RVEs of slightly curved surfaces can be analyzed with negligible surface effects. In the case of an air spring, the curvature of the rolling lobe is large compared to the curvature in the circumferential direction. Hence, the highly loaded areas can be modeled as a parabolic RVE.

2.4 Implementation

3.1 Verification of surface effects and validation

To verify the surface effects, multiple homogeneous second-order RVEs are connected in series. These RVEs are defined by dimensions specified as $$N\,[l\times w \times h]$$, with the number of these RVEs denoted as N, and each having defined length (l), width (w), and thickness (h). These RVEs are loaded with a high curvature of $$K_{11}=0.4\,\mathrm{mm^{-1}}$$. The surface effects refer to disturbances in the peripheral regions of the RVE. On the left side of the illustration 5 (the first image from the left), it is clearly evident that changes in the von Mises stress distribution occur not only in the thickness direction but also in other coordinate directions, indicating disturbances in the peripheral areas. When the RVE is reduced in size (as seen in the third image from the left), the disturbances in the peripheral region decrease despite constant curvature. This becomes particularly evident when the stress distribution changes exclusively in the thickness direction. From this, the following conclusion can be drawn: When analyzing a small RVE with significant curvature, the curvature is indeed significant, but the RVEs are small enough to assume that surface effects are relatively negligible. The smaller the RVE, the more the surface effect is reduced until it converges to a free surface effect. In contrast, when examining large RVEs with minimal curvature, the curvature is less pronounced, but the RVEs are large enough to justify the assumption that surface effects are negligible. In the case of a rotation-based RVE, the stress distribution is only free from thickness any surface effects in the parabolic scenario, regardless of the size of the RVE, as depicted in Figure 6 on the left. This is due to the strict constraints in Equation (8). This emphasizes the statements of the theorems by Minding and Liebmann, which explain the surface effects in elliptical and hyperbolic structures. Only surfaces with a Gaussian curvature of zero are considered “developable”, meaning that the curved surface can be flattened without distortion.

In order to prove the correct implementation of the constraints and the validity of the homogenization, a homogeneous FE-plate made of elastomer is considered. Initially, a FE-plate with dimensions of $$10\times 40\times 1 \,\mathrm{mm^{3}}$$ for length, width, and thickness is modeled, using volume elements (5 elements in the thickness direction). The plate is assigned a soft isotropic material with Hooke's parameters $$E = 42\,\mathrm{MPa},\; \nu = 0.4$$, and a curvature of $$K_{11} = 0.4\,\mathrm{mm^{-1}}$$ is applied. The results of the macroscopic simulation show at a specific location the maximum normal stress in the tensile region of $$\sigma _{{11}\mathrm{max}}=9.32\,\mathrm{MPa}$$ (on the upper surface) and the minimum normal stress of $$\sigma _{{11}\mathrm{min}}=-10.12\,\mathrm{MPa}$$ in the compressive region (on the lower surface). At the same location, an RVE with dimensions of $$1\times 1\times 1\,\mathrm{mm^{3}}$$ is extracted, and the same material and boundary condition parameters are applied. The results of the microscopic simulation of the RVE using both methods are presented in Figure 6. It is shown that the rotation-based RVE provides more accurate results, and the small differences observed are due to numerical differentiation and can be disregarded when using a fine mesh. Instead, the second-order RVE exhibits some deviation, which diminishes as the size is reduced. It is shown that the rotation-based RVE provides more accurate results, and the small deviation in results between the FE plate and the RVE is attributed to numerical differentiation and can be disregarded when using a fine mesh. Instead, the second-order RVE exhibits some deviation, which diminishes as the size of RVE is reduced.

3.2 Cross-layer reinforced composites of an air spring

In this section, shell-like rotation-based RVEs for cord-elastomer composites with two intersecting reinforcement layers are considered. The objective is to analyze the locale stresses (localization) in the composite under realistic loading conditions that occur in a component such as an air spring. This aims to enable efficient optimization of cord-elastomer composites, including their durability, under realistic conditions. Figure 7 (left) shows the geometry of an undeformed air spring bellows with its dimensions in a top view. The RVEs represent sections of the bellows with different periodicity and loading directions. Although the initial geometry of many air spring bellows is cylindrical, the RVEs are assumed to be initially straight due to the large radius compared to the bellows thickness and the high number of cords n. The parameters of the cross-layer reinforced composite are cord angle $$\alpha _C$$, the thickness of the composite h, the spacing of the cords $$t_C$$, and the relative position of the cords in the thickness direction. The RVE features a structured mesh with C3D8H hybrid elements, and the material model assigned to each integration point is assumed to be influenced by its corresponding material region. A hyperelastic model is chosen for the rubber matrix. In particular, the strain energy potential of Neo-Hookean type is used, which is split into deviatoric and volumetric contributions. The second material region that needs to be considered is the cord, which has a filament structure. The cord bundle is considered to possess transversely isotropic behavior with the preferred direction represented by $$\underline{t}_F$$. The total strain energy potential is composed of an anisotropic part and an isotropic part that accounts for shear and volumetric deformations [1]. The next step involves developing a global FE model in ABAQUS using the rebar formulation. In this context, the global model refers to an FE model of the air spring that does not rely on any assumed symmetries. The aim of the global model is to accurately capture all typical macroscopic deformation modes present in the air spring bellows, enabling their transfer to the RVE.

These deformation modes specifically include membrane deformation, curvatures, and transverse shear effects occurring within the matrix between two cord layers (see, ref. [8]), as illustrated in Figure 8. A transfer of the macroscopic deformations to the RVE as PBC allows for a more precise analysis of the local stress distribution. However, the use of multiphase elements inevitably entails a loss of information concerning the interface. Since the interface between the cords and the elastomer often represents preferred locations for crack initiation and, consequently, for later component failure, the analysis of stress in this region is of particular interest. One way to significantly improve the quality of results near the interface is through the application of submodeling techniques. Figure 9 illustrates the distribution of maximum logarithmic strain in both the RVE and the submodel. Based on the simulations, it has been observed that the primary influence in changing the cord angle is a result of the combined effects of curvature and transverse shear. This interaction leads to a significant shearing of the elastomer layer located between the cord layers. The subsequent step involves calculating an effective shell stiffness using a second-order RVE, as depicted in Figure 9. This calculation is necessary for homogenization the effective mechanical behavior of the corresponding shell element and, thus, developing an appropriate constitutive model.

Additionally, all contributions highlighted in red appear as discretization errors. Besides the tension-torsion coupling, the coupling between bending and shearing is particularly interesting. Due to the curvature of the RVE, tension and compression sides are formed. Instead of simply stretching or compressing, the cords can also change their angle in response.