A hybrid approach for the efficient computation of polycrystalline yield loci with the accuracy of the crystal plasticity finite element method

Direct experimental evaluation of the anisotropic yield locus of a given material, representing the zeros of the material's yield function in the stress space, is arduous. It is much more practical to determine the yield locus by combining limited measurements of yield strengths with predictions from numerical models based on microstructural features such as the orientation distribution function (ODF; also referred to as the crystallographic texture). For the latter, several different strategies exist in the current literature. In this work, we develop and present a new hybrid method that combines the numerical efficiency and simplicity of the classical crystallographic yield locus (CYL) method with the accuracy of the computationally expensive crystal plasticity finite element method (CPFEM). The development of our hybrid approach is presented in two steps. In the first step, we demonstrate for diverse crystallographic textures that the proposed hybrid method is in good agreement with the shape of the predicted yield locus estimated by either CPFEM or experiments, even for pronounced plastic anisotropy. It is shown that the calibration of only two parameters of the CYL method with only two yield stresses for different load cases obtained from either CPFEM simulations or experiments produces a reliable computation of the polycrystal yield loci for diverse crystallographic textures. The accuracy of the hybrid approach is evaluated using the results from the previously established CPFEM method for the computation of the entire yield locus and also experiments. In the second step, the point cloud data of stress tensors on the yield loci predicted by the calibrated CYL method are interpolated within the deviatoric stress space by cubic splines such that a smooth yield function can be constructed. Since the produced yield locus from the hybrid approach is presented as a smooth function, this formulation can potentially be used as an anisotropic yield function for the standard continuum plasticity methods commonly used in finite element analysis.

Experimental Study and Failure Mechanism Analysis of Rubber Fiber Concrete under the Compression-Shear Combined Action

In order to examine the compression-shear combined mechanical properties of rubber fiber concrete, an experimental study was carried out on rubber fiber concrete of three different configurations using a material compression-shear testing machine by considering different axial compression ratios. The failure modes and shear stress-strain curves of rubber fiber concrete under different loading conditions were obtained. By comparatively analyzing the mechanical parameters of rubber fiber concrete under different axial compression ratios, the following conclusions were drawn. With the increase of the axial compression ratio, the failure mode in the shear direction gradually developed from a relatively straight crack to a main crack accompanied by a certain amount of axial cracks; meanwhile, the number of concrete slags on the shear failure section was gradually increased and the friction marks were gradually deepened. The addition of rubber particles increased the randomness and discreteness of the concrete upon failure, while fibers inhibited the development of oblique micro-cracks and the dropping of concrete slags. The shear stress of the concrete specimen containing rubber particles was significantly lower than those without rubber particles. Comparatively, fibers showed little effect on the shear stress. As the axial compression ratio increased, the shear stress and shear strain of rubber fiber concrete were gradually increased, but the increasing amplitude of shear stress tended to become flattened. Under the influence of the axial compression ratio, the shear stress of C-0%-0%, C-30%-0%, and C-30%-0.6% was increased by 4.57 times, 3.26 times, and 2.69 times, respectively, suggesting a gradually decreasing trend. At the same time, based on the principal stress space and the octahedral stress space, the compression-shear combined failure criterion was proposed for the three different rubber fiber concretes. The research findings are of great significance to the engineering application and development of rubber fiber concrete.

Multi-level deep drawing simulations of AA3104 aluminium alloy using crystal plasticity finite element modelling and phenomenological yield function

This paper proposed a hierarchical multi-level model to study the crystallographic texture induced mechanical anisotropy of AA3104-H19 aluminium sheet from mesoscale to continuum scale. In the mesoscale, full-field crystal plasticity finite element method (CPFEM) was used to provide both in-plane and out-of-plane yield stresses and plastic potential points in various deformation modes. In the continuum scale, these materials sampling points were used to determine the parameters of two phenomenological yield functions (Yld2000-2d in plane stress space and Yld2004-18p in 3D stress space) using associated flow rule (AFR) and non-associated flow rule (non-AFR). The results indicate that higher accuracy obtained by Yld2000-2d and Yld2004-18p yield functions associated with non-AFR in comparison with AFR. These phenomenological models were successfully implemented into finite element (FE) code using an explicit integration scheme to simulate sheet metal forming. It is found that the 3D Yld2004-18p model involved with both in-plane and out-of-plane anisotropies is superior to 2D Yld2000-2d model which only accounts for in-plane anisotropy.

Thermomechanical deformation of the orthotropic shell taking into account the deformation anisotropy

A variant of the rotation shell in the particular form of a closed circular cylindrical shell, which is often used in the design practice of civil, power and other industrial structures, is considered. The specificity of the considered shell lies in the features of its material, which has a manifestation of dual anisotropy. In particular, this material is orthotropic in structure, and the nature of deformation shows the dependence of stiffness and strength on the type of stress state. The loading of the shell is assumed to be axisymmetric, taking into account the influence of a medium with variable thermal parameters. The temperature difference between the shell surfaces is taken into account here. The statement of the general thermomechanical problem is carried out in an unrelated form, taking into account a certain independence of the problems of thermodynamics and mechanics. Taking into account the limitations of the classical thermomechanical theories of shells made of materials with dual anisotropy and the fact that the known models for such materials have significant drawbacks, the authors used a variant of the normalized stress space. Differential equations of thermoelasticity for a cylindrical shell are obtained, taking into account the complicated thermomechanical properties of its material. Particular solutions with the features of the results of calculating the shell states are illustrated, and their analysis is carried out.

Triaxial Strength Criteria in Mohr Stress Space for Intact Rocks

Conventional triaxial strength criteria are important for the judgment of rock failure. Linear, parabolic, power, logarithmic, hyperbolic, and exponential equations were, respectively, established to fit the conventional triaxial compression test data for 19 types of rock specimens in the Mohr stress space. Then, a method for fitting the failure envelope to all common tangent points of each two adjacent Mohr’s circles (abbreviated as CTPAC) was proposed in the Mohr stress space. The regression accuracy of the linear equation is not as good as those of the nonlinear equations on the whole, and the regression uniaxial compression strength (σc)r, tensile strength (σt)r, cohesion cr, and internal frictional angle φr predicted by the regression linear failure envelopes with the method for fitting the CTPAC in the Mohr stress space are close to those predicted in the principal stress space. Therefore, the method for fitting CTPAC is feasible to determine the failure envelopes in the Mohr stress space. The logarithmic, hyperbolic, and exponential equations are recommended to obtain the failure envelope in the Mohr stress space when the data of tensile strength (σt)t are or are not included in regression owing to their higher R2, less positive x-intercepts, and more accurate regression cohesion cr. Furthermore, based on the shape and development trend of the nonlinear strength envelope, it is considered that when the normal stress is infinite, the total bearing capacity of rock tends to be a constant after gradual increase with decreasing rates. Thus, the hyperbolic equation and the exponential equation are more suitable to fit triaxial compression strength in a higher maximum confining pressure range because they have limit values. The conclusions can provide references for the selection of the triaxial strength criterion in practical geotechnical engineering.

Partial Relaxation of 𝐶0 Vertex Continuity of Stresses of Conforming Mixed Finite Elements for the Elasticity Problem

A conforming triangular mixed element recently proposed by Hu and Zhang for linear elasticity is extended by rearranging the global degrees of freedom. More precisely, adaptive meshes {\mathcal{T}_{1},\ldots,\mathcal{T}_{N}} which are successively refined from an initial mesh {\mathcal{T}_{0}} through a newest vertex bisection strategy, admit a crucial hierarchical structure, namely, a newly added vertex {\boldsymbol{x}_{e}} of the mesh {\mathcal{T}_{\ell}} is the midpoint of an edge e of the coarse mesh {\mathcal{T}_{\ell-1}}. Such a hierarchical structure is explored to partially relax the {C^{0}} vertex continuity of symmetric matrix-valued functions in the discrete stress space of the original element on {\mathcal{T}_{\ell}} and results in an extended discrete stress space: for such an internal vertex {\boldsymbol{x}_{e}} located at the coarse edge e with the unit tangential vector {t_{e}} and the unit normal vector {n_{e}=t_{e}^{\perp}}, the pure tangential component basis function {\varphi_{\boldsymbol{x}_{e}}(\boldsymbol{x})t_{e}t_{e}^{T}} of the original discrete stress space associated to vertex {\boldsymbol{x}_{e}} is split into two basis functions {\varphi_{\boldsymbol{x}_{e}}^{+}(\boldsymbol{x})t_{e}t_{e}^{T}} and {\varphi_{\boldsymbol{x}_{e}}^{-}(\boldsymbol{x})t_{e}t_{e}^{T}} along edge e, where {\varphi_{\boldsymbol{x}_{e}}(\boldsymbol{x})} is the nodal basis function of the scalar-valued Lagrange element of order k (k is equal to the polynomial degree of the discrete stress) on {\mathcal{T}_{\ell}} with {\varphi_{\boldsymbol{x}_{e}}^{+}(\boldsymbol{x})} and {\varphi_{\boldsymbol{x}_{e}}^{-}(\boldsymbol{x})} denoted its two restrictions on two sides of e, respectively. Since the remaining two basis functions {\varphi_{\boldsymbol{x}_{e}}(\boldsymbol{x})n_{e}n_{e}^{T}}, {\varphi_{\boldsymbol{x}_{e}}(\boldsymbol{x})(n_{e}t_{e}^{T}+t_{e}n_{e}^{T})} are the same as those associated to {\boldsymbol{x}_{e}} of the original discrete stress space, the number of the global basis functions associated to {\boldsymbol{x}_{e}} of the extended discrete stress space becomes four rather than three (for the original discrete stress space). As a result, though the extended discrete stress space on {\mathcal{T}_{\ell}} is still a {H(\operatorname{div})} subspace, the pure tangential component along the coarse edge e of discrete stresses in it is not necessarily continuous at such vertices like {\boldsymbol{x}_{e}}. A feature of this extended discrete stress space is its nestedness in the sense that a space on a coarse mesh {\mathcal{T}} is a subspace of a space on any refinement {\hat{\mathcal{T}}} of {\mathcal{T}}, which allows a proof of convergence of a standard adaptive algorithm. The idea is extended to impose a general traction boundary condition on the discrete level. Numerical experiments are provided to illustrate performance on both uniform and adaptive meshes.